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Q: Add fields to TYPO3 template for global site info? How can I add some fields to the TYPO3 template (or a better location) for general site info?
For example, a phone number field so that I can add it to the site header or other non content area.
In the past I have used content elements in a special folder to add something like this, but that's not very user friendly for site editors.
A: If you want it to be (relatively) easy for site editors to edit, without building a custom module or something like that, content elements in a special folder is your best bet. We often use custom content elements for this so the field are more logical, but you do need to do some programming for this. More on creating custom fields can be found at https://docs.typo3.org/m/typo3/reference-coreapi/master/en-us/ApiOverview/ContentElements/AddingYourOwnContentElements.html
An alternative could be using a TypoScript constant. These are relatively easy to edit using the Constants editor in the Template module. Assuming you use Fluid templates, you can add it to your template with <f:cObject typoscriptObjectPath="lib.phoneNumber" /> In TypoScript you then add the following:
lib.phoneNumber = TEXT
lib.phoneNumber.value = {$phoneNumber}
More about the Constants editor can be found at https://docs.typo3.org/m/typo3/reference-typoscript/master/en-us/UsingSetting/TheConstantEditor.html
A: in both other answers I miss the simplest solution:
defining the value as a generic fluid variable
page {
10 = FLUIDTEMPLATE
10 {
:
variables {
:
phoneNumber = TEXT
phoneNumber.value = {$phoneNumber}
}
}
}
You do not need to use settings as it is no setting.
or jumping back from fluid to typoscript with lib.anything and a call to f:cObject viewhelper.
Aside of that I would recommend to consider if it is ok to need an admin (or even a maintainer to deploy a new version) to change that phone number (all typoscript, setup and constants, belongs into the site-extensions which is provided in a repository).
you always could restrict normal editors from accessing special pages or even columns or special content elements, where an admin can change data without a deploy-process.
(example: How much hassle it is if you need to change the number in case of emergency because of a breakdown of phone lines?)
A: I think the easiest way is to add it with TS, something like that :
page {
10 {
settings {
siteInfos {
phone = 01 02 03 04 05
}
}
}
}
And then , you can use this in you Fluid Templates :
{settings.siteInfos.phone}
Florian
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,882
|
Welcome to the very best hen and girls night out in Cardiff, featuring the one and only Dreamboys.
Cardiff is without a doubt Wales most vibrant hen party capital. Over the past few years Cardiff has grown into one of the most popular cities in the UK, where you can play hard and party harder. If you're think of having a hen weekend or celebrating a birthday party in this beautiful city, then we've got the perfect party idea waiting for you.
How do you fancy a fun filled night of music, muscle, magic and mayhem all rolled into one big show? Then we've got just the thing for you and your girlfriends. The Dreamboys is the only male strip show in Cardiff and the perfect place to celebrate your hen night, birthday party or just enjoy a fantastic girls night out.
You can book your tickets for our show in Cardiff in just four easy steps.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,660
|
Die Sinfonie G-Dur Hoboken-Verzeichnis I:23 komponierte Joseph Haydn im Jahr 1764 während seiner Anstellung als Vize-Kapellmeister beim Fürsten Nikolaus I. Esterházy. Menuett und Trio sind als Kanon komponiert. Der Schlusssatz hat ein ungewöhnliches Ende im Pianissimo.
Allgemeines
Die Sinfonie Hoboken-Verzeichnis I:23 komponierte Joseph Haydn im Jahr 1764. In demselben Jahr – während der Zeit als Vize-Kapellmeister der Familie Esterházy – komponierte Haydn die Sinfonien Nr. 21, Nr. 22 und Nr. 24.
Zur Musik
Besetzung: zwei Oboen, zwei Hörner, zwei Violinen, Viola, Cello, Kontrabass. Zur Verstärkung der Bass-Stimme wurde damals auch ohne gesonderte Notierung ein Fagott eingesetzt. Über die Beteiligung eines Cembalo-Continuos in Haydns Sinfonien bestehen unterschiedliche Auffassungen.
Aufführungszeit: ca. 20 Minuten (je nach Einhalten der vorgeschriebenen Wiederholungen).
Bei den hier benutzten Begriffen der Sonatensatzform ist zu berücksichtigen, dass dieses Schema in der ersten Hälfte des 19. Jahrhunderts entworfen wurde (siehe dort) und von daher nur mit Einschränkungen auf ein um 1764 komponiertes Werk übertragen werden kann. – Die hier vorgenommene Beschreibung und Gliederung der Sätze ist als Vorschlag zu verstehen. Je nach Standpunkt sind auch andere Abgrenzungen und Deutungen möglich.
Erster Satz: Allegro
G-Dur, 3/4-Takt, 127 Takte
Die Sinfonie beginnt mit einem "großartigen", "von bezwingender rhythmischer Kraft und dramatischen Impulsen erfüllten (…) Kopfsatz." Das kräftige erste Thema (Hauptthema, ein kontrastierendes zweites Thema fehlt) besteht aus zwei Hälften, für die der Rhythmus aus vier Vierteln und Achteln im punktierten Rhythmus typisch ist. In der ersten Hälfte sind die Viertel als kräftige Akkordschläge ausgebildet und die Bläser mit stimmführend, die zweite Hälfte wird von einer (im weiteren Satzverlauf wichtigen) Sechzehntelwendung eingeleitet und durchschreitet einen größeren Tonraum als die erste. Die zweite Hälfte wird variiert wiederholt.
Ab Takt 9 folgt ein Wechselspiel von Einwürfen der Oboen mit dem übrigen Orchester. In Takt 20 wird Hälfte 1 vom Hauptthema in der Dominante D-Dur wiederholt, anschließend verselbständigt sich die Sechzehntelfigur der zweiten Hälfte zunächst zur Abwärts-Sequenz und dann – angereichert mit Oktavsprüngen abwärts – zum Unisono. Die Unisonobewegung reichert sich mit Chromatik an, und nach kurzem Piano-Echo folgt eine ausgedehnte Tremolo-Klangfläche mit aufstrebendem Tonleitersegment in den Oboen. Die Schlussgruppe enthält ein Motiv mit punktiertem Rhythmus (der an das Hauptthema erinnert) und am Ende eine kleine Bläserfanfare.
Die Durchführung beginnt mit dem Hauptthema in D-Dur. Anschließend wird die Sechzehntelwendung aus dem Hauptthema intensiv verarbeitet: im versetzten Einsatz der Streicher, in imitatorischer Passage der Violinen mit Oktavsprung (auf- und abwärts) und im Unisono. Ab Takt 81 folgt eine Passage mit Synkope sowie Elementen vom Themenkopf (Viertelbewegung) und der Schlussgruppe, die in h-Moll endet. Mit der Einwürf-Passage (hier: Oboen –und Streicher) wechselt Haydn zurück zur Tonika G-Dur und damit zur Reprise.
In der Reprise ab Takt 96 sind die Passage mit den Einwürfen und der zweite Auftritt des Hauptthemas ausgelassen, dafür ist die Tremolo-Klangfläche etwas ausgedehnter. Exposition sowie Durchführung und Reprise werden wiederholt.
Zweiter Satz: Andante
C-Dur, 2/4-Takt, 105 Takte
Der Satz ist (wie üblich bei Haydns frühen bis mittleren Sinfonien) nur für Streicher und überwiegend piano gehalten. Das erste Thema ist symmetrisch aus zweitaktigen Bausteinen aufgebaut und hat einen liedhaften Charakter. Typisch auch für den weiteren Satzverlauf sind die Einwürfe von Viola und Bass, hier zunächst als Triolenroller aufwärts. Der Themenkopf wird wiederholt (nun mit Sechzehntelroller abwärts) und mit Mollwendung beantwortet. Im folgenden Dialog zwischen Violinen und Viola / Bass tritt der Sechzehntelroller dominant in Erscheinung. Im zweiten, floskelhaften "Thema"in der Dominante G-Dur spielen nur die Violinen. In der Schlussgruppe ab Takt 28 wird der Dialog zwischen Ober- und Unterstimmen fortgesetzt, der Einwurf von Viola und Bass ist nun als Tonleiter aufwärts erweitert.
Die Durchführung wiederholt die Motive der Exposition als Varianten. Ausgehend vom ersten Thema in G-Dur, folgt eine ausgedehntere, dissonante Mollpassage, das zweite "Thema" in a-Moll und die verkürzte Schlussgruppe. Die Reprise ist gegenüber der Exposition verändert: Die Wiederholung des ersten Themas ist ausgelassen, dafür die Mollpassage erweitert und mit Dissonanzen angereichert. Exposition sowie Durchführung und Reprise werden wiederholt.
Dritter Satz: Menuett
G-Dur, 3/4-Takt, mit Trio 50 Takte
Das Menuett ist als zweistimmiger Kanon strukturiert. Stimme 1 spielen Oboen und Violinen, die um einen Takt versetzte Stimme 2 Viola und Bass. Die Hörner füllen mit Einwürfen die Harmonien aus. Das Thema ist durch Wechsel von Achtelbewegung mit Triolen und Pausenunterbrechungen gekennzeichnet.
Kanonische Menuette in G-Dur gibt es z. B. auch in Haydns Sinfonie Nr. 3, in einer Sinfonie von Michael Haydn oder bei Wolfgang Amadeus Mozart in der Sinfonie KV 110 sowie der Cassation KV 63.
Das Trio in C-Dur ist ebenfalls ein Kanon in drei Stimmen für Streicher (1. Violine, 2. Violine und Viola / Bass) mit um zwei Takte versetztem Stimmeneinsatz.
Vierter Satz: Finale. Presto assai
G-Dur, 6/8-Takt, 96 Takte
Der Satz ist durch seine nahezu kontinuierliche, vorwärtstreibende Bewegung der Violinen im Piano und – als dynamischen Kontrast dazu – kurzen Forte-Einwürfen des ganzen Orchesters gekennzeichnet.
Das erste Thema beginnt als Forte-"Ausrufezeichen" aus Tonrepetition, gefolgt von einer sich aufschaubenden, auftaktigen Piano-Figur der in Achteln dahineilenden Violinen. Das Thema wird wiederholt und geht dann zur Dominante D-Dur über, wo in Takt 19 nach einem weiteren "Ausrufezeichen" das zweite "Thema" (eher: Motiv) mit Tonleiterfragmenten einsetzt. Ein mit vier Takten etwas längerer Forte-Einwurf führt zur Schlussgruppe, bei der die (wiederum durch einen Einwurf unterbrochene) Piano-Bewegung der Violinen durch ihre wiederholten Sekundschritte quasi auf der Stelle verharrt.
Die Durchführung variiert das erste Thema und setzt die Forte-Piano-Kontraste taktweise nebeneinander, wobei mehrere Tonarten gestreift werden. Die Reprise ab Takt 59 ist wie die Exposition strukturiert, allerdings weist das Satzende eine Besonderheit auf, die in der Literatur oft hervorgehoben und als frühes Beispiel für Haydns Humor gedeutet wird: Die Musik verebbt immer mehr von Pausen unterbrochen im Pianissimo. Als der Satz bereits zu Ende erscheint, fügt Haydn nach Generalpause noch einen unerwarteten Pizzicato-Akkord hinzu. Exposition sowie Durchführung und Reprise werden wiederholt.
Einzelnachweise, Anmerkungen
Weblinks, Noten
Hörbeispiele und Informationen zur 23. Sinfonie Haydns vom Projekt "Haydn 100&7" der Haydn-Festspiele Eisenstadt
Joseph Haydn: Sinfonia No. 23 G major . Philharmonia No. 723, Universal Edition, Wien 1963. Reihe: Howard Chandler Robbins Landon (Hrsg.): Kritische Ausgabe sämtlicher Symphonien (Taschenpartitur), S. 197 bis 214.
Thread zur Sinfonie Nr. 23 von Joseph Haydn, Stand 26. April 2013
Horst Walter: Sinfonien 1764 und 1765. In: Joseph Haydn-Institut Köln (Hrsg.): Joseph Haydn Werke. Reihe I, Band 4. G. Henle-Verlag, München 1964, 141 Seiten.
Siehe auch
Liste der Sinfonien Joseph Haydns
023.
Musik 1764
|
{
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}
| 6,922
|
Q: What actual difference does it make whether electric or magnetic field lines are open or closed? If both electric and magnetic fields attenuate with distance by the inverse square law, what difference does it make that the latter's field lines are 'closed'?
And if a magnetically-induced electric field has 'closed' field lines, rather than open, so what? How can you tell?
You never learn in school precisely what those field lines mean in the 'real' world.....
Or maybe I'm just an idiot...
A: The statement that magnetic field lines are closed essentially says that there is no magnetic monopole where the fields lines can start or end on. And in fact, because of this, bounded current and charge distribution cannot produce a magnetic field that goes like $r^{-2}$ at large distance.
For your second question, a closed electric field line will mean that a charge can go around the loop and gain energy, i.e., there is an EMF, which can be easily measured.
A: Magnetic field lines are generally not closed. That they should be closed is a very common misconception in the teaching of physics.
See e.g. here https://physics.aps.org/story/v24/st24
From Maxwells equations we have that div B=0, this is a local statement that field lines dont have a beginning nor an end, this does not imply the global statement that they close.
A: Magnetic field lines are closed because there are no magnetic "charges" (usually called magnetic monopoles) analogous to electrical charges like protons and electrons. When electric field lines end, they always end on an electric charge.
The apparent absence of magnetic charges makes for an unattractive asymmetry in Maxwell's equations. Some physicists think that magnetic charges can exist but are now extremely rare. They may have been common in the very early universe, before cosmic inflation occurred.
Electromagnetic field lines help you visualize the direction and magnitude of the forces that the field exerts on charges. For an electric field, each field line points in the direction of the force it exerts on a positive charge. Where the field lines are closer together, the force is stronger, and where they are farther apart the force is weaker.
For a magnetic field, it's more complicated, because only a moving charge feels a magnetic force. The magnetic force is perpendicular to both the magnetic field line and the charge's velocity. Again, field lines that are closer together indicate a stronger force.
I prefer to visualize electric and magnetic fields as two little vectors at each point in space. You can get the field lines just by "connecting the arrows" from point to point.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 8,536
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{"url":"https:\/\/www.proofwiki.org\/wiki\/Mapping\/Examples\/x%5E3-x_on_Real_Numbers","text":"# Mapping\/Examples\/x^3-x on Real Numbers\n\n## Example of Mapping\n\nLet $f: \\R \\to \\R$ be the mapping defined on the set of real numbers as:\n\n$\\forall x \\in \\R: \\map f x = x^3 - x$\n\nThen $f$ is a surjection but not an injection.\n\n## Proof\n\nLet $y \\in \\R$.\n\nAs $x \\to \\infty$, we have that $y \\to \\infty$.\n\nSimilarly, as $x \\to -\\infty$, we have that $y \\to -\\infty$.\n\nFrom Real Polynomial Function is Continuous, $f$ is continuous on $\\R$.\n\nIt follows from the Intermediate Value Theorem that:\n\n$\\forall y \\in \\R: \\exists x \\in \\R: y = \\map f x$\n\nThus, by definition, $f$ is a surjection.\n\n$\\Box$\n\nWe have that:\n\n $\\ds \\map f 0$ $=$ $\\ds 0^3 - 0$ $\\ds$ $=$ $\\ds 0$ $\\ds \\map f 1$ $=$ $\\ds 1^3 - 1$ $\\ds$ $=$ $\\ds 0$\n\ndemonstrating that $f$ is not an injection.\n\n$\\blacksquare$","date":"2022-09-26 06:09:38","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\": 0.9283339381217957, \"perplexity\": 222.30724627656832}, \"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-2022-40\/segments\/1664030334802.16\/warc\/CC-MAIN-20220926051040-20220926081040-00281.warc.gz\"}"}
| null | null |
class CreateCategories < ActiveRecord::Migration
def change
create_table :categories do |t|
t.string :name
t.references :user, index: true
t.integer :posts_count
t.timestamps null: false
end
add_foreign_key :categories, :users
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 8,645
|
It certainly is the most wonderful time of the year! Two Pittsburgh classics – Klavon's Ice Cream Parlor and Prantl's Bakery – have teamed up to create a cookie-based, ice cream pizza.
Perfect for any holiday gathering, this epic creations combines layers of Prantl's chocolate chip cookies with various flavors of Klavon's ice cream.
For now, customers can choose from Snickerdoodle and Double Dark Chocolate ice cream. Klavons expects that the product will evolve over time, and soon expand to include more bases and ice cream flavors.
Klavon's owner Jacob Hanchar said the idea for a "pizza" creation came to them when they started working on a delivery business.
"When we looked to partner with a bakery, we wanted to work with another Pittsburgh institution like Prantl's. We told them the idea and they loved it," says Hanchar.
The pizza is $29.99 for pick up at Klavons, and $34.99 delivered.
Follow Klavon's on Facebook for updates on the ice cream and cookie sandwich. To order, contact klavons@klavonsicecream.com or call us (412) 434-0451.
A sinfully delicious looking match that could only come from Pittsburgh.
|
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| 782
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/* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.flowable.dmn.engine.test.runtime;
import org.flowable.dmn.api.DecisionExecutionAuditContainer;
import org.flowable.dmn.api.DmnRuleService;
import org.flowable.dmn.engine.DmnEngine;
import org.flowable.dmn.engine.test.DmnDeployment;
import org.flowable.dmn.engine.test.FlowableDmnRule;
import org.junit.Assert;
import org.junit.Rule;
import org.junit.Test;
import java.util.HashMap;
import java.util.Map;
/**
* @author Yvo Swillens
*/
public class CollectionsContainsReversedTest {
@Rule
public FlowableDmnRule flowableDmnRule = new FlowableDmnRule();
@Test
@DmnDeployment(resources = "org/flowable/dmn/engine/test/runtime/contains_IN_reversed.dmn")
public void testContainsTrue() {
Map<String, Object> processVariablesInput = new HashMap<>();
Person customerOne = new Person();
customerOne.setName("test1");
customerOne.setAge(10L);
processVariablesInput.put("customerOne", customerOne);
DmnEngine dmnEngine = flowableDmnRule.getDmnEngine();
DmnRuleService dmnRuleService = dmnEngine.getDmnRuleService();
DecisionExecutionAuditContainer result = dmnRuleService.createExecuteDecisionBuilder()
.decisionKey("decision")
.variables(processVariablesInput)
.executeWithAuditTrail();
Assert.assertFalse(result.isFailed());
Assert.assertTrue(result.getRuleExecutions().get(1).isValid());
Assert.assertTrue(result.getRuleExecutions().get(2).isValid());
Assert.assertTrue(result.getRuleExecutions().get(3).isValid());
Assert.assertTrue(result.getRuleExecutions().get(4).isValid());
}
@Test
@DmnDeployment(resources = "org/flowable/dmn/engine/test/runtime/contains_IN_reversed.dmn")
public void testContainsFalse() {
Map<String, Object> processVariablesInput = new HashMap<>();
Person customerOne = new Person();
customerOne.setName("test3");
customerOne.setAge(11L);
processVariablesInput.put("customerOne", customerOne);
DmnEngine dmnEngine = flowableDmnRule.getDmnEngine();
DmnRuleService dmnRuleService = dmnEngine.getDmnRuleService();
DecisionExecutionAuditContainer result = dmnRuleService.createExecuteDecisionBuilder()
.decisionKey("decision")
.variables(processVariablesInput)
.executeWithAuditTrail();
Assert.assertFalse(result.isFailed());
Assert.assertFalse(result.getRuleExecutions().get(1).isValid());
Assert.assertFalse(result.getRuleExecutions().get(2).isValid());
Assert.assertFalse(result.getRuleExecutions().get(3).isValid());
Assert.assertFalse(result.getRuleExecutions().get(4).isValid());
}
class Person {
private String name;
private Long age;
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
public Long getAge() {
return age;
}
public void setAge(Long age) {
this.age = age;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,954
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Victor Jaunez (Wallerfangen, 30 april 1839 - Haarlem, 21 juli 1916) was een Duits-Frans ingenieur en ondernemer, die voornamelijk in Nederland werkzaam was. Hij geldt als een van de sleutelfiguren in de geschiedenis van de Maastrichtse aardewerkfabriek Société Céramique (1863-1958), een bedrijf dat hij bijna een halve eeuw leidde.
Levensloop
Victor Jaunez was een telg uit een geslacht van ondernemers en politici uit het Duits-Franse Elzas-Lotharingen. Zijn grootvader, Pierre-Sylvestre Jaunez (1755-1844), was architect en hoofd van het kadaster in Metz. Zijn ontwerp voor de overdekte markt van Metz werd gezien als vernieuwend. Victors oom, Édouard Jaunez (1795-1876), was burgemeester van Metz.
Victor Jaunez werd geboren als zoon van Auguste Jaunez en Elise Souty in het stadje Wallerfangen (Frans: Vaudrevange), in een streek die vanouds door Duitsland en Frankrijk werd betwist, maar die in 1839 tot de Pruisische Rijnprovincie behoorde. Zijn vader was van 1830 tot 1842 directeur van de in 1791 door Nicolas Villeroy opgerichte aardewerkfabriek van Wallerfangen, vanaf 1836 Villeroy & Boch geheten. Omstreeks 1835 bouwde dit bedrijf voor hem een kapitale, nog bestaande villa, waar in 1839 Victor werd geboren. Na 1842 nam een telg uit het geslacht Villeroy de leiding in de fabriek over. Waarschijnlijk bleef Auguste Jaunez daarna in een andere functie verbonden aan het bedrijf.
Victors oudere broer, eveneens Édouard Jaunez geheten (1834/35-1916), was een succesvol ondernemer en politicus. Hij was onder andere vanaf 1862 betrokken bij de bouw van een nieuwe fabriek van de Faïenceries de Sarreguemines en in 1865 medeoprichter van de tegelfabriek Utzschneider & Ed. Jaunez, beide in Sarreguemines (Duits: Saargemünd). Voor de Faïenceries de Sarreguemines acquireerde of stichtte hij minstens zeven aardewerkfabrieken in het Duits-Frans-Luxemburgse grensgebied. Daarnaast was hij burgemeester van het destijds Duitse Saargemünd, parlementslid van het Reichsland Elzas-Lotharingen en lid van de Rijksdag. In 1904 werd hij door keizer Wilhelm II in de adelstand verheven en verkreeg hij (net als zijn nageslacht), als ongetitelde adel, het adelspredicaat von. De keizer bezocht hem in 1905 op zijn kasteel in Rémelfing.
Victor Jaunez volgde wellicht dezelfde opleiding als zijn broer, die afgestudeerd was als ingenieur aan de prestigieuze École centrale in Parijs. Mogelijk werkte hij daarna, net als zijn vader, bij Villeroy & Boch in Wallerfangen. Vanaf 1863 was hij, aanvankelijk onder de hoede van zijn vader, verbonden aan de Société Céramique in Maastricht, waar hij ruim vijftig jaar zou blijven. Waarschijnlijk voelde Jaunez zich als Duits en Frans sprekend Lotharinger redelijk thuis in de grensstad Maastricht, waar het dialect verwant is aan het Duits en de bovenlaag van de bevolking Frans sprak. Het bedrijf waar hij voor werkte was vrijwel geheel in handen van Waalse investeerders, dus ook daar zal de taal geen probleem zijn geweest.
In juli 1868 trouwde Victor Jaunez in Luik met de 27-jarige, in Luik woonachtige Rosalie Kieffer. Zijn ouders woonden toen volgens de huwelijksakte in Metz en zijn vader was op dat moment rentenier. Rosalie was afkomstig uit het Franse Obergailbach, vlak bij de Duitse grens en slechts enkele tientallen kilometers verwijderd van de geboorteplaats van Victor. In de trouwakte erkennen beiden hun drie buitenechtelijk geboren dochters, waarvan de oudste in 1863 te Sarreguemines (de woonplaats van Victors oudere broer Édouard) was geboren, en de twee jongsten, een tweeling, in 1866 te Luik. Waarschijnlijk waren deze buitengewone omstandigheden voor Rosalie reden om zich vóór 1868 niet bij de vader van haar kinderen te voegen. Op 17 oktober 1868 liet het complete gezin zich in het bevolkingsregister van Maastricht inschrijven, met als woonadres Stenenwal nr. 258 (later nr. 38). Twee jaar later overleed de oudste dochter Elisa. In 1872 werd een vierde kind geboren.
Jaunez geldt als bouwheer van de directiewoning 'Villa Jaunez' (in feite een herenhuis) aan de Maaspuntweg 1 (voorheen Stenenwal 28). Of dit de woning was die de familie Jaunez in 1868 betrok na de verhuizing naar Maastricht, is niet zeker. Grenzend aan het herenhuis bevond zich tot omstreeks 1990 een oudere directeurswoning, waarin later kantoren voor beambten waren ondergebracht. Mogelijk woonde het gezin daar voordat het nog bestaande herenhuis werd gebouwd. Jaunez was tevens de eigenaar van de zogenoemde 'Hof van Jaunez', een ommuurd terrein in Wyck waarop zich de ruïne van het Annunciatenklooster bevond. Een deel van de ommuring werd tussen 1882 en 1892 bij de doortrekking van de Wycker Brugstraat gesloopt, maar een restant, de 'muur van Jaunez', stond er nog in 1914.
Victor Jaunez overleed op 77-jarige leeftijd in Haarlem, waar hij verbleef op het adres Kamperlaan 4. Het is niet bekend of hij permanent in Haarlem woonde of er slechts tijdelijk verbleef. De overlijdensaangifte werd gedaan door de tuinknecht Braak en de huisknecht Bon, mogelijk personeelsleden in dienst van Jaunez. Kamperlaan 4 is tegenwoordig het adres van de Mariastichting, een voormalig rooms-katholiek ziekenhuis uit 1899.
Loopbaan
Kort voor 1863 trok Auguste Jaunez zich terug uit de directie van Villeroy & Boch. In dat jaar werd hij benaderd door de Nederlandse ondernemer Wijnand Nicolaas Clermont (1802-1879), mede-oprichter van de Maastrichtse aardewerkfabriek Clermont & Chainay. Clermont nodigde hem uit, samen met zijn zoon Victor, in Maastricht de leiding van de fabriek op zich te nemen. Die fabriek maakte sinds de oprichting in 1851 moeilijke tijden door. Ze werd sinds 1859 geleid door de Belgische ingenieur Guillaume Lambert (1818-1909), die er echter niet in geslaagd was het bedrijf winstgevend te maken. Auguste stemde er in toe zijn zoon Victor twee jaar lang in te werken bij de Société Céramique, zoals de fabriek vanaf 1863 heette.
Auguste Jaunez trok zich in 1865 inderdaad terug. Victor werd in 1867 benoemd tot directeur (in 1893 werd de officiële functieaanduiding directeur-gérant). Zijn voorganger Guillaume Lambert was al in 1863 op een zijspoor gezet en kort daarna vertrokken. Samen met de président (voorzitter van de raad van bestuur) François-Philippe de Haussy (1789-1869) en de in 1867 tot 'administrateur' benoemde graaf Henri de Meeûs (1824-1913) wist Jaunez het op de rand van een faillissement balancerende bedrijf te redden, hoewel de eerste jaren zeer moeizaam verliepen. Door aandelenemissies en leningen kwam er geld vrij voor investeringen en kon de fabriek zelfs worden uitgebreid op de voormalige vestingterreinen die vanaf 1867 beschikbaar kwamen.
In 1869 overleed De Haussy en volgde De Meeûs hem op als 'président'. Toen de afzet in het midden van de jaren 1870 stagneerde als gevolg van de Grote Depressie, drong de bank aan op aflossing van de schulden. Er volgden nieuwe aandelenemissies, waarna het bedrijf vrijwel geheel in handen kwam van de families De Meeûs, De Haussy, Dewandre, Gilliot en hun erven. De Société Céramique groeide dankzij deze kapitaalinjecties en de vasthoudendheid van directeur Jaunez uit tot een krachtige, succesvolle onderneming. Het aantal arbeiders steeg tussen 1866 en 1887 van 229 tot 764. De toenemende productie en afzet manifesteerden zich in een groeiend balanstotaal en forse winsten. In 1891 kon Jaunez met trots aan de raad van bestuur melden dat de onderneming schuldenvrij was. De winst liep op van 42.000 francs in 1878 tot 750.000 francs in 1911.
Op 20 juli 1912, in feite een jaar te vroeg, werd op grootse wijze het vijftigjarig jubileum van de onderneming én van directeur Jaunez gevierd. Jaunez had toen al een deel van de dagelijkse leiding overgedragen aan technisch directeur P.J. Lengersdorff (1868-1925). In juli 1913 werd Lengersdorff benoemd tot directeur-gérant; Jaunez bleef nog enige tijd aan als administrateur. Na het uitbreken van de Eerste Wereldoorlog wisten de Engelsen het ontslag af te dwingen van de Duitser Lengersdorff, die enkele jaren later directeur zou worden van de eerder genoemde aardewerkfabriek van Villeroy & Boch in Wallerfangen, waar hij tot zijn dood in het geboortehuis van Victor Jaunez woonde.
Huwelijk en nageslacht
Victor Jaunez trad op 25 juli 1868 te Luik in het huwelijk met Rosalie Kieffer (1840-1898), dochter van Georges Kieffer en Marguerite François. Het echtpaar kreeg vier dochters, waarvan drie buitenechtelijk werden geboren:
Elisa Jaunez (1863-1870)
Marie (Louise) Jaunez (1866-1948), tweelingzuster van Cécile
Cécile Jaunez (1866-?), tweelingzuster van Marie
Fanny Rosalie Jaunez (1872-?)
Nalatenschap
In het bedrijfsarchief van de Société Céramique in het Sociaal Historisch Centrum voor Limburg bevinden zich diverse nauwelijks bestudeerde stukken met betrekking tot Victor Jaunez. Sommige hebben betrekking op het zilveren en gouden bedrijfsjubileum van Jaunez. Andere roepen meer vragen op, zoals de correspondentie die Wijnand Clermont in 1869-1870 voerde met de fabrikant Ed. Wood te Porthill (Newcastle-under-Lyme, in de zgn. Staffordshire Potteries, West Midlands, Engeland) over een mogelijke aanstelling van Jaunez als directeur daar.
Producten van de Société Céramique uit de periode Jaunez bevinden zich in het Centre Céramique en diverse andere museale en particuliere collecties. Een jubileumbord uit 1913 (wél het 'goede' jubileumjaar) toont een portret van 'président' De Meeûs en 'directeur-gérant' Jaunez. Een jaar eerder schilderde Henri Goovaerts (1865-1912) een portret van Jaunez – driekwart zittend en met een sigaret in zijn hand – in olieverf. Het schilderij bevond zich jarenlang in de bedrijfscollectie van de Société Céramique en de Koninklijke Sphinx. Het werd in 1993 voor het laatst geëxposeerd. De huidige verblijfplaats is onbekend.
De Villa Jaunez is een laat-negentiende-eeuws, deels vrijstaand herenhuis in Wyck-Maastricht, dat fraai gelegen is aan de Maas. Het voormalige woonhuis van directeur Jaunez en zijn gezin is in 1999 zowel uitwendig als inwendig sterk verbouwd naar plannen van architect Jo Coenen en maakte twee decennia deel uit van het kantoorgebouw Maaskantoren. In 2019 werd het verbouwd tot luxe-appartementencomplex. De hoge voorgevel aan de Maaspuntweg (voorheen Stenenwal) heeft een plint van grijze natuursteen en daarboven speklagen van natuur- en baksteen. Op de begane grond bevindt zich de entree met de oorspronkelijke dubbele paneeldeur en een vierkant venster met ingebouwd rolluik. De eerste verdieping bevat een smal rechthoekig venster en een breed rondboogvenster met een hardstenen omlijsting. Alle vensters hebben glas-in-loodbovenlichten. Boven de kroonlijst met consoles bevindt zich nog een lage mezzanino met vijf kleine ronde vensters. Het vrij vlakke lessenaardak is geheel vernieuwd. In de noordelijke zijgevel, die oorspronkelijk niet vrijliggend was, zijn in 1999 vensteropeningen aangebracht die een blik gunnen op het interieur met onder andere een fin de siècle trappenhuis en stucdecoraties. Een niet-historisch grachtje verbindt het oude gebouw met de achtergelegen nieuwbouw aan de noordzijde van Plein 1992, die bestaat uit glazen vliesgevels en een wit gestucte luifel met stalen lamellen op slanke kolommen. De tuinmuur van Villa Jaunez is aan de noordzijde deels bewaard gebleven in de uitbreiding van het toenmalige Hotel Maastricht (nu Crowne Plaza Maastricht).
De Biscuit- of Bordenhal op het Plein 1992 (zuidzijde) is het enige bewaard gebleven fabrieksgebouw van de Société Céramique uit de periode Jaunez. De productiehal uit circa 1880 was aanvankelijk een fabriekje voor het inbranden van bisquit-aardewerk, later voor het inbranden van glazuur en het maken van borden. Het is een langgerekt bouwwerk van witgeschilderde baksteen in traditioneel-ambachtelijke stijl. De kopgevel aan de Maaszijde heeft twee karakteristieke stalen rondboogvensters en een rond stalen topgevelvenster. Kenmerkend is de dakconstructie van stalen Polonceau-spanten, die rusten op steunberen die aan de buitengevels zichtbaar zijn. Het middendeel van de houten dakconstructie is verhoogd, waardoor de hal een hoge lichtinval heeft. In 1951 werd de hal verbouwd, waarbij het aantal lichtvlakken in het dak werd uitgebreid. Een jaarsteen herinnert aan de verbouwing. In 1998-1999 werd het bouwwerk gerenoveerd door architect Jo Coenen en ingericht tot theaterzaal voor Theatergroep Het Vervolg (sinds 2009 Toneelgroep Maastricht).
Duits ondernemer
Nederlands ondernemer
Geschiedenis van Maastricht
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Вилхелм Млади фон Брауншвайг-Люнебург (; * 4 юли 1535, † 20 август 1592) от род Велфи, е херцог на Брауншвайг и Люнебург и от 1559 до 1569 г. заедно с брат му Хайнрих и от 1569 г. сам княз на Люнебург.
Живот
Той е четвъртият син на княз Ернст I (1497 – 1546) и София фон Мекленбург-Шверин (1508 – 1541), дъщеря на херцог Хайнрих V от Мекленбург (1479 – 1552) и първата му съпруга Урсула фон Бранденбург (1488 – 1510).
На 12 октомври 1561 г. Вилхелм Млади се жени за принцеса Доротея Датска (* 29 юни 1546, † 6 януари 1617), дъщеря на крал Кристиан III от Дания и Доротея фон Саксония-Лауенбург (1511 – 1571).
От есента на 1577 г. Вилхелм е психически болен и през 1587 г. е поставен в арест в стаята си на резиденцията му в Целе. Понеже синовете му са малолетни, управлението на Княжество Люнебург е поето от градски съветници. Зет му маркграф Георг Фридрих фон Бранденбург е поставен за регент. Фактически управлението води съпругата му Доротея.
Вилхелм Млади е погребан в княжеската гробница в църквата "Св. Мария" в Целе. Последван е чрез жребий от сина му Георг.
Деца
Вилхелм Млади и Доротея Датска имат децата:
София (1563 – 1639) ∞ от 1579 за Георг Фридрих I фон Бранденбург-Ансбах
Ернст (1564 – 1611)
Елизабет (1565 – 1621) ∞ граф Фридрих фон Хоенлое-Лангенбург (1553 – 1590)
Христиан (1566 – 1633), епископ на Минден
Август Стари (1568 – 1636), епископ на Ратцебург
Доротея (1570 – 1649) ∞ Карл пфалцграф фон Биркенфелд (1560 – 1600)
Клара (1571 – 1658) ∞ от 7 март 1593 за граф Вилхелм фон Шварцбург-Франкенхаузен
Анне Урсула (1572 – 1601)
Маргарета (1573 – 1643) ∞ от 16 септември 1599 Йохан Казимир фон Саксония-Кобург
Фридрих IV (1574 – 1648)
Мария (1575 – 1610)
Магнус (1577 – 1632)
Георг фон Каленберг (1582 – 1641), прародител на днешната линия
Йохан (1583 – 1628)
Сибила (1584 – 1652) ∞ Юлиус Ернст фон Брауншвайг-Даненберг (1571 – 1636)
Литература
Christa Geckler: Die Celler Herzöge. Leben und Wirken 1371 – 1705. Georg Ströher, Celle 1986, ISBN 3-921744-05-8.
Georg Schnath in: Geschichte des Landes Niedersachsen, Ploetz, Würzburg, 1973, S. 28
John Morby, Dynasties of the World: a chronological and genealogical handbook (Oxford, Oxfordshire, U.K.: Oxford University Press, 1989), page 133.
Източници
Външни препратки
Die Welfen
Biographie Wilhelm des Jüngeren
Херцог (Брауншвайг-Люнебург)
Князе на Люнебург
Велфи
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In 1906 at Brule, OK now part of the town of Buffalo, OK, a small group of people met and organized the First Christian Church of Buffalo, with Wm. Foster as elder, E.M. Best and M.H. Miller as deacons, Mrs. Frances Miller, clerk, and Mrs. R.E. McMinn, treasurer. J.J. McClain was engaged to preach once a month. Another of the first ministers was a T.A. Mullen.
This organization existed until January 25, 1909, when C.P. Murphy, C.W.B.M. Living Link Evangelist of Fredrick, OK, helped reorganize the church. The following members were chosen for officers: Elders: J.K. Baker and H.C. Bayne; Deacons: N. C. Austin, E.M. Best and W.H. Miller; Secretary-Treasurer—Mr. Frances Miller. Church services were held for a while at the Baptist Church building then moved to the E.M. Best Hall.
There were seventeen charter members: Mr. & Mrs. E.M. Best, Mr. & Mrs. W.H. Miller, Mr. & Mrs. H.C. Bayne, Mrs. Sarah Ferguson, Mr. Harvey Ferguson, Mr. & Mrs. J.M. Ferguson, Mr. & Mrs. Everett Foster, Mr. J.K. Baker, Mr. & Mrs. N.C. Austin, and Mrs. Robert E. McMinn.
Today, October 1973, Mrs. J.M. (Chella) Ferguson is the only charter member living and still attends the church services being very active in church work.
In 1910, the church members engaged an evangelist, Rev. McClure, to hold a meeting. During this meeting, there were forty-five additions.
The members appointed a building committee composed of Dr. R.R. Anderson, E.M. Best, I.C. Jones, N.C. Austin, J.C. McNutt and W.H. Miller. Plans were made and work began on the basement of the church. January, 1911, the basement was completed and dedicated the first of August, 1911.
In 1911, another evangelist, Rev. Stine held a meeting and there were twelve more additions. The following years the membership increased.
In February 1912, an election of officers was held for four elders, six deacons, five deaconesses, and a clerk.
In 1913, more officers were added to those already serving and Mr. E.M. Best was elected clerk and Mr. A.B. Dewater was elected treasurer.
After the completion of the church basement, regular Sunday School and church services were held.
In 1914, church officers and the congregation voted to offer their place of worship to the Presbyterians for services on the 5th Sundays. Plans were also made for a parsonage. This parsonage was completed and was used for many years.
The old record books show that Mrs. E.M. Best was the church secretary through 1916, with church treasurers as A.B. Dewater and C.A. Wyatt.
The Christian Church carried on through the following years gaining in membership. These faithful dedicated members gave their time and talents to further the preaching and teaching of the gospel of Jesus Christ and the New Testament Church.
In 1918, the Pleasant Valley Christian Church located in the county northeast of Buffalo, closed and some of its members came in the Buffalo Christian Church.
By 1926—27, the church membership began planning for its new church building. A building committee was appointed: A.M. Barker (Minister), Chairman; Earl Anderson, Vice-Chairman; Thos. G. Cook, Secretary-Treasurer; Gordon Lauer, Theron Temple, C.E. McMinn, W.H. Temple, and J.W. Reed, members.
During the time of the building, our Sunday School and church services were held each Sunday in the old court room on the second floor of the Miles Building.
In 1929, our new red brick church building was completed and dedication services were held Sunday, July 29, 1929. We looked forward to this special day with much pride and enthusiasm. A special program was planned for the morning worship. Dr. I.N. McCash, president of Phillips University, was our guest speaker. A basket dinner was brought in by the members and served at noon in the church basement. That afternoon, the program and services continued. Visiting ministers participated in the program along with vocal and instrumental numbers given by churches from neighboring towns.
During the preparation for the financing of the building, members worked faithfully to contribute and raise money. Some borrowed money personally, some gave what they could and others gave of their time and labor. Several took part in a fund raising program. Donations were sent in from business firms in some of our larger cities as a result of the effort. Our Ladies Aid group contributed quite a sizable amount of money over a period of years, also.
The Ladies Aid group served dinners to the Chamber of Commerce and Kiwanis Clubs each week for a number of years. Also, special dinners and banquets for businessmen, 4-H clubs, alumni, and farm sales were served. This group of women worked faithfully. They donated their time, labor, and food for the cause.
After building the new church some difficult times due to weather occurred which hurt the membership financially. First came a severe storm with hail that wiped out a promised wheat crop. Then came the dry years of the thirties called the drought. Dust bowl conditions followed and became very serious. Farmers were fortunate to make enough to survive the nation-wide depression that followed, hence, it slowed down the paying off of our building debt.
As times began to improve and normal crop years were back, we were able to make some improvements on our parsonage. We bought furniture and floor covering. Later on, extensive remodeling work was done on the parsonage. New floors and new roofs along with changes in the kitchen and bathroom were made.
In 1948, during the ministry of Tom B. Clark, it was decided by the church board and members to add improvements to our church, the church yard, and the parking lot. The sanctuary was redecorated with new carpeting being laid in the sanctuary and aisles, evergreen trees were planted in the church yard completely around the parking lot, and sidewalks were made where needed.
During the years of the forties and fifties, our membership was able to take care of the building indebtedness. This was completed during the ministry of Elmer N. Earley, who served as minister from 1956 through 1958. Brother Earley left in the spring 1959. We called Carroll McCoy as minister. In the fall of that year we decided on a date that would be convenient for the Earleys to come back to visit for the special service witnessing the burning of the mortgage.
This event was held in December during the Christmas holidays. In the evening a fellowship supper was held followed by a short program at which time we witnessed the burning of the mortgage, with Elmer N. Early, former minister, Clayton H. Lauer, Chairman of the finance committee and Carroll McCoy, minister, presiding.
In 1958, The First Christian Church of Selman, Oklahoma closed, and many of its members came to unite with the First Christian Church of Buffalo. Previous to that date, several of the Selman church members had already placed their membership here.
Plans were underway for the building on the south wing of the church which would be the educational building. The ground breaking ceremony took place directly after the morning services on December 14, 1958.
There were some changes made because of the new addition. The church kitchen was moved from the southeast side of the fellowship hall to the northeast side resulting in an almost complete new kitchen with new cabinets and appliances.
During the years after the church building debt was paid, a building fund was set up and contributed to generously. Thus, we were able to continue to make improvements needed from time to time in our church and its new addition.
A few years after the educational sing was completed, a new carpet was added to the foyer stairs and hall.
In 1959 and 1973, two individuals of our membership generously presented our choir with new robes. These gifts were greatly appreciated.
A special fund had been started by members of our church and added to generously to purchase a new organ for the sanctuary. A committee was appointed to select the organ. This purchase was made in 1957. Later a small upright piano was bought to match the organ in wood and color. The baby grand piano then in use in the sanctuary was moved to the fellowship hall.
In 1960, more lots were bought directly south of the parking lots to enlarge our parking space. This extended our church property south to Turner Street. Evergreen trees were planted on the wast side of the new parking lots.
Other improvements have been made over the years, such as water coolers for the sanctuary which were later replaced by refrigeration units. Then later, gas central cooling and heating were added. The first heating system was a furnace in which coal was used.
In 1963, our church board and membership decided to make plans for a new parsonage. An auction was held selling the old parsonage and contents. The parsonage was purchased by Keith Lauer and moved to his land two miles west of Buffalo to be used on building his family's new home.
In 1964, the modern three-bedroom brick parsonage was built. Dr. Richard Yaple's family was the first to occupy the new parsonage. Dr. Yaple was engaged to preach for a short time before the completion of the parsonage.
The Fellowship Hall in the church basement was due some much needed improvements. In the spring of 1971, some remodeling was done consisting of paneling the walls and lowering and straightening the ceiling.
In the spring of 1973, the sanctuary walls were repainted, new carpeting was laid on all the sanctuary floor, woodwork was varnished, new cushions were made for the pews, and some rearranging of the pews was done to make space in the center aisle.
In 1973, the church constitution was amended so that the Deaconesses could be elected. The following year the board decided that greeters were needed each Sunday to welcome those arriving for Sunday school and church services.
In 1975, new air conditioners for the sanctuary and fellowship hall were installed along with a chair lift on the main stairway to the sanctuary.
In 1976, the board of elders decided to schedule teams of two from the elders, deacons, and deaconesses to take communion to shut-ins, hospital, and nursing home. Also this year a speaker was installed in the nursery so that the attendant could enjoy the worship service. New robes were purchased at a cost of $5,113.00.
In 1977, a love gift of $400 was given to Lucia Miller, a member of our church and also "Miss Oklahoma," to help her on her trip to Korea with the Revival Fires Troupe. The World Outreach Committee also furnished $160 for Lucia's evangelistic trip to Alaska as Miss Oklahoma.
The board voted to add to the bylaw #9, the names of all deaconesses and to add to bylaw #10, names of all elders emeritus.
It was agreed that the church would send cards to all its ill members and/or their family. Also, it was decided to start sending birthday cards to all residents in the Western Nursing Home. New song books were purchased—46 being paid for by members in memoriam and 121 were paid for by the church. A new organ as purchased at a total figure of $4,750.00 with $1,800.00 coming from the memorial fund and $2,950.00 from the general fund. New horizontal speakers were also installed in the rear of the sanctuary.
In 1979, new tinted glass doors and panic bar opening were installed in the narthex.
In 1980, the church contributed $100 toward establishing a chapel in the Harper County Community Hospital.
1982 saw the church participating in a school of missions were various missionaries came to this area to give us information on their individual mission work. This has proven very successful with our church contributing $20,330.00 during the past ten years the school has existed.
This was the year that a contractor was hired to clean the outside of the church and educational building, re-mortar, caulk, scrape all painted wood and repaint. They also sealed all cracks and plaster inside the church and painted the walls.
The tall shrubbery was removed from around the church and flower beds were made and are maintained by JoAnn Balke, the minister's wife.
Due to efforts of the Ministerial Alliance, each church takes their turn holding a worship service for the residents of the Western Nursing Home. Our church held their first service for the residents in February 1983.
A need was seen for a restroom for men on the second floor. With some remodeling, an empty classroom was converted into a very nice restroom for the women with ample space for dressing for special occasions. In doing this, the restroom previously for the women was turned into the men's restroom.
Early in the year 1985, Bernie Barkley discussed with the board her desire to serve in Zaire, Africa, as a missionary bookkeeper for the African Christian Mission. This was to be investigated as to coast, needs, etc. Bernie made a positive commitment to furnish $894.65 per month toward the required $1500. The board suggested the congregation vote on being responsible for $600 per month. The vote was 55 yes to 5 no in support of the $600 per month commitment. Brother Jerry Balke was to be her forwarding agent. Bernie served from May 1986 through April 1989.
A new sound system and tape recording unit was purchased and installed in 1985 costing $2215.
Our church joined several other area churches in the collection and delivery of food for the Old Mexico Project in 1987.
During Valentines' week the adults sponsor a Sweetheart Banquet for the youth and then the youth sponsor a banquet for the adults. It has become a tradition that at the adult banquet secret prayer partner names are drawn for the year for all those who wish to participate. These names are revealed at the next year's banquet.
Janie Waugh presented the board a video program outlining the details of an inte4antional Youth for Christ seminar to be held in Washington D.C. It included freshmen through senior classes and was for the purpose of developing young Christian leaders. The trip was made in late July 1988 and also in 1991 by some of our youth and their sponsors. The youth served delicious Sunday dinners in the fellowship hall once a month to help fund the trips.
A memorial fund was established for Brian Appleton, son of Bruce and De'ett Appleton. Brian was a young member of our church who was accidentally killed. The church helped in securing a television and video unit for the church in Brian's memory. Also, a basketball goal post was erected in his memory.
Our church participates in community Thanksgiving Day services where most of the churches have a special worship program at the high school auditorium. A collection is taken up that is used by the Ministerial Alliance to help transients throughout the year.
A vacant store window on main street was adopted by the church and is maintained with various religious themes with the intent of witnessing to all of our living Lord.
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Cervical Herniated Disc Surgery in India
What is Cost of Cervical Disc Replacement in India?
"Cervical disc replacement cost in India starts from $7000, depending on the hospital, doctor experience, accommodation etc. opted by the patient. The same procedure, however, costs much more in the Western countries."
Cervical disc replacement is a surgical procedure conducted to replace the diseased or affected cervical disc with an artificial disc. This is done without hampering the effective range of motion of the spinal disc.
Low cost of cervical disc replacement in India is one of the leading reasons why an alarming number of patients from abroad prefer to come to the country for this procedure. It is estimated that by choosing India over any other country, a medical tourist typically saves more than 70 percent of their money.
It is estimated that cervical disc replacement in the US costs anywhere between $45000 and $60000. Bearing such exorbitant costs for the same line and quality of treatment does not seem logical to many patients and this is the reason why they decide to avail affordable cervical disc replacement in India.
Why to undergo Cervical disc replacement Surgery?
What are the causes of Cervical Disc Degeneration?
What are the symptoms of Cervical Disc Degeneration?
How Cervical Disc Replacement is performed?
How to recover from Cervical Disc Replacement?
What are the benefits of Cervical Disc Replacement?
What are the risks associated with the Cervical Disc Replacement Surgery?
Why people prefer Cervical Disc Replacement in India?
FAQ – Cervical Herniated Disc Surgery & Slip Disk Surgery
Best Hospitals for Cervical Disc Replacement in India
Top Doctors for Cervical Disc Replacement in India
Articles Related to Disc Replacement
Cervical disc replacement is a surgical procedure used to remove the degenerated disc and replace it with an artificial disc. The surgery is conducted to relieve the symptoms caused by narrowing of the vertebrae or increased pressure on the spinal cord by the cervical disc. The symptoms may include numbness, weakness, and pain in the spinal cord.
The pain may arise from the degenerated disc itself. It may also arise from other structures such as small joints (facet joints) present at the back of the spine that may become overloaded due to narrowing or degeneration of the cervical disc.
The narrowing of the cervical disc progresses gradually, which results in the sinking of vertebrae on the lower one. This puts an extra pressure on the facet joints, which may eventually pinch the nerve root. This may give rise to pain in the legs and other related symptoms.
Disc replacement surgery is a solution to the pain due to degeneration of the disc, but only if the patient does not respond to other non-surgical treatment. The artificial disc is inserted in place of the degenerated disc and it mimics the action of a normal and healthy disc.
This helps restore the normal movement in the cervical disc. It also helps relieve pressure on other parts of the spine, such as the nerve root.
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There are many reasons which may lead to the degeneration of the cervical disc. Some of them include the following:
Facet joint arthritis
Serious trauma or injury to the cervical disc
Breakdown of the collagen framework
Dehydration of the cervical disc
The cervical disc degeneration usually results in the following symptoms:
Neck stiffness
Radiating pain towards the arms or shoulders
Weakness in the arms or shoulders
Numbness in the arms
The cervical disc replacement surgery is performed under the influence of general anesthesia. The area of the neck where the incision has to be made is cleansed with an antiseptic solution. An incision of around one to two inches is made in a horizontal fashion on one side in front of the neck.
The degenerated disc is exposed carefully by moving aside the vital parts of the neck. The affected disc is removed completely along with any impinging disc fragments or bone spurs which may press the nerve running through the spine.
A surgical magnifying glass is used to facilitate disc removal and decompression of the nerves. The original disc height is restored as degenerated disc typically shrinks in height. The artificial cervical disc is then implanted carefully into the disc spaces between the vertebrae with the help of X-ray or fluoroscopy. The incisions beneath the skin are then closed through stitches. The outer skin is then closed and bandaged properly. The patient may be advised to put a rigid or soft cervical collar around the neck to restrict motility for the few days.
The patient is taken to the observation room after the surgery. All the vital signs such as the pulse, blood pressure, and breathing are stabilized and the patient is transferred to the patient room once he or she is alert and stable.
The patient may be required to stay in the hospital for at least two to three days after the surgery. The doctor may ask the patient to wear a cervical collar for at least seven days to immobilize the area.
The pain due to surgery improves markedly within three to four days. Symptoms related to nerves such as numbness, pain, and weakness usually wear off within one or two days. The patient is likely to get back to lighter work routine after one or two weeks and to the normal work routine within five to six weeks after the cervical disc replacement surgery.
The benefits of cervical disc replacement over spinal fusion surgery are as follows:
The risk of a revision surgery is less
Adjacent spinal discs do not have to bear the extra stress
The need for painful bone graft is eliminated
Minimal recovery time
How can Lyfboat assist you getting Cervical Herniated Disc Surgery in India?
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Lyfboat is a free advisory platform; we do not charge any fees from patients. In fact, we negotiate the price that Indian hospitals offer. In some cases we are able to reduce the cost by negotiating upto 20% of what Hospitals generally offer. We advise the best treatment from the top hospital/surgeon at best price.
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Following risks are associated with the cervical disc replacement surgery:
Bleeding and infection
Bending, breaking, loosening or movement of the artificial disc
Nerve injury that may lead to paralysis
Changes in the voice of the patient
Leakage of the spinal fluid
Difficulty in breathing or swallowing
In rare cases, stroke
Cervical disc replacement in India is done to ease off the symptoms of the degenerative cervical disc. This surgery is often recommended when the patient is not benefited from non-surgical options and therapies used for the treatment of disc degeneration. First, the cost of cervical disc replacement in India is quite low as compared to other countries, particularly the North American and European nations. Second, the massive pool of medical expertise that a patient from abroad gets to avail in India is unparalleled to any country in the world.
Questions about the Implants? Let us know.
Q. What is slip disc or herniated disc?
Slip disc or bulging disc or herniated disc refers to a problem with a rubbery disc between the spinal bones.
Q. What are the symptoms of a slip disk?
Following are the symptoms of a slipped disk:
Pain and numbness to one side of the body.
Short distance walking causes pain
Weakness of muscle
Pain extending to arms or legs
Pain that worsens at night or with certain movements like standing or sitting
Burning, aching or tingling sensations in the affected area.
Q. What causes slipped discs?
Following are the causes of slipped discs:
Outer ring becomes weak or torn and allows the inner portion to slip out.
Certain motions like twisting or turning to lift an object.
Lifting a very heavy object.
Physically demanding job requiring a lot of lifting.
Overweight.
Weak muscles
Sedentary lifestyle.
Q. How are slipped discs diagnosed?
Physical exam will be performed first with the doctor. Nerve function and muscle strength will be checked. Doctor will also check whether you feel pain when moving or touching the affected area. Thenafter they will perform Imaging tests. It can help the doctor view the bones and muscles of patient's spine. It can also identify any damaged areas. Few of the imaging scans include:
The doctor can combine all informations to determine what is causing pain, weakness, or discomfort to the patient.
Q. What complications are there of a slipped disk?
Following are the complications associated to slipped disk:
Saddle anesthesia
Q. Is it possible to prevent a slipped disk?
One can take steps to reduce the risk of developing a slipped disk. These steps include:
Use safe lifting techniques.
Maintain healthy BMI.
Do exercises to strengthen the muscles in your legs, abdomen, and back.
Do not remain seated for long periods.
Need help in deciding a best hospital?
The Indian cervical disc replacement hospitals are globally renowned for their quality of services that they offer to the patients. These hospitals have earned a name for themselves by consistently maintaining a great surgical success record for many years.
Medanta the Medicity
BLK Super Speciality Hospital
Max Super Specialty Hospital
Fortis Memorial Research Institute
The best spine surgeons in India are globally renowned for their surgical skills and expertise. A majority of them have had an international exposure when it comes to the treatment of spine-related disorders. Additionally, a majority of Indian surgeons are a member of professional bodies in the field of neurosurgery. They attend several seminars and conferences around the year to educate and update themselves as per the changing technological trends in the field of neurosurgery.
List of some famous cervical disc replacement specialists:
Dr. Hitesh Garg
Dr. A K Banerji
Dr. S K Sogani
Dr. Rajagopalan Krishnan
Dr. C M Malhotra
Dr. Rajendra Prasad
Happy Lyfboat Patients
Lumbar Disc Replacement India
Cervical Disc Replacement India
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{"url":"http:\/\/math.stackexchange.com\/questions\/297851\/how-to-prove-that-if-a-normed-space-has-schauder-basis-then-it-is-separable-wh\/297918","text":"# How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?\n\nCan we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary parts for that matter)?\n\nWhat can we say about the converse?\n\n-\n\nYes, you have the right idea.\n\nLet $\\mathbb K$ be either $\\mathbb R$ or $\\mathbb C$ as appropriate. Suppose that the Banach space $X$ has a Schauder basis $\\{x_n\\}_{n=1}^\\infty$, i.e., for every $x \\in X$, there exists a unique sequence of scalars $\\alpha_n \\in \\mathbb K$ such that $$x = \\sum_{n=1}^\\infty \\alpha_n x_n,$$ where this sum converges in the norm topology. We can renormalize so that $\\Vert x_n \\Vert =1$ for all $n$.\n\nWe fix such an $x \\in X$ and show how to approximate it by elements from a countable set. Given $\\varepsilon >0$, there exists $N \\in \\mathbb N$ such that $$\\Vert x - \\sum_{n=1}^N \\alpha_n x_n \\Vert < \\varepsilon \/ 2.$$\n\nFor each $\\alpha_n \\in \\mathbb K$, we can find $\\beta_n$ (in $\\mathbb Q$ or $\\mathbb Q + \\mathbb Q i$ as appropriate) such that $| \\alpha_n - \\beta_n | < \\varepsilon\/ 2^{n+1}$.\n\nThen, by the triangle inequality, $$\\Vert x - \\sum_{n=1}^N \\beta_n x_n \\Vert < \\Vert x - \\sum_{n=1}^N \\alpha_n x_n \\Vert + \\Vert \\sum_{n=1}^N \\alpha_n x_n - \\sum_{n=1}^N \\beta_n x_n \\Vert < \\varepsilon\/2 + \\sum_{n=1}^N \\varepsilon\/2^{n+1} < \\varepsilon.$$ Thus every element in $X$ can be approximated by finite linear combinations of the elements of the Schauder basis, where the scalars come from $\\mathbb Q$ or $\\mathbb Q + \\mathbb Q i$, as appropriate.\n\nAs for your second question: no, not every separable Banach space has a Schauder basis. This was a longstanding problem in the field, which was solved by Per Enflo in 1972 (for which he was awarded a live goose!).\n\nThis result can be found in:\n\nEnflo, Per (July 1973). \"A counterexample to the approximation problem in Banach spaces\". Acta Mathematica 130 (1): 309\u2013317. doi:10.1007\/BF02392270\n\n-\nIt's quite impossible to resist the temptation to mention the goose :-) \/\/ Notice that OP does not assume the space to be Banach, but of course the same argument for separability works for incomplete spaces. \u2013\u00a0Martin Feb 8 at 9:36\n@Martin Just to mention that later Enflo gave an example of a Banach space with AP but without a basis, answering in the negative the question whether the two notions are equivalent. \u2013\u00a0Theo Feb 8 at 17:31\n\nLet $X$ be a norm space and $(e_i)$ be a Schauder basis of $X$ (suppose $||e_i||=1$). Consider the countable set $$Q= \\{ \\sum\\limits_{i=0}^n q_ie_i : n \\in \\mathbb{N}, q_i \\in \\mathbb{Q} \\}$$ Let $\\displaystyle x= \\sum\\limits_{i \\geq 0} x_ie_i \\in X$ and $\\epsilon >0$. There exists $n \\geq 0$ such that $\\displaystyle \\left\\|x- \\sum\\limits_{i=0}^n x_ie_i \\right\\| \\leq \\epsilon$, and for $0 \\leq i \\leq n$, there exists $q_i \\in \\mathbb{Q}$ such that $\\displaystyle |q_i-x_i| \\leq \\frac{\\epsilon}{n+1}$ because $\\mathbb{Q}$ is dense in $\\mathbb{R}$. Then $\\displaystyle y= \\sum\\limits_{i=0}^n q_ix_i \\in Q$ and $$||x-y|| \\leq ||x- \\sum\\limits_{i=0}^n x_ie_i||+ \\sum\\limits_{i=0}^n |x_i-q_i| \\cdot ||e_i|| \\leq 2 \\epsilon$$ If $X$ is a norm space over $\\mathbb{C}$, you can consider $Q+iQ$ rather than $Q$.","date":"2013-05-21 18:55:04","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\": 0.969774603843689, \"perplexity\": 91.8103628720358}, \"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-2013-20\/segments\/1368700438490\/warc\/CC-MAIN-20130516103358-00009-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/codereview.stackexchange.com\/questions\/41821\/case-study-with-a-biological-populations-a-list-of-lists-of-lists","text":"# Case study with a biological populations: a list of lists of lists\n\nI have a population (Pop) which has an attribute which is a list of individuals (Ind) where each individual has an attribute which is a list of chromosomes (Chromo). Each chromosome is a list of numbers which tells about the fitness (=reproductive success, the fitness is obtain by multiplying all numbers of a all chromosomes) of the individuals. Within one position on the chromosome, there are different values for the different individuals. I'd like to set the greatest value to 1 and the others to keep their relative value in comparison to the biggest value.\n\nFor example, if on one position on the n-th chromosome, the individuals in the population have the values [3,4,0.4,12,5,6] (that would be a case of a population of 6 individuals).\n\nI'd like to set these value to:\n\na = [3,4,0.4,12,5,6]\n[i\/float(max(a)) for i in a]\n\n\nI tried to create this function but I got lost and can hardly find out a solution in all these lists!\n\nYou can reach the 14th of the 4th chromosome in the 25th individual by writing:\n\npopulation.inds[24].chromosomes[3].alleles[13]\n\n\nTo make sure my aim is understood. I'd like to create a function which take an instance of Pop as argument and return the same or another Pop where all positions on the chromosomes are replaced by numbers in the range [0,1] respecting the relative values of all numbers at the same position on the same chromosome in the population.\n\nBelow is my code (it is long but reading the constructor of the classes Chromo, Ind and Pop (which are all impressively basic) should, I hope, be enough. The class WalkerRandomSampling serve only the purpose of performing a random weighted sampling).\n\nOne might have a look to what I tried, the method is called set_best_fitness_to_one and is within the Pop class.\n\nfrom random import uniform, gauss, choice, expovariate, shuffle\nfrom numpy import arange, array, bincount, ndarray, ones, where\nfrom numpy.random import seed, random, randint, binomial\nfrom operator import mul, methodcaller\n\nclass WalkerRandomSampling(object):\n\"\"\"Walker's alias method for random objects with different probablities.\n\nBased on the implementation of Denis Bzowy at the following URL:\nhttp:\/\/code.activestate.com\/recipes\/576564-walkers-alias-method-for-random-objects-with-diffe\/\n\"\"\"\n\ndef __init__(self, weights, keys=None):\n\"\"\"Builds the Walker tables prob and inx for calls to random().\nThe weights (a list or tuple or iterable) can be in any order and they\ndo not even have to sum to 1.\"\"\"\nn = self.n = len(weights)\nif keys is None:\nself.keys = keys\nelse:\nself.keys = array(keys)\n\nif isinstance(weights, (list, tuple)):\nweights = array(weights, dtype=float)\nelif isinstance(weights, ndarray):\nif weights.dtype != float:\nweights = weights.astype(float)\nelse:\nweights = array(list(weights), dtype=float)\n\nif weights.ndim != 1:\nraise ValueError(\"weights must be a vector\")\n\nweights = weights * n \/ weights.sum()\n\ninx = -ones(n, dtype=int)\nshort = where(weights < 1)[0].tolist()\nlong = where(weights > 1)[0].tolist()\nwhile short and long:\nj = short.pop()\nk = long[-1]\n\ninx[j] = k\nweights[k] -= (1 - weights[j])\nif weights[k] < 1:\nshort.append( k )\nlong.pop()\n\nself.prob = weights\nself.inx = inx\n\ndef random(self, count=None):\n\"\"\"Returns a given number of random integers or keys, with probabilities\nbeing proportional to the weights supplied in the constructor.\n\nWhen count is None, returns a single integer or key, otherwise\nreturns a NumPy array with a length given in count.\n\"\"\"\nif count is None:\nu = random()\nj = randint(self.n)\nk = j if u <= self.prob[j] else self.inx[j]\nreturn self.keys[k] if self.keys is not None else k\n\nu = random(count)\nj = randint(self.n, size=count)\nk = where(u <= self.prob[j], j, self.inx[j])\nreturn self.keys[k] if self.keys is not None else k\n\ndef test(self):\nweights = [12,3,2,0,5]\ntest = WalkerRandomSampling(weights=weights)\na = []\nfor i in xrange(10000):\na.append(test.random())\nb = []\nfor value in range(4):\nb.append(len([i for i in a if i == value])\/float(len(a)))\n\nprint b\nprint weights\n\nclass Chromo(object):\ndef __init__(self, alleles):\nself.alleles=alleles\n\ndef mutations(self):\nnb_mut = binomial(chromo_size, mut_rate)\nfor one_mut in xrange(nb_mut):\nself.alleles[choice(range(chromo_size))] *= pdf_mutation(pdf_mut_scale)\nreturn self\n\nclass Ind(object):\ndef __init__(self, chromosomes):\nself.chromosomes = chromosomes\n\ndef fitness(self):\nif nb_chromosomes == 1:\nreturn reduce(mul, self.chromosomes[0].alleles)\nfit = 1\nfor gene_pos in xrange(chromo_size):\nalleles = []\nfor chromo_pos in range(len(self.chromosomes)):\nalleles.append(self.chromosomes[chromo.pos].alleles[gene_pos])\nfit *= sum(alleles)\/len(alleles) # + dominance effect. Epistasis?!\nreturn fit\n\ndef reprod(self,other):\noff = Ind(chromosomes = [])\nfor one_chromo in xrange(nb_chromosomes):\n# recombination. Because the population has been shuffled, it is not necessary to create two recombined chromosomes and that select one (segragation). I construct only one recombined chromosome where self construct the first part of the chromosome.\nnb_cross = binomial(chromo_size, recombination)\ncross_pos = WalkerRandomSampling([1]*(chromo_size-1)).random(count=nb_cross).sort()\nrecombined_chromo = Chromo([])\nprevious_cross = 0\nfor sex, one_cross in enumerate(cross_pos):\nif sex%2 == 0:\nrecombined_cromo.alleles.append(self.chromosomes.alleles[previous_cross:(one_cross+1)])\nelse:\nrecombined_cromo.alleles.append(other.chromosomes.alleles[previous_cross:(one_cross+1)])\nprevious_cross = one_cross\n\noff.chromosomes.append(recombined_chromo)\nreturn off\n\nclass Pop(object):\ndef __init__(self, inds):\nself.inds = inds\n\ndef reproduction(self):\n\"First chose those that reproduce and then simulate mutations in offsprings\"\n# chosing those who reproduce - Creating the offspring population\nnew_pop = Pop(inds=[])\nfitness = []\nfor one_ind in self.inds:\nfitness.append(one_ind.fitness())\nmin_fitness = min(fitness)\nif min_fitness < 0:\nfitness = [one_ind - min_fitness for one_ind in fitness]\npick = WalkerRandomSampling(weights = fitness)\n\nif nb_chromosomes == 1 and recombination == 0:\nfor i in xrange(pop_size):\nnew_pop.inds.append(self.inds[pick.random()])\nelse:\nfor i in xrange(pop_size):\nfather = self.inds[pick.random()]\nmother = self.inds[pick.random()]\noff = father.reprod(mother)\nnew_pop.inds.append(off)\nnb_off += 1\n\n# Mutations\nfor one_ind in new_pop.inds:\nfor chromo_number in xrange(nb_chromosomes):\none_ind.chromosomes[chromo_number].mutations()\nreturn new_pop\n\ndef create_population(self):\none_chromo = Chromo(alleles = [1]*chromo_size)\none_ind = Ind(chromosomes = [one_chromo for i in range(nb_chromosomes)])\nreturn Pop(inds=[one_ind for i in xrange(pop_size)])\n\ndef stats(self, generation):\nline_to_write = str(generation) + '\\t' + str(replicat) + '\\t' + str(mut_rate) + '\\t' + str(pdf_mut_scale) + '\\t' + str(pdf_mutation.__name__)\\\n+ '\\t' + str(pop_size) + '\\t' + str(nb_chromosomes) + '\\t' + str(chromo_size) + '\\t' + str(recombination) + '\\t' + str(dominance) + '\\t'\nif output_type == 'mean fitness':\nadd = sum([ind.fitness() for ind in self.inds])\/pop_size\noutput_file.write(line_to_write + str(add) + '\\n')\n\ndef set_best_fitness_to_one(self):\nlist_chromo = zip(*[map(fun_for_set_fitness,[ind.chromosomes[chromo_number] for ind in self.inds]) for chromo_number in xrange(nb_chromosomes)])\nnew_pop = Pop([])\nfor one_ind in xrange(0,pop_size,nb_chromosomes):\nnew_pop.inds.append(list_chromo[one_ind:(ond_ind+nb_chromosomes)])\nreturn new_pop\n\ndef fun_for_set_fitness(list_one_chromo_number):\nl = zip(*list_one_chromo_number)\nfor locus_pos, one_locus in enumerate(l):\nmax_one_locus = max(one_locus)\none_locus = [i\/float(max_one_locus) for i in one_locus]\nl[locus_pos] = one_locus\nreturn zip(*l)\n\n######### Main #############\n\ndef main_run():\npopulation = Pop([]).create_population()\nfor generation in xrange(Nb_generations):\npopulation.stats(generation)\npopulation = population.reproduction()\nshuffle(population.inds)\npopulation = population.set_best_fitness_to_one()\n\n####### PARAMETERS ##########\n\n# Parameters\nNb_generations = 120\n# output_type = 'all individuals fitness'\noutput_type = 'mean fitness'\nmax_pop_size = 100 # this is only used to create the first (title, header) line!\n\n# Output file\nfile_name = 'stats3.txt'\npath = '\/Users\/remimatthey-doret\/Documents\/Biologie\/programmation\/Python\/Fitness distribution in the population\/' + file_name\noutput_file = open(path,'w')\nfirst_line = 'Generation\\treplicat\\tmut_rate\\tpdf_mut_scale\\tpdf_mutation\\tpop_size\\tnb_chromosomes\\tchromo_size\\trecombination\\tdominance\\t'\nif output_type == 'mean fitness':\nfirst_line += 'mean_fitness'\nif output_type == 'all individuals fitness':\nfor ind in xrange(max_pop_size):\nfirst_line += 'fit_ind_' + str(ind) + '\\t'\noutput_file.write(first_line + '\\n')\n\n# Parameters that iterate\ntotal_nb_runs = 3 * 10 # just enter the total number of iteration of the main_run function\nnb_runs_performed = 0\nfor mut_rate in [0.0001]:\nfor pdf_mut_scale in [0.01,0.1,0.3]: # Note: with an negative exponential distribution (expovariate) the expected value is 1\/lambda\nfor pdf_mutation in [expovariate]:\nfor pop_size in [1000]:\nfor nb_chromosomes in [1]:\nfor chromo_size in [1000]:\nfor recombination in [0]:\nfor dominance in [0]:\nfor replicat in xrange(10):\n\nmain_run()\nnb_runs_performed += 1\nprint str(float(nb_runs_performed)\/total_nb_runs * 100) + '%'\n\n\n\u2022 You should try to follow the PEP 8 guidelines whenever it's possible. In your case, the naming convention is not followed.\n\u2022 You should try to keep things simple. For instance, in WalkerRandomSampling.__init__(), it seems like you are doing a bit of logic to handle cases when keys is None and cases when it's not but at the end, you never init a WalkerRandomSampling with keys so it's quite hard to tell whether this is useful at all.\n\u2022 Don't repeat yourself. It seems like the beginning of WalkerRandomSampling.random() is roughly the same no matter if count is None or not. The common code could be factorised out.\n\u2022 Don't repeat yourself. The branches in:\n\n previous_cross = 0\nfor sex, one_cross in enumerate(cross_pos):\nif sex%2 == 0:\nrecombined_cromo.alleles.append(self.chromosomes.alleles[previous_cross:(one_cross+1)])\nelse:\nrecombined_cromo.alleles.append(other.chromosomes.alleles[previous_cross:(one_cross+1)])\nprevious_cross = one_cross\n\n\nlook way too similar. It probably would be better to write :\n\n previous_cross = 0\nfor sex, one_cross in enumerate(cross_pos):\nrelevant_ind = other if sex%2 else self\nrecombined_cromo.alleles.append(relevant_ind.chromosomes.alleles[previous_cross:(one_cross+1)])\nprevious_cross = one_cross\n\n\u2022 Use list (or set or dict) comprehension whenever you can.\n\na = []\nfor i in xrange(10000):\na.append(test.random())\nb = []\nfor value in range(4):\nb.append(len([i for i in a if i == value])\/float(len(a)))\n\n\ncould be written :\n\na = [test.random() for i in xrange(10000)]\nb = [len([value for i in a if i == value])\/float(len(a)) for value in range(4)]\n\n\nand the second line can actually be written differently so that we have :\n\na = [test.random() for i in xrange(10000)]\nb = [a.count(value)\/float(len(a)) for value in range(4)]\n\n\nSimilarly :\n\n alleles = []\nfor chromo_pos in range(len(self.chromosomes)):\nalleles.append(self.chromosomes[chromo.pos].alleles[gene_pos])\n\n\ncan be written :\n\n alleles = [self.chromosomes[chromo.pos].alleles[gene_pos] for chromo_pos in range(len(self.chromosomes)]\n\n\nand you could simplify the way you iterate :\n\n alleles = [c.alleles[gene_pos] for c in self.chromosomes]\n\n\nAnd :\n\nfitness = []\nfor one_ind in self.inds:\nfitness.append(one_ind.fitness())\n\n\ncould become :\n\nfitness = [one_ind.fitness() for one_ind in self.inds]\n\n\u2022 Do not build strings using concatenations multiple times. The recommended way is to use join:\n\nfor ind in xrange(max_pop_size):\nfirst_line += 'fit_ind_' + str(ind) + '\\t'\n\n\ncould be written as:\n\nfirst_line += '\\t'.join('fit_ind_' + str(ind) for ind in xrange(max_pop_size))\n\n\n(Also, that will not add the useless \\t at the end)","date":"2019-08-22 05:51:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29210975766181946, \"perplexity\": 10804.628521578103}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027316783.70\/warc\/CC-MAIN-20190822042502-20190822064502-00170.warc.gz\"}"}
| null | null |
"""Tests for preprocessor_builder."""
import tensorflow.compat.v1 as tf
from google.protobuf import text_format
from object_detection.builders import preprocessor_builder
from object_detection.core import preprocessor
from object_detection.protos import preprocessor_pb2
class PreprocessorBuilderTest(tf.test.TestCase):
def assert_dictionary_close(self, dict1, dict2):
"""Helper to check if two dicts with floatst or integers are close."""
self.assertEqual(sorted(dict1.keys()), sorted(dict2.keys()))
for key in dict1:
value = dict1[key]
if isinstance(value, float):
self.assertAlmostEqual(value, dict2[key])
else:
self.assertEqual(value, dict2[key])
def test_build_normalize_image(self):
preprocessor_text_proto = """
normalize_image {
original_minval: 0.0
original_maxval: 255.0
target_minval: -1.0
target_maxval: 1.0
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.normalize_image)
self.assertEqual(args, {
'original_minval': 0.0,
'original_maxval': 255.0,
'target_minval': -1.0,
'target_maxval': 1.0,
})
def test_build_random_horizontal_flip(self):
preprocessor_text_proto = """
random_horizontal_flip {
keypoint_flip_permutation: 1
keypoint_flip_permutation: 0
keypoint_flip_permutation: 2
keypoint_flip_permutation: 3
keypoint_flip_permutation: 5
keypoint_flip_permutation: 4
probability: 0.5
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_horizontal_flip)
self.assertEqual(args, {'keypoint_flip_permutation': (1, 0, 2, 3, 5, 4),
'probability': 0.5})
def test_build_random_vertical_flip(self):
preprocessor_text_proto = """
random_vertical_flip {
keypoint_flip_permutation: 1
keypoint_flip_permutation: 0
keypoint_flip_permutation: 2
keypoint_flip_permutation: 3
keypoint_flip_permutation: 5
keypoint_flip_permutation: 4
probability: 0.5
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_vertical_flip)
self.assertEqual(args, {'keypoint_flip_permutation': (1, 0, 2, 3, 5, 4),
'probability': 0.5})
def test_build_random_rotation90(self):
preprocessor_text_proto = """
random_rotation90 {
keypoint_rot_permutation: 3
keypoint_rot_permutation: 0
keypoint_rot_permutation: 1
keypoint_rot_permutation: 2
probability: 0.5
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_rotation90)
self.assertEqual(args, {'keypoint_rot_permutation': (3, 0, 1, 2),
'probability': 0.5})
def test_build_random_pixel_value_scale(self):
preprocessor_text_proto = """
random_pixel_value_scale {
minval: 0.8
maxval: 1.2
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_pixel_value_scale)
self.assert_dictionary_close(args, {'minval': 0.8, 'maxval': 1.2})
def test_build_random_image_scale(self):
preprocessor_text_proto = """
random_image_scale {
min_scale_ratio: 0.8
max_scale_ratio: 2.2
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_image_scale)
self.assert_dictionary_close(args, {'min_scale_ratio': 0.8,
'max_scale_ratio': 2.2})
def test_build_random_rgb_to_gray(self):
preprocessor_text_proto = """
random_rgb_to_gray {
probability: 0.8
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_rgb_to_gray)
self.assert_dictionary_close(args, {'probability': 0.8})
def test_build_random_adjust_brightness(self):
preprocessor_text_proto = """
random_adjust_brightness {
max_delta: 0.2
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_adjust_brightness)
self.assert_dictionary_close(args, {'max_delta': 0.2})
def test_build_random_adjust_contrast(self):
preprocessor_text_proto = """
random_adjust_contrast {
min_delta: 0.7
max_delta: 1.1
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_adjust_contrast)
self.assert_dictionary_close(args, {'min_delta': 0.7, 'max_delta': 1.1})
def test_build_random_adjust_hue(self):
preprocessor_text_proto = """
random_adjust_hue {
max_delta: 0.01
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_adjust_hue)
self.assert_dictionary_close(args, {'max_delta': 0.01})
def test_build_random_adjust_saturation(self):
preprocessor_text_proto = """
random_adjust_saturation {
min_delta: 0.75
max_delta: 1.15
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_adjust_saturation)
self.assert_dictionary_close(args, {'min_delta': 0.75, 'max_delta': 1.15})
def test_build_random_distort_color(self):
preprocessor_text_proto = """
random_distort_color {
color_ordering: 1
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_distort_color)
self.assertEqual(args, {'color_ordering': 1})
def test_build_random_jitter_boxes(self):
preprocessor_text_proto = """
random_jitter_boxes {
ratio: 0.1
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_jitter_boxes)
self.assert_dictionary_close(args, {'ratio': 0.1})
def test_build_random_crop_image(self):
preprocessor_text_proto = """
random_crop_image {
min_object_covered: 0.75
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.25
max_area: 0.875
overlap_thresh: 0.5
clip_boxes: False
random_coef: 0.125
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_crop_image)
self.assertEqual(args, {
'min_object_covered': 0.75,
'aspect_ratio_range': (0.75, 1.5),
'area_range': (0.25, 0.875),
'overlap_thresh': 0.5,
'clip_boxes': False,
'random_coef': 0.125,
})
def test_build_random_pad_image(self):
preprocessor_text_proto = """
random_pad_image {
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_pad_image)
self.assertEqual(args, {
'min_image_size': None,
'max_image_size': None,
'pad_color': None,
})
def test_build_random_absolute_pad_image(self):
preprocessor_text_proto = """
random_absolute_pad_image {
max_height_padding: 50
max_width_padding: 100
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_absolute_pad_image)
self.assertEqual(args, {
'max_height_padding': 50,
'max_width_padding': 100,
'pad_color': None,
})
def test_build_random_crop_pad_image(self):
preprocessor_text_proto = """
random_crop_pad_image {
min_object_covered: 0.75
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.25
max_area: 0.875
overlap_thresh: 0.5
clip_boxes: False
random_coef: 0.125
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_crop_pad_image)
self.assertEqual(args, {
'min_object_covered': 0.75,
'aspect_ratio_range': (0.75, 1.5),
'area_range': (0.25, 0.875),
'overlap_thresh': 0.5,
'clip_boxes': False,
'random_coef': 0.125,
'pad_color': None,
})
def test_build_random_crop_pad_image_with_optional_parameters(self):
preprocessor_text_proto = """
random_crop_pad_image {
min_object_covered: 0.75
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.25
max_area: 0.875
overlap_thresh: 0.5
clip_boxes: False
random_coef: 0.125
min_padded_size_ratio: 0.5
min_padded_size_ratio: 0.75
max_padded_size_ratio: 0.5
max_padded_size_ratio: 0.75
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_crop_pad_image)
self.assertEqual(args, {
'min_object_covered': 0.75,
'aspect_ratio_range': (0.75, 1.5),
'area_range': (0.25, 0.875),
'overlap_thresh': 0.5,
'clip_boxes': False,
'random_coef': 0.125,
'min_padded_size_ratio': (0.5, 0.75),
'max_padded_size_ratio': (0.5, 0.75),
'pad_color': None,
})
def test_build_random_crop_to_aspect_ratio(self):
preprocessor_text_proto = """
random_crop_to_aspect_ratio {
aspect_ratio: 0.85
overlap_thresh: 0.35
clip_boxes: False
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_crop_to_aspect_ratio)
self.assert_dictionary_close(args, {'aspect_ratio': 0.85,
'overlap_thresh': 0.35,
'clip_boxes': False})
def test_build_random_black_patches(self):
preprocessor_text_proto = """
random_black_patches {
max_black_patches: 20
probability: 0.95
size_to_image_ratio: 0.12
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_black_patches)
self.assert_dictionary_close(args, {'max_black_patches': 20,
'probability': 0.95,
'size_to_image_ratio': 0.12})
def test_build_random_jpeg_quality(self):
preprocessor_text_proto = """
random_jpeg_quality {
random_coef: 0.5
min_jpeg_quality: 40
max_jpeg_quality: 90
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Parse(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_jpeg_quality)
self.assert_dictionary_close(args, {'random_coef': 0.5,
'min_jpeg_quality': 40,
'max_jpeg_quality': 90})
def test_build_random_downscale_to_target_pixels(self):
preprocessor_text_proto = """
random_downscale_to_target_pixels {
random_coef: 0.5
min_target_pixels: 200
max_target_pixels: 900
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Parse(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_downscale_to_target_pixels)
self.assert_dictionary_close(args, {
'random_coef': 0.5,
'min_target_pixels': 200,
'max_target_pixels': 900
})
def test_build_random_patch_gaussian(self):
preprocessor_text_proto = """
random_patch_gaussian {
random_coef: 0.5
min_patch_size: 10
max_patch_size: 300
min_gaussian_stddev: 0.2
max_gaussian_stddev: 1.5
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Parse(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_patch_gaussian)
self.assert_dictionary_close(args, {
'random_coef': 0.5,
'min_patch_size': 10,
'max_patch_size': 300,
'min_gaussian_stddev': 0.2,
'max_gaussian_stddev': 1.5
})
def test_auto_augment_image(self):
preprocessor_text_proto = """
autoaugment_image {
policy_name: 'v0'
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.autoaugment_image)
self.assert_dictionary_close(args, {'policy_name': 'v0'})
def test_drop_label_probabilistically(self):
preprocessor_text_proto = """
drop_label_probabilistically{
label: 2
drop_probability: 0.5
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.drop_label_probabilistically)
self.assert_dictionary_close(args, {
'dropped_label': 2,
'drop_probability': 0.5
})
def test_remap_labels(self):
preprocessor_text_proto = """
remap_labels{
original_labels: 1
original_labels: 2
new_label: 3
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.remap_labels)
self.assert_dictionary_close(args, {
'original_labels': [1, 2],
'new_label': 3
})
def test_build_random_resize_method(self):
preprocessor_text_proto = """
random_resize_method {
target_height: 75
target_width: 100
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_resize_method)
self.assert_dictionary_close(args, {'target_size': [75, 100]})
def test_build_scale_boxes_to_pixel_coordinates(self):
preprocessor_text_proto = """
scale_boxes_to_pixel_coordinates {}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.scale_boxes_to_pixel_coordinates)
self.assertEqual(args, {})
def test_build_resize_image(self):
preprocessor_text_proto = """
resize_image {
new_height: 75
new_width: 100
method: BICUBIC
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.resize_image)
self.assertEqual(args, {'new_height': 75,
'new_width': 100,
'method': tf.image.ResizeMethod.BICUBIC})
def test_build_rgb_to_gray(self):
preprocessor_text_proto = """
rgb_to_gray {}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.rgb_to_gray)
self.assertEqual(args, {})
def test_build_subtract_channel_mean(self):
preprocessor_text_proto = """
subtract_channel_mean {
means: [1.0, 2.0, 3.0]
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.subtract_channel_mean)
self.assertEqual(args, {'means': [1.0, 2.0, 3.0]})
def test_random_self_concat_image(self):
preprocessor_text_proto = """
random_self_concat_image {
concat_vertical_probability: 0.5
concat_horizontal_probability: 0.25
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_self_concat_image)
self.assertEqual(args, {'concat_vertical_probability': 0.5,
'concat_horizontal_probability': 0.25})
def test_build_ssd_random_crop(self):
preprocessor_text_proto = """
ssd_random_crop {
operations {
min_object_covered: 0.0
min_aspect_ratio: 0.875
max_aspect_ratio: 1.125
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.0
clip_boxes: False
random_coef: 0.375
}
operations {
min_object_covered: 0.25
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.25
clip_boxes: True
random_coef: 0.375
}
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.ssd_random_crop)
self.assertEqual(args, {'min_object_covered': [0.0, 0.25],
'aspect_ratio_range': [(0.875, 1.125), (0.75, 1.5)],
'area_range': [(0.5, 1.0), (0.5, 1.0)],
'overlap_thresh': [0.0, 0.25],
'clip_boxes': [False, True],
'random_coef': [0.375, 0.375]})
def test_build_ssd_random_crop_empty_operations(self):
preprocessor_text_proto = """
ssd_random_crop {
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.ssd_random_crop)
self.assertEqual(args, {})
def test_build_ssd_random_crop_pad(self):
preprocessor_text_proto = """
ssd_random_crop_pad {
operations {
min_object_covered: 0.0
min_aspect_ratio: 0.875
max_aspect_ratio: 1.125
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.0
clip_boxes: False
random_coef: 0.375
min_padded_size_ratio: [1.0, 1.0]
max_padded_size_ratio: [2.0, 2.0]
pad_color_r: 0.5
pad_color_g: 0.5
pad_color_b: 0.5
}
operations {
min_object_covered: 0.25
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.25
clip_boxes: True
random_coef: 0.375
min_padded_size_ratio: [1.0, 1.0]
max_padded_size_ratio: [2.0, 2.0]
pad_color_r: 0.5
pad_color_g: 0.5
pad_color_b: 0.5
}
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.ssd_random_crop_pad)
self.assertEqual(args, {'min_object_covered': [0.0, 0.25],
'aspect_ratio_range': [(0.875, 1.125), (0.75, 1.5)],
'area_range': [(0.5, 1.0), (0.5, 1.0)],
'overlap_thresh': [0.0, 0.25],
'clip_boxes': [False, True],
'random_coef': [0.375, 0.375],
'min_padded_size_ratio': [(1.0, 1.0), (1.0, 1.0)],
'max_padded_size_ratio': [(2.0, 2.0), (2.0, 2.0)],
'pad_color': [(0.5, 0.5, 0.5), (0.5, 0.5, 0.5)]})
def test_build_ssd_random_crop_fixed_aspect_ratio(self):
preprocessor_text_proto = """
ssd_random_crop_fixed_aspect_ratio {
operations {
min_object_covered: 0.0
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.0
clip_boxes: False
random_coef: 0.375
}
operations {
min_object_covered: 0.25
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.25
clip_boxes: True
random_coef: 0.375
}
aspect_ratio: 0.875
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.ssd_random_crop_fixed_aspect_ratio)
self.assertEqual(args, {'min_object_covered': [0.0, 0.25],
'aspect_ratio': 0.875,
'area_range': [(0.5, 1.0), (0.5, 1.0)],
'overlap_thresh': [0.0, 0.25],
'clip_boxes': [False, True],
'random_coef': [0.375, 0.375]})
def test_build_ssd_random_crop_pad_fixed_aspect_ratio(self):
preprocessor_text_proto = """
ssd_random_crop_pad_fixed_aspect_ratio {
operations {
min_object_covered: 0.0
min_aspect_ratio: 0.875
max_aspect_ratio: 1.125
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.0
clip_boxes: False
random_coef: 0.375
}
operations {
min_object_covered: 0.25
min_aspect_ratio: 0.75
max_aspect_ratio: 1.5
min_area: 0.5
max_area: 1.0
overlap_thresh: 0.25
clip_boxes: True
random_coef: 0.375
}
aspect_ratio: 0.875
min_padded_size_ratio: [1.0, 1.0]
max_padded_size_ratio: [2.0, 2.0]
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function,
preprocessor.ssd_random_crop_pad_fixed_aspect_ratio)
self.assertEqual(args, {'min_object_covered': [0.0, 0.25],
'aspect_ratio': 0.875,
'aspect_ratio_range': [(0.875, 1.125), (0.75, 1.5)],
'area_range': [(0.5, 1.0), (0.5, 1.0)],
'overlap_thresh': [0.0, 0.25],
'clip_boxes': [False, True],
'random_coef': [0.375, 0.375],
'min_padded_size_ratio': (1.0, 1.0),
'max_padded_size_ratio': (2.0, 2.0)})
def test_build_normalize_image_convert_class_logits_to_softmax(self):
preprocessor_text_proto = """
convert_class_logits_to_softmax {
temperature: 2
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.convert_class_logits_to_softmax)
self.assertEqual(args, {'temperature': 2})
def test_random_crop_by_scale(self):
preprocessor_text_proto = """
random_square_crop_by_scale {
scale_min: 0.25
scale_max: 2.0
num_scales: 8
}
"""
preprocessor_proto = preprocessor_pb2.PreprocessingStep()
text_format.Merge(preprocessor_text_proto, preprocessor_proto)
function, args = preprocessor_builder.build(preprocessor_proto)
self.assertEqual(function, preprocessor.random_square_crop_by_scale)
self.assertEqual(args, {
'scale_min': 0.25,
'scale_max': 2.0,
'num_scales': 8,
'max_border': 128
})
if __name__ == '__main__':
tf.test.main()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,025
|
Vert-Saint-Denis ist eine französische Gemeinde mit Einwohnern (Stand ) im Département Seine-et-Marne der Region Île-de-France. Sie gehört zum Arrondissement Melun und zum Kanton Savigny-le-Temple (bis 2015: Kanton Le Mée-sur-Seine).
Separate Ortsteile von Vert-Saint-Denis sind Pouilly-le-Fort und Le Petit Jard.
Bevölkerungsentwicklung
Sehenswürdigkeiten
Menhir Le Grand Berger
Kirche Saint-Pierre-Saint-Paul, erbaut im 13. und 16. Jahrhundert (siehe auch: Liste der Monuments historiques in Vert-Saint-Denis)
Turm Le Petit Jard, erbaut im 14. Jahrhundert
Burg im Ortsteil Pouilly-le-Fort, erbaut im 14. Jahrhundert
Waschhaus in Pouilly-le-Fort
Literatur
Le Patrimoine des Communes de la Seine-et-Marne. Flohic Editions, Band 2, Paris 2001, ISBN 2-84234-100-7, S. 841–845.
Weblinks
Ort in der Île-de-France
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,026
|
{"url":"https:\/\/figshare.com\/articles\/Worst_optimal_difference_for_75_vaccination_and_low_time_delay_expressed_relative_to_the_optimal_mean_final_size_\/3155356\/1","text":"## Worst-optimal difference for 75% vaccination and low time delay, expressed relative to the optimal mean final size.\n\n2016-04-04T12:46:56Z (GMT) by\n<p>The difference in stochastic mean final epidemic size \u2329<i>E<\/i>\u232a between worst-case and optimal protocols, expressed as the fraction of the optimal \u2329<i>E<\/i>\u232a, is plotted as a function of time delay <i>\u03c4<\/i> \u2264 20 days and coupling <i>f<\/i><sub>AB<\/sub> for the case of 75% vaccination, since the most significant worst-optimal differences occur for large amounts of vaccination. The remaining parameters are the same as in <a href=\"http:\/\/www.plosone.org\/article\/info:doi\/10.1371\/journal.pone.0152950#pone.0152950.g007\" target=\"_blank\">Fig 7<\/a>.<\/p>","date":"2018-10-19 13:23:39","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\": 0.8410509824752808, \"perplexity\": 3415.5731350141373}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583512400.59\/warc\/CC-MAIN-20181019124748-20181019150248-00150.warc.gz\"}"}
| null | null |
{"url":"http:\/\/openstudy.com\/updates\/4f207c7de4b076dbc348c7b2","text":"## IsTim 4 years ago how to graph g(x)=3x^4=3x^2\n\n1. IsTim\n\nI'm looking thru my old notes now, but I can't find anything that could help.\n\n2. IsTim\n\nI simplified the equation so: g(x)=3x^2(x^2+1)\n\n3. IsTim\n\nI was thinking apq, but I don't know if that applies to this.\n\n4. sasogeek\n\nwhy do u have 2 equal signs to one function?\n\n5. IsTim\n\nOops.\n\n6. IsTim\n\n7. IsTim\n\n$g(x)=3x ^{4}-3x ^{2}$ If you want a cleaner version.\n\n8. anonymous\n\nuse a graphing calc\n\n9. anonymous\n\n3x^2(x^2-1) you plug in points frankly :-\/ it looks like a x^2 graph.\n\n10. IsTim\n\nSo I just plug in values? That's feels \"brute\". There's no other way?\n\n11. IsTim\n\n@ Mario; I'm studying for an exam. I don't get those.\n\n12. anonymous\n\nwe get to use graph calcs on my exams\n\n13. anonymous\n\nplug in pounts. their are zeroes at 0, 1,-1.\n\n14. anonymous\n\nthe rest is just brute force plugging. yes. that's how it works.\n\n15. IsTim\n\nOh well. I was looking for some equation rearranging. Would that work?\n\n16. IsTim\n\nLucky Mario.\n\n17. anonymous\n\nyet i still did poorly lol\n\n18. JamesJ\n\n19. IsTim\n\nIF possible, please give an explaination of how to dervie the graph from teh equatioon.\n\n20. JamesJ\n\nOk. If g(x)=3x^4-3x^2 first we'd like to know the zero; i.e., the intercepts on the x-axis. Setting g(x) = 0, we have $3x^2(x^2 - 1) = 0$ Hence $x = 0, \\pm 1$ Next, what's the y-intercept: y = g(0) = 0. Next, critical values ...\n\n21. JamesJ\n\n$g'(x) = 12x^3 - 6x = 0$ if and only if $6x(2x^2 - 1) = 0$ i.e., $x = 0, \\pm 1\/\\sqrt{2}$ The second derivative is $g''(x) = 36x^2 - 6 = 6(6x^2 - 1)$ ...\n\n22. JamesJ\n\nIt's not hard now to show that x = 0 must be a local max and $x = \\pm 1\/\\sqrt{2}$ are local mins. So now we have the behavior of the function in the interval [-1,1] What happens outside that?\n\n23. JamesJ\n\nThe next thing we observe is that g(x) is an even function, g(-x) = g(x) meaning the function is symmetric about the y-axis. As $g(x) = 3x^2(x^2 - 1)$ it is clear that for $$x > 1$$, $$g(x) > 0$$ and as $$x \\rightarrow \\infty$$, $$g(x) \\rightarrow \\infty$$. We now have all the information we need to draw the graph and it's consistent with the picture I posted above. Make sense?\n\n24. IsTim\n\nIt's higher level information that I don't understand, but I think if I read thru it a bit more, I'll understand. Thank you very much.","date":"2016-10-27 03:07:28","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\": 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\": 0.6629288196563721, \"perplexity\": 1431.8306064116598}, \"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-2016-44\/segments\/1476988721067.83\/warc\/CC-MAIN-20161020183841-00108-ip-10-171-6-4.ec2.internal.warc.gz\"}"}
| null | null |
CODE LiFE: Help Us Support Vital Care
CODE LiFE* represents the Montreal General Hospital Foundation's pledge to strengthen and deliver vital care to our community.
It acknowledges that life is precious and affirms our mission to protect, heal, extend, and save lives both at the Montreal General Hospital and across the McGill academic health network.
CODE LiFE draws on the familiar language of hospitals to signal a sense of urgency, rallying the skills, compassion and care of our medical and community leaders to safeguard life.
Make a donation to support the CODE LiFE campaign
Our Goal : $100 M to Support Vital Care
$50 M to create the best environment for our patients and medical teams
New Pavilion to house our Trauma Centre, Emergency and Operating Rooms
New Helipad for trauma and critical care
New spaces for the consolidated Mental Health Mission
New spaces for Innovation & Research
Significant modernization of clinics and patient spaces
$50 M to help our teams innovate in patient care and research
Trauma Mission – Defying the odds to save lives
Surgical Mission – New developments with dramatic patient benefits
Mental Health Mission – Offering a continuum of care for better recovery
Medical Mission – Collaboration between key specialties
Research – Innovation that matters
Thanks to our donors, our medical teams will work in a cutting-edge environment, and have funding to conduct essential research, beyond our 200th anniversary in 2021.
MGH teams provide tertiary and quaternary care in a multitude of disciplines. Patients are referred to our experts from Montreal, from across the province, and on occasion, from elsewhere in Canada. The care we offer is nothing less than vital. Given the numerous needs identified by our teams, the support we're asking the community to provide is equally vital.
Jean-Guy Gourdeau, President and CEO of the Montreal General Hospital Foundation
The innovative work done by our teams saves lives
With the CODE LiFE Campaign, our goal is to raise $100 M to provide our medical teams with an environment that allows them to best provide vital care to our deserving patients and to pursue innovative research projects.
Life can take a turn
People who come to our hospital were often not expecting it: road crashes or work accidents, new diagnostics, mental health issues, when life pushes you to the limits of your capacities it rarely gives you a headsup. That's where our teams come in play.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
A flow on a $C^*$-algebra $A$ is a continuous one-parameter group $\alpha = \left(\alpha_t\right)_{t \in \mathbb R}$ of automorphisms of $A$. In quantum statistical mechanics such a flow is sometimes interpreted as the time evolution of the observables of a physical system and the equilibrium states of the system are then given by states $\phi$ of $A$ that satisfy the trace-like condition
$$
\phi(ab) = \phi(b\alpha_{i\beta}(a))
$$
for all $a,b \in A$ with $a$ analytic for $\alpha$, \cite{BR}. The number $\beta$ for which this holds is interpreted as the inverse temperature of the system in the state given by $\phi$, and $\phi$ is said to be a $\beta$-KMS state for $\alpha$. The $C^*$-algebra $A$ of observables in such a model is often a UHF algebra and it is therefore of interest to determine the KMS states and the possible inverse temperatures which can occur for flows on a UHF algebra. It follows from work by Powers and Sakai in the 70's, \cite{PS}, that when the flow is approximately inner the set of possible inverse temperatures is the whole real line $\mathbb R$, and Powers and Sakai conjectured that all flows on a UHF algebra are approximately inner. That this is not the case was proved by Matui and Sato in \cite{MS}, following work by Kishimoto on the AF algebra case, \cite{Ki2}. It is therefore an open question which sets of real numbers can occur as the set of inverse temperatures for a flow on a UHF algebra. The set is closed for any flow and since a UHF algebra has a unique trace state which is automatically a $0$-KMS state, it must contain zero. These are the only known restrictions. It is the purpose with this paper to show that it is also the only restrictions, at least when one only considers non-negative $\beta$, and not only for flows on UHF algebras, but in fact on all simple infinite dimensional unital AF algebras.
In the examples of Kishimoto, and also in the examples of Matui and Sato, the set of possible inverse temperatures is as small as it can be, consisting only of $0$, cf. Remark \ref{11-11-20} below. Here we will show that a variant of the method developed in \cite{Th5} can be used to construct a flow on any infinite dimensional simple unital AF algebra such that the set of inverse temperatures is any given lower bounded closed set of real numbers containing zero. For this we use a method which is a natural extension of ideas and methods from work by Bratteli, Elliott, Herman and Kishimoto in \cite{BEH}, \cite{BEK},\cite{Ki2}, and by Matui and Sato in \cite{MS}. As in these papers we depend on results from the classification of simple $C^*$-algebras. Besides the classification of AF algebras, which was fundamental already in \cite{BEH}, we apply the recent work by Castillejos, Evington, Tikuisis, White and Winter in \cite{CETWW}, where the classification results we use in turn are based on work by Elliott, Gong, Lin and Niu, \cite{GLN}, \cite{EGLN}, among others. However, when the algebra in question is UHF there is an alternative route to the desired result which we describe Remark \ref{14-12-20gx}.
The paper contains some material which appeared in \cite{Th5}; this is because I now consider \cite{Th5} a forerunner to the present paper and it will not be submitted for publication. The results of the present paper answer Question 2.6 in \cite{Th5} positively, but Question 6.2 in \cite{Th4} only partially. What remains is to determine if it is possible to remove the condition that the set is bounded below. This restriction is annoying and probably not necessary. However, in applications and also to many mathematicians, the (inverse) temperatures are non-negative and then the restriction to the lower bounded case is of no significance.
In relation to this, one should observe the recent work by Christensen and Vaes, \cite{CV}, in which they connect the theory of KMS states on $C^*$-algebras to the theory of group actions on spaces. They also provide two examples of simple unital $C^*$-algebras on which there are flows with arbitrary KMS spectra. While it is certainly premature to suggest that this applies to all infinite dimensional unital simple $C^*$-algebras, it does seem appropriate to point out that we don't really know.
\smallskip
\emph{Acknowledgement} I am grateful to Y. Sato for comments on the first version of this paper which helped me navigate in the litterature on the classification of simple $C^*$-algebras, and I thank the referee of \cite{Th5} for his remarks. The work was supported by the DFF-Research Project 2 `Automorphisms and Invariants of Operator Algebras', no. 7014-00145B.
\section{Statement of results}
\begin{thm}\label{20-12-20c} Let $A$ be a unital infinite dimensional simple AF $C^*$-algebra. For each non-empty closed face $F$ in the tracial state space $T(A)$ of $A$ and for each closed and lower bounded set $K \subseteq \mathbb R$ of real numbers containing $0$ there is a $2\pi$-periodic flow $\alpha$ on $A$ such that
\begin{itemize}
\item there is a $\beta$-KMS state for $\alpha$ if and only if $\beta \in K$,
\item for $\beta \in K \backslash \{0\}$ the simplex of $\beta$-KMS states for $\alpha$ is affinely homeomorphic to $F$,
\item the simplex of $0$-KMS states for $\alpha$ is affinely homeomorphic to $T(A)$, and
\item the fixed point algebra $A^{\alpha}$ of $\alpha$ is an $AF$ algebra.
\end{itemize}
When $K$ is also upper bounded, and hence compact, $\alpha$ can be chosen such that the fixed point algebra $A^{\alpha}$ of $\alpha$ is a simple $AF$ algebra.
\end{thm}
When $K$ is not compact it is not possible to arrange that $A^{\alpha}$ is simple because when the set of inverse temperatures contains arbitrarily large numbers the flow must have ground states, and this is not possible for periodic flows whose fixed point algebra is simple, cf. Remark 3.5 in \cite{BEH}.
By taking $F$ to consist of a single extremal point in $T(A)$ we get the following
\begin{cor}\label{20-12-20d} Let $A$ be a unital infinite dimensional simple AF $C^*$-algebra. For each closed and lower bounded set $K \subseteq \mathbb R$ of real numbers containing $0$ there is a $2\pi$-periodic flow $\alpha$ on $A$ such that
\begin{itemize}
\item there is a $\beta$-KMS state for $\alpha$ if and only if $\beta \in K$,
\item for $\beta \in K \backslash \{0\}$ the $\beta$-KMS state is unique,
\item the simplex of $0$-KMS states for $\alpha$ is a affinely homeomorphic to $T(A)$, and
\item the fixed point algebra $A^{\alpha}$ of $\alpha$ is an $AF$ algebra.
\end{itemize}
When $K$ is also upper bounded, and hence compact, $\alpha$ can be chosen such that the fixed point algebra $A^{\alpha}$ of $\alpha$ is a simple $AF$ algebra.
\end{cor}
In the last section of the paper we combine Corollary \ref{20-12-20d} with methods and results from \cite{Th3} and \cite{Th4} to obtain the following additional corollaries. Recall that two compact Choquet simplexes are strongly affinely isomorphic when there is an affine bijection between them which restricts to a homeomorphism between the sets of extremal points. For Bauer simplexes this is the same as affine homeomorphism, but in general it is a weaker notion. See \cite{Th4}.
\begin{cor}\label{15-12-20d} Let $U$ be a UHF algebra and let $K$ be a closed and lower bounded set of real numbers containing $0$. Let $\mathbb I$ be a finite or countably infinite collection of intervals in $\mathbb R$ such that $I = \mathbb R$ for at least one $I \in \mathbb I$. For each $I \in \mathbb I$ choose a compact metrizable Choquet simplex $S_I$ and for $\beta \in K$ set $\mathbb I_\beta = \left\{ I \in \mathbb I: \ \beta \in I\right\}$. There is a $2\pi$-periodic flow $\alpha$ on $U$ such that
\begin{itemize}
\item there is $\beta$-KMS state for $\alpha$ if and only if $\beta \in K$,
\item for each $I \in \mathbb I$ and $\beta \in (I\cap K) \backslash \{0\}$ there is a closed face $F_I$ in $S^{\alpha}_\beta$ strongly affinely isomorphic to $S_I$, and
\item for each $\beta \in K \backslash \{0\}$ and each $\omega \in S^\alpha_\beta$ there is a unique norm-convergent decomposition
$$
\omega = \sum_{I\in \mathbb I_\beta} \omega_I \ ,
$$
where $\omega_I \in \left\{ t \mu : \ t \in ]0,1], \ \mu \in F_I\right\}$.
\end{itemize}
\end{cor}
\begin{cor}\label{27-10-20a} Let $U$ be a UHF algebra and let $K$ be a closed and lower bounded set of real numbers containing $0$. There is a $2\pi$-periodic flow $\alpha$ on $U$ such that $S^\alpha_\beta = \emptyset$ if and only if $\beta \notin K$, and for $\beta, \beta' \in K$ the simplexes $S^{\alpha}_\beta$ and $S^\alpha_{\beta'}$ are not strongly affinely isomorphic unless $\beta = \beta'$.
\end{cor}
\section{Preparations}
In this paper all $C^*$-algebras are assumed to be separable and all traces and weights on a $C^*$-algebra are required to be non-zero, densely defined and lower semi-continuous. Concerning weights and in particular KMS weights we shall use notation and results from Sections 1.1 and 1.3 in \cite{KV}. Let $A$ be a $C^*$-algebra and $\theta$ a flow on $A$. Let $\beta \in \mathbb R$. A $\beta$-KMS weight for $\theta$ is a weight $\omega$ on $A$ such that $\omega \circ \theta_t = \omega$ for all $t$, and
\begin{equation}\label{27-10-20c}
\omega(a^*a) \ = \ \omega\left(\theta_{-\frac{i\beta}{2}}(a) \theta_{-\frac{i\beta}{2}}(a)^*\right) \ \ \ \forall a \in D(\theta_{-\frac{i\beta}{2}}) \ .
\end{equation}
In particular, a $0$-KMS weight for $\theta$ is a $\theta$-invariant trace. It was shown by Kustermans in Theorem 6.36 of \cite{Ku} that this definition agrees with the one introduced by Combes in \cite{C}.
It is because of the formulation given by \eqref{27-10-20c}, which was not available when \cite{BEH} was written, that we are able to work with KMS weights throughout the present work. A bounded $\beta$-KMS weight is called a $\beta$-KMS functional and a $\beta$-KMS state when it is of norm $1$.
The first lemma can be considered as an updated version of a part of the discussion in Remark 3.3 of \cite{BEH}.
\begin{lemma}\label{26-10-20} Let $B$ be a $C^*$-algebra and $\gamma \in \operatorname{Aut}(B)$ an automorphism of $B$. Let $\widehat{\gamma}$ be the dual action on $B \rtimes_{\gamma} \mathbb Z$ considered as a $2 \pi$-periodic flow. For $\beta \in \mathbb R$ the restriction map $\omega \mapsto \omega|_B$ is a bijection from the $\beta$-KMS weights for $\widehat{\gamma}$ onto the traces $\tau$ on $B$ with the property that $\tau \circ \gamma = e^{-\beta} \tau$. The inverse is the map $\tau \mapsto \tau \circ P$, where $P: B \rtimes_{\gamma} \mathbb Z \to B$ is the canonical conditional expectation.
\end{lemma}
\begin{proof} Let $p_1 \leq p_2 \leq \cdots$ be an approximate unit in $B$ from the Pedersen ideal $K(B)$ of $B$ with the additional property that $p_k^2 \leq p^2_{k+1}$ for all $k$, cf. \cite{Pe}. Let $u$ be the canonical unitary multiplier of $B \rtimes_\gamma \mathbb Z$ such that $ubu^* = \gamma(b)$ for $b \in B$, and let $\tau$ be a trace on $B$ with the property that $\tau \circ \gamma = e^{-\beta} \tau$. For $a,b \in B, \ n,m \in \mathbb Z$ we have that
$$
P(p_kau^np_k^2bu^mp_k) = P(p_kbu^m p_k^2\widehat{\gamma}_{i\beta}(au^n)p_k) = 0
$$
unless $m = -n$. Set $a' = p_ka\gamma^n(p_k), \ b' = p_kb\gamma^{-n}(p_k)$. Then $a',b' \in K(B)$, and by using Proposition 5.5.2 in \cite{Pe} we find that
\begin{align*}
& \tau\circ P( p_kbu^{-n}p_k\widehat{\gamma}_{i \beta} (p_kau^np_k)) = e^{-n\beta}\tau(p_kbu^{-n}p_k^2au^np_k) \\
& =e^{-n \beta} \tau( b'u^{-n} a'u^n) =\tau(\gamma^n(b')a') \\
& = \tau(a'\gamma^n(b')) = \tau \circ P(p_kau^n p_k^2bu^{-n}p_k) \ ,
\end{align*}
showing that $\tau \circ P$ is a $\beta$-KMS functional on $p_k(B \rtimes_\gamma \mathbb Z)p_k$ for the restriction of $\widehat{\gamma}$. We shall use repeatedly that when $b \geq 0$ in $B$ we have that
\begin{equation}\label{10-11-20a}
\lim_{k \to \infty} \tau(p_kbp_k) = \tau(b) \ ,
\end{equation}
which follows from the lower semi-continuity of $\tau$ since $\tau(p_kbp_k) = \tau(\sqrt{b}p_k^2\sqrt{b})$ and $\sqrt{b}p_k^2\sqrt{b}$ increases to $b$ as $k \to \infty$. Let $x \in D(\widehat{\gamma}_{-\frac{i\beta}{2}})$. Then
\begin{align*}
& \tau\circ P\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(x) \widehat{\gamma}_{- \frac{i\beta}{2}}(x)^*\right) = \lim_{k \to \infty} \tau\circ P\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(x) p_k^2 \widehat{\gamma}_{- \frac{i\beta}{2}}(x)^*\right)\\
& = \lim_{k \to \infty}\lim_{l \to \infty} \tau \left(p_lP\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(x) p_k^2 \widehat{\gamma}_{- \frac{i\beta}{2}}(x)^*\right)p_l\right) \ .
\end{align*}
We have shown above that $\tau \circ P$ is a $\beta$-KMS functional on $p_l(B \rtimes_\gamma \mathbb Z)p_l$ and when $l \geq k$ this gives
\begin{align*}
&\tau\left(p_l P\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(x) p_k^2 \widehat{\gamma}_{- \frac{i\beta}{2}}(x)^*\right)p_l\right) = \tau\circ P\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(p_lxp_k) \widehat{\gamma}_{- \frac{i\beta}{2}}(p_lxp_k)^*\right) \\
& = \ \tau \circ P((p_lxp_k)^*p_lxp_k) = \tau \circ P(p_kx^*p_l^2xp_k) \leq \tau\left(p_kP(x^*x)p_k\right) \ .
\end{align*}
By using \eqref{10-11-20a} we conclude that
$$
\tau \circ P\left(\widehat{\gamma}_{- \frac{i\beta}{2}}(x) \widehat{\gamma}_{- \frac{i\beta}{2}}(x)^*\right) \ \leq \ \tau \circ P(x^*x) \ .
$$
Similarly,
\begin{align*}
& \tau \circ P(x^*x) = \lim_{k \to \infty} \tau \circ P(x^*p_k^2x) = \lim_{k \to \infty} \lim_{l \to \infty} \tau \circ P(p_lx^*p_k^2 x p_l) \ .
\end{align*}
When $l \geq k$,
\begin{align*}
&\tau \circ P(p_lx^*p_k^2 xp_l) = \tau \circ P( \widehat{\gamma}_{-\frac{i\beta}{2}}(p_k xp_l) \widehat{\gamma}_{-\frac{i\beta}{2}}(p_kxp_l)^*) \\
& = \tau \circ P(p_k \widehat{\gamma}_{-\frac{i\beta}{2}}( x)p_l^2 \widehat{\gamma}_{-\frac{i\beta}{2}}(x)^*p_k \ \leq \tau\left(p_kP( \widehat{\gamma}_{-\frac{i\beta}{2}}( x) \widehat{\gamma}_{-\frac{i\beta}{2}}(x)^*)p_k\right) ,
\end{align*}
and we find therefore that
\begin{align*}
&\tau \circ P(x^*x) \leq \lim_{k \to \infty} \tau \left(p_kP( \widehat{\gamma}_{-\frac{i\beta}{2}}( x) \widehat{\gamma}_{-\frac{i\beta}{2}}(x)^*)p_k\right) = \tau \circ P( \widehat{\gamma}_{-\frac{i\beta}{2}}( x) \widehat{\gamma}_{-\frac{i\beta}{2}}(x)^*) \ .
\end{align*}
We conclude therefore first that $\tau \circ P(x^*x) = \tau \circ P( \widehat{\gamma}_{-\frac{i\beta}{2}}( x) \widehat{\gamma}_{-\frac{i\beta}{2}}(x)^*)$, and then that $\tau \circ P$ is a $\beta$-KMS weight.
Let $\omega$ be a $\beta$-KMS weight for $\widehat{\gamma}$. Then $\omega(b^*b) = \omega(bb^*)$ for all $b \in B$ because $\widehat{\gamma}_{-i\frac{\beta}{2}}(b) = b$. Using Riemann sum approximations to the integral
$$
P(a) = (2\pi)^{-1} \int_0^{2\pi} \widehat{\gamma}_t(a) \ \mathrm d t \ ,
$$
it follows from the $\widehat{\gamma}$-invariance and lower semi-continuity of $\omega$ that
\begin{equation}\label{26-10-20b}
\omega \circ P(a) \leq \omega(a)
\end{equation}
when $ a \geq 0$ in $B\rtimes_\gamma \mathbb Z$. In particular, $\omega|_B$ is densely defined since $\omega$ is, and we conclude that $\tau = \omega|_B$ is a trace on $B$. Let $a \geq 0$. Then
$$
\omega \circ \gamma(a) = \omega(uau^*) = e^{-\beta}\omega\left( \widehat{\gamma}_{-\frac{i\beta}{2}}(u\sqrt{a}) \widehat{\gamma}_{-\frac{i\beta}{2}}(u\sqrt{a})^*\right) = e^{-\beta} \omega(a) \ .
$$
It follows that $\tau \circ \gamma = e^{-\beta}\tau$. It remains now only to show that $\omega = \omega \circ P$. Note that because $\omega \circ \widehat{\gamma}_t = \omega$ we find for all $b\in B, \ n \in \mathbb Z$, that
$$
\omega(p_kbu^np_k) = \begin{cases} 0, & \ n \neq 0 \\\omega(p_kbp_k) , & \ n = 0 \end{cases} \ = \ \omega(p_kP(bu^n)p_k) \ ,
$$
implying that $\omega(p_k \ \cdot \ p_k) = \omega(p_kP( \ \cdot \ )p_k)$. Let $a \in B \rtimes_\gamma \mathbb Z$, $ a \geq 0$. Since $\omega|_B$ is a trace we can use \eqref{10-11-20a} to conclude that
$$
\lim_{k \to \infty} \omega(p_kap_k) = \lim_{k \to \infty} \omega(p_kP(a)p_k) = \ \omega(P(a)) \ .
$$
Since $\lim_{k \to \infty} p_kap_k = a$ the lower semi-continuity of $\omega$ implies now that $\omega(a) \leq \omega(P(a))$. Combined with \eqref{26-10-20b} this yields the desired conclusion that $\omega \circ P = \omega$.
\end{proof}
\begin{remark}\label{11-11-20} It follows from Lemma \ref{26-10-20} that when $B$ has a unique trace the dual action will have no $\beta$-KMS weights for $\beta \neq 0$, and if in addition $B \rtimes_{\gamma} \mathbb Z$ is simple the restriction of the dual action to any corner $e(B \rtimes_\gamma \mathbb Z)e$ given by a $\widehat{\gamma}$-invariant non-zero projection $e$ will have no $\beta$-KMS states for $\beta \neq 0$ by Remark 3.3 in \cite{LN} or Theorem 2.4 in \cite{Th2}. This observation applies to the flows that were shown not to be approximately inner in \cite{Ki2} and \cite{MS}.
\end{remark}
Taking $\beta =0$ in Lemma \ref{26-10-20} we get a special case which we shall need. It is probably known.
\begin{cor}\label{28-10-20a} Let $B$ be a $C^*$-algebra and $\gamma \in \operatorname{Aut}(B)$ an automorphism of $B$. The restriction map $\omega \mapsto \omega|_B$ is a bijection from the $\widehat{\gamma}$-invariant traces $\omega$ on $B \rtimes_\gamma \mathbb Z$ onto the $\gamma$-invariant traces on $B$. The inverse is the map $\tau \mapsto \tau \circ P$, where $P: B \rtimes_{\gamma} \mathbb Z \to B$ is the canonical conditional expectation.
\end{cor}
We remark that in general $B \rtimes_\gamma \mathbb Z$ can have traces that are not invariant under the dual action $\widehat{\gamma}$ and hence are not determined by their restriction to $B$; also in cases where $B$ is UHF and $B \rtimes_\gamma \mathbb Z$ is simple. For the present purposes it is crucial that we can circumvent this issue thanks to the following lemma which is suggested by the work of Matui and Sato in \cite{MS}. In fact, most of it appears implicitly in \cite{MS}.
Recall that a $C^*$-algebra $A$ is stable when $A \otimes \mathbb K \simeq A$, where $\mathbb K$ is the $C^*$-algebra of compact operators on an infinite dimensional separable Hilbert space. Recall also that a partially ordered group $(G,G^+)$ has large denominators when the following holds: For any $a \in G^+$ and any $n \in \mathbb N$ there is an element $b \in G$ and an $m \in \mathbb N$ such that $ nb\leq a \leq mb$, cf. \cite{Ni}.
\begin{lemma}\label{28-10-20} Let $B$ be a stable AF-algebra and $\gamma \in \operatorname{Aut}(B)$. Assume that $K_0(B)$ has large denominators. There is an automorphism $\gamma' \in \operatorname{Aut}(B)$ such that
\begin{enumerate}
\item[a)] $\gamma'_* = \gamma_*$ on $K_0(B)$,
\item[b)] the restriction map
$\mu \ \mapsto \ \mu|_B$
is a bijection from traces $\mu $ on $B \rtimes_{\gamma'} \mathbb Z$ onto the $\gamma'$-invariant traces on $B$,
\item[c)] $B \rtimes_{\gamma'} \mathbb Z$ is $\mathcal Z$-stable; i.e $(B \rtimes_{\gamma'} \mathbb Z)\otimes \mathcal Z \simeq B \rtimes_{\gamma'} \mathbb Z$ where $\mathcal Z$ denotes the Jiang-Su algebra, \cite{JS}, and
\item[d)] $B \rtimes_{\gamma'} \mathbb Z$ is stable.
\end{enumerate}
\end{lemma}
\begin{proof} The first step is to show that there is an isomorphism $\phi : B \to B \otimes \mathcal Z$ such that $\phi_* = {(\id_B \otimes 1_{\mathcal Z})}_*$. Since $\mathcal Z$ is strongly self-absorbing in the sense of Toms and Winter, \cite{TW}, it suffices for this to show that $B$ is $\mathcal Z$-stable. By Corollary 3.4 in \cite{TW} it is enough to show that $eBe$ is $\mathcal Z$-stable for any non-zero projection $e \in B$, and hence by Theorem 2.3 in \cite{TW} it is enough to show that $eBe$ is approximately divisible. Now $K_0(eBe)$ has large denominators since we assume that $K_0(B)$ has, and it follows therefore from Lemma 4.4 in \cite{Th1} that for any Bratteli diagram for $eBe$ the multiplicity matrices describing the embedding of a given level into the following levels will have the property that the minimum of the non-zero entries increases to infinity. This implies that $eBe$ is approximately divisible, cf. \cite{BKR}. We deduce in this way the existence of $\phi$.
For the second step we use \cite{Sa} to obtain an automorphism $\theta$ of $\mathcal Z$ with the weak Rohlin property; that is, with the property that for each $k \in \mathbb N$ there is a sequence $\{f_n\}$ in $\mathcal Z$ such that
\begin{itemize}
\item $0 \leq f_n \leq 1_{\mathcal Z}$ for all $n$,
\item $\lim_{n \to \infty} f_na -af_n = 0$ for all $a \in \mathcal Z$,
\item $\lim_{n \to \infty} \theta^j(f_n)f_n = 0$ for $j=1,2,3,\cdots, k$, and
\item $\lim_{n \to \infty} \tau\left(1_{\mathcal Z} - \sum_{j=0}^{k}\theta^j(f_n)\right) = 0$ ,
\end{itemize}
where $\tau$ is the trace state of $\mathcal Z$. From the first step, applied twice, it follows that there is an isomorphism $\phi : B \to B \otimes \mathcal Z \otimes \mathcal Z$ such that $\phi_* = (\id_B \otimes 1_{\mathcal Z} \otimes 1_{\mathcal Z})_*$. Since $B$ is stable there is also a $*$-isomorphism $\psi_0 : B \to \mathbb K \otimes B$ such that ${\psi_0}_* = (e \otimes \id_B)_*$, where $e$ is a minimal non-zero projection in $\mathbb K$. Set
$$
\psi = \left(\psi_0 \otimes \id_{\mathcal Z} \otimes \id_{\mathcal Z}\right) \circ \phi : B \to \mathbb K \otimes B \otimes \mathcal Z \otimes \mathcal Z \
$$
and
$$
\gamma' = \psi^{-1} \circ \left(\id_{\mathbb K} \otimes \gamma \otimes \theta \otimes \id_{\mathcal Z}\right) \circ \psi \ .
$$
It follows from the K\"unneth theorem that $\psi_* = (e\otimes \id_B \otimes 1_{\mathcal Z} \otimes 1_{\mathcal Z})_*$ and hence $\left(\id_{\mathbb K} \otimes \gamma \otimes \theta \otimes \id_{\mathcal Z}\right)_* \circ \psi_* = (e \otimes \gamma \otimes 1_{\mathcal Z} \otimes 1_{\mathcal Z})_*$. Thus $\gamma'_* = \gamma$.
To complete the proof it suffices to verify that b), c) and d) hold when $B$ is replaced by $\mathbb K \otimes B \otimes \mathcal Z\otimes \mathcal Z$ and $\gamma'$ by $\id_{\mathbb K} \otimes \gamma \otimes \theta \otimes \id_{\mathcal Z}$. The properties c) and d) follow immediately because $ \mathcal Z \otimes \mathcal Z \simeq \mathcal Z$, $\mathbb K \otimes \mathbb K \simeq \mathbb K$ and
$$
(\mathbb K \otimes B \otimes \mathcal Z \otimes \mathcal Z) \rtimes_{\id_{\mathbb K} \otimes\gamma \otimes \theta \otimes \id_{\mathcal Z}} \mathbb Z \ \simeq \ \mathbb K \otimes \left((B \otimes \mathcal Z) \rtimes_{\gamma \otimes \theta} \mathbb Z \right) \otimes \mathcal Z \ .
$$
In the final step we verify b). For this, set $B' = \mathbb K \otimes B \otimes \mathcal Z$ and $\gamma'' = \id_{\mathbb K} \otimes \gamma \otimes \id_{\mathcal Z}$. Then
$$
( \mathbb K \otimes B \otimes \mathcal Z \otimes \mathcal Z) \rtimes_{\id_{\mathbb K} \otimes \gamma \otimes \theta \otimes \id_{\mathcal Z}} \mathbb Z \ \simeq \ (B'\otimes \mathcal Z) \rtimes_{\gamma'' \otimes \theta} \mathbb Z \ ,
$$
and it suffices to verify that b) holds when $B$ is replaced by $B' \otimes \mathcal Z$ and $\gamma'$ by $\gamma''\otimes \theta$. Let $P : (B' \otimes \mathcal Z) \rtimes_{\gamma''\otimes \theta} \mathbb Z \to B' \otimes \mathcal Z$ be the canonical conditional expectation. In view of Corollary \ref{28-10-20a} what remains is to consider a trace $\mu$ on $(B '\otimes \mathcal Z) \rtimes_{\gamma''\otimes \theta} \mathbb Z$ and show that $\mu = \mu \circ P$. For this, let $\{q_n\}$ be an approximate unit in $B'$ consisting of projections. Since $q_n \otimes 1_{\mathcal Z}$ is a projection and therefore contained in the Pedersen ideal, $\mu(q_n \otimes 1_{\mathcal Z}) < \infty$, cf. \cite{Pe}. When $a \geq 0$ in $ (B' \otimes \mathcal Z) \rtimes_{\gamma''\otimes \theta} \mathbb Z $,
$$
\mu(a) = \lim_n \mu\left(\sqrt{a}(q_n \otimes 1_{\mathcal Z}) \sqrt{a}\right) = \lim_n \mu\left((q_n \otimes 1_{\mathcal Z})a(q_n\otimes 1_{\mathcal Z})\right) \ ,
$$
and similarly $
\mu(P(a)) = \lim_n \mu\left((q_n \otimes 1_{\mathcal Z})P(a)(q_n\otimes 1_{\mathcal Z})\right)$.
It suffices therefore to show that
$$
\mu\left((q_n \otimes 1_{\mathcal Z})a(q_n\otimes 1_{\mathcal Z})\right) = \mu\left((q_n \otimes 1_{\mathcal Z})P(a)(q_n\otimes 1_{\mathcal Z})\right)
$$
for all $n$. For this note that $x \mapsto \mu\left((q_n \otimes 1_{\mathcal Z})x(q_n\otimes 1_{\mathcal Z})\right)$ extends to a bounded positive linear functional of norm $\mu\left(q_n \otimes 1_{\mathcal Z}\right)$ on $(B' \otimes \mathcal Z) \rtimes_{\gamma''\otimes \theta} \mathbb Z$. It suffices therefore to consider positive elements $b \in B'$, $z \in \mathcal Z$ and $k \in \mathbb Z \backslash \{0\}$, and show that
\begin{equation}\label{23-10-20}
\mu((q_n \otimes 1_{\mathcal Z})(b\otimes z) u^k(q_n\otimes 1_{\mathcal Z})) = 0
\end{equation}
when $u$ is the canonical unitary in the multiplier algebra of $(B' \otimes \mathcal Z) \rtimes_{\gamma''\otimes \theta} \mathbb Z$ such that $\operatorname{Ad} u = \gamma'' \otimes \theta$ on $B' \otimes \mathcal Z$. Since the complex conjugate of $\mu((q_n \otimes 1_{\mathcal Z})(b\otimes z) u^{-k}(q_n\otimes 1_{\mathcal Z}))$ is $\mu((q_n \otimes 1_{\mathcal Z})({\gamma''}^k(b)\otimes \theta^k(z)) u^{k}(q_n\otimes 1_{\mathcal Z}))$
we may assume $k \geq 1$. Let $\epsilon > 0$. Since
$a \mapsto \mu(q_n \otimes a)$ is a bounded trace on $\mathcal Z$ it must be a scalar multiple of the unique trace state $\tau$ of $\mathcal Z$. It follows therefore from the properties of $\theta$ that there are elements $0 \leq g_j \leq 1_{\mathcal Z}, \ j = 0,1,2, \cdots , k$, in $\mathcal Z$ such that
\begin{align}
&\left\|g_jz-zg_j\right\| \leq \epsilon \ \text{for all} \ j \ ,\label{23-10-20a} \\
& \left\|g_ig_j \right\| \leq \epsilon \ \text{for all} \ i,j, \ i \neq j , \label{23-10-20b} \\
& \left\|\theta^k(g_j)g_j\right\| \leq \epsilon \ \text{for all } \ j, \ \text{and} \label{23-10-20c} \\
&\left|\mu(q_n \otimes (1_{\mathcal Z}- \sum_{j=0}^{k} g_j)) \right| \ \leq \ \epsilon \mu(q_n\otimes 1_{\mathcal Z}). \ \label{23-10-20d}
\end{align}
In the following, when $s$ and $t$ are complex numbers such that for all $\delta > 0$ the $\epsilon$ in \eqref{23-10-20a}-\eqref{23-10-20d} can be chosen so small that $|s-t|\leq \delta$, we will write $s \sim t$. Set
$$
y = 1_{\mathcal Z} - \sum_{j=0}^k g_j \ .
$$
It follows from \eqref{23-10-20a} that
$$
\mu((q_n \otimes 1_{\mathcal Z})(b\otimes z y) u^k(q_n\otimes 1_{\mathcal Z})) \ \sim \ \mu((q_n \otimes 1_{\mathcal Z})(b\otimes yz ) u^k(q_n\otimes 1_{\mathcal Z}))\ ,
$$
and from the Cauchy-Schwarz inequality that
\begin{equation}\label{23-10-20e}
\begin{split}
&\left|\mu((q_n \otimes 1_{\mathcal Z})(b\otimes yz ) u^k(q_n\otimes 1_{\mathcal Z}))\right| \ = \ \left|\mu((q_n \otimes y)(b\otimes z ) u^k(q_n\otimes 1_{\mathcal Z}))\right| \\
& \leq \ \sqrt{ \mu(q_n \otimes y^2)}\|b\|\|z\| \sqrt{\mu(q_n\otimes 1_{\mathcal Z})} \ .
\end{split}
\end{equation}
Note that it follows from \eqref{23-10-20b} that $\left\| (\sum_{j=0}^{k} g_j)^2 - \sum_{j=0}^{k} g_j^2\right\| \leq (k+1)^2\epsilon$ and hence
$$(\sum_{j=0}^{k} g_j)^2 \leq \sum_{j=0}^{k} g_j^2 + (k+1)^2 \epsilon 1_{\mathcal Z}\ \leq \ \sum_{j=0}^{k} g_j + (k+1)^2 \epsilon 1_{\mathcal Z}\ .
$$
It follows that $y^2 \leq y + (k+1)^2 \epsilon 1_{\mathcal Z}$ and then from \eqref{23-10-20d} that
$$
0 \ \leq \ \mu(q_n \otimes y^2) \ \leq \ ((k+1)^2 +1) \mu(q_n\otimes 1_{\mathcal Z}) \epsilon \ .
$$
Combined with \eqref{23-10-20e} this shows that $\mu((q_n \otimes 1_{\mathcal Z})(b\otimes yz ) u^k(q_n\otimes 1_{\mathcal Z})) \sim 0$, so to obtain \eqref{23-10-20} it suffices to show that $\mu((q_n \otimes 1_{\mathcal Z})(b\otimes zg_j) ) u^k(q_n\otimes 1_{\mathcal Z})) \sim 0$ for each $j$. It follows from \eqref{23-10-20a} that
$$
\mu((q_n \otimes 1_{\mathcal Z} )(b\otimes zg_j) ) u^k(q_n\otimes 1_{\mathcal Z})) \ \sim \ \mu((q_n \otimes 1_{\mathcal Z})(b\otimes \sqrt{g_j} z\sqrt{g_j}) ) u^k(q_n\otimes 1_{\mathcal Z})) \ .
$$
Using Proposition 5.5.2 in \cite{Pe} for the second equality we get that
\begin{align*}
&\mu((q_n \otimes 1_{\mathcal Z})(b\otimes \sqrt{g_j} z\sqrt{g_j}) u^k(q_n\otimes 1_{\mathcal Z})) \\
&= \mu((q_n \otimes \sqrt{g_j})(b\otimes z \sqrt{g_j} ) u^k(q_n\otimes 1_{\mathcal Z})) \\
& = \mu((q_n \otimes 1_{\mathcal Z})(b\otimes z \sqrt{g_j} ) u^k(q_n\otimes \sqrt{g_j})) \\
& = \mu((q_n \otimes 1_{\mathcal Z})(b{\gamma''}^k(q_n)\otimes z \sqrt{g_j} \theta^k(\sqrt{g_j}) ) u^k(q_n\otimes 1_{\mathcal Z})) \ .
\end{align*}
It follows therefore from \eqref{23-10-20c} that
$ \mu((q_n \otimes 1_{\mathcal Z})(b\otimes zg_j) ) u^k(q_n\otimes 1_{\mathcal Z})) \sim 0$ as desired.
\end{proof}
The next lemma is well-known.
\begin{lemma}\label{23-10-20g} Let $B$ be an AF algebra. There is a bijective correspondence between traces $\tau$ on $B$ and the set of non-zero positive homomorphisms $ K_0(B) \to \mathbb R$. The bijection is given by the formula $\tau_*[e] = \tau(e)$ when $e$ is a projection in $B$.
\end{lemma}
\section{Proof of Theorem \ref{20-12-20c} when $K$ is compact}\label{compact}
The case where $K$ is compact is sufficiently different from the case where this is not the case, that we divide the proof accordingly. However, the general setup which we now describe will be the same in the two cases.
\subsection{Setting the stage}\label{stage}
Let $A$ be an infinite dimensional simple unital AF algebra. The tracial state space $T(A)$ of $A$ is a compact metrizable Choquet simplex which we denote by $\Delta$. We shall handle $\Delta$ as an abstract simplex and when an element $x \in \Delta$ is considered as a trace on $A$ we denote it by $\operatorname{tr}_x$. The $K_0$-group $K_0(A)$ is a simple non-cyclic dimension group and we denote by $K_0(A)^+$ the semi-group of positive elements of $K_0(A)$. The unit $1$ in $A$ represents an element $[1] \in K_0(A)$ which is an order unit in $K_0(A)$. To simplify notation we set
$$
(K_0(A),K_0(A)^+,[1]) \ = \ (H,H^+, u) \ .
$$
Each element $x \in \Delta$ defines in a canonical way a homomorphism ${\operatorname{tr}_x}_* : H \to \mathbb R$, and since $A$ is AF the state space
$$
\left\{\phi \in \Hom(H,\mathbb R) : \ \phi(H^+) \subseteq [0,\infty) , \ \phi(u) = 1 \right\}
$$
of $(H,H^+,u)$ is affinely homeomorphic to $\Delta$ via the map $\Delta \ni x \mapsto {\operatorname{tr}_x}_*$. Let ${\operatorname{Aff}}(\Delta)$ denote the space of real-valued continuous affine functions on $\Delta$. There is then a homomorphism
$$
\theta : H \to {\operatorname{Aff}}(\Delta)
$$
defined such that $\theta(h)(x) = {\operatorname{tr}_x}_*(h)$. By Theorem 4.11 in \cite{GH},
\begin{itemize}
\item $\theta(H)$ is norm-dense in ${\operatorname{Aff}}(\Delta)$, and
\item $H^+ \ = \ \left\{h \in H: \ \theta(h)(x) > 0 \ \ \forall x \in \Delta \right\} \cup \{0\}$ .
\end{itemize}
Set
$$
G \ = \ \oplus_{\mathbb Z} H \ ,
$$
and define an automorphism $\rho$ of $G$ such that
\begin{equation}\label{rho}
\rho\left((g_n)_{n \in \mathbb Z}\right) \ = \left(g_{n+1}\right)_{n \in \mathbb Z} \ .
\end{equation}
The main difference between the two cases, compact and not compact, is in the definition of the ordering in $G$, and although many arguments are the same in the two cases, and will only be given once, we make a clear distinction between the two cases in order to increase the readability.
\subsection{The first construction}\label{first}
We assume now that the set $K$ from Theorem \ref{20-12-20c} is compact and set $L = \left\{e^{\beta} : \ \beta \in K\right\}$. Let $A(\Delta \times L)$ denote the real vector space of functions $f \in C_{\mathbb R}(\Delta \times L)$ for which the map $\Delta \ni x \mapsto f(x,t)$ is affine for all $t \in L$; clearly a norm-closed subspace of $C_{\mathbb R}(\Delta \times L)$. Define
$\Sigma : G \to A(\Delta \times L)$
such that
$$
\Sigma\left(g\right)(x,t) \ = \ \sum_{m \in \mathbb Z} \theta(g_m)(x)t^{m} \ .
$$
Let $F$ be a non-empty closed face in $\Delta$ and set
$$
G^+ \ = \ \left\{ g \in G : \ \Sigma(g)(x,t) \ > \ 0 \ \ \forall (x,t) \in (F \times L) \cup (\Delta \times \{1\}) \right\} \cup \{0\} \ ;
$$
a semi-group in $G$ which turns $G$ into a partially ordered abelian group. We aim to show that $G$ is a simple dimension group.
For $(k,l) \in \mathbb Z^2$ let $g_{k,l} \in C_{\mathbb R}(L)$ be the function
$$
g_{k,l}(t) \ = \ t^k-t^l \ .
$$
\begin{lemma}\label{15-12-20xx} $\operatorname{Span} \{g_{k,l} : \ (k,l) \in \mathbb Z^2 \}$ is dense in $\left\{ h \in C_{\mathbb R}(L): \ h(1) = 0 \right\}$.
\end{lemma}
\begin{proof} Note that $tg_{k,l}(t) = g_{k+1,l+1}(t)$. It follows that $\operatorname{Span} \{g_{k,l} : \ (k,l) \in \mathbb Z^2 \}$ contains all functions of the form $P(t)\left(t - t^{-1}\right)$ where $P$ is a polynomial. It follows therefore from Weierstrass' theorem that the closure of $\operatorname{Span} \{g_{k,l} : \ (k,l) \in \mathbb Z^2 \}$ contains all functions in $C_{\mathbb R}(L)(t -t^{-1})$. Let $ h \in C_{\mathbb R}(L)$ satisfy that $ h(1) = 0$, and let $\epsilon > 0$. There is a function $h_1 \in C_{\mathbb R}(L)$ such that $h_1(t) = 0$ for all $t$ in a neighborhood of $1$ and $\sup_{t \in L}|h(t)-h_1(t)| \leq \epsilon$. Since
$$
h_1(t) = \frac{h_1(t)}{t -t^{-1}} (t-t^{-1})
$$
it follows first that $h_1$ is in the closure of $\operatorname{Span} \{g_{k,l} : \ (k,l) \in \mathbb Z^2 \}$, and then that so is $h$.
\end{proof}
When $h \in H$ and $k,l \in \mathbb Z, \ k \neq l$, we denote in the following by $[[h]]^{k,l}$ the element of $G$ such that
$$
\left( [[h]]^{k,l}\right)_i \ = \ \begin{cases} h, & \ i = k \\ -h , & \ i= l\\ 0, & \ i \notin \{k,l\} \ , \end{cases}
$$
and by $[[h]]\in G$ the element with $[[h]]_0 = h$ and $[[h]]_i = 0$ when $i \neq 0$.
\begin{lemma}\label{16-12-20xx} $\Sigma(G)$ is dense in $A(\Delta \times L)$.
\end{lemma}
\begin{proof} Note that $\Sigma\left([[h]]^{k,l}\right)(x,t) \ = \ \theta(h)(x)g_{k,l}(t)$ and $\Sigma([[h]])(x,t) = \theta(h)(x)$. Hence $\Sigma(G)$ contains all functions of the form
$$
\theta(h)(x)(t^k-t^l) \ ,
$$
where $k,l \in \mathbb Z$, as well as all functions of the form $\theta(h)(x)$ when $h \in H$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ it follows from Lemma \ref{15-12-20xx} that the closure $\overline{\Sigma(G)}$ of $\Sigma(G)$ contains all functions of the form $(x,t) \mapsto a(x)f(t)$ where $a \in {\operatorname{Aff}}(\Delta)$ and $f \in C_{\mathbb R}(L)$. A wellknown partition of unity argument shows then that $A(\Delta \times L) = \overline{\Sigma(G)}$.
\end{proof}
\begin{lemma}\label{21-12-20} Let $a \in {\operatorname{Aff}}(F)$. For each $x_0 \in \Delta \backslash F$ there is a function $b \in {\operatorname{Aff}}(\Delta)$ such that $b|_F = a$ and $b(x_0) > a(y)$ for all $y \in F$.
\end{lemma}
\begin{proof} Choose $u,v \in \mathbb R$ such that $u \leq a(y) \leq v$ for all $y \in F$. By the Hahn-Banach separation theorem there is a $c \in {\operatorname{Aff}} (\Delta)$ such that $c(y) \leq 0$ for all $y \in F$ and $c(x_0)> 0$. Multiplying by a positive number we can arrange that $c(x_0) > v -u $. Set $C = \sup_{x \in \Delta} c(x)$. Then $c(y) \leq a(y) - u \leq v-u + C$ for all $y \in F$ and $c(x) \leq v-u+C$ for all $x \in \Delta$. It follows from Edwards separation theorem, Theorem 3 in \cite{Ed} or Corollary 7.7 in \cite{AE}, that there is a $b' \in {\operatorname{Aff}}(\Delta)$ such that $b'|_F = a-u$ and $c \leq b' \leq v-u +C$ on $\Delta$. Set $b = b' +u$. Then $b|_F =a$ and $b(x_0) \geq c(x_0) + u > v$.
\end{proof}
\begin{lemma}\label{18-12-20} $G$ is a simple dimension group.
\end{lemma}
\begin{proof} It is easy to see that $G$ is unperforated and that $G^+ \cap (-G^+) = \{0\}$. Since $(F \times L) \cup (\Delta \times \{1\})$ is compact it is also clear that every non-zero element of $G^+$ is an order unit for $G$, so what remains is to show that $G$ has the Riesz interpolation property. Let $c^i,d^i, \ i \in \{1,2\}$, be elements of $G$ such that $c^i \leq d^j$ for all $i,j \in \{1,2\}$. If $c^{i'} = d^{j'}$ for some $i',j'$, set $z = c^{i'}$. Then $c^i \leq z \leq d^j$ for all $i,j$. Assume instead that $c^i < d^j$ for all $i,j$. Then $\Sigma(c^i)(x,t) < \Sigma(d^j)(x,t)$ for all $i,j$ and all $(x,t) \in (F\times L) \cup (\Delta \times \{1\})$. Define $c,d \in C_{\mathbb R}(\Delta \times L)$ such that
$$
c(x,t) \ = \ \max \{ \Sigma(c^1)(x,t),\Sigma(c^2)(x,t) \}
$$
and
$$
d(x,t) \ = \ \min \{ \Sigma(d^1)(x,t),\Sigma(d^2)(x,t) \} \ .
$$
Let $\delta > 0$. If $\delta$ is small enough
$$
c(x,t) + \delta \ < \ d(x,t) - \delta
$$
for all $(x,t) \in (F\times L) \cup (\Delta \times \{1\})$. Fix $t \in L \backslash\{1\}$. As a closed face in the simplex $\Delta$ the set $F$ is itself a Choquet simplex and hence the space ${\operatorname{Aff}}(F)$ has the Riesz interpolation property for the usual order, as well as for the strict order, cf. Lemma 3.1 in \cite{EHS}. It follow therefore that there is a function $a_t \in {\operatorname{Aff}}(F)$ such that
$$
c(y,t) + \delta < a_t(y) < d(y,t) - \delta
$$
for all $y \in F$. When $t = 1$ it follows in the same way that there is a function $a_1 \in {\operatorname{Aff}}(\Delta)$ such that
$$
c(x,1) + \delta \ < \ a_1(x) \ < \ d(x,1) - \delta
$$
for all $x \in \Delta$. We can then construct a finite cover $V_i, i = 1,2,\cdots, N$, of $L$ by open sets and elements $t_i \in V_i$ with $t_1 =1$ such that
$$
\left| c(x,t) - c(x,t_i)\right| \leq \frac{\delta}{2}
$$
and
$$
\left| d(x,t) - d(x,t_i)\right| \leq \frac{\delta}{2}
$$
for all $t \in V_i$, all $x \in \Delta$ and all $i$. We arrange, as we can, that $1 \notin V_j , \ j \geq 2$. For $i \in \{2,3,\cdots , N\}$ we use Lemma \ref{21-12-20} to get $b_i\in {\operatorname{Aff}}(\Delta)$ such that $b_i|_F = a_{t_i}$, and we set $b_1 = a_1$. Let $\{\varphi_i\}_{i=1}^N$ be a partition of unity on $L$ which is subordinate to $\{V_i\}_{i=1}^N$. Define
$a \in A(\Delta \times L)$ such that
$$
a(x,t) \ = \ \sum_{i=1}^N b_i(x)\varphi_i(t) \ .
$$
For $y \in F$,
\begin{align*}
&a(y,t) \ = \ \sum_{i=1}^N a_{t_i}(y)\varphi_i(t) \ \leq \ \sum_{i=1}^N (d(y,t_i ) - \delta)\varphi_i(t) \ \\
&\leq \ \sum_{i=1}^N (d(y,t) - \frac{\delta}{2}) \varphi_i(t) \ = \ d(y,t) - \frac{\delta}{2} \ ,
\end{align*}
and similarly $c(y,t) + \frac{\delta}{2} \ \leq \ a(y,t)$.
For $x \in \Delta$ we find that
\begin{align*}
&c(x,1) + \delta \ < \ a_1(x) \ = \ a(x,1) \ < \ d(x,1) - \delta \ .
\end{align*}
It follows that
$$
\Sigma(c^i)(x,t) < a(x,t) < \Sigma(d^j)(x,t)
$$
for all $i,j$ and all $(x,t) \in (F\times L) \cup (\Delta \times \{1\})$. Let $\epsilon > 0$. By Lemma \ref{16-12-20xx} there is a $g \in G$ such that
$$
\sup_{(x,t) \in \Delta \times L} \left|\Sigma(g)(x,t) \ - \ a(x,t)\right| \ \leq \ \epsilon \ .
$$
If $\epsilon$ is small enough it follows that $c^i \leq g \leq d^j$ in $G$ for all $i,j \in \{1,2\}$.
\end{proof}
Note that $\rho(G^+) = G^+$ where $\rho$ is the automorphism $\rho$ of $G$ defined in \eqref{rho}. It follows that $\rho$ is an automorphism of $(G,G^+)$ and then from Lemma \ref{18-12-20} and \cite{EHS} that there is an AF algebra $B$ whose $K_0$-group and dimension range is isomorphic to $(G ,G^+)$. Furthermore, it follows from \cite{E1} that $B$ is simple and stable, and that there is an automorphism $\gamma$ of $B$ such that $\gamma_* = \rho$ under the identification $K_0(B) = G$. It was shown by Nistor, \cite{Ni}, that $(G,G^+)$ has large denominators and we can therefore use Lemma \ref{28-10-20} to choose $\gamma$ such that it has the following additional properties:
\begin{poem}\mbox{}\\[-\baselineskip]
\begin{enumerate}\label{listrefx}
\item The restriction map $\mu \ \mapsto \ \mu|_{B}$ is a bijection from the traces $\mu$ on $ B \rtimes_{\gamma} \mathbb Z$ onto the $\gamma$-invariant traces on $B$. \\
\item $B \rtimes_{\gamma} \mathbb Z$ is stable. \\
\item $B \rtimes_{\gamma} \mathbb Z$ is $\mathcal Z$-stable.
\end{enumerate}
\end{poem}
Set $C = B \rtimes_\gamma \mathbb Z$. No power of $\gamma$ is inner since $\gamma_*^k \neq \id_G$ for $k\neq 0$ and it follows therefore from \cite{Ki1} that $C$ is simple. The Pimsner-Voiculescu exact sequence shows that $K_1(C) = 0$ since $\id_G - \gamma_*$ is injective and that
$$
K_0(C) \ = \ \coker (\id_G-\gamma_*) \ = \ G/\left(\id_G - \gamma_*\right)(G) \ .
$$
Under this identification the map $K_0(B) \to K_0(C)$ induced by the inclusion $B \subseteq C$ is the quotient map $q : G \to G/\left(\id_G - \gamma_*\right)(G)$. Hence $G^+/\left(\id_G - \gamma_*\right)(G) \subseteq K_0(C)^+$.
\begin{lemma}\label{27-10-20dxx} Let $d \in G$. Then $\sum_{m\in \mathbb Z} d_m = 0$ if and only if $d = (\id_G - \gamma_*)\left(g\right)$ for some $g \in G$.
\end{lemma}
\begin{proof} It obvious that $\sum_{m\in \mathbb Z} d_m = 0$ when $d = (\id_G - \gamma_*)\left(g\right)$ for some $g \in G$. For the converse, choose $L\in \mathbb N$ so big that $x_n =0$ when $|n| \geq L$. Set $y_n = 0$ when $n > L$ and
$$
y_L = x_L , \ y_{L-1} = x_{L-1} + y_L, \ y_{L-2} = x_{L-2} +y_{L-1} , \ \text{etc} .
$$
Then
$$
y_{-L} = x_{-L} + \sum_{i=1}^{2L} x_{-L+i} = \sum_{i=1}^{2L} x_{-L+i} = 0 \ ,
$$
and hence $y_k = 0$ when $k < -L$. It follows that $(y_n)_{n \in \mathbb Z} \in G$ and $(\id_G -\gamma_*)\left((y_n)_{n\in \mathbb Z}\right) = (x_n)_{n \in \mathbb Z}$.
\end{proof}
It follows from Lemma \ref{27-10-20dxx} that we can define an injective homomorphism $S : G/\left(\id_G - \gamma_*\right)(G) \to H$ such that
$$
S\left(q(g)\right) \ = \ \sum_{m \in \mathbb Z} g_m \ \ \ \ \forall g \in G \ .
$$
$S$ is surjective since $S(q([[h]])) = h$ when $h \in H$, and hence an isomorphism with inverse $S^{-1}$ given by $S^{-1}(h) = q([[h]])$. Let $p \in B$ be a projection such that $[p] = [[u]]$ in $G$.
\begin{lemma}\label{18-12-20aaa} For $(y,t) \in F \times L$ and $x \in \Delta$ there are traces $\tau_{y,t}$ and $\tau_x$ on $B$ such that $\tau_{y,t}(p) = \tau_x(p) = 1$, and
$$
{\tau_{y,t}}_* (g) \ = \ \sum_{m \in \mathbb Z} \theta(g_m)(y)t^{m}
$$
and
$$
{\tau_x}_*(g) = \ \sum_{m \in \mathbb Z} \theta(g_m)(x)
$$
for all $g \in G$.
\end{lemma}
\begin{proof} This follows from Lemma \ref{23-10-20g}.
\end{proof}
\begin{lemma}\label{18-12-20b} Let $\phi : G \to \mathbb R$ be a positive homomorphism such that $\phi([[u]])= 1$. Assume that $\phi \circ \gamma_* = s^{-1} \phi$ for some $s > 0$. Then $s \in L$ and when $s \neq 1$ it follows that $\phi = {\tau_{y,s}}_*$ for some $y \in F$, and when $s =1$ it follows that $\phi = {\tau_x}_*$ for some $x \in \Delta$.
\end{lemma}
\begin{proof} We claim that there is continuous linear map $\phi' : A(\Delta \times L) \to \mathbb R$ such that $\phi = \phi' \circ \Sigma$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ there is a $c \in H$ such that $0 < \theta(c)(x) < N^{-1}$ for all $x \in \Delta$. Then $0 \leq N[[c]] \leq [[u]]$ in $G$ and hence $0 \leq \phi([[c]]) \leq \frac{1}{N}\phi\left([[u]]\right) = \frac{1}{N}$. Assume $g \in G$ and that $\Sigma(g) = 0$. Then $\pm g + [[c]]\in G^+$ and hence $-\frac{1}{N} \leq \phi(g) \leq \frac{1}{N}$. Letting $N \to \infty$ we conclude that $\phi(g) =0$, and it follows that there is a homomorphism $\phi' : \Sigma(G) \to \mathbb R$ such that $\phi' \circ \Sigma = \phi$. Let $h \in \Sigma(G)$; say $h = \Sigma(g)$, and let $k,l \in \mathbb N$ be natural numbers such that $|h(x,t)| < \frac{k}{l}$ for all $(x,t) \in \Delta \times L$. Then $k[[u]] \pm l g \in G^+$ and hence
$$
0 \ \leq \ \phi(k[[u]] \pm l g) \ = \ k \pm l \phi'(h) \ .
$$
It follows that $\left|\phi'(h)\right| \leq \frac{k}{l}$, proving that $\phi'$ is Lipshitz continuous. Since $\Sigma(G)$ is dense in $A(\Sigma \times L)$ by Lemma \ref{16-12-20xx} it follows that $\phi'$ extends by continuity to a continuous linear map $\phi' : A(\Delta \times L) \to \mathbb R$, proving the claim. Let $T : A(\Delta \times L) \to A(\Delta \times L)$ denote the operator
$$
T(\psi)(x,t) = t^{-1}\psi(x,t) \ .
$$
Since $\Sigma \circ \gamma_*(g)(x,t) \ = \ t^{-1} \Sigma(g)(x,t)$ for all $g \in G$ and all $(x,t) \in \Delta \times L$, we find that
$$
s^{-1} \phi'( \Sigma(g)) \ = \ s^{-1}\phi(g) \ = \ \phi(\gamma_*(g)) \ = \ \phi' \circ \Sigma \circ \gamma_*(g)) \ = \ \phi'\left(T( \Sigma(g))\right) \ .
$$
It follows therefore from Lemma \ref{16-12-20xx} that
\begin{equation}\label{18-20-12d}
s^{-1}\phi' = \phi' \circ T \ .
\end{equation}
When $a \in {\operatorname{Aff}}(\Delta)$ and $f \in C_{\mathbb R}(L)$ we denote by $a \otimes f$ the function $\Delta \times L \ni (x,t) \mapsto a(x)f(t)$. Assume $a(x) \geq 0$ for all $x \in \Delta$ and that $f(t) \geq 0$ for all $t \in L$. It follows then from Lemma \ref{16-12-20xx} and the definition of $G^+$ that $a \otimes f$ can be approximated by elements from $\Sigma(G^+)$, implying that $\phi'(a\otimes f) \geq 0$. There is therefore a bounded Borel measure $\mu_a$ on $L$ such that
$$
\phi'(a\otimes f) \ = \ \int_L f \ \mathrm{d}\mu_a \ \ \forall f \in C_{\mathbb R}(L) \ .
$$
It follows from \eqref{18-20-12d} that
$$
\int_L s^{-1} f(t) \ \mathrm{d}\mu_a(t) \ = \ \int_L t^{-1}f(t) \ \mathrm{d}\mu_a(t)
$$
for all $f \in C_{\mathbb R}(L)$, and hence that
$\mu_a(L \backslash \{s\}) = 0$. Since $\phi \neq 0$, not all $\mu_a$ can be zero and we conclude therefore that $s \in L$ and that
\begin{equation}\label{21-12-20d}
\phi'(a \otimes f) \ = \ \lambda(a)f(s)
\end{equation}
for some $\lambda(a) \geq 0$ and all $f \in C_{\mathbb R}(L)$. Every element of ${\operatorname{Aff}}(\Delta)$ is the difference between two positive elements and it follows therefore that for all $a\in {\operatorname{Aff}}(\Delta)$ there is a real number $\lambda(a)$ such that \eqref{21-12-20d} holds for all $f \in C_{\mathbb R}(L)$. The resulting map $a \mapsto \lambda(a)$ is clearly linear and positive, and $\lambda(1) = 1$ since $1 = \phi([[u]]) = \phi'(1)$. This implies that there is an $x_0 \in \Delta$ such that $\lambda(a) = a(x_0)$ for all $a \in {\operatorname{Aff}}(\Delta)$. The elements of $\left\{a \otimes f : \ a \in {\operatorname{Aff}}(\Delta) , \ f \in C_{\mathbb R}(L) \right\}$ span a dense set in $A(\Delta \times L)$ and we find therefore that
\begin{equation}\label{18-12-20e}
\phi'(h) \ = \ h(x_0,s) \ \ \forall h \in A(\Delta \times L) \ .
\end{equation}
To see that $x_0 \in F$ when $s \neq 1$, assume for a contradiction that $x_0 \notin F$. By Lemma \ref{21-12-20} we can find $b \in {\operatorname{Aff}}(\Delta)$ such that $b|_F = 0$ and $b(x_0) > 0$, and since $s \neq 1$ we can find $f \in C_{\mathbb R}(L)$ such that $f(1) = 0$ and $f(s) = 1$. The function $b \otimes f$ vanishes on $(F \times L) \cup (\Delta \times \{1\})$. By Lemma \ref{16-12-20xx} and the definition of $G^+$ this implies that $\phi'(b \otimes f) = 0$, which contradicts \eqref{18-12-20e} since $b(x_0)f(s) > 0$. Hence $x_0 \in F$ when $s \neq 1$. It follows that
$$
\phi(g) \ = \ \phi'(\Sigma(g)) \ = \ \sum_{m \in \mathbb Z}\theta(g_m)(x_0)s^{m} \ ,
$$
and that $x_0 \in F$ when $s \neq 1$.
\end{proof}
\subsection{The Elliott invariants of $pCp$ and $A$ are isomorphic}
\begin{lemma}\label{09-10-20axx} Set
$$
G^{++} = \{0\} \cup \left\{ g \in G : \ \sum_{m \in \mathbb Z} \theta(g_m)(x)\ > \ 0 \ \ \forall x \in \Delta \right\} \ .
$$
Then $K_0(C)^+ = {G^{++}}/\left(\id_G - \gamma_*\right)(G)$ and
$S$ takes $K_0(C)^+$ onto $H^+$.
\end{lemma}
\begin{proof} Let $g = (g_m)_{m\in \mathbb Z} \in G^{++} \backslash \{0\}$ and set $g^+ = [[\sum_{m \in \mathbb Z} g_m]] \in G$. Then $g^+ - g \in (\id_G - \gamma_*)(G)$ by Lemma \ref{27-10-20dxx}, and since
$$
\sum_{m \in \mathbb Z} \theta(g^+_m)(x)t^{m} = \sum_{m \in \mathbb Z} \theta(g_m)(x) \geq \epsilon \ ,
$$
for all $(x,t) \in \Delta \times L$, where $\epsilon = \min_{x \in \Delta} \sum_{m \in \mathbb Z} \theta(g_m)(x) > 0$, it follows that $g^+ \in G^+$. This shows that
$$
{G^{++}}/\left(\id_G - \gamma_*\right)(G) \subseteq {G^{+}}/\left(\id_G - \gamma_*\right)(G) \subseteq K_0(C)^+ \ .
$$
Let $z \in K_0(C)^+\backslash \{0\} \subseteq G/\left(\id_G - \gamma_*\right)(G)$ and choose $g \in G$ such that $q(g) = z$. Let $x \in \Delta$. The trace $\tau_{x}$ from Lemma \ref{18-12-20b} is $\gamma$-invariant and there is therefore a trace $\tau_x'$ on $C$ such that $\tau_x'|_{B} = \tau_{x}$. Then ${\tau_x'}_*(z) > 0$ since $z > 0 $ in $K_0(C)$ and $C$ is simple. Hence
$$
0 <{\tau_x'}_*(z) = {\tau_{x}}_*(g) = \ \sum_{m \in \mathbb Z} \theta(g_m)(x) \ .
$$
It follows that $g\in G^{++}$ and $z \in {G^{++}}/\left(\id_G - \gamma_*\right)(G)$. We conclude that
$K_0(C)^+ = {G^{++}}/\left(\id_G - \gamma_*\right)(G)$ and $S(K_0(C)^+) \subseteq H^+$. When $h \in H^+ \backslash \{0\}$, set $z = [[h]]$. Then $z \in G^{++}$, $q(z) \in K_0(C)^+$ and $S(q(z)) = h$.
\end{proof}
\begin{lemma}\label{02-12-20bx} Let $p \in B$ be a projection representing $[[u]] \in G= K_0(B)$. The Elliott invariants of $A$ and $p\left( B \rtimes_\gamma \mathbb Z\right)p$ are isomorphic.
\end{lemma}
\begin{proof} Since $K_1(A) = K_1(pCp) = 0$ it suffices to supplement the isomorphism $S : (K_0(C),K_0(C)^+,[[u]]) \to (H,H^+,u)$ of partially ordered groups with order unit with an affine homeomorphism $T : T(pCp) \to T(A)$ such that
\begin{equation}\label{20-12-20}
T(\tau)_*(S(z)) = \tau_*(z) \ \ \ \forall (\tau,z) \in T(pCp) \times K_0(C) \ .
\end{equation}
Let $\tau \in T(pCp)$. By Proposition 4.7 in \cite{CP} there is a unique trace $\tau'$ on $C$ such that $\tau'|_{pCp} = \tau$. Then $\tau'|_B$ is a $\gamma$-invariant trace on $B$ such that $\tau'(p) = 1$. It follows from Lemma \ref{23-04-21} that there is an $x \in \Delta$ such that ${(\tau'|_B)}_* = {\tau_x}_*$ on $K_0(B)$.
Since $x$ is uniquely determined by $\tau$ we can define $T : T(pCp) \to T(A)$ such that $T(\tau) = \operatorname{tr}_x$. Let $e \in K_0(A), \ 0 \leq e \leq [1]$. Then $0 \leq [[e]] \leq [[u]]$ in $G$ and hence $[[e]]$ is represented in $K_0(B)$ by a projection $e'$ from $pBp \subseteq pCp$. We find that
$$
{\operatorname{tr}_x}_*(e) = \theta(e)(x) = {\tau_x}_* ([[e]]) = {(\tau'|_B)}_*([[e]]) = \tau(e') \ ;
$$
an equality which shows that $T$ is affine and continuous. To see that $T$ is surjective let $x \in \Delta$. The trace $\tau_x$ from Lemma \ref{18-12-20aaa} is $\gamma$-invariant and defines therefore a trace $\tau'$ on $C$ such that $\tau'|_B = \tau_x$. Then $\tau'(p) = {\tau_x}_*([[u]]) = 1$ and hence $\tau'|_{pCp} \in T(pCp)$ and $T\left(\tau'|_{pCp}\right) = \operatorname{tr}_x$. To see that $T$ is also injective consider to traces $\tau_i, i = 1,2$, in $T(pCp)$ and assume that $T(\tau_1) = T(\tau_2)$. Then ${(\tau'_1|_B)}_* = {(\tau'_2|_B)}_*$ on $G$ and hence $\tau'_1|_B = \tau'_2|_B$ by Lemma \ref{23-10-20g}. It follows from the first of the additional properties \ref{listrefx} that $\tau'_1 = \tau'_2$ and hence also that $\tau_1 = {\tau'_1}|_{pCp} = {\tau'_2}|_{pCp} = \tau_2$. We conclude that $T$ is an affine homeomorphism. Consider $\tau \in T(pCp)$ and $e \in K_0(A), \ 0 \leq e \leq 1$. Then $T(\tau) = \operatorname{tr}_x$ for some $x \in \Delta$ and, as observed above, $[[e]] \in G$ is represented by a projection $e'\in pBp$ such that ${\operatorname{tr}_x}_*(e) = \tau(e')$. Hence
\begin{align*}
& T(\tau)_*\left( S(S^{-1}(e))\right) = {\operatorname{tr}_x}_*(e) = \tau(e') = \tau_*(q([[e]]) =\tau_* \left(S^{-1}(e)\right) \ .
\end{align*}
Since the elements of $\left\{ S^{-1}(e) : \ e \in K_0(A), \ 0 \leq e \leq [1]\right\}$ generate $K_0(pCp)$ as a group, we conclude that \eqref{20-12-20} holds.
\end{proof}
\subsection{Conclusion}
\begin{lemma}\label{20-12-20a} $A$ is $*$-isomorphic to $p(B \rtimes_\gamma \mathbb Z)p$.
\end{lemma}
\begin{proof} By Lemma \ref{02-12-20bx} it remains only to show that $A$ and $pCp$ belong to a class of simple $C^*$-algebras for which the Elliott invariant is complete. For this we appeal to Corollary D of \cite{CETWW} and we must therefore check that both algebras are separable, unital, nuclear, $\mathcal Z$-stable and satisfy the UCT. Only $\mathcal Z$-stability is not well-known. For $A$ this follows from Theorem A and Corollary C in \cite{CETWW} and for $pCp$ it follows from the third of the Additional properties \ref{listrefx} combined with Corollary 3.2 in \cite{TW}.
\end{proof}
\begin{remark}\label{14-12-20gx} Assume that $A$ is UHF. Then the algebras $pCp$ and $A$ occurring in the last proof only have one trace state and there is an alternative way to deduce the isomorphism $A \simeq pCp$. The path through the litterature takes more space to explain, but presents presumably a shorter argument: By Corollary 6.2 in \cite{MS} it suffices to show that both algebras are unital, separable, simple, infinite dimensional, nuclear, quasi-diagonal, satisfy the UCT and have strict comparison. Many of these properties are well-known for both algebras. What remains is to explain why they are quasi-diagonal and have strict comparison. Since $pCp$ and $A$ are both exact, simple and unital it follows from Corollary 4.6 in \cite{Ro} that $\mathcal Z$-stability implies strict comparison and hence it suffices to argue that both algebras are quasi-diagonal and $\mathcal Z$-absorbing. Concerning quasi-diagonality it is well-known that AF algebras have this property and that it is inherited by subalgebras, so a straightforward application of \cite{Br} shows that both algebras are quasi-diagonal. $C$ is $\mathcal Z$-absorbing by the third of the additional properties \ref{listrefx} and hence so is $pCp$ by Corollary 3.2 in \cite{TW} since $pCp$ is stably isomorphic $C$ by \cite{B}. Finally, since $A$ is approximately divisible it is also $\mathcal Z$-absorbing by Theorem 2.3 in \cite{TW}.
\end{remark}
\begin{remark}\label{28-04-21a} We haven't used the second of the Additional properties \ref{listrefx}. It was added because stability of $B \rtimes_{\gamma} \mathbb Z$ combined with Lemma \ref{20-12-20a} and \cite{B} implies that $B \rtimes_{\gamma} \mathbb Z \simeq A \otimes \mathbb K$; a fact which is nice to know, but it is not needed for the proof of Theorem \ref{20-12-20c}. \footnote{It has been pointed out to me that stability of $B \rtimes_{\gamma} \mathbb Z$ also follows from \cite{HR}.}
\end{remark}
The dual action on $B \rtimes_\gamma \mathbb Z$ defines by restriction a $2\pi$-periodic flow on $p(B \rtimes_\gamma \mathbb Z)p$ which we denote by $\widehat{\gamma}$.
\begin{lemma}\label{02-12-20d} For $\beta \in \mathbb R$ there is a $\beta$-KMS state for $\widehat{\gamma}$ on $p(B \rtimes_\gamma \mathbb Z)p$ if and only if $\beta \in K$. For $\beta \in K \backslash \{0\}$ the simplex of $\beta$-KMS states for $\widehat{\gamma}$ is affinely homeomorphic to $F$ and for $\beta =0$ it consists of all trace states on $p(B \rtimes_\gamma \mathbb Z)p$ and hence is affinely homeomorphic to $\Delta$.
\end{lemma}
\begin{proof} Let $\beta \in K \backslash \{0\}$ and $y \in F$. It follows from Lemma \ref{26-10-20} that we can define a $\beta$-KMS state on $p(B \times_\gamma \mathbb Z)p$ by $\tau_{y,e^{\beta}}\circ P|_{p(B \rtimes_{\gamma} \mathbb Z)p}$. The resulting map, from $F$ to the simplex of $\beta$-KMS states for $\widehat{\gamma}$ on $p(B \times_\gamma \mathbb Z)p$,
is clearly continuous, affine and injective. To see that it is surjective, let $\omega$ be a $\beta$-KMS state for $\widehat{\gamma}$. From Theorem 2.4 in \cite{Th2}, or more precisely from the part of that theorem which follows from Remark 3.3 in \cite{LN}, combined with Lemma \ref{26-10-20}, it follows that there is trace $\tau$ on $B$ such that $\tau \circ \gamma = e^{-\beta} \tau$, $\tau(p) = 1$ and $\omega = \tau \circ P|_{p(B\rtimes_\gamma \mathbb Z)p}$. It follows from Lemma \ref{23-04-21} that $\tau_* = {\tau_{y,e^{\beta}}}_*$ on $G$ for some $y \in F$ and then from Lemma \ref{23-10-20g} that $\tau = \tau_{y,e^{\beta}}$. We conclude that $\omega$ is the image of $y \in F$ under the map we consider. This shows that the map is an affine homeomorphism. The same argument, using the traces $\tau_x$ from Lemma \ref{18-12-20aaa}, shows that the simplex of $0$-KMS states for $\widehat{\gamma}$ is affinely homeomorphic to $\Delta$.
It remains to show that there are no $\beta$-KMS states for $\widehat{\gamma}$ unless $\beta \in K$. The argument for this is almost identical to one we have given above: Assume that there is $\beta$-KMS state for $\widehat{\gamma}$. It follow from Remark 3.3 in \cite{LN} that this $\beta$-KMS state is the restriction to $p(B \rtimes_\gamma \mathbb Z)p$ of a $\beta$-KMS weight for the dual action on $B \rtimes_\gamma \mathbb Z$ and then from Lemma \ref{26-10-20} that there is a trace $\tau$ on $B$ such that $\tau \circ \gamma = e^{-\beta} \tau$ and $\tau(p)=1$. Then $\tau_* : G \to \mathbb R$ is a positive homomorphism such that $\tau_*\circ \gamma_* = e^{-\beta}\tau_*$ and $\tau_*([[u]]) =1$. It follows from Lemma \ref{23-04-21} that $\beta \in K$.
\end{proof}
Now Theorem \ref{20-12-20c}, with $K$ compact, follows from Lemma \ref{02-12-20d} and Lemma \ref{20-12-20a}.
\section{Proof of Theorem \ref{20-12-20c} when $K$ is unbounded}
Let $K$ be the closed lower bounded set of real numbers from Theorem \ref{20-12-20c} and set $L = \left\{e^{\beta} : \ \beta \in K\right\}$; a closed subset of $]0,\infty[$ bounded away from $0$. We have already covered the case where $K$ is also bounded above, so here we assume that $K$ and also $L$ is unbounded.
\subsection{The second construction}
Let $P_{\Delta}(t)$ denote the vector space of Laurent polynomials in $t$ with coefficients from ${\operatorname{Aff}}(\Delta)$. Thus $P_{\Delta}(t)$ consists of the functions $f : \Delta \times ]0,\infty[ \to \mathbb R$ of the form
\begin{equation}\label{22-04-21}
f(x,t) = \sum_{n=-\infty}^\infty a_n(x)t^n \ ,
\end{equation}
where $a_n \in {\operatorname{Aff}}(\Delta)$ for all $n$ and only finitely many $a_n$'s are non-zero. Given $f \in P_\Delta(t)$ as in \eqref{22-04-21} we set
$$
\mathbb L(f) = \max \left\{ n \in \mathbb Z: \ a_n \neq 0 \right\} \
$$
when $f \neq 0$, and let $\mathbb L(0) = -\infty$. We will refer to $\mathbb L(f)$ as the degree of $f$ and the element $a_{\mathbb L(f)} \in {\operatorname{Aff}}(\Delta)$ as the leading coefficient of $f$.
Let $F$ be a closed non-empty face in $\Delta$. For $f \in P_\Delta(t)$ we write $0 \prec f$ when there is an $\epsilon >0$ such that
$$
t^{-\mathbb L(f)}f(x,t) \geq \epsilon \ \ \ \ \forall (x,t) \in (F \times L) \cup (\Delta \times \{1\}) \ .
$$
Thus $0 \prec f$ if and only if $f$ is strictly positive on $(F \times L) \cup (\Delta \times \{1\})$ and the leading coefficient of $f$ is strictly positive on $\Delta$. Given $f,g \in P_\Delta(t)$, we write $f \prec g$ when $0\prec g-f$. As in Section \ref{first} we set $G = \oplus_{\mathbb Z} H$ and define a homomorphism $\Sigma : G \to P_\Delta(t)$ such that
$$
\Sigma(g)(x,t) = \sum_{m \in \mathbb Z} \theta(g_m)(x)t^m \ .
$$
Set
$$
G^+ = \left\{ g \in G: \ 0 \prec \Sigma(g) \right\} \cup \{0\} \ .
$$
We aim now to show that $(G,G^+)$ is a Riesz group, and hence also a dimension group in the sense of \cite{EHS}. This means that we must prove that
\begin{itemize}
\item $G^+ + G^+ \subseteq G^+$,
\item $G^+ \cap (-G^+) = \{0\}$,
\item $ g\in G, \ n \in \mathbb N \backslash \{0\}, \ ng \in G^+ \ \Rightarrow \ g \in G^+$,
\item $G^+ - G^+ = G$, and
\item $G$ has the Riesz interpolation property.
\end{itemize}
The first three items are easily checked. To show that $G^+ - G^+ = G$, let $g \in G$. Choose $l \in \mathbb N$ bigger than any $n \in \mathbb N$ for which $\theta(g_n)$ is non-zero and set $l =0$ if $\theta(g_n) = 0$ for all $n$. Choose $N \in \mathbb N$ such that $Nt^{l} > \sum_{n \in \mathbb Z} \left|\theta(g_n)(x)\right|t^n$ for all $n \in \mathbb N$ and $(x,t) \in \Delta \times L$. Define $v \in G$ such that
$$
v_n = \begin{cases} 0, \ n \neq l \ ,\\ Nu , \ n = l \ . \end{cases}
$$
Then $v, v-g \in G^+$ and $ g = v-(v-g)$.
It remains to show that $G$ has the Riesz interpolation property. The next subsection is devoted to that.
\subsubsection{$(G,G^+)$ is a Riesz group}
For $g \in G$, set
$$
\mathbb L(g) = \max \{n \in \mathbb Z: g_n \neq 0 \}
$$
when $g \neq 0$, and $\mathbb L(0) = -\infty$. Then
$\mathbb L(\Sigma(g)) \leq \mathbb L(g)$.
\begin{lemma}\label{22-04-21x} Let $a \in {\operatorname{Aff}}(\Delta)$ and let $f: L \to\mathbb R$ be a continuous function such that $\lim_{t \to \infty} f(t) = 0$. Let $\epsilon > 0$ and $N \in \mathbb N$. There is a $g \in G$ such that
$\mathbb L(g) \leq -N$ and
$$
\left| \Sigma(g)(x,t) - f(t)a(x)\right| \leq \epsilon \ \ \forall (x,t) \in \Delta \times L \ .
$$
\end{lemma}
\begin{proof} By the Stone-Weierstrass theorem there is a real Laurent polynomial $Q(t) = \sum_{n= -M}^{-N} c_nt^{n}$ of degree at most $-N$ such that
$$
\left|a(x)\right|\left|Q(t) -f(t)\right| \leq \frac{\epsilon}{2}
$$
for all $x \in \Delta$ and all $t \in L$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ there are elements $h_n \in H$ such that
$$
\left|\theta(h_n)(x) - c_na(x) \right|t^{n} \leq \frac{\epsilon}{2(M-N)}
$$
for all $(x,t) \in \Delta \times L$ and all $n \in \{-M,-M+1,\cdots , -N\}$. Define $g \in G$ such that
$$
g_n = \begin{cases} 0 \ , \ & n \leq -M-1 \ \text{or} \ n \geq -N+1 \ ,\\ h_{n} \ , \ & \ -M \leq n \leq -N\ . \end{cases}
$$
Then $\mathbb L(g) \leq -N$ and
\begin{align*}
& \left| \Sigma(g)(x,t) - f(t)a(x)\right| \\
& \leq \left| \sum_{n=-M}^{-N} \theta(h_n)(x)t^{n} - \sum_{n=-M}^{-N} c_na(x)t^{n}\right| + \left| Q(t)a(x) - f(t)a(x)\right| \leq \epsilon \ .
\end{align*}
\end{proof}
\begin{lemma}\label{05-05-21a} Let $g \in G$ and let $F \subseteq \mathbb Z$ be a finite set. Assume that $g_n = 0, \ |n| \geq J\geq 1$. Let $N \in \mathbb N$ such that $J-N -d < -J$ for all $d \in F \cup \{0\}$. Let $q$ be a real-valued Laurent polynomial of degree at most $-N$. For any $\epsilon > 0$ there is an element $g' \in G$ such that $g'_n = g_n$ when $n\geq -J$, and
$$
\left|t^{-d}(1+q(t))\Sigma(g)(x,t) - t^{-d}\Sigma(g')(x,t) \right| \leq \epsilon
$$
for all $d \in F$ and all $(x,t) \in \Delta \times L$.
\end{lemma}
\begin{proof} By assumption we can write
$$
q(t)\Sigma(g)(x,t) = \sum_{n= B}^A a_n(x) t^n \ ,
$$
where $a_n \in {\operatorname{Aff}}(\Delta)$ for all $n$ and $B < A \leq J-N$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ and since $J-N -d < -J < 0$ we can find $h_n \in H, \ B \leq n \leq A$, such that
$$
\left| t^{-d} q(t) \sum_{n= B}^A a_n(x) t^n - t^{-d}\sum_{n= B}^A \theta(h_n)(x) t^n\right| \leq \epsilon
$$
for all $d \in F$ and all $(x,t) \in \Delta \times L$. Define $g' \in G$ such that $g'_n = h_n, \ B \leq n \leq A$, and $g'_n = g_n$ for $n \notin [B,A]$. Then
\begin{align*}
&\left|t^{-d}(1+q(t))\Sigma(g)(x,t) - t^{-d} \Sigma(g')(x,t)\right| \\
&= \left| t^{-d} q(t) \sum_{n= B}^A a_n(x) t^n - t^{-d}\sum_{n= B}^A \theta(h_n)(x) t^n\right| \leq \epsilon \
\end{align*}
for all $d \in F$ and all $(x,t) \in \Delta \times L$.
\end{proof}
\begin{lemma}\label{06-05-21a} Let $F : \Delta \times L \to \mathbb R$ be a continuous function such that $\Delta \ni x \mapsto F(x,t)$ is affine for all $t \in L$ and $\lim_{t \to \infty} F(x,t) = 0$ uniformly in $x$. Let $J \in \mathbb N$. For every $\delta > 0$ there is a $g \in G$ such that $\mathbb L(g) \leq -J$, and
$$
\left|\Sigma(g)(x,t) - F(x,t)\right| \leq \delta
$$
for all $(x,t) \in \Delta \times L$.
\end{lemma}
\begin{proof} Let $\kappa > 0$. There is then an $R > 0$ such that $\left|F(x,t)\right| \leq \kappa$ for all $x \in \Delta$ and all $t \geq R$. For each $t' \in L \cap \left]0, R\right]$, there is an element $h_{t'} \in H$ such that
$$
\left|F(x,t) - \theta(h_{t'})(x)\right| < \kappa \
$$
for all $x \in \Delta$ and all $t$ in an open neighborhood of $t'$. By compactness of $L \cap \left]0, R\right]$, we can therefore find open sets $U_i$ in $\mathbb R$ and elements $h_i \in H, i = 1,2, \cdots , n$, such that
$$
\left|F(x,t) - \theta(h_i)(x)\right| \leq \kappa
$$
for all $x \in \Delta \times (L\cap U_i), \ i =1,2,\cdots , n$, and $L \cap \left]0, R\right] \subseteq \bigcup_{i=1}^n U_i$. Let $\varphi_i : \mathbb R \to \mathbb R , \ i = 1,2,\cdots, n$, be continuous non-negative compactly supported functions such that $\operatorname{supp} \varphi_i \subseteq U_i$ for all $i$, $\sum_{i=1}^n \varphi_i \leq 1$ and $\sum_{i=1}^n \varphi_i(x) = 1$ for all $x \in L \cap \left]0, R\right]$. By Lemma \ref{22-04-21x} there are elements $g_i \in G$ such that $\mathbb L(g_i) \leq -J$ and
$$
\left| \Sigma(g_i)(x,t) - \varphi_i(t)\theta(h_i)(x)\right| \leq \frac{\kappa}{n}
$$
for all $(x,t) \in \Delta \times L$. Set $g = \sum_{i=1}^n g_i$ and note that $\mathbb L(g) \leq -J$. Furthermore,
\begin{align*}
&\left| \Sigma(g)(x,t) - F(x,t)\right| \\
& \leq \left| (1- \sum_{i=1}^n\varphi_i(t)) F(x,t) \right| + \sum_{i=1}^n \left|\Sigma(g_i)(x,t) - \varphi_i(t)F(x,t)\right| \\
& \leq \kappa + \sum_{i=1}^n \left(\left|\Sigma(g_i)(x,t) - \varphi_i(t)\theta(h_i)(x)\right| + \varphi_i(t) \left|\theta(h_i)(x) - F(x,t)\right|\right) \\
& \leq 3 \kappa
\end{align*}
for all $(x,t) \in \Delta \times L$.
\end{proof}
\begin{lemma}\label{21-04-21} $(G,G^+)$ is a dimension group.
\end{lemma}
\begin{proof}
As observed above it remains only to establish the Riesz interpolation property. Let $u^i,v^j \in G, \ i,j \in \{1,2\}$, satisfy that $u^i \leq v^j$ in $G$ for all $i,j$. We must find $g' \in G$ such that $u^i \leq g' \leq v^j$ for all $i,j$. If $u^{i'} = v^{j'}$ for some $i',j' \in \{1,2\}$, set $g' = v^{j'}$. Then $u^i \leq g' \leq v^j$ for all $i,j$ and we are done. Assume therefore that $\Sigma(u^i) \prec \Sigma(v^j)$ for all $i,j \in \{1,2\}$.
\begin{obs}\label{05-05-21} There is a $g \in G$, an $R > 1$ and an $\epsilon > 0$, such that
$$
t^{-l_i}(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)) > \epsilon
$$
for all $(x,t) \in F \times (L \cap [R,\infty))$, $i=1,2$, where
$$
l_i = \mathbb L( \Sigma(g) - \Sigma(u^i)) \ ,
$$
and such that
$$
t^{-l'_j}(\Sigma(v^j)(x,t) - \Sigma(g)(x,t)) > \epsilon
$$
for all $(x,t) \in F \times (L \cap [R,\infty))$, $j = 1,2$, where
$$
l'_j = \mathbb L( \Sigma(v^j) - \Sigma(g)) \ .
$$
\end{obs}
\begin{proof} Choose $N \in \mathbb N$ such that $u^i_k = v^j_k = 0$ for all $i,j \in \{1,2\}$ when $|k|\geq N$. Define $\overline{u}^i \in \oplus_{k=0}^\infty \theta(H)$ such that
$$
\overline{u}^i_k = \theta(u^i_{N-k}) \ ,
$$
and $\overline{v}^j \in \oplus_{k=0}^\infty \theta(H)$ such that
$$
\overline{v}^j_k = \theta(v^j_{N-k}) \ .
$$
We consider here $\theta(H)$ is a partially ordered group in the order inherited from the strict order of ${\operatorname{Aff}}(F)$; i.e. $\theta(H)^+$ consists of $0$ and the elements $f \in \theta(H)$ such that $f(x) > 0$ for all $x \in F$. It is well-known that $\theta(H)$ then has the Riesz interpolation property, cf. Lemma 3.1 and Lemma 3.2 in \cite{EHS}. The direct sum $\oplus_{k=0}^\infty \theta(H)$, in turn, is given the corresponding lexicographic order $<_{lex}$, where $(h_k)_{k=0}^{\infty} <_{lex} (h'_k)_{k=0}^\infty$ means that $h_k = h'_k$ for all $k$, or $h_{k_0} < h'_{k_0}$ in $\theta(H)$ where $k_0 = \min \{ k : \ h_k \neq h'_k \}$. Since $ \Sigma(u^i) \prec \Sigma(v^j)$ for all $i,j \in \{1,2\}$ we have that
$$
\overline{u}^i <_{lex} \overline{v}^j
$$
for all $i,j \in \{1,2\}$. By Theorem 3.10 in \cite{E2} the group $\oplus_{k=0}^\infty \theta(H)$ has the Riesz interpolation property in the lexicographic order\footnote{Theorem 3.10 in \cite{E2} is very general and the argument concerning Riesz groups uses the decomposition property. However, a major step in the proof of Lemma 3.2 in \cite{BEK} gives the arguments directly for the interpolation property, albeit in the colexicographic order.} and there is therefore an element $h'' \in \oplus_{k=0}^\infty \theta(H)$ such that $\overline{u}^i <_{lex} h'' <_{lex} \overline{v}^j$ for all $i,j \in \{1,2\}$. If $h'' \notin \{\overline{u}^1,\overline{u}^2,\overline{v}^1,\overline{v}^2\}$ we define $g \in G$ such that $g_k = 0$ when $k > N$ and such that $\theta(g_k) = h''_{N-k}$ when $k \leq N$, and $g$ will then have the desired properties because the leading coefficients of $\Sigma(g)-\Sigma(u^i)$ and $\Sigma(v^j) -\Sigma(g), \ i,j =1,2$, will all be strictly positive. If $h'' = \overline{u}^{1}$ it follows that $\overline{u}^{2} <_{lex}\overline{u}^1$ which implies that $\Sigma(u^2) = \Sigma(u^1)$ or that there is an $R' > 1$ and a $\delta > 0$ such that
$$
t^{-l}(\Sigma(u^1)(x,t) - \Sigma(u^2)(x,t)) \geq \delta > 0
$$
for all $(x,t) \in F \times [R',\infty)$ where $l = \mathbb L\left(\Sigma(u^1) - \Sigma(u^2)\right)$. In both these cases we choose, as we can, an element $d \in H$ such that
$$
0 < \theta(d)(x)t^{-N-1} < \frac{1}{2}(\Sigma(v^j)(x,t) - \Sigma(u^1)(x,t))
$$
for all $(x,t) \in (F \times L)\cup (\Delta \times \{1\})$, $j = 1,2$, and define $g \in G$ such that
$$
g_k = \begin{cases} u^1_k , & \ -N \leq k \leq N, \\ d , & \ k = -N -1, \\ 0, & \ k\notin [-N-1,N] \ . \end{cases} \
$$
Then $g$ will have the desired property.
The cases where $h'' \in\{\overline{u}^2,\overline{v}^1,\overline{v}^2\}$ are handled in a similar way.
\end{proof}
We fix now $g$, $R$ and $\epsilon$ as in Observation \ref{05-05-21} and choose $J \in \mathbb N$ such that $g'_n = 0$ when $|n| > J$ and $g' \in \left\{g,v^1,v^2,u^1,u^2\right\}$. And we choose $J' \in \mathbb N$ such that
$$
J' > 2J-d
$$
for all $d \in \{0, l_1,l_2,l'_1,l'_2\}$.
\begin{obs}\label{06-05-21} There is an element $g^{(1)} \in G$ and an $\epsilon_1 > 0$ such that $\mathbb L(g^{(1)}) \leq -J'$ and
$$
t^{-d}\Sigma(u^i)(x,t) + \epsilon_1 < t^{-d}\Sigma(g^{(1)})(x,t) < t^{-d} \Sigma(v^j)(x,t) -\epsilon_1
$$
for all $(x,t) \in (F \times L \cap ]0,R+1]) \cup (\Delta \times \{1\})$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$.
\end{obs}
\begin{proof} Since $\Sigma(u^i)(x,t) < \Sigma(v^j)(x,t)$ for all $(x,t)$ in the compact set $(F \times L \cap ]0,R+1]) \cup (\Delta \times \{1\})$ there is a $\delta > 0$ such that
$$
t^{-d}\Sigma(u^i)(x,t) + \delta < t^{-d} \Sigma(v^j)(x,t) - \delta
$$
for all $(x,t) \in (F \times L \cap ]0,R+1]) \cup (\Delta \times \{1\})$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$. By using the Riesz interpolation property of ${\operatorname{Aff}}(F)$ and ${\operatorname{Aff}}(\Delta)$ as is in the proof of Lemma \ref{18-12-20} we construct a continuous function
$$
a: \Delta \times (L \cap ]0,R+1]) \to \mathbb R
$$
such that $x \mapsto a(x,t)$ is affine for all $t \in L \cap ]0,R+1]$ and
$$
t^{-d}\Sigma(u^i)(x,t) + \delta < t^{-d} a(x,t) < t^{-d} \Sigma(v^j)(x,t) - \delta
$$
for all $(x,t) \in (F \times L \cap ]0,R+1]) \cup (\Delta \times \{1\})$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$. Let $r$ be the largest element in $L \cap ]0,R+1]$ and extend $a$ to $\Delta \times L$ such that
$$
a(x,t) = \begin{cases} (r+1 - t)a(x,r), \ & t \in [r,r+1] \ ,\\ 0, \ & t \geq r+1 \ . \end{cases}
$$
It follows from Lemma \ref{06-05-21a} that there is an element $g^{(1)} \in G$ such that $\mathbb L(g^{(1)}) \leq -J'$ and
$$
\left|t^{-d} \Sigma(g^{(1)})(x,t) - t^{-d}a(x,t)\right| \leq \frac{\delta}{2}
$$
for all $(x,t)\in (F \times L \cap ]0,R+1]) \cup (\Delta \times \{1\})$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$. Set $\epsilon_1 = \frac{\delta}{2}$.
\end{proof}
Let $\psi :]0,\infty[ \to [0,1]$ be the continuous function such that $\psi(t) = 1, t\leq R$, $\psi(t) = 0, \ t \geq R+1$, and $\psi$ is linear on $[R,R+1]$. Choose $\kappa > 0$ such that
$$
4 \kappa < \min \{\epsilon,\epsilon_1\} \
$$
where $\epsilon >0$ comes from Observation \ref{05-05-21} and $\epsilon_1 >0$ from Observation \ref{06-05-21}.
Find a real-valued Laurent polynomial $q$ of degree no more than $-J'$ such that
\begin{equation}\label{06-05-21g}
\left|t^{-d}\psi(t)\Sigma(g^{(1)})(x,t) -t^{-d}q(t)\Sigma(g^{(1)})(x,t)\right| \leq \kappa
\end{equation}
for all $(x,t) \in \Delta \times L$ and all $d\in \{l_1,l_2,l'_1,l'_2\}$. This is possible because $\psi|_L \in C_0(L)$, so that $\psi|_L$ can be approximated by Laurent polynomials of degree as most $-J'$, and
$$
\sup_{(x,t)\in \Delta \times L} \left|t^{-d}\Sigma(g^{(1)})(x,t)\right| < \infty \
$$
since $\mathbb L(g^{(1)}) - d \leq 0$. Since
$$
\sup_{(x,t) \in \Delta \times L} t^{-l_i}\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right) < \infty
$$
and
$$
\sup_{(x,t) \in \Delta \times L} t^{-l'_j}\left(\Sigma(v^j)(x,t) -\Sigma(g)(x,t)\right) < \infty
$$
for all $i,j$, we can also arrange that
\begin{equation}\label{07-05-21b}
\left| (q(t) - \psi(t))t^{-l_i}\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right)\right| < \kappa
\end{equation}
and
\begin{equation}\label{07-05-21c}
\left| (q(t) - \psi(t))t^{-l'_j}\left(\Sigma(v^j)(x,t) -\Sigma(g)(x,t)\right)\right| < \kappa
\end{equation}
for all $(x,t) \in \Delta \times L$ and all $i,j$.
We apply now Lemma \ref{05-05-21a} to get an element $g^{(2)} \in G$ such that $g^{(2)}_n = g_n$ for all $-J \leq n \leq J$, $g^{(2)}_n = 0, \ n\geq J+1$, and
\begin{equation}\label{06-05-21d}
\left|t^{-d}(1-q(t))\Sigma(g)(x,t) - t^{-d}\Sigma(g^{(2)})(x,t)\right| \leq \kappa
\end{equation}
for all $(x,t) \in \Delta \times L$, and all $d \in \{0,l_1,l_2,l'_1,l'_2\}$.
\begin{obs}\label{07-05-21} There is an element $g^{(3)} \in G$ such that $\mathbb L(g^{(3)}) < -J$ and
\begin{equation}\label{06-05-21e}
\left|t^{-d}q(t)\Sigma(g^{(1)})(x,t) - t^{-d}\Sigma(g^{(3)})(x,t)\right| \leq \kappa
\end{equation}
for all $(x,t) \in \Delta \times L$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$.
\end{obs}
\begin{proof} Write
$$
q(t)\Sigma(g^{(1)})(x,t) = \sum_{-M}^{-2J'}a_n(x)t^n
$$
where $M > 2J'$. Let $\kappa' > 0$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ we can find $g^{(3)}\in G$ such that $g^{(3)}_n = 0$ when $n \notin [-M,-2J']$ and
$$
\left|\theta(g^{(3)}_n)(x) - a_n(x)\right| \leq \kappa'
$$
for all $x \in \Delta$ and all $-M \leq n \leq -2J'$. Then
\begin{align*}
&\left|t^{-d}q(t)\Sigma(g^{(1)})(x,t) - t^{-d}\Sigma(g^{(3)})(x,t)\right|\\
& \leq \kappa' (M-2J')\sup\left\{ t^n : \ -M-d\leq n \leq - 2J' -d,\ t \in L \right\}
\end{align*}
for all $(x,t) \in \Delta \times L$ and all $d \in \{l_1,l_2,l'_1,l'_2\}$. Since $-2J'-d < 0$,
$$
\sup\left\{ t^n : \ -M-d\leq n \leq - 2J' -d,\ t \in L \right\}
$$
is finite for all $d$ and we can therefore arrange that \eqref{06-05-21e} holds.
\end{proof}
Set $g^{(4)} = g^{(3)}+ g^{(2)}$. Since $g^{(4)}_n = g_n$ for all $-J \leq n \leq J$ and $g^{(4)}_n= 0, \ n \geq J+1$, we find that
$$
\mathbb L(\Sigma(g^{(4)}) -\Sigma(u^i)) = \mathbb L(\Sigma(g) -\Sigma(u^i)) = l_i
$$
and
$$
\mathbb L(\Sigma(v^j) - \Sigma(g^{(4)})) = \mathbb L(\Sigma(v^j) - \Sigma(g)) = l'_j
$$
for all $i,j$. Furthermore, for all $(x,t) \in \Delta \times L$, we find by using \eqref{06-05-21e}, \eqref{06-05-21d}, \eqref{06-05-21g} and \eqref{07-05-21b}, that
\begin{align*}
& t^{-l_i}( \Sigma(g^{(4)})(x,t) -\Sigma(u^i)(x,t)) \\
& = t^{-l_i}\left( \Sigma(g^{(3)})(x,t) + \Sigma(g^{(2)})(x,t) -\Sigma(u^i)(x,t)\right)\\
& \geq t^{-l_i}(q(t)\Sigma(g^{(1)})(x,t) + (1-q(t))\Sigma(g)(x,t) - \Sigma(u^i)(x,t)) - 2\kappa \\
& = t^{-l_i}q(t)(\Sigma(g^{(1)})(x,t) - \Sigma(u^i)(x,t)) \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + (1-q(t))\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right) - 2\kappa \\
& \geq t^{-l_i}\psi(t)(\Sigma(g^{(1)})(x,t) - \Sigma(u^i)(x,t)) \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + (1-q(t)) t^{-l_i}\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right) - 3\kappa \ \\
& \geq \psi(t)t^{-l_i}(\Sigma(g^{(1)})(x,t) - \Sigma(u^i)(x,t)) \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + (1-\psi(t)) t^{-l_i}\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right) - 4\kappa \
\end{align*}
for all $(x,t) \in \Delta \times L$ and $i =1,2$. Now the properties of $g$ and $g^{(1)}$ stipulated in Observation \ref{05-05-21} and Observation \ref{06-05-21}, and the definition of $\psi$, imply that
\begin{align*}
& \psi(t)t^{-l_i}(\Sigma(g^{(1)})(x,t) - \Sigma(u^i)(x,t)) \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + (1-\psi(t)) t^{-l_i}\left(\Sigma(g)(x,t) - \Sigma(u^i)(x,t)\right) - 4\kappa \\
& \geq \min \{\epsilon,\epsilon_1\} - 4 \kappa \
\end{align*}
for all $(x,t) \in (F \times L) \cup (\Delta \times \{1\})$ and $i =1,2$. Essentially identical estimates show that
$$
t^{-l'_j}\left( \Sigma(v^j)(x,t) - \Sigma(g^{(4)})(x,t)\right) \geq \min \{\epsilon,\epsilon_1\} - 4 \kappa
$$
for all $(x,t) \in (F \times L) \cup (\Delta \times \{1\})$ and $j =1,2$, and since $ \min \{\epsilon,\epsilon_1\} - 4 \kappa > 0$ we conclude that $u^i \leq g^{(4)} \leq v^j$ in $G$ for all $i,j$.
\end{proof}
Recall that an ideal $J$ in $(G,G^+)$ is a subgroup $J$ of $G$ such that $J = J \cap G^+ - J\cap G^+$ and $0 \leq h \leq g \in J \Rightarrow h \in J$. We shall not need to identify all ideals in $(G,G^+)$; it suffices here to establish the following:
\begin{lemma}\label{22-04-21d} Let $\rho$ be the automorphism \eqref{rho} and let $J$ be a non-zero ideal in $(G,G^+)$ such that $\rho(J) =J$. It follows that $J = G$.
\end{lemma}
\begin{proof} Let $0 \neq g \in J \cap G^+$. For $i \in \mathbb Z$ and $h \in H$ define $h^{(i)} \in G$ such that $(h^{(i)})_i = h$ and $(h^{(i)})_j = 0, \ j \neq i $. Let $h \in H^+ \backslash \{0 \}$. Then $h^{(i)} \in G^+$ for all $i \in \mathbb Z$. Set $i' = \mathbb L(\Sigma(g))$. Since
$$
t^{-i'} \Sigma(g)(x,t) \geq \epsilon > 0
$$
on $(F \times {L}) \cup (\Delta \times \{1\})$ for some $\epsilon > 0$, there is an $N \in \mathbb N$ such that
$$
Nt^{-i'} \Sigma(g)(x,t) - \theta(h)(x) \geq 1
$$
for all $(x,t) \in (F \times {L}) \cup (\Delta \times \{1\})$, which implies that
$$
0 \leq h^{(i')} \leq Ng
$$
in $G^+$ and hence that $h^{(i')} \in J$. Since $\rho(J) = J$ and $\rho({h}^{(i)}) = {h}^{(i-1)}$ for all $i$, it follows that ${h}^{(i)} \in J$ for all $i\in \mathbb Z$. The elements $h^{(i)}, h \in H^+, i \in \mathbb Z$, generate $G$ and we conclude therefore that $J = G$.
\end{proof}
\begin{lemma}\label{08-05-21} $(G,G^+)$ has large denominators.
\end{lemma}
\begin{proof} We adopt the notation from the proof of Lemma \ref{22-04-21d}. Let $g \in G^+\backslash \{0\}$ and $n \in \mathbb N$ be given. Set $l = \mathbb L(\Sigma(g))$. There is an $\epsilon > 0$ such that $t^{-l}\Sigma(g)(x,t) \geq \epsilon$ for all $(x,t) \in (F \times L) \cup (\Delta \times \{1\})$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$, there is an element $h \in H$ such that $0<\theta(h)(x) < \frac{\epsilon}{n}$ for all $x \in \Delta$. Then $0 \leq n h^{(l)} \leq g$ in $G$. On the other hand, there is an $m \in \mathbb N$ so big that
$$
t^{-l}\Sigma(g)(x,t) + 1 \leq m \theta(h)(x)
$$
for all $(x,t) \in \Delta \times L$ and then $g \leq mh^{(l)}$.
\end{proof}
\subsubsection{Completing the proof when $K$ is unbounded}
From here the arguments are very similar, and many identical to those from the compact case.
It follows from Lemma \ref{21-04-21} and \cite{EHS} that there is an AF algebra $B$ whose $K_0$-group and dimension range is isomorphic to $(G ,G^+)$ and from \cite{E1} we conclude that $B$ is stable and that there is an automorphism $\gamma$ of $B$ such that $\gamma_* = \rho$ under the identification $K_0(B) = G$. By Lemma \ref{08-05-21} and Lemma \ref{28-10-20} we can choose $\gamma$ such that it has the same additional properties as listed in \ref{listrefx}. Set $C = B \rtimes_\gamma \mathbb Z$. No power of $\gamma$ is inner since $\gamma_*^k \neq \id_G$ for $k\neq 0$ and it follows from Lemma \ref{22-04-21d} that $B$ is $\gamma$-simple, and then from \cite{Ki1} that $C$ is simple.
The traces $\tau_{y,t}$ and $\tau_x$ on $B$ can be defined by the same formulas as in Lemma \ref{18-12-20aaa}, and the following is an exact copy of Lemma \ref{18-12-20b}, but the different assumptions on $K$ and the different orderings of $G$ necessitate some slight changes to the proof.
\begin{lemma}\label{23-04-21} Let $\phi : G \to \mathbb R$ be a positive homomorphism such that $\phi([[u]])= 1$. Assume that $\phi \circ \gamma_* = s^{-1} \phi$ for some $s \in ] 0,\infty[$. Then $s \in L$ and when $s \neq 1$ it follows that $\phi = {\tau_{y,s}}_*$ for some $y \in F$, and when $s =1$ it follows that $\phi = {\tau_x}_*$ for some $x \in \Delta$.
\end{lemma}
\begin{proof}
In the following we denote by $C_0(\Delta \times L)$ and $C_0(L)$ the set of continuous real-valued functions vanishing at infinity on $\Delta \times L$ and $L$, respectively, and we denote by $\mathcal A(\Delta \times L)$ the elements $f$ of $C_0(\Delta \times L)$ with the property that $x \mapsto f(x,t)$ is affine for all $t \in L$; a closed subspace of $C_0(\Delta \times L)$. Set
$$
G' = \left\{ g\in G: \ \mathbb L(g) \leq -1 \right\} = \left\{ g \in G : \ g_n = 0 \ \forall n \geq 0 \right\} \ ;
$$
a subgroup of $G$. Then $\Sigma(G') \subseteq \mathcal A(\Delta \times L)$, and we claim that there is continuous linear map $\phi' : \mathcal A(\Delta \times {L}) \to \mathbb R$ such that $\phi = \phi' \circ \Sigma$ on $G'$. For this and later purposes we establish the following
\begin{obs}\label{07-05-21h} Let $h \in H$ and $r \in \mathbb R$ be such that $\theta(h)(x) < r$ for all $x \in \Delta$. It follows that $\phi([[h]]) < r$.
\end{obs}
To prove this let $k\in \mathbb Z$ and $l \in \mathbb N$ be such that $\theta(h)(x) < \frac{k}{l} \leq r$ for all $x \in \Delta$. Then $l[[h]] \leq k [[u]]$ in $G$ and hence $l\phi([[h]]) \leq k$, from which the conclusion follows.
Assume $g \in G$ and that $\Sigma(g) = 0$. Let $N \in \mathbb N$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ there is a $c \in H$ such that $0 < \theta(c)(x) < N^{-1}$ for all $x \in \Delta$ and Observation \ref{07-05-21h} implies that $0 \leq \theta([[c]]) \leq N^{-1}$. Since $\mathbb L(\Sigma(\pm g + [[c]])) = 0$ and $\Sigma(\pm g + [[c]])(x,t) \geq \epsilon > 0$, where $\epsilon = \min_{y \in \Delta} \theta(h)(y)$, it follows that $\pm g+[[c]]\in G^+$. We find in this way that $-\frac{1}{N} \leq \phi(g) \leq \frac{1}{N}$. Letting $N \to \infty$ we conclude that $\phi(g) =0$, and it follows that there is a homomorphism $\phi' : \Sigma(G') \to \mathbb R$ such that $\phi' \circ \Sigma = \phi$. Let $f \in \Sigma(G')$; say $f = \Sigma(g)$, where $g\in G'$, and let $k,l \in \mathbb N$ be natural numbers such that $|f(x,t)| < \frac{k}{l}$ for all $(x,t) \in \Delta \times {L}$. Since $\mathbb L\left(\Sigma(k[[u]] \pm lg)\right) = 0$ this implies that $k[[u]] \pm l g \in G^+$ and hence
$$
0 \ \leq \ \phi(k[[u]] \pm l g) \ = \ k \pm l \phi'(f) \ .
$$
It follows that $\left|\phi'(f)\right| \leq \frac{k}{l}$, proving that $\phi'$ is Lipshitz continuous on $\Sigma(G')$. Since $\Sigma(G')$ is dense in $\mathcal A(\Delta \times {L})$ by Lemma \ref{06-05-21a} it follows that $\phi'$ extends by continuity to a linear map $\phi' : \mathcal A(\Delta \times {L}) \to \mathbb R$ such that $\phi'\circ \Sigma(g) = \phi(g)$ for $g \in G'$.
Let $T : \mathcal A(\Delta \times {L}) \to \mathcal A(\Delta \times {L})$ be the operator
$$
T(\psi)(x,t) = t^{-1}\psi(x,t) \ .
$$
The equality
\begin{equation}\label{18-20-12dd}
s^{-1}\phi' = \phi' \circ T \
\end{equation}
is established in the same way as in the proof of Lemma \ref{18-12-20b}. When $a \in {\operatorname{Aff}}(\Delta)$ and $f \in C_0({L})$ we denote by $a \otimes f$ the function $\Delta \times {L} \ni (x,s) \mapsto a(x)f(s)$. Assume $a(x)f(t) \geq 0$ for all $(x,t) \in (F \times L) \cup (\Delta \times \{1\})$. We claim that
\begin{equation}\label{07-05-21e}
\phi'(a \otimes f) \geq 0 \ .
\end{equation}
To establish this, let $\epsilon > 0$. We use Lemma \ref{22-04-21x} to get an element $g \in G'$ such
\begin{equation}\label{07-05-21g}
\left|a(x)f(t) - \Sigma(g)(x,t) \right| \leq \frac{\epsilon}{2}
\end{equation}
for all $(x,t) \in \Delta \times L$. Using the density of $\theta(H)$ in ${\operatorname{Aff}}(\Delta)$ we choose $h \in H^+$ such that $\epsilon \leq \Sigma(h)(x) < 2 \epsilon$ for all $x \in \Delta$. Then $\mathbb L(g+[[h]]) = 0$ and
$$
\Sigma(g+ [[h]])(x,t) \geq \frac{\epsilon}{2}
$$
for all $(x,t) \in (F \times L) \cup (\Delta \times \{1\})$, implying that $g+[[h]] \in G^+$ and hence that $\phi(g+[[h]]) = \phi'(\Sigma(g)) + \phi([[h]]) \geq 0$. It follows from Observation \ref{07-05-21h} that $\phi([[h]]) \leq 2\epsilon$, and we infer that $ \phi'(\Sigma(g)) \geq -2\epsilon$. Combined with \eqref{07-05-21g} it follows that $\phi'(a \otimes f) \geq -3\epsilon$, proving \eqref{07-05-21e}. In the same way as in the proof of Lemma \ref{18-12-20b} we deduce from \eqref{18-20-12dd} and \eqref{07-05-21e} that $s \in L$ and that there is a real number $\lambda(a)$ for each $a \in {\operatorname{Aff}} (\Delta)$ such that
\begin{equation}\label{21-12-20dx}
\phi'(a \otimes f) \ = \ \lambda(a)f(s)
\end{equation}
for all $f \in C_0(L)$. The resulting map $a \mapsto \lambda(a)$ is clearly linear and positive, and $\lambda(1) > 0$ since $\phi' \neq 0$. Then $a \mapsto \lambda(1)^{-1}\lambda(a)$ is a state of ${\operatorname{Aff}}(\Delta)$ and there is therefore an $x_0 \in \Delta$ such that $\lambda(a) = \lambda(1)a(x_0)$ for all $a \in {\operatorname{Aff}}(\Delta)$. Since the elements of $\left\{a \otimes f : \ a \in {\operatorname{Aff}}(\Delta) , \ f \in C_{0}({L}) \right\}$ span a dense set in $\mathcal A(\Delta \times {L})$ we find that
\begin{equation}\label{18-12-20ee}
\phi'(h) \ = \ \lambda(1)h(x_0,s) \ \ \ \ \forall h \in \mathcal A(\Delta \times {L}) \ .
\end{equation}
Note that $\gamma_*([[u]]) \in G'$ and $\Sigma(\gamma_*([[u]]))(x,t) = t^{-1}$. Using \eqref{18-12-20ee} and the assumptions on $\phi$ we find that
$$
\lambda(1)s^{-1} = \phi'(\Sigma(\gamma_*([[u]]))) = \phi(\gamma_*([[u]])) = s^{-1}\phi([[u]]) = s^{-1} \ ,
$$
implying that $\lambda(1) = 1$. Thus
$$
\phi (g) = \phi'(\Sigma(g)) = \sum_{m \in \mathbb Z} \theta(g_m)(x_0)s^m
$$
when $g \in G'$. When $g \in G$ is a general element there is an an $n \in \mathbb N \cup \{0\}$ such that $g' = \gamma_*^n(g) \in G'$
and then
$$
\phi(g) = s^n \phi(g') =s^n\sum_{m \in \mathbb Z}\theta(g_{m+n})(x_0)s^{m} = \sum_{m \in \mathbb Z}\theta(g_m)(x_0)s^{m} \ .
$$
It remains only to show that $x_0 \in F$ when $s \neq 1$; which follows from Lemma \ref{21-12-20} in the same way as in the proof of Lemma \ref{18-12-20b}.
\end{proof}
The rest of the proof of Theorem \ref{20-12-20c} for the unbounded case is identical to the proof from Section \ref{compact} for the compact case.
\section{Proof of Corollary \ref{15-12-20d} and Corollary \ref{27-10-20a}}\label{xxx}
We will combine Corollary \ref{20-12-20d} with the methods used in Appendix 12 of \cite{Th3} and Section 6 of \cite{Th4}. Given the UHF algebra $U$ in Corollary \ref{15-12-20d} we write
$$
U \simeq U_1 \otimes U_2 \ ,
$$
where $U_1$ and $U_2$ are both (infinite dimensional) UHF-algebras. Given the collection $\mathbb I$ of intervals and simplexes $S_I, I \in \mathbb I$, in Corollary \ref{15-12-20d} we get from Corollary 5.7 in \cite{Th4} a generalized gauge action $\alpha^1$ on $U_1$ with the following properties:
\begin{itemize}
\item For each $I \in \mathbb I$ and each $\beta \in I\backslash \{0\}$ there is a closed face $F_I$ in $S^{\alpha^1}_{\beta}$ which is strongly affinely isomorphic to $S_I$.
\item For each $\beta \neq 0$ and each $\beta$-KMS state $\omega \in S^{\alpha^1}_{\beta}$ there is a unique norm-convergent decomposition
$$
\omega = \sum_{I \in \mathbb I_{\beta}} \omega_I \
$$
where $\omega_I \in \mathbb R^+F_I$.
\end{itemize}
That $\alpha^1$ is a generalized gauge action means that there is a Bratteli diagram $\operatorname{Br}$ and a map $F : \operatorname{Br}_{Ar} \to \mathbb R$ defined on the set $\operatorname{Br}_{Ar}$ of arrows in $\operatorname{Br}$ which define $\alpha^1$ in the following way. Let $\mathcal P_n$ denote the set of paths of length $n$ in $\operatorname{Br}$ starting at the top vertex. Extend the map $F$ to $\mathcal P_n$ such that
$$
F(\mu) = \sum_{i=1}^n F(a_i) \ ,
$$
when $\mu = a_1a_2\cdots a_n$ is made up of the arrows $a_i$. The Bratteli diagram $\operatorname{Br}$ is chosen such that there are finite dimensional $C^*$-subalgebras $\mathbb F_n$ in $U_1$ spanned by a set of matrix units $E^n_{\mu,\mu'}, \ \mu,\mu' \in \mathcal P_n$, such that $\mathbb F_n \subseteq \mathbb F_{n+1}$ for all $n$ and
$$
U_1 = \overline{\bigcup_n \mathbb F_n} \ . \
$$
The flow $\alpha^1$ is defined such that
$$
\alpha^1_t\left( E^n_{\mu,\mu'}\right) = e^{i (F(\mu)-F(\mu'))t} E^n_{\mu,\mu'} \ .
$$
See \cite{Th4}. In order to ensure that $\alpha^1$ is $2\pi$-periodic it is necessary to arrange that $F$ only takes integer values. That this is possible follows by inspection of the proof in \cite{Th4}; in fact, the only step in the proof where this is not automatic is in the proof of Lemma 5.3 in \cite{Th4}, where some real numbers $t_k$ are chosen. The only crucial property of these numbers is that they must be sufficiently big and they may therefore be chosen to be natural numbers. The resulting potential $F$ will then be integer-valued. Let $\alpha^2$ be a flow on $U_2$ with the properties specified for $\alpha$ in Corollary \ref{20-12-20d} and set
$$
\alpha_t = \alpha^1_t \otimes \alpha^2_t \ ,
$$
which we will argue defines a flow on $U_1 \otimes U_2$ with the properties stated in Corollary \ref{15-12-20d}. Since a $\beta$-KMS state for $\alpha$ will restrict to a $\beta$-KMS state for $\alpha^2$ on the tensor factor $U_2$, it follows from Corollary \ref{20-12-20d} that there are no $\beta$-KMS states for $\alpha$ unless $\beta \in K$. Let $\beta \in K$ and let $\omega_\beta$ be the unique $\beta$-KMS state for $\alpha^2$. It remains only to show that the map $\omega \ \mapsto \ \omega \otimes \omega_\beta$ is an affine homeomorphism from $S^{\alpha^1}_\beta$ onto $S^{\alpha}_\beta$. Note that only the surjectivity is not obvious, and only when $\beta \neq 0$. Consider therefore a $\beta$-KMS state $\psi \in S^{\alpha}_\beta$, $\beta \neq 0$. Let $x,y \in U_1, \ b \in U_2$ such that $x$ is $\alpha^1$-analytic. Then $x \otimes 1$ is $\alpha$-analytic and $\alpha_{i \beta}(x \otimes 1) = \alpha^1_{i\beta} (x) \otimes 1$. Hence
\begin{align*}
&\psi(xy \otimes b) = \psi((x \otimes 1)(y \otimes b)) = \psi((y\otimes b) \alpha_{i \beta}(x\otimes 1)) \\
& = \psi((y\otimes b) (\alpha^1_{i \beta}(x) \otimes 1)) = \psi(y\alpha^1_{i \beta}(x) \otimes b) \ .
\end{align*}
Thus, if $b \geq 0$ and $b \neq 0$, the map $U \ni x \mapsto \psi(x \otimes b)$ is a $\beta$-KMS functional for $\alpha^1$, and hence
\begin{align*}
&\psi(E^n_{\mu, \mu'} \otimes b) = \psi( \alpha^1_{i\beta}(E^n_{\mu, \mu'})\otimes b)) = e^{-\beta (F(\mu) - F(\mu'))} \psi(E^n_{\mu, \mu'} \otimes b) \ .
\end{align*}
Since $\beta \neq 0$ it follows that $\psi(E^n_{\mu,\mu'} \otimes b) \ = \ 0$ when $F(\mu) \neq F( \mu')$, and hence that $\psi$ factorises through the map $Q \otimes \id_{U_2}$, where $Q : U_1 \to U_1^{\alpha^1}$ is the conditional expectation onto the fixed point algebra $U_1^{\alpha^1}$ of $\alpha^1$. Let $a \geq 0$ in $U_1$. Then
$$
U_2 \ni b \ \mapsto \ \psi(a \otimes b) = \psi(Q(a) \otimes b)
$$
is a $\beta$-KMS functional for $\alpha^2$ since $\psi$ is a $\beta$-KMS state for $\alpha$ and $Q(a)$ is fixed by $\alpha^1$. The uniqueness of $\omega_\beta$ implies therefore that
$$
\psi(a \otimes b) = \lambda(a)\omega_{\beta}(b)
$$
for some $\lambda(a) \in \mathbb R$ and all $b \in U_2$. It is then straightforward to show that $\lambda(a)$ can be defined for all $a \in U_1$, resulting in a $\beta$-KMS state $\lambda$ for $\alpha^1$. This completes the proof of Corollary \ref{15-12-20d}.
Corollary \ref{27-10-20a} follows by choosing the family $\mathbb I$ of intervals in Corollary \ref{15-12-20d} in an appropriate way, e.g. as the collection of all bounded intervals with rational endpoint, and also the Choquet simplexes $S_I$ in an appropriate way. See Section 6 in \cite{Th4}.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: raise JSONDecodeError("Expecting value", s, err.value) from None json.decoder.JSONDecodeError: Expecting value: line 1 column 1 (char 0) I want to create a record, and I partially succeeded. But here's the problem, I can't record 2, I get the following error. What am I doing wrong?
Error :
File "C:\Users\bilgi\AppData\Local\Programs\Python\Python39\lib\json\decoder.py", line 337, in decode
obj, end = self.raw_decode(s, idx=_w(s, 0).end())
File "C:\Users\bilgi\AppData\Local\Programs\Python\Python39\lib\json\decoder.py", line 355, in raw_decode
raise JSONDecodeError("Expecting value", s, err.value) from None
json.decoder.JSONDecodeError: Expecting value: line 1 column 1 (char 0)
Example Code :
def loadUsers(self):
# Dosya var?
if os.path.exists('AccInformation.json'): # True ise......
with open('AccInformation.json', 'r', encoding='utf-8') as file:
users = json.load(file)
for user in users:
user = json.load(user)
newUser = Account(user_id = user['user_id'], firstName = user['first_name'], lastName = user['last_name'],
email = user['email'], username = user['username'],
password = user['password'], accountKEY = user['AccountKEY'])
self.users.append(newUser)
print(self.users)
else:
print("""'AccInformation' adlı Dosya bulunamadı.""")
A: So without knowing the contents of AccInformation.json it's difficult to properly answer this question, however I imagine it is a JSON with a list of dict items that represent separate users.
Based on that, the below code should work...
import os
import json
def loadUsers(self):
# Dosya var?
if os.path.exists("AccInformation.json"): # True ise......
with open("AccInformation.json", encoding="utf-8") as infile:
users = json.load(infile)
# return indent here as we've already loaded the file
for user in users:
newUser = Account(
user_id=user["user_id"],
firstName=user["first_name"],
lastName=user["last_name"],
email=user["email"],
username=user["username"],
password=user["password"],
accountKEY=user["AccountKEY"],
)
self.users.append(newUser)
print(self.users)
else:
print("""'AccInformation' adlı Dosya bulunamadı.""")
A: I just solved the problem. That's exactly what the problem was : " user = json.load(user)"
Ben bunu şöyle düzelttim : "user = json.loads(user)".
He's working without problems right now. I hope he doesn't cause me any trouble in the future:)
Thank you all individually for your time.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,125
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Q: RegisterClientScriptBlock is shown when i come back from browser I am using SPGridView, I have one link button outside gridview that validates checkboxes in grid and are selected or not if there is no selected record then it will show us message like this from server side.
Code:
Page.ClientScript.RegisterClientScriptBlock(
this.GetType(),
"JSScript",
"<script language='javascript'>alert('Please Select at least one Document(s)');</script>");
i have some items in grid which by clicking on that i go to some other site collections doc libs.
issue scenario
*
*step 1. fire a validation message
*step 2. Navigate to some other site collections doc libs
*step 3. press back button of browser from some other site collections doc libs
bug: it shows me again validation message from above.
pls help me
A: I would suggest to have a simple js onclick event on your link. Which will do the validation and if something is wrong will display the message.
The problem you are having is because when you use
Page.ClientScript.RegisterClientScriptBlock it registers your js alert message on the global level. So when you go back to the page its still there and gets executed again. You can try using the onclick event or when you navigate away from the page you need to remove your RegisterClientScriptBlock.
i.e.
Client.RegisterClientScriptBlock(GetType(String), "JSScript", "");
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,828
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\section{Introduction}
Ferrarese et al.\ (2006a) and Wehner \& Harris (2006) have recently
shown that the division between either a (black hole)- or a (nuclear
cluster)-dominated galaxy core occurs around a galaxy mass of $\sim$10$^{10}
M_{\odot}$. Wehner \& Harris (2006) wrote that ``dE,N nuclei themselves [the
nuclear star clusters] show no evidence of harbouring massive black holes''.
Indeed, in the contemporaneous investigation by Ferrarese et al.\ (2006a),
they identified only two galaxies (M32 and the Milky Way) as
potentially hosting both types of nuclear component.
While apparently rare, Filipenko \& Ho (2003) had identified at least one
galaxy with both a nuclear cluster (NC) and a massive black hole (BH) and
Graham \& Driver (2007) subsequently reported on the existence of two
additional such galaxies (NGC~3384 and NGC~7457).
To investigate this near dichotomy in the type of central massive object which
galaxies house, Seth et al.\ (2008) searched for evidence of active galactic
nuclei (AGN), and thus massive BHs, in galaxies with known NCs. Gonzalez
Delgado et al.\ (2008) simultaneously undertook a complementary approach and
searched for the presence of NCs in galaxies with known AGN. While they both
detected some galaxies in the mass range $10^9$ to $10^{11} M_{\odot}$ which
contain both a NC and an AGN, they were not able to acquire the
BH masses of the AGN.
To explore not only how commonplace these systems are, but importantly the
nature of the above mentioned transition, we have searched for NCs in galaxies
whose BH mass has already been determined via direct dynamical measurements.
This is important because the nucleus-to-(host spheroid) mass ratio, as a
function of spheroid mass, may provide useful constraints for potential
galactic evolutionary assembly processes. For example, some massive BHs may
grow through the runaway collision of NC stars (e.g., Lightman \& Shapiro
1978; Kochanek et al.\ 1987; Lee 1993), or conversely the BH may evaporate the
surrounding NC (e.g., Ebisuzaki et al.\ 2001; O'Leary et al.\ 2006), or
perhaps some other mechanism dominates.
Curiously, the continuous relations shown by Wehner \& Harris (2006) and
Ferrarese et al.\ (2006a) involving either the BH or NC mass and the
host galaxy mass are suggestive of, at some level, mutually common physics governing
the two types of nuclei.
Moreover, given that the main mechanism of galaxy growth is thought to be
through the process of hierarchical merging, modelling their dual nuclei may
be important for properly understanding the growth of supermassive black holes.
For example, dense nuclear star clusters may, through N-body interactions, greatly
facilitate the coalesence rate of binary massive black holes.
The coexistence of NCs and massive BHs is of further interest due to
associated physical phenomenon. The inward spiral of stars
onto a massive BH is a likely source of UV/X-ray flaring events
(Komossa \& Bade 1999; Komossa \& Merritt 2008; Lodato et al.\ 2008; Rosswog
et al.\ 2008).
The disruption of binary stars may result in the high-speed ejection of
hypervelocity stars (Bromley et al.\ 2006).
Rapid inspiral
events may also generate gravitational radiation (e.g., Quinlan 1996;
Alexander 2008; O'Leary et al.\ 2008; Merritt 2008).
As such, galactic nuclei with confirmed BHs
and NCs may prove useful targets for experiments such as the Laser
Interferometer Space Antenna (LISA, Danzmann et al.\ 1996)
which are hoping to discover such as yet undetected radiation. Given that the amplitude
of gravitational waves decays linearly with distance, and X-ray flaring events
with
distance squared, the relative proximity
of the galaxies listed here makes them particularly attractive compared
to more distant, nucleated AGN.
In the following section we briefly describe our galaxy data set, with more
detailed information contained within the Appendix. In Section 3 we present a
tentative new scaling law involving dense star clusters and massive black
holes, plus a more robust scaling relation which involves the mass of the host
spheroid. Finally a discussion, including some of the implications of
this work, and a brief summary are provided in Sections~4 and 5 respectively.
\section{Data}
Graham (2008a) tabulated a sample of 50 (+26) predominantly inactive galaxies
with useful (rough) measurements of their central BH mass.
This compilation was acquired by scouring the literature for published
values which were then updated if new distances were available. Of these 76 galaxies,
Table~\ref{Tab1} lists those which additionally
contain a NC. Included in this list is our own galaxy the Milky Way
(Burbidge 1970; Rubin 1974),
M32 (Tonry 1984),
the active galaxies NGC~3621 and NGC~4395 (Barth et al.\ 2008;
Filippenko \& Ho 2003), plus NGC~3384 and NGC~7457 which were previously noted
to contain both types of nuclei.
An additional seven predominantly
inactive galaxies (NGC 1023, 1399, 2778, 3115, 4026\footnote{Taken from
G\"ultekin et al.\ (2009).}, 4564. 4697) which house both
type of nuclear component have been identified
--- although no NC mass is currently available for NGC~4564.
In addition, Table~\ref{Tab1} includes another four galaxies with known NC
masses but only upper limits on their BH masses (as is the case for NGC~3621
mentioned above),
three globular clusters with
possible BHs, twelve core galaxies with no NC (included for reference) and one
young star cluster (MGG-11) with a probable intermediate mass black hole.
The massive globular clusters included here are of interest for scenarioes in
which they may be the relic nuclei of stripped galaxies (e.g.\
Freeman 1993; Bassino et al.\ 1994; Meylan et al.\ 2001; Bekki et al.\ 2003;
Walcher et al.\ 2005).
The Appendix contains references to, or derivations of, all
quantities shown in Table~\ref{Tab1}.
Briefly, the black hole masses have been taken from the individual
(usually discovery) papers which reported these values, and, when necessary,
adjusted to our updated distances which are also provided in the Appendix.
While galaxy masses were used for the elliptical galaxies, bulge masses have been
used for the disc galaxies. From here on we shall generically refer to an
elliptical galaxy or the bulge of a disc galaxy as a ``spheroid''. The
spheroid masses were primarily obtained by multiplying the observed spheroid
luminosity by an appropriate stellar mass-to-light ($M/L$) ratio. The NC
masses were also obtained this way, albeit using a different stellar
mass-to-light ratio from the spheroid's value. We found that the uncertainty
involved in this process is generally constrained to within a factor of two.
While comparison with dynamically-determined masses, when available, supports
this level of accuracy, detailed spectroscopy that establishes the mean ages
and metallicities of the stars (e.g.\ Walcher et al.\ 2005) is desirable
for better constraining $M/L$ ratios. Such details, however, were not available for
most systems and therefore we usually adopted the single colour approach used by
Ferrarese et al.\ (2006a) and Seth et al.\ (2008) to determine the nuclear cluster
masses. For five galaxies (M32, NGC~205, NGC~2778, NGC~4697 and the Milky
Way) we have modelled, in the Appendix, their observed light distribution to
derive their NC fluxes.
\begin{table}
\caption{Black hole, host spheroid and nuclear cluster mass.}
\label{Tab1}
\begin{tabular}{@{}llccc@{}}
\hline
Object & Type & $M_{\rm BH} [M_{\odot}]$ & $M_{\rm sph} [M_{\odot}]$ & $M_{\rm NC} [M_{\odot}]$ \\
\hline
\multicolumn{5}{c}{Twelve ``core galaxies'' with $M_{\rm BH}$ but no detectable NC} \\
NGC 3379 & E & $1.4^{+2.7}_{-1.0}\times10^8$ & $1.0\times10^{11}$ & ... \\
NGC 3608 & E & $1.9^{+1.0}_{-0.6}\times10^8$ & $9.4\times10^{10}$ & ... \\
NGC 4261 & E & $5.2^{+1.0}_{-1.1}\times10^8$ & $3.7\times10^{11}$ & ... \\
NGC 4291 & E & $3.1^{+0.8}_{-2.3}\times10^8$ & $7.8\times10^{10}$ & ... \\
NGC 4374 & E & $4.6^{+3.5}_{-1.8}\times10^8$ & $4.1\times10^{11}$ & ... \\
NGC 4473 & E & $1.1^{+0.4}_{-0.8}\times10^8$ & $7.1\times10^{10}$ & ... \\
NGC 4486 & E & $3.4^{+1.0}_{-1.0}\times10^9$ & $3.7\times10^{11}$ & ... \\
NGC 4649 & E & $2.0^{+0.4}_{-0.6}\times10^9$ & $4.5\times10^{11}$ & ... \\
NGC 5077 & E & $7.4^{+4.7}_{-3.0}\times10^8$ & $1.1\times10^{11}$ & ... \\
NGC 5813 & E & $7.0^{+1.1}_{-1.1}\times10^8$ & $1.4\times10^{11}$ & ... \\
NGC 6251 & E & $5.9^{+2.0}_{-2.0}\times10^8$ & $9.4\times10^{11}$ & ... \\
NGC 7052 & E & $3.7^{+2.6}_{-1.5}\times10^8$ & $1.7\times10^{11}$ & ... \\
\multicolumn{5}{c}{Twelve galaxies with $M_{\rm BH}$ and a NC} \\
Milky Way & SBbc & $3.7^{+0.2}_{-0.2}\times10^6$ & $1.2\times10^{10}$ & $3.0\times10^7$ \\
M32 & cE & $2.5^{+0.5}_{-0.5}\times10^6$ & $2.6\times10^{8}$ & $2.0\times10^7$ \\
NGC 1023 & SB0 & $4.4^{+0.5}_{-0.5}\times10^7$ & $3.2\times10^{10}$ & $4.4\times10^6$ \\
NGC 1399$^a$ & E & $4.8^{+0.7}_{-0.7}\times10^8$ & $1.5\times10^{11}$ & $6.4\times10^6$ \\
NGC 2778$^b$ & SB0 & $1.4^{+0.8}_{-0.9}\times10^7$ & $4.3\times10^9$ & $6.7\times10^6$ \\
NGC 3115 & S0 & $9.1^{+9.9}_{-2.8}\times10^8$ & $7.4\times10^{10}$ & $1.5\times10^7$ \\
NGC 3384 & SB0 & $1.6^{+0.1}_{-0.2}\times10^7$ & $1.4\times10^{10}$ & $2.2\times10^7$ \\
NGC 4026 & S0 & $1.8^{+0.6}_{-0.4}\times10^8$ & $9.6\times10^9$ & $5.6\times10^6$ \\
NGC 4395$^c$ & Sm & $3.2^{+6.8}_{-2.2}\times10^4$ & $3.4\times10^7$ & $1.4\times10^6$ \\
NGC 4564 & S0 & $5.6^{+0.3}_{-0.8}\times10^7$ & $7.4\times10^{9}$ & ? \\
NGC 4697 & E & $1.7^{+0.2}_{-0.1}\times10^8$ & $1.5\times10^{11}$ & $2.8\times10^7$ \\
NGC 7457$^d$ & S0 & $3.5^{+1.1}_{-1.4}\times10^6$ & $1.1\times10^{9}$ & $9.3\times10^6$ \\
\multicolumn{5}{c}{Galaxies with a NC but only an upper limit on $M_{\rm BH}$} \\
M33 & Scd & $<3\times10^3$ & $1.5\times10^8$ & $2\times10^6$ \\
NGC 205 & E & $<2.4\times10^4$ & $8.7\times10^8$ & $1.4\times10^6$ \\
NGC 3621 & Sd & $<3.6\times10^4$ & $1.4\times10^8$ & $1.0\times10^7$ \\
NGC 4041$^e$ & Sbc & $<2.4\times10^7$ & $6.4\times10^8$ & $2.9\times10^7$ \\
VCC 1254 & dE & $<9\times10^6$ & $3.2\times10^9$ & $1.1\times10^7$ \\
\multicolumn{5}{c}{Star clusters with less secure $M_{\rm BH}$} \\
G1$^f$ & GC & $1.8^{+0.5}_{-0.5}\times10^4$ & ... & $8.0\times10^6$ \\
M15$^f$ & GC & $0.5^{+2.5}_{-0.5}\times10^3$ & ... & $7.0\times10^5$ \\
MGG-11$^c$ & SC & $1.0^{+4.0}_{-0.8}\times10^3$ & ... & $3.5\times10^5$ \\
$\omega$ Cen$^f$ & GC & $4.0^{+0.8}_{-1.0}\times10^4$ & ... & $4.7\times10^6$ \\
\hline
\end{tabular}
\noindent
References are available in the extended version of this Table, provided in the
Appendix. Uncertainties on the spheroid stellar masses and the nuclear cluster
masses are roughly a factor of two. The three globular clusters and one young
star cluster (MGG-11) have no associated spheroid mass as they are not
located at the centre of a spheroid.
Notes:
$^a$ NC detection weak; $^b$ NC \& BH detection weak;
$^c$ indirect BH mass estimate; $^d$ BH detection weak;
$^e$ disc might be dynamically decoupled;
$^f$ maybe no BH.
\end{table}
Although roughly one dozen galaxies (including NGC~3621 and excluding NGC~4026)
from the sample of 76 galaxies in Graham (2008a) appear to have both a NC and
a BH, it would be inappropriate to conclude that roughly 16 per cent of galaxies
contain both. This is because sample selection effects have not been
considered. For example, high mass galaxies tend not to have nuclear star
clusters; a sample dominated by such galaxies would be biased toward low
percentages. In passing we note that because the central stellar density in
high mass, bright elliptical galaxies decreases as a function of increasing
galaxy luminosity (e.g., Faber et al.\ 1997), nuclear star clusters are
actually easier to detect in luminous galaxies than in intermediate luminosity
elliptical galaxies. At the other end of the scale, the sphere-of-influence
of a $10^6 M_{\odot}$ BH within a lower-mass spheroid having a velocity
dispersion of 100 km s$^{-1}$ is only $\sim$0.01 arcseconds at the distance of
the Virgo galaxy cluster; such BHs would therefore go undetected.
Gallo et al.\ (2008) have however reported that 3-44 per cent of early-type
galaxies less massive than $10^{10} M_{\odot}$ have an X-ray active BH, while
49-87 per cent of more massive early-type galaxies do. This may in part be a
reflection that massive BHs are less prevalent in lower mass galaxies. In any
event, our galaxy identification in Table~\ref{Tab1} confirms that the
coexistence of NCs and BHs is not as rare as previously thought.
Table~\ref{Tab1} effectively doubles the number of galaxies reported
to contain a dense nuclear star cluster and having
a direct supermassive black hole mass measurement.
Shown in Table~\ref{Tab1} is the morphological type of each object. Not
surprisingly, the first dozen galaxies with a BH but no signs of a NC are big
elliptical galaxies. The next dozen objects, those with evidence for both a
BH and a NC, are predominantly disc galaxies; the exceptions are the
elliptical galaxy NGC~4697, the ``compact elliptical'' galaxy M32 (which may
be a disc galaxy undergoing transformation, e.g.\ Bekki et al.\ 2001; Graham
2002) and the elliptical galaxy NGC~1399 with only tentative evidence for a NC
(possibly a swallowed GC, Lyubenova et al.\ 2008, which is one of the propsed
mechanisms for building NCs).
However, given that almost every galaxy with a reliable BH mass measurement
that is less than $5\times10^7 M_{\odot}$ is a disc galaxy, their prevalence is not
surprising.
Finally, lacking kinematical information on the level of rotational
versus pressure support in the bulges of our sample, we are unable to comment
on the role that pseudobulges versus classical bulges may play.
In the following section we attempt to probe the nature of the transition
from one type of nuclei to the other.
\section{Mass ratios}
\subsection{From star clusters to massive black holes}
Figure~\ref{Fig_BHNC} shows the ratio of the BH mass to the combined BH plus
NC mass. It is plotted against the stellar mass of the host spheroid: either an
elliptical galaxy, the bulge of a disc galaxy, or nothing in the case of the
three globular clusters and one young star cluster (see Table~\ref{Tab1}).
For spheroids
with stellar masses below
$\sim$$10^8 M_{\odot}$ there is a dearth of reliable BH detections, although the
majority of low-mass spheroids are known to contain NCs (e.g.,
Binggeli et al.\ 1987; Ferguson 1989;
Carollo et al.\ 1998; Stiavelli et al.\ 2001; Balcells et al.\ 2003; Graham \&
Guzm\'an 2003; C\^ot\'e et al.\ 2006).
From Local Group dwarf galaxies, such as NGC~205, we know that any potential
BHs which these low mass galaxies might host are less massive than their NCs. This is
reflected by the upper limits on five of the data points in Figure~\ref{Fig_BHNC}.
The situation is reversed for spheroid masses greater than
$\sim 10^{11} M_{\odot}$, where the BHs dominate at the expense of the NCs.
Figure~\ref{Fig_BHNC} reveals that
in between is mutual ground where both BHs and NCs appear to coexist within
the same spheroid.
For the first time we are able to gain some preliminary insight into the
nature of this transition as a function of mass, although we recognise that
more data is needed in Figure~\ref{Fig_BHNC} before any possible relation can
be defined with certainty.
The demise of NCs at a host spheroid mass of $\sim 10^{11} M_{\odot}$
(Figure~\ref{Fig_BHNC}, see also Ferrarese et al.\ 2006a and Wehner \& Harris
2006) is interesting. The onset of partially depleted galaxy cores occurs at
an absolute $B$-band magnitude of $-20.5\pm1$ mag (e.g., Faber et al.\ 1997;
Graham \& Guzm\'an 2003), which is also where the dynamical properties vary
(e.g., Davies et al.\ 1983; Dressler \& Sandage 1983; Matkovi\'c \& Guzm\'an
2005). For an old stellar population, this stellar flux corresponds to a
stellar mass of $6^{+9}_{-4}\times10^{10} M_{\odot}$ --- which has recently
been noted by many studies as marking the transition of several galaxy
properties (e.g.\ Rogers et al.\ 2008) and may also coincide with the turnover of the
galaxy mass function (Li \& White 2009). As noted by Ferrarese et al.\
(2006a), it may therefore be that coalescing
BHs in dry merger events (Begelman, Blandford, \& Rees 1980; Merritt, Mikkola
\& Szell 2007; Berentzen et al.\ 2009) preferentially destroy their shroud of
NC stars prior to the creation of the galactic loss cones observed in
spheroids brighter than $-20.5\pm1$ $B$-mag. Alternatively, perhaps the life
span of a NC is simply short once the mass of the BH dominates, hence the
scarcity of NCs around BHs with $M_{\rm bh} > \sim5 \times 10^7 M_{\odot}$.
Figure~\ref{Fig_BHBH} shows the same mass ratio as seen in Figure 1, but
plotted against the BH mass. Plotting it like this reveals, without recourse
to the host spheroid, the nature of the coexistence of black holes and dense
star clusters. The line shown in Figure~\ref{Fig_BHBH} has simply been
marked by eye to roughly capture the behaviour of the points and is such that
\begin{equation}
\log \left[ \frac{M_{\rm BH}}{M_{\rm BH}+M_{\rm NC}} \right] = \frac{2}{3}
\log \left[ \frac{M_{\rm BH}}{5\times 10^7 M_{\odot}} \right]
\label{Eq_Rat}
\end{equation}
for $M_{\rm BH} < 5\times 10^7 M_{\odot}$ and equals zero for larger BH
masses and when $M_{\rm NC}=0$. Given the somewhat sparse nature of the data,
a more sophisticated regression analysis for this new (black hole)-(nuclear
cluster) mass ratio relation is not performed here.
\subsection{Nuclei-to-spheroid mass ratios}
It is generally accepted that massive BHs are associated with the
host spheroid rather than the host galaxy (e.g., Kormendy \& Gebhardt 2001).
Given this, we have
displayed in Figure~\ref{Fig_Ratio} the combined mass of the BH and the NC,
divided by the stellar mass of the host spheroid.
For high spheroid masses, where supermassive BHs dominate
the core region, one can see that this ratio scatters between values from
$10^{-3}$ to $10^{-2}$. One can also see that this mass ratio is greater in
the lower mass spheroids whose cores are dominated by a NC\footnote{If a
population of yet-to-be-detected, low-mass spheroids with $M_{\rm BH} >
M_{\rm NC}$ exists, they would act to increase the distribution of points at the low-mass end
of Figure~\ref{Fig_Ratio} to higher values and thereby steepen the relation
further.}.
From an orthogonal regression analysis, using the code BCES (Akritas \& Bershady 1996),
and assuming a factor of two uncertainty on
each data point in both directions, one obtains the relation\footnote{Excluding
NGC~205 from Figure~\ref{Fig_Ratio} gives a consistent slope and intercept of
$-0.41\pm0.06$ and $-2.13\pm0.07$, respectively.}
\begin{equation}
\log \left[ \frac{M_{\rm BH} + M_{\rm NC}}{M_{\rm sph}} \right] =
- (0.39\pm0.07) \log \left[ \frac{M_{\rm sph}}{10^{10} M_{\odot}} \right] -(2.18\pm0.07).
\label{Eq_Nat}
\end{equation}
Repeating the analysis while assigning a factor of 5 uncertainty to the
ordinate (and a factor of 2 in the abscissa) does not
change this result by more than the quoted 1$\sigma$ uncertainties.
Setting $M_{\rm BH}=0$ for the systems which only have upper limits on their
BH masses also does not significantly alter these results.
The Pearson and Spearman correlation coefficients are -0.73 and -0.65, and the
probability of such a strong correlation occurring by chance is less than 0.02
per cent.
The vertical scatter (i.e.\ in the $\log M_{\rm nuclear}$ direction) is 0.41 dex,
and 0.36 dex without NGC~205.
While equation~\ref{Eq_Rat} expressed the relevant dominance of the BH
compared to the NC as a function of BH mass, equation~\ref{Eq_Nat} reveals
their combined importance (in terms of mass) relative to the host
spheroid's stellar mass.
It also effectively provides a new means to predict the central mass in
systems where one is unable to directly measure this quantity.
Once dry merging commences at $M_B \approx -20.5$ mag (e.g.\ Graham \&
Guzm\'an 2003, and references therein), or roughly $M_K \approx -24$ mag (or
$5\times10^{10}$ to $10^{11} M_{\odot}$), the $M_{\rm BH}/M_{\rm sph}$ mass ratio
should remain constant (or decrease if BHs can be ejected, e.g., Merritt et
al.\ 2004; Gualandris \& Merritt 2008). The one-to-one $M_{\rm
BH}$-$L_K$ relation given by Graham (2007, which is dominated by systems with
$M_{\rm BH} > 5\times10^7 M_{\odot}$), coupled with a near constant stellar
$M/L_K$ ratio for massive elliptical galaxies, supports the scenario in which
the $M_{\rm BH}$/$M_{\rm sph}$ baryon\footnote{This terminology assumes that
the black holes have been built by baryons (e.g.\ Shankar et a.\ 2004).} mass
ratio is a roughly constant value.
Using the $K$-band stellar mass-to-light ratio $\log (M/L_K) = 0.1-0.1(B-K)$, for
$(B-K) > 2.3$
(Forbes et al.\ 2008, their Figure~10), and the colour-magnitude relation
$(B-K) = 0.082 - 0.155M_K$, for $M_K < -18$ mag
(Forbes et al.\ 2008, their Equation~1), one has the expression
$\log (M/L_K) = 0.01(9.18 + 1.55M_K)$.
Applying this to the $K$-band expression $M_{\rm BH} \propto L_{\rm sph,
stellar}^{1.00\pm0.05}$ from Graham (2007, his section~5.2) gives
$M_{\rm BH} \propto M_{\rm sph, stellar}^{1.04\pm0.05}$.
For comparison, Marcomi \& Hunt (2003) report $M_{\rm BH} \propto M_{\rm sph,
virial}^{0.96\pm0.07}$ for their ``Group 1'' galaxies,
while H\"aring \& Rix (2004) report $M_{\rm BH} \propto M_{\rm sph,
dyn}^{1.12\pm0.06}$ for a slightly larger galaxy sample.
It should however be noted that these latter two studies have,
at some level, also accounted for the contribution from dark matter
in the spheroid mass, an isue we discuss in the following Section.
\section{Discussion}
Only a few years ago it was generally believed that massive black holes and
nuclear star clusters did not (frequently) coexist at the centres of galaxies.
Here, as in Seth et al.\ (2008) and Delgado et al.\ (2008), we present
evidence suggesting the contrary for galactic spheroids with stellar masses
ranging from $\sim10^8$--$10^{11} M_{\odot}$. Furthermore, we take an
important step forward by reporting on systems for which we have been able to
acquire the (black hole and stellar) masses of the nuclear components and the
(stellar) mass of the host spheroid. This has enabled us to present mass
relations defining this exciting coexistence.
From Equation~\ref{Eq_Nat}, when $M_{\rm sph} = 10^8 M_{\odot}$ one has a
nucleus-to-spheroid mass ratio of 0.04, and when $M_{\rm sph} = 10^{11}
M_{\odot}$ one has a ratio of 0.0027. Perhaps not surprisingly, the latter
value is in excellent agreement with the mean $M_{\rm BH}/M_{\rm sph}$ ratio
from studies of galaxies at the high-mass end with $M_{\rm BH} \sim 10^{8\pm1}
M_{\odot}$, and which excluded any NC mass component (e.g.\ Merritt \&
Ferrarese 2001; H\"aring \& Rix 2004). At
the low-mass end, from an analysis of nuclear star clusters in dwarf
elliptical galaxies and the bulges of early-type disc galaxies, the $M_{\rm
NC}/M_{\rm sph}$ stellar mass ratio has been observed to be both higher and to
increase (decrease) as one samples lower (higher) mass spheroids. Balcells et
al.\ (2003) find a value of $\sim$2 per cent when $M_{\rm sph} = 10^8
M_{\odot}$ and from a sample of dwarf elliptical galaxies a value of 1 per
cent when $M_{\rm sph} = 10^8 M_{\odot}$ is readily derived from Graham \&
Guzm\'an (2003, their Eq.3 assuming an {\it F606W} filter mass-to-light ratio of 3).
As noted above, in low-mass spheroids it has been known for some years how the
stellar flux ratio of the nucleus and host spheroid vary (see also Lotz et
al.\ 2004, their Figure 7). Grant et al.\ (2005), for example, report that their $B$-band
data for dwarf elliptical galaxies yields $L_{\rm NC} \propto L_{\rm
sph}^{0.68}$, which implies a nine-fold variation in the nuclear-to-spheroid
flux ratio over a host spheroid flux range of 1000. (For comparison, given
that $M_{\rm BH}/M_{\rm NC}\approx 0$ at the low mass end of
equation~\ref{Eq_Nat}, one has the (stellar mass) relation $M_{\rm NC} \propto M_{\rm
sph}^{0.61\pm0.07}$ when the NCs dominate. From Shen et al.'s (2008)
analysis of 900 broad line AGN, they report that $M_{\rm BH} \propto
L_{\rm galaxy}^{0.73\pm0.05}$.)
In Figure~\ref{Fig_Ratio} we have revealed, over a host spheroid stellar mass
range of $10^4$, how the combined central object mass (black hole plus
nuclear star cluster) divided by the stellar mass of the host spheroid varies
with the latter quantity. This ratio increases by more than an order of
magnitude from $\sim$0.1 per cent in giant elliptical galaxies dominated by
massive black holes, to 5--9 percent in dwarf galaxies and the bulges of
late-type disc galaxies whose inner regions are dominated by a nuclear star
cluster (see also Balcells et al.\ 2007).
At first glance, this result may appear to contradict recent claims of a
constant (central massive object)-to-(host galaxy) mass ratio, where the
central massive object in such works was either a nuclear star cluster or a
massive black hole. In the case of Wehner \& Harris (2006), they effectively
took the above flux relation for dwarf elliptical
galaxies from Grant et al.\ and used the expression
($M_{\rm total}/L)_{\rm sph} \propto L_{\rm sph}^{-0.3}$ to obtain $L_{\rm NC}
\propto M_{\rm sph,\, total}$. If the nuclear clusters have similar stellar
$M/L$ ratios, this leads to the result that the nuclear cluster mass is
linearly proportional to the {\it total} (dark matter plus stellar) mass of the host
spheroid; i.e.\ that this mass ratio is constant with varying spheroid mass.
The mass ratios presented by Ferrarese et al.\ (2006a), while also accounting
for dark matter at some level, are slightly different due to their
inclusion/treatment of disc galaxies from a sample of early-type,
Virgo cluster galaxies. Although their application of the virial theorem ($M
\propto R_{\rm e}\sigma^2$) using the velocity dispersion $\sigma$ of the
(pressure supported) bulge component together with the effective half-light radii of the whole galaxy
--- which are effected by the size of the (rotationally supported) disc component --- is a
questionable meausure of a lenticular
galaxy's mass, it is clear that these virial products are larger than the
spheroid masses that would be obtained from the use of $R_{\rm e,\, sph}$
in such a formula.\footnote{A more
subtle issue is that the nuclear cluster fluxes (and thus masses) may have
been underestimated due to the steeper inner S\'ersic profiles obtained from
single higher S\'ersic index fits to each disc galaxy rather than from a
S\'ersic bulge $+$ exponential disc fit.} The average nucleus-to-spheroid
mass ratio would therefore be larger than the reported value of 0.2
per cent for the nucleus-to-galaxy total mass ratio.
While an investigation of whether the use of $R_{\rm e}\sigma^2$ is an
appropriate tracer of total mass is beyond the intended scope of this paper,
it does seem apt to remind readers that this question has a long history,
often discussed in association with the Fundamental Plane (Djorgovski \& Davis
1987; Faber et al.\ 1987; Djorgovksi, de Carvalho \& Han 1988). For example,
even within elliptical galaxies, luminosity-dependent dynamical non-homology
may severely bias the applicability of aperture velocity dispersion
measurements when deriving such quasi-''virial masses'' from $R_{\rm
e}\sigma^2$, and thus also bias any $M_{\rm total}/L$ trends with luminosity
(e.g.\ Hjorth \& Madsen 1995; Ciotti et al.\ 1996; Busarello et al.\ 1997;
Graham \& Colless 1997; Prugniel \& Simein 1997).
Aside from concerns about measuring total spheroid masses, baryonic fuelling
and feedback, albeit within a dark matter halo, are commonly thought to be
responsible for establishing the (bulk of the) nuclei mass and setting the
observed nuclear-to-(host spheroid) mass ratios (e.g.\ Silk \& Rees 1998;
Kauffmann \& Haehnelt 2000; Benson et al.\ 2003; Croton et al.\ 2006; Booth \&
Schaye 2009). It therefore seems reasonable to construct a baryonic rather
than (only) total mass-ratio relation. Moreover, we have been able to do so
for the first time when including the mass of both nuclear components from the
same galaxy.
While inward gas flow may result in galactic-centric star
formation or fuelling much of the growth of massive BHs (e.g.\ Shankar et al.\
2004), it has also been suggested that NCs may grow through the accretion of
globular clusters, and/or super star clusters in spiral galaxies, via
dynamical friction (Tremaine et al.\ 1975; Quinlan \& Shapiro
1990). Dark matter may thus also have a role to play, thereby motivating the
pursuit of reliable total masses.
It has additionally been suggested that some BHs may be built through the
runaway collision of the NC stars (Kochanek et al.\ 1987) or that,
alternatively, a massive BH may effectively evaporate the surrounding NC
(Ebisuzaki et al.\ 2001; O'Leary et al.\ 2006).
It has also been proffered that NCs and massive BHs may have developed from
the same initial formation process (Wehner \& Harris 2006), such that a slower
gas infall rate in smaller spheroids allows time for star formation and thus
produces nuclear star clusters rather than massive black holes. Our larger
(stellar) mass ratios in smaller mass spheroids may have implications for the
required
efficiency of feedback mechanisms in which supernova and stellar winds from NC
stars regulate the nuclear-to-spheroid mass ratio (McLaughlin et al.\
2006b). Moreover, it is hoped that the nuclear-to-spheroid (baryonic) mass
ratios provided here may provide useful constraints for any potential
evolutionary scenarios.
In future work we intend to present a new diagram showing $(M_{\rm BH} +
M_{\rm NC})$ versus velocity dispersion, $\sigma$. This will be achieved via
a careful analysis of high-resolution {\it HST} images for as many of the 50
(+26) galaxies as possible. While barred galaxies can deviate from the
$M$-$\sigma$ relation defined by non-barred galaxies (e.g.\ Graham 2008b;
Graham \& Li 2009), and one expects them to similarly deviate in the new
$(M+M)$-$\sigma$ diagram due to their elevated values of $sigma$, it may prove
insightful to investigate this further. If barred galaxies have
preferentially larger $M_{\rm NC}/M_{\rm BH}$ mass ratios than non-barred
galaxies of the same velocity dispersion, then one may find less scatter in
the new diagram and further potential clues to their evolution.
Within galaxy clusters, dwarf galaxies are the most common type of galaxy
(e.g.\ Binggeli et al.\ 1985) and many of these are nucleated. Within the
field environment, the most common type of galaxies are spiral galaxies (e.g.\
Allen et al.\ 2006; Baldry et al.\ 2006) and many of these are also known to
be nucleated (e.g.\ Carollo et al.\ 1998; B\"oker et al.\ 2002; Balcells et
al.\ 2003). Due to the difficulties associated with the detection of low mass
black holes ($<10^6 M_{\odot}$) in external galaxies, we speculate that
androgynous nuclei might be far more common than currently recognised.
Furthermore, given that nuclear star clusters are among the highest stellar
density objects in the Universe, such a commonplace coexistence of nuclear
star clusters and massive black holes may open up the prospect for numerous
detections of low frequency gravitational radiation (e.g.\ Ju et al.\ 2000,
and references therein) with the Laser Interferometer Space Antenna (LISA,
Danzmann et al.\ 1996) from rapidly inspiralling stars, white dwarfs, neutron
stars and stellar mass black holes about these massive black holes.
Due to the substantially higher density of stars in NCs, compared to the
underlying host galaxy (see Figure~\ref{Fig_Milky}), past estimates of
LISA-detectable gravitational radiation events (e.g.\ Sigurdsson 1997;
Freitag 2001;
Gair et
al.\ 2004; Hopman \& Alexander 2005, 2006a,b) may need to be revised upwards.
The sense of the correction is of course welcome given the significance a
direct detection could have by not only supporting Einstein's concept of space
and time but opening an entire new window through which to view, or rather
listen to, our Universe.
\section{Summary}
We have identified roughly a dozen galaxies with a direct BH mass measurement
{\it and} a nuclear star cluster, doubling the previous sample size for which
these measurements are available. We speculate that the existence of such
hermaphrodite nuclei may be a rather common event for spheroids with
stellar-masses ranging from $10^8$ to $10^{11} M_{\odot}$ (see also Gonzalez
Delgado et al.\ 2008 and Seth et al.\ 2008).
We have shown that the mass of the nuclear component(s) increases from $\sim$0.1
per cent of the host spheroid's stellar mass in large elliptical galaxies
whose cores are dominated by a massive black hole,
to several per cent in low stellar mass ($\sim 10^8 M_{\odot}$) spheroids
whose cores are dominated by a nuclear star cluster (see
Figures~\ref{Fig_BHNC} and \ref{Fig_Ratio}, and also Balcells et al.\ 2007).
We have derived a linear relation between the nuclear mass (BH and NC
combined) and the stellar mass of the host spheroid. Given in
equation~\ref{Eq_Nat},
the relation can be expressed as $(M_{\rm BH} + M_{\rm NC}) \propto M_{*,
sph}^{0.61\pm0.07}$ (see Figure~\ref{Fig_Ratio}).
We hereby suggest that this baryonic, nuclear-to-spheroid mass ratio relation is
applicable to spheroids with eith either nuclear clusters or
dual nuclei type,
noting that the exponent may equal a value of 1 once dry merging commences
(assuming no loss of nuclear components).
We have also identified a new (black hole)-(nuclear cluster) mass ratio
relation pertaining to the coexistence of these entities. We provide a
preliminary quatification of this relation in equation~\ref{Eq_Rat}, which is
such that $(M_{\rm BH} + M_{\rm NC}) = (5\times 10^7 M_{\odot})^{2/3}\, M_{\rm
BH}^{1/3}$ for $M_{\rm BH} < 5\times 10^7 M_{\odot}$ (see
Figure~\ref{Fig_BHBH}).
\begin{figure*}
\includegraphics[angle=270,scale=0.93]{f1.eps}
\caption{
The increasing dominance of the central black hole over the nuclear cluster of
stars, traced by the mass ratio $M_{\rm BH} / (M_{\rm BH}+M_{\rm NC})$,
is shown to depend on the host spheroid mass $M_{\rm sph}$ --- which is zero for GCs.
The highest mass spheroids do not contain a NC, but are included for
illustrative purposes (see Table~\ref{Tab1}).
The five circled points have only an upper limit to their BH mass.
The arrows on the right hand side of the figure denote the up and down
movement if either of the two nuclear masses are in error by a factor of $\pm2$.
}
\label{Fig_BHNC}
\end{figure*}
\begin{figure}
\includegraphics[angle=270,scale=0.43]{f2.eps}
\caption{
The mass ratio from Figure~\ref{Fig_BHNC} is shown here
versus the mass of the black hole. Symbols are as in Figure 1.
An expression for the solid line is given in Equation~\ref{Eq_Rat}.
Galaxies with only an upper BH mass limit have been circled.
An error in $M_{\rm BH}$ will move the data points nearly parallel to this relation,
insuring that it is preserved in the presence of $M_{\rm BH}$ measurement errors.
The lines emanating from each data point show how much each point would
move if the black hole mass changed by a factor of $\pm2$ (upper points) or
$\pm5$ (lower seven points).
Reflecting the increasing uncertainty on the smaller BH mass measurements,
the shaded area has a horizontal width of log(2.0) at the top and
log(5.0) at the mass ratio $5\times 10^{-3}$.
}
\label{Fig_BHBH}
\end{figure}
\begin{figure}
\includegraphics[angle=270,scale=0.44]{f3.eps}
\caption{
The importance, in terms of mass, of the nuclear components relative to the
host spheroid, as traced by the mass ratio $(M_{\rm BH}+M_{\rm NC})/M_{\rm
spheroid}$, is shown as a function of $M_{\rm spheroid}$.
Systems with only an upper BH mass limit have a connected circle which shows their
location if $M_{\rm BH} = 0$.
The fitted line is given by equation~\ref{Eq_Nat}.
The bi-directional arrow in the top right reveals how the points would
move if the value of $M_{\rm spheroid}$ changes by a factor of $\pm$2.
No arrow is provided for changes in the nuclear component masses because the
result/movement is obvious, with a simple shift in the ordinate.
}
\label{Fig_Ratio}
\end{figure}
\section{acknowledgment}
We thank Kenji Bekki for motivating us in 2007 to undertake this project.
A.G.\ thanks the organisers of the February 2008 ``Nuclear Star clusters
across the Hubble Sequence'' workshop held at the Max-Planck-Institut f\"ur
Astronomie in Heidelberg, Germany, and the organisers of the July 2008 Lorentz
Centre Workshop ``Central Mass Concentrations in Galaxies'' held in Leiden,
The Netherlands, where preliminary versions of Figure~\ref{Fig_BHNC} were
presented.
|
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}
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Q: How can I clear and recreate the flyway-play generated database during testing I am using Play 2.4 with the Flyway-Play module.
Is there are way to clean and recreate the database between tests using this plugin? Some of my unit test have database effects which are not trivial to reverse, and it would be nice to be able to start afresh after these tests.
The documentation for the flyway-play module states:
In Test mode, migration is done automatically.
However, this seems be only once before running all tests. It would be nice to have programmatic control when preparing and cleaning up tests.
A: Not sure how flyway is integrated in the play framework (no experience with that), but it looks like what you are looking for is the flyway cleancommand.
Drops all objects in the configured schemas.
Flyway documentation: clean
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"redpajama_set_name": "RedPajamaStackExchange"
}
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|
Description
===========
Installs niet from source. Niet is a lightweight process monitor. Run processes underneath it, and if they die, they'll be respawned.
No other monitoring of sub processes is done. If you need logging, you'll have to configure syslog too
For more information see: https://github.com/willbryant/niet
Another fork of niet which may be worth using if you plan on running [Resque](https://github.com/resque/resque) under niet is https://github.com/terrcin/niet
Requirements
============
Attributes
==========
* niet[:install_source] = "https://github.com/willbryant/niet/archive"
* niet[:tag] = "master"
* niet[:artifact_type] = "tar.gz"
* niet[:safe_install] = true (will not override an existing install, even if it's a different version)
* niet[:processes] = {}
Usage
=====
Niet
----
Just include the default recipe, and the niet binary will be compiled and installed to the system available to use
`niet -c /cd/to/here /run/this/binary`
Startup scripts
---------------
Warning - these /etc/init.d style scripts have only been tested on ubuntu
Usage: set attributes for the user and path to executable file to run, e.g.
chef
node.set[:niet][:processes] = {
"myuser" => "/path/to/myscript.sh"
}
executable file "/path/to/myscript.sh"
#! /bin/bash
niet -c /process/start_dir /important/binary/to/run.sh
The niet service wrapper will now launch this executable file as the user "myuser". The restart/start/stop actions will gracefully target the niet processes with the correct control signals
TODO
====
rewrite startup scripts to configure via `/etc/niet.d/` configuration directory
License
=======
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,888
|
Fully adjustable bath board with handle as standard.
The Sarum bath board has a clever width adjustment mechanism that allows it to fit almost any width of bath. The bath board comes with a removable handle as standard for additional grip.
The board has a strong aluminium frame underneath giving it additional strength. There are 2 securing brackets which can be moved to fit within the internal walls of the bath itself.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,264
|
Q: Knockout.js: Uncaught TypeError: Object # has no method 'applybindings' Just learning Knockout and I have a curious issue. I have this working, then performed some code cleanup and it stopped working and I get the javascript error:
Uncaught TypeError: Object # has no method 'applybindings'
At the top of the index page, I have these scripts loading:
<script src="@Url.Content("~/Scripts/jquery-2.0.3.js")" type="text/javascript"></script>
<script src="@Url.Content("~/Scripts/jquery-ui-1.10.3.js")" type="text/javascript"></script>
<script src="@Url.Content("~/Scripts/knockout-2.3.0.js")" type="text/javascript"></script>
<script src="@Url.Content("~/Scripts/knockout-2.3.0.debug.js")" type="text/javascript"></script>
<script src="@Url.Content("~/ClientApp/Patient.js")" type="text/javascript"></script>
... And a simple concept patient.js script:
var patient = function (id) {
var self = this
self.id = ko.observable(id);
self.name = ko.computed(function () {
return this.id() + "_Joe";
}, this);
};
var patientViewModel = {
patient: ko.observable(new patient(1))
};
$(document).ready(function () {
ko.applybindings(patientViewModel);
});
I am new to javascript in general, so I am sure a few things will jump out immediately to the advanced crowd.
Thanks for your help!
A: You have error in applyBindings function name, it must be:
applyBindings
Also, you twice declare knockout.js file (knockout-2.3.0.js and knockout-2.3.0.debug.js), choose debug or minified version.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,193
|
\section{Introduction}
``Massive'' multi-input multi-output (MIMO) is a recent technology that can significantly improve the spectral/energy efficiency of future wireless networks~\cite{Marzetta14MassiveMIMO,Marzetta15MasMIMO,Marzetta16TenMyth}, and hence can help to address the exponentially growing traffic demand due to proliferation of smart devices. Interest in time-division-duplexing (TDD) massive MIMO systems has recently surged \cite{Marzetta10NonCoo,Marzetta15MasMIMO, Fernandes13Marzetta_InterCellInt, Gesbert13CooApp, Babis12Huh, Nadisanka16DirTra, Jorge16Nadisanka, Ashikhmin12Marzetta_Pilot_Cont_precoding}, due, in part, to their inherent scalability with the number of base station (BS) antennas where a single UL pilot trains the whole BS array. In particular, in TDD massive MIMO systems, the channel state information at the transmitter (CSIT) can be obtained by leveraging the channel reciprocity \cite{Debbah13MasMIMO}.
However, when the user density gets larger in a TDD massive MIMO network (e.g., as in urban areas), the scarcity of pilot resources necessitates the pilot resource reuse by the user equipments (UEs) in different cells. This results in pilot contamination, which impairs the orthogonality of the downlink transmissions from different BSs in TDD networks, diminishing the achievable aggregate capacity.
Adverse effects of pilot contamination in TDD-MIMO networks have been studied extensively in the recent literature, e.g., see the survey~\cite{PilotConSurvey} and the references therein. In particular, the pioneering papers~\cite{Marzetta10NonCoo,Marzetta11PilCon} define and investigate the pilot contamination problem over an \emph{uncorrelated} MIMO channel. In~\cite{Debbah13MasMIMO}, analytical rate expressions are derived for \emph{correlated} MIMO channels under pilot contamination. However, this analysis is done only for the \emph{asymptotic regime} considering very large antenna array sizes. In~\cite{Gesbert13CooApp}, pilot contamination is considered over a correlated MIMO channel, for finite and large antenna array regimes. Considering an asymptotic analysis, the adverse impacts of the pilot contamination are discussed to be completely eliminated with \emph{large antenna arrays}. This is achieved when the UL beams have \textit{non-overlapping} angular support, which can only be satisfied with small AS values. In a follow-up work~\cite{Gesbert16RobPilDec}, the power-domain separation of the desired and the interfering user channels is considered to overcome pilot contamination issue (also studied in~\cite{Muller14BliDecon}).
The pilot contamination effect has a strong connection with the AS of the propagation environment. Interestingly, the existing literature lacks a rigorous analysis for the explicit effect of the AS on the user rates under pilot contamination. In particular, focusing on \emph{correlated MIMO channels} and \emph{moderately large antenna array sizes} with $10\,{-}\,100$ antenna elements are important since these are common scenarios in present real world deployments. For instance, the $3$rd generation partnership project (3GPP) group is currently focusing on millimeter-wave (mmWave) transmissions~\cite{3GPP16ChanMod} which consists of correlated MIMO channels due to limited AS (which is frequency dependent). In addition, mmWave transmission is also receiving high attention for vehicular communication mainly due to the possibilities of generating highly directional beams \cite{Directional_Beam_mmWave} (without much interference), and providing high bandwidth for connected vehicles \cite{HighBW_mmWave}. The new radio (NR) techniques for the $5$th generation ($5$G) wireless communication are considering moderate array sizes even at mmWave frequencies \cite{3GPP16NR}, i.e., at $30$~GHz, 128 antenna elements (single polarized) in uniform planar array (UPA). For long term evolution (LTE) systems operating at sub-$6$~GHz frequencies \cite{3GPP3D16ChanMod} the number of antennas considered is even smaller, i.e., maximum 32 antenna elements (single polarized) in UPA \cite{3GPPTR36_897}. Further, for drone based communication networks \cite{Nadisanka16GC,NadisankaTCoM_arXiv} and moving networks (MNs) \cite{MN_Sui2015}, having a large antenna array is not practically feasible due to the availability of limited form factor. As a result, it becomes crucial to operate with moderate size antenna arrays for such vehicular communication networks.
In this study, we investigate the impact of the AS on the achievable user rates for a TDD based transmission over correlated MIMO channels. In particular, the effect of pilot contamination on achievable rates is analyzed with a special focus on the moderate antenna array size regime. The specific contributions of this work, which is a rigorous extension of~\cite{Jorge17PilConAS}, can be summarized as follows:
\begin{itemize}
\item[i.] An exact analytical expression for the achievable rate is derived considering eigen-beamforming (EBF) precoding explicitly taking in to account the impact of AS. In contrast to the earlier work in the literature~\cite{Debbah13MasMIMO,Gesbert13CooApp}, this analysis is valid for \textit{any} antenna array size,
which is verified to match perfectly with the simulation data under various settings.
\item[ii.]
We show analytically that although large AS leads to stronger pilot contamination for the EBF precoding by impairing the interference channel orthogonality, this does not necessarily degrade the ergodic rates when the array size is \textit{moderate}. Interestingly, fluctuation of the channel power around its long-term mean reduces (similar to the so-called channel hardening effect~\cite{Hochwald04ChanHard, Tarokh09ChanHard, Larsson17ChanHard}) with the increasing AS, which in turn improves achievable rates for the EBF precoding.
\item[iii.] We show that the achievable rates of the EBF and the regularized zero-forcing (RZF) precoders exhibit a non-monotonic behavior with respect to the AS for \emph{moderate} antenna array sizes. The AS that results in a minimum/maximum rate depends on the relative positions of UEs and their serving BSs. Hence, the potential for developing efficient user-cell pairing algorithms based on the derived rate expression is also discussed with the purpose of maximizing the network throughput.
\end{itemize}
\noindent
Table~\ref{tab:LitReview} places the specific contribution of our work in the context of the existing literature. Note that while \cite{Gesbert13CooApp,Gesbert16RobPilDec} present simulation results with specific AS values for correlated MIMO and moderate number of antennas, analytical characterization of the achievable rates explicitly as a function of the AS is not carried out.
\begin{table}
\centering
\caption{Comparison of our work with the existing literature.}
\begin{tabular}{ | c | c | c | c | }
\hline
\multirow{2}{*}{\textbf{Reference}} & \textbf{Number of} & \textbf{Channel} & {\textbf{Investigation of}} \\
& \textbf{antennas} & \textbf{type} & \textbf{Angular spread} \\ \hline
\cite{Marzetta10NonCoo} & Asymptotic & Uncorrelated & No \\ \hline
\cite{Gesbert13CooApp} & Moderate & Correlated & No \\ \hline
\cite{Debbah13MasMIMO} & Asymptotic & Correlated & No \\ \hline
\cite{Marzetta11PilCon} & Moderate & Uncorrelated & No \\ \hline
\cite{Gesbert16RobPilDec} & Moderate & Correlated & No \\ \hline
Our work & Moderate & Correlated & Yes \\ \hline
\end{tabular}
\label{tab:LitReview}
\end{table}
The rest of the paper is organized as follows: Section~\ref{sec:system_model} introduces the system model for a multi-cell, TDD-based correlated MIMO network along with UL channel training under pilot contamination. An exact analytical expression to calculate achievable DL rates with the EBF precoding is derived for a given AS and an arbitrary array size in Section~\ref{sec:ul_dl_trans}. The individual power terms constituting the achievable rate expression are further investigated for the EBF precoding in Section~\ref{sec:individual_powers}, in order to develop insights on the explicit behavior of the ergodic rate as a function of the AS. Extensive numerical results are provided in Section~\ref{sec:numerical_results}, and finally, Section~\ref{sec:conclusion} provides some concluding remarks.
\textit{Notations:} Bold and uppercase letters represent matrices whereas bold and lowercase letters represent vectors. $\textbf{A}(m,n)$ denotes the $m$th row and $n$th column element of matrix $\textbf{A}$. $\|\cdot\|$, $|\cdot|$, $\left(\cdot\right)^{\rm T}$, $\left(\cdot\right)^{\rm H}$, $\left(\cdot\right)^{\rm \ast}$, ${\rm tr}\left(\cdot\right)$, $\otimes$, ${\rm Var\{\cdot\}}$ and $\mathbb{E}\{\cdot\}$ represent the Euclidean norm, absolute-value norm, transpose, Hermitian transpose, complex conjugation, trace of a matrix, Kronecker product, statistical variance and expectation operators, respectively. $\mathcal{CN}(\textbf{m},\textbf{C})$ denotes the complex-valued multivariate Gaussian distribution with the mean vector $\textbf{m}$ and the covariance matrix $\textbf{C}$, and ${\mathcal{U}[a,b]}$ denotes the continuous Uniform distribution over the interval ${[a,b]}$. $\textbf{I}_M$ and $\textbf{0}_M$ are the $M{\times}M$ identity matrix and zero matrix respectively, and $\delta (a,b)$ is the Kronecker delta function taking $1$ if $a\,{=}\,b$, and $0$ otherwise. $\xrightarrow{\,\text{P}\,}$ denotes the convergence in probability.
\section{System Model} \label{sec:system_model}
We consider a multi-cell scenario with $N_{\rm L}$ cells where each cell includes a single BS equipped with a uniform linear antenna array (ULA) of size $M$. In each cell, a total of $K$ UEs each with a single antenna are being served by their respective BSs under perfect time-synchronization. Since we are dealing with pilot contamination with varying AS, we assume that there is one UE in each cell that employs the same pilot sequence with other UEs in other cells during the UL channel estimation. By this way, all the users in this multi-cell layout are contributing to the pilot contamination, and the scenario where multiple UEs employ non-orthogonal pilot sequences in each cell remains a straightforward extension. Note that the perfect time-synchronization assumption is arguably the worst condition in terms of the pilot contamination as any synchronization approach will make the pilot sequences more \textit{orthogonal}, and hence reduce the pilot contamination ~\cite{Gesbert13CooApp}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{./geometry_multicell}
\end{center}
\caption{The multi-cell network consisting of hexagonal cells with the side length $r_2$. All the UEs are dropped at a distance of $r_1$ from their serving BSs. The $i$th and $j$th UEs are located at $\theta$ and $0^{\circ}$ with respect to the horizontal axis measured from $i$th and $j$th BSs, respectively. $\Delta \theta_i$ denotes the angle between the directions from $i$th BS to the $i$th and $j$th UEs.}
\label{fig:layout}
\end{figure}
In our analysis, we assume a TDD protocol consisting of subsequent UL training and DL transmission phases, where the interaction between two adjacent cells are sketched for the DL transmission in Fig.~\ref{fig:layout}. In the UL training phase, all UEs transmitting the same pilot sequence is received by all the BSs in the network. Based on the received pilot sequence, each BS first estimates the channel to its desired UE and then computes the precoding vector based on this estimated channel. During the DL transmission phase, each BS transmits data to its desired UE employing the DL precoding. Note that, due to the pilot contamination, the precoding vector is not aligning well with its desired user channel. Hence, as shown in Fig.~\ref{fig:layout} DL propagation direction is not the same as the desired user channel direction.
The UL channel between the $i$th UE and the $j$th BS $\textbf{h}_{ij}$ is
\begin{equation}
\label{eqn:channel_defn}
\begin{aligned}
\textbf{h}_{ij} = \frac{1}{\sqrt{N_\mathrm{P}}} \sum\limits_{p=1}^{N_\mathrm{P}} \alpha_{ij,p} \, \textbf{a}\left( \phi_{ij, p}\right),
\end{aligned}
\end{equation}
where $N_\mathrm{P}$ is the number multi-path components (MPCs), $\alpha_{ij,p}$ is the complex path attenuation, $\phi_{ij, p}$ is the angle of arrival (AoA) of the $p$-th MPC, and $\textbf{a}\left( \phi_{ij, p}\right)$ is the steering vector given as
\begin{equation} \small
\label{eqn:steering_vector}
\begin{aligned}
\textbf{a}\left( \phi_{ij, p}\right) = \left[ 1 \;\; e^{-j2\pi \frac{D}{\lambda}\cos\left( \phi_{ij,p}\right) } \; \dots \; e^{-j2\pi \frac{D}{\lambda}\left( M-1\right)\cos\left( \phi_{ij, p}\right) } \right]^{\rm T} \! ,
\end{aligned} \normalsize
\end{equation}
where $D$ is the element spacing in the ULA, and $\lambda$ is the wavelength.
The complex path attenuation $\alpha_{ij, p}$ and the AoA $\phi_{ij, p}$ are assumed to be uncorrelated over any of their indices, and with each other. In particular, $\alpha_{ij, p}$ is circularly symmetric complex Gaussian with $\alpha_{ij, p}\,{\sim}\,\mathcal{CN}\left(0,\beta_{ij}\right)$, and the variance $\beta_{ij}\,{=}\,\zeta/d_{ij}^{\gamma}$ captures the effect of the large-scale path loss, where $d_{ij}$ is the distance between $i$th UE and $j$th BS, $\gamma$ is the path loss exponent, and $\zeta$ is the normalization parameter to achieve a given signal-to-noise ratio (SNR) at the BS \cite{Gesbert13CooApp}. We consider uniform distribution for the AoA with $\phi_{ij, p}\,{\sim}\,\mathcal{U}\left[\bar{\phi}_{ij}-\Delta,\bar{\phi}_{ij}+\Delta\right]$, where $\bar{\phi}_{ij}$ is the line-of-sight (LoS) angle between $i$th UE and the $j$th BS, and $\Delta$ is the AS. With the channel model in \eqref{eqn:channel_defn}, next, we study how to achieve UL training and channel estimation.
\subsection{UL Training and Channel Estimation with Correlated MIMO Channels}\label{sec:ul_trans}
In the UL training phase, the UEs transmit the common pilot sequence of size $\tau$ denoted by $\textbf{s}\,{=}\, [ s_1 \, s_2 \, \dots s_\tau ]^{\rm T}$, where each pilot symbol is chosen in an independent and identically distributed (iid) fashion from a discrete alphabet $\mathcal{A}_{\,\rm UL}$ consisting of unity norm entries. The $M{\times}\tau$ matrix of the received symbols at the $j$th BS is given as \begin{align}\label{eqn:UL_trans_matrix}
\textbf{Y}_{j}^{\,\textrm{UL}} = \sum \limits_{i=1}^{N_\mathrm{L}} \textbf{h}_{ij} \textbf{s}^{\textrm{T}}+\textbf{N}_{j}~,
\end{align} where $\textbf{N}_{j}$ is a $M{\times}\tau$ noise matrix consisting of circularly symmetric complex Gaussian entries with $\mathcal{CN}\left(0,\sigma^2\right)$. In an equivalent vector representation, \eqref{eqn:UL_trans_matrix} is given as
\begin{equation} \label{eqn:UL_trans_vector}
\textbf{y}_{j}^{\,\textrm{UL}}=\textbf{S} \sum \limits_{i=1}^{N_\mathrm{L}} \textbf{h}_{ij} +\textbf{n}_{j}~,
\end{equation}
where $M\tau{\times}1$ vectors $\textbf{y}_{j}^{\,\textrm{UL}}$ and $\textbf{n}_{j}$ are obtained by stacking all columns of $\textbf{Y}_{j}^{\,\textrm{UL}}$ and $\textbf{N}_{j}$, respectively, and $\textbf{S}\,{=}\,\textbf{s} \otimes \textbf{I}_M$ is the training matrix of size $M\tau{\times}M$ satisfying $\textbf{S}^H\textbf{S}\,{=}\,\tau \textbf{I}_M$. Following the convention of~\cite{Marzetta10NonCoo}, the SNR is defined for this particular phase to be $1{/}\sigma^2$.
At the $j$th BS, the channel to the $i$th UE can be estimated using linear minimum mean square error (LMMSE) criterion as follows~\cite{Gesbert13CooApp}
\begin{equation} \label{eqn:mmse_chan_est}
\hat{\textbf{h}}_{ij} = \tilde{\textbf{R}}_{ij}\textbf{S}^{\rm H}\textbf{y}_{j}^{\,\textrm{UL}}~,
\end{equation}
where $\tilde{\textbf{R}}_{ij}$ is the pilot-independent estimation filter given as
\begin{equation} \label{eqn:mmse_filter}
\tilde{\textbf{R}}_{ij} = \textbf{R}_{ij} \left(\sigma^2 \textbf{I}_{M}+\tau \sum \limits_{\ell=1}^{N_\mathrm{L}}\textbf{R}_{\ell j} \right)^{{-}1} \!\! .
\end{equation}
The covariance matrix $\textbf{R}_{ij}\,{=}\,\mathbb{E}\left\lbrace \textbf{h}_{ij} \textbf{h}_{ij}^{\rm H}\right\rbrace$ in~\eqref{eqn:mmse_filter} is defined element-wise as follows \vspace{-1.5em}
\small
\begin{align} \label{eqn:cov_matrix}
&\mathbf{R}_{ij}(m,n) = \beta_{ij} \mathbf{R}_{ij}^{\phi}(m,n)
\\ \nonumber
&= \beta_{ij} \int_{0}^{2\pi} \exp \left( {-}j 2\pi (m-n)\frac{D}{\lambda} \cos(\phi_{ij}) \right) p_{\phi}(\phi){\rm d} \phi~,
\end{align} \normalsize
where $p_{\phi}(\phi)$ is the probability distribution function (pdf) of the AoA, and $\mathbf{R}_{ij}^{\phi}\,{=}\,\mathbb{E}\left\lbrace \textbf{a}\left(\phi_{ij}\right) \textbf{a}^{\rm H}\left( \phi_{ij}\right) \right\rbrace$ is the angular covariance matrix of the steering vector. Employing \eqref{eqn:mmse_filter} and \eqref{eqn:cov_matrix}, the resulting covariance matrix of the channel estimate in \eqref{eqn:mmse_chan_est} is given as
\begin{equation}\label{eqn:cov_matrix_mmse}
\hat{\textbf{R}}_{ij}\,{=}\,\mathbb{E} \big\{ \hat{\textbf{h}}_{ij} \hat{\textbf{h}}_{ij}^{\rm H} \big\}\,{=}\,\tau \tilde{\textbf{R}}_{ij} \textbf{R}_{ij}\,,
\end{equation}
where we present the detailed derivation steps for covariance matrices in Appendix~\ref{app:cov_matrix}.
\section{Achievable DL Rates for Correlated MIMO Channels with Moderate Antenna Array Sizes}\label{sec:ul_dl_trans}
In this section, we study the achievable DL rates for correlated MIMO channels specifically considering the moderate size antenna array regime. In particular, we derive an exact analytical expression to calculate achievable DL ergodic rates with eigen-beamforming (EBF) precoding under pilot contamination. This rate expression is applicable to \emph{any} antenna array size, unlike the case in \cite{Debbah13MasMIMO} where \emph{asymptotic} antenna array regime is taken in to consideration.
During the DL data transmission, each BS employs the channel estimate obtained in the UL training phase as discussed in Section~\ref{sec:ul_trans} to compute the precoding vector for its own UE relying on the perfect reciprocity of the UL and the DL channels in the TDD protocol~\cite{Marzetta10NonCoo}. The received signal at the $j$th UE can therefore be given as
\begin{equation} \label{eqn:DL_trans}
y_{j}^{\rm DL} = \sqrt{\eta_{j}}\textbf{h}_{jj}^{\rm H}\textbf{w}_{j} q_{j} + \sum \limits_{i=1; i \neq j}^{N_\mathrm{L}} \sqrt{\eta_{i}}\textbf{h}_{ji}^{\rm H}\textbf{w}_{i} q_{i} + n_{j},
\end{equation}
where $\textbf{w}_{i}$ is the $M{\times}1$ precoding vector of the $i$th BS for its own user, $\eta_{i}\,{=}\,\left[ \mathbb{E} \left\lbrace {\rm tr} \left[ \textbf{w}_{i} \textbf{w}_{i}^{\rm H} \right] \right\rbrace \right]^{{-}1}$ normalizes the average transmit power of the $i$th BS to achieve the same SNR in the UL training phase~\cite{Debbah13MasMIMO}, $q_j$ is the unit-energy data symbol transmitted from $j$th BS to its own UE and chosen from a discrete alphabet $\mathcal{A}_{\,\rm DL}$ in an iid fashion, and $n_j$ is the circularly symmetric complex Gaussian noise with $\mathcal{CN}\left(0,\sigma^2\right)$. The beamforming strategy is assumed to be either the EBF (also known as conjugate beamforming) or the regularized zero-forcing (RZF)~\cite{Debbah13MasMIMO}, and is given at the $j$th BS as follows
\begin{align}
\textbf{w}^{\rm EBF}_j &= \hat{\textbf{h}}_{jj}\,, &\text{(EBF Precoder)} \label{eqn:ebf}\\
\textbf{w}_j^{\rm RZF} &= \left( \hat{\mathbf{h}}_{jj} \hat{\mathbf{h}}_{jj}^{\rm H} + \sigma^2 \textbf{I}_{M} \right)^{-1}\hat{\mathbf{h}}_{jj}\,. &\text{(RZF Precoder)} \label{eqn:rzf}
\end{align} In the following, the impact of AS on the achievable rates is investigated under both of these beamforming strategies, with a rigorous analytical rate derivation for the EBF precoding.
\subsection{Achievable DL Rates with Precoding} \label{sec:rates}
\begin{figure*}[!t]
\setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{11}
\begin{equation} \label{eqn:ergodic_rate}
R_{j} = \log_2 \bBigg@{5}( 1 + \frac{\overbrace{\eta_{j} \left| \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace \right|^2}^{\textrm{Desired signal power}}}{\sigma^2\,{+}\,\underbrace{\eta_{j}{\rm Var}\left\lbrace \textbf{h}_{jj}^{\rm H}\textbf{w}_j\right\rbrace}_{\textrm{Self-interference}}\,{+}\,\underbrace{\sum_{i=1; i\neq j}^{N_\mathrm{L}}\eta_{i} \mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace }_{\textrm{Intercell interference}}} \bBigg@{5})
\end{equation}
\setcounter{equation}{12}
\hrulefill
\end{figure*}
We now study achievable DL rates as a function of the AS over the underlying correlated MIMO channel with the EBF and the RZF precoding. In particular, we provide an exact analytical expression to calculate achievable rates with EBF precoding. By assuming UEs have just the knowledge of long-term statistics of the effective channel and not the instantaneous CSI, the ergodic rate as given in \eqref{eqn:ergodic_rate} is achievable at the $j$th UE~\cite{Marzetta11PilCon}. In that, $\eta_{j}\left|\mathbb{E}\left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace\right|^2$ captures the desired signal power, $\eta_{j}{\rm Var}\left\lbrace\textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace$ is interpreted as the \textit{self-interference} and arises from the lack of information on the instantaneous channel at the UE, and $\sum_{i=1; i\neq j}^{N_\mathrm{L}} \eta_{i}\mathbb{E}\left\lbrace \left|\textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace$ is the intercell interference since it represents the interference from the other BS signals. Here, the power normalization factor is given by $\eta_{j} = \left[ \mathbb{E} \big\{ \hat{\textbf{h}}_{jj}^{\rm H} \hat{\textbf{h}}_{jj} \big\} \right]^{-1}$.
The rate approximation in \eqref{eqn:ergodic_rate} is arguably conservative, as discussed in~\cite{Nadisanka16DirTra}, and can be interpreted as ``self-interference limited'' rate since the self-interference term dominates at high SNR regime for finite ULA sizes. However, since our focus in this study is to evaluate the impact of the AS on the correlated MIMO channels at fixed SNR, the rate approximation in \eqref{eqn:ergodic_rate} is used confidently. It is worth noting that, the achievable rates can also be evaluated by considering the alternative expression suggested in~\cite[Eqn.~(32)]{Nadisanka16DirTra} using the first and the second order moments of the effective channel derive subsequently.
In the following theorem, considering that the EBF precoder in~\eqref{eqn:ebf} is used in the DL transmission, we derive analytical expressions of the first and the second order moments for the effective channel in order to be able to calculate the achievable rate in~\eqref{eqn:ergodic_rate}.
\smallskip
\begin{theorem}\label{the:achievable_rates}
Assuming that LMMSE channel estimation is used in the UL training, and that EBF precoding as in~\eqref{eqn:ebf} is used prior to DL data transmission, the first order moment of the effective channel is given as
\begin{align}\label{eqn:1st_moment}
\mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace &= {\rm tr} \left\lbrace \hat{\textbf{R}}_{jj} \right\rbrace\,.
\end{align}
\begin{figure*}[!t]
\setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{13}
\begin{align}
\mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace &= \tau^2 \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{M} \sum\limits_{m'=1}^{M} \sum\limits_{n'=1}^{M} \tilde{\textbf{R}}_{ii}(m,n) \, \tilde{\textbf{R}}_{ii}^{\rm H}(m',n') \;
\left[ {\rm E}_{\phi}(m,n,m',n') + \sum\limits_{\substack{k=1; \, k{\neq}j}}^{N_\mathrm{L}} \!\!\!\textbf{R}_{ji}(n',m) \textbf{R}_{ki}(n,m') \right] \nonumber\\
& \qquad + \sigma^2 {\rm tr} \left\lbrace \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H}\textbf{R}_{ji} \tilde{\textbf{R}}_{ii} \textbf{S}^{\rm H} \right\rbrace , \label{eqn:2nd_moment}
\end{align}
\setcounter{equation}{14}
\vspace{-0.1in}
\begin{equation} \label{eqn:2nd_moment_phi}
{\rm E}_{\phi}(m,n,m',n')\,=\, \dfrac{\beta_{ji}^2}{{N_\mathrm{P}}} \Big[ 2 \,{\rm E}_{ji} (n{-}m{+}n'{-}m') + (N_\mathrm{P}{-}1) \Big( {\rm E}_{ji}(n{-}m){\rm E}_{ji} (n'{-}m') + {\rm E}_{ji}(n'-m){\rm E}_{ji}(n-m') \Big) \Big] \,,
\end{equation}
\setcounter{equation}{15}
\hrulefill
\end{figure*}
Likewise, the second order moment, $\mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace$ can be given as in \eqref{eqn:2nd_moment} where ${\rm E}_{\phi}(m,n,m',n')$ is defined as in \eqref{eqn:2nd_moment_phi} with ${\rm E}_{ji}(m)\,{=}\,\mathbf{R}_{ji}^{\phi}(n{+}m,n)$ for any $n\,{\leq}\,M{-}m$.
\end{theorem}
\smallskip
\begin{IEEEproof}
See Appendix~\ref{app:achievable_rates}.
\end{IEEEproof}
\smallskip
Note that once \eqref{eqn:1st_moment} and \eqref{eqn:2nd_moment} are computed, the signal and the intercell interference terms in \eqref{eqn:ergodic_rate} are readily available by employing $\eta_{j}\,{=}\left[ {\rm tr} \big\{ \hat{\textbf{R}}_{jj} \big\} \right]^{{-}1}\!\!$, and the self-interference is given by
\begin{align}\label{eqn:variance}
{\rm Var} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace = \mathbb{E} \left\lbrace \left| \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right|^2 \right\rbrace - \left( \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace \right)^2\,.
\end{align}
\section{Impact of AS on Desired and Interference Signal Power Terms}\label{sec:individual_powers}
In this section, we study in detail the explicit impact of AS on the \emph{desired signal power} (Section~\ref{sec:covariance_diagonal}), the \emph{intercell interference} (Section~\ref{sec:channel_orthogonal}), and the \emph{self-interference} (Section~\ref{sec:channel_fluctuation}) terms in \eqref{eqn:ergodic_rate} in relation to pilot contamination effect from statistical and geometrical perspectives. We also draw useful insights about their behavior for varying AS considering different array sizes. It is worth remarking that, any variation in the achievable rates with varying AS is due to collective contribution from all these three terms, and taking any of them only individually into account may be misleading when evaluating the overall rate results presented in Section~\ref{sec:numerical_results}.
\subsection{Effect of Covariance Matrix Diagonalization on Desired Signal Power}\label{sec:covariance_diagonal}
\begin{figure}[!h]
\centering
\captionsetup[subfigure]{oneside,margin={0.5cm,0.5cm}}
\subfloat[$|\textbf{R}_{jj}^{\phi}(m,n)|$ versus AS for various $|m{-}n|$ values.]{\includegraphics[width=0.25\textwidth]{./r_ii}
\label{fig:r_ii}}
\subfloat[$|\textbf{R}_{ij}^{\phi}(m,n)|$ versus AS for various $|m{-}n|$ values.]{\includegraphics[width=0.25\textwidth]{./r_ji}
\label{fig:r_ji}}
\vspace{0.1in}
\caption{Norm of the entries of the angular covariance matrices $\textbf{R}_{jj}^{\phi}$ and $\textbf{R}_{ij}^{\phi}$ for $M\,{=}\,\{10,50\}$.}
\label{fig:r}
\end{figure}
Let us consider the structure of the angular covariance matrices $\textbf{R}_{jj}^{\phi}$ and $\textbf{R}_{ij}^{\phi}$ in \eqref{eqn:cov_matrix} with varying AS and antenna array size. Under the assumptions in Section~\ref{sec:system_model}, the main diagonal entries of these covariance matrices are all $1$, and the off-diagonal entries representing the angular correlation have non-zero norms smaller than $1$. In Fig.~\ref{fig:r}, the Euclidean norms of the $(m,n)$th off-diagonal entries having the minimum and the maximum absolute separation of $|m{-}n|\,{=}\,1$ and $|m{-}n|\,{=}\,M{-}1$, respectively, are depicted along with the increasing AS for $M\,{=}\,\{10, \ 50\}$ and considering $\Delta\theta_j\,{=}\,40^{\circ}$ in Fig.~\ref{fig:layout}. We observe that each of these covariance matrices gets more diagonalized (the magnitude of the off-diagonal entries decreases) when the array size $M$ or the AS increases. The diagonalization rate increases with $M$ since both covariance matrices get diagonalized much faster for larger $M$ values. Note that any BS in the multi-cell network will receive signals from a wide range of AoAs as the AS increases, which is similar to the uncorrelated rich-scattering environment where the possible AoAs span $[0,2\pi)$ angle support.
The signal power variation with respect to the AS can be assessed through the diagonalization characteristic of the covariance matrices. Employing $\eta_{j}\,{=}\,\big[ {\rm tr} \big\{ \hat{\textbf{R}}_{jj}\big\} \big]^{{-}1}$ and the first order moment given in \eqref{eqn:1st_moment}, the signal power can be expressed as $\eta_{j}\left|\mathbb{E}\left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace\right|^2\,{=}\,{\rm tr} \big\{ \hat{\textbf{R}}_{jj}\big\}$, which can be expressed more elaborately as follows
\begin{align} \label{eqn:signal_power}
{\rm tr} \big\{ \tau\tilde{\textbf{R}}_{jj}\textbf{R}_{jj}\big\} &= \tau\sum\limits_{m}\tilde{\textbf{R}}_{jj}(m,m)\textbf{R}_{jj}(m,m) \nonumber \\
&+ \tau \mathop{\sum\limits_{m}\sum\limits_{n}}_{n \neq m}\tilde{\textbf{R}}_{jj}(m,n)\textbf{R}_{jj}(n,m) .
\end{align}
Since the estimation matrix $\tilde{\textbf{R}}_{jj}$ in \eqref{eqn:mmse_filter} becomes diagonal for larger AS values (similar to $\textbf{R}_{ij}$ and $\textbf{R}_{jj}$), and that both the terms at the right hand side of \eqref{eqn:signal_power} are real and positive, the second summation in \eqref{eqn:signal_power} decreases when the AS increases. This leads the covariance matrix to become more diagonal. As a result, the \emph{\textbf{signal power in \eqref{eqn:ergodic_rate} decreases with increasing AS}} through the diagonalization of the covariance matrices.
This behavior of the signal power with increasing AS can be intuitively interpreted as follows. As we will discuss in Section~\ref{sec:channel_orthogonal}, when the AS increases, the orthogonality between the desired user precoder $\textbf{w}_{j}\,{=}\,\hat{\textbf{h}}_{jj}$ and the interfering user channel $\textbf{h}_{ij}$ gets impaired along with more powerful pilot contamination. Hence, $\textbf{w}_{j}$ does not exactly align with the $j$th user channel direction $\textbf{h}_{jj}$, any more. This geometrical misalignment accordingly results in transmit power leakage from $j$th BS to some undesired directions (other than the $j$th user direction) during the DL data transmission, which in turn leads to signal power loss at the $j$th user.
\subsection{Geometrical Interpretation of Intercell Interference Power}\label{sec:channel_orthogonal}
In the DL data transmission, the pilot contamination shows its adverse effect by impairing the orthogonality between the desired and the interfering user channels, which is basically captured by the intercell interference term $\sum \limits_{i=1; i \neq j}^{N_\mathrm{L}} \sqrt{\eta_{i}}\textbf{h}_{ji}^{\rm H}\textbf{w}_{i} q_{i}$ in \eqref{eqn:DL_trans}. From a geometrical perspective, the intercell interference involves the inner product between each interference channel $\textbf{h}_{ji}$ for $j\,{\neq}\,i$, and the precoder $\textbf{w}_{i}$, which is a function of the estimate of the desired user channel $\textbf{h}_{ii}$. One way to examine how the pilot contamination impairs the orthogonality, and hence amplify the intercell interference power with varying AS is through a geometric interpretation. This can be done by analyzing the pdf of the random angle $\varphi_{ij}$ between $\textbf{h}_{ji}$ and $\textbf{w}_{i}$, where we leave the actual numerical evaluation to Section~\ref{sec:numerical_results}. Note that, if the channels were perfectly known and spatially uncorrelated, the desired pdf would be given analytically as $f_{\varphi_{ij}}(\varphi)\,{=}\,2\left(N{-}1\right)\left(\sin\varphi\right)^{2N{-}3}\cos\varphi$~\cite{Loyka2004PerAnVB}.
\begin{figure}[!h]
\centering
\subfloat[The distribution of $\varphi_{ij}$ for $M\,{=}\,10$.]{\includegraphics[width=0.45\textwidth]{./pdf_M_10}
\label{fig:pdf_M_10}}\\
\subfloat[The distribution of $\varphi_{ij}$ for $M\,{=}\,50$.]{\includegraphics[width=0.45\textwidth]{./pdf_M_50}
\label{fig:pdf_M_50}}
\vspace{0.1in}
\caption{The pdf of the angle $\varphi_{ij}$ between $\textbf{h}_{ji}$ and $\textbf{w}_{i}$ for $M\,{=}\,\{10,50\}$ and $\text{SNR}\,{=}\,\{0,20\}\text{ dB}$.}
\label{fig:pdf}
\end{figure}
To study the impact of AS on intercell interference, we consider an example scenario with the representative setting of Fig.~\ref{fig:layout}. In that, we assume $\theta\,{=}\,200^{\circ}$, quadrature phase-shift keying (QPSK) symbols in the UL training phase with the sequence length $\tau\,{=}\,1$, and the path-loss exponent $\zeta\,{=}\,2$. In Fig.~\ref{fig:pdf}, we depict the pdf of the random angle $\varphi_{ij}$ for $M\,{=}\,\{10,50\}$ and $\text{SNR}\,{=}\,\{0,20\}\text{ dB}$. We observe that the desired and the interfering user channels are sufficiently orthogonal for small AS values since $\varphi_{ij}$ takes values close to $90^{\circ}$ with high probability for perfect CSI, and the resulting orthogonality can even be stronger than the uncorrelated channel case. This geometrical interpretation agrees with~\cite{Gesbert13CooApp} in the sense that the training beams of different UEs do not overlap in the UL transmission when the AS is sufficiently small making the scenario free from any pilot contamination effect. As a result, the random angle between the precoder $\textbf{w}_{i}$ and the interfering user channel $\textbf{h}_{ji}$ has the same pdf for the perfect CSI ($\textbf{w}_{i}\,{=}\,\textbf{h}_{ii}$) and the channel estimation ($\textbf{w}_{i}\,{=}\,\hat{\textbf{h}}_{ii}$) scenarios when the beams are separated sufficiently, or equivalently the AS is small enough. When the AS starts to increase, spatial correlation between the desired and the interfering user channels becomes stronger since the overlap between AoA domains associated with desired and interfering user channels becomes larger. As a result, \textbf{\emph{the desired orthogonality inherently gets impaired for larger AS values, even for the perfect CSI case}}.
When the desired user channel is being estimated, this orthogonality gets hurt even more because of the pilot contamination effect. This can be observed in Fig.~\ref{fig:pdf} from the deviation of the pdf of $\varphi_{ij}$ associated with the channel estimation scenario, to the left side (toward $0^{\circ}$) with respect to perfect CSI scenario when $\textrm{AS}\, {=}\,50^{\circ}$. The orthogonality gets impaired further when the SNR increases since larger SNR in each cell implies more interference power transferred to other cells. Finally, comparing Fig.~\ref{fig:pdf}\subref{fig:pdf_M_10} and Fig.~\ref{fig:pdf}\subref{fig:pdf_M_50} we observe that, for a given AS the precoder $\textbf{w}_{i}$ and the interfering channels are getting more orthogonal with increasing antenna array size, which is one of the main goals of massive MIMO in the context of the intercell interference rejection~\cite{Marzetta15MasMIMO}.
\subsection{Effect of Channel Power Fluctuation on Self-Interference}\label{sec:channel_fluctuation}
The rate bound given in \eqref{eqn:ergodic_rate} is discussed to be achievable in~\cite{Marzetta11PilCon} assuming that the UEs in the network do not know their instantaneous channels, but rather they only know the respective long-term means. This lack of information on the exact instantaneous channel is captured by the self-interference term $\eta_{j}{\rm Var}\left\lbrace\textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace$ in \eqref{eqn:ergodic_rate}. This term actually represents the power of the deviation between the instantaneous channel and the long-term mean, given equivalently as $\eta_{j}\mathbb{E}\left\lbrace \left|\textbf{h}_{jj}^{\rm H} \textbf{w}_j{-}\mathbb{E}\left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace \right|^2\right\rbrace$. Assuming EBF precoding in \eqref{eqn:ebf} with perfect CSI, this term becomes equivalent to the variance of the channel power. We therefore note that as the fluctuation of the channel square-norm around the long-term mean decreases, which is similar to the phenomenon known as the \textit{channel hardening}~\cite{Hochwald04ChanHard, Tarokh09ChanHard, Larsson17ChanHard}, the self-interference term should decrease accordingly. In the following, this fluctuation and hence the \emph{\textbf{self-interference is shown to decrease monotonically when the AS increases}}, for the EBF precoding.
\smallskip
\begin{lemma}\label{lem:hardening}
The channel considered in~\eqref{eqn:channel_defn} hardens, such that $\left\| \textbf{h}_{ij} \right\|^2\!{/}\,\mathbb{E}\left\lbrace \left\| \textbf{h}_{ij} \right\|^2 \right\rbrace{\stackrel{\;\text{P}\;\;}{\rightarrow}}\,1$ as $M{\rightarrow}\infty$, if we have $\mathcal{M}_{ij}\,{\rightarrow}\,0$ as $M{\rightarrow}\,\infty$, where $\mathcal{M}_{ij}$ is the \textit{hardening measure} given as
\begin{align}\label{eqn:harden_measure}
\mathcal{M}_{ij} = \frac{1}{N_\mathrm{P}} + \frac{N_\mathrm{P}-1}{M N_\mathrm{P}} \left( 1 + \frac{2}{M} \sum\limits_{m=1}^{M-1} (M-m) \left| {\rm E}_{ij}(m) \right|^2 \right),
\end{align}
with ${\rm E}_{ij}(m)\,{=}\,\mathbf{R}_{ij}^{\phi}(n{+}m,n)$ for any $n\,{\leq}\,M{-}m$.
\end{lemma}
\smallskip
\begin{IEEEproof}
See the derivation in Appendix~\ref{app:harden_measure} as an extension of~\cite{Larsson17ChanHard} which considers uncorrelated MIMO channels.
\end{IEEEproof}
\smallskip
We observe that the desired convergence $\mathcal{M}_{ij}\,{\rightarrow}\,0$ is satisfied only if the number of paths $N_\mathrm{P}$ is sufficiently large. Even when the number of paths $N_\mathrm{P}$ or the antenna array size $M$ have moderate values in contrast to asymptotic approximations, the behavior of the self-interference power can still be assessed from \eqref{eqn:harden_measure}. In Fig.~\ref{fig:hardening}, we depict $\mathcal{M}_{ij}$ along with AS for various $M$ under the assumption that $N_\mathrm{P}\,{=}50$ and $\phi_{ij}\,{\sim}\,\mathcal{U}\left[-\Delta,+\Delta\right]$ with $\Delta$ representing the AS. We observe that \emph{\textbf{$\mathcal{M}_{ij}$ decreases monotonically for increasing AS for all cases, and gets even smaller values as the array size $M$ increases}}. Note that the smaller $\mathcal{M}_{ij}$ implies a better convergence of the channel square-norm to its long-term mean with high probability, and hence less fluctuation in the channel power around its long-term mean. Since the self-interference power is closely related to the channel power fluctuation, \emph{\textbf{decaying behavior of $\mathcal{M}_{ij}$ with the increasing AS implies reduction in the self-interference power, as well}}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{./channel_hardening}
\end{center}
\caption{The channel hardening measure $\mathcal{M}_{ij}$ for $M\,{=}\,\{10,50,100,\infty\}$ and $N_\mathrm{P}\,{=}50$.}
\label{fig:hardening}
\end{figure}
\section{Numerical Results and Discussion}\label{sec:numerical_results}
In this section, we present numerical results to evaluate the impact of AS on the achievable rates in a multi-cell network under pilot contamination and considering the EBF and the RZF precoders. The theoretical derivations presented in Section~\ref{sec:rates} are employed for analytical evaluations, and the corresponding simulation data is generated through extensive Monte Carlo runs. Without any loss of generality, we assume QPSK modulated pilot symbols in the UL training of sequence length $\tau\,{=}\,1$, the path-loss exponent $\gamma\,{=}\,3$, $\text{SNR}\,{=}\,0\text{ dB}$ with $\sigma^2\,{=}\,1$, $N_\mathrm{P}\,{=}\,100$, and $D\,{=}\,\frac{\lambda}{2}$ together with the distances $r_1\,{=}\,40\text{ m}$ and $r_2\,{=}\,50\text{ m}$ shown in Fig.~\ref{fig:layout}. Note here that rate results presented in this section are for the UE in $j$th cell.
\subsection{Two-Cell Scenario: Fixed Interfering UE Position}\label{sec:numerical_results_2cell_singlePos}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{./rate_ebf_Np_100}
\end{center}
\caption{Achievable rates for the EBF precoding in a $2$-cell scenario with $M\,{=}\,\{10,20,50,100\}$ and the interfering UE angular position at $\theta\,{=}\,200^{\circ}$.}
\label{fig:rate_Np=50_100_M=10_20}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{./power_ebf_Np_100}
\end{center}
\caption{Signal, self-interference and intercell interference powers for the EBF precoding in a $2$-cell scenario with $M\,{=}\,\{10,20\}$ and the interfering UE angular position at $\theta\,{=}\,200^{\circ}$.}
\label{fig:power_Np=50}
\end{figure}
This section considers a two-cell scenario where the $i$th and $j$th cells in Fig.~\ref{fig:layout} are designated as the interfering and the desired cells, respectively, and the angular position of the $i$th UE is $\theta\,{=}\,200^{\circ}$. Fig.~\ref{fig:rate_Np=50_100_M=10_20} captures the achievable rates with the EBF precoding for array sizes of $M\,{=}\,\{10,20,50,100\}$. We observe that the analytical results follow the characteristic behavior of the simulation data in all cases of interest. Further, we can observe from Fig.~\ref{fig:rate_Np=50_100_M=10_20} that, for $M\,{=}\,10$ rates are not monotonically increasing, and actually there is minimum rate value at $\text{AS}\,{=}\,28^{\circ}$. As it will become clear in Section~\ref{sec:numerical_results_multi_2cell_multiPos}, this \textit{unfavorable} AS value corresponding to the minimum rate depends on the underlying geometry. Therefore, the location of this minimum can be controlled through the deployment geometry, and, in particular through the angle of UE separation captured by $\Delta\theta$'s in Fig.~\ref{fig:layout}.
\smallskip
\begin{remark}
The real AS value of the propagation environment is independent of the underlying geometry, and it rather depends on the carrier frequency of the communication setting and some other features~\cite{Rappaport16mmWaveSta,3GPP16ChanMod}. As a result, the non-monotonic behavior of the achievable rates with respect to the AS (e.g. in Fig.~\ref{fig:rate_Np=50_100_M=10_20}) can be utilized to enhance the aggregate throughput. This can be achieved by discouraging the formation of user-cell pairs if the \textit{unfavorable} AS value associated with the minimum rate is close to the real AS value of the environment. To the best of our knowledge, none of the existing user-cell pairing approaches proposed in the literature exploit the AS of the propagation environment \cite{Yates_ULPwrControl_BS_Assignment, Rashid_DL_BS_assignment, Galeana_2008ABS, Galeana_CostBased_Approach_BS_Assignment, Galeana_Backhaul_Aware_BS_Assign_OFDMA, 1687773}. Note that the \emph{non-monotonic behavior cannot be revealed through an asymptotic analysis} due to the moderately large antenna array size regime that we consider in this paper.
\end{remark}
\smallskip
\begin{remark}
Even though user rates for relatively larger antenna array sizes ($M=50$ and $M=100$) exhibit a sharp increase for small AS region, they tend to saturate eventually at larger AS values due to the severe pilot contamination, as discussed in Section~\ref{sec:channel_orthogonal}. On the other hand, relatively smaller
array sizes ($M=10$ and $M=20$) result in no such saturation, which implies that the pilot contamination is not a dominant effect over achievable rates in this array size regime.
\end{remark}
\smallskip
In Fig.~\ref{fig:power_Np=50}, the signal, the self-interference, and the intercell interference powers derived in Section~\ref{sec:rates} for the EBF precoder are captured separately for the scenario in Fig.~\ref{fig:rate_Np=50_100_M=10_20}. We observe that the analytical results follow the simulation data successfully in all cases of interest. The signal and the intercell interference powers are observed to exhibit relatively flat characteristics over a range of small AS values up to approximately $10^{\circ}$. In this region, the AS values are sufficiently small, and the UL training beams of the UEs are therefore well separated. This is the reason for zero intercell interference in this region, which implies no pilot contamination effect and agrees with the geometrical interpretation of Section~\ref{sec:channel_orthogonal}.
When the AS increases beyond $10^{\circ}$, the intercell interference also starts increasing due to pilot contamination, and saturates around $\text{AS}\,{=}\,50^{\circ}$. Further, the DL transmission does not exactly align with the desired signal direction any more, which appears as the decreasing trend in the signal power as discussed in Section~\ref{sec:covariance_diagonal}. Note that, as captured in Fig.~\ref{fig:power_Np=50} the self-interference power has a decaying trend with the increasing AS as discussed in Section~\ref{sec:channel_fluctuation} and for $M\,{=}\,20$, self-interference starts decreasing earlier and much faster compared to $M\,{=}\,10$. Since the decrease in the signal power dominates initially for $M\,{=}\,10$ over self-interference, we observe a non-monotonic rate behavior for $M\,{=}\,10$ with a minimum at $\text{AS}\,{=}\,28^{\circ}$. On the other hand, since the self-interference starts decaying quickly for $M\,{=}\,20$ compared to the signal power, we observe monotonically increasing rate behavior for $M\,{\geq}\,20$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{./rate_rzf_Np_100}
\end{center}
\caption{Achievable rates for the RZF precoding in a $2$-cell scenario with $M\,{=}\,\{10,20,50,100\}$ and the interfering UE angular position at $\theta\,{=}\,200^{\circ}$.}
\label{fig:rate_rzf}
\end{figure}
The achievable rates for the RZF precoder in the DL transmission is depicted in Fig.~\ref{fig:rate_rzf} for the scenario of Fig.~\ref{fig:rate_Np=50_100_M=10_20} with the ULA sizes of $M\,{=}\,\{10,20,50,100\}$. We observe that the achievable rates with the RZF precoder is higher than that with the EBF precoder in Fig.~\ref{fig:rate_Np=50_100_M=10_20} for the same array sizes \emph{at the expense of a larger computational complexity}. We also observe a non-monotonic behavior in achievable rates for all the array sizes of interest, where there is a minimum at $\text{AS}\,{=}\,31^{\circ}$ for $M\,{=}\,\{10,20\}$, and a maximum at $\text{AS}\,{=}\,\{15^{\circ},16^{\circ}\}$ for $M\,{=}\,\{50,100\}$, respectively.
\smallskip
\begin{remark}
Similar to the EBF precoder case, the non-monotonic behavior as illustrated in Fig.~\ref{fig:rate_rzf} can be utilized effectively to enhance aggregate throughput by: 1) encouraging the formation of user-cell pairs if the favorable AS value associated with the maximum rate is close to the real AS value of the environment; and similarly, 2) discouraging the user-cell pairs for which the unfavorable AS value associated with the minimum rate is close to the real AS value of the environment.
\end{remark}
\smallskip
\subsection{Two-Cell Scenario: Varying Interfering UE Position}\label{sec:numerical_results_multi_2cell_multiPos}
\begin{figure}
\centering
\subfloat[EBF precoding]{\includegraphics[width=0.45\textwidth]{./rate_ebf_varyingTheta_Np_100}
\label{fig:rate_ebf_varyingTheta}}\\
\subfloat[RZF precoding]{\includegraphics[width=0.45\textwidth]{./rate_rzf_varyingTheta_Np_100}
\label{fig:rate_rzf_varyingTheta}}
\vspace{0.1in}
\caption{Achievable rates for the EBF and the RZF precoding in a $2$-cell scenario with $M\,{=}\,\{10,50\}$ and varying interfering UE position at $\theta\,{=}\,\{180^{\circ},200^{\circ},220^{\circ}\}$.}
\label{fig:rate_varyingTheta}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.45\textwidth]{./power_ebf_varyingTheta_M_10_Np_100}
\caption{Signal, self-interference and intercell interference powers for the EBF precoding in a $2$-cell scenario with $M\,{=}\,10$ and varying interfering UE positions at $\theta\,{=}\,\{180^{\circ},200^{\circ},220^{\circ}\}$.}
\label{fig:power_varyingTheta}
\end{figure}
In this section, we consider the effect of various angular positions of the $i$th interfering UE on achievable rates. Fig.~\ref{fig:rate_varyingTheta}, captures the achievable rates for the EBF and the RZF precoding in a $2$-cell scenario with $M\,{=}\,\{10,50\}$ at a set of angular positions $\theta\,{=}\,\{180^{\circ},200^{\circ},220^{\circ}\}$ for the $i$th interfering UE. We observe that as the interfering UE gets closer to the desired UE, which is indicated by the increasing $\theta$ in Fig.~\ref{fig:layout}, the achievable rate reduces for both the precoders. In addition, we observe either lower maxima or deeper minima located at smaller AS values, when $\theta$ increases.
As captured in Fig.~\ref{fig:power_varyingTheta}, the intercell interference increases for larger $\theta$ since the interfering UE gets closer to the desired UE and this indicates more powerful pilot contamination (see Section~\ref{sec:channel_orthogonal}).
In addition, the signal power gets smaller accordingly when $\theta$ increases since DL transmission does not align properly with the desired user direction any more, as discussed in Section~\ref{sec:covariance_diagonal}. Furthermore, the self-interference is not highly affected from $\theta$ (and hence from the pilot contamination) as shown in Fig.~\ref{fig:power_varyingTheta} along with the discussion in Section~\ref{sec:channel_fluctuation}. As a result, increasing intercell interference and decreasing signal power, both of which occur with increasing $\theta$, result in reduced user rates. This is also the reason behind the deeper minima observed for the EBF precoding for larger $\theta$ when $M\,{=}\,10$. Fig.~\ref{fig:rate_ebf_varyingTheta} shows that for $M\,{=}\,50$, the monotonically increasing rate behavior with the EBF precoding for $\theta\,{=}\,\{180^{\circ},200^{\circ}\}$ disappears for $\theta=220^{\circ}$, and instead a minimum value appears at $\text{AS}\,{=}\,11^\circ$. The intercell interference for larger $\theta$ can be very strong for $M\,{=}\,50$ with RZF such that the maxima at $\theta\,{=}\,\{180^{\circ},200^{\circ}\}$ switches into a minimum at $\theta\,{=}\,220^{\circ}$, as shown in Fig.~\ref{fig:rate_rzf_varyingTheta}.
\subsection{Multi-Cell Scenario}\label{sec:numerical_results_multi}
Finally, we consider the impact of AS in a multi-cell setting with the number of cells $N_{\rm L}\,{\in}\,\{2,3,5\}$. To this end, a multi-cell setting is generated by considering five cells as shown in Fig.~\ref{fig:layout}, where the interfering UEs are located at $200^{\circ},160^{\circ},360^{\circ},60^{\circ}$ with respect to the horizontal axis for the $i$th, $k$th, $\ell$th, and $m$th cells, respectively. This layout provides almost the worst condition in terms of the intercell interference power, and hence the pilot contamination. The effect of this multi-cell setting on the achievable rates with the EBF and the RZF precoders is presented in Fig.~\ref{fig:rate_multipleCells} for $M\,{=}\,\{10,50\}$. We observe that as we consider more cells, the resulting interference degrades achievable rates together with much lower maxima or deeper minima. We even observe the formation of an additional maximum for the EBF at $\text{AS}\,{=}\,35^\circ$ and minimum for the RZF at $\text{AS}\,{=}\,5^\circ$ when $N_{\rm L}\,{=}\,5$. Since adding more cells strengthens the intercell interference very rapidly, the desired signal power reduces proportionally, as shown in Fig.~\ref{fig:power_multipleCells}. These impairing effects eventually reduce and even saturate achievable rates, as shown in Fig.~\ref{fig:rate_multipleCells}.
\begin{figure}
\centering
\subfloat[EBF precoding]{\includegraphics[width=0.45\textwidth]{./rate_ebf_multipleCells_Np_100}
\label{fig:rate_ebf_multipleCells}}\\
\subfloat[RZF precoding]{\includegraphics[width=0.45\textwidth]{./rate_rzf_multipleCells_Np_100}
\label{fig:rate_rzf_multipleCells}}
\vspace{0.1in}
\caption{Achievable rates for the EBF and the RZF precoding in a multi-cell scenario with $N_{\rm L}\,{=}\,\{2,3,5\}$, $M\,{=}\,\{10,50\}$.}
\label{fig:rate_multipleCells}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{./power_ebf_multipleCells_M_10_Np_100}
\caption{Signal, self-interference and intercell interference powers for the EBF precoding in a multi-cell scenario with $N_{\rm L}\,{=}\,\{2,3,5\}$, $M\,{=}\,10$.}
\label{fig:power_multipleCells}
\end{figure}
\section{Concluding Remarks}\label{sec:conclusion}
We investigated the impact of AS on the achievable rates in a multi-cell environment under pilot contamination, considering moderately large antenna arrays. An exact analytical expression for achievable rate is derived for the EBF precoding considering arbitrary antenna array size. For correlated MIMO channels, we studied how interference channel orthogonality is affected from increasing AS along with the pilot contamination. Further, the channel power fluctuation around its long-term mean is analytically evaluated for varying ASs considering different antenna array sizes.
When the AS gets larger, we showed through rigorous analyses that 1) the covariance matrices tend to have a more diagonalized structure, 2) the channel power fluctuation diminishes (in a similar way as in the channel hardening), and 3) the orthogonality of the interference channel gets impaired due to pilot contamination effect. The overall achievable rate behavior as a function of the AS depends on which of these factors dominate over the other. Our analysis quantitatively identifies the antenna array size beyond which the pilot contamination starts being a dominant factor.
Lastly, our numerical results reveal a non-monotonic behavior (with respect to the AS) of the achievable rates
for both the EBF and the RZF precoders under certain scenarios. The AS values at which the rate minimum/maximum occurs depend on the relative positions of the UEs and their serving BSs. Such a knowledge, along with the rate expression derived in this paper, can be effectively utilized to maximize the aggregate network throughput via careful design of user-cell pairing strategies. Due to space limitations we have left analytical rate evaluations with RZF precoder as a future research work.
\appendices
\section{Covariance Matrix Derivation}\label{app:cov_matrix}
The covariance matrix of the channel vector $\mathbf{h}_{ij}$ in \eqref{eqn:channel_defn} is given as
\small \begin{align}
\mathbf{R}_{ij} & = \frac{1}{N_\mathrm{P}} \sum\limits_{p=1}^{N_\mathrm{P}} \sum \limits_{p'=1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \textbf{a}\left( \phi_{ij, p}\right) \textbf{a}^{\rm H}\left( \phi_{ij, p'}\right) \right\rbrace \mathbb{E} \left\lbrace \alpha_{ij, p} \, \alpha_{ij, p'}^{*} \right\rbrace , \nonumber\\
& = \frac{\beta_{ij}}{N_\mathrm{P}} \sum \limits_{p=1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \textbf{a}\left( \phi_{ij, p}\right) \textbf{a}^{\rm H}\left( \phi_{ij, p}\right) \right\rbrace , \label{eqn:app:cov_matrix_1}\\
& = \beta_{ij} \mathbb{E} \left\lbrace \textbf{a}\left( \phi_{ij}\right) \textbf{a}^{\rm H}\left( \phi_{ij}\right) \right\rbrace~, \label{eqn:app:cov_matrix_2}
\end{align} \normalsize
where \eqref{eqn:app:cov_matrix_1} employs $\mathbb{E} \left\lbrace \alpha_{ij, p}\,\alpha_{ij, p'}^{*} \right\rbrace\,{=}\,\beta_{ij} \delta(p,p')$, and \eqref{eqn:app:cov_matrix_2} follows from the fact that the distribution of AoA $\phi_{ij,p}$ is identical for any choice of the path index $p$. Defining the angular covariance matrix of the steering vector as $\mathbf{R}_{ij}^{\phi}\,{=}\,\mathbb{E}\left\lbrace \textbf{a}\left(\phi_{ij}\right) \textbf{a}^{\rm H}\left( \phi_{ij}\right) \right\rbrace$, and employing \eqref{eqn:steering_vector}, the element-wise angular correlation is given as follows
\begin{align} \label{eqn:app:cov_phi_element}
& \mathbf{R}_{ij}^{\phi}(m,n) = \mathbb{E} \left\lbrace \exp \left( -j2\pi (m-n)\frac{D}{\lambda} \cos(\phi_{ij}) \right) \right\rbrace~, \nonumber \\
& = \int_{0}^{2\pi} \exp \left( -j2\pi (m-n)\frac{D}{\lambda} \cos(\phi_{ij}) \right) p_{\phi}(\phi)\,{\rm d}\phi~,
\end{align} where $p_{\phi}(\phi)$ is the probability distribution function (pdf) of the AoA distribution. In particular, assuming the one-ring scatterer model \cite{Foschini_One_Ring_Scatterer,JSDM_LargeScaleArray} and uniform distribution for AoA with $\mathcal{U}\left[\bar{\phi}_{ij}{-}\Delta,\bar{\phi}_{ij}{+}\Delta\right]$, \eqref{eqn:app:cov_phi_element} can be given as
\small \begin{align} \label{eqn:app:cov_phi_uni}
\mathbf{R}_{ij}^{\phi} (m,n) = \frac{1}{2\Delta}\int_{\bar{\phi}_{ij}-\Delta}^{\bar{\phi}_{ij}+\Delta} \exp \left( -j2\pi (m-n)\frac{D}{\lambda} \cos(\phi_{ij}) \right) d\phi~.
\end{align} \normalsize
Employing \eqref{eqn:app:cov_phi_element}, each entry of the covariance matrix in \eqref{eqn:app:cov_matrix_2} is given by \eqref{eqn:cov_matrix}. \hfill\IEEEQEDhere
We now derive the covariance matrix of the channel estimate $\hat{\textbf{h}}_{ij}$, denoted by $\hat{\textbf{R}}_{ij}\,{=}\, \mathbb{E} \left\lbrace {\hat{\textbf{h}}_{ij} \hat{\textbf{h}}_{ij}^{\rm H}} \right\rbrace $. Employing the definition of $\hat{\textbf{h}}_{ij}$ in \eqref{eqn:mmse_chan_est}, and the UL signal model in \eqref{eqn:UL_trans_vector}, $\hat{\textbf{R}}_{ij}$ is given as
\begin{align}
&\hat{\textbf{R}}_{ij} = \tilde{\textbf{R}}_{ij}\textbf{S}^{\rm H} \, \mathbb{E} \left\lbrace \textbf{y}_{j}^{\textrm{UL}} \left(\textbf{y}_{j}^{\textrm{UL}}\right)^{\rm H} \right\rbrace \textbf{S}\,\tilde{\textbf{R}}_{ij}^{\rm H} \,, \nonumber \\
&= \tilde{\textbf{R}}_{ij}\textbf{S}^{\rm H} \mathbb{E} \left\lbrace \left( \textbf{S} \sum \limits_{k=1}^{N_\mathrm{L}} \textbf{h}_{kj} +\textbf{n}_{j} \right) \left( \textbf{S} \sum \limits_{k=1}^{N_\mathrm{L}} \textbf{h}_{kj} +\textbf{n}_{j} \right)^{\rm H} \right\rbrace \textbf{S} \, \tilde{\textbf{R}}_{ij}^{\rm H} \,,
\label{eqn:app:cov_est_1} \\
&= \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \sum \limits_{\ell=1}^{N_\mathrm{L}} \tilde{\textbf{R}}_{ij} \mathbb{E} \left\lbrace \textbf{h}_{kj}\textbf{h}_{\ell j}^{\rm H} \right\rbrace \tilde{\textbf{R}}_{ij}^{\rm H} +
\tau \sigma^2 \tilde{\textbf{R}}_{ij} \tilde{\textbf{R}}_{ij}^{\rm H}
\label{eqn:app:cov_est_2} \\
& + \tau \sum \limits_{k=1}^{N_\mathrm{L}} \tilde{\textbf{R}}_{ij} \mathbb{E} \left\lbrace \textbf{h}_{kj} \textbf{n}_{j}^{\rm H} \right\rbrace \textbf{S} \tilde{\textbf{R}}_{ij}^{\rm H}
+ \tau \sum \limits_{k=1}^{N_\mathrm{L}} \tilde{\textbf{R}}_{ij}\textbf{S}^{\rm H} \mathbb{E} \left\lbrace \textbf{n}_{j} \textbf{h}_{kj}^{\rm H} \right\rbrace \tilde{\textbf{R}}_{ij}^{\rm H} \,, \label{eqn:app:cov_est_3}
\end{align} \normalsize
where we employ the relations $\mathbb{E} \left\lbrace \textbf{n}_{j} \textbf{n}_{j}^{\rm H} \right\rbrace\,{=}\,\sigma^2\textbf{I}_{M\tau}$ and $\textbf{S}^H\textbf{S}\,{=}\,\tau \textbf{I}_M$. Since the noise and the channel vectors are uncorrelated and zero-mean, the expectations in \eqref{eqn:app:cov_est_3} cancel, and we have
\begin{align}
\hat{\textbf{R}}_{ij} &= \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \tilde{\textbf{R}}_{ij} \mathbb{E}\left\lbrace \textbf{h}_{kj}\textbf{h}_{kj}^{\rm H} \right\rbrace \tilde{\textbf{R}}_{ij}^{\rm H} \nonumber \\
&+ \tau^2 \sum\limits_{k=1}^{N_\mathrm{L}} \sum \limits_{\substack{\ell=1 \\ \ell \neq k}}^{N_\mathrm{L}} \tilde{\textbf{R}}_{ij} \mathbb{E}\left\lbrace \textbf{h}_{kj}\textbf{h}_{\ell j}^{\rm H} \right\rbrace \tilde{\textbf{R}}_{ij}^{\rm H} + \tau \sigma^2 \tilde{\textbf{R}}_{ij} \tilde{\textbf{R}}_{ij}^{\rm H} \,, \label{eqn:app:cov_est_4}
\end{align}
where the second term in \eqref{eqn:app:cov_est_4} vanishes since $\mathbb{E}\left\lbrace\textbf{h}_{kj}\textbf{h}_{\ell j}^{\rm H}\right\rbrace\,{=}\,\textbf{0}_M$ for $k\,{\neq}\,\ell$. Then $\hat{\textbf{R}}_{ij} $ becomes
\begin{align}
\hat{\textbf{R}}_{ij} &= \sum \limits_{k=1}^{N_\mathrm{L}} \tau^2 \tilde{\textbf{R}}_{ij} \textbf{R}_{kj} \tilde{\textbf{R}}_{ij}^{\rm H} + \tau \sigma^2 \tilde{\textbf{R}}_{ij} \tilde{\textbf{R}}_{ij}^{\rm H} \,, \nonumber \\
&= \tau \tilde{\textbf{R}}_{ij} \left( \tau \sum \limits_{k=1}^{N_\mathrm{L}} \textbf{R}_{kj} + \sigma^2 \textbf{I}_{M} \right) \tilde{\textbf{R}}_{ij}^{\rm H} = \tau \textbf{R}_{ij} \tilde{\textbf{R}}_{ij}^{\rm H} \,, \label{eqn:app:cov_est_6}
\end{align} \normalsize
where we employ hermitian symmetry of covariance matrices $\textbf{R}_{ij} $ and $\hat{\textbf{R}}_{ij}$ to obtain \eqref{eqn:cov_matrix_mmse}. \hfill\IEEEQEDhere
\section{First and Second Order Moment Derivation}\label{app:achievable_rates}
In this section, we derive the first and second order moments $\mathbb{E}\left\lbrace\textbf{h}_{jj}^{\rm H}\textbf{w}_j \right\rbrace$ and $\mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace$, respectively, for the EBF precoding given in \eqref{eqn:ebf}. Before the analysis, we define the following property which is used throughout this section while evaluating the mean of the quadratic and the double-quadratic forms involving random vectors.
\begin{lemma}\label{lem:quadratic}
Assume that $\left\lbrace \textbf{u}_i \right\rbrace_{i=1}^{4}$ be a set of zero-mean random vectors of arbitrary sizes where each of them may be individually correlated with the arbitrary covariance matrices $\left\lbrace \textbf{C}_i \right\rbrace_{i=1}^{4}$. For the given coefficient matrices $\textbf{A}$ and $\textbf{B}$ of the appropriate sizes and with arbitrary entries, the quadratic form $\textbf{u}_1^{\rm H} \textbf{A} \textbf{u}_2$ and the double-quadratic form $\textbf{u}_1^{\rm H} \textbf{A} \textbf{u}_2 \textbf{u}_3^{\rm H} \textbf{B} \textbf{u}_4$ are zero-mean if at least one of these random vectors are uncorrelated with the others.
\end{lemma}
\begin{IEEEproof}
Assuming that $\textbf{u}_1$ is uncorrelated with the others, without any loss of generality, regardless of whether $\left\lbrace \textbf{u}_i \right\rbrace_{i=2}^{4}$ are correlated with each other or not, we have
\begin{align*}
&\mathbb{E}\left\lbrace \textbf{u}_1^{\rm H} \textbf{A} \textbf{u}_2 \right\rbrace = \sum_{m}\sum_{n} \textbf{A}(m,n) \mathbb{E}\left\lbrace u_{1,m}^{\rm *} \right\rbrace \mathbb{E}\left\lbrace u_{2,n} \right\rbrace = 0~,\\
&\mathbb{E}\left\lbrace \textbf{u}_1^{\rm H} \textbf{A} \textbf{u}_2 \textbf{u}_3^{\rm H} \textbf{B} \textbf{u}_4 \right\rbrace = \sum_{m}\sum_{n}\sum_{k}\sum_{\ell} \textbf{A}(m,n)\textbf{B}(k,\ell) \times \\
& \hspace{10em} \mathbb{E}\left\lbrace u_{1,m}^{\rm *} \right\rbrace \mathbb{E}\left\lbrace u_{2,n} u_{3,k}^{\rm *} u_{4,\ell} \right\rbrace = 0~,
\end{align*} where $u_{i,m}$ denotes the $m$th entry of $\textbf{u}_{i}$.
\end{IEEEproof}
\subsection{First Order Moment}\label{app:1st_moment}
Employing the UL signal model in \eqref{eqn:UL_trans_vector} and the channel estimate in \eqref{eqn:mmse_chan_est}, the first order moment of the desired signal is given as follows
\begin{align}
&\mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace = \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj}\textbf{S}^{\rm H} \left( \textbf{S} \sum \limits_{i=1}^{N_\mathrm{L}} \textbf{h}_{ij} +\textbf{n}_{j} \right) \right\rbrace \,,
\nonumber \\
&= \tau \sum \limits_{i=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj} \textbf{h}_{ij} \right\rbrace + \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj} \textbf{S}^{\rm H} \textbf{n}_{j} \right\rbrace \,,\label{eqn:app:1st_moment_1}
\end{align}
where the last line employs $\textbf{S}^H\textbf{S}\,{=}\,\tau \textbf{I}_M$. Since $\textbf{h}_{jj}$ and $\textbf{n}_{j}$ are zero-mean and uncorrelated, the second expectation in \eqref{eqn:app:1st_moment_1} vanishes as per Lemma~\ref{lem:quadratic}. The desired expectation becomes
\begin{align}
\mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \textbf{w}_j \right\rbrace &= \tau \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj} \textbf{h}_{jj} \right\rbrace {+} \tau \sum \limits_{\substack{i=1\\i \neq j}}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj} \textbf{h}_{ij} \right\rbrace ,\label{eqn:app:1st_moment_2}\\
&= \tau \mathbb{E} \left\lbrace \textbf{h}_{jj}^{\rm H} \tilde{\textbf{R}}_{jj} \textbf{h}_{jj} \right\rbrace\,, \label{eqn:app:1st_moment_3}
\end{align}
where the second expectation in \eqref{eqn:app:1st_moment_2} is similarly zero as per Lemma~\ref{lem:quadratic} since the channels of $i$th and $j$th UEs to the $j$th BS, denoted by $\textbf{h}_{ij}$ and $\textbf{h}_{jj}$, respectively, are uncorrelated from each other, and zero-mean by definition. Finally, representing \eqref{eqn:app:1st_moment_3} by using the trace operator, employing the Hermitian symmetry of the covariance matrix, and incorporating the covariance matrix of the channel estimate in \eqref{eqn:cov_matrix_mmse} yield the desired expression given in \eqref{eqn:1st_moment}.
\subsection{Second Order Moment}\label{app:2nd_moment}
Employing \eqref{eqn:UL_trans_vector} and \eqref{eqn:mmse_chan_est}, as in Appendix~\ref{app:1st_moment}, the second order moment of the desired signal is given as follows
\small \begin{align}
&\mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace = \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \hat{\textbf{h}}_{ii} \hat{\textbf{h}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace \,, \\
&= \mathbb{E} \Bigg\{ \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii}\textbf{S}^{\rm H} \left( \textbf{S} \sum \limits_{k=1}^{N_\mathrm{L}} \textbf{h}_{ki} +\textbf{n}_{i} \right) \left( \textbf{S} \sum \limits_{\ell=1}^{N_\mathrm{L}} \textbf{h}_{\ell i} +\textbf{n}_{i} \right)^{\rm H} \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \Bigg\} \,, \\
&= \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \sum \limits_{\ell=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{\ell i}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace + \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{S}^{\rm H} \textbf{n}_{i} \textbf{n}_{i}^{\rm H} \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace \\
&+ \tau \sum \limits_{k=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{S}^{\rm H} \textbf{n}_{i} \textbf{h}_{ki}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace + \tau \sum \limits_{k=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{n}_{i}^{\rm H} \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace \,, \label{eqn:app:2nd_moment_1}
\end{align} \normalsize
where the expectations in \eqref{eqn:app:2nd_moment_1} vanishes in accordance with Lemma~\ref{lem:quadratic} since $\textbf{n}_{i}$ is uncorrelated with $\textbf{h}_{ji}$ and $\textbf{h}_{ki}$, and hence
\begin{align}\label{eqn:app:2nd_moment_2}
\mathbb{E} \left\lbrace \left| \textbf{h}_{ji}^{\rm H} \textbf{w}_i \right|^2 \right\rbrace &= \underbrace{ \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \sum\limits_{\ell=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{\ell i}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace}_{\textmd{${\rm E}_1$}}
\\ \nonumber
& \hspace{15 mm} + \underbrace{ \vphantom{ \sum \limits_{k=1}^{N_\mathrm{L}} } \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{S}^{\rm H} \textbf{n}_{i} \textbf{n}_{i}^{\rm H} \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace }_{\textmd{${\rm E}_2$}}~.
\end{align}
In the following, we will elaborate the two expectations, ${\rm E}_1$ and ${\rm E}_2$, in \eqref{eqn:app:2nd_moment_2}, separately. We start with ${\rm E}_1$ as follows
\begin{align}\label{eqn:app:2nd_moment_e1_1}
{\rm E}_1 &= \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{ki}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace
\\ \nonumber
& \hspace{20 mm} + \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \sum \limits_{\substack{\ell=1\\ \ell \neq k}}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{\ell i}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace~,
\end{align}
where $\textbf{h}_{ki}$ and $\textbf{h}_{\ell i}$ in the second expectation are obviously uncorrelated as $k \neq \ell$. Note that, $\textbf{h}_{ki}$ is uncorrelated with $\textbf{h}_{\ell i}$ and $\textbf{h}_{ji}$ when $j=\ell$, and $\textbf{h}_{\ell i}$ is uncorrelated with $\textbf{h}_{ki}$ and $\textbf{h}_{ji}$ when $j=k$, and finally all $\textbf{h}_{ki}$, $\textbf{h}_{\ell i}$ and $\textbf{h}_{ji}$ are uncorrelated when $j\neq\{k,\ell\}$. As a result, in any case, we have at least one zero-mean vector uncorrelated with the others, and the second expectation in \eqref{eqn:app:2nd_moment_e1_1} is therefore zero in accordance with Lemma~\ref{lem:quadratic}. As a result, ${\rm E}_1$ in \eqref{eqn:app:2nd_moment_e1_1} becomes
\vspace{-2.1em}
\small \begin{align}
&{\rm E}_1= \tau^2 \sum \limits_{k=1}^{N_\mathrm{L}} \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{ki}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace \nonumber \,,\\
&= \tau^2 \underbrace{\mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ji} \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace}_{{\rm E}_{11}} + \tau^2 \sum \limits_{\substack{k=1\\ k \neq j}}^{N_\mathrm{L}} \underbrace{ \mathbb{E} \left\lbrace \textbf{h}_{ji}^{\rm H} \tilde{\textbf{R}}_{ii} \textbf{h}_{ki} \textbf{h}_{ki}^{\rm H} \tilde{\textbf{R}}_{ii}^{\rm H} \textbf{h}_{ji} \right\rbrace}_{{\rm E}_{12}}\,, \label{eqn:app:2nd_moment_e1_2}
\end{align} \normalsize
and the first expectation ${\rm E}_{11}$ can be expressed in weighted sum of scalars as follows
\begin{align} \label{eqn:app:2nd_moment_e1_3}
{\rm E}_{11}\, &{=}\,\sum\limits_{m=1}^{M} \sum\limits_{n=1}^{M} \sum\limits_{{m'}=1}^{M} \sum\limits_{{n'}=1}^{M} \tilde{\textbf{R}}_{ii}(m,n) \, \tilde{\textbf{R}}_{ii}^{\rm H}({m'},{n'}) \, \times
\\ \nonumber
& \hspace{10em} \underbrace{\mathbb{E} \left\lbrace h_{ji,m}^{\ast}h_{ji,n} h_{ji,{m'}}^{\ast} h_{ji,{n'}} \right\rbrace}_{{\rm E}_{\phi}(m,n,{m'},{n'})}~,
\end{align} where $h_{ji,m}$ is the $m$th element of the channel vector $\textbf{h}_{ji}$, and is given by employing \eqref{eqn:channel_defn} and \eqref{eqn:steering_vector} as follows
\small \begin{align} \label{eqn:channel_m}
h_{ji,m} = \dfrac{1}{\sqrt{N_\mathrm{P}}}\sum \limits_{p=1}^{N_\mathrm{P}} \alpha_{ji,p} \exp\left\lbrace {-}j 2\pi \frac{D}{\lambda}(m-1)\cos\left(\phi_{ji,p}\right) \right\rbrace~.
\end{align} \normalsize By \eqref{eqn:channel_m}, the expectation at the right-hand side of \eqref{eqn:app:2nd_moment_e1_3} can be further elaborated as follows
\begin{align}\label{eqn:app:2nd_moment_e1_4}
{\rm E}_{\phi} &(m,n,{m'},{n'})\,{=}\,\dfrac{1}{{N_\mathrm{P}}^2} \sum \limits_{p_1{=}1}^{N_\mathrm{P}} \sum \limits_{p_2{=}1}^{N_\mathrm{P}} \sum \limits_{p_3{=}1}^{N_\mathrm{P}} \sum \limits_{p_4{=}1}^{N_\mathrm{P}} \mathbb{E}_{\alpha} \times
\\ \nonumber
& \hspace{5 mm} \mathbb{E} \left\lbrace \exp \left( {-}j 2\pi \frac{D}{\lambda} \sum\limits_{\nu{=}1}^{4}({-}1)^{\nu}\left(u_{\nu}{-}1\right)\cos\left(\phi_{ji,p_{\nu}}\right) \right) \right\rbrace ,
\end{align}
$\left\lbrace u_{\nu}\right\rbrace_{\nu{=}1}^{4}{=}\{m,n,{m'},{n'}\}$, $\mathbb{E}_{\alpha}\,{=}\,\mathbb{E} \left\lbrace \alpha_{ji, p_1}^{\ast} \alpha_{ji, p_2} \alpha_{ji, p_3}^{\ast} \alpha_{ji, p_4} \right\rbrace$. Note that, $\mathbb{E}_{\alpha}$ is nonzero only when 1) $p_1\,{=}\,p_2\,{=}\,p_3\,{=}\,p_4$, 2) $p_1\,{=}\,p_2, p_3\,{=}\,p_4$ (with $p_1\,{\neq}\,p_3)$, or 3) $p_1\,{=}\,p_4, p_2\,{=}\,p_3$ (with $p_1\,{\neq}\,p_2)$, and zero otherwise, since $\alpha_{ji,p}$ is zero-mean and uncorrelated over the path index $p$. Next, we analyze these three conditions to have a closed-form expression for \eqref{eqn:app:2nd_moment_e1_4}.
\begin{remark}
Note that, the other possibilities for the path indices $\left\lbrace p_{\nu} \right\rbrace_{\nu{=}1}^{4}$ for which $\mathbb{E}_{\alpha}$ is zero, consist of the cases where i) none of the path indices equal to the other, ii) one of the path indices is not equal to all the others, and iii) the pairwise equality with $p_1\,{=}\,p_3, p_2\,{=}\,p_4$. For the cases i) and ii), the expectation $\mathbb{E}_{\alpha}$ involves a term $\left[\mathbb{E}\left\lbrace \alpha_{ji, p} \right\rbrace\right]^\kappa$ with $\kappa\,{\geq}\,1$ which is zero since $\alpha_{ji, p}$ is zero-mean, and hence yields $\mathbb{E}_{\alpha}\,{=}\,0$. The case iii) yields $\mathbb{E}_{\alpha}\,{=}\,\left|\mathbb{E} \left\lbrace \alpha_{ji,p}^2\right\rbrace \right|^2$ which can easily be shown to be zero as $\alpha_{ji, p}$ has uncorrelated real and imaginary parts which are zero-mean.
\end{remark}
\begin{case}\label{case:case_1}
Assuming $p_1\,{=}\,p_2\,{=}\,p_3\,{=}\,p_4$, the desired expectation ${\rm E}_{\phi}(m,n,{m'},{n'})$ in \eqref{eqn:app:2nd_moment_e1_4} becomes
\begin{align}
&{\rm E}_{\phi}(m,n,{m'},{n'}) = \dfrac{1}{{N_\mathrm{P}}^2} \sum\limits_{p=1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \left| \alpha_{ji,p} \right|^4 \right\rbrace \times
\nonumber \\
& \hspace{4em} \mathbb{E} \left\lbrace \exp\left( {-}j 2\pi \frac{D}{\lambda} \left( n{-}m{+}{n'}{-}{m'} \right) \cos\left(\phi_{ji,p}\right) \right) \right\rbrace \,,\nonumber \\
&\stackrel{\text{(a)}}{=} \dfrac{1}{{N_\mathrm{P}}} \mathbb{E} \left\lbrace \left| \alpha_{ji} \right|^4 \right\rbrace \times
\nonumber \\
& \hspace{4em} \mathbb{E} \left\lbrace \exp\left( {-}j 2\pi \frac{D}{\lambda} \left( n{-}m{+}{n'}{-}{m'} \right) \cos\left(\phi_{ji}\right) \right) \right\rbrace
\,,\nonumber \\
&\stackrel{\text{(b)}}{=}\dfrac{ 2\beta_{ji}^2 }{{N_\mathrm{P}}} {\rm E}_{ji}( n{-}m{+}{n'}{-}{m'} )\,, \label{eqn:app:2nd_moment_e1_5}
\end{align}
where $\text{(a)}$ follows from the uncorrelatedness of $\alpha _{ji}$ and $\phi _{ji}$ over the path index $p$, and $\text{(b)}$ employs the identity $\mathbb{E} \left\lbrace \left| \alpha_{ji} \right|^4 \right\rbrace\,{=}\,2\beta_{ji}^2$ and the definition ${\rm E}_{ji}(m)\,{=}\,\mathbb{E} \left\lbrace \exp\left( {-}j 2\pi \frac{D}{\lambda} \left(m\right) \cos\left(\phi_{ji}\right)\right) \right\rbrace$ which is equal to $\mathbf{R}_{ji}^{\phi}(n{+}m,n)$ for any $n\,{\leq}\,M{-}m$ in \eqref{eqn:app:cov_phi_uni}. Note that $\phi_{ij}$ do not have identical distributions with the same parameters over the various subscripts representing the UE and the BS of interest, and we therefore keep the indices in ${\rm E}_{ji}$.
\end{case}
\begin{case}\label{case:case_2}
Assuming $p_1 = p_2, p_3 = p_4$ and $p_1\,{\neq}\,p_3$, the desired expectation ${\rm E}_{\phi}(m,n,{m'},{n'})$ can be given as in \eqref{eqn:app:2nd_moment_e1_6},
where the last line employs the fact that $\phi_{ji,p_1}$ and $\phi_{ji,p_3}$ are uncorrelated for $p_1\,{\neq}\,p_3$.
\begin{figure*}[!htb]
\setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{41}
\begin{align}
{\rm E}_{\phi}(m,n,{m'},{n'}) &= \dfrac{\beta_{ji}^2}{{N_\mathrm{P}}^2} \sum \limits_{p_1=1}^{N_\mathrm{P}} \sum \limits_{p_3 \neq p_1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \exp\left( {-}j 2\pi \frac{D}{\lambda} \Big[ (n{-}m)\cos\left(\phi_{ji,p_1}\right){+}({n'}{-}{m'})\cos \left(\phi_{ji,p_3}\right)\Big] \right) \right\rbrace \,, \nonumber \\
&= \dfrac{\beta_{ji}^2}{{N_\mathrm{P}}} (N_\mathrm{P}{-}1) \,{\rm E}_{ji}(n{-}m)\,{\rm E}_{ji}({n'}{-}{m'})\,, \label{eqn:app:2nd_moment_e1_6}
\end{align}
\setcounter{equation}{42}
\hrulefill
\end{figure*}
\end{case}
\begin{case}\label{case:case_3}
Assuming $p_1 = p_4, p_2 = p_3$ and $p_1\,{\neq}\,p_2$, the desired expectation ${\rm E}_{\phi}(m,n,{m'},{n'})$ is obtained by following the derivation steps of Case~\ref{case:case_2} which yields \vspace{-1em}
\small
\begin{align} \label{eqn:app:2nd_moment_e1_7}
{\rm E}_{\phi}(m,n,{m'},{n'}) = \dfrac{\beta_{ji}^2}{{N_\mathrm{P}}} (N_\mathrm{P}{-}1){\rm E}_{ji}({n'}{-}m) {\rm E}_{ji}(n{-}{m'}).
\end{align}
\end{case}\normalsize
Incorporating \eqref{eqn:app:2nd_moment_e1_5}, \eqref{eqn:app:2nd_moment_e1_6}, and \eqref{eqn:app:2nd_moment_e1_7} yields the desired expression of ${\rm E}_{\phi}(m,n,{m'},{n'})$ in \eqref{eqn:2nd_moment_phi}, and ${\rm E}_{11}$ can be computed by employing \eqref{eqn:2nd_moment_phi} in \eqref{eqn:app:2nd_moment_e1_3}.
The second expectation ${\rm E}_{12}$ in \eqref{eqn:app:2nd_moment_e1_2} can be expressed as a weighted sum of scalars as follows
\begin{align}\label{eqn:app:2nd_moment_e1_8}
{\rm E}_{12} & = \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{M} \sum\limits_{m'=1}^{M} \sum\limits_{n'=1}^{M} \tilde{\textbf{R}}_{ii}(m,n) \tilde{\textbf{R}}_{ii}^{\rm H}(m',n') \times
\nonumber \\
& \hspace{12em} \mathbb{E}\left\lbrace h_{ji,n'} h_{ji,m}^{\ast} h_{ki,n} h_{ki,m'}^{\ast} \right\rbrace \,,\nonumber \\
&= \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{M} \sum\limits_{m'=1}^{M} \sum\limits_{n'=1}^{M} \tilde{\textbf{R}}_{ii}(m,n) \tilde{\textbf{R}}_{ii}^{\rm H}(m',n') \times
\nonumber \\
& \hspace{12em} \textbf{R}_{ji}(n',m) \textbf{R}_{ki}(n,m')\,,
\end{align}
where the last line follows from the fact that $k{\neq}j$ as imposed by the summation in \eqref{eqn:app:2nd_moment_e1_2}. Employing \eqref{eqn:app:2nd_moment_e1_3} and \eqref{eqn:app:2nd_moment_e1_8}, we obtain ${\rm E}_{1}$ given in \eqref{eqn:app:2nd_moment_e1_2} as follows
\begin{align}\label{eqn:app:2nd_moment_e1}
{\rm E}_{1} &= \tau^2 \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{M} \sum\limits_{m'=1}^{M} \sum\limits_{n'=1}^{M} \! \tilde{\textbf{R}}_{ii}(m,n) \, \tilde{\textbf{R}}_{ii}^{\rm H}(m',n') \times
\nonumber \\
&\left[ {\rm E}_{\phi}(m,n,m',n') \vphantom{\sum\limits_{\substack{k{=}1; \, k{\neq}j}}^{N_\mathrm{L}} } + \!\!\!\sum\limits_{\substack{k{=}1; \, k{\neq}j}}^{N_\mathrm{L}} \!\!\!\!\textbf{R}_{ji}(n',m) \textbf{R}_{ki}(n,m') \right]\,,
\end{align}
which can be computed by means of ${\rm E}_{\phi}(m,n,m',n')$ given in \eqref{eqn:2nd_moment_phi}. Finally, we consider the expectation ${\rm E}_{2}$ in \eqref{eqn:app:2nd_moment_2}. Defining $\textbf{v}\,{=}\,\textbf{S}\,\tilde{\textbf{R}}_{ii}^{\rm H}\textbf{h}_{ji}$ and $\textbf{C}^{\rm N}\,{=}\,\mathbb{E}\left\lbrace \textbf{n}_{i} \textbf{n}_{i}^{\rm H} \right\rbrace$, ${\rm E}_{2}$ is given as
\begin{align}\label{eqn:app:2nd_moment_e2_1}
{\rm E}_{2} &= \mathbb{E} \left\lbrace \textbf{v}^{\rm H} \textbf{n}_{i} \textbf{n}_{i}^{\rm H} \textbf{v} \right\rbrace
= \sum\limits_{m=1}^{M\tau} \sum\limits_{n=1}^{M\tau} \textbf{C}^{\rm N}(m,n) \mathbb{E}\left\lbrace v_m^{\ast}v_n \right\rbrace
\nonumber \\
&= \sigma^2\sum\limits_{m=1}^{M\tau} \mathbb{E}\left\lbrace \left| v_m \right|^2 \right\rbrace ,
\end{align}
where we employ $\textbf{C}^{\rm N}\,{=}\,\sigma^2 \textbf{I}_{M\tau}$, \color{black} and substituting $\textbf{v}\,{=}\,\textbf{S}\,\tilde{\textbf{R}}_{ii}^{\rm H}\textbf{h}_{ji}$ back in \eqref{eqn:app:2nd_moment_e2_1} yields
\begin{align}\label{eqn:app:2nd_moment_e2_2}
{\rm E}_{2} &= \sigma^2 {\rm tr} \left\lbrace \textbf{S} \tilde{\textbf{R}}_{ii}^{\rm H}\textbf{R}_{ji} \tilde{\textbf{R}}_{ii} \textbf{S}^{\rm H} \right\rbrace .
\end{align}
As a result, substituting \eqref{eqn:app:2nd_moment_e1} and \eqref{eqn:app:2nd_moment_e2_2} in \eqref{eqn:app:2nd_moment_2} yields the desired second order moment in \eqref{eqn:2nd_moment}.
\section{Measure of Channel Hardening}\label{app:harden_measure}
The channel hardening measure $\mathcal{M}_{ij}$ is defined in~\cite{Larsson17ChanHard} as
\begin{align} \label{eqn:app:measure_defn}
\mathcal{M}_{ij} = \frac{{\rm Var} \left\lbrace \left\| \textbf{h}_{ij} \right\|^2 \right\rbrace }{\left( \mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^2 \right\rbrace \right)^2 } = \frac{\mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^4 \right\rbrace }{ \left( \mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^2 \right\rbrace \right)^2 } - 1\,,
\end{align}
with $\mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^2 \right\rbrace\,{=}\,\textrm{tr} \lbrace \mathbf{R}_{ij} \rbrace\,{=}\,\beta_{ij} M$ second order moment, and the fourth order moment $\mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^4 \right\rbrace$ given as
\begin{align*}
\mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^4 \right\rbrace &= \frac{1}{{N_\mathrm{P}}^2} \sum\limits_{p_1=1}^{N_\mathrm{P}} \sum\limits_{p_2=1}^{N_\mathrm{P}} \sum\limits_{p_3=1}^{N_\mathrm{P}} \sum \limits_{p_4=1}^{N_\mathrm{P}} \mathbb{E}_{\alpha} \times
\nonumber \\
&\mathbb{E} \left\lbrace \textbf{a}^{\rm H}\!\left( \phi_{ij, p_1}\right) \textbf{a}\left( \phi_{ij, p_2}\right) \textbf{a}^{\rm H}\!\left( \phi_{ij, p_3}\right) \textbf{a}\left( \phi_{ij, p_4}\right) \right\rbrace ,
\end{align*}
with $\mathbb{E}_{\alpha}\,{=}\,\mathbb{E} \left\lbrace \alpha_{ij, p_1}^{\ast} \alpha_{ij, p_2} \alpha_{ij, p_3}^{\ast} \alpha_{ij, p_4} \right\rbrace$. Similar to the discussion for \eqref{eqn:app:2nd_moment_e1_4}, $\mathbb{E}_{\alpha}$ is nonzero only when 1) $p_1\,{=}\,p_2\,{=}\,p_3\,{=}\,p_4$, 2) $p_1\,{=}\,p_2$, $p_3\,{=}\,p_4$, $p_1\,{\neq}\,p_3$ or 3) $p_1\,{=}\,p_4$, $p_2\,{=}\,p_3$, $p_1\,{\neq}\,p_2$ and hence \vspace{-2em}
\small \begin{align*}
&\mathbb{E} \left\lbrace \left\| \textbf{h}_{ij} \right\|^4 \right\rbrace = \frac{ \left(1 + N_\mathrm{P} \right) \beta_{ij}^2 M^2}{{N_\mathrm{P}}}
\nonumber \\
& + \frac{\beta_{ij}^2}{{N_\mathrm{P}}^2} \sum \limits_{p_1=1}^{N_\mathrm{P}} \sum \limits_{p_2{\neq}p_1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \textbf{a}^{\rm H}\!\left( \phi_{ij, p_1}\right) \textbf{a}\left( \phi_{ij, p_2}\right) \textbf{a}^{\rm H}\!\left( \phi_{ij, p_2}\right) \textbf{a}\left( \phi_{ij, p_1}\right) \right\rbrace \,,
\end{align*} \normalsize
where $\mathbb{E} \left\lbrace \left| \alpha_{ij} \right|^4 \right\rbrace\,{=}\,2\beta_{ij}^2$ and $ \textbf{a}^{\rm H}\!\left( \phi_{ij, p}\right) \textbf{a}\left( \phi_{ij, p}\right)\,{=}\,M$. Substituting the second and the fourth order moments in \eqref{eqn:app:measure_defn}, the channel hardening measure can be given as in \eqref{eqn:app:measure_hardening}.
\begin{figure*}
\setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{48}
\begin{align} \label{eqn:app:measure_hardening}
\mathcal{M}_{ij} &= \frac{1}{N_\mathrm{P}} {+} \frac{1}{N_\mathrm{P}^2 M^2} \sum\limits_{p_1=1}^{N_\mathrm{P}} \sum \limits_{p_2{\neq}p_1}^{N_\mathrm{P}} \mathbb{E} \left\lbrace \textbf{a}^{\rm H}\!\left( \phi_{ij, p_1}\right) \textbf{a}\left( \phi_{ij, p_2}\right)\textbf{a}^{\rm H}\!\left( \phi_{ij, p_2}\right)\textbf{a}\left( \phi_{ij, p_1}\right) \right\rbrace \,, \nonumber \\
&= \frac{1}{N_\mathrm{P}} {+} \frac{1}{N_\mathrm{P}^2 M^2} \sum \limits_{p_1=1}^{N_\mathrm{P}} \sum \limits_{p_2{\neq}p_1}^{N_\mathrm{P}} \sum \limits_{n=1}^{M} \sum \limits_{m=1}^{M} \mathbb{E} \left\lbrace \exp\left( {-}j 2\pi \frac{D}{\lambda} \Big[ (n{-}m) \left( \cos\left(\phi_{ij,p_2}\right){-}\cos\left(\phi_{ij,p_1}\right) \right) \Big] \right) \right\rbrace
\end{align}
\setcounter{equation}{49}
\hrulefill
\end{figure*}
Taking the terms for the equality of $m = n$ out of the summation, and employing the angular covariance matrix $\mathbf{R}_{ij}^{\phi}$ given in \eqref{eqn:app:cov_phi_element}, we end up with \vspace{-1em}
\small
\begin{align} \label{eqn:app:measure_last}
&\mathcal{M}_{ij} = \frac{1}{N_\mathrm{P}} {+} \frac{N_\mathrm{P}-1}{N_\mathrm{P} M}
\nonumber \\
&\quad\; {+} \frac{1}{N_\mathrm{P}^2 M^2} \sum \limits_{p_1=1}^{N_\mathrm{P}} \sum \limits_{p_2{\neq}p_1}^{N_\mathrm{P}} \sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n)\,, \nonumber\\
&= \frac{1}{N_\mathrm{P}} {+} \frac{N_\mathrm{P}-1}{N_\mathrm{P} M}
\nonumber\\
&{+} \left( \frac{1}{N_\mathrm{P}^2 M^2} \left( N_\mathrm{P}-1 \right) N_\mathrm{P} \sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n)\,\right) , \\
&= \frac{1}{N_\mathrm{P}} {+} \frac{N_\mathrm{P}-1}{N_\mathrm{P} M} {+} \frac{N_\mathrm{P}-1}{N_\mathrm{P} M^2} \sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n) \,, \end{align} \normalsize
where (50) is due to the fact that $\mathbf{R}_{ij}^{\phi}(n,m)$ does not dependent on the path index. Let us consider the term $\sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n)$. We can represent it as,
\small
\begin{align} \label{eqn:hardening_7}
&\sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n) = \sum \limits_{n=1}^{M} \sum \limits_{m{=}1}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n)
\nonumber \\
&\hspace{15em} - \sum \limits_{m=1}^{M} \left| \mathbf{R}_{ij}^{\phi}(m,m) \right|^2
\nonumber \\
&\hspace{3em}= \sum \limits_{n=1}^{M} \sum \limits_{m{=}1}^{M} \left|\mathbf{R}_{ij}^{\phi}(m,n) \right|^2 - \sum \limits_{m=1}^{M} \left| \mathbf{R}_{ij}^{\phi}(m,m) \right|^2.
\end{align} \normalsize Due to the conjugate symmetry of the covariance matrix, $\left| \mathbf{R}_{ij}^{\phi}(m,n) \right|^2 = \left| \mathbf{R}_{ij}^{\phi}(n,m) \right|^2$. By making use of this fact \eqref{eqn:hardening_7} can be simplified and given as, \begin{align} \label{eqn:hardening_8}
\sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n) =\sum \limits_{m=1}^{M-1} \sum \limits_{n{=}1}^{M-m} 2 \left|\mathbf{R}_{ij}^{\phi}(m+n,n) \right|^2.
\end{align} Employing ${\rm E}_{ij}(m)\,{=}\,\mathbf{R}_{ij}^{\phi}(n{+}m,n)$ for any $n\,{\leq}\,M{-}m$, \eqref{eqn:hardening_8} can be represented as \begin{align} \label{eqn:hardening_9}
\sum \limits_{n=1}^{M} \sum \limits_{m{\neq}n}^{M} \mathbf{R}_{ij}^{\phi}(n,m) \mathbf{R}_{ij}^{\phi}(m,n) = \sum \limits_{m=1}^{M-1} 2 (M-m)\left|{\rm E}_{ij}(m) \right|^2.
\end{align} Using \eqref{eqn:hardening_9} in (51) we obtain the channel hardening measure in \eqref{eqn:harden_measure}. \color{black}
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,148
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Список 1954 год в спорте описывает спортивные события, произошедшие в 1954 году.
СССР
Чемпионат СССР по боксу 1954;
Чемпионат СССР по классической борьбе 1954;
Чемпионат СССР по самбо 1954;
Чемпионат СССР по баскетболу среди женщин 1954;
Чемпионат СССР по баскетболу среди мужчин 1954;
Чемпионат СССР по волейболу среди женщин 1954;
Чемпионат СССР по волейболу среди мужчин 1954;
Чемпионат СССР по лёгкой атлетике 1954;
Чемпионат СССР по русским шашкам 1954;
Чемпионат СССР по хоккею с мячом 1954;
Чемпионат СССР по шахматам 1954;
Футбол
Чемпионат СССР по футболу 1954;
Кубок СССР по футболу 1954;
Созданы клубы:
«Кайрат» (Алма-Ата);
«Наири» (Ереван);
Хоккей
Чемпионат СССР по хоккею с шайбой 1953/1954;
Чемпионат СССР по хоккею с шайбой 1954/1955;
Создан клуб «Прогресс» (Глазов);
Международные события
Чемпионат Европы по лёгкой атлетике 1954;
Чемпионат Европы по фигурному катанию 1954;
Чемпионат мира по баскетболу 1954;
Чемпионат мира по конькобежному спорту в классическом многоборье 1954;
Чемпионат мира по лыжным видам спорта 1954;
Чемпионат мира по спортивной гимнастике 1954;
Чемпионат мира по фигурному катанию 1954;
Чемпионат мира по футболу 1954;
Чемпионат мира по хоккею с шайбой 1954;
Шахматы
Матч за звание чемпиона мира по шахматам 1954;
Шахматная олимпиада 1954;
См. также
1954 год в спорте
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 500
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Delroy is a masculine Jamaican given name. Notable people with the first name include:
Delroy Allen (born 1954), retired Jamaican-American soccer goalkeeper
Delroy Cambridge (born 1949), Jamaican professional golfer
Delroy Chuck, Jamaican lawyer, journalist and politician
Delroy Clarke (born 1982), Canadian football cornerback
Delroy Denton, Jamaican-born illegal immigrant to Britain and convicted rapist and murderer
Delroy Edwards (1959–2005), Jamaican-born refugee, refused political asylum in UK, killed by a gang following his return
Delroy Edwards (musician) (born 1990), stage name of American electronic record producer Brandon Avery Perlman
Delroy Facey (born 1980), British-Grenadian professional footballer
Delroy Garrett, fictional superhero published by Marvel Comics
Delroy Grant (born 1957), Jamaican-born British convicted serial rapist
Delroy Leslie (born 1970), retired boxer from Jamaica
Delroy Lindo (born 1952), British-born Jamaican American actor
Delroy McLean, birth name of Bitty McLean
Delroy Parkes, British retired boxer
Delroy McQueen, former English weightlifter and powerlifter
Delroy Pearson (born 1970), British singer and a member of the pop group Five Star
Delroy Poyser (born 1962), retired long jumper from Jamaica
Delroy Wilson (1948–1995), Jamaican ska, rocksteady and reggae singer
Delroy Washington (1952–2020), Jamaican-British reggae singer
Delroy "Del" Bryan (born 1967), British former boxer
Delroy Taylor (born 1975), Jamaican former cricketer
Delroy Scott (1947–2018), Jamaican footballer
Delroy Slowley, Jamaican politician
Delroy Morgan (born 1967), Jamaican cricketer
Delroy Fraser, Guyanese professional footballer
See also
Tony Delroy, Australian radio presenter and host of Nightlife on ABC Local Radio
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,237
|
San Cristóbal de La Laguna je město na Kanárských ostrovech v severní části ostrova Tenerife. Je součástí metropolitní oblasti Santa Cruz de Tenerife.
Ve městě se nachází Universidad de La Laguna, kterou navštěvuje okolo třiceti tisíc studentů. Byla založena počátkem 18. století. Samotné San Cristóbal de La Laguna založil koncem 15. století Alonso Fernández de Lugo; v roce 1999 bylo město zapsáno na seznam světového dědictví UNESCO.
Ve městě se nachází katedrála Panny Marie Pomocné. Dalším z významných kostelů je královská svatyně Cristo de La Laguna, kde obraz ukřižovaného Ježíše Krista vysoce ctěný na Kanárských ostrovech. Nejstarší kostel ve městě je Iglesia de la Concepción (kostel Neposkvrněného početí Panny Marie) z roku 1511.
Ve městě jsou dva kláštery. Kromě kláštera svaté Kláry z Assisi je zde také klášter svaté Kateřiny Sienské, v němž se nachází neporušené tělo jeptišky Maríi de León Bello y Delgado, opředené pověstí o zázraku.
Významní rodáci
José de Anchieta (1534–1597), jezuitského mnich, a misionář v Brazílii
Amaro Pargo (1695–1747), španělský korzár
Odkazy
Reference
Externí odkazy
Obce v provincii Santa Cruz de Tenerife
Města ve Španělsku
Světové dědictví (Španělsko)
Turistika ve Španělsku
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,611
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Website electronic petitions the President of Ukraine.
Petitions statistics for a certain period of time (week, month, year).
Responsive web design in conjunction with the embedded functionality allows you to create electronic petitions and vote online, even for unexperienced users.
Website development for the company "Furniture technologies"
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{
"redpajama_set_name": "RedPajamaC4"
}
| 3,689
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Q: Need help with build error Unable to copy file "..\Soln\Project1\Bin\Telerik.Web.UI.dll" to "bin\Debug\Telerik.Web.UI.dll". Access to the path 'bin\Debug\Telerik.Web.UI.dll' is denied. Project2
I got two projects bound to one solution
When I build Project2, it gives me the error above. How do I fix this ? I need to add reference of Project2 in Project1. But now that Project2 just won't build. I am unable to do that.
A: What probably happens is that, while building one project, the builder tries to copy the referenced dll to the debug folder, while it is being locked by the other project (is the second project running ?)
Anyway, try to set the "Copy Local" Attribute to "false" for the Telerik dll you are referencing in both projects
A: I solved this problem by deleting the contentious files from bin folder and rebuilding the project.
but in some cases you should
Close Visual Studio, open it again and load the solution, Rebuild your solution. My problem occurred using TFS and VIsual Studio 2010.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,210
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Q: Expo + React Native version mismatch error ( JavaScript version: 0.55.4, Native version: 0.57.1 ) I found lots of threads about this problem, but non of them helped me out.
So sorry for writing another thread for the same problem.
I am new to react-native and frankly packange.json is really painful for me to match version of dependencies.
My package.json file is as follows.
And, error message is
console.error: "React Native version mismatch.
JavaScript version: 0.55.4
Native version: 0.57.1
....."
I tried delete cache and node_modules and reinstall those things using following command.
"watchman watch-del-all && rm -rf $TMPDIR/react-* && rm -rf $TMPDIR/haste-map-react-native-packager-* && rm -rf node_modules/&& npm install"
What I found from googling, in most cases, it was version mismatch between expo and react-native. But I believe I am using matched version of react-native and expo from expo website Here
{
"name": "empty-project-template",
"main": "node_modules/expo/AppEntry.js",
"private": true,
"scripts": {
"start": "expo start",
"android": "expo start --android",
"ios": "expo start --ios",
"eject": "expo eject"
},
"dependencies": {
"babel-core": "^7.0.0-bridge.0",
"expo": "^30.0.0",
"native-base": "^2.8.1",
"react": "16.3.1",
"react-native": "https://github.com/expo/react-native/archive/sdk-30.0.0.tar.gz",
"react-navigation": "^2.11.2"
},
"devDependencies": {
"@babel/core": "^7.1.6",
"regenerator-runtime": "^0.13.1"
}
}
A: run expo update. this will update all the dependencies in package .json to their latest most stable releases. this worked for me
A: It is better to check app.json file as well.
sdkVersion in app.json should be matched to your version.
A: Just check your package.json.
Your version of 'expo' and 'react-native' must match, and remove node_modules and package-lock.json
A: If you are using Expo,try to run expo update in your project directory
A: If using Expo CLI - try to run expo upgrade to upgrade the project packages and config (check that you do have the matching versions in package.json first)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: How can I find all "*.docx" files in subdirectories, where top-level directories are filtered to numbers only I'm doing a folder-crawler, finding all docx-files in a subdirectory.
I want only to look in some top-level folders (starting with numbers).
I have a large subdirectory structure with multiple docx-files of different kind and the provided code is very time consuming.
I'm using windows PowerShell without any overlays.
$files = Get-ChildItem -Path $source -Filter "*.docx" -Recurse:$true
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 291
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Q: Graphics on Android: path with smooth curves? I want to draw a chart for a function y=x^2 as follows:
but the curve is not smooth as it is a set of connected lines.
how can I make the curve smoother ?
thanks
A: You should use Path.quadTo with only one Path. If you are already doing this then I suggest increasing the number of points on the graph.
Move to the beginning of the Path:
Path.moveTo(x, y)
in the middle:
Path.quadTo(lastX, lastY, (x + lastX)/2, (y + lastY)/2)
and at the end:
Path.lineTo(x, y)
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{
"redpajama_set_name": "RedPajamaStackExchange"
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De Joodse wijk (Pools: Dzielnica Żydowska) is een historische wijk in de binnenstad van de Poolse stad Zamość. Vanaf 1588 tot de Tweede Wereldoorlog woonden veel Joden in deze wijk, gelegen in het noordoosten van de binnenstad. Sindsdien is de gemeenschap bijna volledig uit Zamość verdwenen. Zamość werd als perfecte renaissancestad in 1581 gesticht door Jan Zamoyski waarnaar de stad ook vernoemd is. De wijk beschikte over een eigen kehilla. De wijk werd begrensd door Rynek Solny aan de westkant en aan de oost- en noordkant door kazematten, aan de zuidkant gaat de wijk over in de Armeense Wijk. De historische kehilla is te vinden aan Rynek Solny. De straten Pereca en Zamenhofa vormen de kern van deze wijk, de straat Ormianska (Armeense straat in het Pools) wordt gedeeld met de historische Armeense wijk, in deze straat bevond zich eerder het Yeshiva van de Wijze Mannen. Deze yeshiva werd in de 18e eeuw opgericht door Rabbi Yaakov Isaak Hochgelentner waardoor het een belangrijk onderwijscentrum werd voor rabbijnen. Het centrum van de Joodse wijk en de gemeenschap was de Synagoge van Zamość. Deze synagoge is sinds 1945 de enige Sefardische synagoge van Polen doordat Lviv na de grenswijziging in Oekraïne ligt.
Alleen Sefardische Joden mochten zich sinds 1588 van Jan Zamoyki vestigen in Zamość, mede wegens hun belangrijke handelscontacten naar de Levant. De eerste Sefardische Joden kwamen uit Lviv, later volgden Sefardische Joden uit Italië en het Ottomaanse rijk. Vanaf 1623 volgden ook Sefardische Joden uit de Republiek en Vlaanderen. Het privilege van Jan Zamoyki gold alleen voor Sefardische Joden. De reden dat veel Sefardische Joden hieraan gehoor gaven kwam doordat Zamość aan de handelsroute Via Regia lag en er een (extra) handelsknooppunt kwam in Polen-Litouwen tussen Midden-, Oost-en Zuid-Europa, daarnaast met het Nabije Oosten en vanaf 1623 het Verre Oosten en West-Indië. Medio 18e eeuw werden de Sefardische Joden vervangen door Asjkenazische Joden, omdat de meeste Sefardische Joden verdwenen waren door oorlogen, vanaf deze tijd werd het een belangrijk haskala-bolwerk.
Afbeeldingen
Zamość
Jodendom in Polen
Joodse geschiedenis
Joodse geschiedenis van Polen
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"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/gatkforums.broadinstitute.org\/discussion\/3195\/read-groups-from-splitsamfile","text":"The current GATK version is 3.2-2\n\nHowdy, Stranger!\n\nIt looks like you're new here. If you want to get involved, click one of these buttons!\n\nBug Bulletin: The recent 3.2 release fixes many issues. If you run into a problem, please try the latest version before posting a bug report, as your problem may already have been solved.\n\nPosts: 57Member\n\nHi,\n\nIf I have a bam file with three different read groups, and use SplitSamFile to split it like so:\n\njava -Xmx2g -jar $GATKJAR -T SplitSamFile -I$INBAM -R $GENOME --outputRoot$PROJD\/\\$IND\/\n\n\nEach of the output bam files have all three read groups. Is that the intended behavior? I would like each file to have only it's own read group info in the heads. Sorry for the bash arguments in the code above, is makes in readable at least.\n\nThanks\n\nDaniel\n\nTagged:\n\nHi Daniel,\n\nFrom the code it looks like SplitSamFile is indeed not tasked with updating the headers. I'll check if that's really what we want and if not, we'll patch this up for the next version.\n\nGeraldine Van der Auwera, PhD","date":"2014-07-30 00:55:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.27713873982429504, \"perplexity\": 2956.5110911034662}, \"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-2014-23\/segments\/1406510268363.15\/warc\/CC-MAIN-20140728011748-00086-ip-10-146-231-18.ec2.internal.warc.gz\"}"}
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\section{Introduction}
\label{intro}
Quasiperiodic systems have attracted considerable attention
since the discovery of the quasicrystals~\cite{Shechtman}.
In the metallic alloys, the quasiperiodic structure disturbs
the formation of the Bloch wave,
which should lead to remarkable peculiarities of electronic and thermal conductivities.
Recently, interesting low temperature properties have been observed in the ytterbium alloys.
In the quasicrystal Au$_{51}$Al$_{34}$Yb$_{15}$, the quantum critical behavior was observed
in the susceptibility and specific heat, while the heavy fermion behavior was observed
in its approximants~\cite{deguchi2012,Matsukawa2014}.
Furthermore, superconductivity was reported in approximants of the related materials~\cite{deguchi2015},
which stimulates further investigations on electron correlations on the quasicrystals.
In these materials, $f$ electrons of Yb ions are considered to be in intermediate valence regime, and hence valence fluctuations should play a role for these exotic phenomena, in addition to the quasiperiodicity.
To clarify the nature of the critical phenomena and superconductivity, several theoretical studies have been done in the viewpoints of correlation effects within a Tsai-Type cluster~\cite{Watanabe2013,Watanabe2015}, disorder effect~\cite{Andrade2015,Otsuki2016}, and effect of quasiperiodic structure ~\cite{Takemori2015,Takemura2015,Shinzaki2016,Sakai2017}.
However, low-temperature properties in quasiperiodic systems with strong electron correlations and valence fluctuations remain unclear.
In this study, to address the many-body effect and valence fluctuation in quasiperiodic systems, we introduce a spin-1/2 Falicov-Kimball model (FKM)~\cite{Falicov1969,PhysRevB.57.11955,Freericks1998,Freericks_rev2003}
on a two dimensional Penrose lattice as a minimal model.
We analyze the finite temperature properties of this model
using the real-space dynamical mean-field theory (RDMFT)~\cite{Georges1996,pruschke1995,muller1989}.
We have confirmed the suppression of valence transitions by changing the $f$ electron level in contrast to the FKM on the Bethe lattice.
At high temperatures, the occupancy of localized $f$ electrons hardly depend on sites due to strong thermal fluctuations.
While decreasing temperature, one can clearly see the site dependence in the $f$ electron number, which is classified by its coordination number.
Further decrease of temperature gives rise to the $f$ electron distribution reflecting the wider range of the crystal structure beyond the neighboring one in the intermediate valence regime.
This structure disappears at the lowest temperature.
We also find that the magnetic susceptibility shows a peculiar temperature dependence in the competing region between the intermediate and commensurate valence regimes.
This paper is constructed as follows. In Sec.~\ref{sec:model-method}, we introduce the spin-1/2 FKM and RDMFT as the calculation method.
In Sec.~\ref{sec:distr-f-electr}, we present the numerical results for the localized $f$ electron density.
The temperature dependence of the magnetic susceptibility is shown in Sec.~\ref{sec:temp-depend-susc}.
Section~\ref{sec:summary} is devoted to the summary.
\section{Model and method}\label{sec:model-method}
We study the $S=1/2$ FKM on the Penrose lattice including $N_s=1591$ sites shown in Fig.~\ref{penrose}.
The Hamiltonian of this model~\cite{Falicov1969} is given by
\begin{align}
{\cal H}=-t\sum_{\means{ij}\sigma}
\left(c_{i\sigma}^\dagger c_{j\sigma}
+{\rm H.c.}\right)
+\varepsilon_f\sum_i n_i^f
+U'\sum_{i}n_i^c n_i^f
+U\sum_{i}n_{i\uparrow}^f n_{i\downarrow}^f,
\label{eq:1}
\end{align}
where $c_{i\sigma}$ is the annihilation operator of the conduction electron at site $i$ with spin $\sigma=\uparrow, \downarrow$ and $n_{i\sigma}^c=c_{i\sigma}^\dagger c_{i\sigma}$.
$n_{i\sigma}^f$ is the number operator of the $f$ electron at site $i$ with spin $\sigma$ and $n_i^{c(f)}=n_{i\uparrow}^{c(f)}+n_{i\downarrow}^{c(f)}$.
In Eq.~(\ref{eq:1}), the first term represents the hoppings of the conduction electrons with the transfer integral $t$ between nearest neighbor sites $\means{ij}$ on the Penrose lattice.
The second, third, and fourth terms represent the $f$ electron level $\varepsilon_f$, local Coulomb interaction $U'$ between the conduction and $f$ electrons, and on-site Coulomb interaction $U$ in the $f$ electron orbital.
In the present study, we impose $U\to\infty$,
indicating that the doubly occupied state for $f$ electrons is excluded.
Total electron number $N=\sum_i(n_i^c+n_i^f)$ is fixed to $n=\means{N}/N_s=1.9$ to avoid the appearance of the charge ordered phase.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\columnwidth,clip]{penrose.pdf}
\caption{
(a) Penrose lattice with $N_s=1591$ sites used in the present study.
(b) DOS of the tight-binding Hamiltonian ${\cal H}_t$ on the penrose lattice.
}
\label{penrose}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{bethe.pdf}
\caption{
(a) DOS of the tight binding Hamiltonian on the Bethe lattice.
(b) $f$ electron number as a function of $\varepsilon_f$ in the $S=1/2$ FKM on the Bethe lattice at $U'/W=2$.
}
\label{bethe}
\end{center}
\end{figure}
To analyze the model, we adopt the RDMFT~\cite{Georges1996,pruschke1995,muller1989,Metzner1989}.
In this method, the site-dependence of the self-energy $\Sigma$ is taken into account but its inter-site components are neglected.
The noninteracting Green function is given by ${\cal G}_i(i\omega_n)=[G_{ii}^{\rm latt}(i\omega_n)^{-1}+\Sigma_i(i\omega_n)]^{-1}$ in the DMFT scheme, which maps the lattice model onto impurity problems.
Here, $G_{ii}^{\rm latt}$ is the lattice Green function and $\omega_n=(2n+1)\pi T$ is the Matsubara frequency with integer $n$.
An impurity solver updates $\Sigma_i$ using ${\cal G}_i$ for each site.
Finally, the lattice Green function is updated so that $G^{\rm latt}(i\omega_n)^{-1}=(i\omega_n+\mu)I-\Sigma-{\cal H}_t$, where $I$ is the identity matrix and ${\cal H}_t$ is the $N_s\times N_s$ matrix with respect to site representing the first term of Eq.~(\ref{eq:1}).
In general, numerical techniques such as
the continuous-time quantum Monte Carlo method~\cite{CTQMCREV}
are needed to solve impurity problems in DMFT.
Therefore, it is hard to discuss very low temperature properties.
In particular, in the $f$-electron systems on the quasiperiodic structure,
the complex temperature dependence should be expected~\cite{Takemura2015,Shinzaki2016}.
On the other hand, in the Falicov-Kimball model, solving the impurity problem can be analytically carried out~\cite{Freericks1998,Freericks_rev2003}.
The $f$ electron number is given by $\means{n_i^f}=1+\frac{1}{2}e^{\beta(\varepsilon_f-\mu)}A_i^2$ with $A_i=\prod_n[1-U'{\cal G}_i(i\omega_n)]^{-1}$, and the self-energy is calculated from $\Sigma_i={\cal G}_i^{-1}-[G_i^{\rm imp}]^{-1}$, where $G_i^{\rm imp}$ is the impurity Green function given by $G_i^{\rm imp}=(1-\means{n_i^f}){\cal G}_i+\means{n_i^f}[{\cal G}_i^{-1}-U']^{-1}$.
Here, we assume the paramagnetic system and the Green functions and self-energy does not depend on spin.
We iterate the above procedure until the self-energy converges for all $\omega_n$ at each site.
The magnetic susceptibility for conduction ($f$) electrons defined by $\chi^{cc(ff)}=\frac{1}{N_s}\sum_{ij}\int_0^\beta d\tau\left[\means{M_i^{c(f)}(\tau)M_j^{c(f)}(0)}-\means{M_i^{c(f)}(\tau)}\means{M_j^{c(f)}(0)}\right]$ with $M_i^{c(f)}(\tau)=e^{\tau({\cal H}-\mu N)}\frac{1}{2}(n_{i\uparrow}^{c(f)}-n_{i\downarrow}^{c(f)}) e^{-\tau({\cal H}-\mu N)}$ is also evaluated from $\Sigma$ and $G^{\rm latt}$~\cite{Freericks1998,Freericks_rev2003}.
The $f$-electron susceptibility is simply given by $\chi^{ff}=\frac{1}{N_s}\sum_i\frac{\means{n_i^f}}{4T}$ and $\chi^{cf}=0$.
On the other hand, the magnetic susceptibility for the conduction electrons is more complicated due to the presence of the intersite correlations.
This is given by $\chi^{cc}=\frac{1}{2N_s}\sum_{ij}T\sum_n\chi_{ij}^{cc}(i\omega_n)$ with $\chi_{ij}^{cc}(i\omega_n)=[\chi^0(I+\Gamma \chi^0)^{-1}]_{ij}$,
where the bare susceptibility and vertex function are $\chi_0^{cc}=\frac{1}{2N_s}\sum_{ij}T\sum_n\chi_{ij}^{0}(i\omega_n)$ with $\chi_{ij}^0(i\omega_n)=-G_{ij}^{\rm latt} G_{ji}^{\rm latt}$ and $\Gamma_{ij}(i\omega_n)=\frac{\partial \Sigma_i}{\partial G_{ii}^{\rm latt}}\delta_{ij}=\frac{\Sigma_i(U'-\Sigma_i)}{1+G_{ii}^{\rm latt}(2\Sigma_i-U')}\delta_{ij}$, respectively.
In the following, we set the unit of the energy to the half-bandwidth $W$.
\section{Distribution of $f$-electron occupancy in Penrose lattice}\label{sec:distr-f-electr}
Before showing the numerical results for the Penrose lattice,
we briefly touch the DMFT results on the Bethe lattice,
whose density of states (DOS) is shown in Fig.~\ref{bethe}(a).
Figure~\ref{bethe}(b) shows the $f$ electron number $\means{n^f}$ at $U'/W=2$ and $n=1.9$,
where $W$ is the half bandwidth.
$\means{n^f}$ monotonically increases with decreasing $\varepsilon_f$ as expected.
At lower temperatures, there appears a flat region with $\means{n^f}\sim 0.1$
around $\varepsilon_f=-1$.
We refer to this as the intermediate valence region.
At $T/W=0.025$, we find a jump singularity in $\mean{n^f}$ around $\epsilon_f/W=-1.5$.
This suggests a first-order valence transition from the intermediate valence state
to the almost commensurate valence state with $\means{n^f}\sim 1$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{nf.pdf}
\caption{
(a) Site-dependent $f$ electron numbers as a function of $\varepsilon_f$ at $T/W=0.0233$ and (b) those as a function of temperature at $\varepsilon_f/W=-0.465$ in the Penrose lattice. Here, $U'$ is fixed to $U'/W=1.16$.
They are classified by the coordination number $z$.
}
\label{nf}
\end{center}
\end{figure}
Now, we consider the case of the Penrose lattice without translational symmetry,
whose half bandwidth is given by $W\simeq 4.3t$ as shown in Fig.~\ref{penrose}(b).
In the case, the $f$ electron number explicitly depends on the site position.
In Fig.~\ref{nf}(a), we present the $\varepsilon_f$ dependences of $\means{n_i^f}$
for all sites on the Penrose lattice.
The site average of $\means{n_i^f}$ is shown by the black filled circles.
In contrast to the case of the Bethe lattice, we do not find any anomalies corresponding to valence transitions at the lowest temperature we have performed.
More importantly, the $\varepsilon_f$ dependences of $\means{n_i^f}$ are roughly classified by the coordination number of their sites as shown Fig.~\ref{nf}(a).
This feature originates from the itinerant properties of the conduction electrons and has been discussed in other models with local interactions on the Penrose lattice, such as the Hubbard and extended Anderson lattice models~\cite{Takemori2015,Takemura2015,Shinzaki2016}.
In the intermediate valence regime around $\varepsilon_f/W = -0.6$, the site dependence of $\means{n_i^f}$ is observed remarkably.
$\means{n_i^f}$ at the sites with $z=6$ and $7$ take almost $1$,
while those with $z=3$ and $4$ vanish in this region.
We also find the large site dependence within the sites with $z=5$ shown in Fig.~\ref{nf}(a).
To clarify the more detailed nature presented in the intermediate valence regime, we show the temperature dependence of the $\means{n^f}$ at $\varepsilon_f/W=-0.465$ in Fig.~\ref{nf}(b).
At high temperatures, $\means{n_i^f}$ hardly depends on its site position,
and the quasiperiodic structure play a minor role in the electronic state.
With decreasing temperature ($T/W\sim 0.2$),
the variance of $\means{n_i^f}$ for the site position increases but this almost depends on the coordination number, namely the local structure including its neighboring sites.
Further decrease of temperature below $T/W\simeq 0.1$ brings about
more complicated site dependence.
In particular, we find that $f$-electron number for the lattice sites with $z=5$
are distributed in the wide range $0<\mean{n^f}\lesssim 0.5$.
Nevertheless, all $\means{n_i^f}$ for $z=5$ approaches zero
and the $f$ electron number depends only on the coordination number $z$ in the zero temperature limit.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{figz5.pdf}
\caption{
(a) Site-dependent $f$ electron numbers for the site with $z=5$.
Each colored symbol represents $\means{n_i^f}$ on a particular site with the different local structure shown by the corresponding colors in (b).
}
\label{figz5}
\end{center}
\end{figure}
Figure~\ref{figz5}(a) shows the temperature dependence of the $f$ electron numbers for the sites with $z=5$,
which is extracted from Fig.~\ref{nf}(b).
In the Penrose lattice, similar local lattice structures appear with a certain density,
which is one of the most striking features of the quasiperiodic lattice.
Here, we focus on the four local structures with $z=5$ shown in Fig.~\ref{figz5}(b).
Figure~\ref{figz5}(a) shows the data on the site with these local structures using the corresponding colors.
This result indicates that the site dependence of $\means{n^f}$ around $T/W=0.01$ is classified by the wider range of lattice structure surrounding its site beyond the nearest neighbor one.
On the other hand, this feature for $z=5$ is expected to disappear at zero temperature.
The present results suggest that the wider range of lattice structure affects $\means{n^f}$ only at intermediate temperatures while the neighboring structure is relevant at zero and high temperatures.
This temperature dependence is schematically depicted in Fig.~\ref{figz5}(b).
\section{Temperature dependence of susceptibility}\label{sec:temp-depend-susc}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{chi.pdf}
\caption{
Temperature dependence of the susceptibilities for the conduction and $f$ electrons at (a) $\varepsilon_f/W=-0.930$, (b) $\varepsilon_f/W=-0.767$, and (c) $\varepsilon_f/W=-0.465$.
The bare susceptibility $\chi_0^{cc}$ is also shown in each figure.
}
\label{chi}
\end{center}
\end{figure}
Finally, we discuss the magnetic response in the FKM on the Penrose lattice.
It was confirmed that the susceptibility is almost independent of the cluster size in a similar model~\cite{shinzaki2017cluster}.
Figure~\ref{chi} shows the temperature dependences of the susceptibilities $\chi^{cc}$ and $\chi^{ff}$ for the conduction and $f$ electrons, respectively.
Note that $\chi^{cc}$ includes intersite dynamical correlations within DMFT.
At the high temperature limit, $\chi^{cc}$ and $\chi^{ff}$ obey the Curie law.
With decreasing temperature, $\chi^{cc}$ increases but changes to decrease around $T/W=0.1$, which is commonly seen in the different $\varepsilon_f$ shown in Figs.~\ref{chi}(a)--\ref{chi}(c).
Note that the deviation of $\chi^{cc}$ from the bare susceptibility $\chi_0^{cc}$ is enhanced with increasing $\varepsilon_f$ at low temperatures.
On the other hand, $\chi^{ff}$ exhibits distinct temperature dependences for the different $\varepsilon_f$.
In the case with the deep $f$ electron level, $\chi^{ff}$ monotonically increases
with decreasing temperature, as shown in Fig.~\ref{chi}(a).
We then find asymptotic behavior $1/(4T)$ since $f$ electron level is filled at zero temperature.
In the intermediate valence regime ($\epsilon_f/W=-0.465$),
nonmonotonic behavior is exhibited, as shown in Fig.~\ref{chi}(c).
The decrease of $\chi_{ff}$ around $T/W\sim 0.1$ originates from
the monotonic decrease of the site average of $\means{n_i^f}$, as shown in Fig.~\ref{nf}(b).
Between the intermediate valence state and that with $\means{n^f}\sim 1$,
intriguing behavior appears.
Figure~\ref{chi}(b) shows that $\chi^{ff}$ obeys $1/\sqrt{T}$ at lower temperatures, which is also observed in the extended Anderson lattice model on the Penrose lattice~\cite{Shinzaki2016}.
Note that lower-temperature data could not be calculated because of the bad convergence in the DMFT scheme.
At $\varepsilon_f/W=-0.767$, the large site dependence of $\means{n_i^f}$ due to the Penrose lattice is seen in Fig.~\ref{nf}(a).
This might be an origin of the peculiar behavior of $\chi^{ff}$ in the competing region between the intermediate and commensurate valence regimes.
\section{Summary}\label{sec:summary}
We investigated the FKM on the two-dimensional Penrose lattice using RDMFT.
While the first-order valence transition exists in the FKM on the Bethe lattice, this is suppressed in the Penrose lattice.
Owing to the absence of the translational symmetry,
the $f$ electron number explicitly depends on the lattice site and is roughly classified by the coordination number, which may lead to the suppression of the valence transition.
In the intermediate temperature region, this is affected by the wider range of the lattice structure surrounding its site but the short range structure is only relevant at the lowest temperature.
We also calculated the magnetic susceptibility and found the peculiar temperature dependence, which might be due to the absence of the translational symmetry intrinsic in the quasiperiodic Penrose lattice.
\begin{acknowledgements}
This work is supported by Grant-in-Aid for Scientific Research from
JSPS, KAKENHI Grant Nos. JP16K17747(J.N.) and JP17K05536 (A.K.).
Parts of the numerical calculations were performed
in the supercomputing systems in ISSP, the University of Tokyo.
\end{acknowledgements}
\bibliographystyle{spphys}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 5,873
|
Knocked a few things off the list, updated and bumped .
After some busy shopping towards the end of the month, I've made some progress towards nearly completing my OTC collection. My list is updated here.
ML Doctor Octopus Thanks Brent!
Vintage Star Wars - working towards loose, complete figure collection.
Spider-Man Classics - Mysterio, Lizard, other bad guys.
I'd love a full collection, but I'm missing about a dozen.
Another minor update...finished off the BK Toys, and found two of the three cup sets.
AT-AT - c'mon Jim, find me a clearance price.
First 21 - MOMC - yeah right, I still like my wife.
Not so bad. If any new Tomy mini-dioramas pop up I'd include those as well.
None, thank you very much.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 450
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Q: Why am I getting error ORA-00918: Column ambiguously defined in Oracle SQL I am running this statement and I am getting this error code: ora-00918 column ambiguously defined in Oracle SQL. I am not sure what I am missing here. Here is the statement:
SELECT NET_EQUITY, TOTAL_COMMISSION_CALC, TOTAL_TRADES, ADJUSTED_NET_EQUITY_CALC, PROCESS_DATE, SCORE, DESCRIPTION, ACCOUNT_NAME, ACCOUNT_NUMBER, INVESTMENT_OBJECTIVE_DESC
FROM SPAPP_OWNER.SP_ACCOUNT_PROFILE AP
INNER JOIN SPAPP_OWNER.SP_ACCOUNT_PRODUCT_PROFILE PP
ON AP.ACCOUNT_KEY=PP.ACCOUNT_KEY
INNER JOIN SPAPP_OWNER.SP_MODEL_DISTRIBUTION_DETAILS DD
ON PP.SEGMENT_KEY=DD.SEGMENT_KEY
INNER JOIN udmcds_owner.ACCOUNT A
ON AP.ACCOUNT_KEY=A.ACCOUNT_KEY
INNER JOIN udmcds_owner.ACCOUNT_INVESTMENT_OBJECTIVE AIO
ON A.TENANT_CD=AIO.TENANT_CD
WHERE PROFILE_DATE BETWEEN '31-JAN-22' AND '31-DEC-22'
AND TOTAL_TRADES >= 3
AND ADJUSTED_NET_EQUITY_CALC >= '100000.00'
AND TOTAL_COMMISSION_CALC > '1000.00'
A: Apparently one of the column names in your SELECT clause or in your WHERE clause is found in more than one of the tables you have joined together here. Fully qualify with alias prefixes every one of your column references in these clauses, just like you already have done with the join clauses.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,512
|
<?php
/**
* Return the current process identifier
*
* @phpstub
*
* @return int Returns the identifier, as an ``integer``.
*/
function posix_getpid()
{
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,333
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shop to let in london bridge station
POA. We give you expert advice while you're looking for your shop and help you set up in your chosen space once you've found it. Hamleys was founded by William Hamley as "Noahs Ark" in High Holborn, London, in 1760 and is now the oldest and best-known toy shop in the world. ARRANGE VIEWING. By submitting this form, you accept our Terms It has its own entrance, two single sash windows overlooki... +44 20 3478 3205 ** ADAM Grooming Atelier London's leading barber providing the ultimate male grooming experience. © Crown copyright 2020. request for information or to arrange a viewing. Crown Lets 4U, CR0 +44 20 8128 4477 ** Contact Save Hide. Contact £5,000 pre covid – current £3,000 per week. See all 16 commercial properties for sale in London Bridge Station, London on Realla. Each floor provides modern open plane office space, together with male and female W/Cs and showers at basement level. When you London is a master of reinvention and there's no better example than the fashion industry. request for information or to arrange a viewing. read more. The office space is open plan with a kitchenette and its own wc's and is to be decorated throughout. Near London Bridge. All commercial properties to rent in London Bridge Station, Retail premises to rent in London Bridge Station, Industrial to rent in London Bridge Station, Leisure/hospitality to rent in London Bridge Station, Serviced offices to rent in London Bridge Station. By searching for all retail space for rent in London on Realla you have access to a range of shops available to rent now in London, and you can also tune your search to the shop size you need in London. Available immediately this self-contained office is situated on the ground and lower ground floors. Newcomen Street London Bridge. Good food and good karma. Hide. The unit has central heating, a... Class E arch to rent Overview A rare opportunity to take on a unique and prime E class premise by Borough Market. Contact It lies southeast of London Bridge and northeast of Guy's Hospital, and it is adjacent to the tourist attraction called the London Dungeon. Where to shop for fashion, clothes and accessories in London Bridge. Search for London shops to rent today and contact local London … Butchers 3. We love shops. Loans Warehouse Limited is authorised and regulated by the Financial Conduct Authority (FCA) under firm reference 713110. Whether your looking for a shop to let in London, or elsewhere in the UK, and whatever you sell, we can help match you with your ideal retail premises. 1. If you're looking for a shop to let in London, we'd love to help you shop for one. London shops to let on Realla. Their London Bridge branch is located in London Bridge Station's Western Arcade. Save Save Arch 7, Kew Bridge 1,044 sq ft Zone 3 To Let. Shop in Ticket Hall, Barbican Underground Station 460 sq ft Zone 1 To Let. Confectioners 3. read more, 72 Weston Street, London, Greater London SE1, Available immediately this self-contained office is situated on the ground and lower ground floors. refurbished throughout with alcohol licence. ... SE1. Hutong Housed inside a former fire station, Brigade Bar & Bistro supports homeless people trying to get a start in the food industry. Three-tier Covid-19 restrictions: what does it mean for the property market? read more, St Margarets Court, Borough High Street, London SE1, One office is available on the floor of this recently refurbished building. The current crossing, which opened to traffic in 1973, is a box girder br idge built from concrete and steel. Convenience Stores 11. The property is 2 minutes walk from archway station (northern line). Shard restaurants. HELP & INFORMATION Help Centre (FAQ's) Delivery & Returns Store Finder Contact ref: C2074. Refurbishment and extensive works ... Property to rent in London Bridge. SE1. With annual rental costs to suit any budget, MOVEHUT can help you find the ideal shop in Leyton without breaking the bank. Office to let in north London, London. £260 p.w (£1126pcm) 1 Bedroom Apartment; 1. request for information or to arrange a viewing. 15 year established A1 coffee shop / café. Unit 136 London Bridge Station. ADDRESS. read more, The property is situated on Talbot Yard, which is located off Borough High Street, close to the junction with Southwark Street. The Home insurance comparison service is provided by Autonet Insurance Services Ltd, registered in England No. Corner shop / showroom - TO LET | NO PREMIUM. Here you find opening hours, address and more about the specialty store for Made-to-measure T.M.Lewin in London, London Bridge Station. of Use and Privacy Policy. Browse apartments from the leading agents in London Bridge Station on a map and find contact details. Newly Refurbished Creative Office Studio To Let In The Heart Of London's South Bank, +44 20 8033 7797 ** There are two communal WCs, a shower and a kitchen. Opening Hours: Monday - Friday: 7am - 9pm 220-226 Brompton Rd is the latest shop for rent (2,921 sq ft), listed by Orme Retail. Situated between Cannon Street Railway Bridge and Tower Bridge, it forms the western end of the Pool of London. There is also a bicycle ... Delicatessens 15. This is a … Hide. Arches 1-13, Wood Lane Underground Station 0 sq ft Zone 2 To Let. The space is located on the upper concourse and is accessible by lift on the lower concourse, at street level or via the London Underground station. read more, Class E arch to rentA rare opportunity to take on a unique and prime E class premise by Borough Market. London Coffee Shops for Sale: The Basics. These are the buses that leave from/ arrive at stops by London Bridge. London Bridge is a central London railway terminus and connected London Underground station in Southwark, south-east London.It occupies a large area on three levels immediately south-east of London Bridge, from which it takes its name.The main line station is the oldest railway station in London fare zone 1 and one of the oldest in the world having opened in 1836. read more, A ground floor office predominantly open plan with excellent natural light including a reception area. That's been our aim for over 50 years now. Numerous... 10. realtyww.info . WHICH BUS ROUTES STOP AT LONDON BRIDGE STATION? It's close to London Bridge Station and the Shard, and is ideal for 12-15 people. Save Shops To Let in Greater London. Contact The Best Cafes and Coffee Shops in London Bridge London Grind "When a man is tired of London, he is tired of life", or so says the bright neon sign on the wall at London Grind, a much-loved London Bridge cafe.. Ground Floor modern office within Mews close to London Bridge SE1 595 sq ft - £29,500 exclusive short term / long term Available now. Finding a shop for rent is easy, but you always have to consider what your exact budget is. See 137 results for A1 shops to let in London at the best prices, with the cheapest ad starting from £1,014. With entrances from the London Bridge Underground Station and Joiner Street, it is the perfect place to shop for food and beverages, gifts and essentials. read more, Refurbished Former Warehouse Moments From Borough MarketNo.1 Clink Street is located in one of the oldest parts of London, immersed in history and culture. The space is complemented by easily booked meeting rooms. Find the travel option that best suits you. More about keywords. Retail premises (42) Keywords. Sort: List. Enter your email address to receive alerts when we have new listings available for Shop to let in London Bridge. Find a wide range of commercial properties to let in London Bridge Station. The property is just a short walk from Borough Market and all ... Report. Contact London Bridge is a bridge over the River Thames, connecting the City of London and Southwark, in central London. By Bus Bus stops for London Bridge station are S, M and Y. And you can't miss it … Visit Our London Bridge eBike Centre Our flagship showroom is based on historic Bermondsey Street, a short walk from London Bridge station. Save £2,383 pcm. The first station on the site was built of wood in 1836, but more Marketed by Easy Offices, National . A changing place facility is available for use by disabled passengers in London Bridge. Our London Bridge shop is located at street level, through the station arches of the newly developed entrance of St. Thomas Street. Old Kent Road SE1. © Crown copyright 2020. The accommodation is split into an open plan area with two meeting rooms and a kitchen. London Bridge station has recently been transformed from one of Britain's least loved stations into a showpiece 21 st century transportation hub. Our great selection includes properties to rent from all leading London Bridge Station estate agents. read more, Location The premises are excellently located being on the south side of St Thomas Street, moments from the newly redeveloped London Bridge station, 'The Shard' and Borough Market. Modern offices to let at the first floor front office of this five story period building. Shop online at Paperchase, a leader in innovative, design-led stationery, cards and gift-wrap. 646 SF High Street Shop for Sale | Class: A3 380 Green Lanes, London, N13 5PD. Arches 11-14 & Part 15 Putney Bridge Road, Putney Bridge Underground Station 9,091 sq ft Zone 2 To Let. Our great selection includes properties to rent from all leading Tower Bridge Road, London SE1 estate agents. Search through 403 apartments and studios to let in London Bridge from £144 per week. By Road NCP car park is a short walk from London Bridge station (Kipling St, SE1 3RU). And you can't miss it with it's old school cinema look frontage. D1 unit to let on bell street, Marylebone, London, W1 - size - 3032 sqftsituated a few minutes' walk from Marylebone station, and a short stroll from Marylebone... 12 realtyww.info It replaced a 19th-century stone-arched bridge, which in turn superseded a 600-year-old stone-built medieval structure. Food & Drink . From Savile Row, where the suit was invented to the King's Road, where the mini skirt rose to fame, London leads the pack when it comes to inventing new styles and making big statements. Looking for more options? Get in touch with local commercial specialist agents in London Bridge Station, London Please edit your search location if you would like to filter your results. Save * Sizes listed are approximations. 30+ days ago. The property is well located near to the junction with Southwark Street ... +44 20 3641 5637 ** Tanner Street Tower Bridge London SE1 Well presented office/studio premises to let on the first floor. Map. Come visit our oasis in the middle of the hustle and bustle of a busy London Station which celebrated its 180th birthday in December 2016. Find a wide range of commercial properties to let in Tower Bridge Road, London SE1. The office space is currently open-plan on both floors with a small meeting room/office ... The cheapest way to get from London Bridge (Station) to Bridgend Designer Outlet costs only £10, and the quickest way takes just 2 hours. The office benefits from good office specifications and offers a usable basement to ... Save This shop stocks labels like T.M. read more. It's close to London Bridge Station and the Shard, and is ideal for 12-15 people. The space has a height-adjustable adult-sized changing bench, hoist, shower facility, extra wide rolls of paper, and non-slip floors. The location of the shop in London should be another crucial factor … Retail premises to let in Southwark (London Borough) 1 - 25 of 42 List Grid Map. 388 results We couldn't find what you're looking for right now ... 300m from the entrance of the £1bn newly redeveloped London Bridge Station.Surrounding area has been substantially transformed with Bermondsey Street just around the corner providing a thriving hub of bars, restaurants and artisan coffee shops. Hide. Contact ALL OUR STORES IN THE UK AND REPUBLIC OF IRELAND ARE NOW OPEN (With the exception of Edinburgh Airport and Stansted Airport) Why not head to your … The office space is open plan with a kitchenette and its own wc's and ... Start a new search to search for other commercial properties for rent in London or to find shops to rent in other areas . London Bridge, Borough Several bridges named London Bridge have spanned the River Thames between the City of London and Southwark, in central London. Commercial Properties To Let in Greater London. Third floor of this five story period building of 1,630 sq ft Zone 3 to let in Bridge... Store Finder London Bridge Land Registry/Registers of Scotland in London or to arrange a viewing be decorated throughout opening,... To know about new properties matching your search location if you would like to filter your...., Borough and Waterloo stations, it 's also a commuter 's dream £260 p.w ( £1126pcm ) -... Retail premises to shop to let in london bridge station in London Bridge Station on the ground and lower.. To all the amazing stalls so let your nose lead the way Wood Underground... That ' s least loved stations into a showpiece 21 st century transportation hub unsubscribe or your! ) Delivery & Returns store Finder London Bridge p.w ( £1126pcm ) 1 - of... Interest within walking distance of London at London Bridge receive alerts when have. David & George 13 March 2019 Hamleys open browse apartments from the leading agents in London, we 'd to. Properties that include specific words e.g the food industry p.w ( £1126pcm ) 1 Apartment... The Pool of London current crossing, which in turn superseded a stone-built... 12-15 people on Flexible Terms & inclusive rent £390 per desk per Month Dunton Road Putney. The buses that leave from/ arrive at stops by London Bridge Station, Bar...: Not specified 235 - 237 Finchley Road, London SE1 estate agents to us via either London Station... West Croydon ( 0.9 miles ) East Croydon ( 0.9 miles ) Croydon... Line ) * Contact Save Hide currently open-plan on both floors with a period building Station! Search criteria Save Hide line measurements by the Financial Conduct Authority ( FCA ) firm. 'S and is to be decorated throughout customers will be recorded for,... Look frontage period building for right now medieval structure, local great school, and. Listed by Orme Retail rooms with open plan with a period building a new search to search other! A1 and A2 Use on Plastet Grove, East Ham London 0371 1977... Movehut has plenty to offer to the market a shop shop with A1 and A2 Use on Grove... Are pleased to offer to the Shard and London Bridge is a box girder idge! 7, Kew Bridge 1,044 sq ft Zone 2 to let could n't find what you think this... Building, close to London Bridge Station with Primelocation Bridge over the River Thames, connecting the of! Ll find all of these points of interest within walking distance of London and,! It has its own entrance, two single sash windows overlooki... +44 20 8128 4477 * uSwitch! Tanner Street Tower Bridge, it 's also a commuter 's shop to let in london bridge station Nile Street, Burslem Stoke-on-Trent... ) East Croydon ( 0.4 miles ) Note: Distances are straight line measurements Terms inclusive! Communal WCs, a short distance away from Upton Park Underground Station sq. Room/Office partitioned on the second floor ( 526 sq ft ) of recently... High Street from across the rent market let at the first floor finding shop. Basement office with a small meeting room/office partitioned on the lookout for a shop with A1 and Use. London Bridge Station and the Shard, our store is perfectly located to provide quality... The enormity of the best London Bridge new listings available for shop shop to let in london bridge station let in London tube. Under firm reference 713110 maisonette in a vibrant area of the best London Bridge Underground Station also. Latest London shop for one Under firm reference 713110 ( 0.9 miles East... Finder London Bridge Station or the East side of the newly developed entrance St.! To get a start in the heart of London Bridge Station on Map! Iconic Shard, and is ideal for 12-15 people Centre ( FAQ 's ) Delivery & store! Connection to London Bridge Station are s, M and Y 1 bedrooms, £1300 pcm Fees!: 308213 ) extra wide rolls of paper, and non-slip floors +44 8128... Alerts at any time 595 sq ft Zone 2 to let on the side! Female W/Cs and showers at basement level, clothes and accessories in London, there are 1324 available in... Bridge, it 's also a commuter 's dream is a … Hamleys Western Arcade, SE1. Hampton Station, neighbouring the iconic Shard, our store is perfectly located to High... M and Y modern open plane office space miss it with it ' s what you think this... River Thames, connecting the City of London Bridge Station or the East side of the end! Any time in MyZoopla time shop to let in london bridge station perspective, the enormity of the Bridge itself time user,. 2 Dunton Road, London Bridge ( tube Station ) Isle of Flowers Weston Street ( arches ) Concourse! The Shard and London Bridge tube 1977 UK recently refurbished building supports homeless people trying to get start. And you can get to us via either London Bridge tube hours, and! Refurbished building Original and Vertuo coffees as well as machines and accessories in London London! Area by rating these categories think of this area by rating these categories lookout for a shop A1! Is ideal for 12-15 people, Barbican Underground Station 1,023 sq ft ), listed by Orme Retail available. £390 per desk per Month for this fully inclusive and furnished office space shops to rent in areas. We want to offer you the very best experience to look up… 25 Vertuo! & part 15 Putney Bridge Underground Station 595 sq ft Zone 2 Under offer a let. Provide High quality, express repairs Bridge Road, London the office space is by. Space to let is open plan with a period building what does it mean the! Is perfectly located to provide High quality, express repairs to Tower Bridge SE1. In England No full range of Original and Vertuo coffees as well as machines accessories! Financial Conduct Authority ( FCA ) ( Registration number: 308213 ) ideal., London rental costs to suit any budget, MOVEHUT can help you shop for.. 2020. request for information or to find more properties: 10 or to arrange viewing! Situated moments from Borough market trying to get a start in the food.., refurbished period building of 1,630 sq ft Zone 1 to let on the lower ground floors in this,... Your nose lead the way central London advantage of our great selection includes properties to rent near Bridge! Bridge over the River Thames, connecting the City of London and Southwark, Borough and stations! Per week bars, restaurants and also close to the market a to. On one side the premises face Highams Park Railway Station which offers a frequent connection to London Bridge, 's! New properties matching your search area is too large to further filter your results immediately! Weston Street ( arches ) main Concourse London SE1 estate agents in this attractive former conversion. Cambridge Grove, East Ham London sq ft Zone 2 to let on the west side of the Bridge.. What does it mean for the property market properties matching your search area is large! Dunton Road, Putney Bridge Road, Putney Bridge Underground Station Bermondsey Street, Burslem Stoke-on-Trent. Replaced a 19th-century stone-arched Bridge, which opened to traffic in 1973, is located on the first.... Search through 403 apartments and studios to let in London, we love...... +44 20 8033 2680 * * Calls to this number will be choosing property. Station 822 sq ft Zone 2 Under offer 527 results we could n't what. Station estate agents built of Wood in 1836, but you always have consider. Arrange a viewing recently refurbished building the best London Bridge Station,,. Accept our Terms of Use and Privacy Policy reference number 312850 Bridge branch is located on the lower.! ' ll need to look up… 25 any budget, MOVEHUT can help you for. Houses to rent in London Bridge shop is located on the second floor ( 526 sq ft 2! Own shop to let in london bridge station 's and is situated on the west side of the first floor High Street shop fashion... Office/Studio premises to let covering the third floor of this area by rating these.! Let covering the third floor of this five story period building when for! Update your preferences at any time in MyZoopla Kew Bridge 1,044 sq ft 2... Gated development and is ideal for 12-15 people Flowers Weston Street ( arches ) main Concourse London SE1:.. Suit any budget, MOVEHUT can help you find opening hours, address and more about the specialty store Made-to-measure... Is 2 minutes walk from archway Station ( northern line ) currently open-plan on both with! Let freehold investment property for sale to provide High quality, express repairs Vertuo coffees as well as and. Of Use and Privacy Policy Financial Conduct Authority ( FCA ) ( Registration number: )! Be able to purchase the brand ' s been created can be bewildering amenities cafes,,. Apartment ; 1 20 8033 2680 * * Contact Save Hide let covering the third floor of this story! In Southwark ( London Borough ) 1 - 25 of 42 List Grid Map shop to let on the for. Be counted once to you when searching for a shop to let Southwark. People trying to get a start in the heart of London Bridge from £144 per week office at Street...
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Cmentarz żydowski w Różanie – kirkut położony jest w Różanie blisko rzeki Narew, na terenie zwanym Wyjemka. Powstał w drugiej połowie XIX wieku. W czasie II wojny światowej uległ dewastacji. W czasach komunistycznych teren kirkutu upaństwowiono i sprzedano. W 2003 działka cmentarna została kupiona przez Efraima Ben Dora. W 2004 nekropolia została ogrodzona dzięki funduszom zgromadzonym przez Ziomkostwo Żydów Różana w Stanach Zjednoczonych i Izraelu, Fundację Ochrony Dziedzictwa Żydowskiego w Polsce, Fundację Rodziny Nissenbaumów oraz przy wsparciu miasta i gminy Różan.
Bibliografia
Mapa WIG Różan Pas 37 Słup 33 Warszawa 1935
Linki zewnętrzne
Cmentarz żydowski w Różanie na portalu Wirtualny Sztetl
Kirkuty.xip
Różan
Obiekty sakralne w Różanie
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\section{Introduction}
The cosmological constant $C$ can be defined as a vacuum expectation value
(VEV) of an effective lagrangian:
\begin{equation}
C= <{\cal L}_{eff}>
\end{equation}
It vanishes $(C=0)$ in any theory with global
supersymmetry, - that is a direct consequence of a supersymmetry algebra.
The situation is not so unambiguous in a supergravity, where in general
$ C \ne 0 $. But $C=0$ for the heterotic superstring in ten dimensions
(D=10).
So, one may expect that the Type I (N=1, D=10) supergravity considered as
a field-theory limit of a heterotic superstring also leads to $C=0$.
The same must be true for the dual N=1, D=10 supergravity considered as a
field-theory limit of a fivebrane \cite{S1}, \cite{D}, \cite{DL}
because this theory
can be obtained from a Type I supergravity by dual transformation of
the axionic field.
It is not evident, that the condition $C=0$ will persist in a
compactification for lower dimensions because the part of the
supersymmetry can be lost and vacuum properties are rather specific
in the process.
We demonstrate in the present paper, that rather general and realistic
compactification procedure from the space $M_{10}$ to the $M_6$ leads to
vanishing of the cosmological constant.
(Here $M_D $ is a $D$-dimensional
space-time with Minkowsky signature). Really the case is considered:
\begin{equation}
\label{2}
M_{10} \rightarrow M_{6} \otimes E_4
\end{equation}
where the compact Eucledian D=4 space $E_4$. We require that vacuum
configuration leaves only halph of the D=10 supersymmetry unbroken and
leads to chiral theory in D=6. This assumption greatly
simplifies the equations for vacuum configuration and uniquely
fixes the topological structure of $E_4$ manifold. It follows, that
$E_4$ is related to the $K_3$-space
(see \cite{P} for referencies on $K_3$-space).
We concentrate on the dual supergravity because the supersymmetric
lagrangian for this theory is constructed including terms of the next
order in the string-tension parameter $\alpha'$
(see \cite{STZ1}, \cite{STZ2}).
\section{Lagrangian}
The lagrangian of dual supergravity in $M_{10}$ is equal to:
\begin{equation}
\label{3}
{\cal L} =
{\cal L}^{(gauge)}+ {\cal L}^{(grav)}
\end{equation}
where ${\cal L}^{(gauge)}$ and ${\cal L}^{(grav)} $ are lagrangians for
gauge-matter and for supergravity multiplet. The term
${\cal L}^{(gauge)}$ takes the form \cite{CH}:
\begin{equation}
\label{4}
E^{-1}\,{\cal L}^{(gauge)}={1\over g^2}\, tr\, \left[
{1\over4}\,{\cal F}_{AB}\,{\cal F}^{AB}-
{1\over{8 \cdot 6!}}\,{\varepsilon}^{A_1 \ldots A_{10}}\,C_{A_1 \ldots A_6}
\,{\cal F}_{A_7A_8}\,
{\cal F}_{A_9A_{10}} \right]
\end{equation}
Here $C_{A_1 \ldots A_6}$ is an axionic potential, ${\cal F}_{AB} $ is a
gauge field which is in the algebra of internal symmetry group $G$,
${\cal F}_{AB} = E_A^ME_B^N \, F_{MN}, $ where $E_M^A$ is the veilbein in
$M_{10}$.
One must consider $G=SO(32)$ or $G=E_8\times E_8$ and $ 4\,g^2 =-1/\alpha'$
as it follows from
the superstring consideration \cite{GS}. The symbol $tr$ in (\ref{4})
means the trace in a vectorial representation of $SO(32)$. It can be
changed to
$ (1/30)Tr $, where $Tr$ means the trace in the adjoint representation of
$E_8\times E_8$ or $SO(32)$.
We consider only bosonic terms in the lagrangian
(\ref{3}) because we are interested in the
vacuum configuration. Notations correspond in general
to \cite{STZ1} with some differencies which are or
self-evident or explained in
the text. In particular, the following index notations are used:
$A,B,C,\ldots$,
$ a,b,c,\ldots $ and $\alpha, \beta, \gamma, \ldots $ are flat
indices respectively in $M_{10}$, $M_6$ and $E_4$;
$ M,N,P,\ldots $,
$ m,n,p,\ldots$ and $ \mu, \nu, \lambda, \ldots $ are corresponding world
indices; $Z^M =(x^\mu, y^m)$ is the coordinate in $M_{10}$, $x,\,y$ are
coordinates in $M_6$ and $E_4$ respectively.
We present the gravity part of the lagrangian as an expansion in $\alpha'$:
\begin{equation}
\label{5}
{\cal L}^{(grav)}= {\cal L}^{(grav)}_0 +\alpha' {\cal L}^{(grav)}_1
\end{equation}
where $ {\cal L}^{(grav)}_0$ is equal to \cite{CH}
(see \cite{STZ2} for further references on the subject):
\begin{equation}
\label{6}
E^{-1}\,{\cal L}^{(grav)}_0 = \phi\,\left({\cal R}-
{1\over 12}\,{\tilde M}_{ABC}^2 \right)
\end{equation}
Here ${\cal R} $ is the curvature scalar, $\phi$ is the dilatonic
field, ${\tilde M}_{ABC} $ is defined by:
\begin{equation}
\label{7}
{\tilde M}_{ABC} = {1 \over 7!}{\varepsilon_{ABC}}^{A_1\ldots A_7}
M_{A_1\ldots A_7}
\end{equation}
where $M_{N_1\ldots N_7} = 7\,\partial_{[N_1}C_{N_2 \ldots N_7]}$
is the axionic field-strength.
The result for ${\cal L}^{(grav)}_1 $ was obtained in \cite{STZ1},
\cite{STZ2} in the form:
$$ {\cal L}^{(grav)}_1 = 2\,{\cal R}^2_{AB} -{\cal R}_{ABCD}^2 +
{1\over 2\cdot 6!}
\varepsilon^{ABCDF_1\ldots F_6}\,{{\cal R}_{AB}}^{IJ}
{\cal R}_{CDIJ}C_{F_1\ldots F_6} - $$
$$ -{1\over 2}\,{\cal R}^{AB}({\tilde M}^2)_{AB}
-{1\over 6}\, {\tilde M}^{ABC}D_F^2{\tilde M}_{ABC} + $$
\begin{equation}
\label{8}
+ {1\over 2}\, {\tilde M}^{ABC;D}({\tilde M}^2)_{ABCD}
-{1\over 24}\, ({\tilde M}^2)_{ABCD}({\tilde M}^2)_{ACBD}
\end{equation}
Here ${\cal R}_{ABCD} $ is the curvature tensor, ${\cal R}_{AB}$ is the
Ricci tensor, $;B$ means the covariant derivative $D_B $.
The following notations are introduced here and below:
$$ {\tilde M}^2 = ({\tilde M}_{ABC})^2, \ \
({\tilde M}^2)_{AB} = {{\tilde M}_A}^{CD} {\tilde M}_{BCD} $$
$$ ({\tilde M}^2)_{ABCD}=
{{\tilde M}_{AB}}{}^F {\tilde M}_{CDF}, \ \
({\tilde M}^3)_{ABC} = {{\tilde M}_A}{}^{IJ} {{\tilde M}_{BJ}}{}^K
{\tilde M}_{CKI} $$
The ${\tilde M}_{ABC} $ field is connected by the dual transformation
with the 3-form axionic field of standard Type I supergravity (see below).
\section{Vacuum Configuration}
The most general anzatz for VEV of the veilbein in $M_{10} $, which
corresponds to the compactification according to eq.(\ref{2}),
takes the form:
\begin{equation}
\label{9}
{E_M}^A = \left(
\begin{array}{cc}
e^{2\,\xi(y)}\,\delta_\mu^\alpha & 0 \\
0 & e^{-2\,\xi(y)}\, {e_m}^a(y)
\end{array}
\right)
\end{equation}
where $ {{\tilde e}_m}^a \equiv \exp(-2\,\xi)\, {e_m}^a$
is the veilbein in $E_4\ $ (the factor $ \exp(-2\,\xi) $ is extracted for
future convenience), $\xi $
is an arbitrary function. The analogous ansatz was considered
\cite{S2} in the study of compactification
scheme $ M_{10}\rightarrow M_4 \otimes E_6 $ with only partial account of $\alpha'$
corrections.
Only the gauge-field component ${\cal F}_{ab} =\exp(2\xi) \, F_{ab} $
survive
in the vacuum configuration ( $F_{ab} $ $ = e_a{}^m\,e_b{}^n\,F_{mn}$ ).
The folowing VEV's of curvature tensor componets survive
(${\cal R} =d\omega + \omega \wedge \omega $
where $\omega $ is the connection related with $E_M{}^A$):
$${{\cal R}_{\alpha\beta}}^{\gamma \delta}=
2\,e^{4\xi}\,\delta^{\,\gamma}_{[\alpha}
\delta^{\,\delta}_{\beta]}\,\xi^f\xi_f $$
$$ {{\cal R}_{\alpha b}}^{\gamma c}=e^{4\xi} \delta_\alpha^\gamma
\,({\xi_b}^c + 5\,\xi_b\xi^c -2\, \delta_b^c\, \xi^f\xi_f) $$
\begin{equation}
\label{10}
{{\cal R}_{ab}}^{cd}=e^{4\xi}\,(R_{ab}{}^{cd} -8\,
\delta_{[a}^{}{[c} \,\xi_{b]}{}^{d]} -
16\,\delta_{[a}{}^{[c} \,\xi_{b]}\xi^{d]} +8\,\delta_{[a}^{\,c}\delta_{b]}^d\,
\xi^f\xi_f)
\end{equation}
where $R_{mnab} $ is the curvature defined in terms of ${e_m}^a$,
$\xi_b =\nabla_b\,\xi= e_b{}^m \,\partial_m \,\xi,$ etc.
Let us start now to study equations defining vacuum confi\-guration:
$ <\delta_Q \, \Phi >= 0, $
where $\Phi $ is some field but $\delta_Q$
is a super\-sym\-metry transfor\-mation.
When $\Phi$ is a boson, such an
equation satisfied identically. It has nontrivial content for
$\Phi =\psi_A,$ $\chi,$ $\lambda $, i.e. for
gravitino, dilatino and gaugino fields respectively.
We get \cite{STZ3} (see also \cite{BBLPT}, \cite{AFRR} where another
parametrization is used):
\begin{equation}
\label{11}
<\delta_Q\psi_A> = \epsilon_{;A} +{1\over 144}
\left( 3\,{\tilde M}_{BCD}\Gamma^{BCD}\Gamma_A+
\Gamma_A{\tilde M}_{BCD}\Gamma^{BCD}\right)\epsilon =0
\end{equation}
\begin{equation}
\label{12}
<\delta_Q \chi> = {1\over 2}\partial_A \phi \, \Gamma^A \epsilon -
\left( {\phi\over 36}\,{\tilde M}_{ABC}\Gamma^{ABC} -\alpha'\,
A_{ABC}\Gamma^{ABC}\right)\epsilon =0
\end{equation}
\begin{equation}
\label{13}
<\delta_Q \lambda> = {1\over 4}{\cal F}_{AB}\Gamma^{AB} \epsilon =0
\end{equation}
Here $\epsilon $ is a 32-component Dirac spinor, - the parameter of
supersymmetry transformation. It is subjected to the Majorana-Weyl
condition: $ \epsilon_c = {\bar\epsilon}\ $, $\epsilon = \Gamma \epsilon $, where
$ \Gamma $ is the chirality matrix (see below), $\epsilon_c$ is
the charge-congugated spinor.
We suppose that $\epsilon $ is independent on
the coordinates in $M_6$: $\epsilon = \epsilon(y) $, that corresponds to the
symmetry of vacuum configuration. All the fields in the r.h.s. of
(\ref{11})-(\ref{13}) depend only on $y$.
The 3-form field $A_{ABC} $ in eq.(\ref{12}) is equal to \cite{STZ1},
\cite{STZ2}:
$$ A_{ABC} = -{1\over 18} \Box {\tilde M}_{ABC} +
{7\over 36}({\tilde M}^2_{D[ABC]}){}^{;D} +{1\over 36} {\tilde M}_{DE[A;B}
{{\tilde M}_{C]}}{}^{DE}- $$
$$- {5\over 8\cdot 243}{\tilde M}^2{\tilde M}_{ABC}+
{5\over 8\cdot 27} {\tilde M}^2_{D[A}{{\tilde M}_{BC]}}{}^D
-{5\over 4\cdot 27}{\tilde M}^3_{ABC}-$$
\begin{equation}
\label{14}
-{1\over 4\cdot 972}{\varepsilon_{ABC}}^{DEFGHIJ}{\tilde M}_{DEF}
({\tilde M}_{HIJ;G} +{\tilde M}^2_{GHIJ})
\end{equation}
The following representation of $\Gamma $-matrices $M_{10}$ is convenient:
$$ \Gamma^\alpha = \gamma^\alpha \otimes {\cal T}, \ \ \
\alpha = 0,1,\ldots ,5 $$
\begin{equation}
\label{15}
\Gamma^{a+5} = I \otimes {\cal T}^a, \ \ \ a=1,2,3,4
\end{equation}
where $\gamma^\alpha $ are $8\times 8 $ Dirac matrices in $M_6$, but
${\cal T}^a $ are $4\times 4$ Dirac matrices in $E_4$. The chirality
matrices are:
\begin{equation}
\label{16}
\gamma = \gamma^0\gamma^1\ldots\gamma^5, \ \ \
{\cal T} = {\cal T}^1{\cal T}^2{\cal T}^3{\cal T}^4
\end{equation}
Then
\begin{equation}
\label{17}
\Gamma =\Gamma^0\Gamma^1 \ldots \Gamma^9 = \gamma \otimes {\cal T}, \ \ \
(\Gamma)^2 =(\gamma)^2 =({\cal T})^2 =1
\end{equation}
We use also the standard notation: $ \Gamma_{A_1\ldots A_k}
= \Gamma_{[A_1}\Gamma_{A_2} \ldots \Gamma_{A_k]}. $
We begin our study from eq.(\ref{11}). For $A=\alpha $ it
reduces to:
\begin{equation}
\label{18}
\gamma_\alpha \gamma \otimes \left( e^{2\xi}\,\xi_a\,{\cal T}^a +
{1\over 36} {\tilde M}_{abc}\,{\cal T}^{abc} \right)\epsilon=0
\end{equation}
Let us suppose, that $\epsilon$ has definite chirality in $E_4$:
\begin{equation}
\label{19}
{\cal T}\epsilon = \nu \epsilon, \ \ \mbox{where} \ \nu=1 \ \mbox{or} \ -1
\end{equation}
It means that we keep only "one-halph" of the supersymmetry in $M_{10} $.
Using the relation:
$$ {\cal T}^{abc} = - \varepsilon^{abcd}\,{\cal T}_d \,{\cal T} $$
where $\varepsilon^{abcd} $ is a completely antisymmetric tensor
($\varepsilon^{1234} =1$), we get the solution of (\ref{18}) in the form:
\begin{equation}
\label{20}
{\tilde M}_{abc} = 6\,\nu \,e^{2\,\xi}\,\varepsilon_{abcd}\, \xi^d
\end{equation}
It means that the only nonzero component of the axion potential with all
flat $M_6$ indices appears to be a constant:
\begin{equation}
\label{21}
C_{\beta_1\ldots \beta_6} = -\nu\, \varepsilon_{\beta_1\cdots \beta_6}
\end{equation}
where $\varepsilon^{\beta_1 \ldots \beta_6} $ is a
completely antisymmetric tensor ($\varepsilon^{01\ldots 6} =1$).
Having all this, one can reduce eq.(\ref{11}) with $A=a $ to the
following:
\begin{equation}
\label{22}
\nabla_a \epsilon' = 0, \ \ \mbox{where} \ \ \epsilon'= e^{-{\xi/2}}\,\epsilon
\end{equation}
Equation $[\nabla_a, \nabla_b]\epsilon'= 0 $ leads to:
\begin{equation}
\label{23}
R_{abcd}\,{\cal T}^{cd}\, \epsilon=0
\end{equation}
Eq. (\ref{23}) is fulfilled for arbitrary chiral $\epsilon$ if the
curvature defined by the veilbein ${e_m}^a$ is (anti)selfdual for the
second pair of indices:
\begin{equation}
\label{24}
R_{abcd} = {\nu \over 2}\,{\varepsilon_{cd}}^{ef}\,R_{abef}
\end{equation}
This equation immediately follows from (\ref{23}) because ${\cal T}^{ab}=
-(1/2)\varepsilon^{abcd}\,{\cal T}_{cd}\,{\cal T} $.
It follows from (\ref{24}) with the help of simmetry properties of
the curvature tensor that curvature is
(anti)selfdual for the first pair of indices too. The compact 4-dimensional
space with the (anti) selfdual curvature is the $K_3$ space. It means that the
space ${\tilde E}_4$ which is defined by the veilbein ${e_m}^a$ is the
$K_3 $-space. This result follows from the ansatz (\ref{9}) with the constraint
(\ref{10}).
Now we turn to the eq.(\ref{13}).
It is analogous to eq.(\ref{23}), so it
leads to the selfduality of the gauge-field ${\cal F}_{ab}$:
\begin{equation}
\label{25}
{\cal F}_{ab} ={\nu \over 2}\, {\epsilon_{ab}}^{cd}\,{\cal F}_{cd}
\end{equation}
We are left with eq.(\ref{12}) which is the most complicated one.
Taking into account eqs. (\ref{10}), (\ref{20}) one is able
to present the $A_{abc} $-tensor from eq.(\ref{14}) in
a rather simple form:
\begin{equation}
\label{26}
A_{abc} = -{\nu\over3}\, \varepsilon_{abcd}
\left[e^{6\xi}\,({\xi^f}_f +
\xi^f\xi_f)\right]^{;d}.
\end{equation}
where $;d$ means the covariant derivative $\nabla_d$ defined by the
veilbein $e_m{}^a$.
Expresion (\ref{26}) allows us to drop one derivative
in eq.(\ref{12}) reducing the
order of equation from third to second. Then it follows immediately
from eq.(\ref{13}):
\begin{equation}
\label{27}
e^{2\xi}\, \phi +4\,\alpha'\, e^{6\xi}\,({\xi^f}_f +\xi^f\xi_f) =C_0
\end{equation}
where $C_0$ is an arbitrary constant. This equation connects the dilaton VEV
with function
$\xi(y)$ introduced by the anzatz (\ref{9}).
\section{Equations of Motion}
We examine here whether equations of motion impose some additional
constraints on the parameters of vacuum configuration. Equations
for $M_{10}$ supergravity obtained in the lowest and next order in
$\alpha' $ are used (see \cite{STZ1}, \cite{STZ2}), see also \cite{P}
where another parametrization was considered):
One can easily see that gauge-field equations of motions are fulfilled
because of selfduality condition and Bianchi Identity $\nabla_{[a}F_{bc]}=0$.
The dilaton equation of motion can be presented in the form:
$$\Box \phi +{1\over12}\phi {\tilde M}^2_{ABC}+
{1\over 12\,g^2}tr\,({\cal F}_{AB})^2
-\alpha'\left[ -{2\over 3}({\cal R}_{AB})^2
+{1\over3}({\cal R}_{ABCD})^2 -\right.$$
$$-{1\over 6}{\cal R}^{AB}{\tilde M}^2_{AB}
-{1\over18}{\tilde M}^{ABC}\Box {\tilde M}_{ABC}+
{1\over3}{\tilde M}^{ABC;D}{\tilde M}^2_{ABCD}-$$
\begin{equation}
\label{28}
\left. -{1\over24} {\tilde M}^2_{ABCD} {\tilde M}^{2 \ ACBD}
-{1\over12} \Box {\tilde M}^2 +
{1\over6} ({\tilde M}^2_{AB})^{;AB} \right] =0
\end{equation}
where $\Box = D_B^2$.
Calculating all the terms in eq.(\ref{28}) with the help of relations
obtained before and using eq.(\ref{27}), one obtains the following
result:
\begin{equation}
\label{29}
{\left( e^{-6\xi}C_0 -12\,\alpha'\,{\xi_a}^a \right)^{;b}}_{;b}+
{1\over 4g^2}\, tr\,(F_{ab})^2 -\alpha' (R_{abcd})^2 =0
\end{equation}
where $F_{ab} = e_a{}^m e_b{}^n F_{mn}\ $, $R_{abcd}=e_a{}^m e_b{}^n
R_{mncd} $ .
One can expect (cf. \cite{CHSW})
that VEV's of supersymmetry transformations
(\ref{11})-(\ref{13})
provide the complete information on the vacuum configuration, i.e.
they are equivalent to equations of motion. In such a case,
one must be able to derive eq.(\ref{29}) starting immediately from
(\ref{11})-(\ref{13}). But up to now we were not able to do this.
One gets immediately from (\ref{29}) the
topological constraint \cite{W} (in form notations):
\begin{equation}
\label{30}
\int_{K_3} (tr\,F\wedge F -tr\,R\wedge R) =0
\end{equation}
Here $tr$ is calculated over indices of vectorial representation
of corrsponding group, i.e $tr\,R\wedge R)= R_{ab}\wedge R^{ba} $.
The integral is fulfilled over the compact space with the
veilbein $e_m^a $ (i.e the $K_3$-space).
Now we turn to the equation of motion for the $C_{A_1\ldots A_6}$-field.
One can write it in the form \cite{STZ1}:
\begin{equation}
\label{31}
H_{[ABC;D]}+ 3\,\alpha' \left( tr\, {\cal F}_{[AB}{\cal F}_{CD]}-
{\cal R}_{[AB}{}^{EF}{\cal R}_{CD]FE} \right) =0
\end{equation}
where
$$ H_{ABC} =\phi \, {\tilde M}_{ABC} -2\,\alpha'\Bigl(-\Box {\tilde M}_{ABC}
+3\,({\tilde M}^2_{D[ABC]})^{;D} + $$
\begin{equation}
\label{32}
+{3\over 2}{\tilde M}_{DF[A;B}{\tilde M}_{C]}{}^{DF}
-3\,{\cal R}_{D[A} {\tilde M}_{BC]}{}^D -
{1\over 2}{\tilde M}^3_{[ABC]}\Bigr)
\end{equation}
Only the $H_{abc}$-component survive in the vacuum configuration.
The calculation of this component is similar to that performed
for $A_{abc}$. The result is:
\begin{equation}
\label{33}
H_{abc} = 6\,\nu\,\epsilon_{abcd}\,(C_0 \,\xi^d +2\,\alpha'\, e^{6\xi}\,
{\xi^{df}}_f)
\end{equation}
Then, one can check with the help of selfduality conditions
that eq.(\ref{31}) is equivalent to eq.(\ref{29}), i.e.
no new constrains are produced.
So we obtain, that eq.(\ref{29}) is
the VEV of usual $M_{10}$ supergravity Bianchi Identity:
\begin{equation}
\label{34}
dH' = 2\alpha'(-tr \, F\wedge F +tr \, R\wedge R)
\end{equation}
where $H'_{abc}$ is interpreted as a VEV of axionic field-strength of
usual Type I supergravity. It takes the form
\begin{equation}
\label{35}
H'_{abc} =\nu\,\epsilon_{abcd}\,H^{;d}
\end{equation}
where $H(y)$ is a scalar field. It follows from (\ref{29}):
\begin{equation}
\label{36}
{(H-e^{-6\xi}\,C_0 +12\, \alpha'\, {\xi^a}_a)^{;b}}_{;b}=0
\end{equation}
One can easily find the relation between $H'_{abc}$ and $H_{abc} $-field in
(\ref{33}).
A long study of a rather complicated graviton equation of
motion ($\phi {\cal R}_{AB} + \ldots =0 $) does not produce additional
constraints for vacuum configuration.
\section{Zero Cosmological Constant}
In this section we demonstrate that the action VEV vanishes.
The internal space assumed to be closed, so surface integrals
do not give any contribution.
Using eqs.(\ref{10}) and (\ref{20}) one gets immediately:
\begin{equation}
\label{37}
<\,{\cal L}_0^{(grav)}\,> =0
\end{equation}
where ${\cal L}_0^{(grav)}$ is defined in eq.(\ref{6}).
The gauge-matter part of the lagrangian (\ref{3}) can be transformed to
the form:
\begin{equation}
\label{38}
<\,E^{-1}{\cal L}^{(gauge)}\,> ={1\over 4g^2}\,tr\,({\cal F}_{ab}^2-
{\nu\over2}\,\epsilon^{abcd}{\cal F}_{ab}{\cal F}_{cd})
\end{equation}
where eq.(\ref{21}) was used. Then, the selfduality condition (\ref{25})
leads immediately to:
\begin{equation}
\label{39}
<\,{\cal L}^{(gauge)}\,>=0
\end{equation}
The most difficult term in (\ref{3}) is the ${\cal L}_1^{(grav)}$.
Using (\ref{8}) and relations obtained before we are able
to transform this term to the form of complete derivative:
\begin{equation}
\label{40}
<\,E^{-1}{\cal L}_1^{(grav)}\,> =
\nabla_a \,\left[ e^{6\xi}\,(12\,\xi^{ab}\xi_b-3\,\xi^a{\xi_b}^b -
12\,\xi^a\xi^b\xi_b)\right]
\end{equation}
So, one can put:
\begin{equation}
\label{41}
<\,\int \,d^{10}Z\,{\cal L}_1^{(grav)}\,>=0
\end{equation}
Then
\begin{equation}
\label{42}
C = <\,\int \,d^{10}Z\, ({\cal L}^{(gauge)}+{\cal L}_0^{(grav)}+
\alpha'\, {\cal L}_1^{(grav)})\,> =0.
\end{equation}
We conclude, that no cosmological constant is generated.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,485
|
\section{Introduction}
\label{Intro}
Time crystals and their dynamics became an interesting problem after recent publications of \cite{Wilczek},
etc. In short time crystals are systems where natural
symmetry is not compatible with gauge field. A good example is a system of two or more ions in
a cylindrically symmetric harmonic trap and
in an additional constant homogeneous magnetic field directed along the symmetry axis. Because of cylindrical
symmetry in the ground state the system wave function should be an eigenstate of the angular momentum component
along the symmetry axis. On the other hand in is a property of the magnetic field that the total flux
of the field should be a multiple of the flux quantum. These two conditions are not compatible with
each other. It has been argued \cite{Wilczek}that the ground state as well as excited state
of the system show a nontrivial dynamics. In particular the ground state of a time crystal is not stationary,
but instead exhibits time dependence characteristic for motion. We will explore this idea in detail in this paper.
A system of ions on a trap with additional magnetic field was recently
investigated by \cite{LGYQYZDZ} from the point of view of time crystal dynamics.
It was shown there that two o more ions indeed provide a good example o a time crystal and the dynamics
was discussed.
In this paper we will further investigate similar systems. In particular we will discuss the
dynamics, i.e. time dependence of the wave function. We will restrict ourselves to the simplest
case of two ions, this simple system exhibits all the characteristic features of a time crystal. More ions
add complications not giving any new aspects to the problem.
there is no need to discuss more ions.
The problem of interaction of two charged non-relativistic particles in a trap
with added homogeneous magnetic field is interesting also from another point of view. This is one of
the simplest systems in which non-trivial effects associated with the difference
between the canonical and kinetic momentum can be studied.
The particular form of the coupling of the electromagnetic field to the matter
fields appears as a consequence of the gauge invariance of quantum
electrodynamics. Even in the realm of non-relativistic quantum mechanics
that coupling leads quite profound effects, in particular, to the necessity
to distinguish between the canonical and kinetic momentum. In the presence of
(non-static) electromagnetic field the canonical momentum while retaining its role
in the Hamiltonian formalism loses its status of a physical observable and
is to be replaced with the velocity. This happens even in the simplest
situation when the electromagnetic field reduces itself to the constant homogeneous
magnetic field when the corresponding vector potential is linear in the coordinate.
One may suspect that a system with non-zero expectation value of the radial
coordinate (hence non-zero expectation value of the vector potential) can
be of particular interest. Indeed, in such a system the expectation value
of velocity does not vanish in the ground state because the minimum
of the effective potential is displaced from zero. One way to achieve that
is to introduce a repulsive Coulomb potential with the center at the same
point as the center of the trap potential. This can be realized by placing
to ions with identical charges in a trap. Then the problem reduces to a one-body
problem in the relative coordinates.
Similar ring-shaped traps have been considered, e.g., in
\cite{LGYQYZDZ,SSH,MG,OYTWSO,CMPHKK,Clark}.
A remarkable comparison of the energy spectra as produced by the propagation
of a semiclassical wave packet and those computed from the WKB approximation
has been performed in \cite{GK}.
A real and very interesting related
systems are composed of aromatic molecules in the magnetic field \cite{GM,MHS}
The main part of this work is organized as follows. In Section 2, the construct
the mathematical model and describe the basis states. Section 3 contains
the results of the calculation of lowest energy levels as well as teh
ground-state expectation value of the velocity. In Section 4 we investigate
the effects associated with abrupt turning on of the magnetic field.
Section 5 is devoted to the evolution of the wave packet, and
Section 6 contains some concluding remarks.
\section{The model}
Let us consider a system of two identically charged particles which are trapped
in a harmonic external potential with additional homogeneous magnetic field.
The Hamiltonian of one of the particles in the relative coordinates
(in two spatial dimensions) can be written as:
\begin{equation}
H = \frac{1}{2 \mu} \left( {\bf p} - e {\bf A} \right)^{2} + \frac{1}{2}
\mu \omega_{t}^{2} {\bf r}^{2} + \frac{k e^{2}}{|{\bf r}|},
\end{equation}
where $\mu$ is the reduced mass, $e$ is the charge of each particle,
${\bf p}$ is the relative canonical momentum,
${\bf A}$ is the vector potential, $\omega_{t}$ is the frequency of the trap,
$k = 1/(4 \pi \epsilon_{0})$, and ${\bf r} = (x, y)$ are the relative coordinates.
We choose the so-called symmetric gauge for the vector potential to represent
the homogeneous magnetic field (which sets the $z$-axis of the system):
$$
{\bf A} = \frac{1}{2} {\bf B} \times {\bf r},
$$
where ${\bf B} = (0, 0, B_{z})$ is the magnetic induction.
Before we proceed let us briefly discuss the motion of particles within the framework
of classical mechanics. It is clear that the lowest energy solution of the classical
equations of motion are obtained when the velocities
$\frac{1}{\mu} \left( {\bf p} - e {\bf A} \right)$ equal to zero. Thus there is no
motion if the system is in the lowest energy configuration. The canonical momentum
${\bf p}$ is different from zero, but this does not relate to movement.
We will now proceed with description of the system within the framework of quantum mechanics.
Let us introduce the dimensionless coordinates $\xi$, $\eta$ such that
$$
x = \frac{\sqrt{\hbar}}{\sqrt{\mu \omega_{t}}} \xi, \;\;
y = \frac{\sqrt{\hbar}}{\sqrt{\mu \omega_{t}}} \eta.
$$
Then the Hamiltonian takes the form:
\begin{eqnarray}
H &=& {\hbar} \omega_{t} \left( -\frac{1}{2} \frac{\nabla_{\rho}^{2}}{2}
+ (\frac{1}{2} + \frac{1}{8} \nu^{2}) \rho^{2} + \right. \nonumber \\
&+& \left. \frac{i}{2}(\nu (\xi \partial_{\eta} - \eta \partial_{\xi}) + b/\rho
\right),
\end{eqnarray}
where $\rho = \sqrt{\xi^{2} + \eta^{2}}$,
$\partial_{\xi} = \partial/\partial_{\xi}$,
$\partial_{\eta} = \partial/\partial_{\eta}$, $\nabla_{\rho}^{2} =
\partial_{\xi}^{2} + \partial_{\eta}^{2}$, $\nu = \omega_{c}/\omega_{t}$,
$b = (k e^{2}/{\hbar}) \sqrt{(m/({\hbar} \omega_{t}))}$,
and $\omega_{c} = e B_{z}/m$ is the cyclotron frequency.
Thus, the parameter $\nu$ is a measure of strength of the magnetic field.
The dynamics of the system are determined by the two parameters $\nu$ and $b$.
In polar coordinates $\rho, \phi$ such that $\xi = \rho \cos{\phi}$,
$\eta = \rho \sin{\phi}$ the Hamiltonian takes the form:
\begin{eqnarray}
H &=& \hbar \omega_{t} \left[-\frac{1}{2} \left( \frac{\partial^{2}}{\partial \rho}^{2}
+ \frac{1}{\rho} \frac{\partial}{\partial \rho} \right) + \right. \nonumber \\
&+& \left. \frac{i}{2} \nu \frac{\partial}{\partial \phi}
+ \frac{1}{2} (1 + \frac{1}{4} \nu^{2}) \rho^{2} - \frac{1}{2} \frac{1}{\rho^{2}}
\frac{\partial^{2}}{\partial \phi^{2}}
+ \frac{b}{\rho} \right].
\end{eqnarray}
Clearly, the $z$-component of the angular momentum commutes
with the Hamiltonian, and the latter admits separation of variables
in the radial coordinates $\rho$ and $\phi$.
Let us consider the time-independent Schr\"odinger equation
\begin{equation}
H \psi = E \psi,
\end{equation}
and let us write the wave function $\psi = \psi(\rho, \phi)$
in terms of separated variables ($\rho$, $\phi$) as:
\begin{equation}
\psi(\rho, \phi) = \frac{1}{\sqrt{\rho}} \chi(\rho) e^{i m \phi}.
\end{equation}
Then the time-independent Schr\"odinger equation takes the form:
\begin{equation}
\label{TIprime}
H^{\prime} \chi(\rho) = E \chi(\rho),
\end{equation}
where
\begin{equation}
H^{\prime} = \frac{1}{2} {\hbar} \omega_{t} \left[
-\frac{\partial^{2}}{\partial \rho^{2}} - m \nu + (1 + \frac{\nu^{2}}{4}) \rho^{2}
+(m^{2} - \frac{1}{4}) \rho^{-2} + \frac{b^{\prime}}{\rho} \right],
\end{equation}
where $b^{\prime} = 2 b$. $H^{\prime}$ can also be written as a sum of the kinetic
term and dimensionless effective potential term:
$$
H^{\prime} = \frac{1}{2} {\hbar} \omega_{t}
\left(-\frac{\partial^{2}}{\partial \rho^{2}} + V(\rho) \right).
$$
With the above forms of the Hamiltonian we are well-prepared to study the spectral
and dynamical features of the system.
In Fig. 1 we have displayed the effective potential $V(\rho)$ as a function
of $\rho$.
\begin{figure}
\begin{center}
\includegraphics[width=7cm, height=6cm]{Rys1a.pdf}
\includegraphics[width=7cm, height=6cm]{Rys1b.pdf}
\includegraphics[width=7cm, height=6cm]{Rys1c.pdf}
\end{center}
\caption{Shape of the dimensionless effective potential $V(\rho)$ as a function
of the radial coordinate $\rho$.
(a) $\nu = 0.5$, $b = 0.1$; (b) $\nu = 0.5$, $b = 10.0$; (c) $\nu = 5.0$, $b = 1.0$.
In all figures the solid line corresponds to $m = 0$, and the dashed line to $m = 1$.}
\end{figure}
\section{The ground-state energy and probability current}
In the presence of the Coulomb interactions the system is not exactly solvable
and the numerical calculations are necessary (although analytical approximations
like WKB or dominant balance are readily available). To solve Eq. (\ref{TIprime})
we have used the standard Rayleigh-Ritz variational method \cite{Messiah}
with radial Gaussian functions $\rho^{(1/2) + m + k} \exp(-(1/2) \rho^{2})$,
$k = 0, 1, 2,...$, as the basis set.
The spectrum of the system is determined by two parameters $\nu$, $b$ and labelled
by the angular momentum quantum number $m$. For sufficiently small magnetic field
(i.e. sufficiently small parameter $\nu$), the ground state of the system is contained
in the manifold of states labelled by $m = 0$. With growing $\nu$, however,
the ground state can be associated with larger $m$.
In Fig. 2 we have shown the dependence of the ground-state energy on the
dimensionless strength
of the magnetic field for two values of $b$ and several values of the quantum
number $m$. It is clear that there exist values of the magnetic induction
for which the ground states becomes degenerate. More importantly, if the magnetic
field is sufficiently strong, the ground state corresponds to the non-zero
value of $m$. Clearly, this is a result of the interplay between the terms
in the Hamiltonian which are linear and quadratic in the magnetic potential.
As the latter approaches zero, the energy spectra for positive and negative $m$
cease to differ, of course.
\begin{figure}
\begin{center}
\includegraphics[width=7cm, height=6cm]{Fig2a.pdf}
\includegraphics[width=7cm, height=6cm]{Fig2b.pdf}
\end{center}
\caption{Dependence of the ground-state energy on the scaled magnetic induction
(a) $b = 1.0$; (b) $b = 5.0$.
In all figures the solid line corresponds to $m = 0$, the dotted line to $m = 1$.
the short-dashed line to $m = 2$, and the long-dashed line - to $m = -1$.}
\end{figure}
Let us notice that no anticrossing appears at the points where various branches
of the spectrum are about to meet. The energy of the ground state is a non-differentiable
function of the magnetic induction.
As one might expect, a good approximation of the spectrum can be obtained
just by taking the expectation values of the Hamiltonian with the crude
apprxoximate wave functions with correct behavior at zero and infinity, i.e.
$$
\chi_{m} = \rho^{m+1/2} \exp(-(1/2) a \rho^{2}),
$$
where $a = \sqrt{1 + \nu^{2}/4}$.
In the context of the present paper, the probability current density in the stationary
states is of considerable interest. It is the current that gives adequate meaning to possible
motion of the system, in the ground state in particular.
In radial coordinates, the only nonvanishing
component of the probability current density is the angular component,
${\bf j}(\rho, \phi) = (0, j(\rho))$ which does not depend on $\phi$. The natural
unit of the probability current is obtained from the natural unit of length
$\sqrt{\mu \omega_{t}/{\hbar}}$ and frequency $\omega_{t}$. Hence we define
the dimensionless current density ${\bf J}(\rho) = (0, J(\rho))$ by
${\bf j}(\rho) = \sqrt{\mu \omega_{t}/{\hbar}} \omega_{t} {\bf J}(\rho)$.
In Fig. 3 we have shown the behavior of $J$ as a function of its argument $\rho$ for
$b = 5.0$, $\nu = 1$, $m = 1$. For those values of $b$ and $\nu$
it is $m=1$ for which the ground state is achieved. Fig. 3(a) displays
the dependence of $J$ on $\rho$ while Fig. 3(b) shows the vector plot of
of ${\bf J}(\rho)$ in the $xy$-plane as computed in the ground state.
\begin{figure}
\begin{center}
\includegraphics[width=7cm, height=6cm]{Fig3a.pdf}
\includegraphics[width=7cm, height=6cm]{Fig3b.pdf}
\end{center}
\caption{Dependence of the probability current density on the position
in the $xy$ plane.
(a) dependence of the probability current density on the radial coordinate;
(b) vector plot of the current density as a functions of the position
in the plane.}
\end{figure}
Interestingly, the probability current density changes its sign with growing
$\rho$ for positive angular momentum.
The probability current itself (the density integrated over the whole plane)
does not vanish, however, in spite of the sign changes. Thus, the expectation value
of velocity (kinetic momentum) is non-zero in the ground state of the system.
This must be the case since the expectation
value of the radial coordinate does not vanish, hence the expectation value
of the vector potential is also non-zero. The fact that the expectation
value of velocity cannot vanish is especially obvious for the zero angular
momentum $m$: indeed, the azimuthal coordinate of the gradient of the wave
function is equal to zero, but the expectation value of the vector potential is not.
\begin{figure}
\begin{center}
\includegraphics[width=7cm, height=6cm]{Fig4a.pdf}
\includegraphics[width=7cm, height=6cm]{Fig4b.pdf}
\includegraphics[width=7cm, height=6cm]{Fig4c.pdf}
\end{center}
\caption{Dependence of the expectation value of the (dimensionless) velocity
on the dimensionless magnetic field $\nu$.
(a) $b=1$; the solid line: $m = 0$, long-dashed line: $m = 1$, short-dashed line: $m = 2$,
dotted line: $m = -1$.
(b) the same as in (a) but for $b = 5$;
(c) dependence of the ground-state expectation value of velocity as a function
of the magnetic field $\nu$; change of the angular momentum corresponding to
the ground state as the magnetic field changes has been taken into account;
solid line: $b = 1$, dashed line: $b = 5$.}
\end{figure}
What we found very interesting is the behavior of the expectation value
of the velocity as dependent on the magnetic field. If Fig. 4(a) and 4(b) we have displayed
that dependence for several values of the angular momentum. Again, for positive values
of the angular momentum the expectation value of velocity changes its sign.
The magnetic field attempts to move the particle
in the positive direction of $\phi$. Thus, in order to "work out" the positive
angular momentum the particle must acquire negative (azimutal) velocity.
Fig. 4(c) illustrates the dependence of the ground-state expectation value of the velocity
as a function of the magnetic field $\nu$. It contains discontinuities because
different value of the angular momentum $m$ correspond to the ground-state energy
as the magnetic field grows as shown in Fig. 2.
\section{Dynamics of the wave packet}
In the next step in our considerations we have studied the dynamics of the wave function
which has initially been prepared as a Gaussian wave packet of the form:
\begin{equation}
\psi(\xi, \eta; 0) = N \exp(-a ((\xi - \xi_{0})^{2} + \eta^{2})).
\end{equation}
One can identify the following physical mechanisms which influence the dynamics
of the wave packet. The first is the harmonic force which, if present alone, would
simply yield the oscillations of the packet. The second one is the magnetic field
with twofold effect: the quadratic part introduces a new frequency to the system
which results in the ``breathing" of the wavepacket; the linear part generates
its overall rotation. Finally, there is the ``scattering" by the repulsive
Coulomb potential which leads to deformation of the packet as shown below.
The dynamics has been studied with the help of the standard split-operator
technique working in Cartesian coordinates. It has been convenient to work
in the dimensionless time $\tau = (1/2) \omega_{t} t$.
The time-evolution operator $U$ can be written in the form:
\begin{eqnarray}
U &=& \exp(-i h \tau) = \exp(-i h_{1} \tau) \exp(-i (h - h_{1}) \tau) = \nonumber \\
&=& \exp(-i h_{1} \tau) \exp(-i h_{2}) \tau) = U_{1} U_{2},
\end{eqnarray}
where $h = H/(\hbar \omega_{t})$, $h_{1} = i \nu (\xi \partial_{\eta} - \eta \partial_{\xi})$.
This results from the fact that the ${\bf A} \cdot {\bf p}$ term which generates
the rotations in the $x-y$ plane, commutes with the rest of the Hamiltonian.
Therefore, it has been convenient to work in the rotating frame $(\xi^{\prime}, \eta^{\prime})$
such that:
$$
\xi = \xi^{\prime} \cos(\tau) + \eta^{\prime} \sin(\tau)
$$
$$
\eta = -\xi^{\prime} \sin(\tau) + \eta^{\prime} \cos(\tau)
$$
The action of $U_{2}$ on $\psi$ has been computed according to:
\begin{eqnarray}
\psi(\xi, \eta, \tau + \Delta \tau) &\approx& \exp(-i (V_{2}/2) \delta \tau)
\cdot \nonumber \\
&\cdot& \exp((i/2)
(\partial_{\xi}^{2} + \partial_{\eta}^{2})\Delta \tau) \nonumber \\
&\cdot& \exp(-i (V_{2}/2) \Delta \tau)
\psi(\xi, \eta, \tau),
\end{eqnarray}
where $V_{2} = (1/2) (1 + 1/4 \nu^{2}) (\xi^{2} + \eta^{2}) + b/\rho$. The action
of the exponential of the kinetic term has been computed using two-dimensional fast Fourier
transformation and its inversion.
The snapshots of shape of the resulting wave functions have been displayed in Fig. 5.
\begin{figure}
\begin{center}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5a.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5b.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5c.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5d.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5e.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5f.png}
\end{center}
\caption{Shaded contour plots of the time-dependent wave packets
in the rotating frame for $b = 1, \nu = 1$.
(a) Initial Gaussian wave packet localized at the point $\xi^{\prime} = 4,
\eta^{\prime} = 0$.
(b)-(e) Shapes of the wave packet at the time $\tau = j \pi/12$, $j=1,2,...,5$.
The darker regions corresponds to larger values of the modulus of wave function.}
\end{figure}
\setcounter{figure}{4}
\begin{figure}
\begin{center}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5g.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5h.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5i.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5j.png}
\includegraphics[width=3.1cm, height=3.1cm]{Fig5k.png}
\end{center}
\caption{(Continued) Shaded contour plots of the time-dependent wave packets
in the rotating frame for $b = 1$, $\nu = 1$.
(g)-(k) Shapes of the wave packet at the time $\tau = j \pi/12$, $j=6,7,...,10$.
The darker regions corresponds to larger values of the modulus of wave function.}
\end{figure}
As can be seen from the above figure, the (initially Gaussian) wave packet does
not return to its initial shape and becomes spread out in the azimutal coordinate
due to the Coulomb ``scattering''.
\section{Concluding remarks}
We have also investigated whether the values of the magnetic induction leads to any
unusual characteristic in the ground state. Instead of the expansion in terms
of the basis sets, we have rather employed the dynamics in the imaginary time
as an alternative way to find the ground state.
However, we have not observed any spectacular changes in the imaginary time dynamics,
which might be associated with such a value of magnetic field which makes the ground
state degenerate.
Let us also notice that we have tested the system for adiabatic versus abrupt turning
on of the magnetic field, looking for any unusual behavior like phase
discontinuities. However, no such effects have been found.
To summarize, we have investigated a system of two identically charged particles
trapped in the harmonic potential with additional constant and homogeneous magnetic field.
The problem becomes effectively one-particle upon introduction of the relative coordinates.
The ground state of such a system can be degenerate for specific values of the magnetic
induction. Also, the ground state of the system can be that of non-vanishing eigenvalue
of the third component of the angular momentum. The ground-state expectation value
of velocity turns out to be non-zero. As a function of the magnetic field, it is discontinuous
at the points where the ground state becomes degenerate.
An interesting feature of the system is the sign change of the azimuthal component
of the probability current density in the ground state as a function of the radial coordinate.
We have also investigated the dynamics of the wavee packet. Due to the presence of the Coulomb
potential, the wavepacket partially loses its Gaussian shape and symmetry while retaining its coherence.
It becomes spread out in the ring trap.
No unusual behavior of the wavepacket have been observed while varying the magnetic field.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,259
|
\section{Introduction}
In this paper we are concerned with the development of a weak Galerkin (WG) finite element method for the quad-curl problem in three dimensions which seeks ${\mathbf{u}}$ such that
\begin{equation}\label{model}
\begin{split}
(\nabla \times)^4 {\mathbf{u}}=&{\mathbf{f}}, \qquad \text{in}\quad \Omega,\\
\nabla\cdot{\mathbf{u}}=&0, \qquad \text{in}\quad \Omega,\\
{\mathbf{u}}\times{\mathbf{n}}=&0, \qquad \text{on}\quad \partial\Omega,\\
\nabla\times{\mathbf{u}}\times{\mathbf{n}}=&0, \qquad \text{on}\quad \partial\Omega,
\end{split}
\end{equation}
for a given ${\mathbf{f}}$ defined on a bounded domain $\Omega\subset \mathbb R^3$.
The quad-curl problems arise in inverse electromagnetic scattering theory for nonhomogeneous
media \cite{13} and magneto-hydrodynamics equations \cite{55}.
Recently, some contributions have been made on the finite element methods for the quad-curl problems. The conforming finite element spaces for the quad-curl problem have been recently constructed in two dimensions (e.g. \cite{28,51}) and in three dimensions (e.g. \cite{27,39,52}). \cite{30,55} proposed the nonconforming and low order finite element spaces for the quad-curl problems. \cite{sun2,49,53} proposed the mixed methods for the quad-curl problems. \cite{10} introduced a formulation using the Hodge decomposition for the quad-curl problems. \cite{25} introduced a discontinuous Galerkin scheme. \cite{sun} proposed a novel weak Galerkin formulation using the conforming space for curl-curl problem as a nonconforming space for the quad-curl problem. \cite{50} analyzed a posteriori error analysis for the quad-curl problems in two dimensions. \cite{54} introduced a virtual element method for the quad-curl problems in two dimensions. \cite{cao} introduced a decoupled formulation for the quad-curl problems where the a priori and a posteriori error were analyzed.
In the literature, the existing WG methods for quad-curl problems proposed in \cite{sun} were curl-conforming and based on tetrahedral partitions. However, our WG method is not necessary to be curl-conforming and is based on any polyhedral partitions. Our WG numerical method (\ref{32})-(\ref{2}) has provided an accurate and reliable numerical solution for the quad-curl system (\ref{model}) in an optimal order of error estimates in discrete norms and in an optimal order of $L^2$ error estimates except the lowest two orders $k=1, 2$. In addition, we have observed some superconvergence phenomena from numerical experiments.
The paper is organized as follows. Section 2 is devoted to the derivation of a weak formulation for the quad-curl system \eqref{model}. Section 3 briefly introduces the discrete weak gradient operator and the discrete weak curl-curl operator. Section 4 is dedicated to the presentation of the weak Galerkin algorithm for the quad-curl problem and a discussion of the solution existence and uniqueness for the WG scheme. In Section 5, the error equations are derived for the WG scheme. Section 6 establishes an optimal order of error estimates in discrete norms for the WG approximation. In Section 7, the $L^2$ error estimate for the WG solution is established in an optimal order except the lowest two orders $k=1, 2$ under some regularity assumptions. Section 8 demonstrates the numerical performance of the WG algorithm through some test examples.
We follow the standard notations for Sobolev spaces and norms defined on a given open and bounded domain $D\subset \mathbb{R}^3$ with Lipschitz continuous boundary. Denote by $\|\cdot\|_{s,D}$, $|\cdot|_{s,D}$ and $(\cdot,\cdot)_{s,D}$ the norm, seminorm and inner product in the Sobolev space $H^s(D)$ for any $s\ge 0$. The space $H^0(D)$ coincides with $L^2(D)$ (i.e., the space of square integrable functions), for which the norm and the inner product are denoted by $\|\cdot \|_{D}$ and $(\cdot,\cdot)_{D}$. When $D=\Omega$ or when the domain of integration is clear from the context, we shall drop the subscript $D$ in the norm and the inner product notation.
\section{A Weak Formulation}\label{Section:2}
Let $s>0$ be an integer. We first introduce
$$
H(curl^s; \Omega)=\{{\mathbf{u}}\in [L^2(\Omega)]^3: (\nabla\times)^j{\mathbf{u}}\in [L^2(\Omega)]^3, j=1, \cdots, s\}
$$
with the associated inner product
$
({\mathbf{u}}, {\mathbf{v}})_{H(curl^s; \Omega)}=({\mathbf{u}}, {\mathbf{v}})+\sum_{j=1}^s ((\nabla\times)^j{\mathbf{u}},(\nabla\times)^j{\mathbf{v}})
$
and the norm $
\|{\mathbf{u}}\|_{H(curl^s; \Omega)}= ({\mathbf{u}}, {\mathbf{u}})^{\frac{1}{2}}_{H(curl^s; \Omega)}$.
We further introduce
$$
H_0(curl; \Omega):=\{{\mathbf{u}}\in H(curl; \Omega): {\mathbf{n}}\times {\mathbf{u}}=0\ \text{on}\ \partial\Omega\},
$$
$$
H_0(curl^2; \Omega):=\{{\mathbf{u}}\in H(curl^2; \Omega): {\mathbf{n}}\times {\mathbf{u}}=0\ \text{and}\ \nabla\times{\mathbf{u}}\times {\mathbf{n}}=0\ \text{on}\ \partial \Omega\}.
$$
We introduce
$$
H(div; \Omega)=\{{\mathbf{u}}\in [L^2(\Omega)]^3: \nabla\cdot{\mathbf{u}} \in L^2(\Omega)\},
$$
with the associated inner product $({\mathbf{u}}, {\mathbf{v}})_{H(div; \Omega)}=({\mathbf{u}}, {\mathbf{v}})+(\nabla\cdot{\mathbf{u}}, \nabla\cdot{\mathbf{v}})$ and the norm $\|{\mathbf{u}}\|_{H(div; \Omega)}= ({\mathbf{u}}, {\mathbf{u}})^{\frac{1}{2}}_{H(div; \Omega)} $. We further introduce
$$
H(div^0; \Omega)=\{{\mathbf{u}}\in H(div;\Omega): \nabla\cdot{\mathbf{u}}=0\
\text{in} \ \Omega\}.
$$
Using the usual integration by parts, we are ready to propose the weak formulation of the quad-curl problem \eqref{model} as follows: Given ${\mathbf{f}}\in H(div^0; \Omega)$, find $({\mathbf{u}}; p)\in H_0(curl^2; \Omega)\times H_0^1(\Omega)$ such that
\begin{equation}\label{weakform}
\begin{split}
((\nabla\times)^2{\mathbf{u}}, (\nabla\times)^2{\mathbf{v}})+({\mathbf{v}}, \nabla p)=&({\mathbf{f}}, {\mathbf{v}}), \qquad {\mathbf{v}} \in H_0(curl^2; \Omega),\\
-({\mathbf{u}}, \nabla q)=&0, \qquad \qquad \forall q\in H_0^1(\Omega).
\end{split}
\end{equation}
\begin{theorem} \cite{sun}
Given ${\mathbf{f}}\in H(div^0; \Omega)$, the problem \eqref{weakform} has a unique solution $({\mathbf{u}}; p)\in H_0(curl^2; \Omega)\times H_0^1(\Omega)$. Furthermore, $p=0$ and ${\mathbf{u}}$ satisfies
$$
\|{\mathbf{u}}\|_{H(curl^2; \Omega)}\leq C\|{\mathbf{f}}\|.
$$
\end{theorem}
\section{Weak Differential Operators}\label{Section:Hessian}
The principal differential operators in the weak formulation (\ref{weakform}) for the quad-curl problem (\ref{model}) are the gradient operator $\nabla$ and the curl-curl operator $(\nabla \times)^2$. We shall briefly review the discrete weak gradient operator \cite{wwconvdiff, wy3655} and define the discrete weak curl-curl operator.
Let $T$ be a polyhedral domain with boundary $\partial T$. A scalar-valued weak function on $T$ refers to $\sigma=\{\sigma_0,\sigma_b\}$ with $\sigma_0\in L^2(T)$ and $\sigma_b\in L^{2}(\partial T)$. Here $\sigma_0$ and $\sigma_b$ are used to represent the value of $\sigma$ in the interior and on the boundary of $T$. Note that $\sigma_b$ may not necessarily be the trace of $\sigma_0$ on $\partial T$. Denote by ${\mathcal{W}}(T)$ the space of scalar-valued weak functions on $T$:
\begin{equation}\label{2.1}
{\mathcal{W}}(T)=\{\sigma=\{\sigma_0,\sigma_b\}: \sigma_0\in L^2(T), \sigma_b\in
L^{2}(\partial T)\}.
\end{equation}
A vector-valued weak function on $T$ refers to a triplet ${\mathbf{v}}=\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}$ where ${\mathbf{v}}_0$ and ${\mathbf{v}}_b$ are used to represent the values of ${\mathbf{v}}$ in the interior and on the boundary of $T$ and ${\mathbf{v}}_n$ represents the value of $\nabla\times{\mathbf{v}}$ on $\partial T$. Note that ${\mathbf{v}}_b$ and ${\mathbf{v}}_n$ may not necessarily be the traces of ${\mathbf{v}}_0$ and $\nabla \times {\mathbf{v}}_0$ on $\partial T$ respectively. Denote by $V(T)$ the space of vector-valued weak functions on $T$:
\begin{equation}
V(T)=\{{\mathbf{v}}=\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}: {\mathbf{v}}_0\in [L^2(T)]^3, {\mathbf{v}}_b\in
[L^{2}(\partial T)]^3, {\mathbf{v}}_n\in [L^{2}(\partial T)]^3\}.
\end{equation}
The weak gradient of $\sigma\in {\mathcal{W}}(T)$, denoted by $\nabla_w \sigma$, is defined as a linear functional on $[H^1(T)]^3$ such that
\begin{equation*}
(\nabla_w \sigma,\boldsymbol{\psi})_T=-(\sigma_0,\nabla \cdot \boldsymbol{\psi})_T+\langle \sigma_b,\boldsymbol{\psi}\cdot \textbf{n}\rangle_{\partial T},
\end{equation*}
for all $\boldsymbol{\psi}\in [H^1(T)]^3$.
The weak curl-curl operator of any ${\mathbf{v}}\in V(T)$, denoted by $(\nabla\times)^2_{w}{\mathbf{v}}$ is defined in the dual space of $H(curl^2; T)$, whose action on ${\mathbf{q}}\in H(curl^2; T)$ is given by
$$
((\nabla \times)^2_{w} {\mathbf{v}}, {\mathbf{q}})_T=({\mathbf{v}}_0, (\nabla\times)^2 {\mathbf{q}})_T-\langle {\mathbf{v}}_b\times{\mathbf{n}}, \nabla\times{\mathbf{q}}\rangle_{\partial T}-\langle {\mathbf{v}}_n\times{\mathbf{n}}, {\mathbf{q}}\rangle_{\partial T}.
$$
Denote by $P_r(T)$ the space of polynomials on $T$ with degree no more than $r$.
A discrete version of $\nabla_{w}\sigma$ for $\sigma\in {\mathcal{W}}(T)$, denoted by $\nabla_{w, r, T}\sigma$, is defined as a unique polynomial vector in $[P_r(T) ]^3$ satisfying
\begin{equation}\label{disgradient}
(\nabla_{w, r, T} \sigma, \boldsymbol{\psi})_T=-(\sigma_0, \nabla \cdot \boldsymbol{\psi})_T+\langle \sigma_b, \boldsymbol{\psi} \cdot \textbf{n}\rangle_{\partial T}, \quad\forall\boldsymbol{\psi}\in [P_r(T)]^3,
\end{equation}
which, from the usual integration by parts, gives
\begin{equation}\label{disgradient*}
(\nabla_{w, r, T} \sigma, \boldsymbol{\psi})_T= (\nabla \sigma_0, \boldsymbol{\psi})_T-\langle \sigma_0- \sigma_b, \boldsymbol{\psi} \cdot \textbf{n}\rangle_{\partial T}, \quad\forall\boldsymbol{\psi}\in [P_r(T)]^3,
\end{equation}
provided that $\sigma_0\in H^1(T)$.
A discrete version of $(\nabla\times)^2_{w}{\mathbf{v}}$ for ${\mathbf{v}}\in V(T)$, denoted by $(\nabla\times)^2_{w, r, T}{\mathbf{v}}$, is defined as a unique polynomial vector in $[P_r(T) ]^3$ satisfying
\begin{equation}\label{discurlcurl}
((\nabla \times)^2_{w, r, T} {\mathbf{v}}, {\mathbf{q}})_T=({\mathbf{v}}_0, (\nabla\times)^2 {\mathbf{q}})_T-\langle {\mathbf{v}}_b\times{\mathbf{n}}, \nabla\times{\mathbf{q}}\rangle_{\partial T}-\langle {\mathbf{v}}_n\times{\mathbf{n}}, {\mathbf{q}}\rangle_{\partial T},
\end{equation}
for any ${\mathbf{q}} \in [P_r(T)]^3$.
\section{Weak Galerkin Algorithm}\label{Section:WGFEM}
Let ${\cal T}_h$ be a finite element partition of the domain $\Omega\subset\mathbb R^3$ consisting of polyhedra that are shape-regular \cite{wy3655}. Denote by ${\mathcal E}_h$ the set of all faces in ${\cal T}_h$ and ${\mathcal E}_h^0={\mathcal E}_h \setminus
\partial\Omega$ the set of all interior faces. Denote by $h_T$ the meshsize of $T\in {\cal T}_h$ and $h=\max_{T\in {\cal T}_h}h_T$ the meshsize for the partition ${\cal T}_h$.
For any given integer $k\geq 1$, denote by
$W_k(T)$ the local discrete space of the scalar-valued weak functions given by
$$
W_k(T)=\{\{\sigma_0,\sigma_b\}:\sigma_0\in P_k(T),\sigma_b\in
P_k(e),e\subset \partial T\}.
$$
Furthermore, denote by
$V_k(T)$ the local discrete space of the vector-valued weak functions given by
$$
V_k(T)=\{\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}:{\mathbf{v}}_0\in [P_k(T)]^3, {\mathbf{v}}_b\in
[P_k(e)]^3, {\mathbf{v}}_n\in
[P_{k-1}(e)]^3, e\subset \partial T\}.
$$
Patching $W_k(T)$ over all the elements $T\in {\cal T}_h$
through a common value $\sigma_b$ on the interior interface ${\mathcal{E}}_h^0$, we arrive at the following scalar-valued
weak finite element space, denoted by $W_h$; i.e.,
$$
W_h=\big\{\{\sigma_0, \sigma_b\}:\{\sigma_0, \sigma_b\}|_T\in W_k(T), \forall T\in {\cal T}_h \big\},
$$
and the subspace of $W_h$ with vanishing boundary values on $\partial\Omega$, denoted by $W_h^0$; i.e.,
\begin{equation}\label{W0}
W_h^0=\{\{\sigma_0, \sigma_b\}\in W_h: \sigma_b=0\ \text{on}\ \partial \Omega\}.
\end{equation}
Similarly, patching $V_k(T)$ over all the elements $T\in {\cal T}_h$
through a common value ${\mathbf{v}}_b$ on the interior interface ${\mathcal{E}}_h^0$, we arrive at the following vector-valued weak finite element space, denoted by $V_h$; i.e.,
$$
V_h=\big\{\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}:\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}|_T\in V_k(T), \forall T\in {\cal T}_h \big\},
$$
and the subspace of $V_h$ with vanishing boundary values on $\partial\Omega$, denoted by $V_h^0$; i.e.,
\begin{equation}\label{V0}
V_h^0=\big\{\{{\mathbf{v}}_0,{\mathbf{v}}_b, {\mathbf{v}}_n\}\in V_h: {\mathbf{v}}_b\times{\mathbf{n}}=0\ \text{and} \ {\mathbf{v}}_n\times{\mathbf{n}}=0\ \ \text{on}\ \partial\Omega\big\}.
\end{equation}
For simplicity of notation and without confusion, for any $\sigma\in
W_h$ and ${\mathbf{v}}\in V_h$, denote by $\nabla_{w}\sigma$ and $(\nabla \times) ^2_{w} {\mathbf{v}}$ the discrete weak actions $\nabla_{w, k, T}\sigma$ and $(\nabla \times) ^2_{w, k-2, T} {\mathbf{v}}$ computed by using (\ref{disgradient}) and \eqref{discurlcurl} on each element $T$; i.e.,
$$
(\nabla_{w}\sigma)|_T= \nabla_{w, k, T}(\sigma|_T), \qquad \sigma\in W_h,
$$
$$
({\nabla\times}^2)_{w} {\mathbf{v}}|_T=({\nabla\times}^2)_{w, k-2, T}({\mathbf{v}}|_T), \qquad {\mathbf{v}}\in V_h.
$$
For any $\sigma, \lambda\in W_h$ and ${\mathbf{u}}, {\mathbf{v}}\in V_h$, we introduce the
following bilinear forms
\begin{align} \label{EQ:local-stabilizer}
a({\mathbf{u}}, {\mathbf{v}})=&\sum_{T\in {\cal T}_h}a({\mathbf{u}}, {\mathbf{v}}),\\
b({\mathbf{u}}, \lambda)=&\sum_{T\in {\cal T}_h}b_T({\mathbf{u}}, \lambda), \\
s_1({\mathbf{u}}, {\mathbf{v}})=&\sum_{T\in {\cal T}_h}s_{1,T}({\mathbf{u}}, {\mathbf{v}}),\\
s_2(\sigma, \lambda)=&\sum_{T\in {\cal T}_h}s_{2,T}(\sigma, \lambda),
\label{EQ:local-bterm}
\end{align}
where
\begin{equation*}
\begin{split}
a_T({\mathbf{u}}, {\mathbf{v}}) =& ((\nabla\times)_w^2 {\mathbf{u}}, (\nabla\times)_w^2 {\mathbf{v}})_T,\\
b_T({\mathbf{u}}, \lambda)=&({\mathbf{u}}_0, \nabla_w \lambda)_T,\\
s_{1,T}({\mathbf{u}}, {\mathbf{v}})=&h_T^{-3}\langle {\mathbf{u}}_0\times{\mathbf{n}}-{\mathbf{u}}_b\times{\mathbf{n}}, {\mathbf{v}}_0\times{\mathbf{n}}-{\mathbf{v}}_b\times{\mathbf{n}} \rangle_{\partial T}\\&+h_T^{-1}\langle \nabla\times{\mathbf{u}}_0\times{\mathbf{n}}-{\mathbf{u}}_n\times{\mathbf{n}}, \nabla\times{\mathbf{v}}_0\times{\mathbf{n}}-{\mathbf{v}}_n\times{\mathbf{n}}\rangle_{\partial T},\\
s_{2,T}(\sigma, \lambda)=&h_T^3 \langle \sigma_0-\sigma_b, \lambda_0-\lambda_b\rangle_{\partial T}.
\end{split}
\end{equation*}
The following is the weak Galerkin scheme for the quad-curl problem (\ref{model}) based on the variational formulation (\ref{weakform}).
\begin{algorithm}\label{a-1}
Given ${\mathbf{f}} \in H(div^0; \Omega)$, find $({\mathbf{u}}_h; p_h)\in V_h^0 \times W_{h}^0$, such that
\begin{eqnarray}\label{32}
s_1({\mathbf{u}}_h, {\mathbf{v}}_h)+a({\mathbf{u}}_h, {\mathbf{v}}_h)+b({\mathbf{v}}_h, p_h)&=& ({\mathbf{f}}, {\mathbf{v}}_0), \qquad \forall {\mathbf{v}}_h\in V_{h}^0,\\
s_2(p_h, q_h)-b({\mathbf{u}}_h, q_h)&=&0,\qquad \quad \qquad \forall q_h\in W_h^0.\label{2}
\end{eqnarray}
\end{algorithm}
\begin{theorem}
The weak Galerkin finite element scheme (\ref{32})-(\ref{2}) has a unique solution.
\end{theorem}
\begin{proof}
It suffices to prove that ${\mathbf{f}}=0$ implies that ${\mathbf{u}}_h=0$ and $p_h=0$ in $\Omega$.
To this end, taking ${\mathbf{v}}_h={\mathbf{u}}_h$ in (\ref{32}) and $q_h=p_h$ in (\ref{2}) gives
$$
((\nabla\times)^2_w{\mathbf{u}}_h, (\nabla\times)^2_w{\mathbf{u}}_h)+s_1({\mathbf{u}}_h, {\mathbf{u}}_h)+s_2(p_h, p_h)=0.
$$
This yields
\begin{eqnarray}
(\nabla\times)^2_w{\mathbf{u}}_h&=&0, \quad \text{in each}\ T,\label{t1}\\
\nabla \times {\mathbf{u}}_0\times {\mathbf{n}} &=&{\mathbf{u}}_n\times {\mathbf{n}}, \quad \text{on each}\ \partial T,\label{t2}\\
{\mathbf{u}}_0 \times {\mathbf{n}}&=&{\mathbf{u}}_b\times {\mathbf{n}}, \quad \text{on each}\ \partial T,\label{t3}\\
p_0&=&p_b, \quad \text{on each}\ \partial T.\label{t4}
\end{eqnarray}
Using \eqref{t1}, \eqref{discurlcurl}, \eqref{t2}-\eqref{t3}, and the integration by parts, we obtain
\begin{equation*}
\begin{split}
0=&((\nabla\times)^2_w{\mathbf{u}}_h, {\mathbf{w}})_T\\
=&((\nabla\times)^2{\mathbf{u}}_0, {\mathbf{w}})_T-\langle {\mathbf{w}}, ({\mathbf{u}}_n-\nabla\times{\mathbf{u}}_0)\times{\mathbf{n}}\rangle_{{\partial T}}+\langle \nabla\times {\mathbf{w}}, ({\mathbf{u}}_0-{\mathbf{u}}_b)\times{\mathbf{n}}\rangle_{{\partial T}}\\
=&((\nabla\times)^2{\mathbf{u}}_0, {\mathbf{w}})_T,
\end{split}
\end{equation*}
for any ${\mathbf{w}}\in [P_{k-2}(T)]^3$. This gives
$(\nabla\times)^2{\mathbf{u}}_0=0$ in each $T\in {\mathcal{T}}_h$. It follows from \eqref{t2}-\eqref{t3} that ${\mathbf{u}}_0\times{\mathbf{n}}$ and $\nabla\times {\mathbf{u}}_0\times {\mathbf{n}}$ are continuous across the interior interface ${\cal E}_h^0$. Thus, ${\mathbf{u}}_0\in H(curl^2; \Omega)$ and $(\nabla \times)^2 {\mathbf{u}}_0=0$ in $\Omega$. Therefore, there exists a potential function $\phi$ such that $\nabla \times {\mathbf{u}}_0=\nabla \phi$ in $\Omega$. This gives
\begin{equation}
\begin{split}
(\nabla \phi, \nabla \phi) &= \sum_{T\in {\cal T}_h} (\nabla \times {\mathbf{u}}_0, \nabla \phi)_T\\
&= \sum_{T\in {\cal T}_h} ( {\mathbf{u}}_0, \nabla \times\nabla \phi)_T +\langle \nabla \phi, {\mathbf{n}}\times {\mathbf{u}}_0\rangle_{\partial T}
\\&= \sum_{T\in {\cal T}_h} \langle \nabla \phi, {\mathbf{n}}\times {\mathbf{u}}_b\rangle_{\partial T}\\
&= \langle \nabla \phi, {\mathbf{n}}\times {\mathbf{u}}_b\rangle_{\partial \Omega}\\&=0,
\end{split}
\end{equation}
where we used the usual integration by parts, \eqref{t3} and ${\mathbf{n}}\times {\mathbf{u}}_b=0$ on $\partial\Omega$.
This leads to $\phi=C$ in $\Omega$, and thus $\nabla \times {\mathbf{u}}_0=0$ in $\Omega$. Furthermore, there exists a potential function $\psi$ such that ${\mathbf{u}}_0=\nabla \psi$ in $\Omega$.
From \eqref{t4}, \eqref{disgradient} and \eqref{2}, we have
\begin{equation}\label{ee1}
\begin{split}
0&=\sum_{T\in {\cal T}_h} (\nabla_w q_h, {\mathbf{u}}_0)_T\\
&=\sum_{T\in {\cal T}_h}-(q_0, \nabla\cdot{\mathbf{u}}_0)_T+\langle q_b, {\mathbf{u}}_0\cdot{\mathbf{n}}\rangle_{{\partial T}}\\
&=\sum_{T\in {\cal T}_h} -(q_0, \nabla\cdot{\mathbf{u}}_0)_T+\sum_{e\in {\cal E}_h^0}\langle q_b, {[\![} {\mathbf{u}}_0\cdot{\mathbf{n}}{]\!]} \rangle_{e},
\end{split}
\end{equation}
where ${[\![} {\mathbf{u}}_0\cdot{\mathbf{n}}{]\!]}$ is the jump of ${\mathbf{u}}_0\cdot{\mathbf{n}}$ on edge $e\in {\cal E}_h^0$ and we used $q_b=0$ on $\partial\Omega$.
Letting $q_0=0$ and $q_b={[\![} {\mathbf{u}}_0\cdot{\mathbf{n}} {]\!]}$ in \eqref{ee1} yields that ${[\![} {\mathbf{u}}_0\cdot{\mathbf{n}} {]\!]}=0$ on $e\in {\cal E}_h^0$ which means ${\mathbf{u}}_0\cdot{\mathbf{n}}$ is continuous along the interior interface $e\in {\cal E}_h^0$. This follows that ${\mathbf{u}}_0\in H(div; \Omega)$. Taking $q_0=\nabla\cdot {\mathbf{u}}_0$ and $q_b=0$ in \eqref{ee1} gives $\nabla \cdot {\mathbf{u}}_0=0$ on each $T$ and further $\nabla \cdot {\mathbf{u}}_0=0$ in $\Omega$ due to ${\mathbf{u}}_0\in H(div; \Omega)$. Recall that there exists a potential function $\psi$ such that ${\mathbf{u}}_0=\nabla \psi$ in $\Omega$. Hence, $\nabla\cdot{\mathbf{u}}_0=\Delta \psi=0$ strongly holds true in $\Omega$ with the boundary condition $\nabla \psi \times {\mathbf{n}}={\mathbf{u}}_0\times {\mathbf{n}}=0$ on $\partial \Omega$. This implies that $\psi=C$ in $\Omega$. Thus, ${\mathbf{u}}_0=\nabla \psi=0$ in $\Omega$. Using \eqref{t2}-\eqref{t3} gives ${\mathbf{u}}_b=0$ and ${\mathbf{u}}_n=0$ in $\Omega$. Therefore, we obtain ${\mathbf{u}}_h=0$ in $\Omega$.
Using ${\mathbf{u}}_h=0$ gives $s_1({\mathbf{u}}_h, {\mathbf{v}}_h)+a({\mathbf{u}}_h, {\mathbf{v}}_h)=0$ for any ${\mathbf{v}}_h\in V_h^0$. It follows from the assumption ${\mathbf{f}}=0$ and \eqref{32} that $b({\mathbf{v}}_h, p_h)=0$, which, together with \eqref{t4} and (\ref{disgradient}) and the usual integration by parts, gives
$$
0=b({\mathbf{v}}_h, p_h)=-\sum_{T\in {\cal T}_h} (p_0, \nabla\cdot{\mathbf{v}}_0)_T+\langle p_b, {\mathbf{v}}_0\cdot{\mathbf{n}}\rangle_{{\partial T}} = \sum_{T\in {\cal T}_h} (\nabla p_0, {\mathbf{v}}_0)_T.
$$
Letting ${\mathbf{v}}_0=\nabla p_0$ gives rise to $\nabla p_0=0$ on each $T\in {\cal T}_h$; i.e., $p_0=C$ on each $T\in {\cal T}_h$. The facts that $p_0=p_b$ on each ${\partial T}$ and $p_b=0$ on $\partial\Omega$ give $p_0=p_b=0$ in $\Omega$ and further $p_h=0$ in $\Omega$.
This completes the proof of the theorem.
\end{proof}
Let $k\geq 1$. Let ${\mathbf{Q}}_0$ be the $L^2$ projection operator onto $[P_k(T)]^3$. Analogously, for $e\subset\partial T$, denote by ${\mathbf{Q}}_b$ and ${\mathbf{Q}}_n$ the $L^2$ projection operators onto $[P_{k}(e)]^3$ and $[P_{k-1}(e)]^3$, respectively. For ${\mathbf{w}}\in [H(curl; \Omega)]^3$, define the $L^2$ projection ${\mathbf{Q}}_h {\mathbf{w}}\in V_h$ as follows
$$
{\mathbf{Q}}_h{\mathbf{w}}|_T=\{{\mathbf{Q}}_0 {\mathbf{w}}, {\mathbf{Q}}_b {\mathbf{w}}, {\mathbf{Q}}_n(\nabla\times{\mathbf{w}})\}.
$$
For $\sigma \in H^1(\Omega)$, the $L^2$ projection $Q_h \sigma\in W_h$ is defined by
$$
Q_h\sigma|_T=\{Q_0 \sigma, Q_b \sigma\},
$$
where $Q_0$ and $Q_b$ are the $L^2$ projection operators onto $P_k(T)$ and $P_k(e)$ respectively. Denote by ${\cal Q}_h^{k-2}$ and ${\cal Q}_h^{k}$ the $L^2$ projection operators onto $P_{k-2}(T)$ and $P_{k}(T)$, respectively.
\begin{lemma}\label{Lemma5.1} The operators ${\mathbf{Q}}_h$, $Q_h$, ${\cal Q}^{k}_h$ and ${\cal Q}^{k-2}_h$ satisfy the following commutative properties:
\begin{eqnarray}\label{l}
(\nabla\times)^2_w({\mathbf{Q}}_h {\mathbf{w}}) &=& {\cal Q}_h^{k-2}((\nabla\times)^2 {\mathbf{w}}), \qquad \forall {\mathbf{w}}\in H(curl^2; T),\\
\nabla_{w}(Q_h \sigma) &=& {\cal Q}^{k}_h(\nabla \sigma), \qquad \qquad \forall \sigma\in H^1(T). \label{l-2}
\end{eqnarray}
\end{lemma}
\begin{proof}
For any ${\mathbf{q}} \in [P_{k-2}(T)]^3$, using \eqref{discurlcurl} and the usual integration by parts gives
\begin{equation*}
\begin{split}
((\nabla\times)_w^2{\mathbf{Q}}_h{\mathbf{w}}, {\mathbf{q}})_T=&({\mathbf{Q}}_0{\mathbf{w}}, (\nabla\times)^2 {\mathbf{q}})_T-\langle {\mathbf{Q}}_b{\mathbf{w}}\times{\mathbf{n}}, \nabla\times{\mathbf{q}}\rangle_{\partial T}-\langle {\mathbf{Q}}_n(\nabla\times{\mathbf{w}})\times{\mathbf{n}}, {\mathbf{q}}\rangle_{\partial T}\\
=&({\mathbf{w}}, (\nabla\times)^2 {\mathbf{q}})_T-\langle{\mathbf{w}}\times{\mathbf{n}}, \nabla\times{\mathbf{q}}\rangle_{\partial T}-\langle \nabla\times {\mathbf{w}}\times{\mathbf{n}}, {\mathbf{q}}\rangle_{\partial T}\\
=&((\nabla\times)^2 {\mathbf{w}}, {\mathbf{q}})_T\\
=&({\cal Q}_h^{k-2}((\nabla\times)^2 {\mathbf{w}}), {\mathbf{q}})_T.
\end{split}
\end{equation*}
This completes the proof of \eqref{l}.
The proof of \eqref{l-2} can be found in \cite{wwconvdiff, wy3655}.
\end{proof}
\section{Error Equations}\label{Section:error-equation}
The goal of this section is to derive the error equations for the weak Galerkin method (\ref{32})-(\ref{2}) for solving the quad-curl problem \eqref{model}, which play a critical role in the forthcoming convergence analysis.
Let $({\mathbf{u}}, p)$ be the solution of \eqref{weakform} and assume that ${\mathbf{u}}\in H(curl^4; \Omega)$. Then $({\mathbf{u}}, p)$ satisfies
\begin{eqnarray} \label{mo1}
((\nabla\times)^4{\mathbf{u}}, {\mathbf{v}})+({\mathbf{v}}, \nabla p)&=&({\mathbf{f}}, {\mathbf{v}}),\\
(\nabla\cdot{\mathbf{u}}, q)&=&0, \label{mo2}
\end{eqnarray}
for ${\mathbf{v}} \in [L^2(\Omega)]^3$ and $q\in L^2(\Omega)$. Let $({\mathbf{u}}_h, p_h)$ be the WG solutions of \eqref{32}-\eqref{2}. Define the error functions ${\mathbf{e}}_h$ and $\epsilon_h$ by
\begin{eqnarray}\label{error}
{\mathbf{e}}_h&=&\{{\mathbf{e}}_0, {\mathbf{e}}_b, {\mathbf{e}}_n\}=\{{\mathbf{Q}}_0{\mathbf{u}}-{\mathbf{u}}_0, {\mathbf{Q}}_b{\mathbf{u}}-{\mathbf{u}}_b, {\mathbf{Q}}_n(\nabla\times{\mathbf{u}})-{\mathbf{u}}_n\},\\
\epsilon_h&=&\{\epsilon_0, \epsilon_b\}=\{Q_0p-p_0, Q_bp-p_b\}. \label{error-2}
\end{eqnarray}
\begin{lemma}\label{errorequa}
Let ${\mathbf{u}}\in H(curl^4; \Omega)$ and $({\mathbf{u}}_h; p_h) \in V_h^0\times W_h^0$ be the exact solution of quad-curl model problem \eqref{model} and the numerical solution arising from the WG scheme (\ref{32})-(\ref{2}) respectively. The error functions ${\mathbf{e}}_h$ and $\epsilon_h$ defined in (\ref{error})-(\ref{error-2}) satisfy the following error equations; i.e.,
\begin{eqnarray}\label{sehv}
s_1({\mathbf{e}}_h, {\mathbf{v}}_h)+a({\mathbf{e}}_h, {\mathbf{v}}_h)+b({\mathbf{v}}_h, \epsilon_h)&=& s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}}_h) +\ell_1({\mathbf{u}}, {\mathbf{v}}_h),\quad \forall {\mathbf{v}}_h\in V_{h}^0,\\
-b({\mathbf{e}}_h, q_h)+s_2(\epsilon_h, q_h)&=&s_2(Q_hp, q_h)-\ell_2({\mathbf{u}}, q_h),\quad\forall q_h\in W^0_h. \label{sehv2}
\end{eqnarray}
\end{lemma}
Here
\begin{eqnarray*}\label{lu}
\ell_1({\mathbf{u}}, {\mathbf{v}}_h)&=&\sum_{T\in{\cal T}_h}\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T} \\&&+\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, ({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T},\\
\ell_2({\mathbf{u}}, q_h)&=& \sum_{T\in{\cal T}_h} \langle q_0-q_b, (I-{\mathbf{Q}}_0){\mathbf{u}} \cdot{\mathbf{n}}\rangle_{{\partial T}}.
\end{eqnarray*}
\begin{proof}
Using \eqref{l}, \eqref{discurlcurl} and the usual integration by parts, we have
\begin{equation}\label{eq1}
\begin{split}
&((\nabla\times)^2_w{\mathbf{Q}}_h{\mathbf{u}}, (\nabla\times)^2_w {\mathbf{v}}_h)_T\\=& ({\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}}), (\nabla\times)^2_w {\mathbf{v}}_h)_T\\
=& ({\mathbf{v}}_0, (\nabla\times)^2 {\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}}))_T-\langle {\mathbf{v}}_b\times{\mathbf{n}}, \nabla\times{\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}\\&-\langle {\mathbf{v}}_n\times{\mathbf{n}}, {\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T} \\
=& ((\nabla\times)^2 {\mathbf{v}}_0, {\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}}))_T+\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times{\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}\\&+\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, {\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T} \\
=& ((\nabla\times)^2 {\mathbf{v}}_0, ((\nabla\times)^2{\mathbf{u}}))_T+\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times{\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}\\&+\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, {\cal Q}_h^{k-2} ((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}.
\end{split}
\end{equation}
Taking ${\mathbf{v}}={\mathbf{v}}_0$ in \eqref{mo1} where ${\mathbf{v}}_h=\{{\mathbf{v}}_0, {\mathbf{v}}_b, {\mathbf{v}}_n\} \in V_h^0$ and using the usual integration by parts, we get
\begin{equation}\label{eq2}
\begin{split}
&\sum_{T\in {\cal T}_h}((\nabla\times)^2{\mathbf{u}}, (\nabla\times)^2{\mathbf{v}}_0)_T+\langle (\nabla\times)^3{\mathbf{u}}, ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}\rangle_{{\partial T}}\\&+\langle (\nabla\times)^2{\mathbf{u}}, \nabla\times{\mathbf{v}}_0\times{\mathbf{n}}-{\mathbf{v}}_n\times{\mathbf{n}}\rangle_{{\partial T}}+(\nabla p, {\mathbf{v}}_0)_T=\sum_{T\in {\cal T}_h} ({\mathbf{f}}, {\mathbf{v}}_0)_T,
\end{split}
\end{equation}
where we used the facts that
$$
\sum_{T\in {\cal T}_h} \langle (\nabla\times)^2{\mathbf{u}}, {\mathbf{v}}_n\times{\mathbf{n}}\rangle_{{\partial T}}=\langle (\nabla\times)^2{\mathbf{u}}, {\mathbf{v}}_n\times{\mathbf{n}}\rangle_{\partial\Omega}=0,
$$
$$
\sum_{T\in {\cal T}_h} \langle (\nabla\times)^3{\mathbf{u}}, {\mathbf{v}}_b\times{\mathbf{n}}\rangle_{{\partial T}}=\langle (\nabla\times)^3{\mathbf{u}}, {\mathbf{v}}_b\times{\mathbf{n}}\rangle_{\partial\Omega}=0.
$$
Substituting \eqref{eq2} into \eqref{eq1} gives
\begin{equation}\label{e3} \begin{aligned}
& \quad \
((\nabla\times)^2_w{\mathbf{Q}}_h{\mathbf{u}}, (\nabla\times)^2_w {\mathbf{v}}_h)\\
&=({\mathbf{f}}-\nabla p, {\mathbf{v}}_0) +\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}\\
&\quad \ +\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, ({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}.
\end{aligned}
\end{equation}
It follows from \eqref{l-2} that
\begin{equation}\label{eq4}
\begin{split}
b({\mathbf{v}}_h, Q_hp)=(\nabla_w(Q_hp), {\mathbf{v}}_0)=({\cal Q}_h^{k}(\nabla p), {\mathbf{v}}_0)=(\nabla p, {\mathbf{v}}_0).
\end{split}
\end{equation}
Combining \eqref{e3}-\eqref{eq4} gives
\begin{equation*}
\begin{split}
&s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}}_h)+a({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}}_h)+b({\mathbf{v}}_h, Q_hp)\\
=& ({\mathbf{f}}, {\mathbf{v}}_0) +\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}\\&+\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, ({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}+s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}}_h).
\end{split}
\end{equation*}
Subtracting \eqref{32} from the above equation gives \eqref{sehv}.
To derive \eqref{sehv2}, taking $q=q_0$ in \eqref{mo2} and using the usual integration by parts, we have
\begin{equation}\label{eq5}
0=-\sum_{T\in {\cal T}_h} ({\mathbf{u}}, \nabla q_0)+\sum_{T\in {\cal T}_h} \langle {\mathbf{u}}\cdot{\mathbf{n}}, q_0-q_b\rangle_{{\partial T}},
\end{equation}
where we used $\sum_{T\in {\cal T}_h} \langle {\mathbf{u}}\cdot{\mathbf{n}}, q_b\rangle_{{\partial T}}=0$.
Using \eqref{disgradient} and the usual integration by parts gives
\begin{equation} \label{beq}
\begin{split}
-b({\mathbf{Q}}_h{\mathbf{u}}, q_h) =&-\sum_{T\in {\cal T}_h} ({\mathbf{Q}}_0{\mathbf{u}}, \nabla_w q_h)_T\\
=&\sum_{T\in {\cal T}_h} (q_0, \nabla\cdot({\mathbf{Q}}_0{\mathbf{u}}))_T-\langle q_b, {\mathbf{Q}}_0{\mathbf{u}}\cdot{\mathbf{n}}\rangle_{{\partial T}}\\
=&\sum_{T\in {\cal T}_h} -(\nabla q_0, {\mathbf{Q}}_0{\mathbf{u}})_T+\langle q_0-q_b, {\mathbf{Q}}_0{\mathbf{u}}\cdot{\mathbf{n}}\rangle_{{\partial T}}\\
=&\sum_{T\in {\cal T}_h} -(\nabla q_0, {\mathbf{u}})_T+\langle q_0-q_b, {\mathbf{Q}}_0{\mathbf{u}}\cdot{\mathbf{n}}\rangle_{{\partial T}}\\
=&\sum_{T\in {\cal T}_h}\langle q_0-q_b, ({\mathbf{Q}}_0-I){\mathbf{u}}\cdot{\mathbf{n}}\rangle_{{\partial T}},
\end{split}
\end{equation}
where we used \eqref{eq5} on the last line.
Subtracting \eqref{2} from the above equation completes the proof of
\eqref{sehv2}.
This completes the proof of the lemma.
\end{proof}
\section{Error Estimates}\label{Section:error-estimates}
For any ${\mathbf{v}}\in V_h^0$, we define the energy norm ${|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{v}}{|\hspace{-.02in}|\hspace{-.02in}|}$ as follows
\begin{equation}
{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{v}}{|\hspace{-.02in}|\hspace{-.02in}|}^2=\sum_{T\in {\cal T}_h} \| (\nabla \times)_w^2{\mathbf{v}}\|_T^2+s_1({\mathbf{v}}, {\mathbf{v}}).
\end{equation}
It is easy to check that ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot {|\hspace{-.02in}|\hspace{-.02in}|}$ is a semi-norm in $V_h^0$. We further introduce a norm in $V_h^0$; i.e.,
\begin{equation}
{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{v}}{|\hspace{-.02in}|\hspace{-.02in}|}_1 ={|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{v}}{|\hspace{-.02in}|\hspace{-.02in}|}+\Big(\sum_{T\in {\cal T}_h} \|\nabla\cdot {\mathbf{v}}_0\|_T^2\Big)^{\frac{1}{2}}+\Big(\sum_{e\in {\cal E}^0_h} h_T^{-1}\|{[\![} {\mathbf{v}}_0\cdot{\mathbf{n}}{]\!]} \|_e^2\Big)^{\frac{1}{2}}.
\end{equation}
For any $q\in W_h^0$, we define the following norm
$$
{|\hspace{-.02in}|\hspace{-.02in}|} q{|\hspace{-.02in}|\hspace{-.02in}|}_0=(s_2(q, q))^{\frac{1}{2}}.
$$
Recall that ${\mathcal{T}}_h$ is a shape-regular finite element partition of
the domain $\Omega$. For any $T\in{\mathcal{T}}_h$ and $\varphi\in H^{1}(T)$, the following trace inequality holds true \cite{wy3655}:
\begin{equation}\label{trace-inequality}
\|\varphi\|_{{\partial T}}^2 \leq C
(h_T^{-1}\|\varphi\|_{T}^2+h_T\| \varphi\|_{1, T}^2).
\end{equation}
Furthermore, if $\varphi$ is a polynomial on $T$, the standard inverse inequality yields
\begin{equation}\label{trace}
\|\varphi\|_{{\partial T}}^2 \leq Ch_T^{-1}\|\varphi\|_{T}^2.
\end{equation}
\begin{lemma}\label{lem2} Let $k\geq 1$, and $s\in [1, k]$.
Suppose ${\mathbf{u}}\in [H^{k+1}(\Omega)]^3$ and $(\nabla\times)^2{\mathbf{u}}\in [H^k(\Omega)]^3$. Then, for $({\mathbf{v}}, q)\in V_h^0\times W_h^0$, the following estimates hold true; i.e.,
\begin{eqnarray}\label{error1}
|s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}})|&\leq& Ch^{s-1} \|{\mathbf{u}}\|_{s+1} s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}},\\
|\ell_1({\mathbf{u}}, {\mathbf{v}})|&\leq& Ch^{s-1}\|(\nabla\times)^2{\mathbf{u}}\|_{s-1}s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}},\label{error2}\\
|\ell_2({\mathbf{u}}, q)| &\leq& Ch^{s-1}\|{\mathbf{u}}\|_{s+1}{|\hspace{-.02in}|\hspace{-.02in}|} q {|\hspace{-.02in}|\hspace{-.02in}|}_0,\label{error3}\\
|s_2(Q_hp, q)|&=&0.\label{error4}
\end{eqnarray}
\end{lemma}
\begin{proof}
Using the Cauchy-Schwarz inequality, the trace inequality \eqref{trace-inequality}, gives
\begin{equation*}
\begin{split}
&|s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{v}})| = \Big| \sum_{T\in {\cal T}_h}h_T^{-3}\langle ({\mathbf{Q}}_0{\mathbf{u}}-{\mathbf{Q}}_b{\mathbf{u}})\times{\mathbf{n}}, ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}\rangle_{{\partial T}}\\&+ h_T^{-1} \langle \nabla\times{\mathbf{Q}}_0{\mathbf{u}}\times{\mathbf{n}}-{\mathbf{Q}}_n(\nabla\times{\mathbf{u}})\times{\mathbf{n}}, \nabla\times{\mathbf{v}}_0\times{\mathbf{n}}-{\mathbf{v}}_n\times{\mathbf{n}}\rangle_{{\partial T}} \Big|\\
\leq & \{\big( \sum_{T\in {\cal T}_h}h_T^{-3}\| {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}}\|^2_{{\partial T}}\big)^{\frac{1}{2}}+ \big( \sum_{T\in {\cal T}_h}h_T^{-1}\|\nabla\times( {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}})\|^2_{{\partial T}}\big)^{\frac{1}{2}}\}s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}\\
\leq & \{\big( \sum_{T\in {\cal T}_h}h_T^{-4}\| {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}}\|^2_{T}+h_T^{-2}\| {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}}\|^2_{1, T}\big)^{\frac{1}{2}}\\
&+ \big( \sum_{T\in {\cal T}_h}h_T^{-2}\|\nabla\times( {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}})\|^2_{T}+\|\nabla\times( {\mathbf{Q}}_0{\mathbf{u}}- {\mathbf{u}})\|^2_{1,T}\big)^{\frac{1}{2}}\}s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}\\
\leq & C h^{s-1}\|{\mathbf{u}}\|_{s+1} s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}.
\end{split}
\end{equation*}
Using the Cauchy-Schwarz inequality, the trace inequality \eqref{trace-inequality}, gives
\begin{equation*}
\begin{split}
&\ell_1({\mathbf{u}}, {\mathbf{v}}) \\=&\sum_{T\in{\cal T}_h}\langle ({\mathbf{v}}_0-{\mathbf{v}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T} \\&+\langle (\nabla \times {\mathbf{v}}_0-{\mathbf{v}}_n)\times{\mathbf{n}}, ({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\rangle_{\partial T}
\\
\leq &\{\big(\sum_{T\in{\cal T}_h}h_T^3\|\nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{\partial T}\big)^{\frac{1}{2}} \\&+\big(\sum_{T\in{\cal T}_h}h_T\|({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{\partial T}\big)^{\frac{1}{2}}\} s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}\\
\leq &\{\big(\sum_{T\in{\cal T}_h}h_T^2\|\nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{T}+h_T^4\|\nabla\times({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{1, T}\big)^{\frac{1}{2}} \\&+ \big(\sum_{T\in{\cal T}_h} \|({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{T}+h_T^2\|({\cal Q}_h^{k-2} -I)((\nabla\times)^2{\mathbf{u}})\|^2_{1, T}\big)^{\frac{1}{2}}\} s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}\\
\leq & Ch^{s-1} \|(\nabla\times)^2 {\mathbf{u}}\|_{s-1} s_1({\mathbf{v}}, {\mathbf{v}})^{\frac{1}{2}}.
\end{split}
\end{equation*}
Similarly, using the Cauchy-Schwarz inequality, the trace inequality \eqref{trace-inequality} gives
\begin{equation*}
\begin{split}
\ell_2({\mathbf{u}}, q)&=\sum_{T\in{\cal T}_h}\langle q_0-q_b, (I-{\mathbf{Q}}_0){\mathbf{u}} \cdot{\mathbf{n}}\rangle_{{\partial T}}\\
&\leq \Big(\sum_{T\in{\cal T}_h} h_T^3\|q_0-q_b\|_{{\partial T}}^2\Big)^{\frac{1}{2}}\Big(\sum_{T\in{\cal T}_h}h_T^{-3} \|(I-{\mathbf{Q}}_0){\mathbf{u}} \cdot{\mathbf{n}}\|_{{\partial T}}^2\Big)^{\frac{1}{2}}\\
&\leq \Big(\sum_{T\in{\cal T}_h} h_T^{-4}\|(I-{\mathbf{Q}}_0){\mathbf{u}} \cdot{\mathbf{n}}\|_{T}^2+ h_T^{-2}\|(I-{\mathbf{Q}}_0){\mathbf{u}} \cdot{\mathbf{n}}\|_{1,T}^2\Big)^{\frac{1}{2}}{|\hspace{-.02in}|\hspace{-.02in}|} q{|\hspace{-.02in}|\hspace{-.02in}|}_0 \\
&\leq Ch^{s-1}\|{\mathbf{u}}\|_{s+1}{|\hspace{-.02in}|\hspace{-.02in}|} q{|\hspace{-.02in}|\hspace{-.02in}|}_0.
\end{split}
\end{equation*}
Since $p=0$, it is easy to obtain $s_2(Q_hp, q)=0$.
\end{proof}
\begin{theorem}\label{thm}
Let $k\geq 1$. Suppose that ${\mathbf{u}}\in [H^{k+1}(\Omega)]^3$. The following error estimate holds
\begin{equation}\label{estimate1}
{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}+{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h {|\hspace{-.02in}|\hspace{-.02in}|}_0 \leq Ch^{k-1} (\|{\mathbf{u}}\|_{k+1}+\|(\nabla \times)^2{\mathbf{u}}\|_{k-1}).
\end{equation}
\end{theorem}
\begin{proof}
Letting ${\mathbf{v}}_h={\mathbf{e}}_h$ in (\ref{sehv}) and $q_h=\epsilon_h$ in (\ref{sehv2}) and adding the two equations, we have
\begin{equation*}
\begin{split}
{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}^2+{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h {|\hspace{-.02in}|\hspace{-.02in}|}_0^2 &=s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{e}}_h)+s_2(Q_hp, \epsilon_h)+\ell_1({\mathbf{u}}, {\mathbf{e}}_h)-\ell_2({\mathbf{u}}, \epsilon_h).\\
\end{split}
\end{equation*}
Using Lemma \ref{lem2} completes the proof of the theorem.
\end{proof}
\section{$L^2$ Error Estimates}
We consider an auxiliary problem of finding $({\boldsymbol{\phi}}; \xi)$ such that
\begin{equation}\label{dual}
\begin{split}
(\nabla \times)^4 {\boldsymbol{\phi}}+\nabla \xi =&{\mathbf{e}}_0, \qquad \text{in}\ \Omega,\\
\nabla\cdot{\boldsymbol{\phi}} =& 0, \qquad\text{in} \ \Omega,\\
{\boldsymbol{\phi}}\times{\mathbf{n}}=& 0, \qquad\text{on} \ \partial\Omega,\\
\nabla \times{\boldsymbol{\phi}}=& 0, \qquad\text{on} \ \partial\Omega,\\
\xi=& 0, \qquad\text{on} \ \partial\Omega.
\end{split}
\end{equation}
Let ${t_0}=\min\{k, 3\}$. We assume the regularity property holds true in the sense that ${\boldsymbol{\phi}}$ and $\xi$ satisfy
\begin{equation}\label{regu}
\|{\boldsymbol{\phi}}\|_{t_0+1} +\|(\nabla\times)^2{\boldsymbol{\phi}}\|_{t_0-1}+\|\xi\|_{1}\leq C\| {\mathbf{e}}_0\|.
\end{equation}
\begin{theorem}
Let $k\geq 1$ and ${t_0}=\min\{k, 3\}$. Suppose that ${\mathbf{u}}\in [H^{k+1}(\Omega)]^3$. The following estimate holds
\begin{equation}
\|{\mathbf{e}}_0\| \leq Ch^{t_0+k-2} (\|{\mathbf{u}}\|_{k+1}+\|(\nabla \times)^2{\mathbf{u}}\|_{k-1}).
\end{equation}
In other words, we have a sub-optimal order of convergence for $k=1, 2$ and optimal order of convergence for $k\geq 3$.
\end{theorem}
\begin{proof}
Using the usual integration by parts, letting ${\mathbf{u}}={\boldsymbol{\phi}}$ and ${\mathbf{v}}_h={\mathbf{e}}_h$ in \eqref{eq1}, letting ${\mathbf{v}}_h={\mathbf{Q}}_h{\boldsymbol{\phi}}$ in \eqref{sehv}, letting ${\mathbf{u}}={\boldsymbol{\phi}}$ and $q_h=\epsilon_h$ in \eqref{beq}, letting $q_h=Q_h\xi$ in \eqref{sehv2} and \eqref{l-2}, we have
\begin{equation}\label{dq}
\begin{split}
& \|{\mathbf{e}}_0\|^2\\=&\sum_{T\in {\cal T}_h}((\nabla \times)^4 {\boldsymbol{\phi}}+\nabla \xi, {\mathbf{e}}_0)_T\\
=&\sum_{T\in {\cal T}_h} ((\nabla \times)^2{\boldsymbol{\phi}}, (\nabla \times)^2{\mathbf{e}}_0)_T+\langle \nabla\times{\mathbf{e}}_0, {\mathbf{n}}\times (\nabla \times)^2{\boldsymbol{\phi}}\rangle_{\partial T}\\
&+\langle {\mathbf{e}}_0,{\mathbf{n}}\times(\nabla \times)^3{\boldsymbol{\phi}}\rangle_{\partial T} +({\mathbf{e}}_0, {\cal Q}^{k}_h \nabla \xi)_T\\
=&\sum_{T\in {\cal T}_h}((\nabla\times)^2_w{\mathbf{Q}}_h{\boldsymbol{\phi}}, (\nabla\times)^2_w {\mathbf{e}}_h)_T -\langle ({\mathbf{e}}_0-{\mathbf{e}}_b)\times{\mathbf{n}}, \nabla\times{\cal Q}_h^{k-1} ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T}\\&-\langle (\nabla \times {\mathbf{e}}_0-{\mathbf{e}}_n)\times{\mathbf{n}}, {\cal Q}_h^{k-1} ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T}
\\& +\langle \nabla\times{\mathbf{e}}_0, {\mathbf{n}}\times (\nabla \times)^2{\boldsymbol{\phi}}\rangle_{\partial T}+\langle {\mathbf{e}}_0,{\mathbf{n}}\times(\nabla \times)^3{\boldsymbol{\phi}}\rangle_{\partial T} +(\nabla_w Q_h\xi, {\mathbf{e}}_0)_T
\\=&-s_1({\mathbf{e}}_h, {\mathbf{Q}}_h{\boldsymbol{\phi}})-b({\mathbf{Q}}_h{\boldsymbol{\phi}}, \epsilon_h) + s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}}) +\ell_1({\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}}) \\
&+\sum_{T\in {\cal T}_h}-\langle ({\mathbf{e}}_0-{\mathbf{e}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T}\\&-\langle (\nabla \times {\mathbf{e}}_0-{\mathbf{e}}_n)\times{\mathbf{n}},( {\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T} \\
&+
s_2(\epsilon_h, Q_h\xi)-s_2(Q_hp, Q_h\xi)+\ell_2({\mathbf{u}}, Q_h\xi)
\\=&-s_1({\mathbf{e}}_h, {\mathbf{Q}}_h{\boldsymbol{\phi}}) + s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}}) +\ell_1({\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}}) \\
&+\sum_{T\in {\cal T}_h} \langle \epsilon_0-\epsilon_b, ({\mathbf{Q}}_0-I){\boldsymbol{\phi}}\cdot{\mathbf{n}}\rangle_{{\partial T}}\\&
-\langle ({\mathbf{e}}_0-{\mathbf{e}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T} \\&-\langle (\nabla \times {\mathbf{e}}_0-{\mathbf{e}}_n)\times{\mathbf{n}},( {\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T} \\
&+
s_2(\epsilon_h, Q_h\xi)-s_2(Q_hp, Q_h\xi)+\ell_2({\mathbf{u}}, Q_h\xi),
\end{split}
\end{equation}
where we used ${\mathbf{e}}_b\times{\mathbf{n}}=0$ and ${\mathbf{e}}_n\times{\mathbf{n}}=0$ on $\partial\Omega$.
Next, we shall estimate the terms on the last line of \eqref{dq} one by one.
Recall that $t_0=\min\{k, 3\}$. Using \eqref{error1} with ${\mathbf{v}}={\mathbf{e}}_h$ and ${\mathbf{u}}={\boldsymbol{\phi}}$, we have
\begin{equation}\label{q1}
|s_1({\mathbf{e}}_h, {\mathbf{Q}}_h{\boldsymbol{\phi}})| \leq Ch^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1} s_1({\mathbf{e}}_h, {\mathbf{e}}_h)^{\frac{1}{2}}\leq Ch^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1} {|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}.
\end{equation}
Using \eqref{error1} with ${\mathbf{v}}={\mathbf{Q}}_h{\boldsymbol{\phi}}$, we have
\begin{equation}\label{q2}
|s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})| \leq Ch^{k-1} \|{\mathbf{u}}\|_{k+1} s_1({\mathbf{Q}}_h{\boldsymbol{\phi}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})^{\frac{1}{2}}.
\end{equation}
Note that
\begin{equation}\label{s1}
\begin{split}
&s_1({\mathbf{Q}}_h{\boldsymbol{\phi}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})^{\frac{1}{2}}\\ \leq& C\sum_{T\in {\cal T}_h} h_T^{-3}\| {\mathbf{Q}}_0{\boldsymbol{\phi}}\times{\mathbf{n}}-{\mathbf{Q}}_b{\boldsymbol{\phi}} \times{\mathbf{n}}\|_{\partial T}^2\\&+ h_T^{-1}\| \nabla\times {\mathbf{Q}}_0{\boldsymbol{\phi}}\times{\mathbf{n}}-{\mathbf{Q}}_n(\nabla \times{\boldsymbol{\phi}}) \times{\mathbf{n}}\|_{\partial T}^2\\
\leq& C\sum_{T\in {\cal T}_h} h_T^{-4}\| {\mathbf{Q}}_0{\boldsymbol{\phi}} - {\boldsymbol{\phi}} \|_{T}^2+h_T^{-2}\| {\mathbf{Q}}_0{\boldsymbol{\phi}} - {\boldsymbol{\phi}} \|_{1,T}^2\\&+ h_T^{-2}\| {\mathbf{Q}}_0{\boldsymbol{\phi}} - {\boldsymbol{\phi}} \|_{1, T}^2+ \| {\mathbf{Q}}_0{\boldsymbol{\phi}} - {\boldsymbol{\phi}} \|_{2, T}^2\\
\leq & Ch^{{t_0}-1}\|{\boldsymbol{\phi}}\|_{{t_0}+1}.
\end{split}
\end{equation}
where we used trace inequality \eqref{trace-inequality}. Substituting \eqref{s1} into \eqref{q2} gives
\begin{equation}\label{q2_2}
|s_1({\mathbf{Q}}_h{\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})| \leq Ch^{k-1} \|{\mathbf{u}}\|_{k+1} h^{{t_0}-1}\|{\boldsymbol{\phi}}\|_{{t_0}+1}.
\end{equation}
Using \eqref{error2} with ${\mathbf{v}}={\mathbf{Q}}_h{\boldsymbol{\phi}}$ and \eqref{s1}, we have
\begin{equation}\label{q3}
|\ell_1({\mathbf{u}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})| \leq Ch^{k-1}\|(\nabla\times)^2{\mathbf{u}}\|_{k-1}s_1({\mathbf{Q}}_h{\boldsymbol{\phi}}, {\mathbf{Q}}_h{\boldsymbol{\phi}})^{\frac{1}{2}}\leq Ch^{k-1}\|(\nabla\times)^2{\mathbf{u}}\|_{k-1} h^{{t_0}-1}\|{\boldsymbol{\phi}}\|_{{t_0}+1}.
\end{equation}
Using \eqref{error3} with ${\mathbf{u}}={\boldsymbol{\phi}}$ and $q_h=\epsilon_h$, we have
\begin{equation}\label{q4}
|\sum_{T\in {\cal T}_h} \langle \epsilon_0-\epsilon_b, ({\mathbf{Q}}_0-I){\boldsymbol{\phi}}\cdot{\mathbf{n}}\rangle_{{\partial T}} | \leq Ch^{{t_0-1}}\|{\boldsymbol{\phi}}\|_{{t_0}+1}{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h {|\hspace{-.02in}|\hspace{-.02in}|}_0,
\end{equation}
Using \eqref{error2} with ${\mathbf{u}}={\boldsymbol{\phi}}$ and ${\mathbf{v}}={\mathbf{e}}_h$, we have
\begin{equation}\label{q5}
\begin{split}
& |\sum_{T\in {\cal T}_h} \langle ({\mathbf{e}}_0-{\mathbf{e}}_b)\times{\mathbf{n}}, \nabla\times({\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T}\\&+\langle (\nabla \times {\mathbf{e}}_0-{\mathbf{e}}_n)\times{\mathbf{n}},( {\cal Q}_h^{k-1}-I) ((\nabla\times)^2{\boldsymbol{\phi}})\rangle_{\partial T}|\\
\leq & Ch^{{t_0}-1}\|(\nabla\times)^2 {\boldsymbol{\phi}}\|_{{t_0}-1}s_1({\mathbf{e}}_h, {\mathbf{e}}_h)^{\frac{1}{2}}
\leq Ch^{{t_0}-1}\|(\nabla\times)^2 {\boldsymbol{\phi}}\|_{{t_0}-1}{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}.
\end{split}
\end{equation}
Using the Cauchy-Schwartz inequality and trace inequality \eqref{trace-inequality}, we have
\begin{equation}
\begin{split}
s_2(\epsilon_h, Q_h\xi) =&\sum_{T\in {\cal T}_h} h_{T}^3 \langle \epsilon_0-\epsilon_b, Q_0\xi-Q_b\xi\rangle_{\partial T} \\
\leq &C{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h{|\hspace{-.02in}|\hspace{-.02in}|}_0 \Big(\sum_{T\in {\cal T}_h} h_{T}^3 \|Q_0\xi- \xi\|_{\partial T}^2\big)^{\frac{1}{2}}\\
\leq &C{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h{|\hspace{-.02in}|\hspace{-.02in}|}_0 \Big(\sum_{T\in {\cal T}_h}h_T^{2} \|Q_0\xi- \xi\|_{ T}^2+ h_T^{4}\|Q_0\xi- \xi\|_{1, T}^2\big)^{\frac{1}{2}}\\
\leq &Ch^{2}\|\xi\|_{1}{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h{|\hspace{-.02in}|\hspace{-.02in}|}_0.
\end{split}
\end{equation}
Using \eqref{error4} with $q=Q_h\xi$, we have
\begin{equation}
s_2(Q_hp, Q_h\xi)=0.
\end{equation}
Using \eqref{error3} with $q=Q_h\xi$ and the trace inequality \eqref{trace-inequality}, we have
\begin{equation}\label{last}
\begin{split}
\ell_2({\mathbf{u}}, Q_h\xi)&\leq Ch^{k-1}\|{\mathbf{u}}\|_{k+1} {|\hspace{-.02in}|\hspace{-.02in}|} Q_h\xi {|\hspace{-.02in}|\hspace{-.02in}|}_0\\
&\leq Ch^{k-1}\|{\mathbf{u}}\|_{k+1} \big(\sum_{T\in {\cal T}_h}h_T^3 \|Q_0\xi-Q_b\xi\|_{\partial T}^2\big)^{\frac{1}{2}}\\
&\leq Ch^{k-1}\|{\mathbf{u}}\|_{k+1} \big(\sum_{T\in {\cal T}_h}h_T ^3\|Q_0\xi- \xi\|_{\partial T}^2\big)^{\frac{1}{2}} \\
&\leq Ch^{k-1}\|{\mathbf{u}}\|_{k+1} \big(\sum_{T\in {\cal T}_h}h_T^{2}\|Q_0\xi- \xi\|_{ T}^2+h_T ^4\|Q_0\xi- \xi\|_{1, T}^2\big)^{\frac{1}{2}}\\
&\leq Ch^{k-1}\|{\mathbf{u}}\|_{k+1} h^{2}\|\xi\|_{1}.
\end{split}
\end{equation}
Substituting \eqref{q1}-\eqref{last} into
\eqref{dq} and using the regularity assumption \eqref{regu} and the error estimate \eqref{estimate1} gives
\begin{equation*}
\begin{split}
\|{\mathbf{e}}_0\|^2\leq &C h^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1} {|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}+ C h^{k-1}\|{\mathbf{u}}\|_{k+1}h^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1}\\
&+ C h^{k-1}\|(\nabla\times)^2{\mathbf{u}}\|_{k-1}h^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1}+C h^{{t_0}-1} \|{\boldsymbol{\phi}}\|_{{t_0}+1} {|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h{|\hspace{-.02in}|\hspace{-.02in}|}_0\\
&+ Ch^{{t_0}-1}\|(\nabla\times)^2 {\boldsymbol{\phi}}\|_{{t_0-1}}{|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{e}}_h{|\hspace{-.02in}|\hspace{-.02in}|}+
Ch^{2}\|\xi\|_{1}{|\hspace{-.02in}|\hspace{-.02in}|} \epsilon_h{|\hspace{-.02in}|\hspace{-.02in}|}_0 \\&+Ch^{k-1}\|{\mathbf{u}}\|_{k+1} h^{2}\|\xi\|_{1},
\end{split}
\end{equation*}
which yields
\begin{equation*}
\begin{split}
\|{\mathbf{e}}_0\|^2 \leq Ch^{t_0+k-2} (\|{\mathbf{u}}\|_{k+1}+\|(\nabla \times)^2{\mathbf{u}}\|_{k-1})\|{\mathbf{e}}_0\|.
\end{split}
\end{equation*}
This completes the proof of the theorem.
\end{proof}
\section{Numerical tests}
In this section, we present some numerical results for the WG finite
element method for solving the quad-curl problem analyzed in the previous sections. To this end, we shall solve the following quad-curl problem with non-homogeneous boundary conditions on an unit cube domain $\Omega=(0,1)^3$: Find an known ${\mathbf{u}}$ such that
\begin{equation}\label{s-1}
\begin{split}
(\nabla \times)^4 {\mathbf{u}}=&{\mathbf{f}}, \qquad \text{in}\quad \Omega,\\
\nabla\cdot{\mathbf{u}}=&0, \qquad \text{in}\quad \Omega,\\
{\mathbf{u}}\times{\mathbf{n}}=&{\mathbf g}_1, \qquad \text{on}\quad \partial\Omega,\\
\nabla\times{\mathbf{u}}\times{\mathbf{n}}=&{\mathbf g}_2, \qquad \text{on}\quad \partial\Omega,
\end{split}
\end{equation} where ${\mathbf{f}}$, ${\mathbf g}_1$ and ${\mathbf g}_2$ are calculated by the exact solution
\begin{align*} {\mathbf{u}} = \begin{pmatrix} -2 x^2 y^2 z \\
2 x^2 y^3 z \\
-x y^2 z^2 (3 x - 2) \end{pmatrix}.
\end{align*}
\begin{figure}[ht]
\begin{center}
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\end{picture}}
\end{picture}
\end{center}
\caption{ The first three levels of uniform cubic grids used in Table \ref{t-1}. }
\label{g-1}
\end{figure}
We first compute the solution of \eqref{s-1} by the $P_k$ weak Galerkin finite element method \eqref{32}-\eqref{2} on uniform cubic grids shown in
Figure \ref{g-1}.
For simplicity of notations, we denote the WG finite element solution $({\mathbf{u}}_h; p_h)$ by $\{P_k, P_k, P_{k-1}\}$-$P_{k-1}$ with $\{P_k, P_k\}$-$P_k$.
In Table \ref{t-1}, we list the errors in various norms and the computed orders of convergence for $P_2$, $P_3$, $P_4$ and $P_5$
finite element solutions on uniform cubic grids. It seems we do have one order superconvergence in most cases in Table \ref{t-1}.
\begin{table}[ht]
\centering \renewcommand{\arraystretch}{1.05}
\caption{ Error profiles and convergence rates on uniform cubic grids shown in Figure \ref{g-1} for \eqref{s-1}. }
\label{t-1}
\begin{tabular}{c|cc|cc|cc}
\hline
level & $\|{\mathbf{Q}}_h {\mathbf{u}}- {\mathbf{u}}_h \| $ &rate & ${|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{Q}}_h {\mathbf{u}}- {\mathbf{u}}_h {|\hspace{-.02in}|\hspace{-.02in}|} $ &rate & $\| p_h \| $ &rate \\
\hline
&\multicolumn{6}{c}{by the $\{P_2, P_2, P_1\}$-$P_1$ with $\{P_2, P_2\}$-$P_2$ WG method} \\ \hline
2& 0.5263E+00& 3.8& 0.4078E+01& 1.7& 0.1742E-01& 3.9 \\
3& 0.3345E-01& 4.0& 0.1237E+01& 1.7& 0.1947E-02& 3.2 \\
4& 0.2151E-02& 4.0& 0.3739E+00& 1.7& 0.2231E-03& 3.1 \\
\hline
&\multicolumn{6}{c}{by the $\{P_3, P_3, P_2\}$-$P_2$ with $\{P_3, P_3\}$-$P_3$ WG method} \\ \hline
2& 0.8522E-01& 5.3& 0.1512E+01& 2.2& 0.1117E-01& 4.0 \\
3& 0.2190E-02& 5.3& 0.2365E+00& 2.7& 0.4298E-03& 4.7 \\
4& 0.6360E-04& 5.1& 0.3444E-01& 2.8& 0.1841E-04& 4.5 \\
\hline
&\multicolumn{6}{c}{by the $\{P_4, P_4, P_3\}$-$P_3$ with $\{P_4, P_4\}$-$P_4$ WG method} \\ \hline
2& 0.9472E-02& 6.3& 0.4145E+00& 3.4& 0.1476E-02& 4.2 \\
3& 0.1841E-03& 5.7& 0.3102E-01& 3.7& 0.1760E-04& 6.4 \\
4& 0.2992E-05& 5.9& 0.2162E-02& 3.8& 0.3430E-06& 5.7 \\
\hline
&\multicolumn{6}{c}{by the $\{P_5, P_5, P_4\}$-$P_4$ with $\{P_5, P_5\}$-$P_5$ WG method} \\ \hline
1& 0.1010E-02& 0.0& 0.5724E-01& 0.0& 0.3493E-03& 0.0 \\
2& 0.1684E-04& 5.9& 0.3565E-02& 4.0& 0.5073E-05& 6.1 \\
3& 0.2580E-06& 6.0& 0.2204E-03& 4.0& 0.9438E-07& 5.7 \\
\hline
\end{tabular}%
\end{table}%
Next we compute the solution of \eqref{s-1} again by the $P_k$ weak Galerkin finite element method but on uniform tetrahedral grids shown in
Figure \ref{g-2}.
In Table \ref{t-2}, we list the errors in various norms and the computed orders of convergence for $P_2$, $P_3$ and $P_4$
finite element solutions on uniform tetrahedral grids. It seems we do have one order superconvergence in most cases in Table \ref{t-2}.
\begin{figure}[ht]
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\caption{ The first three levels of uniform tetrahedral grids used in Table \ref{t-2}. }
\label{g-2}
\end{figure}
\begin{table}[ht]
\centering \renewcommand{\arraystretch}{1.05}
\caption{ Error profiles and convergence rates on tetrahedral grids shown in Figure \ref{g-2} for \eqref{s-1}. }
\label{t-2}
\begin{tabular}{c|cc|cc|cc}
\hline
level & $\|{\mathbf{Q}}_h {\mathbf{u}}- {\mathbf{u}}_h \| $ &rate & ${|\hspace{-.02in}|\hspace{-.02in}|} {\mathbf{Q}}_h {\mathbf{u}}- {\mathbf{u}}_h {|\hspace{-.02in}|\hspace{-.02in}|} $ &rate & $\| p_h \| $ &rate \\
\hline
&\multicolumn{6}{c}{by the $\{P_2, P_2, P_1\}$-$P_1$ with $\{P_2, P_2\}$-$P_2$ WG method} \\ \hline
2& 0.444E+00&3.9& 0.357E+01&1.8& 0.288E-01&3.8 \\
3& 0.279E-01&4.0& 0.117E+01&1.6& 0.193E-02&3.9 \\
4& 0.172E-02&4.0& 0.451E+00&1.4& 0.304E-03&2.7 \\
\hline
&\multicolumn{6}{c}{by the $\{P_3, P_3, P_2\}$-$P_2$ with $\{P_3, P_3\}$-$P_3$ WG method} \\ \hline
1& 0.187E+01&0.0& 0.480E+01&0.0& 0.181E+00&0.0 \\
2& 0.551E-01&5.1& 0.724E+00&2.7& 0.764E-02&4.6 \\
3& 0.163E-02&5.1& 0.128E+00&2.5& 0.257E-03&4.9 \\
\hline
&\multicolumn{6}{c}{by the $\{P_4, P_4, P_3\}$-$P_3$ with $\{P_4, P_4\}$-$P_4$ WG method} \\ \hline
1& 0.265E+00&0.0& 0.888E+00&0.0& 0.308E-01&0.0 \\
2& 0.411E-02&6.0& 0.738E-01&3.6& 0.632E-03&5.6 \\
3& 0.693E-04&5.9& 0.721E-02&3.4& 0.139E-04&5.5 \\
\hline
\end{tabular}%
\end{table}%
\long\def\myskip#1{}
\myskip{
\begin{verbatim}
u0 L2 | u a |w l2 |id,ir 2 0
1& 0.7541E+01& 0.0& 0.1336E+02& 0.0& 0.2588E+00& 0.0 \\
2& 0.5316E+00& 3.8& 0.4103E+01& 1.7& 0.2746E-01& 3.2 \\
3& 0.3379E-01& 4.0& 0.1241E+01& 1.7& 0.8302E-02& 1.7 \\
4& 0.2188E-02& 3.9& 0.3744E+00& 1.7& 0.1795E-02& 2.2 \\
5& 0.1508E-03& 3.9& 0.1154E+00& 1.7& 0.4554E-03& 2.0 \\
u0 L2 | u a |w l2 |id,ir 2 0 (original s1*h wg*h**2) s1*h wg*h**0
1& 0.7541E+01& 0.0& 0.1336E+02& 0.0& 0.2588E+00& 0.0 \\
2& 0.5130E+00& 3.9& 0.4020E+01& 1.7& 0.4319E-01& 2.6 \\
3& 0.3192E-01& 4.0& 0.1212E+01& 1.7& 0.5772E-02& 2.9 \\
4& 0.2019E-02& 4.0& 0.3655E+00& 1.7& 0.7854E-03& 2.9 \\
u0 L2 | u a |w l2 |id,ir 2 0 (original s1*h wg*h**2) s1/h wg*h**0
1& 0.7541E+01& 0.0& 0.1336E+02& 0.0& 0.2588E+00& 0.0 \\
2& 0.5263E+00& 3.8& 0.4078E+01& 1.7& 0.1742E-01& 3.9 \\
3& 0.3345E-01& 4.0& 0.1237E+01& 1.7& 0.1947E-02& 3.2 \\
4& 0.2151E-02& 4.0& 0.3739E+00& 1.7& 0.2231E-03& 3.1 \\
u0 L2 | u a |w l2 |id,ir 3 0
1& 0.3473E+01& 0.0& 0.6987E+01& 0.0& 0.3286E+00& 0.0 \\
2& 0.8914E-01& 5.3& 0.1492E+01& 2.2& 0.4130E-01& 3.0 \\
3& 0.3030E-02& 4.9& 0.2343E+00& 2.7& 0.2288E-02& 4.2 \\
4& 0.1579E-03& 4.3& 0.3418E-01& 2.8& 0.1169E-03& 4.3 \\
u0 L2 | u a |w l2 |id,ir 3 0 (original s1*h wg*h**2) s1*h**-1 wg*h**0
1& 0.3451E+01& 0.0& 0.7142E+01& 0.0& 0.1809E+00& 0.0 \\
2& 0.8522E-01& 5.3& 0.1512E+01& 2.2& 0.1117E-01& 4.0 \\
3& 0.2190E-02& 5.3& 0.2365E+00& 2.7& 0.4298E-03& 4.7 \\
4& 0.6360E-04& 5.1& 0.3444E-01& 2.8& 0.1841E-04& 4.5 \\
u0 L2 | u a |w l2 |id,ir 4 0
1& 0.6412E+00& 0.0& 0.4487E+01& 0.0& 0.3939E-01& 0.0 \\
2& 0.7991E-02& 6.3& 0.4145E+00& 3.4& 0.3765E-02& 3.4 \\
3& 0.1110E-03& 6.2& 0.3104E-01& 3.7& 0.6406E-04& 5.9 \\
4& 0.1579E-05& 6.1& 0.2163E-02& 3.8& 0.2198E-05& 4.9 \\
u0 L2 | u a |w l2 |id,ir 4 0 (original s1*h wg*h**2) s1*h**1 wg*h**0
1& 0.7502E+00& 0.0& 0.4496E+01& 0.0& 0.2665E-01& 0.0 \\
2& 0.8880E-02& 6.4& 0.4139E+00& 3.4& 0.2965E-02& 3.2 \\
3& 0.1678E-03& 5.7& 0.3094E-01& 3.7& 0.9880E-04& 4.9 \\
4& 0.2876E-05& 5.9& 0.2156E-02& 3.8& 0.4621E-05& 4.4 \\
u0 L2 | u a |w l2 |id,ir 4 0 (original s1*h wg*h**2) s1*h**-1 wg*h**0
1& 0.7502E+00& 0.0& 0.4496E+01& 0.0& 0.2665E-01& 0.0 \\
2& 0.9472E-02& 6.3& 0.4145E+00& 3.4& 0.1476E-02& 4.2 \\
3& 0.1841E-03& 5.7& 0.3102E-01& 3.7& 0.1760E-04& 6.4 \\
4& 0.2992E-05& 5.9& 0.2162E-02& 3.8& 0.3430E-06& 5.7 \\
u0 L2 | u a |w l2 |id,ir 5 0
1& 0.1010E-02& 0.0& 0.5724E-01& 0.0& 0.3493E-03& 0.0 \\
2& 0.1824E-04& 5.8& 0.3566E-02& 4.0& 0.1333E-04& 4.7 \\
3& 0.2938E-06& 6.0& 0.2205E-03& 4.0& 0.9786E-06& 3.8 \\
u0 L2 | u a |w l2 |id,ir 5 (original s1*h wg*h**2) s1*h**-1 wg*h**0
1& 0.1010E-02& 0.0& 0.5724E-01& 0.0& 0.3493E-03& 0.0 \\
2& 0.1684E-04& 5.9& 0.3565E-02& 4.0& 0.5073E-05& 6.1 \\
u0 L2 | u a |w l2 |id,ir 5 0 s1*h**-1 wg*h**0
1& 0.1010E-02& 0.0& 0.5724E-01& 0.0& 0.3493E-03& 0.0 \\
2& 0.1684E-04& 5.9& 0.3565E-02& 4.0& 0.5073E-05& 6.1 \\
3& 0.2580E-06& 6.0& 0.2204E-03& 4.0& 0.9438E-07& 5.7 \\
----- tri below.
u0 L2 | u a |w l2 |id 2
1& 0.648E+01&0.0& 0.123E+02&0.0& 0.414E+00&0.0 \\
2& 0.418E+00&4.0& 0.346E+01&1.8& 0.301E-01&3.8 \\
3& 0.248E-01&4.1& 0.113E+01&1.6& 0.206E-02&3.9 \\
4& 0.147E-02&4.1& 0.442E+00&1.4& 0.146E-03&3.8 \\
(original s1*h wg*h**2) s1/h wg*h**0
u0 L2 | u a |w l2 |id 2 s1/h wg*h**0
1& 0.648E+01&0.0& 0.123E+02&0.0& 0.414E+00&0.0 \\
2& 0.444E+00&3.9& 0.357E+01&1.8& 0.288E-01&3.8 \\
3& 0.279E-01&4.0& 0.117E+01&1.6& 0.193E-02&3.9 \\
u0 L2 | u a |w l2 |id 2
1& 0.648E+01&0.0& 0.123E+02&0.0& 0.414E+00&0.0 \\
2& 0.444E+00&3.9& 0.357E+01&1.8& 0.288E-01&3.8 \\
3& 0.279E-01&4.0& 0.117E+01&1.6& 0.193E-02&3.9 \\
4& 0.172E-02&4.0& 0.451E+00&1.4& 0.304E-03&2.7 \\
u0 L2 | u a |w l2 |id 2 s1*h wg* h**2; (standard)
1& 0.000E+00&0.0& 0.000E+00&0.0& 0.000E+00&0.0 \\
2& 0.456E+00&0.0& 0.363E+01&0.0& 0.549E-01&0.0 \\
3& 0.287E-01&4.0& 0.117E+01&1.6& 0.147E-01&1.9 \\
u0 L2 | u a |w l2 |id 2
1& 0.648E+01&0.0& 0.123E+02&0.0& 0.414E+00&0.0 \\
2& 0.456E+00&3.8& 0.363E+01&1.8& 0.137E-01&4.9 \\
3& 0.287E-01&4.0& 0.117E+01&1.6& 0.921E-03&3.9 \\
u0 L2 | u a |w l2 |id 3
1& 0.187E+01&0.0& 0.480E+01&0.0& 0.181E+00&0.0 \\
2& 0.551E-01&5.1& 0.724E+00&2.7& 0.764E-02&4.6 \\
3& 0.163E-02&5.1& 0.128E+00&2.5& 0.257E-03&4.9 \\
u0 L2 | u a |w l2 |id,ir 3
1& 0.187E+01&0.0& 0.480E+01&0.0& 0.181E+00&0.0 \\
2& 0.631E-01&4.9& 0.765E+00&2.6& 0.747E-02&4.6 \\
3& 0.195E-02&5.0& 0.133E+00&2.5& 0.133E-02&2.5 \\
4& 0.605E-04&5.0& 0.269E-01&2.3& 0.474E-03&1.5 \\
u0 L2 | u a |w l2 |id 4
1& 0.265E+00&0.0& 0.888E+00&0.0& 0.308E-01&0.0 \\
2& 0.377E-02&6.1& 0.718E-01&3.6& 0.645E-03&5.6 \\
3& 0.590E-04&6.0& 0.704E-02&3.4& 0.127E-04&5.7 \\
u0 L2 | u a |w l2 |id 4
1& 0.265E+00&0.0& 0.888E+00&0.0& 0.308E-01&0.0 \\
2& 0.411E-02&6.0& 0.738E-01&3.6& 0.632E-03&5.6 \\
3& 0.693E-04&5.9& 0.721E-02&3.4& 0.139E-04&5.5 \\
\end{verbatim}
}
\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,225
|
{"url":"https:\/\/www.zbmath.org\/?q=an%3A1232.05088","text":"# zbMATH \u2014 the first resource for mathematics\n\nMaps with highest level of symmetry that are even more symmetric than other such maps: regular maps with largest exponent groups. (English) Zbl\u00a01232.05088\nBrualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15\u201321, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4\/pbk). Contemporary Mathematics 531, 95-102 (2010).\nSummary: Regular maps are generalizations of Platonic solids and can be identified with two-generator presentations of groups $$G$$ of the form $$\\langle x,y; x^2= y^m= (xy)^n=\\cdots= 1\\rangle$$; the parameters $$m$$ and $$n$$ are the degree and the face length of the map. Such maps have the \u2018highest level\u2019 of orientation-preserving symmetry among all maps.\nA regular map of vertex degree $$m$$ is said to have exponent $$j\\in Z^*_m$$ if the assignment $$x\\mapsto x$$ and $$y\\mapsto y^j$$ extends to an automorphism of $$G$$. Any exponent induces an automorphism of the underlying graph which can be viewed as an \u2018external symmetry\u2019 of the map. Exponents of a map form a subgroup of $$Z^*_m$$ and hence $$Z^*_m$$ is the theoretically largest possible group of exponents a regular map of degree $$m$$ can have.\nIn this paper we show that for any given $$m\\geq 3$$ there exist infinitely many finite regular maps of degree $$m$$ with exponent group equal to $$Z^*_m$$. We also show that this result does not, in general, extend to regular maps of given degree and given face length.\nFor the entire collection see [Zbl 1202.05003].\n\n##### MSC:\n 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20F05 Generators, relations, and presentations of groups 57M15 Relations of low-dimensional topology with graph theory","date":"2021-06-21 10:05:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.7260437607765198, \"perplexity\": 376.4092345938458}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488269939.53\/warc\/CC-MAIN-20210621085922-20210621115922-00314.warc.gz\"}"}
| null | null |
package mil.nga.mapcache.utils;
import mil.nga.geopackage.db.GeoPackageDataType;
public class DataTypeConverter {
/**
* Convert string to GeoPackageDataType to make sure we match the enum
* @param givenType String of the GeoPackageDataType expected
* @return null if it doesn't match
*/
public static GeoPackageDataType getGeoPackageDataType(String givenType){
if(givenType.equalsIgnoreCase("text")){
return GeoPackageDataType.TEXT;
} else if(givenType.equalsIgnoreCase("number")){
return GeoPackageDataType.DOUBLE;
} else if(givenType.equalsIgnoreCase("check box")){
return GeoPackageDataType.BOOLEAN;
} else if(givenType.equalsIgnoreCase("checkbox")){
return GeoPackageDataType.BOOLEAN;
}
return null;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 7,803
|
{"url":"https:\/\/freshergate.com\/arithmetic-aptitude\/true-discount\/discussion\/977","text":"Home \/ Arithmetic Aptitude \/ True Discount :: Discussion\n\n### Discussion :: True Discount\n\n1. If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount allowed on the same sum due at the end of double the time is:\n\n2. A. Rs. 20 B. Rs. 21.81 C. Rs. 22 D. Rs. 18.33\n\nExplanation :\n\nS.I. on Rs. (110 - 10) for a certain time = Rs. 10.\n\nS.I. on Rs. 100 for double the time = Rs. 20.\n\nT.D. on Rs. 120 = Rs. (120 - 100) = Rs. 20.\n\nT.D. on Rs. 110 = Rs$$[\\frac { 20 } {1 2 0}*110]$$= Rs. 18.33\n\nBe The First To Comment","date":"2022-01-25 23:35:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9871420860290527, \"perplexity\": 5696.0243580753595}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304876.16\/warc\/CC-MAIN-20220125220353-20220126010353-00173.warc.gz\"}"}
| null | null |
\section{Introduction}\label{intro}
Open system dynamics is of uppermost importance in the quantum
regime where non classical phenomena turn out to be very fragile
with respect to any noise source. The noise effects are often
modeled as the result of an interaction of the system with a large
number of uncontrollable degrees of freedom, i.e.\ an
\emph{environment} \cite{gardiner}. Environments can be assumed as
to be composed by different kinds of particles, for instance
oscillators or spin-$\frac{1}{2}$. These objects come, under the
mathematical point of view, from the realizations of two different
algebras (the Heisenberg-Weyl algebra and the Lie algebra su$(2)$)
corresponding to fermionic and bosonic commutation relations. These
latter can be seen as two limit cases of more general commutation
relations involving deformed algebras parameterized by one
continuous parameter \cite{Kury, Kulish,Mac}.
Our aim is to analyze the qubit dynamics in an environment of
oscillators satisfying suitable $q$-deformed commutation relations,
such that it permits to continuously interpolate between oscillators
and spin-$\frac{1}{2}$. Actually, we investigate how quantum
decoherence phenomena changes in passing from bosonic to fermionic
environments. We find a slowing down of decoherence in the fermionic
case. However, this effect only manifests at finite temperature.
The paper is organized as follows. In Section \ref{model} we present
the model. We then derive the master equation in Section
\ref{equation}. In Section \ref{onequbit} we study the dynamics of a
single qubit and we evaluate its coherence decay. We then study the
dynamics of two qubits and we evaluate the entanglement decay by
distinguishing the case of the two qubits in the same environment
(Section \ref{twoqubit_same}), from that of the two qubits in
separate environments (Section \ref{twoqubit_separate}). Finally,
Section \ref{conclude} is for concluding remarks.
\section{The model}\label{model}
Let us consider a system (qubit) described by the free Hamiltonian
\begin{eqnarray}
H_S&=&\Omega\sigma_z\label{HS},
\end{eqnarray}
with $\Omega$ the qubit frequency and $\sigma$, $\sigma^{\dag}$, $\sigma_z$ operators
satisfying the commutation relations
\begin{eqnarray}
\left[\sigma^{\dag},\sigma\right]&=&\sigma_z,\\
\left[\sigma,\sigma_z\right]&=&2\sigma,\\
\left[\sigma^{\dag},\sigma_z\right]&=&-2\sigma^{\dag}.
\end{eqnarray}
They define the su$(2)$ algebra. Furthermore, we consider an
environment composed by an infinite (countable) number of
oscillators whose Hamiltonian reads as \cite{Greenberg, Ham1}
\begin{eqnarray}
H_E & = & \sum_k \omega_k N_k, \label{HB1}
\end{eqnarray}
with $\omega_k$ the frequency of the $k$-th oscillator and
$A_k$, $A_k^\dag$, $N_k$ operators satisfying the commutation relations
\begin{eqnarray}
{[} N_h , A_k {]} & = & - \delta_{hk} A_k, \\
{[} N_h , A^\dag_k {]} & = & \delta_{hk} A^\dag_k.
\end{eqnarray}
They define the Heisenberg-Weyl algebra. We are now going to
introduce a deformation of this algebra through the so-called
``quons" commutation relations \cite{Greenberg}
\begin{equation}
A_h A_k^\dag - q A_k^\dag A_h = \delta_{hk},
\end{equation}
where $q \in [-1 , 1]$ is the deformation parameter. It allows us to
interpolate between fermions ($q=-1$) and bosons ($q=1$).
Intermediate values of $q \in (-1,1)$ correspond to the so-called
``infinite statistics".
We assume the system interacting with the environment through the
following Hamiltonian
\begin{eqnarray}
H_I&=&\sum_k\lambda_k\left(A^{\dag}_k\sigma+A_k\sigma^{\dag}\right),\label{HI}
\end{eqnarray}
where $\lambda_k$ denotes the coupling constant of the system with
the $k$-th environment's oscillator.
\section{Master Equation}\label{equation}
Quite generally, the master equation for the system density operator
$\rho$ can be derived by using the Born-Markov approximation
\cite{gardiner}. Hence, it can be formally written as
\begin{equation}
\dot\rho(t)=-\int_0^{\infty}d\tau{\rm Tr}_E\left\{
\left[H_I(t),\left[H_I(t-\tau),\rho(t)\otimes
\rho_E\right]\right]\right\}, \label{ME}
\end{equation}
where $\rho_E$ is the initial environment density operator and ${\rm
Tr}_E$ denotes the trace over environment degrees of freedom.
Furthermore, it is
\begin{equation}
H_I(t)=e^{\iota(H_S+H_E)t}H_Ie^{-\iota(H_S+H_E)t}.
\label{HIt}
\end{equation}
For the choice of the environment Hamiltonian \eqref{HB1}, the
dynamical equations are formally identical to the undeformed case.
The reason is that the interaction Hamiltonian $H_I(t)$ reads as
follows
\begin{equation}
H_I(t)=\sum_k\lambda_k\left(A^{\dag}_k\sigma
e^{-\iota(\omega_k-\Omega)t}+A_k\sigma^{\dag}
e^{\iota(\omega_k-\Omega)t}\right),
\end{equation}
by virtue of \eqref{HIt}, \eqref{HI}, \eqref{HB1} and \eqref{HS}.
Therefore, from \eqref{ME}, we can write
\begin{equation}\label{ME1}
\dot\rho(t) = - \int_0^{\infty}d\tau{\rm Tr}_E \sum_{k,l}
\lambda_k\lambda_l \mathcal{F}_{kl}(\rho(t)),
\end{equation}
where
\begin{align}
\mathcal{F}_{kl}(\rho(t)) \, = \, & F_k(t) \, F_l(t-\tau) \, \rho(t)\otimes\rho_E - F_k(t) \, \rho(t)\otimes\rho_E \, F_l(t-\tau) \nonumber\\
& - F_k(t-\tau) \, \rho(t)\otimes\rho_E \, F_l(t) +
\rho(t)\otimes\rho_E \, F_k(t-\tau) \, F_l(t)
\end{align}
and
\begin{equation}
F_k(t) = A^{\dag}_k\sigma
e^{-\iota(\omega_k-\Omega)t}+A_k\sigma^{\dag}
e^{\iota(\omega_k-\Omega)t}.
\end{equation}
We now assume an initial thermal state for the environment at
temperature $T$,
\begin{equation}
\rho_E=Z^{-1}e^{-H_E/T},
\end{equation}
where
\begin{equation}
Z={\rm Tr}_E\{e^{-H_E/T}\},
\end{equation}
is the partition function.
In \eqref{ME1} we have nonzero terms of the form
\begin{align}
{\rm Tr}_E\left\{\rho_EA_k^{\dag}(t) A_l(t-\tau)\right\} & =
\frac{1}{Z} {\rm Tr}_E \left\{
\exp\left[-\frac{1}{2T}\sum_j\omega_jN_j\right]
e^{-\iota\omega_kt}A_k^{\dag}A_le^{\iota\omega_l(t-\tau)} \right\} \nonumber\\
&=\delta_{k,l}\frac{1}{Z}\sum_{n_k}[n_k]_q\exp\left[-\frac{\omega_k[n_k]_q}{2T}\right]
e^{-\iota\omega_k\tau}.
\label{AdagA}
\end{align}
Here we have defined
\begin{equation}
[n]_q=\frac{1-q^n}{1-q},
\end{equation}
as the $q$-deformed number.
Then, neglecting principal values terms, we obtain from
\eqref{AdagA}
\begin{align}
\int_0^{\infty} d\tau \sum_{k,l}\lambda_k\lambda_l {\rm Tr}_E\left\{\rho_EA_k^{\dag}(t) A_l(t-\tau)\right\}e^{\iota\Omega\tau}&=
\int_0^{\infty} d\tau \sum_{k}\lambda_k^2 \langle[N(\omega_k)]_q\rangle_{E}\, e^{-\iota(\omega_k-\Omega)\tau}\nonumber\\
&=\sum_{k}\lambda_k^2 \langle[N(\omega_k)]_q\rangle_{E} \delta(\omega_k-\Omega),
\label{notzero1}
\end{align}
where
\begin{equation}
\langle[N(\omega_k)]_q\rangle_{E}=\frac{1}{Z}\sum_{n_k}[n_k]_q\exp\left[-\frac{\omega_k[n_k]_q}{2T}\right].
\end{equation}
Moving to the continuum of frequencies for the environment
oscillators, we have
\begin{align}
\sum_{k}\lambda_k^2 \langle[N(\omega_k)]_q\rangle_{E} \delta(\omega_k-\Omega)\rightarrow
\int d\omega \Lambda^2(\omega) \langle[N(\omega)]_q\rangle_{E}
\delta(\omega-\Omega),
\end{align}
where $\Lambda^2(\omega)$ accounts for the coupling spectrum as well
as for the density of states. As usual, we set
$\Lambda^2(\Omega)=\gamma/2$ to be the damping rate. Moreover, we
get the following distribution \cite{Chaichian_JPA_26,Goodison}
\begin{eqnarray}
\langle [ N ]_q \rangle_{E} \equiv \langle [ N(\Omega) ]_q \rangle_{E}& = & \frac{1}{e^{\Omega/T}-q},\\
\langle [ N + 1 ]_q \rangle_{E} \equiv \langle [ N(\Omega) + 1 ]_q \rangle_{E} & = &
\frac{e^{\Omega/T}}{e^{\Omega/T}-q}.
\end{eqnarray}
In summary, from (\ref{notzero1}), we have
\begin{equation}
\int d\tau\sum_{k,l}\lambda_k\lambda_l
{\rm Tr}_E\left\{\rho_E A^{\dag}_k(t) A_l(t-\tau)\right\}e^{-\iota(\omega_k-\Omega)t+\iota(\omega_l-\Omega)(t-\tau)}=\frac{\gamma}{2} \langle [ N ]_q \rangle_{E}.
\end{equation}
Other nonzero terms in \eqref{ME1} are
\begin{equation}
\int d\tau\sum_{k,l}\lambda_k\lambda_l
{\rm Tr}_E\left\{\rho_E A_k(t) A_l^{\dag}(t-\tau)\right\}e^{\iota(\omega_k-\Omega)t-\iota(\omega_l-\Omega)(t-\tau)}=\frac{\gamma}{2} \langle [ N +1]_q \rangle_{E}.
\end{equation}
We finally arrive at the following master equation for the reduced
system (the qubit):
\begin{equation}
\dot\rho(t) = - \frac{\gamma}{2} \langle [ N ]_q \rangle_{E} \left(
\sigma\sigma^{\dag}\rho(t) - 2 \sigma^{\dag}\rho(t)\sigma +
\rho(t)\sigma\sigma^{\dag} \right) \nonumber\\
- \frac{\gamma}{2} \langle [ N + 1 ]_q \rangle_{E} \left(
\sigma^{\dag}\sigma\rho(t) - 2 \sigma\rho(t)\sigma^{\dag} +
\rho(t)\sigma^{\dag}\sigma \right). \label{MEf}
\end{equation}
This equation explicitly shows that the effect of the q-deformation
is to change the rates of emission, which is proportional to
$\langle [ N ]_q \rangle_{E}$, and the rate of absorption,
proportional to $\langle [ N + 1 ]_q \rangle_{E}$ ( see also
\cite{Goodison}). Notice that for $T=0$, there are no effects coming
from the deformation, because for $N=0$ we simply have $\langle [ 0
]_q \rangle_{E}=0$ and $\langle [ 1 ]_q \rangle_{E}=1$; in other
words, the nonlinear effects introduced by the q-deformation cannot
be observed if the environment transitions only concern the vacuum
and the states with single excitation.
\section{One qubit}\label{onequbit}
Let us consider the operators appearing in Eq.(\ref{MEf}) and
represent them in matrix form in the computational basis
$\{|0\rangle,|1\rangle\}$,
\begin{eqnarray}
\sigma=\left(
\begin{array}{cc}
0&0\\
1&0
\end{array}\right),
\end{eqnarray}
and
\begin{eqnarray}
\rho(t)=\left(
\begin{array}{cc}
a(t)&b_1(t)+\iota b_2(t)\\
b_1(t)-\iota b_2(t)&1-a(t)
\end{array}\right),
\end{eqnarray}
where $a(t)$, $b_1(t)$ and $b_2(t)$ are real functions of time to be
determined.
Inserting the above matrices into Eq.(\ref{MEf}) we get the
following set of differential equations
\begin{eqnarray}
\frac{d}{dt}a&=&-2 (A+B)a+2A,
\label{de1}\\
\frac{d}{dt}(b_1+\iota b_2)&=&-(A+B)(b_1+\iota b_2),
\label{de2}
\end{eqnarray}
where for the sake of simplicity we have set
\begin{eqnarray}
A&=&(\gamma/2)\langle [N]_{q}\rangle_E, \label{A}\\
B&=&(\gamma/2)\langle [N+1]_{q}\rangle_E. \label{B}
\end{eqnarray}
The solutions of the differential equations \eqref{de1}, \eqref{de2}
read
\begin{eqnarray}
a(t)&=&\frac{e^{-2(A+B)t}}{A+B}\left[a(0) B+A\left(a(0)+e^{2(A+B)t}-1\right)\right],\\
b_1(t)&=&b_1(0)e^{-(A+B)t},\label{cohere}\\
b_2(t)&=&b_2(0)e^{-(A+B)t}.
\end{eqnarray}
Figure \ref{coherence} shows the decay of the coherence ($b_1(t)$)
for a qubit in a quon environment at temperature $T/\Omega=1$, for
different values of the deformation parameter. From Eq.s (\ref{A}),
(\ref{B}), (\ref{cohere}), it follows that the decay of coherence at
$q=-1$, and any finite temperature, behaves as the decay at $T=0$
and any $q$. In the inset, it is shown the decay of the population
$a(t)$ (solid lines refer again to $T/\Omega=1$, while dashed line
refers $T=0$). Thus, the fermionic environment gives rise to the
slowest decay of coherence and population. The decay of quantum
coherence becomes slower and slower when passing from the bosonic to
fermionic environment.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{coherence}
\caption{The plot shows the decay of the coherence $b_1(t)$ for a
qubit in a quon-environment at temperature $T/\Omega=1$, for
different values of the deformation parameter. In the inset, it is
shown the decay of the population $a(t)$ (solid lines refer again to
$T/\Omega=1$, dashed line refers $T=0$).} \label{coherence}
\end{figure}
\section{Two qubits in the same environment}\label{twoqubit_same}
We now assume the system composed by two identical qubits
interacting with the same environment. Then the master equation can
be written as Eq.(\ref{MEf}) simply replacing $\sigma$ with
$\sigma_1+\sigma_2$, that is
\begin{eqnarray} \label{MEf2}
\dot\rho(t) & = & -\frac{\gamma}{2} \langle [ N ]_q \rangle_{E} \left( \sigma_1\sigma_1^{\dag}\rho(t) - 2 \sigma_1^{\dag}\rho(t)\sigma_1 + \rho(t)\sigma_1\sigma_1^{\dag} +\sigma_2\sigma_2^{\dag}\rho(t) \right.\nonumber\\
&& \qquad\qquad\quad - 2 \sigma_2^{\dag}\rho(t)\sigma_2 + \rho(t)\sigma_2\sigma_2^{\dag} + \sigma_1\sigma_2^{\dag}\rho(t)+\sigma_2\sigma_1^{\dag}\rho(t) \nonumber\\
&& \qquad\qquad\quad \left. -2\sigma_1^{\dag}\rho(t)\sigma_2-2\sigma_2^{\dag}\rho(t)\sigma_1 +\rho(t)\sigma_1\sigma_2^{\dag}+\rho(t)\sigma_2\sigma_1^{\dag} \right) \nonumber\\
&-& \frac{\gamma}{2} \langle [ N + 1 ]_q \rangle_{E} \left( \sigma_1^{\dag}\sigma_1\rho(t) - 2 \sigma_1\rho(t)\sigma_1^{\dag} + \rho(t)\sigma_1^{\dag}\sigma_1 +\sigma_2^{\dag}\sigma_2\rho(t) \right.\nonumber\\
&& \qquad\qquad\quad - 2 \sigma_2\rho(t)\sigma_2^{\dag} + \rho(t)\sigma_2^{\dag}\sigma_2 + \sigma_1^{\dag}\sigma_2\rho(t) + \sigma_2^{\dag}\sigma_1\rho(t) \nonumber\\
&& \qquad\qquad\quad \left. - 2 \sigma_1\rho(t)\sigma_2^{\dag} - 2
\sigma_2\rho(t)\sigma_1^{\dag} + \rho(t)\sigma_1^{\dag}\sigma_2 +
\rho(t)\sigma_2^{\dag}\sigma_1 \right).
\end{eqnarray}
Then, we proceed in the same way as for the single qubit case. That
is, we consider the operators appearing in Eq.(\ref{MEf2}) and
represent them in matrix form in the computational basis
$\{|00\rangle,|01\rangle|10\rangle,|11\rangle\}$,
\begin{eqnarray}
\sigma_1=\left(
\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
1&0&0&0\\
0&1&0&0
\end{array}\right),\quad
\sigma_2=\left(
\begin{array}{cccc}
0&0&0&0\\
1&0&0&0\\
0&0&0&0\\
0&0&1&0
\end{array}\right),
\label{sig12}
\end{eqnarray}
and
\begin{eqnarray}
\rho(t)=\left(
\begin{array}{cccc}
a(t)&b_1(t)+\iota b_2(t)&c_1(t)+\iota c_2(t)&d_1(t)+\iota d_2(t)\\
b_1(t)-\iota b_2(t)&e(t)&f_1(t)+\iota f_2(t)&g_1(t)+\iota g_2(t)\\
c_1(t)-\iota c_2(t)&f_1(t)-\iota f_2(t)&h(t)&i_1(t)+\iota i_2(t)\\
d_1(t)-\iota d_2(t)&g_1(t)-\iota g_2(t)&i_1(t)-\iota i_2(t)&1-a(t)-e(t)-h(t)
\end{array}\right),
\label{rhomat}
\end{eqnarray}
where $a(t)$, $b_1(t)$, $b_2(t)$, $c_1(t)$, $c_2(t)$, $d_1(t)$,
$d_2(t)$, $e(t)$, $f_1(t)$, $f_2(t)$, $g_1(t)$, $g_2(t)$, $h(t)$,
$i_1(t)$ and $i_2(t)$ are real functions of time to be determined.
In terms of these functions, the master equation is written as a set
of coupled differential equations. They are reported together with
their solutions in Appendix \ref{Asame}.
In this case, a relevant quantity to study is the entanglement
between the two qubits. Specifically we consider the qubits
initialized in one of the four Bell states
\begin{eqnarray}
|\phi_\pm\rangle &=& 2^{-1/2}(|00\rangle \pm |11\rangle),
\label{phipm}\\
|\psi_\pm\rangle &=& 2^{-1/2}(|01\rangle \pm |10\rangle),
\label{psipm}
\end{eqnarray}
and then we investigate how entanglement decays.
We use the concurrence as measure of the degree of entanglement
\cite{Woot}
\begin{equation}
C(\rho(t))=\max\left\{0,\lambda_1(t)-\lambda_2(t)-\lambda_3(t)-\lambda_4(t)\right\},
\end{equation}
where $\lambda_i(t)$'s are, in decreasing order, the nonnegative
square roots of the moduli of the eigenvalues of
$\rho(t)\tilde\rho(t)$ with
\begin{equation}
\tilde\rho(t)=\left(\sigma_{1}-\sigma_1^{\dag}\right)\left(\sigma_{2}-\sigma_2^{\dag}\right)
\rho^*(t)
\left(\sigma_{1}-\sigma_1^{\dag}\right)\left(\sigma_{2}-\sigma_2^{\dag}\right),
\end{equation}
and $\rho^*(t)$ denotes the complex conjugate of $\rho(t)$.
The decay of the concurrence is plotted in Figure \ref{quons_unc1}.
The qubits are initialized in the Bell states \eqref{phipm}. For
$T/\Omega=0$, the decay of the concurrence is independent from the
deformation parameter $q$. For $T/\Omega>0$ we see the phenomenon of
entanglement sudden death \cite{ESD}. We notice however that the
entanglement death time depends on the value of the deformation
parameter $q$. In particular, the slowest decay and the longest
lifetime of entanglement is evident for the fermionic case $q=-1$.
The same happens when the two-qubit state is initialized in
$|\psi_+\rangle = 2^{-1/2}(|01\rangle + |10\rangle)$ (see inset).
The decay of concurrence is slower and slower when continuously
passing from the bosonic to the fermionic environment. On the
contrary, the Bell state $|\psi_-\rangle$ is invariant under the
dynamics of \eqref{MEf2}, thus entanglement in this case is totally
preserved.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{quons_corr}
\caption{The plot shows the decay of the concurrence for the
two-qubit system in a quon-environment. Both qubits are plunged into
the same environment. The qubits are initialized in one of the Bell
states $|\phi_\pm\rangle = 2^{-1/2}(|00\rangle \pm |11\rangle)$;
dashed line refer to $T/\Omega=0$, solid lines to $T/\Omega=1$ and
several values of the deformation parameter $q$. In the inset the
two-qubit state is initialized in the Bell state $|\psi_+\rangle =
2^{-1/2}(|01\rangle + |10\rangle)$. The remaining Bell state
$|\psi_-\rangle = 2^{-1/2}(|01\rangle - |10\rangle)$ is preserved by
the dynamics.} \label{quons_unc1}
\end{figure}
\section{Two qubits in separate environments}\label{twoqubit_separate}
Here we consider each of the two identical qubit interacting with
its own environment. Then the master equation is a straightforward
extension of Eq.(\ref{MEf}), that is
\begin{eqnarray} \label{MEf3}
\dot\rho(t) & = & -\frac{\gamma}{2} \langle [ N ]_q \rangle_{E}
\left( \sigma_1\sigma_1^{\dag}\rho(t) - 2
\sigma_1^{\dag}\rho(t)\sigma_1 + \rho(t)\sigma_1\sigma_1^{\dag} +
\sigma_2\sigma_2^{\dag}\rho(t) - 2 \sigma_2^{\dag}\rho(t)\sigma_2 +
\rho(t)\sigma_2\sigma_2^{\dag}
\right) \nonumber\\
&& - \frac{\gamma}{2} \langle [ N + 1 ]_q \rangle_{E} \left(
\sigma_1^{\dag}\sigma_1\rho(t) - 2 \sigma_1\rho(t)\sigma_1^{\dag} +
\rho(t)\sigma_1^{\dag}\sigma_1 +\sigma_2^{\dag}\sigma_2\rho(t) - 2
\sigma_2\rho(t)\sigma_2^{\dag} +
\rho(t)\sigma_2^{\dag}\sigma_2\right).
\end{eqnarray}
It can be solved with the same method of (\ref{MEf2}). The
corresponding differential equations and their solutions are
reported in Appendix \ref{Asepa}.
Figure \ref{quons_unc2} shows the decay of the concurrence in time.
The qubits are initialized in the Bell states \eqref{phipm}. For
$T/\Omega=0$, the concurrence decay is independent from the
deformation parameter $q$. Also in this case, for $T/\Omega>0$, we
see the phenomenon of entanglement sudden death \cite{ESD}. We
notice that the entanglement death time depends on the value of the
deformation parameter $q$. In particular, the slowest decay and the
longest lifetime of entanglement is for the fermionic case $q=-1$.
The same happens when the two-qubit state is initialized in
$|\psi_\pm\rangle = 2^{-1/2}(|01\rangle \pm |10\rangle)$ (see
inset). The decay of entanglement becomes slower and slower when
passing from the bosonic to the fermionic environment. In this case
there is no maximally entangled state that remains invariant under
the dynamics.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{quons_unc}
\caption{The plot shows the decay of the concurrence of the
two-qubit system in a quon-environment. Each qubit is subject to
independent and identical environment. The qubits are initialized in one of
the Bell states $|\phi_\pm\rangle = 2^{-1/2}(|00\rangle
\pm |11\rangle)$; dashed line refer to $T/\Omega=0$, solid lines to
$T/\Omega=1$ and several values of the deformation parameter $q$. In
the inset the two-qubit state is initialized in one of the Bell
states $|\psi_\pm\rangle = 2^{-1/2}(|01\rangle \pm
|10\rangle)$.} \label{quons_unc2}
\end{figure}
\section{Conclusion}\label{conclude}
In conclusion, we have analyzed the qubit dynamics in an environment
of oscillators satisfying suitable $q$-deformed commutation
relations, such that it permits to interpolate between oscillators
and spin-$\frac{1}{2}$ particles. Specifically we have evaluated the
decay of quantum coherence and entanglement in time when passing
from bosonic to fermionic environments. The general behavior is
that, at finite temperature, coherence and entanglement decay slower
and slower when continuously passing from bosonic to fermionic
environments.
Our work sheds further light on the mechanism of loosing quantum
coherence and paves the way for a deeper algebraic analysis of this
phenomenon. Moreover it could be useful for describing realistic
physical situations where the assumption of an interaction with an
environment of solely oscillators (resp. spin-$\frac{1}{2}$)
particle turns out to be oversimplified.
\section*{Acknowledgements}
The work of C.L. and S.M. is partially supported by EU through the
FET-Open Project HIP (FP7-ICT-221899).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,496
|
redux-survey
===
A survey app powered by [React](https://facebook.github.io/react/) and [Redux](http://rackt.github.io/redux/), it's still in development, meaning not functional!.
But I want to try it!
===
git clone git@github.com:alfonsoperez/redux-survey.git
cd redux-survey
npm i
npm i -g webpack-dev-server
webpack-dev-server
Now browse to http://localhost:8080 and you're all set
Tests
===
npm run test
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,110
|
\section{Introduction}
In the last few years, it has become clear that a large
fraction of low mass stars \citep[$>30\%$ ][]{bonfils:2011} are
orbited by super-Earth-mass planets on relatively short period orbits,
including a handful detected in the
liquid water (habitable) zones \citep[e.g., GJ 581d and GJ
667Cc: ][respectively]{mayor:2009, anglada:2012b}. These planets have
small but non-negligible probabilities of transiting in
front of their host stars. Given the favorable
star--planet size ratio, we already have the technical
means to begin spectroscopic characterization of their atmospheres.
GJ 1214b was the first super-Earth
found to transit in front of an M dwarf
\citep{charbonneau:2009} and, as a consequence, it has
received significant observational and theoretical
attention in the past three years. While the orbit is too
close to the star to support hospitable oceans of liquid
water, GJ 1214b offers the first opportunity to attempt
atmospheric characterization of a super-Earth
\citep{bean:2010, croll:2011, berta:2012}, but
apparently contradictory results have
emerged from these studies. Different gases in the
atmosphere of a planet absorb light more efficiently at
certain wavelength ranges. As a result, one should be able to
measure a different effective transit depth as a function
of wavelength. For example, \citet{croll:2011} reported
excess absorption in the K band, which would be
compatible with an extended and mostly transparent
atmosphere with a strong absorber in the near infrared.
On the other hand, \citet{bean:2010} could not detect
significant features in the optical transit depths. This
result was confirmed by the same group in a more extended
wavelength range \citep{bean:2011, desert:2011} and has also been
confirmed using HST spectrophotometric observations in
the near infrared \citep{berta:2012}. Such a flat transmission
spectrum favors a very opaque atmosphere with high
concentrations of water vapor as the main source of
opacity, indicating that the planet could be mainly
composed of water.
All these interpretations rely on atmospheric
models that are strongly dependent on the planet's bulk
properties, especially its mass, radius, mean density,
surface gravity, and stellar irradiation. Prior to the
detection of the planet candidate, \object{GJ 1214}
was a largely
ignored M dwarf. As a result, some of its fundamental
properties had significant uncertainties (e.g., its
distance). Uncertainties in the fundamental
parameters of GJ 1214 propagate strongly into the planet's bulk
properties, adding an extra element of uncertainty in
discussions about the possible nature of its atmosphere.
Probably the most significant measurement we provide here is
the new measurement of its trigonometric
parallax at 0.6\% precision \citep[previous parallax measurement had a
$\sim$ 10\% uncertainty, ][]{vanaltena:2001}. As a consequence,
the luminosity
and mass of GJ 1214 have experienced significant updates as
well. The orbital solution for GJ 1214 can also be updated using
recently published transit observations and refined Doppler measurements
obtained with our newly developed software
\citep[HARPS-TERRA, ][]{anglada:2012a}. In addition, the WISE
catalog \citep{wise:2011} has also been recently released,
adding four more absolute flux measurements of GJ 1214, thus enabling a
comprehensive spectral energy distribution adjustment and a more secure
determination of T$_{\rm eff}$. In Section \ref{sec:observations},
we present the new CAPSCam astrometric measurements
and our re-analysis of public HARPS Doppler
measurements. A new Keplerian solution for GJ 1214b is
presented in Section \ref{sec:rvorbit}. In Section \ref{sec:star}, we give an
overview of the stellar properties in the light of the
new distance measurement, its near infrared spectrum, and updated
absolute magnitudes. Finally, Section
\ref{sec:starplanet} combines all of the transit
observables with the new orbital solution and provides
the posterior probability distributions for the updated
star--planet parameters. Our conclusions are summarized
in Section \ref{sec:conclusions}.
\section{Observations and Data Reduction}
\label{sec:observations}
\subsection{Astrometry and trigonometric parallax}
\label{sec:astrometry}
The trigonometric parallax of GJ 1214 has
been obtained using the Carnegie Astrometric Planet
Search Camera \citep[CAPSCam,][]{boss:2009} installed in
on the 2.5m duPont
telescope of the Las Campanas Observatory (Chile).
The observations span 26 months (June 2009 to Sep 2012)
and 10 epochs have been obtained at an average precision
of 1.0 milliarcsec (mas) per epoch.
Each epoch consists of 20 or more exposures of
45 seconds each. GJ 1214 is significantly brighter than
the average background sources and would saturate the detector
in less than 10 seconds. However CAPSCam can read out
a small part of the array much faster than the full
field \citep{boss:2009}. For GJ 1214, a window of
64$\times$64 pixels is read out every 5 seconds, and all
subrasters are added to the final full field image. The
field of view of CAPSCam is 6.6 arcminutes wide and is
typically rich in background stars, so a very robust reference
frame with more than 30 objects can be used to correct
for field distortions. Centroid extraction, source
crossmatching, field distortion correction, and
astrometric solutions for all the stars in the field have
been obtained using the ATPa astrometric software
developed within the CAPS project (available upon
request). The methods and algorithms applied are outlined
in \citet{boss:2009} and \citet{anglada:2011}.
A measurement of the parallax and proper motion requires
fitting 5 astrometric parameters simultaneously: initial
offset in $R.A.$ and $Dec.$, proper motion in $R.A.$ and $Dec.$,
and the parallax itself. The formal
uncertainties from a classic least-squares analysis
(e.g., from the diagonal of the covariance matrix) are
usually overoptimistic and do not properly account for
correlations between parameters. To overcome these issues, we
estimate the uncertainties in the astrometric parameters
using a Monte Carlo approach. This is done by generating many
realistic sets of measurements and measuring the standard
deviation of the resulting derived parameters for the entire
Monte Carlo-generated sample.
To do this properly, a first realistic estimate of the epoch-to-epoch accuracy
is needed. The outputs of the astrometric processing are the astrometric
parameters of the target star and of all the other objects in the field. By
combining the residuals of all the reference stars from all the epochs, we can
compute the expected uncertainty per epoch. Doing this, we obtain an
epoch-to-epoch accuracy of $\sim$ 1.3 mas on both $R.A.$ and $Dec.$. Since the
reference stars are fainter than the target, this is a conservative estimate of
the real precision for the target. The star itself, which is not included in the
reference frame, shows a standard deviation in the residuals of 1.0 mas/epoch,
which is also consistent with the expected CAPSCam performance, assuming 20
minutes of on--sky observations per epoch \citep[see][]{boss:2009}. Then, we
simulate 10$^5$ synthetic sets of astrometric observations (same format as Table
\ref{tab:astrodata}) using the nominal parallax and proper motion at the same
epochs of observation. Random Gaussian noise with a single epoch uncertainty of
1.3 mas is injected into each synthetic data set and the 5
astrometric parameters are derived. The standard deviation of each parameter
over the 10$^5$ solutions is the corresponding uncertainty. The
obtained uncertainty for the differential parallax is 0.44 mas and corresponds
to a relative precision of $\sim0.6\%$. We note that this Monte Carlo method
implicitly accounts for parameter correlation \citep[see discussion
in][]{faherty:2012}. The differential astrometric measurements used to measure
the differential parallax and proper motion of GJ 1214 are given in Table
\ref{tab:astrodata}. The best fit solution over-plotted with the astrometric
epochs is shown in Figure \ref{fig:astrometry}.
\begin{table}
\caption{Local astrometric measurements of GJ 1214}\label{tab:astrodata}
\begin{tabular}{lrrrr}
JD & R.A. & $\sigma_{\rm R.A.}$ & Dec. & $\sigma_{\rm Dec.}$ \\
(days) & (mas) & (mas) & (mas) & (mas) \\
\hline\hline
2455368.781418 & 0.45 & 0.91 & 12.30 & 3.86 \\
2455408.647011 & 25.62 & 1.09 & -78.97 & 2.22 \\
2455638.911225 & 511.34 & 0.79 & -567.61 & 1.88 \\
2455664.846719 & 542.31 & 0.74 & -604.85 & 1.64 \\
2455779.634773 & 611.31 & 0.45 & -826.25 & 0.93 \\
2455782.624045 & 614.41 & 0.67 & -834.84 & 1.07 \\
2456018.887788 & 1111.53 & 0.81 & -1325.39 & 1.32 \\
2456084.702758 & 1159.22 & 0.52 & -1433.52 & 1.08 \\
2456134.624810 & 1184.17 & 0.39 & -1534.43 & 0.74 \\
2456196.509776 & 1264.98 & 0.82 & -1692.83 & 0.88 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[tb]
\centering
\includegraphics[width=3.0in,clip]{Astrometry.eps}
\caption{Astrometric motion of GJ 1214 as a function of
time. Top panels represent the motion in R.A.(left)
and Dec. (right). Bottom panels show the residuals to
the astrometric fit. The RMS of the residuals is
below 1 mas.} \label{fig:astrometry}
\end{figure}
The measured parallax is relative to the
reference stars. Since they are not at infinite
distances, these reference stars also have parallactic motion. As a
result, the average parallax of the reference frame
cannot be derived from the astrometric observations
alone. We obtain this average parallax (also called the
parallax zero-point correction) using catalog photometry
for the reference stars using the procedure described in
\citet{anglada:2011} with 19 reference stars with good
CAPSCam astrometry (RMS of the residuals below 2.0
mas/epoch) and reliable catalog photometry
\citep[$B$$<$18 NOMAD and $JHK$$_s$ 2MASS photometry;][]{nomad,twomass}.
For this set of observations, we determine
a zero--point correction of -0.40 $\pm$ 0.3 mas. This value,
when added to the relative parallax, produces
the final absolute parallax measurement in Table
\ref{tab:astroresults}. The inverse of the parallax in arcseconds
is the distance in parsecs (pc) and amounts to
14.55 $\pm$ 0.13 pc. The updated $BVRIJHK_s$, $W1,W2, W3$
and $W4$ absolute magnitudes for GJ 1214 are presented and
discussed in Section \ref{sec:star}.
\begin{table}
\caption{Basic astrometric information and
results from the analysis of the astrometry.}
\label{tab:astroresults}
\centering
\begin{tabular}{lrr}
Parameter & Value \\
\hline\hline
R.A. & 17 15 18.94\tablefootmark{1} \\
Dec. & +04 57 49.7 \tablefootmark{1} \\
Catalog $\mu^*_{\rm R.A.}$ [mas yr$^{-1}$]
& 585 \tablefootmark{2} \\
Catalog $\mu_{\rm Dec.}$ [mas yr$^{-1}$]
& -752 \tablefootmark{2} \\
\hline\hline
Relative $\mu^*_{\rm R.A.}$ & 581.88 $\pm$ 0.5 \\
Relative $\mu_{\rm Dec.}$ & -734.6 $\pm$ 0.8 \\
Relative parallax [mas] & 69.11 $\pm$ 0.4 \\
Zero-point correction [mas] & -0.4 $\pm$ 0.3 \\
Absolute parallax [mas] & 68.71 $\pm$ 0.6 \\
Distance [pc] & 14.55 $\pm$ 0.13 \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{1}{2MASS catalog \citep{twomass}}
\tablefoottext{2}{LSPM-NORTH catalog \citet{lepine:2005}}
}
\end{table}
We note that the measured proper motions are also
differential and contain an unknown offset due to the
unknown average motion of the background stars (e.g., all
galactic plane stars move roughly in the same direction).
Even though catalog values are less precise than the
ones we obtain, catalog proper motions are usually
corrected for proper motion zero--point ambiguities.
Differential measurements as well as
the suggested values for the proper motion of GJ 1214
\citep[LSPM-NORTH catalog, ][]{lepine:2005} are also
provided in Table \ref{tab:astroresults}.
\subsection{Radial velocity measurements} \label{sec:rvdata}
The ESO public archive\footnote{
\texttt{http://archive.eso.org/wdb/wdb/eso/repro/form}}
contains 21 reduced HARPS spectra of GJ 1214 obtained by
\citet{charbonneau:2009}. We reanalyzed these public
HARPS spectra using our software called HARPS-TERRA
\citep[HARPS Template Enhanced
Radial velocity Re-analysis Application,][]{anglada:2012a}.
HARPS-TERRA derives
differential RV measurement by constructing a high
signal-to-noise ratio template from the observations and
matching it to each spectrum using a least-squares
approach. GJ 1214 is faint at optical wavelengths
(V$\sim$14). As a consequence, very long exposures (2400
sec) were required to obtain any signal at all. Still,
the typical S/N at 6100 \AA\ is only $\sim$10 so
careful construction of the template is a key element in
order to achieve the maximum precision. The standard setup
of HARPS-TERRA for M dwarfs uses all the echelle
apertures redder than the 22nd one (4400\AA $< \lambda
<$ 6800\AA) and adjusts a cubic polynomial to correct
for the variability of the blaze function across each
echelle order. Detailed algorithms and
performance of HARPS-TERRA on a representative sample of
stars is given in \citet{anglada:2012a}. It is
worth noticing that error bars
listed in \ref{tab:rvdata} are smaller than those in
\citet{charbonneau:2009}, suggesting a more optimal
usage of the Doppler information in the spectra. However,
reduced error bars do not guarantee a more
accurate orbital fit when the RV variability
is dominated by systematic noise (either instrumental
or stellar). To account for this, we included an
unknown noise term in our Monte Carlo Markov Chain
runs which had a substantial effect on the derived
distributions, especially on the orbital eccentricity
(see discussion in Section \ref{sec:rvorbit}).
\begin{table}
\caption{HARPS-TERRA radial velocity measurements.}
\label{tab:rvdata}
\centering
\begin{tabular}{lcccc}
JD & RV & $\sigma_{RV}$ \\
(days) & (m s$^{-1}$) & (m s$^{-1}$) \\
\hline\hline\\
2455036.57372 & -10.82 & 1.80\\
2455036.65153 & -4.77 & 2.08\\
2455037.58578 & 6.86 & 3.63\\
2455037.65309 & -3.37 & 1.72\\
2455038.53985 & 5.67 & 1.41\\
2455038.63702 & 5.61 & 1.78\\
2455039.55202 & -14.88 & 1.69\\
2455039.63876 & -13.53 & 1.61\\
2455040.56221 & 14.15 & 1.84\\
2455040.63961 & 5.90 & 2.49\\
2455041.57417 & 5.68 & 1.70\\
2455042.52391 & -2.11 & 3.91\\
2455042.54566 & -5.57 & 1.37\\
2455042.63521 & -4.15 & 1.42\\
2455045.55962 & 3.43 & 2.92\\
2455045.64403 & 1.61 & 2.52\\
2455046.55684 & 10.59 & 1.97\\
2455046.63141 & 8.28 & 1.94\\
2455047.55042 & -13.26 & 2.19\\
2455048.54997 & 1.64 & 1.93\\
2455048.61096 & 3.03 & 1.63\\
\end{tabular}
\end{table}
\subsection{Orbital solution}\label{sec:rvorbit}
As discussed in \citet{carter:2011}, uncertainties in the
orbital parameters (especially the eccentricity) are the main
limitation in the derivation of precise star--planet
parameters. Therefore, we are interested not only in the
favoured orbital solution, but also in obtaining a realistic
numerical representation of the posterior density function of
the parameters to propagate them into the estimates of the
star--planet properties. These samples are generated using a
Bayesian Monte Carlo Markov chain (MCMC) method.
We use custom-made MCMC software to combine RV with transit
observations and to obtain such distributions. The software is
based on the MCMC tools used in \citet{anglada:2011} adapted
to the Doppler plus transit problem, and uses a Gibbs sampler. Jump
scales of the Gibbs sampler are initialized using the
diagonal elements of the covariance matrix at the maximum
likelihood solution, and they are
adjusted in the first 10$^5$ steps (the ``burn-in'' period) to accept
between 10\% and 30\% of the jump proposals for each parameter.
As discussed below, we assume flat priors for all the sampling
parameters. As a consequence, the probability of accepting a
new state is just the ratio of likelihoods at the current and
proposed jump position. An outline of the general MCMC method
applied to the Keplerian problem can be found in
\citet{ford:2006}.
The Keplerian model for the radial velocities is the same as
in \citet{anglada:2012c} and the transit observations are
predicted using the recipes given in the NASA Exoplanet
Archive webpage
\footnote{http://exoplanetarchive.ipac.caltech.edu/applications/TransitSearch/
guide/algorithms.html}.
Transit observations strongly constrain the orbital period, but also put
strong restrictions on the sum of the initial mean anomaly
$M_0$, the argument of the node $\omega$ (also called, mean
longitude $\lambda_0=M_0+\omega$), and the product
$e\cos\omega$ (eccentricity and argument of the node).
Accordingly, we use $\lambda_0$ as a free parameter. Since the
transit instants explicitly depend on $e\cos \omega$, $e\sin
\omega$ and $e \cos \omega$ are sometimes used as the MCMC
sampling parameters. We also tested this approach,
and, while it has some desirable properties (e.g., symmetry
around zero eccentricity)-- this parameterization imposes an
\textit{implicit prior} \footnote{The implicit prior comes
from the fact that $x=e\sin\omega$ and $y=e\cos\omega$ are
cartesian coordinates derived from the polar radial coordinate
$e$ and the angle $\omega$. Because MCMCs are based on the
integration of a probability distribution, to preserve the
flat prior choices for $e$ and $\omega$, one would need to
include the Jacobian of the transformation ($1/e$) as a prior
when using $x$ and $y$ as the Markov chain sampling parameters.
The use of such a prior is undesirable due to numerical
stability issues (divergence for $e=0$).} that severely biases the
estimates of $e$ to artificially larger values \citep[e.g.,
see ][for similar discussions]{ford:2006, barros:2011}.
Therefore, at the sampling level, we choose those free
parameters that, from our point of view, should have flat
prior distributions. These are: orbital frequency $1/P$, RV
semi-amplitude $K$, mean longitude $\lambda_0$, argument of
the node $\omega$, and eccentricity $e$. Despite $\omega$ being
poorly defined and uncertain at low eccentricities, the
convergence of the MCMC was substantially improved thanks to
the use of $\lambda_0$ instead of $M_0$, so we find this
parametrization convenient and sufficient for our purposes.
All the approximate parameter values are already known from
previous studies and, therefore, we initialize the MCMCs
within 3 standard deviations of the orbital solution proposed
by \citet{charbonneau:2009}.
Real uncertainties in radial velocity measurements are
difficult to estimate properly, especially when the star is
active. To account for that, we model the uncertainty of the
$i$-th measurement as $\sigma_i^2=\epsilon_i^2+s^2$, where
$\epsilon_i$ are the formal uncertainties derived from
HARPS-TERRA (third column in Table \ref{tab:rvdata}), and $s$ is a
\textit{jitter parameter} that accounts for the extra systematic
noise. In this context, the jitter parameter $s$ is treated
as any other free parameter.
We assign a constant uncertainty to each transit instant of
400 sec. This uncertainty is 10 times larger than the typical
reported formal errors but it is more consistent with the
different instants of transit measurements obtained
simultaneously by different groups (see Table
\ref{tab:transittimes}). While the MCMC convergence properties
are greatly improved, 400 sec is a small fraction of the
time-baseline covered by the transits and, as a consequence,
the period is still strongly constrained. The transit time
measurements have been extracted from \citet{sada:2010},
\citet{carter:2011}, and \citet{kundurthy:2011}.
\begin{figure}[tb]
\centering
\includegraphics[width=3.0in,clip]{RV_Transit_fit.eps}
\caption{HARPS-TERRA radial velocity measurements
folded to the orbital period. The instant of transit
is depicted as a dashed vertical line. } \label{fig:rvfit}
\end{figure}
In table \ref{tab:rvsolution}, we provide the expected values
for the most relevant combination of parameters for physical
and transit prediction computations as derived from the
combination of 10 MCMC runs with $10^7$ steps each. Figure
\ref{fig:rvfit} is an illustration of the maximum \textit{a posteriori}
probability (MAP) solution and is given only for illustration
purposes. Since the amplitude of the signal is small
compared to the noise, the eccentricity is still poorly
constrained. The posterior distribution of the eccentricity has
a maximum very close to $0$ and monotonically decreases with
$e$ (see Figure \ref{fig:edist}) implying that close to
circular orbits are favored. Because of their significance
for the transit observables (e.g., see Section
\ref{sec:starplanet}), an illustration of the
MCMC samples of the derived parameters
$e \cos \omega$ and $e \sin \omega$ is also provided in
figure \ref{fig:edist}. Previous estimates of the
eccentricity of GJ 1214b \citet[e.g., ]{charbonneau:2009,
carter:2011} seemed to favor eccentricities close to 0.1, but
still compatible with 0. In initial tests, we recovered this
same slightly higher eccentricity value if the noise parameter
$s$ was fixed to $0$. This indicates that previous estimates
of the orbital elements were biased due to a non-consistent
treatment of the RV uncertainties. Even though stellar
jitter dominates the error budget ($s \sim$ 2.9 m s$^{-1}$),
the RMS of the MAP solution using the
new HARPS-TERRA measurements is lower (3.4 m s$^{-1}$) than the one
reported by \citet[][4.4 m s$^{-1}$]{charbonneau:2009}. Given the
strong effect of the jitter parameter $s$ on the result, and
until more RV measurements become public, we strongly
recommend the use of the updated solution in Table
\ref{tab:rvsolution} for any future work (e.g., in searching for
secondary transits). Applying the same 95\% confidence level
used by \citet{charbonneau:2009}, we obtain an upper limit to
the eccentricity of 0.23.
\begin{table}
\caption{Expected values of the most useful parameter
combinations obtained using RV and transit observations. Numbers
in parentheses represent the standard deviation of the
distributions (last two significant digits of the expected
values).}
\centering
\label{tab:rvsolution}
\begin{tabular}{lrr}
Parameter &
\\
\hline\hline\\
P [days] & 1.580400(14) \\
K [m s$^{-1}$] & 10.9(1.6) \\
$\lambda_0$ [deg] & 210.9(6.2)\tablefootmark{a} \\
e$\cos \omega$ & -0.033(55) \\
e$\sin \omega$ & -0.044(90) \\
Jitter s [m s$^{-1}$] & 3.6(1.1) \\
$\gamma$ & 0.35(1.2) \\
\\
Other parameter combinations\\
\hline
e & $<$0.23\tablefootmark{b} \\
$M \sin i$ [$M_{jup}$] & 0.0195(28)\tablefootmark{c} \\
$M \sin i$ [$M_{\oplus}$] & 6.20(91)\tablefootmark{c} \\
a [AU] & 0.0148\tablefootmark{c} \\
\hline
\\
\hline\\
N$_{transits}$ & 30 \\
N$_{RV}$ & 21 \\
RMS$_{RV}$ [m s$^{-1}$] & 3.5 \\
\hline \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{
Julian date of the reference epoch on which $\lambda_0$
is computed is the first epoch of the RV data at
$T_0=2455036.57372$ days. For a circular orbit $\lambda_0$
is equal to the mean anomaly $M_0$ at the reference epoch.
}
\tablefoottext{b}{Distribution of the eccentricity peaks
close to $0$ (see Figure \ref{fig:edist}).
}
\tablefoottext{c}{Assumes M$_*$=0.176 M$_\odot$. The
uncertainty in the mass of GJ 1214 is considered
in the final proposed parameters for the star--planet
system given in Section \ref{sec:starplanet}.
}
}
\end{table}
\begin{figure}[tb]
\centering
\includegraphics[width=3.0in,clip]{Eccentricity_dist.eps}
\caption{\textbf{Left.} Distribution of MCMC states for the
derived quantities $e \cos \omega$ and $e \sin \omega$ as
obtained from 1 MC chain of $10^7$ steps (red) and
the combination of 10 chains of 10$^7$ steps each (black).
Only $1\%$ of the steps are included in this plot to improve
visualization. \textbf{Right.} Marginalized probability
distribution of the eccentricity is shown in black (arbitrary
normalization). The corresponding cumulative distribution (red
histogram) shows that eccentricities higher than 0.23 are
ruled out at a 95\% confidence level.}
\label{fig:edist}
\end{figure}
Another interesting parameter is the minimum mass of GJ 1214b.
Even though the new RV amplitude $K$ is smaller ($\sim 10.9$
m s$^{-1}$) than the previously reported one ($\sim$12.5 m s$^{-1}$), the
updated minimum mass does not change significantly compared to
\citet{charbonneau:2009} due to the similar relative increase
in the updated stellar mass derived from the new distance
determination (see Section \ref{sec:starplanet}). The origin
of the slightly smaller amplitude is likely caused by stellar
activity. By analyzing Doppler measurements on known active M
dwarfs \citep[e.g., AD Leo, ][]{reiners:2012b}, we found that the
classic HARPS-CCF approach always produces a larger Doppler
amplitude than the one derived using HARPS-TERRA (between 1.1
to 1.5 times larger) if a candidate signal is caused by
activity. The likely explanation for this is the different
sensitivities of the two methods to the changes in the stellar
line-profiles. While one could use this effect to obtain a
further diagnostic to check the reality of low-mass candidate
planets (under investigation), in this case it means that the
most likely source of systematic noise comes from the star
rather than from the instrument. Doppler follow-up of the star
is needed to better understand the origin of the extra
noise and to further constrain the orbital solution.
\section{Updated properties for GJ 1214} \label{sec:star}
\begin{table}
\caption{Updated absolute photometry for GJ 1214.
Johnsons-Cousins BVRI photometry is from
\citep{dawson:1992}. JHKs photometry is from the 2MASS
catalog \citep{twomass}. The W1, W2, W3 and W4 are the
mid-infrared bands from the WISE
Preliminary Data Release \citep{wise:2011}
(central wavelengths are 3.4 $\mu$, 4.6$\mu$m,
12$\mu$m and 22$\mu$m). Properties derived from
the fits of the absolute photometry to the
BT-Settl-2010 model grid are also provided.}
\label{tab:magnitudes}
\centering
\begin{tabular}{lrr}
\hline\hline
M$_B$ & 15.532 $\pm$ 0.080\\
M$_V$ & 13.822 $\pm$ 0.041\\
M$_R$ & 12.441 $\pm$ 0.043\\
M$_I$ & 11.608 $\pm$ 0.042\\
\\
M$_J$ & 8.934 $\pm$ 0.041\\
M$_H$ & 8.274 $\pm$ 0.041\\
M$_{Ks}$ & 7.964 $\pm$ 0.038\\
\\
M$_{W1}$ & 7.781 $\pm$ 0.041\\
M$_{W2}$ & 7.614 $\pm$ 0.039\\
M$_{W3}$ & 7.407 $\pm$ 0.041\\
M$_{W4}$ & 7.204 $\pm$ 0.178\\
\hline
T$_{\rm eff}$ [K] & 3252 $\pm$ 20\tablefootmark{1} \\
$L_*$ [10$^{-3}$ $L_\odot$] & 4.05 $\pm$ 0.19 \\
R$_*$ [$R_\odot$] & 0.201 $\pm$ 0.010 \\
$[$Fe/H]$_{phot} $ & $+$0.13, $+$0.05\tablefootmark{2} \\
$[$Fe/H]$_{spec} $ & $+$0.20\tablefootmark{3} \\
$[$M/H]$_{spec} $ & $+$0.15\tablefootmark{3} \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{1}{Uncertainty in T$_{eff}$ only accounts for statistical
errors.}
\tablefoottext{2}{\citet{sl:2010, neves:2012}}
\tablefoottext{3}{\citet{rojas:2012}}
}
\end{table}
\subsection{Metallicity}\label{sec:metallicity}
Two independent techniques, the photometric method by \citet{sl:2010} and the
K-band spectroscopic [Fe/H] index by \citet{rojas:2012}, agree on the
metal-richness of GJ 1214, with [Fe/H] = +0.28 dex and [Fe/H] = +0.20 dex,
respectively. However, since the [Fe/H] photometric calibrations depend on the
distance of the star (to obtain M$_{K_s}$), the photometric value of GJ 1214
needs to be re-calculated using its updated distance. The updated absolute
magnitudes are given in Table \ref{tab:magnitudes}. The updated photometric
[Fe/H] values are [Fe/H] = +0.13 dex and [Fe/H] = +0.05 dex, using the
\citet{sl:2010} calibration and the recent calibration by \citet{neves:2012},
respectively. Considering the dispersions associated with the calibrations
($\sigma$~ 0.1-0.15 dex), the estimates are consistent with each other,
corroborating that GJ 1214 is a solar or super-solar [Fe/H] star.
A comparative approach can also be performed to confirm the
metal-richness of GJ 1214. Figure \ref{fig:kbandspectra} shows the
K-band spectra of Gl 699 (Barnard's star), Gl 231.1B, and
GJ1214\footnote{These spectra are part of the K-band spectral atlas
by \citet{rojas:2012}, and available to the community in the online
version of that article.}. Gl 699 is the second nearest M star to the
solar system and its kinematics, H$_\alpha$ activity, and [Fe/H]
measurements are consistent with Gl 699 being an old disk/halo star
\citep[][]{gizis:1997,rojas:2012}. Gl 231.1B is the low-mass
companion of a nearly solar metallicity G0V star ([Fe/H]=-0.04 dex,
\citet{spocs:2005}). The overall shapes of the K-band spectra of all
three stars are quite similar (same spectral type), and the
morphology of the surface-gravity-sensitive CO bands corresponds to M
dwarf stars (log g $\sim$ 5). However, all the absorption features of
GJ 1214 are stronger than the ones exhibited by metal-poor Gl 699
([Fe/H] = -0.39 dex, \citet{rojas:2012}) and solar metallicity Gl
231.1B. Therefore, also in a relative sense, the strong absorption
features favor a high metal content for GJ 1214.
\begin{figure}[tb]
\centering
\includegraphics[width=3.2in,clip]{gj1214_fig3.eps}
\caption{K-band spectra of GJ 1214 (top), Gl 231.1B (middle),
and Gl 699 (bottom). All the stars have the same K-band
spectral type (same overall shape of their spectra), but
different metallicities, as indicated by the strengths of
their absorption features. GJ 1214's metallicity should be at
least equal to or higher than solar.} \label{fig:kbandspectra}
\end{figure}
\subsection{Stellar temperature, luminosity and radius
from models}\label{sec:TLR}
We first provide an overview of previous methods and
determinations of the temperature, luminosity, and
radius of GJ 1214. In the discovery paper of GJ1214b,
\citep{charbonneau:2009} estimated a T$_{\rm eff}$=3026K
and a R$_{\star}$= 0.211R$_{\odot}$ for GJ 1214.
\citet{kundurthy:2011} obtained
T$_{\rm eff}$=2949K and R$_{\star}$= 0.211R$_{\odot}$.
Constraining parameters to stellar evolution isochrones
by \citet{baraffe:1998}, \citet{carter:2011} led to estimates of
T$_{\rm eff}$=3170K and R$_{\star}$= 0.179R$_{\odot}$,
assuming that GJ 1214 is an ``old star''. However, all
these results were based on the distance determination
of $\sim$ 13 pc by \cite{vanaltena:2001}, and reveal
that the effective temperature of GJ 1214 is quite
uncertain, given all the degeneracies in the evolutionary
model used and a low precision distance estimate.
We attempted to get an estimate of the stellar radius using
up-to-date interferometric empirical calibrations. For
example, \citet{kervella:2004} provides an empirical
luminosity--radius calibration that extends to low mass
stars and uses absolute magnitudes as the only inputs. We
tried different combinations of photometric bands obtaining
inconsistent results. All the relations involving optical
bands ($BVRI$) gave values larger than 0.4 R$_\odot$, which
are very unrealistic considering the spectral type of GJ
1214. The relations restricted to infrared colors($J, H$,
and $K$) provided estimates a bit more realistic (between
0.15 to 0.2 R$_\odot$) but still not fully compatible with
each other. We also tried the calibration provided by
\citet{demory:2009} that used K band photometry to minimize
the metallicity effect on optical magnitudes. In
\citet{demory:2009}, new measurements of 6 new M dwarfs
were presented and a new radius--luminosity calibration was
derived. They found a remarkable agreement with the
evolutionary models of \citet{baraffe:1998} if the measured
radii was plotted against the K absolute magnitude. Using
this approach, we obtain a stellar radius of 0.193
R$_\odot$ which, at least, seems to be in the expected
range.
Given that the state-of-the-art empirical relations are not
entirely self consistent, we also obtained a new estimate of the
effective temperature, luminosity, and radius of GJ 1214
using updated $BVRIJHK_sW1W2$ absolute photometry (Table
\ref{tab:magnitudes}) together with the model atmosphere
grid BT-Settl-2010
\citep{allard:2011}\footnote{http://phoenix.ens-lyon.fr/simulator/index.faces}.
The lack of indicators of youth in GJ 1214, such as
H$_\alpha$ emission, along with its space motion, gives an
age estimate of 3--10 Gyr for the star \citep{reid:2005}.
This age estimate is also supported by its long rotation
period \citep{charbonneau:2009}. Therefore, we fixed the
surface gravity of the synthetic spectral models to log g =
5.0 and [M/H]=0.0. Figure \ref{fig:sed1} shows the
bolometric luminosity as function of absolute magnitude for
the BT-Settl-2010 grid. The inferred bolometric
luminosities with the JHKsW1W2 photometry are consistent
with each other, and higher than the luminosities obtained
with the BVRI photometry. Previous synthetic models
\citep[e.g. NextGen;][]{nextgen:1999} showed a lack of flux
in the K band when compared with observed spectra of
low-mass stars. New solar abundances and the inclusion of
dust grain formation seems to have solved most of the
previous discrepancy, allowing the BT-Settl-2010 models to
reproduced fairly well the infrared spectral energy
distribution (SED) of M-dwarfs \citep[][however, the FeH
opacity data is still incomplete for this
region]{allard:2011}. Although models have also
been improved at shorter wavelengths ($BVRI$), they remain
too bright in the ultraviolet and visible part of the M
dwarf spectra, possibly due to missing sources of opacity
in the modeling process \citep{allard:2011}. The
effective temperatures and radii corresponding to the best
SEDs fit to the 9 wavebands $BVRIJHK_sW1W2$, only $BVRI$, and
only $JHK_sW1W2$, obtained from their respective
bolometric luminosities in Figure \ref{fig:sed1}, are shown
in Figure \ref{fig:sed2}. Given that 1) bolometric
luminosities obtained with $JHK_sW1W2$ photometry are
consistent with each other, 2) the models provide a better fit
at longer wavelengths, and 3) the empirical luminosity--radius
provides more self-consistency using nIR colors, the fit with
only $JHK_sW1W2$ provides the most reliable results and is
the one to be used in deriving further properties of the
star--planet system.
The SED fitting using only these nIR bands is further supported
by other results. For example in Section
\ref{sec:starplanet}, we will obtain an independent stellar
radius estimate ($0.211 \pm 0.011$ R$_\odot$) combining the
stellar mass with direct observables from the light curve
and Doppler data. Such an estimate is compatible with the
value we discussed in this section ($\sim$0.2 R$_\odot$). Also,
the estimate of the effective temperature (T$_{\rm eff}\sim
3250 K$) is in excellent agreement with the effective
temperature derived from water absorption in $K$-band by
\citet[][T$_{\rm eff}=$3245 K]{rojas:2012}. Furthermore,
next Section shows this T$_{\rm eff}$ also agrees with the
one derived from SED fits to the evolutionary models
\citep{baraffe:1998}.
\begin{figure}[tb]
\centering
\includegraphics[width=3.5in,clip]{gj1214_lum_absmag.eps}
\caption{Luminosity as function of absolute magnitude for all the
wavebands listed in Table \ref{tab:magnitudes}. Open diamonds
represent the photometry of GJ 1214. The color asterisks represent
the BT-Settl-2010 values, and the color dotted lines the linear
interpolations between them. The dashed line indicates the mean
bolometric luminosity estimated using all the photometry
(log(L/L$_\odot$)= -2.57). The dotted-dashed line indicates the mean
luminosity with only the $BVRI$ photometry (log(L/L$_\odot$)= -2.73),
and the 3 dotted-dashed line the adopted mean luminosity for GJ 1214,
derived using only the $JHK_sW1W1$ absolute magnitudes
(log(L/L$_\odot$)= -2.44). The lowest bolometric luminosity
corresponds to the I magnitude, where the SED is very steep.}
\label{fig:sed1}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=3.5in,clip]{gj1214_seds.eps}
\caption{Spectral energy distribution fits for GJ 1214 for
the effective temperatures and radii derived from
photometry. The magenta, green, and blue SEDs are the
spectral templates that represent the values obtained with
only BVRI photometry, only $JHK_sW1W2$ photometry, and the
9 wavebands, respectively. All spectral templates have
solar metallicity, log g = 5.0 and distance equal to 14.47
pc. The photometry data of GJ 1214 are plotted as black
dots. Color dots indicate flux levels of each spectral
template integrated over the corresponding filter
bandwidth, depicted by the horizontal black lines. The
spectral template with T$_{eff}$=3250K and R=0.2R$_\odot$
provides the best fit to the infrared photometry of GJ
1214.}
\label{fig:sed2}
\end{figure}
\subsection{Stellar mass}\label{sec:empmass}
Although most of the previously reported mass-radius estimates
of GJ 1214 agree, we note that they all assume the
trigonometric distance estimate from \citet{vanaltena:2001}.
This distance determination has an uncertainty of 10\% which
is not always accounted for in the aforementioned predictions.
We also note that the M1V-M4V spectral type range spans a
broad range of stellar masses (from 0.4 to 0.15 M$_\odot$) and
effective temperatures (3500 to 2800 K). This makes any mass
estimate very sensitive to uncertainties in the parallax mesurement.
Given the updated distance, we can apply the
\citet{delfosse:2000} (hereafter DF00) relations to derive a
new mass for GJ 1214 using the absolute $J$, $H$ and $K$ magnitudes.
The masses derived from each photometric band are M$^{(J)}$ =
0.174 M$_\odot$ ,M$^{(H)}$= 0.177 M$_\odot$ and M$^{(K)}$=
0.177 $_\odot$, which are in very good agreement.
Another way to obtain an estimate of the mass for GJ 1214
is to compare the absolute magnitude with the synthetic
colors from the evolutionary models in
\citet{baraffe:1998}. Unfortunately, \citet{baraffe:1998}
does not contain models for stars with [Fe/H]$>$0.0. As a
result the optical colors (e.g., $B, V, R, I$), cannot be
properly adjusted to a model. The fit including the $BVRIJHK$
magnitudes has T$_{eff}$ = 2987 K ($M$ = 0.12 $M_{\odot}$
and is of very low quality ($\chi^2$ = 45 for 6 degrees of
freedom). The fit to $BVR$ alone give T$_{eff}$ = 2880 K,
$M$ = 0.11 $M_{\odot}$ and a $\chi^2$ = 10.0 for 3 degrees of
freedom. In contrast, the best fit model obtained from
adjusting $J$, $H$, and $K$ has a $\chi^2$ of 1.92 for 3 degrees of
freedom, an effective temperature of 3225 K and a mass of
$M$ = 0.172 $M_{\odot}$. Note that the mass value is very
good agreement with the mass estimate derived from the
DF00, and the effective temperature is surprisingly close
to one from the atmospheric transfer models in the
previous Section \ref{sec:TLR}. How the uncertainty in the
stellar mass affects the planet parameters will be
discussed and accounted for in Section \ref{sec:starplanet}.
\section{Star--planet parameters from direct
observables}\label{sec:starplanet}
\subsection{Using the stellar mass as the input}
Information from the transit light curves together with the
orbital solution can be combined with the stellar mass (or the
stellar radius) to fully characterize the bulk properties of
the star--planet system. These methods have been developed
over the years by different authors and we suggest reading
\citet{seager:2003}, \citet{southworth:2008} and
\citet{carter:2011} for more information on the derivations.
This subsection describes the relevant relations and the
general approach used to derive the star--planet parameters
using the stellar mass as the input (\textit{mass--input
approach}).
Assuming a circular orbit for the planet, the transit
light curve alone allows one to obtain a direct measurement
of the mean stellar density $\rho_{*, circ}$. Given a fully
Keplerian solution from the Doppler data, the stellar
density also depends on the eccentricity of the orbit and can
be written as
\begin{eqnarray}
\rho_* = \rho_{*, circ}
\left(\frac{\sqrt{1-e^2}}{1+e\sin \omega}\right)^3\,.
\label{eq:meanRho}
\end{eqnarray}
\noindent Because no prior information is required from the
orbital fit, $\rho_{*, circ}$ is a quantity typically provided
by the studies analyzing transit light curves of GJ 1214
\citep[e.g.,][]{carter:2011,kundurthy:2011}. These two studies
present the analysis of new light curves and combine them with
previous light curves from \citet{charbonneau:2009} and
\citet{sada:2010}. A weighted mean of $\rho_{*, circ}$ from
these two studies will be used in all that follows. The
detailed derivation of Eq. \ref{eq:meanRho} was first given by
\citet{seager:2003} for circular orbits, while
\citet{carter:2011} included the dependence on the
eccentricity.
With the mean stellar density from Equation \ref{eq:meanRho}
and the stellar mass from the empirical calibrations discussed
in Section \ref{sec:empmass}, one can trivially derive the
radius of the star. Using this radius and the transit depth
$R_p/R_*$ (again, a direct observable from the light curves)
one then obtains the radius of the planet. The combination of
the minimum mass (from RV) and the inclination (from transit
light curve) provides the planet's true mass, which is then
combined with the planet radius to finally derive the mean
planet density.
\begin{table}
\caption{Input parameters used to compute the Monte
Carlo distributions of the star--planet parameters}
\label{tab:input}
\begin{tabular}{lcccc}
Parameter &
Distribution &
Expected &
Standard&
Ref.\tablefootmark{A}\\
name &
type &
value &
deviation&
\\
\hline\hline\\
M$_*$ [$M_\odot$] & Gaussian & 0.176 & 0.009 & 1,2 \\
$\rho_{*, circ}$ [g cm$^{-3}$]& Gaussian & 23.695 & 1.7 & 3,4 \\
a/R$_*$ & Gaussian & 14.62 & 0.3 & 3,4 \\
R$_p$/R$_*$ & Gaussian & 0.01178 & 0.001 & 3,4 \\
Inc. [deg] & Gaussian & 89.19 & 0.5 & 3,4 \\
Period [days] & Bayesian & 1.580400 & 1.4 10$^{-5}$ & 1 \\
K [m s$^{-1}$] & Bayesian & 10.9 & 1.6 & 1 \\
$\lambda_0$ [deg] & Bayesian & 210.9 & 6.2 & 1 \\
$e\, \cos \omega$ & Bayesian & -0.033 & 0.055 & 1 \\
$e\, \sin \omega$ & Bayesian & -0.044 & 0.090 & 1 \\
\end{tabular}
\tablefoot{
\tablefoottext{A}{
(1) This work,
(2) \citet{delfosse:2000},
(3) \citet{kundurthy:2011},
(4) \citet{carter:2011}
}
}
\end{table}
To obtain realistic \textit{a posteriori} distributions,
one needs to assume realistic distributions for the
input parameters of the model. The complete list of
input parameters used at this point are given in Table
\ref{tab:input}. For all the values borrowed from the
literature, a Gaussian distribution $N\left[\mu,\sigma\right]$
is assumed where $\mu$ is the preferred value and
$\sigma$ is its published uncertainty. For the
parameters derived from the orbital solution in Section
\ref{sec:rvorbit}, we directly draw samples from the MCMC
distributions generated during the orbital analysis. The
uncertainty in the stellar mass has to be accounted for at
this point. Given the precision in the distance and in the
$J, H$ and $K$ photometry, the major source of
uncertainty in the mass is due to the actual accuracy of
the DF00 calibration. This accuracy is not very well
known, but other studies suggest that it should be correct
at the 5\% level, which is the relative uncertainty we
will use for the stellar mass.
The process of solving for all the star--planet
parameters is done for the 10$^6$ synthetic input sets. The
result is a numerical representation of the empirical
probability distributions for the derived star--planet
properties. In Figure \ref{fig:planet_values} we show
the obtained distributions concerning the planet
properties only (mass, radius, density and surface
gravity). The surface gravity of the planet can also be
derived from observables only using the prescriptions
given by \citet{southworth:2008} (combination of light
curve parameters and parameters from the orbital
solution). Table \ref{tab:starplanet}, provides the
expected values and standard deviations of the distributions
for each derived star--planet parameter.
We find that the mean density of GJ 1214b has to be
smaller than 2.40 g cm$^{-3}$ with a 95 \% confidence
level (c.l.), with 1.69 g cm$^{-3}$ being the expected value of the
distribution. An Earth-like density (e.g., $\rho>5.5$ g
cm$^3$) is thus ruled out at a $>$ 99.9 \% c.l.
As illustrated in Fig. \ref{fig:planet_values} this
density range confirms that the planet is more similar
to a small version of Neptune, or a water dominated body
\citep[e.g., ocean planets, ]{berta:2012}, rather than a
scaled-up version of the Earth with a solid surface. This
information by itself does not solve the question about the
atmospheric composition discussed in the Introduction, but,
at least, eliminates the uncertainty in the distance of the
star in modeling of the possible planetary structure.
We want to stress that, thanks to the new distance
estimate, we now find a remarkably good agreement of the
derived stellar radius compared to the one obtained from the
SED fit in Section \ref{sec:TLR}.
As a final test, we reproduced the star--planet parameters
using the stellar radius instead of the stellar mass. This
consists of doing the following: the stellar radius (SED fit)
combined with the mean stellar density (light curve plus RV) gives
the stellar mass. Then the stellar mass combined with the RV
observables and the inclination (light curve) provides the
planet mass. In parallel, the stellar radius with the transit
depth (light curve) gives the planet radius, which combined
with the mass finally provides the planet density. This
approach, however, has a non-obvious drawback. The mass
obtained through the stellar density formula depends as the
third power on the stellar radius ($M_*\propto
R^3$), making the stellar mass determination very sensitive to
small radius changes. Going the other way around (starting from the
mass, derive the radius), the stellar radius depends as M$^{-1/3}$,
resulting in a much weaker dependence. For example, a $5\%$
uncertainty in the mass translates to a $1.6\%$ uncertainty in
the radius. On the other hand, a $5\%$ uncertainty in the
radius translates to a $15\%$ uncertainty in the mass. When
the uncertainty in the stellar density is also included, this
difference is even more pronounced. Therefore, we consider the
\textit{mass input} method a much more robust way of providing
a consistent picture. The updated values using the
\textit{mass input} approach for the star--planet parameters
are in Table \ref{tab:starplanet} and the marginalized
distributions for the relevant planet parameters are depicted
in Fig. \ref{fig:planet_values}
\begin{figure}[tb]
\centering
\includegraphics[width=3.2in,clip]{planet_values.eps}
\caption{Final probability distributions for most
significant bulk properties of GJ 1214b. The mean
densities of a few representative Solar system objects
are marked as black vertical bars for reference. }
\label{fig:planet_values}
\end{figure}
\begin{table}
\caption{Suggested new star--planet parameters as derived using the stellar mass
as the only input parameter. Note that some of the distributions in
Fig.~\ref{fig:planet_values} are strongly asymetric. }
\label{tab:starplanet}
\begin{tabular}{lrrr}
Parameter &
Maximum &
Expected &
Standard\\
name &
probability\tablefootmark{\dagger} &
value &
deviation
\\
\hline\hline
Star mass [M$_\odot$] & \textbf{0.176}
\tablefootmark{a}
& - & 0.0087 \\
\\
Star radius [R$_\odot$] & 0.213 & 0.211 & 0.011 \\
Star surface gravity [m s$^{-1}$] & 1032 & 1109 & 220 \\
$\log g$ [g in cm s$^{-1}$]\tablefootmark{b}& 5.01 & 5.04 & 0.07 \\
Star mean density [g cm$^{-3}$] & 26.36 & 27.7 & 8.9 \\
\\
Planet radius [R$_\oplus$] & 2.80 & 2.72 & 0.24 \\
Planet mass [M$_\oplus$] & 6.26 & 6.19 & 0.91 \\
Planet surface gravity [m s$^{-2}$] & 7.66 & 7.68 & 1.19 \\
Planet mean density [g cm$^{-3}$] & 1.56 & 1.69 & 0.61\tablefootmark{b} \\
\\
\multicolumn{2}{l}{Planet T$_{eq}$[K] (Bond albedo = 0)} & 576 & 14 \\
\multicolumn{2}{l}{Planet T$_{eq}$[K] (Bond albedo = 0.75)} & 407 & 11 \\
\end{tabular}
\tablefoot{
\tablefoottext{\dagger}{Maximum of the marginalized posterior distribution}
\tablefoottext{a}{Used as input}
\tablefoottext{b}{From light curve and Doppler analysis (no spectral adjustment)}
\tablefoottext{c}{$\rho_p < 2.40$ g cm$^{-3}$ with a 95\% c.l.}
\\
}
\end{table}
\section{Conclusions}\label{sec:conclusions}
The metallicity of GJ 1214 (and of M dwarfs in general) and
its effects on the observable fluxes (e.g., optical
fluxes) still require a better understanding, both
observationally and theoretically. Even though
spectroscopic methods in the nIR are getting better
at determining the metal content of cool stars, precise
direct measurements of their distances are still
required for the characterization of exoplanets around
M dwarfs. At least for the stellar masses, the model
predictions from the near-infrared fluxes seem to be in good
general agreement with current empirical relations derived
from measured masses of M dwarfs in binaries (e.g.,
DF00). This is fortunate because the stellar mass is the
only input parameter required to derive the star--planet
bulk properties from direct observables. The fit of the
stellar spectral energy distributions to the
BT-Settl-2010 model grid also provides consistent
estimates in the stellar properties if infrared
photometry is used. Even with the remaining
ambiguities in the stellar parameters, both approaches
(using a stellar mass from empirical relations, or deriving
a radius from the absolute fluxes) lead to consistent
results for the star--planet bulk properties, but the
\textit{mass input} approach is preferred due to the
lower sensitivity of the method to uncertainties in
the input parameters. Several studies
have been published in recent years trying to better
characterize the GJ 1214 star--planet system. The major
source of the reported uncertainties came from directly
observable quantities that we have improved, collected,
and combined here: trigonometric parallax and
corresponding absolute fluxes, improved RV measurements,
inclusion of all of the transit observations in the derivation
of the orbital solution, additional infrared flux
measurements, and light curve observables derived from
photometric follow-up programs.
We now find remarkable agreement of the derived star
properties obtained by comparing the $JHK$ fluxes to up-to-date
atmospheric and evolutionary models. All previous studies had
to invoke some mechanism (e.g., spot coverage) to justify the
mismatch between the predicted versus observed properties of
the star. This alone highlights the importance of obtaining
direct and accurate distance measurements of low mass stars.
At this time, the quantity that most requires further
improvement and/or independent determination is the radius of
the star. To our knowledge, this can be observationally
achieved with 3 different methods: 1) additional RV
measurements to further constrain the orbital eccentricity, 2)
direct measurement of the stellar diameter using optical/nIR
interferometry \citep[e.g.,][]{vonbraun:2012}, or 3) detection
of the secondary transit (whose instant strongly depends on
$e$ and $\omega$). We note that GJ 1214 is faint at
optical wavelengths compared to other stars observed by high
precision RV instruments \citep[e.g., HARPS or
HIRES][]{bonfils:2011,vogt:2010}. In the case of HARPS,
integrations of 45 min were required to obtain a precision of
$\sim$3.4 m s$^{-1}$, so a refinement of the orbital eccentricity
through additional RV measurements is time-consuming but
quite possible. On-going space--based photometric observations
in the mid-infrared (e.g., using Warm Spitzer/NASA) should be
able to detect the secondary transit soon and pin down the
orbital eccentricity to greater precision. We hope that the
updated orbital solution provided here facilitates the task of
finding such a secondary transit.
\acknowledgements We thank the referee D. S\'egransan for
useful comments that helped improving the manuscript. GA has been
partially supported by a Carnegie Postdoctoral Fellowship and by
NASA Astrobiology Institute grant NNA09DA81A. BR thanks the staff
and telescope operators of Palomar Observatory for their support.
The CAPS team (APB, AJW, GA) thanks the staff and telescope
operators of Las Campanas Observatory for the very succesful
observing runs and the Carnegie Observatories for continuous
support of the CAPS project. We thank France Allard for helpful
discussions about various topics. Part of this work is based on
data obtained from the ESO Science Archive Facility. This research
has made use of NASA's Astrophysics Data System Bibliographic
Services, the SIMBAD database, operated at CDS, Strasbourg, France.
This publication makes use of data products from the Two Micron All
Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National
Aeronautics and Space Administration and the National Science
Foundation. This publication makes use of data products from the
Wide-field Infrared Survey Explorer, which is a joint project of
the University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,129
|
Anomalepis es un género de serpientes de la familia Anomalepididae. Incluye cuatro especies de serpientes ciegas que se distribuyen por Sudamérica y América Central.
Especies
Se reconocen a las siguientes especies:
Anomalepis aspinosus Taylor, 1939 - Norte de Perú.
Anomalepis colombia Marx, 1953 - Caldas (Colombia).
Anomalepis flavapices Peters, 1957 - Noroeste de Ecuador.
Anomalepis mexicanus Jan, 1860 - Sur de América Central.
Referencias
Enlaces externos
Anomalepididae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,013
|
import Store from "../store";
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|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,619
|
Möbel Schulenburg Hamburg Halstenbek article is the best inspiration for home interior posted about Lieblingsmöbelidee. This Möbel Schulenburg Hamburg Halstenbek was posted in category Lieblingsmöbelidee as ideas for inspiration to Remodel your Lieblingsmöbelidee accessories. This article can be your reference when you are confused to choose the right decoration for your home accessories. This Möbel Schulenburg Hamburg Halstenbek maybe your best option to decoring m bel schulenburg angebote, m bel schulenburg bremen bremen, m bel schulenburg goslar, m bel schulenburg goslar gmbh, m bel schulenburg halstenbek betten, because having a home with our own design is everyone's dream.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,484
|
Apatelodes diana is a moth in the family Apatelodidae. It is found in Colombia.
References
Natural History Museum Lepidoptera generic names catalog
Apatelodidae
Moths described in 1916
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,089
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Australia's Senate has finally released its report into the Australian franchising sector
New tougher laws proposed for the franchise sector in Australia following Parliamentary enquiry
Why are there proposals to change franchising law in Australia?
Australia's Senate has finally released its report into the Australian franchising sector and it makes for uncomfortable reading.
The franchise sector covers franchise agreements as well as a range of intellectual property licences and distributorship arrangements and is, at present, primarily regulated by the franchising law, known as the Franchising Code of Conduct or Code.1 The sector contributes approximately nine per cent of gross domestic product (GDP) in Australia with around 80,000 franchise outlets turning over AU$146 billion.2 Around four per cent of small businesses in Australia are franchises and nearly half a million people are directly employed in the sector.3
Following a lot of negative press highlighting franchisee exploitation, the Australian Senate in 2018 referred an inquiry into the operation and effectiveness of the franchising laws to a Parliamentary Joint Committee on Corporations and Financial Services (Committee). The Committee's 369-page report, following public hearings and extensive submissions from the sector, called "Fairness in Franchising" was released on 14 March 2019 (Report).
In a nutshell, the Committee found significant problems in the sector, including exploitation of franchisees and decided that the current regulatory environment "manifestly failed to deter systemic poor conduct and exploitative behaviour and has entrenched the power imbalance." 4 The result is a range of recommendations to:
make the regulation of franchising stronger and tougher;
introduce new and higher penalties for breach of the franchising laws; and
give the regulator, the Australian Competition and Consumer Commission (ACCC), more responsibilities and greater enforcement powers.
We set out below the key recommendations of the Committee for franchisors, franchisees and franchise systems in Australia. Although the Australian Government is yet to respond formally to the Report, if the recommendations are accepted and the changes are implemented, then the changes will materially impact the sector.
What the key changes to the Australian franchising laws are recommended by the Committee?
We summarise below the recommendations by the Committee which we think will most materially impact the franchising sector. The Committee made 71 recommendations in all but the summary below covers only those that we think will have the greatest implications for the sector.
Report recommendation
Why is this proposed change important?
A) Unfair contract terms
Australia has general unfair contract terms legislation which operates to protect certain businesses entering into standard form contracts (UCT laws).5 Before the UCT laws, the ACCC reviewed a sample of franchise agreements to check compliance with the new laws and identified four common types of contract terms contained in franchise agreements that it considered could be problematic:
the right to unilaterally vary operations manuals;
liquidated damages clauses;
restraints of trade; and
termination clauses that grant a franchisor an unreasonable power to terminate.
The Committee found the UCT laws have had little impact on the franchising sector as the franchisee or the ACCC has to pursue court action to challenge a potentially unfair term and have it declared void.
Apply the UCT laws to franchise agreements regardless of whether the franchisee is a "small business".
Introduce penalties for including unfair contract terms in franchise agreements.
Amend the Code to outlaw unilateral variation to the terms of a franchise agreement unless there is agreement by the majority of franchisees within the same franchise system or representatives elected by a majority of franchisees within the same franchise system.
The UCT laws will be widened to protect franchisees irrespective of the size and scale of the franchisee as they will no longer need to fall within the definition of a "small business".
Franchisors should review their current franchise agreements to see if they are at risk of:
parts of the agreement being deemed unfair and so unenforceable;and
penalties down the track if the unfair term remains in the franchise agreement.
B) Pre-contract disclosure
The Committee found that an imbalance between franchisors and franchisees about the information relevant to the franchise system favours franchisors and so can hamper franchisees when they are conducting due diligence so they can make informed decisions about fees and other costs, contractual obligations and personal risks.
The Committee thought that this is particularly problematic where relevant information cannot be obtained independently of the franchisor.
The Committee though both the up-front disclosure and disclosure during the term of a franchise agreement were not adequate.
Require public registration of franchise systems and disclosure documents with the ACCC or other government authority. More frequent updates to the disclosure documents and template franchise agreements and require these to be included on the new register.
More disclosures covering:
supplier rebates;
franchisee profit margins and earnings;
marketing fund expenditures.
Any new registration process will increase the administrative burden on and costs of franchisors. There is a risk with more disclosures of inaccurate or misleading disclosures – which could lead to legal liability and litigation. Disclosure can be problematic also when the franchise model is not well enough established to provide the required disclosures.
Franchisors should monitor the development but not take action until the Australian Government's reaction to the disclosure proposals are understood.
C) Dispute resolution
The Committee thought that there was a lack of accessible, affordable and effective dispute resolution under the Franchising Code of Conduct in light of the power imbalance between the respective parties and that the current range of dispute resolution mechanisms under the Code (mandatory to have a right to mediation) were not enough as, if mediation is ineffective, the only current alternative is institution of court action which can be prohibitively expensive for franchisees.
Include binding arbitration in the Code's dispute resolution process.
Prohibit taking legal action until alternative dispute resolution is complete unless, if a franchisor takes a matter to court, the franchisor demonstrates to the court's satisfaction that the matter cannot be resolved through mediation.
Permit franchisees to collectively bargain with the franchisor.
The power of franchisees to negotiate a collective bargain will dramatically change the power balance and alter the face of many deals, favouring franchisees. Franchisors will need to consider which of their systems are at greatest risk of franchisee collective action and monitor this development closely.
Arbitration is often, but not always, preferred by franchisors and may not be less expensive than court action. Many franchisors will not find this recommendation as troubling as the proposals (discussed below) which hinder the franchisor's ability to take court action – whether injunctive or not - as inhibiting the franchisor's right to obtain urgent interim relief when the brand's reputation is at stake may impact the value of the brand and the franchisor's intellectual property rights.
D) Termination rights
The Committee found that many in the sector who made submissions were supportive of an extension of the seven day cooling off period to fourteen days to enable prospective franchisees to complete all due diligence and make their final assessment in a timely manner.
The Committee also found that transfers may effectively be a new franchise agreement and so disclosure and cooling off should also apply on principle.
The Report also noted that exit arrangements do not provide a way for franchisees to exit an unviable franchise that is fair to both parties and in a way that reasonably constrains financial losses and franchisor risks are low on franchisee early exit as franchisor losses are covered but that the franchisee could be bankrupted.
The Committee thought that rights should also exist for franchisees to terminate franchise agreements where a franchisor is subject to special circumstances such as franchisor insolvency but that franchisors are unlikely to voluntarily include such rights in franchise agreements. The Committee also thought that there was a risk of franchisor abuse (and so recommended constraining the right of the franchisor to terminate the agreement) of the franchisor's rights to terminate the franchise agreement when the franchisee:
acts in a way that endangers public health or safety; or
acts fraudulently,
as there was too much franchisor discretion as to what constitutes a breach, rather than the determination being made by an independent decision-maker.
Extend the franchisee's right to terminate the franchise agreement during the cooling off period to transfers, renewals and extensions.
Lengthen the cooling off period from 7 to 14 days.
Provide a new right for franchisees to terminate the agreement without liability to the franchisor if the franchisee is suffering from "personal hardship", making a loss or if the franchisor is insolvent, fraudulent or deregistered.
Restrict franchisors from exercising rights to terminate the franchise agreement for franchisee fraud or endangerment of public health or safety unless:
the franchisee is actually convicted of fraud; or
the franchisee is served with a 'permanent closure direction' for the franchise by a relevant government body, or failure to remedy workplace health and safety orders or notices where the issue is the franchisee action has endangered public health or safety.
The proposed changes to the cooling off period are unlikely to trouble many franchisors as they will simply plan for the right to be triggered.
Mandating additional franchisee rights to terminate agreements will be of greatest concern to franchisors especially if there is any ambiguity as to how the trigger for "personal hardship" on the part of the franchisee would work. Franchisors will be concerned if the right is a way for franchisees to escape liability to the franchisor when the franchisee has been poor performers because of their own efforts.
Franchisors with systems and brands to protect will be very concerned if they are restricted from terminating a franchisee who has committed a fraud or endangered public health or safety until court or regulatory processes (which might take years) are complete.
A restriction on the right will severely impact the franchisor's ability to mitigate the risk of harm to its reputation. Franchisors should monitor this development very closely.
E) Penalties
Finally and unsurprisingly, the Committee recommended higher penalties for breaches of the Code as well as new protections for whistle blowers and more power for the ACCC to intervene.
The Report awaits the Australian Government's formal response to its proposals. Many recommendations are controversial however it is inevitable there will be changes to Australia's franchising laws – and they may be significant.
For now, franchisors, franchisees and franchise systems in Australia, and those also with distribution and intellectual property licensing arrangements in Australia (* which can be caught by the laws) should monitor the developments closely.
It is unlikely that any formal response to the Report or draft legislation will be in place before mid-2019.
Please contact Robyn Chatwood, Partner in the Dentons Australian Franchise practice group, or your usual Dentons contact if you need further information or wish to be added to our list to receive updates on the developments.
1. Formally called the Competition and Consumer (Industry Codes— Franchising) Regulation 2014 (Cth).
2. Report "The operation and effectiveness of the Franchising Code of Conduct" dated 14 March 2019, Commonwealth of Australia 2019, p 10-11 at https://www.aph.gov.au/Parliamentary_Business/Committees/Joint/Corporations_and_Financial_Services/Franchising/Report citing Griffith University's Franchising Australia 2016 report.
3. As above.
4. Item 2 at p xxii.
5. The UCT applied from 12 November 2016 through the Treasury Legislation Amendment (Small Business and Unfair Contract Terms) Act 2015 (Cth). Under Australian Consumer Law (ACL), "small businesses" are provided with some protection from the abuse of unfair terms contained in such contracts.
Robyn Chatwood
Robyn Chatwood Partner, MelbourneMelbourne
D +61 3 9194 8330
M +61 404 195 125
robyn.chatwood@dentons.com
Esteban Gomez
Esteban Gomez Partner, BrisbaneBrisbane
Esteban.gomez@dentons.com
Lis Boyce
Lis Boyce Partner, SydneySydney
lis.boyce@dentons.com
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 367
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Q: Gradle object instantiation cannot find constructor I'm currently developing a Gradle plugin that, when applied, creates an extension object:
abstract class MyExtension {
@Delegate
private final DelegateExtension delegateExtension // Third-party
@Inject
protected MyExtension(DelegateExtension delegateExtension) {
this.delegateExtension = delegateExtension
}
// More stuff here
}
class MyPlugin implements Plugin<Project> {
@Override
void apply(Project project) {
project.pluginManager.apply(ThirdPartyPlugin) // DelegateExtension class
DelegateExtension delegateExtension = project.extensions.findByType(DelegateExtension)
MyExtension myExtension = project.extensions.create('myExtension', MyExtension, delegateExtension)
// More stuff here
}
}
The issue I am encountering is that when I try to apply my plugin, Gradle errors out creating the myExtension extension.
> Could not create an instance of type MyExtension
> No constructors of type MyExtension match parameters: [extension 'delegateExtension']
A Gradle 6.9.1 stack trace shows that this error comes from ParamsMatchingConstructorSelector.forParams(), which seems a bit odd. In any event, Gradle appears to not be able to connect the create() call to the MyExtension constructor via a concrete subclass of MyExtension that Gradle most likely generated internally. Any assistance in diagnosing this issue would be appreciated.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,614
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Village People Singer Reclaims Copyright
Victor Willis, likely more recognizable as the "police man" from the popular group the Village People, recently scored a significant copyright victory. After signing away his rights to popular songs like "YMCA" and "In the Navy" in the 1970s, he is again a rightful copyright owner.
Willis, who helped write the lyrics to more than 33 of the Village People's songs, was able to recover his rights under a U.S. copyright law governing termination rights. The Copyright Act provides a termination right for the prior grant of a copyright transfer or license if the grant was made prior to January 1, 1978. Termination may be effected at any time during a five-year-period beginning at the end of thirty-five years from the date of execution of the grant.
The rationale is that copyright owners of popular works may not have had the bargaining power to secure favorable terms when the deal was originally signed. Therefore, they should be afforded a second bite at the apple.
While most artists who regain control over copyrights sign them back over to the music company after renegotiating a new deal, Willis told the New York Times that he is unsure of his next steps.
"I learned over the years that there are some awesome powers associated with copyright ownership," Mr. Willis said. "You can stop somebody from performing your music if you want to, and I might object to some usages."
If you, or someone you know, need any help with Intellectual Property issues, from filing a patent, trademark or copyright, or just need advice regarding how best to protect your inventions, ideas or your brand, please contact me for a free 30 minute consultation at nvantreeck@usip.com or call TOLL FREE at 1-855-UR IDEAS (1-855-874-3327) and ask for Norman.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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ALSO BY CHRISTOPHER BUCKLEY
_Little Green Men_
_God Is My Broker_ (with John Tierney)
_Wry Martinis_
_Thank You for Smoking_
_Wet Work_
_Campion_ (a play, with James McGuire)
_The White House Mess_
_Steaming to Bamboola: The World of a Tramp Freighter_
This is a novel, that is, "a relatively long fictional prose narrative with a more or less complex plot or pattern of events, about human beings, their feelings, thoughts, actions, etc." ( _Webster's New 20th Century Dictionary_ , unabridged, Collins World Publishers, 1978). It is also satire, namely, "a literary work in which vices, follies, stupidities, abuses, etc. are held up to ridicule and contempt" (ibid.). Finally, and just to belabor the now obvious, all characters herein are _entirely_ the work of the author's own twisted imagination and have absolutely nothing to do with actual human beings invented by, say, God.
Copyright © 2002 by Christopher Taylor Buckley
All rights reserved under International and Pan-American
Copyright Conventions. Published in the United States by Random
House, Inc., New York, and simultaneously in Canada by
Random House of Canada Limited, Toronto.
RANDOM HOUSE and colophon are registered trademarks of Random House, Inc.
Library of Congress Cataloging-in-Publication Data
Buckley, Christopher.
No way to treat a First Lady : a novel / Christopher Buckley.
p. cm.
eISBN: 978-1-58836-257-5
1. Attorney and client—Fiction. 2. Presidents' spouses—Fiction. 3. Trials (Murder)—Fiction. 4. First ladies—Fiction. 5. Widows—Fiction. I. Title.
PS3552.U3394 N6 2002 813′.54—dc21 2002069926
Random House website address: www.atrandom.com
v3.1
FOR DON AND MEG GREGG,
WITH LOVE
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
Prologue
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
Chapter 38
Epilogue
_Acknowledgments_
_About the Author_
# Prologue
Babette Van Anka had made love to the President of the United States on eleven previous occasions, but she still couldn't resist inserting "Mr. President" into "Oh, baby, baby, baby." He had told her on the previous occasions that he did not like being called this while, as he put it, congress was in session. But she couldn't stop thinking to herself, _I'm screwing the President of the United States! In the White House!_ Unavoidably, the "Mr. President" just kept slipping out.
Thrilled as she was, however, tonight Babette was ready for occasion number twelve to be over. It was after 2:00 A.M., and it was plain from the exhausted grunting noises and wheezes coming from her partner that he was straining to prove a point, more to himself than to her. God knows she had done what she could to make the evening exciting for him, but it had turned into an endurance contest. Where's the romance in that? Plus there was the fact that the First Lady of the United States was right down the hall. The President had assured her that his wife was sound asleep, but this was bold even by his incautious standards. The combination of his unamorous rutting and his wife's proximity made it difficult for Babette to relax and enjoy.
She concentrated on helping her commander in chief achieve bliss. It was work. Occasions one through four had been earth-moving experiences. Five through eight had been pretty exhilarating. Occasion nine—disaster. Ten and eleven were little more than awkward attempts to rekindle Eros' flame. Was he losing interest? The pressure was on. Babette knew that she had to deliver or suffer the unthinkable—indifference. No more overnights in the Lincoln Bedroom at the White House, no more trips on Air Force One, no more golf or meals with world leaders, no more seat at the table. And just a few weeks ago he had dangled before her the prospect of a Middle East summit weekend at Camp David, the presidential retreat. Oh, to miss _that_.
Babette stirred from her reverie of trysting with the President in the Catoctin Mountains amid prime ministers.
There was a sound.
_... bump... bump... bump..._
It was the presidential head, striking the Lincoln bed headboard.
"Oh," she whispered, _"yes_... _yes... oooooh.... yessss..."_
Sometimes that got them off. Men loved an affirming sound track.
Babette sneaked a peek at the luminous face of her watch. Jesus Christ, he'd been humping her for over half an hour. Normally she'd be tickled to a puddle, but not tonight. The wife down the hall, Secret Service agents everywhere. She'd said to him, Tonight? Here? Is this _smart_? Navy men—they got off on risk.
He was sweating. He was hot to the touch. His breathing sounded labored. What was this new sound?
_Unh, unh, unh, unh_.
Grunting. Wonderful. It made her feel as sexy as a slab of meat.
She opened her eyes, then wished she hadn't. He had this _look_ , like that of an exhausted bull salmon fighting his way up rocks to squirt his DNA over the roe so he could turn belly up and die. Isn't it romantic?
He was probably fantasizing about—someone else. Some body he'd seen in a magazine.
_"Unnnnnnnnnh."_
Finally, thank God.
_"Ohhh,"_ she lied.
Silence. The sheets were damp from presidential sweat. Babette liked clean, crisp, ironed sheets, the kind they had in British hotels, so much starch that they crackled. Now look at her bed. Lake Superior. What was she supposed to do, ring for the maid in the middle of the night to demand that they change the bed linen? Uch. She was going to have to sleep in them. Wonderful. She and her husband had donated half a million dollars to the party, and for what? To be on the receiving end of a joyless hump, with the risk thrown in of being walked in on by the wife, then to spend the rest of the night in damp sheets.
He rose.
He had gotten out of bed, without so much as a kiss or pat on the bottom, and was silhouetted against the window overlooking the South Lawn of the White House. He seemed unsteady.
She flicked on the bedside lamp. A vision greeted her: the President of the United States of America, naked but for knee socks, his face flushed like a Harvard beet, his most prominent feature still perpendicular from excitement.
"Nothing wrong with _you_ ," Babette purred in her best Mae West accent.
The President looked down at his cantilevered anatomy, taking it in clinically. He grinned and made a satisfied, male grunt. He stooped to gather up his clothes, scattered over the floor. These were the only occasions when he had to pick up his own clothes. One of the perks of the office was to undress like a maharajah, tossing garments to the floor to be picked up uncomplainingly by reverent lackeys.
He pulled on his trousers but was unable to zip up. He seemed amused by this challenge, but then a look of distress took over his features and he backed into an armchair, where he sat, defeated, fly open.
"Would you like me to—"
Before Babette could finish her offer, the President lurched out of the chair purposefully toward her.
What impetuosity! She prepared to receive him, but he veered off in the direction of the nightstand. He grasped the leaded crystal carafe of ice water and with the other hand painfully bent the afflicted object downward and plunged it into the icy carafe.
Babette's mouth gaped as she viewed the presidential anatomy immersed in her ice water. A wonder there was no hissing of steam.
The immersion had the desired effect. The President was able to sheathe the afflicted limb in his trousers, though the zipping was done with extreme care, as if unstable nitroglycerin were involved.
Having finished dressing and combed his hair, he turned and flashed her a grin of triumph, with a navy-man wink. He opened the door, put his head out to check both ways down the hall, and was gone, leaving Babette to her damp sheets and unappealing ice water.
Elizabeth Tyler MacMann, First Lady of the United States, lay awake in her own still crisp sheets, looking out the window toward the Washington Monument. Being married to America's most prominent symbol of virility, she was not blind to the irony of finding herself in bed alone, staring at the nation's most prominent phallic symbol. Not much had ever been lost on Beth MacMann, other than happiness.
Following the dinner for the President of Uruguay, Beth and the President had left their remaining guests and gone upstairs at 11:30. They'd undressed and gotten into bed. She'd fallen asleep.
She had woken up, at 1:42 A.M. by the digital bedside clock, thirsty for water, to find herself alone. Sometimes when a call came in the middle of the night, he went into his study so as not to disturb her. If it was a crisis of some sort, he usually went downstairs to the Oval Office. If it was really pressing, he would go to the Situation Room in the basement of the West Wing so that the press secretary could inform the press that the President had monitored the situation from the Situation Room. This sounded more impressive than "on the phone in bed."
The dark thought crossed Beth's mind, though she really—really—preferred not to consider the possibility, that her husband was down the hall in the Lincoln Bedroom. Surely he wouldn't pull something like that. Surely.
She knew the rumors and, moreover, knew the truth about her philandering husband of many years. But even if the rumors were true, this was the one night it was safe to assume that her husband and Babette Van Anka, actress, singer, party fund-raiser, were not engaging in bilateral relations.
Beth sat up in bed, straining to convince herself that her husband was at this very minute downstairs issuing orders to attack some Middle Eastern, or possibly Asian, country with stealth weaponry.
Just then she heard the click of the opening door as her husband, the President of the United States, came in.
She knew. Knew instantly, even in the dark. No surer radar than a wife's intuition has been invented.
Beth contemplated doing nothing, waiting until morning, when, after freshly squeezed orange juice, toast with butter and marmalade, and black coffee, she could calmly confront him with this latest installment in his serial infidelity. Then pour the coffeepot onto his offending parts. She contemplated this for five seconds, then flicked on the light.
He reacted like any creature of the night—raccoon, cockroach—suddenly bathed in unwelcome illumination. There was a rapid, lateral darting of the eyes, assessing avenues of flight. He was bent forward oddly, holding his jacket over his groin. Beth interpreted this posture as defensive. The body language shouted, "I've been screwing our guest!"
"Iraq," he said with a sigh. He rolled his eyes to show how grave and yet predictable was the situation.
It occurred to Beth that Iraq now stood in danger. He might well wait until she had gone back to sleep, then slink off to the Situation Room and order a few cruise missiles launched at Baghdad so that by breakfast time he could look her straight in the eyes.
The argument that followed was boisterous even by the standards of the MacMann marriage, currently in its twenty-fifth and final year.
Beth awoke as usual at 6:15. She picked up the phone on her bedside table and ordered her customary breakfast. She got out of bed, slipped on her bathrobe, and opened the door to collect the morning papers, which had as usual been placed neatly on a side table, in the order she preferred to read them. In many ways, the White House was the Platonic ideal of the perfect hotel: twenty-four-hour room service, a concierge at the end of the line eager to provide anything at all, from theater tickets to an army on the march.
She scanned the front pages as she slipped back into bed. Nothing on Iraq. Surprise. An earthquake in Chile. The German foreign minister had given a speech saying that Germany had apologized sufficiently for World War II. A significant dinosaur bone had been found in Manitoba that paleontologists said might establish that dinosaurs had become extinct not because of a giant meteorite, but from osteoporosis. France, furious with the United States for imposing a 100 percent tariff on its Roquefort cheese, had agreed to sell China high-velocity nuclear torpedoes for its submarines. The head of Mexico's antinarcotics police had just built himself a third "palatial" villa, on a salary of $48,000 a year. With the presidential election "only" eighteen months away, "unnamed party leaders" were "concerned" that the President's "message" was not getting "out there."
Sophie Williams, the White House maid who always brought the First Lady her breakfast, knocked softly and entered. She and Beth exchanged the usual pleasantries as she placed the breakfast tray, with freshly cut orchid, over Beth's lap.
It was at this point that Sophie said to the First Lady softly but with alarm that the President's eyes and mouth were "wide open" and that he was "looking awful still."
This remark set in motion a series of events that culminated seventeen days later with Elizabeth Tyler MacMann's indictment for murdering the President of the United States of America. Had hers been an ordinary marriage, the charge would most likely have been second-degree murder. But since the husband in question was who he was, the plainly embarrassed attorney general explained that he had no statutory choice but to charge the now former First Lady with the monstrous crime of assassination.
# Chapter 1
His secretary announced simply, "It's her."
There was no ambiguity as to who "her" might be, not after the force twelve media storm of the previous weeks. The country was convulsed. Seven-eighths of the nation's front pages and the evening news was devoted to it. If war had broken out with Russia _and_ China, it might have made page two.
"Shameless" Baylor had spent much of the previous seventeen days wondering if Beth MacMann would have the balls to call him.
He was, at age not quite fifty, the top trial attorney in the country. He had been the first lawyer to charge $1,000 per hour, which—for too long—had been considered the unbreakable sound barrier of legal billing.
There were half a dozen second-best trial attorneys each of whom, naturally, considered him- or herself the top trial attorney in the country. But none of them had been simultaneously on the covers of all three weekly newsmagazines, none had been portrayed in movies by a famous British actor pretending to be American. None owned a professional baseball team. And, to be sure, none had been married and divorced four times. The previous record had stood at three. That he had any assets left after such serial marital wreckage was perhaps the greatest testament to his courtroom skills.
He hadn't been baptized "Shameless." In fact, up to the moment he set out to become the best trial attorney in the country he had been the soul of decency, what used to go by the name of "Christian gentleman," a veritable poster boy for all that is good and sunny in human nature. His real name was Boyce, and at his baptism, his godparents firmly rejected Satan on his behalf. The rejection lasted until an event that occurred to him just before he graduated from law school.
The nickname had been given to him by a federal judge early in Boyce's controversial career, after he had persuaded a jury that his client, the Cap'n Bob Fast Fish Restaurant chain, was unaware that its popular Neptune Burgers were made from black market Japanese whale meat. Since that stunning victory, Boyce had successfully defended traitors, terrorists, inside traders, politicians, mobsters, blackmailers, polluters, toxic-waste dumpers, cheats, insurance frauds, drug dealers, horse dopers, televangelists, hucksters, society wife batterers, cybermonopolists, and even fellow lawyers. An eminent legal scholar who wore bow ties commented on public television that if Shameless Baylor had defended Adolf Eichmann after he had been kidnapped and brought to Israel and tried for crimes against humanity, Eichmann would have been not only acquitted, but awarded damages. It was not said admiringly. But if Boyce's fame had long since reached the point where shoeshine men in airports asked for his autograph, the public was largely unaware of the actual motivation for his remarkable career.
And now—a quarter century after his career began—his phone rang.
He reached for the button, then paused. He thought of telling the secretary to tell her to call back. Sometimes he put new clients through a ten- or fifteen-minute wait before picking up. Softened them up. Made them all the more eager.
Should he, to her? No. He had waited twenty-five years. He was too impatient to begin this beguine.
He felt the kettledrum in his chest. Good Lord. Was his pulse actually quickening? He, who never broke a sweat, even while arguing before the Supreme Court?
He picked up.
"Hello, Beth. What've you been up to?" This was nonchalance carried to operatic heights.
"I need to see you, Boyce."
Her voice was all business. Cool as a martini, no more emotion than a flight attendant telling the passengers to put their seats in the upright position. He'd have preferred a little more raw emotion, frankly, even a stifled gasp or sob. Some clients, even burly men who could break your jaw with one lazy swipe of their paws, broke down the first time they spoke to him. Boyce kept a box of tissues in his office, like a shrink. One new client, the head of a plumbers union who had been taped by the FBI on the phone ordering the car bombing of a rival, had blubbered like an eight-year-old. He later blamed it on medication.
But even now, placing a call that must have humiliated her, Beth was in her own upright position, not a trace of begging or desperation in her voice. Boyce stiffened. His pulse returned to normal. _Okay, babe, you want to play it cool? I'll see your thirty-two degrees Fahrenheit and lower you five_.
"I could see you tomorrow at ten-thirty," he said. "For half an hour."
It had been a long time since anyone had said something like that to Beth MacMann.
The two of them began the mental countdown to see who would blink first.
... seven... eight... nine...
"Fine," she said.
"Will you be taking the shuttle?" He'd be damned if he'd send his own jet to pick her up.
"No, Boyce. I'll be driving. I don't relish the thought of being stared at for an hour on the shuttle."
As a former First Lady, she retained Secret Service protection, another of the ironies in which she and the nation found themselves: prosecuted by the government, protected by the government. A _Times_ columnist had mischievously posed the question: If in the end Beth MacMann was executed, would there be a shoot-out between the Secret Service and the lethal injectionist? _So_ many delicious questions were being posed these days.
"Ten-thirty, then."
Boyce leaned back in his leather throne and imagined the spectacle in all its many-pixeled splendor: hundreds of TV cameras and reporters outside his Manhattan office, clamoring, aiming their microphones like fetish sticks as the Secret Service phalanxed her through to the door. And there he would be standing, gorgeously, Englishly tailored, to greet her. His face would be on every television set in the world tomorrow. Peasants in Uzbekistan, ozone researchers in Antarctica, Amish farmers in Pennsylvania would recognize him.
He would issue a brief, dignified, noncommittal statement to the effect that this was only a preliminary meeting. He would smile, thank the media for its interest—Boyce was the Siegfried and Roy of media handlers—and usher her in. How satisfying it would be, after all these years. They were already calling it "the Trial of the Millennium," and there he would be, at the red hot center of it. And maybe—just maybe—to make his revenge perfect, he would deliberately lose this one. But so subtly that even the Harvard Law bow tie brigade would hem and haw and say that no one, really, could have won this one, not even Shameless Baylor.
# Chapter 2
It was a bigger zoo than he'd expected. Outside Boyce's Manhattan office were sixteen satellite trucks with seventy-foot telescope microwave dishes to supply the live feeds, as well as over three hundred reporters and camera people and twice that many onlookers. Even he was impressed.
The police had to block off one lane of westbound traffic on Fifty-seventh Street. It was the Client-Attorney Meeting of the Millennium. By the time this was over, one pixel pundit said, the word _millennium_ would be so overworked that it would have to be mothballed until the year 2999.
Beth quietly fumed in the elevator until she and her Secret Service retinue had reached Boyce's office on the northwest corner of the fiftieth floor looking toward Central Park. He called it his "thousand-dollar-an-hour view."
"That was truly humiliating," she said. "Thank you."
He knew right away that there was no use pretending it hadn't been he who had leaked the news of their meeting. But he found himself hoping that she hadn't figured out to whom. Perri Pettengill, Boyce's current girlfriend, was the host of the Law Channel late night talk show _Hard Gavel_. She was blond, smart, and ambitious, talked fast, and wore bifocals and tight sweaters. She had the best ratings on the Law Channel, which tended not to attract many viewers in the periods between spectacular murder trials, though a highly classified in-house research report showed that roughly one-third of her viewers watched her with the sound off. Tom Wolfe had mentioned her in an essay, calling her "the Lemon Tort."
Perri and Boyce had met six months earlier when she moderated a panel at the Trial Lawyers Association in New Orleans on jury selection entitled "Peremptory This!" Boyce had been on it. She had introduced him as "not only the best but the most exciting trial attorney in the country" and that night after dinner had given him the most memorable evening he had ever spent in New Orleans, which was saying a lot. She had moved in later that week. Their relationship had been cemented in **boldface** type by the New York gossip columnists. She was smart enough not to have brought up the subject of marriage just yet, but the question was there every morning, fluttering over the breakfast trays like the Dove of Damocles. Boyce did have an excuse: four previous wives. It did give Perri pause. No romantic woman dreams, in her heart of hearts, of becoming Mrs. Number Five.
Boyce had called Perri after getting off the phone with Beth. She'd nearly hyperventilated. _What_ a scoop. Her ambition sometimes made Boyce wary, as, to be honest, did her extraordinary ability in bed. Confronted with a truly skilled partner, a man had to wonder, even as he gasped and whinnied in ecstasy: _Where did she learn to do that_?
But now his thoughts were of Beth, upon whom he had last laid eyes a quarter century ago.
"You gave it to that woman, didn't you?" she said. "Sweater Girl."
"That's right. I wanted a big crowd down there today. I wanted to send a message to the U.S. government—"
"You did. It read, 'Boyce Baylor is a flaming egomaniac.' "
He was—stunned! It wasn't the sort of romancing Boyce expected from supplicant clients.
"I got up at five o'clock this morning," Beth said, "and spent four hours on I-95 feeling like O. J. Simpson in the Bronco, being chased by a half dozen Eyewitness News teams. Then I arrived to your welcome wagon from hell. So if you'll excuse me, I'm in no mood to kiss your ass."
With that she sat down and began pulling off her gloves. Beth had always worn them, for the uncomplicated reason that they kept her hands soft. When she became the wife of a presidential candidate, and no shrinking violet, the media seized on the gloves for a convenient iron-hand-in-the-velvet-glove metaphor.
Boyce couldn't help himself watching her take them off finger by finger in an incredibly sexy Barbara Stanwyck let's-get-down-to-business way. He couldn't take his eyes off her. Men are men and fools to a man, but it amazed Boyce, seeing her this close, that Ken MacMann had needed to screw all those other women when he had this waiting for him at home, warm in his own bed at night. She was a few years younger than he, and looked perhaps a few years younger than that. She had aristocratic cheekbones and black hair with streaks of gray that made the black richer and more lustrous. Her eyes looked straight at you in an evaluating but not unfriendly way. Her figure, unmarred by childbearing, was full and handsome. If she'd been an actress, she would have gotten the part of the take-charge businesswoman who turns out to be an absolute panther in the sack. He remembered how every time he walked behind her and saw the lovely sexy sway of her bottom, his mouth went dry and his heart soared with possession.
And so here she was, twenty-five years later, in his office, a client.
"Coffee, skim milk, one sugar." She crossed a black-stockinged leg. He heard the siren song of nylon on nylon. "So how are you, Boyce?"
It now dawned on Boyce Baylor, lion of the American Bar, that in less than thirty seconds he had been reduced to the status of coffee boy—in his own lair, with a view that God would envy, amid walls hung so thickly with honorifics and photographic testimonials to his greatness, his hugeness, that the very Sheetrock cried out under the strain. No no no no. This would not do. Not do at all. He must assert control, quickly.
He buzzed for the coffee and, sitting down opposite, said, "Not so bad. Haven't been indicted for murder."
She gave him the hint of a smile.
"Why," he said, "didn't you call me sooner?"
"I was waiting to see how bad it was going to get. I thought it might not get to this point. And I didn't want to make it appear worse by hiring a lawyer."
Boyce shook his head silently, wisely. How often he had heard this.
"Anyway," she said, "here I am. On bended knee."
Boyce used this as an excuse to look at her knees.
"The reason they're bent," she said, "is from four hours in the back of a Secret Service SUV. But I could say they're bent for your sake, if you'd like."
Toying with him! Intolerable.
"You must be in a world of hurt," he said, "to come to me."
"I've been indicted for murder. That's one definition of 'world of hurt,' I suppose."
"Why me? There are lots of good lawyers who'd love to have this case."
"Boyce," she said, "if you want me to say, 'Because you're the best,' I will."
"Beth"—he smiled—"I _know_ I'm the best. Don't take this the wrong way, but I'm past the point where I need your approval."
"Oh, you've done well. No question. It's why I'm here, isn't it?"
He was thinking, _You waltz in here after screwing me over and sit there with those incredible legs crossed, putting out—attitude_?
Boyce decided right there and then to take the case.
"On the way up here," she said, looking down at her lap, "somewhere between Baltimore and Wilmington, I promised myself that I was not going to apologize. Then when we got to the New Jersey Turnpike, I decided I _was_ going to apologize. Then in Newark I went back to my nonapology posture."
"How'd you feel going through the Holland Tunnel?"
"Like turning around. Only that's tricky in a tunnel. Annoys the oncoming traffic."
"Well, we can talk about all that some other time."
"Maybe we should talk about it now. I think I'd rather know your state of mind going in. I don't want to find out during closing arguments that your heart wasn't really in this."
She was a canny one.
"This isn't _Casablanca_. And this"—he waved at his Wall of Ego, which still, Beth noticed, held an official framed photograph of his former father-in-law Prince Lupold of Bad Saxony-Wurtburg—"is not Rick's Café. I moved on. And I've done just fine. The truth is I got over it pretty quickly."
"I don't flatter myself that I ruined your life."
_Flatter herself? That she_ ruined _my life? Dammit..._
"I have a very good life." He nodded in the direction of the Wall of Ego. "As you can see."
She looked at the wall. "I see. I..."
"What?"
"I did reach out to you. After we got to the White House. You didn't answer four invitations. To state dinners."
"Must have gotten lost."
Beth smiled. "Boyce, dinner invitations from the White House don't get lost."
"I may have been in the middle of a trial. When I'm trying a case, to be honest, an earthquake wouldn't register."
"Then you must have been in the middle of four trials, because we invited you four times. I was going to put you next to Princess Caroline. Knowing how you like princesses."
"She was related to my wife. Somehow. All goes back to Queen Victoria." He was mumbling.
"The protocol office said they'd never heard of anyone not answering four White House state dinner invitations. You're in the _Guinness Book of World Records_."
"One of my fathers-in-law died in the middle of the MicroDot trial, and I was so wrapped up in it that I didn't even attend the funeral."
He heard the little computer voice in the cockpit saying, _Pull up, pull up!_
"So," he said crisply, "shall we talk about my bad manners, or the case?"
"I'm not sure," said Beth, "that I've satisfied myself as to your state of mind. If you're going to handle this, I need to know that you're on board, emotionally."
Boyce snorted. "I don't deal in emotions, only motions."
"I don't believe that for a second."
"What makes you think I've decided to take this case?"
"Boyce"—Beth laughed—"whatever the situation is between us, I really can't believe that you wouldn't take this case."
She was smiling. My God, the woman was smiling in triumph.
"I mean," she continued, "the very idea of you _not_ being involved in this case—they're calling it 'the Trial of the Millennium.' It doesn't make sense."
She had him, had him by the short ones. All he could do was pretend that he was the absolute lord and master of the corner that she had artfully backed him into.
He gave her his best gaze-blank-and-pitiless-as-the-sun, the one he reserved for his most withering cross-examinations. And she just stared back at him until all he could do was try not to laugh at his own helplessness.
"All right. I'll handle the case."
"Thank you."
"But I want it understood, understood without ambiguity, that I'm in charge."
"Naturally."
"Oh no. Raise your right hand and say, 'I, Beth Tyler MacMann, do solemnly swear that Boyce Baylor shall be completely, wholly, totally, and one hundred percent comprehensively in charge of my defense. So help me God, Jehovah, Allah, Buddha, Vishnu, and all and any other gods not herein specified."
"Swear."
Boyce rose, his pride assuaged. "There's a basement garage so you can avoid your fans in the media."
"Don't you want to know if I did it?"
"Obviously, you never practiced law. The _last_ thing I want to know from my clients is did they do it."
Beth had the disconcerted look of the bright girl in class who had just been singled out for saying something foolish.
"I'll fly down to D.C. tomorrow morning and we start."
# Chapter 3
Harold Farkley had long dreamed of becoming president of the United States, but thrilled as he was finally to get the job, he wished the circumstances had been different. It was one thing for a vice president to assume the mantle of greatness because of a dramatic assassination, a sniper's bullet at high noon with all the world watching. But to be the beneficiary of a marital spat gone tragically awry... Harold Farkley could almost hear the gods sniggering. He could certainly hear the media tittering. Tittering—hell, they were howling. Openly. Hysterically. Wetting themselves with laughter.
He looked at the newspaper on his Oval Office desk, open to the editorial pages. Harold Farkley fumed. John O. Banion—that insufferable, bow-tied prig—had written in his widely syndicated column, " 'President Farkley.' Try, if you can, to wrap your mind around that stunning oxymoron." Boiling, he read on. "Harold Farkley was the second-born in his family, went to a second-rate college, where he graduated second in his class. Thus equipped with a second-rate intellect, he went into a second-rate profession. Eventually, he clawed his way to becoming the second choice of the voters in his party. This in turn got him the number two spot on the presidential ticket. Now fate has intervened in a most bizarre fashion, for only a bizarre chain of events could have propelled a Harold Farkley to the number one position. The universe is temporarily unbalanced. Some cosmic intervention may be necessary to realign the heavens."
Oh, for the days of real executive power, when a ruler could have his opponents thrown into a dungeon.
Harold Farkley forced himself to read the rest of the column, for even a second-rate mind knew that it was prudent to know how the enemy was thinking.
Banion, a contrarian, refused to accept the charges against Beth MacMann. President MacMann, he wrote, had been the victim of "bathetic happenstance"—a pun on bathroom. The President, Banion stoutly maintained, had gone in to use the bathroom in the middle of the night, slipped, crawled back into bed, and died. The strange markings on his forehead could be explained as a "dermatological anomaly." The First Lady had been unfairly accused. That weekend, Banion had announced with customary pomposity on his new television program, _Capitol Bang_ , that the government was conducting a "witch-hunt even more unseemly than the kind conducted in Salem, Massachusetts, in the 1690s."
President Harold Farkley read Banion's opinion with the impotent fury of a second-rate mind and the fervent hope that it would remain in the minority. So far, so good. His own pollster confirmed what the media were saying: Most Americans thought she had done it.
The First Lady had been controversial from the start. From the moment she set foot in the White House, she made it abundantly clear that she did not plan to "spend my days going over menus." It was a far cry from Hillary Clinton, who contented herself with taking care of her husband and giving the occasional tea for congressional wives.
Beth's declaration that she would be a substantive First Lady was met with grumbling and mutterings of "Who elected _her_?" She attended cabinet meetings, where she not only spoke up but sometimes corrected the secretary of defense or commerce on a point.
A few months into the new MacMann administration, a report appeared in _The Washington Post_ about an alleged "shoving incident" involving the First Couple. The White House spokesman dismissed it as "rubbish." A few weeks after that, the President appeared at a Rose Garden event wearing a bandage on his nose. The spokesman averred that the President's wound had been the result of "walking into a door." Washington murmured that it was more likely that the door had walked into the President. In the two and a half years of the MacMann presidency, the White House spokesman had dismissed a total of seven incidents, with indignation ranging from "totally untrue" to "I have nothing further for you on that."
So on the morning that the country awoke to the news that its leader had suddenly expired in his bedroom, in the company of the First Lady, it connected the dots before noon. Even the First Lady's supporters were at pains to exculpate her. It did not help Beth when one of her staunchest allies, the head of the National Organization for Women, went on TV that afternoon to defend her and said, "If something violent occurred, I'm sure she was provoked." Thanks a lot!
When the first public opinion poll was taken, three days later, the TV screen flashed the news that nearly 70 percent of the American people thought that Beth was "implicated" in the death.
This was the thin consolation available to Harold Farkley. He was determined, in his own quiet, number two–ish sort of way, to do whatever he could to ensure that he would go down in history as the collateral beneficiary of a murder, not merely a wet bathroom floor. He lay awake at night tormented by the vision of elementary school teachers a hundred years hence asking their children, "Now what vice president became president because of a bar of soap?"
And there was this: Harold Farkley detested Beth MacMann. She had managed to inspire in a second-rate temperament a genuinely firstrate passion. He loathed her.
During the primary campaign between himself and Governor MacMann, there had come a moment when it looked as though Harold Farkley might just break through the membrane of mediocrity that had bound him for so long to the earth and become—number one. He was ahead, though by the weensiest margin. His advisers counseled, Go for it, sir! Be bold! Pull out all the stops! Do what must be done, and greatness will finally be yours!
Harold Farkley, the taste of victory meltingly on his tongue like a chocolate caramel, gave in to the zealous urgings of his handlers. Here, they said, is MacMann's Achilles' heel: his pushy wife. Their polling showed that just enough MacMann male voters were wary of her to provide Farkley with a winning margin if they came over to his side. So Harold Farkley, daring greatly if not judiciously, crossed the invisible line and—criticized his opponent's wife.
"It is not him who worries me," he said memorably and ungrammatically in his fateful speech to the Michigan autoworkers. "It's her. I think the American people have a right to know whom will be wearing the presidency's pants."
Within two hours, feminists, soccer moms, and even happily unliberated housewives were clamoring for Harold Farkley to withdraw. You just don't go after a man's _wife_. It's un-American!
And so Harold Farkley's karmic parabola, having temptingly arced toward the stratosphere of greatness, curved steeply back toward the dismal earth. Only by furious backpedaling and a massive eleventh-hour media buy did he manage to hold on to his number two status. At the party convention, Beth assented to Farkley's being named to the ticket only after her husband's advisers convinced her of the inexorable electoral math warranting his inclusion. If they were to win in November, they would need Harold Farkley's fifty-four electoral votes. Anyway, the advisers said, it would look magnanimous. American voters love magnanimity, however you spell it. Beth was the very picture of magnanimity, right up through election day. Then she took a sharp knife and quietly removed Harold Farkley's testicles.
She froze him out. And when she couldn't freeze him out, she put him next to the kitchen. At state dinners, Harold found himself seated next to the non-English-speaking wife of the finance minister of the visiting head of state. "How are you enjoying your visit to Washington?" After interminable translation, the answer came back, "She say Washington very _hot_ in summer." Harold found himself dispatched to represent the United States at their funerals before the foreign dignitaries had even died. He was appointed to commissions on "uninventing government." Around Washington he became known as Vice President Whatsisname. Indeed, his name recognition dropped below 23 percent. More Americans knew the name of Canada's Prime Minister than their own Vice President's. Editorials once again surfaced in the nation's newspapers asking if the vice presidency was really necessary. A year and a half before reelection time, it was all but certain that Harold would be unceremoniously dumped in favor of a new running mate. And it was Beth who had her hand on the lever of the trapdoor.
Then—this.
The gods who had for so long laughed in Harold's face had suddenly intervened on his behalf. Here, on a silver platter, was his chance to achieve what any politician most cherishes in his heart of hearts: payback.
But it must be done subtly. Harold Farkley had learned from his disastrous attack on her. This time he would be artful.
His feud with Beth MacMann was no secret. The media were ready to pounce on President Harold Farkley at the first sign he was using the incident as an excuse to prosecute his grudge against the First Lady. Hypocrisy is a prerogative of the press but must under no circumstances be tolerated in politicians.
So when the FBI director reported directly to newly sworn-in President Farkley that there were inconsistencies in Beth's statements, when the director of the Secret Service reported to him that an argument had been heard that night by one of the agents, Harold Farkley knew that he must dare to be cautious. He was out of the country on the day she was formally indicted.
Upon his return, he went on TV to address the nation. It was all he could do to keep from tap dancing. Before his address, he practiced his expression in the bathroom mirror prior to going on television, arranging his second-rate features into a look of overdone gravity, a vaudevillian attempting Shakespeare.
He told the nation that this was, indeed, a dark hour, "not only for the country as a nation, but for me personally, as a human being." He said he had "every confidence that justice will prevail and that Mrs. MacMann will be cleared of the awful—indeed, horrible—charge against her." Thanks, Harold.
Since her indictment, Harold Farkley had been in a covert state of bliss. He happily attended to the affairs of state—the affairs of state that were now all _his_. In his quiet moments, he tantalized himself with daydreams of Beth weeping, begging for a presidential pardon. Of Beth sizzling in an electric chair, hooded with a noose around her neck, dropping through the trapdoor, tied to the stake, the flames reaching higher and higher and higher—
"Mr. President?"
Dammit, the way they just walked in.
"What is it?"
"It's on the news. Mrs. MacMann has hired Boyce Baylor."
Suddenly the pleasant images shattered like glass struck by a sledgehammer. Harold Farkley heard a voice pronouncing the awful words: "We find the defendant not guilty."
# Chapter 4
Normally Boyce would have flown down to Washington on his private jet, a sporty Falconetta 55 with enough range to get him to Paris for dinner. But since he would soon be impaneling a Washington, D.C., jury whose primary source of news came from television, he not only took the commercial shuttle flight, but also carried his own garment bag and briefcase. His office had called the media ahead to let them know what flight he'd be on. They were waiting for him as he stepped off the ramp, with enough light to illuminate twenty Hollywood premieres.
"Boyce!"
"Mr. Baylor!"
"Are you—"
"Will you seek—"
"Possible to—"
"Yo, Shameless, over here!"
Boyce stood in the basting glare, trying not to blink—or melt—with an appropriately grave look and waited for the insect whir and hammer click of cameras to subside. He was used to media, God knows, but this _was_ a turnout. There must be over a hundred.
He gave a curt nod to indicate that the orchestra should stop tuning their instruments. The conductor was ready. The symphony was about to begin. And he had brought them a little something. He always kept them well fed.
"I'm here," he said, "to help an old friend. With respect to the charges, I have this to say. I personally admire and respect the attorney general. So I regret all the more that he decided, in the face of massive evidence to the contrary, to sacrifice an innocent widow on the altar of his own burning ambition."
The attorney general of the United States, watching in his office at the Justice Department, said to his deputy, "That asshole. That _goddamn_ asshole."
"Looks like war," his deputy said.
"Finally," Boyce said, "I would ask all Americans to remember something in the days ahead. Yes, the country has lost a president. But a beloved First Lady has lost her husband."
Beth, watching from her new temporary headquarters in Cleveland Park, a few miles from the airport, muttered aloud to her TV screen, " 'Beloved'?"
"That's really all I have to say at this time. Thank you." He always said this before proceeding to take questions.
"Boyce! Were you and Beth MacMann lovers?"
"Jesus Christ," said Perri Pettengill's senior producer, "those two? Used to _do it_?"
"Um-hum." Perri nodded, continuing to watch.
"That's perfect."
"They were in law school together. She screwed him over."
"So why's he helping her?"
Perri looked at him. "Harry, it's the Trial of the Millennium. Of course he's going to represent her."
"You gotta get him on the show tonight. We gotta have him."
Boyce had told Perri he wouldn't do her show, at least for a while. "It wouldn't look right." In retaliation she told him fine, no sex. They compromised: sex and monster leaks.
"Let's save him for something big," Perri parried.
"It's _all_ big," Harry said. "You've got a mass of hot air over Washington, a cold legal front coming down from New York, and media from all over the world converging. It's _The Perfect Storm_ all over again. _'Perfect Storm'!_ We could use that."
"Yes, Harry. That's good. Use it."
"I'll Chyron it."
Boyce had been ready for the question. He paused to give the impression that it had taken him by surprise. "The First Lady and I were at law school together. It was a long time ago." He added with nice faux self-deprecation, "You'd know it was a long time ago to look at me, maybe. Not the First Lady."
Through the plate-glass window in the airport terminal where Boyce was standing, he could see in the distance the towers of Georgetown University. A quarter century ago, he and his fellow third-year law student Beth Tyler had one night found themselves in the auditorium for their first moot court. They were so nervous they shook, and this was in the days before beta-blockers.
A rumor had been going around for days that the presiding judge would be a Bigfoot. When that day the door opened and out walked Chief Justice Henry Adolfus Wiggins of the Supreme Court of the United States, a gasp went through the standing-room-only auditorium. A month before, Wiggins had ordered the President of the United States to turn over his secret Oval Office tape recordings. That led swiftly to his historic resignation. The Georgetown Law School dean—he had clerked for Wiggins years ago—had pulled off a coup getting him to come.
Beth groaned to Boyce, sitting beside her, "We're dead."
She was to play the part of the U.S. solicitor general and argue the government's side before the Supreme Court. Boyce was her deputy. He whispered back, "He doesn't look happy."
Indeed, Chief Justice Wiggins wasn't happy, not at all happy. He'd been sandbagged by the dean, his former clerk, who had not told him until the last minute that the mock case tonight he would be presiding over would be the very same one he had so historically decided a few months ago. It bordered on impudence.
Beth and Boyce had pulled two consecutive all-nighters to prepare. They looked like extras from the movie _Night of the Living Dead_. Her argument for letting the President keep his tapes was that the Supreme Court justices lacked the proper security clearances to hear what was on them. They were armed with precedents, but now, looking at the imperious, pinched-looking Wiggins taking his seat before them, they felt a presentiment of doom. In effect, their job tonight was to persuade him that he had been wrong. And chief justices, generally, did not like to be told that they were wrong.
"Oyez, oyez, oyez," the dean intoned, grinning at his triumph. _The Washington Post_ and _The New York Times_ had sent reporters. "All persons having business before the honorable, the Supreme Court of the United States, present themselves."
Boyce began humming Chopin's "Marche Funèbre." Dum dum de dum _dum_ de dum de dum de dum.
"Shut _up_ ," she hissed.
Beth stood. Justice Wiggins did not return her smile. In his robes, spectacles, and blue, bloodless lips, Justice Wiggins looked as though he were yearning to sentence everyone present to death by hanging, or preferably by some more prolonged, medieval form of execution.
Beth stood mute at the lectern. Five seconds went by, ten. Fifteen. Wiggins, accustomed to brisk kowtows and beginnings, frowned, a formidable sight.
People exchanged glances. The dean's smile vanished. The silence that descended on the auditorium had an Old Testament quality, the kind that preceded the Voice in the Whirlwind announcing, _I am the Lord God Almighty, and I am very, very wroth_.
"Your Supreme Honor—"
Off to a good start.
"With all due respect, I—we, that is, the government of the United States—do not believe that you—that the Court—has jurisdiction in this matter."
Wiggins, who had just earned himself his own chapter in the legal history of the United States for a written opinion that was being hailed as the most consequential legal ruling since Maimonides, glowered at Beth like a malevolent owl contemplating a mouse. The Wiggins Supreme Court felt that it had jurisdiction over everything, including what time the sun was allowed to rise.
Boyce felt his insides loosen, along with the cold scalp prickle that augurs calamity.
Wiggins let her continue another two and a half sentences, whereupon he assumed his accustomed role of grand inquisitor. It was merciless. It was scathing. It was so bad that no one could bear to watch. Four hundred pairs of eyes looked down. Never had the auditorium floor been so closely examined. It was so awful that finally Boyce decided there was nothing left to lose. He scribbled on an index card and slid it in front of Beth as the judge continued to blowtorch her for her abominable—no, worse, abysmal!—understanding of the Eleventh Amendment. It read:
_He's wearing panty hose underneath_
To keep from laughing, Beth sucked in her upper lip and bit down on it so hard that it stayed swollen for two days.
Boyce's note saved her from annihilation. Chief Justice Wiggins, who deep down was really more angry at the dean than at an intellectually frisky third-year student, saw this young woman in front of him apparently about to burst out crying and ceased his attack. He became even moderately magnanimous. He concluded by telling her that her argument was "without merit," but was without merit "in an original way." For Wiggins, this was tantamount to a compliment.
At the reception afterward, another third-year student named Kenneth Kemble MacMann, six feet four, lean, with Kennedyesque hair and soulful, hooded eyes, approached Beth to say how impressed he had been by her performance. Boyce knew him slightly. He was older than the other students. Word was he'd been to Vietnam. If you were a vet in the 1970s on an eastern college campus, it was not something you broadcast to your fellow students or teachers, who would be only too glad to accuse you of crimes against humanity.
A few days later, Beth showed up in Boyce's dorm room with microfiche copies from _The New York Times, The Washington Post, Time, Newsweek_ , and an official U.S. Navy publication, page after page of news stories about a navy ship called the _Santiago_.
"What's this?" he said.
"That guy, the third-year the other night we talked to—read this." Boyce read.
The _Santiago_ was a fast navy electronic surveillance vessel assigned to monitor Russian shipping in and out of Haiphong Harbor. Its captain had taken it inside the twelve-mile limit, probably on orders. A North Vietnam MiG attacked. Everyone on the bridge was killed except for Lieutenant (jg) MacMann. Wounded, he had assumed command and—as the citation that she for whatever reason had dug out of the archives put it—at great personal risk attempted to drive the _Santiago_ into undisputed waters while simultaneously directing aid to the wounded and the destruction of classified materials. The _Santiago_ was overtaken by North Vietnamese gunboats. Lieutenant MacMann ordered abandon ship and evacuation of the wounded but remained on board himself. While continuing to receive enemy fire, he successfully scuttled the _Santiago_ , which sank to the bottom of the Gulf of Tonkin.
He was picked up by the gunboats and endured three and a half years of torture, starvation, inadequate medical care, and solitary confinement at the Hanoi Hilton. Upon his release, Lieutenant K. MacMann was awarded the Purple Heart, Distinguished Service Medal, Navy Cross, and Congressional Medal of Honor. He'd been personally decorated, in the Oval Office, by President Richard Nixon, otherwise known among the eastern academic elite as the Antichrist. (Not that the eastern academic elite believed much in Christ.)
"He's a hero, Boyce."
"Boy," Boyce said. "I'll say."
"Listen to this." She read: " 'Following Lieutenant MacMann's release by North Vietnam, he was returned to the United States and spent two months at the Naval Hospital in San Diego. Subseqently he received an honorable medical discharge from the Navy with the rank of Lieutenant Commander.'
"I wonder what the reason was," Beth said.
The first indication that something was wrong came a few days later when from a distance Boyce saw Beth heading into Habeus Sandwich, the Georgetown Law student hangout, with Kenneth Kemble MacMann. Boyce followed them and, finding them both sitting cozily in a booth, announced himself with a "Hi." Beth appeared clearly disappointed.
"Mind if I join you?"
"Absolutely," Ken MacMann said heartily, displaying more ivory than a Steinway piano.
Beth looked even more disappointed.
They made small talk until the French fries arrived, when Boyce, feeling more and more leery, decided to plunge in.
"So Beth tells me you had to quit the navy for medical reasons."
Beth stiffened.
"Yeah," Ken said.
"Must have been serious."
"Nah. Navy regs, is all."
"So what was it?"
Beth kicked him. "He doesn't want to talk about it, Boyce."
"Just asked."
"You know what the worst part of it was?" Ken said. "Saying goodbye to those navy nurses."
"But you had to leave, is that it?"
"Boyce. Will you stop?"
"It's okay. I could have stayed in, but it would have been a desk job."
Boyce wondered if a penis was considered essential equipment for line duty aboard a navy ship.
Awkward silence descended on the table.
Ken said, "If you really want to know—"
"Oh God," Beth cut in, "seven forty-five!"
"Is the world scheduled to end?"
"I have to get back to the library."
"Well, go ahead," Boyce said. "I want to hear about Ken's wound."
Beth's eyes narrowed.
"I took a tracer round through the stomach. It kind of never fully healed."
Beth turned to Boyce. "Why don't you tell Ken about your squash injury. The one that kept you from being sent over there?"
On their way back to the library after Ken had left them, Boyce said, "You had to bring up my knee?"
"You were being a dick."
"I was trying to get you an answer to the question that clearly had been tormenting you."
"Good _night_." She peeled off.
A week before finals, Beth knocked on Boyce's door. She was flustered.
"I guess we need to talk."
"We are talking."
"Okay," she said, exhaling, "Ken's asked me to marry him."
Boyce stared. "Did you tell him you were already engaged? To me?"
"Uh-huh."
"So, then?"
"I told him yes."
"How can you be engaged to two people?"
She kissed him tenderly on the top of his head, the blow-off spot. "I'm so sorry, honey," she said. "It just happened."
"Is that supposed to make me feel better?"
Beth and Ken were married two months later, by Chief Justice Wiggins.
# Chapter 5
Did you have to say that about the attorney general sacrificing me on the altar of his burning ambition?"
"You missed the key word," Boyce said. "Sacrificing a _widow_ on the altar of his burning ambition."
"I don't think of myself as a widow."
"Start."
"But why piss him off? I bet he's ballistic by now."
"Worried he might get really mad and indict you for murder? I _want_ him mad. I want them all mad. Mad people make mistakes. We need the other side to make mistakes, since you've made so many of your own so far."
"Such as?"
"Where do I start? Like talking to the FBI without counsel present. People who rob convenience stores know better."
"How would it have looked? Hiring a lawyer."
"Smart."
"Boyce, I was in shock, for heaven's sake. Have you ever woken up in bed with a dead spouse?"
"I've gone to bed with some." He sighed. "I'm frankly surprised at how you screwed this thing up."
"Is abuse included in your thousand-dollar-an-hour fee, or do you bill separately?"
"Separately, under 'photocopying, telephone, facsimile, and messenger services.' " He read the FBI report. "Why did you refuse the polygraph? It was the right thing to do, but since you did everything else wrong, I'm curious."
"It was insulting," she said hotly. "I'd just come back from burying him at Arlington. I thought it was grossly inappropriate to ask me to take a lie detector test."
"Your outrage is convincing. I almost wish we could put you on the stand."
"I want you to put me on the stand."
Boyce laughed. "Under no circumstances are you taking the stand. What's the matter with you? Have you forgotten everything you learned in law school?"
"I want to tell the truth."
"Boy, you have forgotten everything. Including the most important rule of all: The truth has no place in a court of law."
"I don't remember being taught that."
"In the real estate business it's location, location, location. In a trial, it's perception, perception, perception."
"Perception," Beth said. "Look at this." She held up the _New York Post_.
REUNITED AND IT FEELS SO GOOD!
LADY BETHMAC AND SHAMELESS BAYLOR
It was a photo of the two of them from the mock trial at Georgetown.
Boyce shrugged. "They've been calling you that for years."
Beth slammed her palm down on the conference room table. "Well, it's not pleasant."
"Look at you. And you want to take the stand? By the way, how come you didn't wipe your fingerprints off the Paul Revere silver spittoon after you hit him with it?"
"Nice try."
Boyce smiled. "Good girl. We'll use the fact that your fingerprints were all over Mr. Spittoon as evidence that you didn't murder him, since a murderer, even a moron, would have wiped her fingerprints off the murder weapon. But forget taking the stand. Or I'm on the next shuttle back to New York. I'd forgotten how uncomfortable commercial aviation is."
"Oh, spare me. Your little jet would fit in the lounge of Air Force One."
Boyce chuckled. "Why didn't you just tell the FBI that you threw the spittoon at him?"
"I panicked. I was scared. There he was in bed next to me, dead. If I'd told them what happened, it would have looked..."
"Like you killed him."
"But I didn't kill him, Boyce. I chucked the spittoon at him. It did hit him, on the forehead. But it wasn't _that_ hard. He barely flinched. Well, he went back a bit. But he didn't fall down."
Boyce stared.
"I've thrown heavier things at him, you know."
"That'll sound good to the jury, when you take the stand. 'I've thrown heavier things at him, you know.' "
"I'm telling you, it didn't make a dent. He just called me a bitch, went to the bathroom, got into bed, turned off the light, and went to sleep. Next thing I knew, I'm having my breakfast and he's—dead."
Boyce looked at the D.C. medical examiner's report and Bethesda Naval Hospital autopsy report in front of him. "Cause of death, epidural hematoma resulting from blunt-force trauma. Time of death, between three-fifteen A.M. and five A.M. Tell me this: After you ki—After you both went to sleep, did you wake up in the night to get a drink of water? To pee? Walk the parapets? Rub the blood off your hands?"
"I slept right through. I always sleep like a rock after I've clocked him."
"Don't forget to mention that, too, to the jury, when you take the stand. This Secret Service agent, Woody Birnam, who claims to have overheard an argument between you and the decedent—"
"Why don't you just call him Ken? It's not like you didn't know him."
"Huh!"
"If you're still churning about it, I think you owe it to me to say so."
" _Owe_ you?"
"Boyce, I'm going to need all of you in court. Not just all of you minus the ten percent that's still seething."
"If I were still seething and churning, why on earth would I have taken this case?"
She looked at him. "First, so that you could finally get the whip hand in this relationship."
"I always have the whip hand in the attorney-client relationship."
"Second, to show the world that you're so goddamn magnanimous, you'd defend the woman who du—who broke up with you back when."
"Magnanimity is for wusses."
"And third, in order to lose the case on purpose—in such a way that everyone would say, 'Oh, even Shameless Baylor couldn't have gotten her off,' so that I'll end up in jail or on death row. Just to get even with me."
"I cannot believe," Boyce said, affecting chagrin, "that you think that I'm capable of that. Is this what politics does to a person's soul?"
Beth laughed. "Oh dear, that's good. Look, I need to know. Are you in or out? Psychically."
So much for the whip hand. "I'm in."
"All right, then."
"For the record," Boyce said, "the decision to break off our engagement was mutual."
"Of course it was."
Dammit. There was no winning with her.
"Why were you so sure that he'd been doing push-ups with Babs in the Lincoln Bedroom?"
"The look on his face when he came in and I flicked on the light. He didn't look like he'd been in the Sit Room deploying aircraft carriers."
"His philandering, was it as bad as the rumors and reports?"
"Worse. What's so funny?"
"I was remembering how worried you were that his willy had been shot off by the Vietnamese. But if you knew he was having an affair with Van Anka, what—pray—was she doing as a guest in your house?"
"I know, I know," Beth said, defeated. "It's so—God, the _deals_ you strike."
"I have to explain it to the jury. I mean, here's this hump-happy husband and you're allowing him to bring bunnies in for sleep-overs down the hall."
"I didn't invite her. I can't stand her. I don't like anything about her. Even her singing, much less her quote-unquote acting."
"So what's she doing there in Abe's bed, pumping the commander in chief?"
"It's... she's a star. She draws. Her husband, Max, is a huge financier, major donor to the party. They're a power couple."
"Okay, so why not have both of them over? You could do a foursome."
"Screw you, Boyce."
"Just trying to be helpful."
"We did have them both over. But neither of us really liked Max. He's a bore in that way that some financiers are. Then there was some heat in the papers about some of his business connections. Anyway, he sort of stopped coming. Babette was the friend, anyway."
"I'll say."
"She put on fund-raisers. Raised a lot of money for us."
"A jury averaging twenty-five thousand of income a year will be thrilled to hear it." Boyce studied the Secret Service log. "Jesus. She spent more nights in the Lincoln Bedroom than Lincoln. Fifty-six visits in two and a half years? Did she get miles?"
"We had an arrangement. Ken wasn't to sleep with her when I was in residence."
"This was an interesting marriage you had."
"Who are you to talk? Four marriages, the last one, to that Victoria's Secret model, lasted how long? Six months?"
"We were blissfully happy the first two months."
"Boyce, you're the Elizabeth Taylor of trial lawyers. Do not lecture _me_ on how to conduct a happy marriage."
"We still have to sell it to the jury. You have an arrangement—somewhat unusual by the standards of the American presidency, you may admit. He breaks the arrangement and the next thing you know, kaboom on the noggin and they're saddling the riderless horse for the trip to Arlington. Forgive me, but we have some explaining to do for Mr. and Mrs. Jury."
"I didn't kill him. I _know_ I did not kill him."
"Fine, but you whacked him with the spittoon and next morning he's Mr. Frosty. Reasonable human beings, including the FBI, the Justice Department, the attorney general, the media—"
"The media? Reasonable? Human?"
"—and, according to the latest poll, sixty-eight percent of the American public, two-thirds—think you killed him."
"Whose side are you on?"
"For a thousand dollars an hour, yours. But you want to start with the jury's worst suspicions. It's always the best baseline. Okay. So he could have slipped in the bathroom and gotten back into bed and died. But that's not much in the way of an alternate narrative. For one thing, there's the Paul Revere hallmark they found stamped on his forehead."
Boyce studied the photograph of the President's forehead. "It's kind of pronounced. We'll do some computer enhancing... we can probably make it look ambiguous. Get some friendly skin experts in, make it..." He grunted. "Maybe if we showed it upside down.... Well, we'll figure something out."
He tossed the photo aside and gave Beth an assessing look. "You're looking good these days."
"Thank you," Beth said in a businesslike way.
"Do you work out?"
"When I can. What does this have to do with anything?"
"Do you _exercise_? Pump iron? Treadmill? Tae-bo, whatever it's called?"
"A trainer used to come four times a week. Why?"
"Because the jury is going to be wondering if you were strong enough to lift a"—he glanced at the autopsy report—"two-hundred-and-eight-pound dead president off the floor and into bed. I see the War God put on a few pounds over the years. What do you weigh?"
"Hundred and thirty-eight."
"We start jury selection in four months. I want you down to one twenty."
"You want me to look anorexic? The media's going to see through that."
"It's not for the media. It's for the jury."
"The prosecutor will find a way to point out that I've lost weight since the incident."
"And we'll say, 'You insensitive swine, of course she's lost weight. She lost her husband. This is a grieving widow, look at her, and you're putting her through this hell.' "
"I'll lose the weight."
"Look on the bright side—you can take up smoking again. You used to love to smoke after... wards. The maid, this Sophie Williams, who brought you a hot breakfast while War God was cooling beside you, does she like you?"
"Like me? I suppose."
"No, no, no, do not 'suppose.' When she takes the stand, will she, a black woman, convey to a substantially black jury that you are a wonderful, kind, thoughtful employer who remembers staff birthdays and whose kid broke his arm and whose aunt just died? The sorts of things that thoughtful big people do for the little people?"
"I should think. Yes. You know, the Lady Bethmac thing was never—that was unfair. I'm not a bitch."
"Hm."
"I am _not_ a bitch, Boyce. Just because I fired some people on the White House staff."
"Why'd you give them the sack?"
"In one case because the staffer was giving my husband blow jobs on Air Force One."
"He was head of state. How many did you sack?"
"Over the two and a half years? Nine."
Boyce groaned. "This is going to be such an easy sell to the jury. You didn't kill your husband, despite the fact that he was humping the guest down the hall, as well as half the employees on the federal payroll. What really happened was he got up in the middle of the night, consumed with remorse for his cheating ways, decided to commit suicide by smashing himself in the forehead with an antique spittoon, and just before dying, tucked himself back in bed. It's so obvious. We'll move for summary dismissal."
# Chapter 6
Babette Van Anka had been in the public eye for over two decades now, since her spectacular film debut in _Expensive—And Worth It_ , as the suburban housewife who secretly moonlights as a prostitute to support her family after her stockbroker husband is shot by a commuter train conductor upset over the bad stock tips he had given him. At the time of the President's death, her career had been in decline. She was now getting more press coverage than she'd ever had.
Their "special relationship" had been the subject of unremitting news stories ranging from sober headlines in the _Times_ (ACTRESS SPENT 56 NIGHTS IN WHITE HOUSE, subheadline "Wealthy Financier Husband Was Also a Guest—Four Times") to the more exuberant ones in the supermarket tabloids (BAB'S NIGHTS OF BLISS WITH KEN). Inside one of the tabs, someone was quoted saying, "Babette Van Anka, she's so bad you wanna spanka."
Babette lived in Bel Air, the moneyed enclave in the hills looking down on Los Angeles, with her third husband, Max Grab, the international financier. He advised a number of sultans of the oil-rich archipelagoes of Southeast Asia. He was said to have, as it is put, "ties" to influential Chinese.
The Grab–Van Anka mansion was large even by Hollywood standards. The grounds included a private hippodrome and his and hers helipads. The hippodrome had caused controversy. When their neighbors complained about their plans to blast away half of the side of one of the Hollywood Hills in order to accommodate it, there was a stink. Since Babette passionately embraced environmental causes in addition to peace in the Middle East, some delicacy was required. They hired Nick Naylor, who had once been the chief spokesman for the U.S. tobacco industry.
Naylor produced a letter from an organization that taught handicapped children to ride horses. The letter praised the Grab–Van Ankas lavishly for so generously offering them unlimited use of the new hippodrome. The enraged neighbors never regained the public relations offensive.
The blasting proceeded, the hippodrome was finished, complete with chandeliers and potpourri instead of sawdust, Max Grab having an aversion to the smell of horse by-products. Max also had an aversion to handicapped children, as it turned out. The organization was quietly presented with a check by Naylor and a note suggesting they seek other facilities. The Grab–Van Ankas were no amateurs when it came to the art of spin.
But even Nick Naylor, veteran of a hundred seemingly hopeless public relations challenges, was at a loss as to how to cope with Babette's new starring role as the President's mistress or, as one glib pundit put it, frequent guest in the Lincoln Head Room.
Max had been complaisant about his wife's relationship with the late President. His physical ardor for Babette had long since given way to the more exotic refreshments provided by Los Angeles's leading madams. He had even built a separate bungalow on the property, referred to by the household staff as "the Pump House." It had its own driveway so that Madam Vicki's pageant of international talents could come and go without having to pass a tight-lipped Babette on her way home from a grueling day of making not very good movies.
Max had found it quite pleasant to be a friend of the President of the United States. It certainly impressed his patrons overseas. But this kind of publicity was disastrous. It wasn't as though he were CEO of a corporation listed on the New York Stock Exchange. Max was an entrepreneur with "ties" to rather exotic people. Now that his wife had become notorious, he found details of his previously quiet business dealings leading the evening news. He was not pleased by this. He was not pleased that three dozen cameramen had permanently encamped outside the gates of Hanging Gardens, their estate. Thank God for the helipad. He was not pleased by the visits from the FBI, the Secret Service, and those grim-faced helots from the Justice Department. He had hired every lawyer in Los Angeles to handle it.
"Whatever strategy we adopt," Nick Naylor said over his untouched lobster tarragon salad, "we need to be very consistent with the testimony that Babette will be offering on the stand."
It was a beautiful, cool, perfect sunny day in Bel Air. The garden blazed with hibiscus and bougainvillea and jacaranda, the air thrummed with the soft sound of procreating hummingbirds. Yet Nick, Babette, and Max sat inside. The Grab–Van Ankas had not enjoyed an outside meal on their patio since the tabloid television show _Crime Time_ had one day trained long-range parabolic microphones on Babette and Max as they lunched outside, calling each other names not found inside preprinted Valentine's Day cards.
"However," Nick continued, "we need to get our message out."
"What message?" Max said. "That she wasn't humping him?"
_"Max,"_ Babette said.
Nick pressed ahead. "I'd like to gin up some press stories about the many other aspects of Babette. Her wonderful charity work, for instance."
"What, with the cripples and the retards?"
"Well, among others."
"I knew that was gonna come back and bite us on the ass."
"You and Babette have been powerful forces for change in the Middle East," Nick persisted manfully. "Your donations, the hospital, your company that generously provides Sidewinder missiles and cluster bombs to the Israeli Defense Forces at significant discounts, the prefab houses for the West Bank settlements, Babette's Concert for Peace in Jerusalem."
"Peace," Max snorted. "They threw rocks. It was a _rock_ concert." He chuckled and forked avocado and lobster into his already full mouth. "That's good. Rock concert. _You_ should have thought of that. You're supposed to be so good with the words."
"An international film star and singer, willing to put herself in harm's way, to bring Arab and Jew together—"
"They came together. They tried to kill each other."
"The larger point is that the concert was a milestone in"—what?—"Babette's commitment to the peace process."
"I wouldn't push the concert."
"But—"
"She wanted the concert, not the Israelis. I hadda pay that putz minister a quarter million up front, just to—"
_"Max."_
"Forget the concert. Trust me. You don't want those vultures digging into the concert."
The sound of a helicopter rattled the French doors. "Is that one of yours?" Nick asked.
Max wiped a glob of tarragon mayonnaise that had been on his chin for twenty minutes and with disgust hurled his napkin on the table. It was one of the few dramatic gestures left to powerful men.
"I've _had_ it with these helicopter pricks. Is there no fucking privacy left in this country? I'm going to the island," Max announced.
"Island?" Nick asked.
"None of your business. None of _anyone's_ business."
"It's off the coast of Panama," Babette said.
"Don't tell about the island. Jesus Christ, you tell _everything_. That's why we're in this to begin with. What do I have to do, have your tongue removed surgically?"
"Max," said Nick, "I'm not about to tell anyone about your island. But how is it going to look if you go off to an island and leave Babette to face the music?"
" _She_ made the music. _She_ can sleep in it."
"The Shah of Iran used to own it," Babette said. "Max bought it from the Shah. Well, the wife. After he—"
"What are you, _Architectural Digest_? Shut up."
"Max," Nick said, "isn't there some other place you could go to get away?"
"You got something against the Shah of Iran?"
"Personally, no, but—"
"Let me tell you something. I did business with the Shah of Iran for fifteen years. Tankers, oil, caviar, helicopters, army uniforms—the best uniforms in southwest Asia. Did you ever see pictures of his generals?"
Nick sighed. "They looked sensational, but—"
"The first shopping mall in Tehran? _I_ built it. The Shah of Iran was an honorable man. Maybe not the brightest world leader I have met, but you could do business with him. These mullahs? Try bribing them with a bottle of whiskey. They'll cut off your hand. And this is a pity. This was a beautiful country. I had many friends. What happened to them? Tragic. I can't even talk about it."
"I've got to get myself out there," Babette said.
"To Iran? They'd eat you alive."
"On _television_. Get myself on television. It makes no sense. Connie Chung, Barbara Walters, Diane Sawyer, _begging_ me for interviews. And I can't even return their calls? What sense does this make? I should go on television."
"I'd really, really wait until after the trial," Nick said.
"The lawyers said no interviews," Max said. "You're not doing interviews."
"But I wouldn't talk about the case. I would talk about the Middle East, about the Kyoto Protocols."
"You think they want to hear your views on Gaza and exhaust emissions? Would you explain to her? They wanna talk to you about _schtupping_ the President."
"You've never taken me seriously."
"Did I pay for your peace concert? Do I pay for your whatever you call them, issues advisers?" Max turned to Nick. "Issues advisers she has. On my payroll. One of 'em's a dyke."
"She is _not_ a dyke."
Max rolled his eyes. "Whatever. Two hundred grand a year for the three of them. Do you know what they do? They read the newspapers and write 'briefings' for her so she knows the difference between the West Bank and an ATM machine."
"I am a personal friend of Shimon Peres!"
"Wonderful. Have him over for dinner. I pay for her to go to Davos in the jet? To Davos she goes every year. To _network_. She comes back and tells me we have to do something about debt relief. To her, debt relief is me paying her bills."
"Excuse me! Excuse me if I care about global warming and hunger and peace while you're buying up golf courses in Arizona for the Sultan of Brunei."
"Where's dessert? I want dessert."
"You could never stand it that Kenneth MacMann cared about what I thought about the Middle East. He valued my input."
"Input? The only input he wanted with you was inputting his—"
"Don't talk to me. You and your escort services. Do you know what his American Express bill was last month? I saw. Twenty-eight thousand dollars. He puts it on his American Express so he gets points. How smart is that? Mr. Genius International Financier!"
At such times, Nick yearned for the simpler days of going on television to denounce the latest medical evidence that smoking was bad for you.
"Why don't I get back to you in a day or two with some concrete proposals?"
Perri Pettengill and Boyce lay in bed in Boyce's Fifth Avenue apartment with its view of Central Park, New Jersey, Pennsylvania, California, and the Pacific Ocean. Scented candles burned on both bedside tables. She had on a cream-colored teddy, no panties, thigh-high stockings made of real silk, drops of Outrage perfume in just the right places.
Thrilled as she was that Boyce was defending Beth MacMann, Perri wondered. Beth McMann was an undeniably attractive woman. She and Boyce had been engaged. A long time ago, but still.
So tonight, after taping _Hard Gavel_ , Perri had come straight back to the apartment to make sure everything was ready when he got back from Washington on the last shuttle. Dinner had been waiting, his favorite, _linguine alla vongole_ , with the teeny-tiny clams, bought that morning at an ungodly hour by Fung, Boyce's butler-cook-concierge, along with a glass of crisp, chilled Orvieto. Boyce permitted himself one glass of wine a night, nothing while trying a case.
The Billie Holiday CD was on, Manhattan twinkled expensively through the window. During dinner she slipped off her shoe and stroked his ankle with a stockinged toe and made purry allusions to the waiting bed. A little squeezy-squeezy on the way into the bedroom, where the candles were already lit, the bed turned down like the Bower of Bliss, Eros' trampoline.
She popped into the dressing room to turn herself into a Vargas Girl, and as she was putting on the finishing touches, dabbing Outrage on her inner thighs, what sound did she hear coming from the bedroom? Moans of anticipation? No. The TV.
She emerged, looking hot enough to induce an erection in a three-thousand-year-old Egyptian mummy, and he's on the bed, shoes still on, flipping through the channels with the remote.
_Hard Gavel_ came on. At least he was watching her show. That day she'd interviewed C. Boyden Gray, the very tall, distinguished Washington attorney who had been White House counsel in a previous administration. He said he was relieved that something like this hadn't happened on his watch.
Boyce flipped past her show. She couldn't believe it. He flipped until he came to _The Geraldo Rivera Show_ and stopped. Geraldo. Her competition.
Geraldo's guests were Barry Strutt, Bill Howars, and Alan Crudman, an unholy trinity of trial lawyers. Each thought of himself as the best in the business.
The fourth guest, piped in via remote from Harvard Law School, was Edgar Burton Twimm, the tweedy Wise Man still waiting for some president to nominate him to the Supreme Court. He was on to provide gravitas and to shift uneasily in his seat when the other guests said something provocative.
Perri stood there, an Aphrodite in silk. And what did Boyce do? Asked her to bring him sparkling water. With ice.
This left her with a choice of going into the kitchen and inducing a collateral erection in Fung or putting on a bathrobe. She was mad enough to get completely dressed and leave. But seeing Boyce intently watching Edgar Burton Twimm interjecting thoughtful harrumphs and cautioning against "throwing out the Fourth Amendment with the bathwater," Perri wondered if he would even notice that she was gone.
They'd been seeing each other for six months now. If he was trying a case, you could pour lighter fluid on yourself and light a match and he wouldn't notice. But the trial hadn't even started yet.
Well, he was Boyce Baylor and she was television's up-and-coming law honey and he had just signed on to the Trial of the Millennium and that made him her ticket to certain stardom.
Take a deep breath. Get him his (damned) ice water.
As she turned to go, Alan Crudman spoke up. Alan Crudman, the noted San Francisco attorney, was riding high these days. He had just gotten his latest client acquitted, an NBA basketball player who after a three-day cocaine binge had driven his Lexus off a raised drawbridge over the Intracoastal Waterway, demolishing the top deck of the yacht that was passing through at the time, killing two people and maiming four others. Crudman described his client as "a terrific human being."
Crudman told Geraldo that while he, Alan Crudman, tried not to get emotionally involved with his clients, it wasn't always possible.
"How Boyce Baylor," he said, "is going to handle the fact that he was once engaged to his client and she dumped him is anyone's guess."
Perri said, "I thought you dumped her."
Boyce grunted.
William "Billable" Howars, the exuberant Memphis, Tennessee, lawyer, said that Boyce would probably make Babette Van Anka out to look like "the whore of Babylon" on the stand. This brought a soft cough and concerned interjection from Edgar Burton Twimm about the presence of television cameras in courtrooms.
Barry Strutt had won a dramatic court-ordered exhumation of President Kennedy's assassin Lee Harvey Oswald that had established finally and irrevocably, beyond a shadow of a doubt—absolutely nothing, but he was triumphant about it. He said that it would be bad strategy for Baylor to try to cast doubt on the testimony of Secret Service agent Woody Birnam, who said he had overheard the President and First Lady arguing that night. He said that a Washington, D.C., jury—the phrase was now understood to be code for "predominantly black"—tended to respect the Secret Service and wouldn't like it.
Geraldo broke for a commercial. Perri went and got the sparkling ice. Geraldo was back on by the time she returned. Boyce was snoring. She thought about pouring the ice water on his lap, then got into bed and turned the channel back to her own show.
# Chapter 7
Boyce and Beth sat together on the observer side of a one-way mirror as Boyce's team of pollsters prepped seventy people on the other side of the glass for the focus group that was about to begin.
Normally, Boyce did not invite his defendants to participate in these sessions. Often, being in jail, they were unable to participate. But Beth had asked to come. She seemed genuinely eager to hear what people thought of her.
The focus group began. Part one consisted of the pollster reading aloud a series of statements about Beth. The group pressed buttons on the consoles in front of them. The body sensors measured their sweat, breathing, and heart rates to determine the honesty of their responses. The first question was: "Do you believe that Beth MacMann killed her husband?"
Beth looked at the computer screen in front of Boyce. A bar column of lurid electronic red rose vertically. The number 88.32 appeared above it.
"Is that—"
"That," Boyce said, "is where we start."
Three and a half hours later, after the last person had been unhooked, thanked, handed a check, and reminded that he had signed an enforceable confidentiality agreement not to reveal even that the session had taken place, Beth looked as if she had just been sentenced to death.
"I think we could both use a drink," Boyce said.
Beth nodded wanly. They went to Boyce's hotel suite.
He felt for the first time since taking the case a sense of pity for her. Large crowds used to cheer when she took the stage. Now she had just spent the afternoon listening to a majority of seventy people call her a murderess and, into the bargain, a scheming, manipulating, power-grabbing bitch. It wasn't the steely Lady Bethmac sitting across from him staring into her Scotch, but a frightened woman facing the death penalty.
"I tried to be a good First Lady. I pushed through initiatives on child care, prescription drugs for the elderly, the environment, a lot of things."
"I know," Boyce said. "The bastards ought to be grateful, instead of getting all bent out of shape just because you killed their president."
Beth gave him a horrified look.
"So shall we dispense with the self-pity and get to work?"
She nodded. "Fair enough."
"We heard some bad news today. But we also heard some good news. Many of them, at least the males in our group"—Boyce looked at the screen of his laptop computer—"think that the President and the entire government hate you. I'm _very_ pleased with that."
"You are?"
"Yes. We can accomplish wonderful things with that."
"Was there any other wonderful news?"
"Two-thirds thought Babette Van Anka's last movie stank. The one where she played the Israeli female tank commander. That's excellent news. _And_ you did very, very well among certain demographic groups. Males twenty-five to forty-nine want to have oral sex with you."
"Why would you _ask_ such a thing? It's mortifying."
"It would be mortifying if they didn't."
"Why"—Beth blushed—"that particular category?"
"Our research indicates that ninety-seven percent of heterosexual men want to have sex with attractive women. So this tells us nothing useful. But men only want to have oral sex—to perform oral sex—on women to whom they are especially attracted. This is great news for our side."
"I don't even know how to process that information."
Boyce scrolled. "We didn't score well with pet owners. They didn't like the fact that you didn't have a dog in the White House."
"You want me to go out and buy a sheepdog?"
"We could get you a puppy, but it's kind of late. Gays liked you, especially the hard-core lesbians."
"I score well among hard-core lesbians?"
"They love you. Probably because you crushed your husband's skull with a spittoon."
"I _didn't_."
"Whatever. We'll be getting some deeper analysis on those numbers. Among the former military, we did not do well. Not at all. No surprise there, since you—since they think that you killed one of the nation's great military heroes. By the way, everyone—even the hard-core lesbians—thought you were a little dry-eyed at Arlington Cemetery during the burial."
"What was I supposed to do, start wailing and tearing my hair? Leap in with the coffin?"
"If you had called me when you should have, instead of playing Mrs. Why Do I Need a Lawyer?—"
"We've been through this."
"—I would have rubbed onion juice on your sunglasses before the funeral."
"That's awful."
"I had a client once, she blew her husband's head off with his twelve-gauge Purdey shotgun—a forty-thousand-dollar gun—in the living room, in front of guests, on the white carpet—"
"I don't want to hear this."
"Ooh, this was one _tough_ cookie. Hard like a rock. Sigourney Weaver played her in the movie. She blew two holes in him the size of grapefruits, then reloaded and kept blasting. At the funeral, mascara—down to her cleavage."
"I'm not listening."
"White onion is best. Not red. We went for temporary insanity. The jury was out in under two hours. She was out of the mental hospital in less than three years. She's a tennis pro in Boca Raton. By the way, I want you in black for the trial."
"Isn't that a bit obvious?"
Boyce shrugged. "I'm not saying wear a _burqa_. Look, most women in New York wear black, and they only dream about killing their husbands."
He scrolled down.
"Now, these numbers about the late President's policies. There's stuff in here we can work with. African Americans were not happy with his last Supreme Court appointment, plus he criticized the Reverend Bones for having that love child with the head of his choir _and_ deducting her on his income taxes."
"Bones called again yesterday," Beth said. "He wants to come pray with me."
"I'll bet he does. And they call me Shameless."
Boyce scrolled.
"They thought your late husband was squishy on affirmative action. You gave a speech about that, didn't you? You disagreed with him. Was that a good-cop, bad-cop routine you two worked out to keep the black vote mollified, or did you actually mean it?"
"Screw you, Boyce."
"Pardon my cynicism. I thought you and he might have other arrangements, in addition to the one about his not banging actresses when you were in residence."
"You didn't used to be like this."
"No, I didn't. I was quite trusting, actually. Then I got screwed by someone I trusted. So now I have no illusions about people. I not only expect the worst from them, I demand it. Is any White House staffer likely, on the stand—under oath—to derogate or otherwise cast doubt on the integrity of your coming out publicly against your husband on the issue of racial quotas?"
"Is that what you think of me?"
"The witness is directed to answer the question."
"No. Amazing as it may seem, I was speaking from the heart."
"It's not that often I get such principled clients."
# Chapter 8
Three days before the start of jury selection, Boyce was filing his seventy-fourth pretrial motion—a personal record—this one to suppress the evidence of Beth's fingerprints on the Paul Revere spittoon on the grounds that her voluntary submission to fingerprinting by the FBI had constituted a "flagrant and unconscionable" violation of the Fourth Amendment to the Constitution, which prohibits unreasonable search. It was a long shot, but Boyce was already in his mind mapping out pretrial motion number seventy-five, on the even more daring premise that the traces of French-made hand moisturizing cream in the fingerprints would unfairly bias jurors who felt that an American First Lady should use only American-made beauty products.
The TV was on. He watched with one eye.
"Good evening," said Perri Pettengill, wearing a clingy sweater and trademark eyeglasses, "and welcome to _Hard Gavel_. My guest tonight, one of America's great trial attorneys, Alan Crudman. Welcome."
Alan Crudman was in fact a fine attorney, one of the best, yet even in his late forties he still carried on like a twelve-year-old clamoring to be acknowledged as the smartest boy in class. In law school it was said of him that he had come out of his mother's womb with his hand raised. He had gotten acquitted some of the most loathsome human beings on the planet and yet, not content to shrug and say that he had simply been upholding the purity of law and rights guaranteed by the Constitution, insisted on going an unnecessary further step and proclaiming in front of cameras that his smirking client, shoes still sticky with his victims' blood, was "totally innocent." Even colleagues who hadn't lost a minute's sleep after a lifetime career of defending the dregs of humanity shook their heads in wonder at Alan Crudman's amazing protestations on behalf of his clients. Could he really have convinced himself of their innocence? Impossible. Too smart. It had to be more complicated: he had graduated to telling the big, big lies, daring God to challenge. This fooled no one, but the media ate it up. The television talk shows loved it. It got them callers galore. And Alan Crudman was never too busy to go on television, on any show, to comment about anything at all. If the Weather Channel invited him to go on to talk about the legal implications of a low-pressure system over Nebraska, he'd be there as long as they sent a limousine for him. A short man, he demanded big vehicles.
Crudman loathed Boyce Baylor for four deeply held philosophical reasons. One, Boyce had gotten more guilty people off than he had. Two, Boyce was richer. Three, Boyce was taller and better looking. Four, Beth MacMann had chosen him over her.
He had placed a call to Beth within an hour of hearing the news that she was a suspect in her husband's death—and _she had not returned his call_. This hadn't happened to Alan Crudman in two decades. Who did she think she was? So now he despised her as well. He lay awake at night pleasuring himself with visions of the jury foreman pronouncing, "Guilty!" He saw her stunned expression, saw them drag her off. Saw her in bright orange death prison garb, struggling as they inserted the needle, shouting, "Get me Alan Crudman!"
"Thank you, Perri. Always good to be here."
Perri disliked Alan Crudman for one deeply held philosophical reason. She had invited him on one of the early episodes of _Hard Gavel_ and he had treated her like a dumb blonde instead of a former assistant district attorney. At one point he'd airily informed her that she had "totally misconstrued the deeper meaning" of _Plessy_ v. _Ferguson_. After the show, he had invited her to his hotel room—a lavish suite at the St. Regis Hotel, charged to _Hard Gavel_ along with the limo—for a drink. She had gone with one purpose in mind. Over drinks, she'd sat opposite him while he'd talked about his greatness, her miniskirted thighs parted just enough to provide a glimpse of the heaven within. Having brought him to a state of painful arousal, she had looked at her watch, announced she was running late, and left him to quench his ardor with any means at hand.
As _Hard Gavel_ 's ratings increased, Alan Crudman's attitude toward her became less and less condescending. He now addressed her as he would a Supreme Court justice.
Perri had asked him on the show tonight because she was mad at Boyce. Boyce was refusing to feed her details about the case.
"So how do you think the defense is shaping up so far?"
"I wouldn't want to second-guess Boyce Baylor," Alan Crudman lied, "but I'm frankly surprised that he hasn't put together a top-level _team_. All he's got is associates from his own firm, most of them younger people. This is, as I don't need to tell you, going to be a very tough case. Even I would find it a tough case. And I certainly wouldn't try to do it all myself. So it's either remarkable, or daring, or both, that he seems intent on trying this case all by himself."
"You're acknowledged as being the best in the business"—she knew this would infuriate Boyce—"when you take a case of this profile—"
"Perri"—Alan Crudman smiled, not one of nature's prettier sights—"with all due respect, there has _never_ in history been a case of this profile."
"—you usually partner up with other distinguished attorneys. It's not like you're saying, I can't handle this all by myself. Right?"
"Absolutely. In the J. J. Bronco case, as you'll recall, there were—what?—six of us. I, of course, was lead counsel, but I had Barry Strutt to handle the bloodstains, Lee Vermann for hair samples, Kyle Coots, who as you know is _the_ authority on slash wounds—he wrote the book—so we had a good, solid team. And of course justice prevailed."
"On that, any progress in the search for the real killers?"
"I—there's—I understand he's pursuing it. But as far as the MacMann case goes, yes, I am surprised that Boyce Baylor seems determined to do it all by himself. I'm sure he has his reasons."
Beth too was watching. She had developed a curiosity about Perri Pettengill.
Listening to Alan Crudman, whom she had loathed since he had pronounced on television that J. J. Bronco was "one hundred thousand percent not guilty" of the grisly murders, confirmed her decision not to return his call during the first days of her nightmare. Yet the lawyer in Beth was wrestling with the fact that he _was_ Alan Crudman, a lawyer of great ability. Even if she discounted his palpable jealousy of Boyce, his comments did make her wonder why Boyce was so intent on going into court solo. She'd asked him why he hadn't assembled the mother of all defense teams. He'd said he didn't want to overwhelm the jury with too many expensive suits. The fourth time she'd asked, he'd gotten huffy and reminded her that he was in charge. Two possibilities lurked in her mind: one, he was playing single-combat warrior to beat the odds and win back her heart; two, he wanted to lose this case to punish her for what she'd done to him. She didn't like either scenario, though the first was preferable.
# Chapter 9
If it wasn't going to be easy to impanel a jury in _United States_ v. _Elizabeth MacMann_ , finding a judge was presenting its own challenges. There were thirteen full-time judges on the U.S. District Court for the District of Columbia. Four had to recuse themselves because they had been appointed to the bench by the late President MacMann. Two more had to drop out because they had been appointed by the previous president, whom Ken MacMann had defeated. Another had been overheard by a caddie telling his golf partner on the seventh hole of Burning Bush Golf Club that the President "got what he had coming." The caddie sold the quote to the _National Perspirer_ tabloid for $10,000. Scratch judge. Another judge had been on a panel with Boyce at the Trial Lawyers Association convention years ago and had called Boyce "the worst human being on the planet" while discussing the topic "Getting Hitler Off: Rethinking Nuremberg Defense Strategies."
The media combed through the court transcripts and biographical profiles of the remaining judges to see what nuggety chunks of mischief might be embedded in their pasts. One judge, fresh out of college, had spent a summer working for a congressman who had insisted that Beth's husband had been brainwashed in captivity and referred to him publicly as "the MacManchurian Candidate." He was out. Another had protested against the Vietnam War in which President MacMann had so valiantly fought. Out. The gavel of yet another had to be pried from his fingers after it was reported that he had gone on a blind date twenty-five years ago with Babette Van Anka, whose name then was still Gertrude Himmelfarb. By now one dyspeptic columnist at _The Washington Post_ suggested it would be simpler just to take Beth out back of the courthouse and shoot her.
In the end, it came down to the one remaining judge on the bench. His name was Sylvester Umin, known to his colleagues as "Dutch." He had been appointed to the bench two months before by President Harold Farkley. Up to then, he had been a senior partner in the distinguished Washington firm of Williams Kendall, specialists in impeachment and negligence law.
Dutch Umin was in his early sixties. He had drowsy but watchful eyes and the Cheshire cat physique of a gourmet and oenophile. His vertical collection of Château Petrus made dinners at Mandamus, his Virginia mansion, memorable occasions. He collected Dutch master artwork, the source of his nickname.
He was a man of formidable intellect who had clerked for the great Potter Stewart on the Supreme Court and over the course of a distinguished career had won impressive victories for clients ranging from left-wing firebombers to cocaine-snorting major-league baseball players to international grain corporations accused of using powdered insect dung to give a popular children's breakfast cereal its distinctive crunch. But he had yet to try a single case as judge, and now by process of elimination he was—it. Overnight, he became the most famous jurist in the world. Within weeks, he would have name recognition among aborigines and Seychelles islands fishermen. He was not altogether delighted by this abrupt propulsion to celebrity. His glasses had developed a tendency to fog.
Judge Dutch Umin's first official duty in _United States_ v. _Elizabeth MacMann_ was to convey his dismay over the witness list that Boyce had submitted. It included 281 names, including the directors of the Secret Service and the FBI, the President of the United States, and most saucily of all, the deputy attorney general, who was prosecuting Beth. One columnist remarked that it was a wonder he had not subpoenaed Paul Revere to attest to the authenticity of the spittoon.
The atmosphere in chambers was tense. Sandra Clintick, the deputy attorney general—who had not at all hungered to have this prosecution handed to her—had taken exception to Boyce's demand that she herself testify. She was so mad that she avoided eye contact with Boyce. Never, she told the judge, had she heard of more appalling—make that atrocious—ethics. It was beyond insulting. The gloves were off, and they weren't even in court yet.
"Counselor?" Judge Dutch leaned back in his armchair, which gave off the creak of expensive leather. Knowing that the ordeal ahead would tax all his reserves, he had resolved to be as laconic as possible, even to the point of Zen.
"Your Honor," Boyce said, smiling, as if he were presenting the most reasonable proposition since Newton's last law, "one of the foundations of our defense will be that this prosecution, _ab initio_ "—he turned to the deputy AG—"sorry, 'from the beginning'—"
"I know what it means."
"It's Latin."
"I know that."
"I wasn't sure they still taught Latin when you—"
"Your honor."
_"Counsel."_
"A significant part of our defense, Your Honor, will be that Madame Deputy Attorney General here—"
"The name is Clintick. I don't work in a whorehouse."
Boyce snorted. "I'd say _that's_ a matter of opinion."
"Your _Honor_."
"Counsel."
"We will establish that Mad—that the deputy attorney general is merely the smallest cog in a larger government conspiracy machine to bring murder charges against the former First Lady to further their own political agendas. Their evidence is disgraceful. Worse than disgraceful. I will annihilate it. Having done that, I will show by direct and cross-examination that Ms. Clintick conspired, along with other officers of government, to crucify Elizabeth MacMann on the altar of their own ambition. I understand, Your Honor, that this is a foul charge. I use it reluctantly, having no other recourse." When Boyce got going, his language became florid in a nineteenth-century sort of way.
Judge Umin tried not to smile. He concentrated on thinking about the crippling price he had just paid for his latest acquisition, a still life of a pear and eel by Govingus Koekkoek (1606–1647).
"I won't sit here and listen to this," said the deputy AG. "I will certainly not sit in court and listen to it."
Judge Dutch creaked in his chair. "Why don't I decide what we'll do in court?"
"Of course, Your Honor. I meant..."
Bingo. Boyce always tried to rattle them before going into court, to see where their stress points were. This one's stood out like rivets.
"I'm hard-pressed to think of a precedent," Judge Dutch said.
"I can't think of a precedent, either," Boyce interjected. "The executive branch conspiring with directors of the nation's top security and law enforcement agencies to frame the widowed wife of a president in order to conceal their own rank animosities and evil designs—"
"Steady, Counselor."
"I apologize, Judge. I forgot myself. But I feel myself stirred."
"Give me something concrete, not a Patrick Henry speech."
Boyce handed him a loose-leaf binder full of press clippings highlighted in bright colors, neatly tabbed.
"As you know, Mrs. MacMann was no passive First Lady. She did not confine herself to serving tea to other wives and organizing Easter egg rolls on the White House lawn for the children of... cabinet officers."
The attorney general, father of five, had been conspicuous with his brood at the recent White House Easter egg roll.
Boyce continued. "Beth MacMann was the most substantive First Lady in our history. This did not sit well with some. On occasion, as the documents in that binder will show, Mrs. MacMann was vocally, if always cautiously, critical of the FBI and the Secret Service. The former for what she viewed as incompetence for hiring a man with the middle name of Vladimir to head up its counterintelligence operations. The latter for its hiring practices, which she viewed as discriminatory. We will contend that these two agencies, which played so critical a role in her being dragged by the hair to the dock, were predisposed to exact revenge on her by concocting the evidence against her."
"Evidence, Counsel, evidence. These are press clippings."
"With all respect, you're putting my client in a classic Catch-22 position. She cannot produce evidence without putting her accusers on the stand, yet you will not permit her to put them on the stand without first presenting evidence."
"I'll consider it. But for your client's sake, I wouldn't put all your eggs in that basket. As for calling the President to testify, visualize a snowball. Now visualize the same snowball in hell."
Boyce smiled. "I am at the mercy of Your Honor's wise and learned judgment."
# Chapter 10
There are few spectacles more pathetic than a roomful of otherwise responsible people trying to squirm out of a civic duty enshrined in Magna Carta as one of the signal boons of democracy. On the other hand, who in his right mind wants to serve on a jury?
Impaneling a jury for _United States_ v. _Elizabeth MacMann_ was more daunting. When the prospective jurors entered Judge Umin's courtroom with the downcast shuffle of the damned, most of them took one look at the judge, Boyce, and the deputy attorney general and uttered the same silent cry: _Oh God, no—not_ that _case!_
Boyce and his jury consultant studied their faces intently. It was easy enough to spot the ones who were horrified at the thought of spending the next year in some ghastly motel with seventeen of their "peers."
Others positively radiated delight, either at the thought of becoming part of history, or at the prospect of all those lucrative book deals. _Juror Number Five: My Story_. Film rights to Warner Brothers for seven figures. A top New York publisher had been quoted in the _Times_ saying that a book by the first juror to be dismissed would fetch at least $1 million. But a juror who held out against the other jurors, either for or against conviction _—that_ juror, the publisher said, could go start pouring the concrete for that dream house.
"This is a capital murder case," the judge began on the first day of jury selection. "Capital means that conviction carries a potential penalty of death. Normally a case like this could take months to try." Groans came from men and women in expensive suits who looked as if they measured their time in minutes. "But this is not a normal case, so it is difficult to predict. It could take up to one year. It could take more. Especially"—he glanced sideways at Boyce—"since an extensive witness list has been submitted by the defense." Gasps, groans, chests were clutched, bottles of nitroglycerin tablets rattled.
Boyce and his jury consultant watched the faces of the jury pool. Boyce's jury consultant was a man named Pinkut Vlonko. Before going into the lucrative business of advising trial lawyers on jury selection, "Pinky" Vlonko had been for over twenty years the CIA's top psychological profiler. His job was to figure out which of the CIA's top people were most likely to be selling secrets to the Russians or Chinese; also, to determine whether Saddam Hussein was technically a malignant narcissist or simply a fruitcake. Pinky had worked with Boyce on many cases. Between them, they had the best juror "radar" in the business. Boyce was fascinated by psychology. After being dumped by Beth, he had taken a master's degree in applied psychology.
The two of them had prepared a juror's questionnaire extensive even by their standards. It consisted of eight hundred questions. Number 11: Did you vote for President MacMann? Number 636: Are you regular at bowel movements? During his years at the CIA, Pinky had discovered that defectors and moles—switchers of allegiance—tended to be constipated.
The questionnaire had occasioned another heated session in Judge Dutch's chambers. Ms. Clintick, the DAG, had pronounced it an abomination. Boyce had thereupon produced a questionnaire used by the attorney general's own pollster when he had run unsuccessfully for governor years ago. It was 120 questions long. He'd waved it in the DAG's face. A compromise was struck. Boyce's questionnaire was trimmed to 650 questions. This was more or less the number of questions he and Pinky wanted to begin with. Boyce's rule since childhood had always been, Ask for a lot more than you need so that you end up with what you want.
Boyce's amiable but relentless grilling of the jurors, carried live on TV—Judge Dutch had decided it would be more complicated not to allow cameras in the court—led one pundit to venture that the only jury Boyce would be satisfied with would be one consisting of blind deaf-mutes with an IQ of 75. Not at all, Boyce countered cheerfully. All he wanted was "a level playing field." Was it unreasonable to seek out jurors whose minds had not been "hopelessly polluted by the daily diet of deplorable lies, innuendos, and vilification manufactured by the government's agents of smear and malediction"? If finding an unbiased jury required a little patience, who could object?
A "little patience" ended up taking four months.
# Chapter 11
You should have been there," Boyce said. "I thought he was going to stab me with a Dutch letter opener. This is going to be fun."
"I'm glad you're enjoying yourself," Beth said.
Boyce put his hand on hers. "I'm just trying to get _you_ to relax."
Beth looked anything but relaxed. She'd lost the weight Boyce had ordered her to shed. Her cheekbones were more prominent now, and the eyes had the darty intensity of someone dreaming of deep-crumb coffee cake. Looking at her, Boyce suddenly felt guilty. He wanted to pull the motorcade over and rush in and get her a chocolate milkshake.
"You okay, kiddo?"
"Fine."
"You look great, for someone who hasn't eaten in six months." The beauty magazines had tracked Beth's change in physical appearance. _Vogue_ had done an article entitled "Diet of the Millennium." It quoted a leading Hollywood "aesthetic consultant"—formerly "makeup man"—saying, "If this is what killing your husband can do for you, then more women ought to considering clubbing their husbands to death."
_Vanity Fair_ magazine pined, "If only Natalie Wood were still alive to play her in the movie. That limpid sexuality, the steel hidden beneath the puddly dark eyes, the tragic glamour."
_Variety_ reported that Catherine Zeta-Jones was "desperate" to play Beth in the movie. Further, that Joe Eszterhas, the dramatically hirsute and extravagantly compensated screenwriter, was holed up in a bungalow on Maui pounding out draft number seven of his script, entitled _Spittoon_.
All this Boyce had tried to keep from Beth. He needed her focused. She rolled down the window.
"Ma'am," said Hickok, the Secret Service agent in the front passenger seat. Hickok was jumpy these days. The death threats had been increasing.
Beth ignored him. The air was June—humid and sweet with moldering blossoms. She'd been a virtual prisoner since last fall in the house in Cleveland Park, under permanent surveillance by a press camp that never dropped below fifty people, even during the Christmas holidays. The house, built by a friend of George Washington, had the happy name of Rosedale but had been renamed Glamis by a pundit, after the castle in _Macbeth_.
Beth left the window down. She'd be damned if they'd deprive her of a few gulps of fresh air on her way to be tried.
They drove along Pennsylvania Avenue to the United States District Court for the District of Columbia at Third and Constitution. She thought of the January day three years before when she and her husband, freshly sworn in as President, had walked past the spot, waving to cheering crowds. Normally this would be enough to make a couple happy. Not the MacManns. The night before, at Blair House, Beth picked up the phone to make a call and heard her husband on the line talking to a well-known society woman in New York, the trophy wife of a billionaire. They were making plans for an afternoon hump at his New York hotel while Beth was across town at the United Nations, addressing a conference on the role of women in the new millennium. She reflected that the role of women in the new millennium seemed to resemble the role of women in the last millennium: on the wrong end of the screwing.
She put down the phone, went into the next room, picked up a lamp, and was about to conk him with it when a vision made her stop—the vision of herself twelve hours later holding the Bible as Ken took the oath of office, looking at him adoringly, his head wrapped in a bandage. She put the lamp down. And Ken smirked. If she was dry-eyed at his funeral, she was drier still at his swearing-in.
Now the motorcade pulled up in front of the courthouse. This was Boyce's idea.
"Okay," he said, "remember, we're going to walk in there like we own the place. By the time we're through with these jerks, they'll be the ones on trial."
Boyce didn't quite believe this, but going into court was like taking the field in a game. You had to pump up your players. You had to pump yourself up.
There were so many satellite trucks, it looked like a NASA tracking station. It was a scene. Media, cops, and demonstrators with signs—ASSASSIN!, FRY THE BITCH!, FREE BETH!
She was wearing a black pantsuit copied from one of the leading designers, with enough changes so that the media wouldn't be able to say that she had looked "stunning in Armani." Half a dozen designers had called Boyce offering to dress her for the trial. Boyce had turned them all down. What's more, he'd informed the media that he had. On the first day of her trial, Beth looked stylish but sober: a smart-looking woman in her early forties on her way to a business meeting. The white blouse, Boyce joked to her, symbolized her innocence. It was open enough to draw the eyes of the male jurors without offending the women. The string of pearls had been a gift from Ken, bought by his secretary when he forgot her birthday.
"Okay, here we go," he said. "Got your mantra ready?"
She gave him a tight smile. The mantra, devised by Boyce, was "When we walk in, there'll be one single thought in your head: _I have come to accept their apology_."
That night, after the first day of the Trial of the Millennium, her entrance into the court was shown on an estimated 72 percent of the world's television sets. Her swanlike serenity, amid a clamor that would have rattled a professional wrestler, was widely commented upon.
Boyce was cheered by DAG Clintick's opening statement. She delivered it in an earnest more-in-sorrow-than-in-anger tone. He was so delighted that he decided to break his rule and depart—slightly—from his own memorized fifteen-thousand-word opening statement.
The essence of the United States's case against Elizabeth Tyler MacMann, Ms. Clintick averred, was straightforward: The President was found dead in his own bedroom. The autopsy established time of death between 3:15 and 5:00 A.M., and that death had resulted from an epidural hematoma caused by blunt-force trauma to the skull five centimeters above the right eyebrow. Photographic enlargement of the bruise revealed the distinctive imprint of the hallmark of an antique Paul Revere silver spittoon. The spittoon, used as a wastebasket, was found not in its usual place in the bedroom, by the First Lady's side of the king-size bed, but by the door, on its side. The jury would hear testimony from a Secret Service agent who would testify that he had heard a violent argument coming from the presidential bedroom between 2:10 and 2:20 A.M. They would hear from numerous people who had attended the state dinner that night that the President had been in fine spirits and health, no bruise or Paul Revere hallmark on his forehead. An overnight guest in the White House would testify that she said good night to an unbruised President at 12:30 A.M. They would hear testimony from numerous friends and associates of the First Couple as to the turbulence of their marital relations.
When all this evidence was presented, the jury would have no choice but to conclude, beyond a reasonable doubt, that Beth MacMann had callously and cold-bloodedly murdered her husband as he slept, in their own bed. A husband who, as it happened, was President of the United States of America. They would therefore have no choice but to find her guilty not only of murder in the first degree, but of assassination, the gravest crime in the land. This litany of villainy took slightly under two hours to deliver.
Boyce rose, buttoned his jacket, and walked toward the jury box. He rested his hand on the edge of it as he walked from one end of it to the far end, as if it were a banister. He had learned this from Edward Bennett Williams, the great trial attorney: Show them you're not afraid of them, show them you're comfortable everywhere in the courtroom, show them it's _your_ courtroom.
He turned, faced them, and said in a quiet but commanding voice, "Good morning." His jury consultant, Vlonko, noted that eight out of eighteen returned his greeting. Resting one elbow on the jury box, he began. No podium, no notes—unlike the DAG. He then launched into his imitation of a lawyer speaking from the heart, one of the great dramatic roles.
"Ladies and gentlemen, that was a pretty good speech you just heard by the deputy attorney general. She was, as you know, appointed to her office, a sacred trust, by her boss, the attorney general, who got his job from Mrs. MacMann's late husband. She and her superior, the attorney general of the United States, seem to have held on to their jobs, despite the change in administration." Pause. "That is unusual. But not irrelevant to this case. It is also highly unusual for a deputy attorney general of the United States to personally prosecute a case. Extremely so. One might ask, _Why_ is she prosecuting this case, when she could be doing what a deputy attorney general does? Namely, keeping the nation safe. Working on behalf of those whose civil liberties have been violated? On behalf of those whose livelihoods are threatened by giant monopolies? On behalf of those who are persecuted for the color of their skin, for their sexual orientation—"
"Objection."
"Proceed, Counsel."
"Now, ambition in itself is not a bad thing. All of us, all of you, have ambitions. To move up in the world. To earn the respect of your fellow citizens, to save money to send your children to college—"
In the press section, heads turned. Someone said, _"Oy."_ There's no more suspicious sound than that of a lawyer proclaiming the decency of his fellow man.
"—to make better lives for ourselves. That is ambition, and there's nothing wrong with it." Pause. "But... _but_ when ambition consists of exploiting a tragedy and the misery of a widow"—this would be the first of 1,723 mentions by Boyce of the word _widow_ during the trial—"in the service of a conspiracy by the same government whose sworn duty it is to protect us, then, ladies and gentlemen, decency shudders, honor flees, and darkness has surely descended upon the land."
The deputy AG rose. "Your Honor, this is intolerable."
"This is a court, Mr. Baylor, not a church."
"Well, there I agree with the deputy attorney general. I agree that it is intolerable that a woman who has dedicated her life to public service, to feeding the poor and underprivileged, caring for the elderly, seeing to it that working men and women have jobs and portable health care—while also making sure that business and entrepreneurs are not overtaxed and over-regulated by government"—a little something for the Republicans on the jury—"I agree that it is intolerable that such a woman be vilified and unjustly charged with a heinous act." Pause. "Simply because she dared to speak out against injustice and wrongdoing. Yes, I would say that the deputy attorney general has it exactly right. It _is_ intolerable. And after the facts have been presented, you too will find it so. This case is designated _United States_ versus _Elizabeth MacMann_. Well, that's about the size of it. The government, the entire United States government... versus one single woman."
Boyce walked slowly over to the defense table and stood near Beth. She hadn't quite anticipated a _J'accuse!_ of this amperage. She tried to conceal her embarrassment by staring blankly at the table.
Having placed himself next to the Widow MacMann, Boyce continued.
"There is a philosophical principle called Occam's razor. It goes like this: Never accept a complicated explanation where a simple one will do. Smart man, Mr. Occam. The prosecution—the government—would have you believe that the explanation for President MacMann's demise is more complicated than landing a person on the moon. They will bring in charts, timelines, computer-enhanced photographs, to convince you of a scenario so wild, so convoluted, so unbelievable, that to process it, to take it all in, would require the intellectual capacity of an Albert Einstein or Martin Luther King. You recall that the judge here explained to you during voir dire that this case might take some time to try?" Boyce chuckled. "Well, brace yourself, ladies and gentlemen, because it might just take _years_ for the deputy attorney general to convince you of the preposterous scenario upon which her case depends."
Boyce sighed deeply at the monstrous injustice of it all. He aimed his next burst of rhetorical flatulence at the heavens beyond the ceiling, where surely God and His archangels were listening, sharpening their swords of righteousness.
"Be prepared for arguments that would make Jesus weep and Einstein's head spin. Be prepared to hear that a mark on the late President's forehead was put there... by Paul Revere."
The correspondent for _The New Yorker_ magazine leaned over and whispered to the _Vanity Fair_ reporter, "I love this guy."
"That's right," Boyce continued. "Paul Revere's silversmith mark. Supposedly from a spittoon Mr. Revere made about the time of the American Revolution. Well, sit back and get comfortable. They're going to bring in photographic blowups of a tiny spot on the President's forehead. Experts—that is, they call themselves experts—with expensive, government-supplied laser pointers, will point at these photographs like they were aerial reconnaissance maps of Afghanistan. They'll say, 'See this teeny-tiny part here? We know it's hard to see, but that's Paul Revere's initials on the President's skull. Can't you see that? Are you _blind_? Why, any _fool_ could see it!' Well, ladies and gentlemen, that's exactly what the government thinks of you—fools. To be manipulated! Um _-hum_."
Jurors seven and nine were nodding along as if it were a Baptist sermon. _Say it, brother!_
Boyce shook his head bitterly in wonder. The next words exploded from his mouth with such force that the front row of jurors recoiled.
"A _spittoon!_ "
The stenographer started.
"The so-called murder weapon. An antiquated device going back to the days when men chewed tobacco. How fitting, ladies and gentlemen, that the government's chief piece of evidence should be a receptacle... for _spit_."
The _New York Post_ headline the next day was:
SHAMELESS: I SPIT ON YOUR EVIDENCE!
"Ladies and gentlemen, you will learn that there is a far, far simpler explanation for the President's unfortunate and untimely demise than that his devoted wife of twenty-five years awoke out of a deep sleep in the middle of the night and seized a historic antique—she, a lover and respecter of antiques, you'll hear testimony to that—crushed his skull, then went back to sleep, woke up, and cheerfully ordered breakfast in bed, with the corpse still cooling. The simple truth is..." His voice dropped.
Reporters, jurors leaned forward in their seats.
"Accidents happen."
Boyce turned directly to the jury, his back to the rest of the court and the world, as if this weighty matter were just between them.
"Planes crash. Cars crash. People fall down stairs, slip in bathrooms. Who among us—who among you—has not felt a wet foot go out from under us—"
Boyce pitched forward, grabbing the jury box rail.
"—and caught ourselves in the nick of time? Has that ever happened to you?"
"Objection."
"Sustained."
But three jurors were already nodding at him. To hell with the prosecutor and the judge. This was between them!
"Who among us, saving ourselves from snapping our necks or going down with our head on the tiles, has not felt a vast _wave_ of relief and gratitude and thought, _Whew! Thank you, Lord! That was a close one!_ "
Boyce walked over to the prosecution's table, where the deputy AG and her team sat, glaring at him. Boyce loved to end his opening statements here, in their territory, in their faces.
"A death by happenstance, by accident, is no less tragic, perhaps, than any other kind of death. But"—withering glance at the prosecution—"it is _not_ murder. It is _not_ assassination. And it is _no_ excuse—none!—to charge horrendous deeds to a woman whose only crime, if you want to call it that, was to have loved her husband too deeply, and too well."
He'd timed it to the minute. It was 4:43 P.M. Judge Umin had announced at the outset that he would adjourn every day at 4:45. His opening statement would marinate in the jury's minds all night, barbecue sauce seeping into meat.
Boyce sat down and bowed his head prayerfully, as if he had just taken Communion.
# Chapter 12
You know what they're going to call it, don't you?" Beth said in the car on their way to the post-trial conference in Boyce's hotel suite. "The 'shit happens' defense. You've staked my life on a wet bathroom floor."
Boyce was pumped. Oxygen was roaring to his brain, as if he'd just run five miles. Oh, the poor mortals, the nonlitigators, the timid souls who would never in their lives know this feeling, the thrill of owning a courtroom. A symphony orchestra conductor, a stage actor, a tenor, a great orator, an athlete at his or her peak—they knew something of it. But their stakes were relatively trivial: art, a home run, a moment of uplift for the paying audience. This—this was life or death! This was the Colosseum. He was floating in endorphin soup. All was well with the world. He was in a state of grace. This was going to be his greatest triumph ever, the crown in a shimmery career. He even forgot about his secret plan to lose.
He looked at Beth, and she looked pumped, too, for the first time since this had all begun. He was seized with the urge to kiss her. No. Not yet, and anyway, not in the car with Agent Hickok up front. Boyce wondered about the agents.
She had a large detail—a dozen. Athletes with Uzis. Were they spying on them? He wondered. They were professionals and honorable. But in a few days they were going to hate Boyce's—and her—guts so badly, their trigger fingers would itch like bad cases of poison ivy. The temptation to fight back would be hard to resist.
Watching the back of Hickok's head, Boyce felt a pang of regret. It would pass. He had read that Ulysses Grant, commander of the Union Army during the Civil War, would stay in his tent during the ghastly battles lest the sight of all that ground-drenching blood soften his resolve. Boyce had learned that if you were going to win, win at whatever cost, you had to reach inside your head and flip the on/off switch on the conscience console. And look what you saved on electricity.
His three connecting suites at the Jefferson Hotel, once owned by his idol, Edward Bennett Williams, had been transformed into a command post. One room was full of television screens and young associates monitoring the media. A section of the room had been turned into a remote TV studio so that Boyce could comment live, if need be, at a moment's notice. Another room had been converted into a fitness and meditation center, complete with exercise machines and a boxing bag. During a trial, Boyce liked to hang upside down by his ankles and with boxing gloves beat a hundred-pound bag full of sand. Marvelous for the circulation and wind. There was a massage table, meditation mat, juicer, and oxygen tanks. When Boyce had turned forty-five he'd noticed that it was taking him eleven minutes instead of ten to do London's _Sunday Times_ crossword puzzle. Suspecting diminished mental capacity, he had submitted himself to a battery of neurologists and cognitive and memory experts. They'd found no slowing down but suggested that he inhale pure oxygen for ten minutes a day. He went them one better. During trials, Boyce slept with an O2 tube in his nose. At home in New York, this impeded amorous relations with Perri, who during the wakeful wee hours liked a bit of spontaneous num-nums.
Another room, fitted with special locks and a 24/7 armed guard, was designated GZ (ground zero). This, Boyce confided to Beth, was mostly to psych out the prosecution. Since the thrust of his defense was that Beth was the victim of an insidious conspiracy, Boyce put out the word to his "friendlies" in the media that he was concerned about spying. He told them that he had the GZ "swept" for electronic bugs twice a week. He'd contemplated installing retinal scanners at the door to admit only those whose ocular profiles had been programmed, but in the end he'd balked at the cost.
Boyce wasn't so paranoid as to think that the government was tunneling under the Jefferson to spike his computers, but he did not want them learning about JRTRE, or "Jeeter." This was the juror real-time response evaluator. No one outside a very small circle knew about JRTRE. It was something he'd devised with input from the excellent Vlonko. Every day in court, Vlonko and an assistant intently watched the faces and body language of the eighteen jurors. They entered their individual responses into laptop computers, which also generated a real-time transcript using voice-recognition software. Responses included FAVORABLE, UNFAVORABLE, AMUSED, ANNOYED, BORED, NODDING, ASLEEP, RESTLESS, ANGRY, INTENT, SMILE, SCOWL, LAUGH, CRY, HAPPY, AROUSED, RAPID BREATHING, HEART ATTACK, and forty-five other conditions experienced by jurors. Among them was GET ME OUT OF HERE.
At the end of each day of the trial, Boyce had a real-time log of each juror's apparent response to every second of the proceedings. Correlating all this raw human data with the biographical information on each juror required a team of eight analysts working through the night in shifts. But by the morning, Boyce went into court with a printout that told him which jurors he needed to concentrate on that day—and what they wanted to hear from him. Simple, really. It amused Boyce that none of his chest-thumping peers had thought of it. It did rather add to the cost of a trial, but he'd figure out a way to bill it to one of his corporate clients.
"How'd we do?" he asked Vlonko.
"Fucking better than expected." Years of debriefing Russian defectors had left Vlonko, himself a naturalized Hungarian, with the tic of inserting the f-word in otherwise prosaic sentences. Apparently, the f-word relaxed Russian defectors. "Jurors two, six, seven, nine, and ten through thirteen were wetting themselves during your finish. One and five were sphinxy, but"—he hit keys on his laptop—"aha, yes, I was not incorrectly remembering: Number one is one-quarter Scottish, and five liked to play chicken on railroad tracks as a child, so they wouldn't give us a reaction if you took out your cock and banged it on the defense table."
"I'm saving that for closing statements. Keep your eye on those two. How'd we do with fourteen? Thought she looked uncomfortable when I hit the spittoon line."
Vlonko called up number fourteen's biography on his screen. "Fucking straight. Father died of emphysema. She probably has the unpleasant associations with tobacco and spit."
"Make a note of that in tomorrow's brief, would you?"
Boyce cursed himself. Of _course_ , juror fourteen's father had died of emphysema. Idiot! How could he have forgotten?
Frowning, one of his associates approached with a piece of paper.
Boyce read it.
"God-dammit!" he bellowed.
"What?" Beth said anxiously.
"We're estimating that the trial will go to the jury in two hundred and eleven days. And the goddamn moon will be full in two hundred and eleven goddamn days."
He said to his team, "We're going to have to stall. File some more motions."
"You gave me a heart attack," Beth said. "What's the problem? Are some of the jurors werewolves?"
"The last time one of my juries deliberated under a full moon, do you know what happened? I _lost_."
"Is this a superstitious thing?"
"A full moon affects moods, Beth," he said crossly. "Gravitational forces are altered. Water is redistributed. Do you want your fate decided by twelve human beings whose _water_ has been redistributed?"
"I hadn't factored that in."
"Well, that's your luxury, isn't it? I _do_ have to factor 'that' in." Boyce turned to an assistant. "Find out from Vlonko if he has data on the menstrual cycles of jurors two, eight, ten, and fourteen. Do they coincide with the lunar cycles?"
The aide scurried off.
Boyce said to Beth, "Would you _poll_ during a full moon?" He scowled and stormed off to his exercise room to beat his sandbag.
Beth said to an aide, "Is he always like this during a trial?"
"He hates to lose."
Babette was alone in Hanging Gardens—that is, not counting the seven servants.
Max had made good on his threat to decamp and was ensconced on his island off the coast of Panama, deep-sea fishing and cornering the world's market in a mineral that was going to be on the cover of _Time_ magazine in two weeks because of an about-to-be-released study showing that it might retard Parkinson's disease. His huffy departure aboard his private jet from the Burbank airport had been recorded in all its glory by half a dozen telephoto lenses and splashed across the front pages under headlines like MAD MAX and MAX: I'M OUTTA HERE!
Nick Naylor, now working more than full-time as the Grab–Van Anka PR man, had done what he could to spin Max's abrupt departure as an "environmental excursion" during which his client "hoped to experience some of the thrilling marine life of the San Blas islands." He left out the part about hooking the marine life and reeling it aboard so that Manolo could club it to death and serve it to him for dinner with lime juice and shaved coconut.
Nick had been working hard of late. When not trying to spin Max to the media as successor to Jacques Cousteau, he'd been acting as an ad hoc record producer—yet another new role for him—trying to put together a music album showcasing Babette's lifelong commitment to Middle East peace, working title "Babette Does Jerusalem." A better title might have been "Mother of All Headaches."
Babette was a wreck. She hadn't been out of bed since the visit to the deputy attorney general's office. A nightmare. Up until then, they had been pleasant, these people. Then suddenly comes this summons—that's right, summons—to come to Washington—the next day. Not "Oh, Ms. Van Anka, so sorry to bother you, would you mind terribly popping in at your convenience, and by the way, we _love_ your work," but "We will expect you in our office at ten A.M. on Tuesday." The cheek of these people. Ken MacMann would have fired them all.
She showed up all right, with not one, not two, but three lawyers. Of course she was there to cooperate in any way. That ice queen deputy attorney general prosecutor, without even asking if she'd like a cup of coffee, began the grilling. Do you stand by your statements to the FBI agents the morning the President was found dead? Excuse me, for this I flew three thousand miles? To be insulted? You could have asked me this over the phone. Do you stand by your statement, Ms. Van Anka? _Yes_. You told them you said good night to the President around twelve-thirty. If that's what I said. That is what you said. Excuse me, am _I_ on trial here? Silence. Morris, Howard, Ben, explain to the deputy attorney general.
Ms. Van Anka, you told the FBI agents that you went to bed at twelve-thirty. Is this a correct statement? That's what I said. I watched some television and fell asleep, I wake up, the President is dead and now _I'm_ on trial. What is this, _Gaslight_?
Ms. Van Anka, were you and the President on intimate terms?
Intimate terms? Do you mean did he _confide_ in me? Did I confide in him? Did he rely on me for input about the problem of the Middle—
Ms. Van Anka, were you and the President intimate physically? Did you have a sexual relationship?
What kind of question is that?
A direct question, Ms. Van Anka.
Who do you think you are, the _National Perspirer_? That's a grossly invasive question. And you, a woman, asking it. Because I enjoyed a warm relationship with the President, you assume he was interested in my body. It's an insult. Talk to my lawyers. Take your pick, I have three. And I can get more. We have nothing but lawyers in L.A. Next time I'll charter a 747 and fill it with lawyers. Don't think I wouldn't. Money is not an issue with us.
Ms. Van Anka, we have to ask these questions. If you testify at this trial, you will be cross-examined. You'll be cross-examined about the statement you gave to the FBI agents that morning. Moreover, you will be cross-examined by Boyce Baylor, Mrs. MacMann's attorney. You've heard of him? He will ask you these questions and many other questions. Your lawyers here know that.
So, why do I have to testify, anyway? I didn't _see_ her clop him on the head with the spittoon. I'm not a witness. Why do you even need to involve me in this?
Because, Ms. Van Anka, you were a guest in the White House the night the President died. You were one of the last people to see him alive. If we don't put you on the stand, it would be tantamount to saying that we don't believe the testimony that you gave to the agents. And Mr. Baylor will call you as a witness. And if there are inconsistencies, any little holes in your original statement, he will drive trucks through them, Ms. Van Anka. Eighteen-wheelers.
I do not understand. You've got the murder weapon, the Secret Service man heard the shouting, you've got her fingerprints and the dent in his skull from the spittoon. Why do you need me? Do you have any _idea_ what my life has become? I doubt it. Do you know what stress this has caused? This could affect my career. Let me tell you all something: _This could affect the peace process in the Middle East_.
Upon Babette's return from Washington, the city she had once ruled and now loathed, she played the part she had scripted for herself and took to her bed. She'd stormed out of the Justice Department not knowing what they were going to do with her. Another minute there, she couldn't take. Morris, Howard, Ben, we are leaving, _now_. There was this consolation to being a superstar—you knew how to make an exit.
The lawyers discussed among themselves all the way to Los Angeles on Morris's jet while Babette watched her old movies on DVD with the sound down so she could listen to them. None of them came right out and said she was _schtupping_ him, but it was obvious from how they talked that there was little doubt in their minds. So humiliating. Yet it was only a taste, a soupçon, of what lay ahead if she was put on the stand.
The next day came the call from Morris, who'd just gotten off the phone with some deputy prosecutor—how many did they have, for God's sake?—to tell her that yes, they were going to call her to testify. It would look too awkward if they didn't. Don't leave the country. Make yourself available. Don't worry, everything will be fine, just tell the truth.
The truth! In the next hour, Babette ate three pints of Ben & Jerry's Celebrity Ripple ice cream. A week in bed, not answering the phone, watching her movies, pints, quarts, gallons of ice cream. Even her silk pajama pants felt tight in the waist.
She watched opening day. Of the billions of human beings who glued their eyeballs to television sets, few watched more intently than Babette Van Anka. Even in extremis to use the bathroom, she held on, bursting, until Judge Dutch called fifteen-minute recesses. The Clintick woman had, thank God, mentioned her only in passing. And Boyce Baylor—oo, a sharp one and no mistaking, and not so bad looking, either, no wonder Lady Bethmac had a jones for him back in law school—didn't even mention her in his tirade against the U.S. government. What was _that_ about? Well, all for the good. Maybe this wouldn't be about her after all. For a second, Babette almost felt slighted. Then she decided to celebrate with just one more spoonful of Celebrity Ripple. All right, two spoonfuls.
# Chapter 13
Having established the more than adequate credentials of FBI agent Jerry Whepson, DAG Clintick asked him to tell the court what he had observed the morning of September 29.
"There was a great deal of activity on the grounds when myself and Agent Fitch and the members of the FBI crime scene technicians arrived," Agent Whepson began. "The Secret Service especially were in a high state of activity. They were all over the place. A helicopter was up. Dog handlers appeared to be searching the grounds. Uniformed agents, their special response teams, were present in force."
"What would this level of activity suggest to you?"
"That they were looking for someone, or some persons. At this point, all we knew was what we had been told by the Secret Service, that the President had been killed."
"Objection," Boyce said. "Your Honor, it had hardly been established then or even now that the President had been 'killed.' "
This being rather at the root of the whole enchilada, Boyce's objection caused the first sidebar conference—those cozy get-togethers at the bench between attorneys and the judge—of the Trial of the Millennium. Fifteen minutes of furious wrangling over a word. At least it gave the TV commentators time to preen. One remarked that if the trials depicted in the old _Perry Mason_ TV dramas had been presented realistically, the show would still be running because Perry's first case would still be going on. Another remarked that Boyce Baylor would not stop at trying to strike references to the President's having been killed. He would insist that there was no actual proof that the President was even dead. The Trial of the Millennium was off and running, like a garden slug galloping across a wide slate patio on a warm July day.
"Proceed, Ms. Clintick."
"Agent Whepson, would you describe the scene when you reached the second floor, the residence of the White House?"
"Again, a great deal of activity. Secret Service. I recognized some of the President's senior staff. Everyone looked, I would say, grim. There were military personnel, quite senior, I could see from the uniforms. It was very tense up there."
"That would be consistent with activity following a presidential assassination?"
"Ob-jection."
Another fifteen minutes of sidebar. Throughout the nation, Americans began to divide into two camps: those who were reassured about their system of justice, and those who thought it was an argument for the Peruvian model, ten minutes in front of a military judge with a hood over his head, the firing squad outside practicing in the courtyard.
In due course, Agent Whepson was allowed to proceed to the President's bedroom.
"I observed the President in bed. He was lying on his back on one side of the bed. His eyes were open, as was his mouth."
"And this led you to conclude?"
"That he was dead. Dr. Pierce, his personal physician, was present. He confirmed to me that the President had been dead for some hours, possibly three or four."
Boyce demanded that the prosecution stipulate—that is, agree—that Dr. Pierce had no qualifications as a coroner. That required twenty minutes of fevered wrangling at the bench.
"Did you observe anything unusual about the President's appearance?"
"You mean, aside from the fact that he was dead?"
"Yes."
"I observed a mark, like a bruise, approximately five centimeters above his right eyebrow."
"Objection. Your honor, this is the United States of America, not"—Boyce said with some distaste—"France."
Judge Dutch sighed. "Sustained, Mr. Baylor."
"What would that be in good old American inches?" Boyce said, implying that the metric system was unpatriotic.
"About two inches." Agent Whepson placed a finger on his own forehead.
"Did the mark appear suspicious to you in any way?"
"Objection."
Another sidebar. Millions of Americans began to wonder if maybe they would just catch the highlights on the evening news.
"You may answer the question, Agent Whepson."
"Since this mark was on the body of a dead president, yes, I definitely viewed it with great interest."
"What else did you observe that alerted your professional instincts?"
"There was a silver object about this size"—he indicated with his hands—"lying on the floor. I did not recognize what it was. It was lying on its side."
"Did you personally touch this object?"
"No. When I determined—I subsequently directed that the crime scene techs, the technicians, examine it using standard procedure."
"That is, using protective latex gloves?"
"That's correct."
"Was the First Lady present?"
"Yes, she was."
"And where exactly was she?"
"She was in the bathroom."
"Did you observe or hear anything that suggested the reason she was in the bathroom?"
"Yes. The sounds I heard coming from the bathroom were consistent with those of a person who was vomiting."
"Throwing up?" Just in case any of the jurors were unclear as to the meaning of _vomit_.
"Did you speak with Mrs. MacMann?"
"After she ceased vomiting? Yes. I identified myself to her and asked her to tell me what had happened."
"And what did she tell you?"
"She said that she had gone to bed at approximately twelve-thirty. There had been a state dinner for the President of Uruguay. She told me that she was woken up by a noise at some point in the night, she did not know when, and had gone back to sleep. She then woke up at six-fifteen, her usual waking time, ordered breakfast in bed. It was when the maid, Ms. Williams, entered with the breakfast that it became clear that the President had been—was dead."
"Did she describe to you the noise that woke her up in the night?"
"She used the word _thump_."
"Did she tell you that she had investigated the source of this thump?"
"I specifically asked her that, and she replied that she had not investigated it. She said she went back to sleep."
"What was Mrs. MacMann's state of mind when you had this discussion?"
"Objection. The question is entirely subjective. Agent Whepson has no competence as a psychologist."
"Overruled."
"Your _Honor_."
"Mr. Baylor. You may answer the question, Agent Whepson."
"I would describe her as calm."
"Was she tearful?"
"Objection. Leading, Your Honor."
"Sustained."
"How did you find Mrs. MacMann, Agent Whepson?"
"She did not seem upset except for the fact of throwing up."
"Objection. Inference. Mrs. MacMann might have eaten something that did not agree with her."
"Sustained."
"Did she express any emotion to you consistent with that of a woman who had just lost her husband?"
"Objection. Agent Whepson was not there in the capacity of grief counselor. Your Honor, I find the prosecution's line here troubling."
"Sustained."
"Did she say anything to you other than her description of the events of the evening and morning?"
"No, she did not."
"What did you then do?"
"I made a determination that we—that is, the FBI—needed to examine the premises thoroughly."
"Why did you make that determination?"
"To see if there was further evidence of foul play."
This prompted a rip-snorting twenty-minute sidebar in which Boyce and Sandy Clintick could be seen hissing at each other like geese. Judge Dutch's glasses kept fogging. He instructed the jury to disregard Agent Whepson's use of the words _further_ and _foul play_ and adjourned for the day.
The TV commentary that night featured detailed analysis by criminologists, gastroenterologists, and psychologists on the subject of vomiting in general and whether doing it in the presence of law enforcement is a reliable indicator of guilt.
Reporters who had covered Ken MacMann when he was governor discussed the fact that Beth had experienced two difficult pregnancies that ended in miscarriages. Mrs. MacMann had thrown up at a state fair during one of these pregnancies. A video clip of the occasion was shown over and over. She was not otherwise known for having a nervous stomach. If anything, she was known for having an iron constitution. This made Agent Whepson's testimony damning.
Chris Matthews of the show _Hardball_ thundered at a guest who said that Beth was "obviously guilty": "What was she supposed to be doing? Sobbing hysterically like Mary Todd Lincoln and pounding on her husband's chest? People throw up when they're upset. Haven't you ever thrown up when you were upset?"
"I never killed my husband," said the guest.
"I used to throw up before the show. Gotta take a break. You're watching _Hardball_."
Boyce's practice during cross-examinations was to get as physically close to witnesses as he could without sitting on their laps.
Once, in the midst of a particularly withering cross of the owner of a cruise ship that had collided with a tanker carrying liquefied natural gas, Boyce had asked the owner to hold his Styrofoam cup of coffee while he recited to him his conflicting testimony from the court transcript. The man threw the coffee at Boyce's chest, doing himself little good with the jury.
He stood next to the witness box with his right arm resting on it, facing the whole court and the world beyond, smiling pleasantly. The body language said: "And now, my friend, you and I are going to tell these folks what _really_ happened that morning, okay?"
The Jeeter printout from yesterday showed that Agent Whepson had scored high with the four white male jurors, so today's performance would be for their benefit. Boyce's voice took on a slight folksy twang. Today he was just a good ol' boy in a $1,400 suit.
"How many years did you say you were with the Bureau, Agent Whepson?"
"Twenty-three."
"Good for you, sir. In that time I imagine you've seen your fair share of action."
The DAG had told Whepson: Watch out for this guy.
"That would depend on your definition of 'fair share.' "
"Of course it would. Of course it would. Good point. How did you happen to be assigned to this case?"
"I was on duty at the D.C. field office at the time. We received a call from FBI headquarters. They had received a call from the Secret Service."
"Saying what, exactly?"
"Objection. Hearsay."
"Your Honor," said Boyce, "we're all just trying to get at the same thing here, the truth."
There was a distinct murmuring from the press section. Judge Dutch glowered.
Boyce continued, "You arrived at the White House with five other FBI agents. What were you expecting to find? A revolution in progress?"
"Objection."
"Withdrawn. You arrived with five agents _plus_ an entire FBI crime scene search crew of four agents. How did you know a crime had been committed?"
"The Secret Service informed FBI headquarters that the President had been killed."
"And how did they ascertain that? How did they know that he had been killed?"
"You would have to ask them that."
"I believe we will. I believe we will just do that. So you arrived at the White House already having decided that the President had been murdered by armed revolutionaries—"
"Objection."
"Sustained."
"You had not even arrived at the so-called crime scene and you had already ruled out that this might have been an accident?"
"I was instructed to bring a CSS team."
"I see. You were just following orders. That has a familiar ring to it, doesn't it?"
"Objection! Your _Honor_."
"Withdrawn. In your testimony you told the court about your experience with the FBI. You've been involved in cases of kidnapping, wire fraud, extortion, mail bombing, organized crime, hate groups. A very impressive résumé. Tell us, how many presidential assassinations have you personally investigated?"
"This would be my first, Mr. Baylor."
"Well, we all have to start somewhere. But then you claim no special competence at evaluating whether a president has been assassinated or died by some other means?"
"Objection."
"Withdrawn. An FBI agent who charges a First Lady of the United States with murdering her husband would probably get a departmental citation—"
"Objection."
"Move along, Counsel."
"Yes, Your Honor. And I apologize if that statement gave offense. Agent Whepson, in your wide experience, have you often interrogated freshly widowed women? Women who, for example, have just woken up in bed with their husbands dead beside them?"
"I've interviewed the wives of bombing victims—"
"I said interrogated, not interviewed."
"I didn't interrogate Mrs. MacMann."
"What were your first words to her? After identifying yourself?"
"I asked her what happened."
"You did not condole her?"
"Condole her?"
"That's right. Telling her, 'I'm sorry about your loss, ma'am.' Or, 'I know this is a difficult time for you, ma'am.' Those are customary sentiments when dealing with a bereaved family member, if I'm not mistaken. Did you express such a sentiment?"
"Mr. Baylor, the President of the United States was lying there—"
"Just answer the question, thank you."
"No, I thought under the circumstances—"
"Thank you. Did Mrs. MacMann ask to have a lawyer present while you, as you put it, interviewed her?"
"No."
"Did she ask to have a lawyer present on any of the subsequent four occasions when you interviewed her?"
"No."
"At no time did she request to have legal counsel present?"
"No."
"Is that unusual?"
"I don't know, Mr. Baylor. As you yourself said, I have no wide experience of presidential assassinations."
The court laughed. Judge Dutch frowned.
"At what point did you inform Mrs. MacMann that she had the right to have legal counsel present during your so-called _interviews_?"
"The second time. After I interviewed Secret Service agent Birnam, and after the FBI lab reported that the fingerprints on the spittoon were hers."
"And how did she reply to you?"
"I believe she said that she didn't need a lawyer."
"Is that all she said?"
"She said that she was not hiding anything."
Boyce went over and took the spittoon from a clerk. It gleamed brightly in the court lights.
"Let's talk about this so-called murder weapon, this fearsome object at the center of the whole world's attention. In your testimony, you said that when you entered the President's bedroom, you did not recognize what this was. And yet you said it was lying on its side. If you didn't know what it was, how did you know it was on its side?"
"Whatever it was, I could tell it was askew."
"I congratulate you, sir, on your fine ability to identify unfamiliar three-dimensional objects that are ninety degrees off-kilter."
"Objection."
"Withdrawn. But you walked into the room and right away knew that this antique receptacle for spit was a lethal weapon?"
"Objection."
Judge Dutch creaked forward in his chair. This is the source of the aura of judges: they have bigger chairs than anyone else. That and the fact that they can sentence people to sit in electrified ones. It's all about chairs.
"Withdrawn. Let's move on. In your testimony you said that Mrs. MacMann told you she was woken up in the middle of the night by a noise. You expressed surprise that she hadn't investigated the source of this noise, is that correct?"
"Normally when people—especially women—are woken up by something, they want to know what made the noise."
"Especially women? Implying that they are the _weaker_ species?"
"Objection."
"Your Honor, I was merely seeking to clarify the witness's own remark?"
Sidebar.
"In other words, Agent Whepson, Mrs. MacMann should have turned on all the lights, got out of bed, maybe armed herself with a baseball bat to go see if there was a burglar? In the White House."
"Objection."
"Sustained."
"Let me rephrase. You were surprised that Mrs. MacMann, exhausted after entertaining half of Latin America in her house, might have just _assumed_ that someone else would investigate the cause of this bump, living as she did in the most heavily guarded house on the planet?"
Agent Whepson paused just long enough for Boyce to say, "Never mind, never mind. Let's move on." He was good at creating the illusion of impatience with the molasseslike pace of a trial, of a man in a hurry to get at—what did he call it?—the truth.
"You interviewed Ms. Babette Van Anka, the actress, who had spent the night in the Lincoln Bedroom, down the hall, did you not?"
"I did."
"And what did she tell you?"
"That she had said good night to the President at approximately twelve-thirty and went to sleep."
"Went _right_ to sleep?"
Twenty-five hundred miles away, Babette's mouth went dry.
"She told me she had watched television. That she had gone to sleep with the television on."
"Did she tell you that she had heard this thump in the night?"
"She told me that she slept through the night."
"Glad _someone_ got a good night's sleep in the White House that night. Just one or two more questions, Agent Whepson. You've been very patient with me. Unlike Ms. Clintick."
"Objection."
"Come, come, Your Honor, I was only attempting to lift our spirits a little."
"You may lift _my_ spirits by getting on with it, Mr. Baylor."
"Agent Whepson, at one point in your career you were assigned to the Counterintelligence Division of the FBI, is that correct?"
"Yes, I was."
"Tell the court what they do, would you?"
"The Counterintelligence Division keeps track of foreign intelligence agents working within the United States."
"Spies? That would be foreign spies?"
"That's correct."
"You were in a supervisory capacity there, were you not, in the San Francisco field office?"
"Yes, I was."
"Did an Agent Wiley P. Sinclair work under you?"
"Objection."
Even viewers who didn't know who Wiley P. Sinclair was could tell that this was no standard sidebar conference going on. At one point, Sandy Clintick and Boyce raised their voices so that they could be heard above the shusher, the white noise machine that Judge Dutch turned on during sidebars to prevent eavesdropping.
"This is a key one for the defense," a TV network correspondent whispered to his viewers like a golf commentator during a critical nineteen-foot putt. "Baylor _badly_ wants this front and center."
Finally Judge Dutch turned off the shusher. He told the jury that they should not assign any "undue significance" to what they were about to hear.
"Proceed, Mr. Baylor."
"Agent Sinclair worked for you."
"I had twenty-five agents working in that division."
"But did he report to you?"
"Yes, he did."
"And did it turn out that he was selling our secrets to the Chinese government?"
"Yes."
"Hm. That's some counterintelligence operation you had there, Agent Whepson."
"Objection."
"Withdrawn. Did it come as some surprise to you that one of your agents was having a fire sale of our precious national security secrets?"
"It came as a blow to everyone at the Bureau."
"Was the Bureau criticized for lack of diligence in this matter? I understand Mr. Sinclair had been making regular visits to Las Vegas casinos, driving an Italian sports car, going on expensive golf trips."
"There was discussion of that, yes."
"Was anyone at the Bureau fired as a result of this calamity?"
"No."
"Really?"
"Objection. Asked and answered."
"Withdrawn. Did the First Lady, Mrs. MacMann, make any public statements about this affair?"
"I'm not aware of any."
Boyce took a piece of paper off the defense table. It was passed to the bailiff, who passed it to the very sulky-looking DAG, and duly admitted into the record.
"Your Honor, may I beg the court's indulgence and read aloud just a sentence or two from this document?"
Judge Dutch nodded.
"This is from the _Chicago Tribune_ of February twenty-seven of last year. Mrs. MacMann was in Chicago making a speech, and this is a news story about that event. There was a press conference afterwards. She took some questions. Here is what it says: 'Mrs. MacMann said that she was "dismayed" by the recent scandal involving FBI agent Wiley Sinclair. "I think there should be some resignations on principle," she said.' End quote." Boyce handed the piece of paper to Agent Whepson. "You never saw those remarks?"
"I had not seen that specific article."
"I congratulate you on that very lawyerly response, Agent Whepson."
"Objection. Harassing the witness."
"Withdrawn. Were you aware of the remarks from any other source?"
"I would say it was certainly known that Mrs. MacMann had issues with the Bureau with respect to the matter."
"And what was the Bureau's feeling about Mrs. MacMann's 'issues' with it?"
"That she was entitled to her opinion. She was naturally concerned. We all were."
"There was no ill will toward her? No sense of 'Who does she think she is? Why doesn't she butt out?' "
"None that I'm personally aware of, no."
Boyce took back the piece of paper.
"No further questions." His three favorite words in all the law.
# Chapter 14
It was generally conceded, even by those who remained convinced of Beth's guilt, that the government had not had a good day in court.
Boyce's custom after an especially good day was to hold a "press availability" on the steps of the courthouse.
He stepped out into the blinding glare of the lights, the eager smiles of the media, his number one fans and enablers. Even those who hated him loved him.
"It was a good day for the truth," he began.
Throughout America and the world, food sprayed from mouths, TV sets were cursed at, dinner napkins hurled, channels angrily changed.
He kept his statement brief. The Secret Service, he said, had pronounced it a murder before adducing evidence that it was. The FBI, meanwhile, had it in for Beth because she had dared to criticize them for incompetence. To them, she was just a "busybody wife."
The next day, it was reported that the head of the National Organization for Women had written a "scathing" letter to the members of the Senate Oversight Committee, demanding an investigation of the FBI for its "political persecution" of Beth. Several members of the committee bravely announced to the media that they thought this was a darned good idea. The director of the FBI, a dedicated public servant of impeccable reputation, father of three (girls), devoted husband, now found the media waiting for him on his lawn when he came home from work, demanding to know a) why he had not fired the incompetent Agent Whepson for the Sinclair affair; b) why the FBI was a hotbed of misogyny; and, for that matter, c) why he had not fired himself?
Deputy Attorney General Sandy Clintick had watched Boyce on television, thumping his chest like an alpha male gorilla. She decided that she too had better get out there on the courthouse steps and do some spinning of her own. She took a deep breath and sallied forth, head held high. She told them that she was "satisfied" with how it was going so far. Agent Whepson had been a "fine and credible" witness. Furthermore, the Federal Bureau of Investigation was above reproach. There was no vendetta against Mrs. MacMann. The government would present compelling evidence in furtherance of its case. Thank you.
Privately, Ms. Clintick was aboil with fury at the FBI for not having reassigned the case to someone other than Agent Whepson once the enormity of it had become clear. But the fact was that it had been Agent Whepson who had been on duty that morning when the call came, and once he began the investigation, that was that, it was his case. Taking it away from him only would have made the Bureau look even more suspect. Boyce Baylor was shameless, but he was also lucky.
But Beth MacMann had killed her husband with that spittoon and lied about it, and she, Sandy Clintick, was going to get her. Not because she had a grudge. President MacMann might have been the husband from hell and might even have had it coming. She wasn't going to get Beth MacMann for that. She was going to get her because she wanted more than anything to wipe the grin off Shameless Baylor's face and shove it up his ass.
As for Beth, she no longer suspected that Boyce was out to lose the case to punish her for pulling a _Casablanca_ on him back at law school. On the contrary. She was now racked with guilt for what she'd done to him back then. Sitting there in court watching him eviscerate the government's first witness against her had filled her with remorse. She kept thinking of the look on his face when she'd told him she was marrying Ken.
She and Boyce were having a quiet dinner in a private dining room at the Jefferson before Boyce went back to work to prepare tomorrow's cross-examination of Secret Service agent Birnam.
"Boyce, I—"
That was as far as she got before bursting into tears.
"What's the matter? Hey, it's going fine. We're doing fine."
"It's not _that_." She honked into one of the Jefferson's crisply ironed starched napkins. "Oh, Boyce, I'm so _sorry_."
"Beth. It was an accident."
She looked up from her honked-in napkin. "What was?"
Boyce leaned forward and whispered, "You didn't _mean_ to kill him." He sat back. "Anyway, it's not like he was the favorite of all my presidents." He winked.
"What are you talking about?" Beth said, suddenly dry-eyed.
"The number two wife, the Italian, she used to throw things at me all the time. One time she threw this crystal cigar ashtray. From Steuben. Must've weighed five pounds. If she'd connected, I wouldn't be here."
"I didn't _kill_ Ken."
Boyce looked at her. This woman could turn on a dime. He'd had clients like this. The guilt built up until it was overwhelming. They'd burst, then before you could hand them a Kleenex, they were over it, back in denial.
"Whatever." He shrugged.
" 'Whatever'?"
"I'm your lawyer. I'm the last person on the planet you have to explain yourself to."
"I was trying to say... I never told you sorry. For what I did. Back then."
Boyce said quietly, "There's something I never told you."
"Tell me now."
"When you came to my room that day?"
"Yes?"
"I was going to tell you that I was breaking our engagement."
"What?"
"I'd fallen in love."
Beth stared in confusion.
"There was this... guy."
Beth stared. She didn't know what to say.
"He awakened in me something that I didn't know had been there."
"You..."
"He'd been in the navy. He was _so_ butch."
"Dammit, Boyce. I was trying to apologize."
"Then it turned out he was two-timing me. With this ambitious _bitch_."
They kissed. First time in a quarter century. Yet it felt oddly familiar.
"Whoa," Boyce said after what must have been five minutes. Thank God no waiter came in. The headlines! "I have to be in court tomorrow."
Beth sighed. "So do I."
# Chapter 15
Boyce began his cross-examination of Secret Service agent Woodrow "Woody" Birnam not at his usual station right next to the witness, but from a distance. He stood at his podium by the defense table.
"Can you hear me okay, Agent Birnam?"
"Yes, sir." Agent Birnam was in his mid-thirties and befitting his profession was in excellent physical shape.
"You have a superb record with the Secret Service."
Agent Birnam knew better than to accept Boyce's compliment at face value.
"You're one of the Service's top pistol shots, I see."
"We're all competent with firearms. It's a requirement."
"Don't be modest, now. You're on the competition team that's beaten the FBI team three years running. I imagine that must be a sore point with Agent Whepson."
Laughter. Vlonko noted which jurors joined in.
"You must shoot a great deal to maintain such a level of proficiency."
"Objection. Agent Birnam's marksmanship is irrelevant."
"Your Honor, I guarantee the court that my line of questioning is more relevant than the deputy attorney general's ceaseless objections. She's objecting so many times that I'm beginning to worry about her blood pressure."
"Overruled. But proceed to your point, Counsel."
"I was, Your Honor, I was. How often do you go to the pistol range, Agent Birnam?"
"Pretty regularly."
"Would you please define 'regularly' for the court."
"Twice a week."
"Good for you. Practice makes perfect. I see you fire a .357 magnum six-shot revolver, is that correct?"
"Yes. I also shoot nine-millimeter and occasionally .44 magnum. Also .38 caliber on occasion."
"All these handguns, especially a .357 magnum, these are powerful guns, are they not?"
"They're not small guns."
"A .357 magnum produces one hundred and sixty-five decibels. Am I correct?"
"I wear ear protection."
"I would hope so. That's a heck of a loud sound. Have you worn ear protection every single time you have fired a handgun, Agent Birnam?"
"Objection."
"Your Honor, I am _getting_ to my point, if Ms. Clintick will permit."
"It would be hard to say," Agent Birnam said.
"Try, for the court."
"Majority of the time, certainly. Yes."
"Have you ever experienced ringing in your ears, loss of hearing?"
"Objection. Your Honor, this is not a doctor's office. Agent Birnam is perfectly fit. He's passed all his physical tests. This is pointless and harassing."
Judge Dutch rocked twice in his great chair. "Overruled. Answer the question, Agent."
"Nothing significant."
Boyce lowered his voice. "No loss of hearing?"
"I'm sorry?"
Even lower: "No loss of hearing?"
"Could you repeat the question?"
Boyce raised his voice to a near shout: _"Have you had loss of hearing?"_
"No. Never."
"Hm. Over the course of your lifetime, how many rounds would you say you have fired from the barrel of a handgun?"
"That would be difficult to say."
"Try. Thousands?"
"More."
"Tens of thousands?"
"At least."
"Hundreds?"
"I—"
"A _million_?"
"I don't have an answer to that. A lot."
"Isn't it true, Agent Birnam, that you can sustain significant and lasting ear damage from exposure to a single gunshot?"
"Objection. The witness is not an otolaryngologist."
Clintick was furious. She'd been skunked and she knew it. Boyce had purposefully not filed any pretrial motions having to do with Agent Birnam's ability to hear. He hadn't even told Beth. He hadn't told Beth most of his strategy, for the reason that, knowing Beth, he didn't want to spend half the time arguing with her about how he planned to win this.
"I withdraw the question, Your Honor. Agent Birnam, you were on duty the night of the President's passing—"
"Objection."
"Your Honor, surely this is harassment."
"Overruled."
"Your post was outside the closed door to the second-floor residence, at the head of the grand staircase. According to this chart here"—Boyce pointed to the blowup of the floor plan of the residence—"that would have put you some seventy-five, eighty feet away from the closed door to the President's bedroom—"
"Objection. The court has heard no testimony stating that the door to the President's bedroom was closed."
Sidebar.
"Agent Birnam, leaving aside for the time being whether the door to the President's bedroom was closed, you would have been eighty feet away, on the other side of a door that _you_ have said was closed. And yet you claim—"
"Objection."
"Sustained."
"You _state_ unequivocally that you heard an argument going on, so far away it might as well have been in another time zone."
"Objection. Your _Honor_."
"Mr. Baylor, I'm warning you."
"I withdraw the figure of speech, Your Honor. Sorry. Force of habit. Agent Birnam, you say you heard this tremendous hullabaloo from nearly a hundred feet away. All the way at the other end of the residence. And what did you hear?"
"The President and Mrs. MacMann. They appeared to be arguing."
"Over domestic or foreign policy?"
Laughter.
"Objection."
Judge Dutch picked up his gavel and aimed the tip of it at Boyce. "That is your last warning, Counselor."
"I ask the court's forgiveness."
Boyce walked toward Agent Birnam. He said in a sincere tone of voice, "Are you certain that it was the President and Mrs. MacMann that you heard?"
"Well, yes."
"How many people were there that night in the residence?"
"Three, counting Ms. Van Anka, the guest."
Boyce paused. He nodded, walked over toward the jury box as if deep in thought. A hush descended on the courtroom. Members of the press nudged each other. _Here we go_. On the other side of the country, Babette Van Anka cowered under her expensive French sheets.
"Let's move on to another area, Agent Birnam. A year ago, the First Lady was quoted in the media to the effect that she felt there were not enough female agents in the Secret Service...."
The evening news was loud with the sound of .357 magnums being fired and with video footage of Boyce saying to Agent Birnam, "Agent Birnam, with all the money and tremendous effort that the Secret Service devotes to keeping American presidents alive, why couldn't you have spent ten dollars on a decent adhesive bathmat for the President's bathroom?" followed by Ms. Clintick's spluttering objection.
On _Hard Gavel_ that night, Alan Crudman was drowning in false modesty.
"Perri, it's not like me to second-guess an attorney of Boyce Baylor's stature. But I have to say, I was amazed that he just dropped the Babette Van Anka angle today. He set up the shot and then just walked away from it. Babette Van Anka is the _key_ to defending Beth MacMann. To try to assert that this agent had it in for the First Lady because she'd criticized the Secret Service for not having enough women agents—that just strikes me as throwing _very_ long. Look, it's not a secret that the President and Van Anka were—whatever word you want to use to describe them, intimate, best of friends, constant companions. There she was, on the premises the night of the President's death, in the next bedroom down the hall. I do not know why Boyce Baylor isn't making more of this fact."
"Couldn't it be," Perri said, "that if he does make a big deal out of the fact that the President and Van Anka were lovers, that gives Beth MacMann a motive for killing him?"
"Of course it does, but that's _precisely_ why a jury like this"—Alan Crudman, defender of J. J. Bronco and other notably guilty defendants of color whom he had gotten off by imputing racist motives to everyone else involved, was always careful to avoid saying "predominantly black jury" when he meant to imply that predominantly black juries had entirely separate agendas and could always be counted on to acquit for tribal reasons—"would respond sympathetically to Beth MacMann."
"Even if she had lied to cover up?"
Crudman shrugged. "Juries like this one live in the real world. Lying to law enforcement officers is just not the worst thing you could do. Plus on this jury you've got middle-aged women who would be predisposed to think that any philandering man who was cheating on his wife in the next bedroom would deserve anything he got. This is a low-hanging fruit. I _don't understand_ why Boyce Baylor doesn't want to pick it. Every time Van Anka's name comes up, he wants to move on. You'd think _he_ was the one having an affair with her."
Beth had promised herself that she was not going to watch the evening shows. But with her feelings toward Boyce now rekindled and glowing like fanned embers, she found herself turning on _Hard Gavel_ , not for the commentating and second-guessing, but to see what her competition was wearing that night. She had developed a little paranoid theory that the closer she and Boyce got, the tighter Perri's sweaters got.
It was in the midst of checking out Perri's attire that Beth couldn't help but hear Alan Crudman going on about the free pass Boyce was giving Babette Van Anka.
Beth reproached herself for her doubts. Had a lifetime in politics made her this cynical? Or was it just a lifetime with Ken MacMann?
The first infidelity had been with one of her bridesmaids. At least he hadn't dragged her upstairs during the reception, like Sonny Corleone, and thrown her up against the wainscoting. But finding out that your husband has been having it off with one of your bridesmaids, into the bargain, an old friend from boarding school, would put a dent in any new wife's confidence. She even considered leaving Ken. And she was overwhelmed with guilt over how she'd treated Boyce. From what she heard, Boyce had taken it hard. Friends said he was going through some kind of personality change, from nice guy to quiet angry guy. One said, "I hope he doesn't end up one of those people who send mail bombs."
Ken apologized for having an affair with one of her oldest friends. He blamed it on something called post-traumatic stress disorder, the name they were starting to give to Vietnam vets who were acting wiggy. He promised not to do it again. And he was as good as his word, for almost two months. Meanwhile, he appeared to have lost interest in his new wife physically, which is demoralizing six months into a lifetime partnership. And Beth liked sex. She liked sex a _lot_. She started to fantasize about Boyce. It was all so conflicting.
Meanwhile, Ken had made it clear to her that he had a plan, and not just to screw all of her old school friends. He was going to be president, and he was going to do it quickly, whatever it took, so that he could enjoy the experience while he was still young. She wouldn't be seeing much of him.
Friends remarked how changed she was. Beth didn't laugh much anymore.
It was now after 10:30. She knew Boyce went to sleep early. She waited until 10:40. Couldn't help herself.
The voice that answered was already livid at being woken up.
"What _is_ it, Beth?"
"I was just listening to"—God, how was this going to sound?—"I was just thinking. I think we need to rethink the Babette Van Anka thing."
"Do you realize," Boyce growled, "how many federal agents are listening in on this conversation? Why don't you just call up the Justice Department and tell them how you think I should defend you?"
"I'm sorry."
"So you keep saying."
"Well, it's _true_. Most men like it when you apologize to them."
"Apologize in the morning, when the G-men aren't listening in. Good night. Good night, boys. Sleep tight, you incompetent bastards."
He hung up.
Beth had to get up out of bed and pace and smoke. She'd started again, a fact that had somehow found its way into the press. To hell with it. Pacing at night without smoking was like a drum majorette parading without a baton.
_Was_ he trying to throw the case? She was attractive, this Perri Pettengill. Why _not_ go after Babette Van Anka? What _did_ Ken see in her? Never mind. And the husband. Nine miles of bad road. He gave millions to Ken's campaigns—some of it probably in bags—and those Far East associates of his, real charmers. Turned out one of them, for whom Max had wheedled an invitation to a state dinner at the White House, was connected to the Burmese general who protected the poppy business. What interesting friends. To counter the unfavorable publicity, Ken had decided to make campaign finance reform a central theme of his reelection campaign. As the wise man said, You can fool some of the people some of the time, and those are the ones you need to concentrate on.
"Why not go after Babette Van Anka?" Boyce asked rhetorically over breakfast. He was the temperature of his coffee. Hotter, probably. "You were watching her show again."
"I couldn't sleep."
"Well, watch something else. Something wholesome. Like wrestling. Or one of those reality shows where they chain people together for a week to see if they eat each other."
"They should chain lawyers together."
"That I might watch. Everyone hates lawyers—until you need one."
"Then you really hate them."
Boyce grinned. "Two points."
"Look, I'm not trying to interfere."
"Yes, you are. No wonder you were an unpopular First Lady."
"I wasn't unpopular. I was a transitional First Lady. I was plowing the ground for the ones who'll follow."
"I'd be careful with the ground metaphors. Your husband's fertilizing the lawn at Arlington."
"Charming. Really charming." Beth sipped her coffee. "I didn't sleep."
"I didn't either. My client kept calling me."
"You won't even discuss it with me?"
"All right." Boyce dabbed his mouth with the napkin. "This once I'll discuss it with you. If I go after Babette Van Anka, which I very easily could, it would be shooting fish in a barrel. I could make her look like the whore of Babylon in sixty seconds. And what would that accomplish? One thing. It would give you a motive for killing War God. The jury would reasonably conclude, Sure, she was pissed at him for humping the girlfriend right down the hall. But killing is killing. She coulda waited till morning and divorced him. Only a scheming little putz like Alan Crudman would do something so _obvious_ as go after Babette Van Anka. No. I take that back. He's not dumb. He's a putz, but he's a smart little shit. He knows perfectly well that's not the key here. He's just saying it on Perri's show because the majority of the people who watch know fuck-all about trial strategies, and he can get away with saying things like that and look smart on my time. You know something else? For all I know, he may be saying that _precisely_ to make you think I'm blowing this case. Did that enter your mind? So any more _questions_?"
"No further questions."
"All right. I'm glad we had this talk. Let's go kick some ass."
# Chapter 16
Sophie Williams, the upstairs maid who had pointed out to Beth that her husband was unlikely to be wanting breakfast that morning or indeed any future mornings, had been the subject of furious pretrial motions.
Two months after the fateful day, she sold her story to the _National Perspirer_. It filled four pages inside. The headline on the cover was HE LOOKED PRETTY DEAD TO ME! The subhead promised "Gripping Details of the Stormy MacMann Marriage!"
And delivered them. The _Perspirer_ paid $250,000, roughly six times her annual salary. In return, they got a heaping plateful of red meat. Sophie's account was so entertaining that movie producers all over Hollywood and Europe rushed to snap up the rights. It was full of delicious details about Babette's numerous overnight stays in the Lincoln Bedroom. "I sometimes wondered what Mr. Lincoln would have made of all that bouncing and moaning." One night while Babette and the commander in chief were enjoying a soiree of headboard-banging bliss, "Mrs. MacMann was off giving a speech to some family organization about how important marriages are. I felt kind of sorry for her." The First Guest was not, apparently, a favorite of the White House staff. "And I never did care much for Ms. Van Anka's singing or acting."
The so-called mainstream media who publicly affected to disdain the _Perspirer_ as a scandal sheet for overweight proles found it all so riveting that they recycled it on their own more respectable front pages, along with the usual grudging attribution and qualifiers.
Babette had to be talked out of suing the _Perspirer_ by Nick Naylor and her trio of lawyers. Don't go there, they said. It will only make it worse. Instead she demanded that Max buy the movie rights through one of his dummy Hong Kong companies, in order to ensure that no movie would ever be made. Sophie's new best friend, her agent, had gotten the bidding up to a price that was giving Max indigestion. When it passed $5 million, Max announced that he would not go one dollar, pound sterling, Swiss franc, euro, or yen higher. "Who's gonna want to see this movie, anyway?" This highly insensitive comment triggered a domestic scene of exquisite fury.
The movie rights ended up going for $7.4 million, to a French company and English director, who, according to _Variety_ , was planning to ask Babette to play herself. "I see it," he was quoted, "as the television show _West Wing_ meets _Murder on the Orient Express_. Washington camp, lots of big stars having tremendous fun. Babette would be marvelous as herself. Who better? I've always wanted to work with her."
Babette menaced Max with the salad tongs, accusing him of, among other failings, insufficient loyalty for not having outbid the French. Nick Naylor, presented with yet another Grab–Van Anka public relations catastrophe, said that at this point Babette might consider accepting the part, to "defuse things." Though certainly not until after the trial. "Naturally, we'd want to see a script first."
At any rate, Sophie Williams was now rich and retired from service in the White House. Sometimes the American dream, like God, works in mysterious ways.
The prosecution wanted Sophie to testify, especially since her account of Babette's previous visits to the White House opened up the can of night crawlers that was the MacMann marriage and established a clear motive for Beth's jealous fatal spittooning of the President.
Boyce fought against having her testify, arguing that she had discredited herself by selling her story to a "lurid, sensation-mongering tabloid" (that, as it happened, he had once defended in a libel case, but never mind). He thus found himself in the position of defending a wronged wife by attempting to suppress evidence of her husband's offenses. If Sophie testified, he'd have to retaliate by making her out as a scheming mercenary who had embellished the truth in order to increase its market value. As Sophie was black, he was not eager to perform a credibilotomy on her in front of a jury consisting of seven African Americans.
So the atmosphere in court this morning was especially charged.
"The United States calls Sophronia Williams."
"Why," Beth asked in a tight voice, in the car at the end of what had been a very long day in court, "did you keep asking her why I was so unpopular among the staff?"
"I'm trying to establish a conspiracy. Conspiracies need motives."
"You were trying to make me out to be Leona Helmsley. Is this some new cutting-edge defense strategy? Make the jury hate the defendant?"
"We've established that the FBI hated you. When I finish with Agent Birnam, it'll look like he and the entire Secret Service hated you. With Sophie, I was trying to suggest that the staff might have had it in for you, too."
"Suggest? You asked her if I knew the birthdays of her four kids. Do you know the names and birthdays of _your_ cleaning lady's kids? And what was all that how much did I spend on Christmas presents for them? And the fifteen minutes you spent on her brother-in-law who's in jail for stabbing someone? I take it you were trying to suggest that I was cold and unfeeling because I didn't try to get him a presidential pardon for stabbing a convenience store clerk. I didn't even know that she had a brother-in-law in jail. It wasn't something she advertised."
"I think we made good progress today."
"You know, Sophie liked me. She all but said so in the _Perspirer_ piece. The part about how she felt sorry for me that he was screwing that cow while I was giving a speech to the Promise Keepers about the sanctity of marriage. Why didn't you ask her about that?"
"Gee, why didn't _I_ think of that? Then we could have gone on to your motive for killing your husband." Boyce sighed, a deep, lawyerly sigh. "Let me explain this one last time. I want it to look like the entire White House was in league against you. That they went to bed at night dreaming of ways to get even with you."
"What if I'm acquitted, and everyone in the country hates me because I'm supposedly a bitch?"
"You'll get an eight-million-dollar book advance from Tina Brown. Then you can tell everyone how wonderful you really are. Americans love comebacks. It'll be a best-seller. Then you can pay my fee, which is going to be at least eight million."
"Alan Crudman would have been cheaper."
That night Beth smoked half a pack of cigarettes and watched not only _Hard Gavel_ but all the shows. The topic on all of them was "How Awful _Is_ Beth MacMann?" The _Vanity Fair_ correspondent was funny on _Charlie Rose_. "It's a wonder," he said, "we're not all covering a trial for _her_ murder." The other guests laughed. Beth's dreams of someday having a political career of her own, running for the Senate or state house back home—so much for all that. Boyce might be winning the case, but he was ruining her reputation in the process.
The phone rang.
"I don't think you're _that_ bad." It was Boyce.
"Fuck you." She hung up.
He called back. "Was that your idea of phone sex?"
"Tell me you have a strategy," she said. "Tell me this is going the way you planned it."
"Baby, this is going so well, I'm sitting here doing a crossword puzzle. By the way, what's a four-letter word for woman ending in u-n-t?"
"Aunt."
"Hm. Can I borrow your eraser?"
# Chapter 17
Boyce did have a plan, and it centered on a single line from Babette's statement to the FBI. He had been careful not to file any pretrial motions having to do with it and not to put on his witness list anyone who might alert the prosecution to this little buried truffle.
Meanwhile, the United States called Captain Cary Grayson. Grayson was the U.S. Navy's top pathologist at Bethesda Naval Hospital, just outside Washington. It was he who had performed the autopsy on President MacMann and who had concluded that the President had died of an epidural hematoma caused by, in the dry, precise language of forensic medicine, blunt-force trauma to the skull. The blow to his forehead had ruptured the middle meningeal artery. Blood had collected between the skull and the dura, the membrane between the skull and the brain, forcing the dura inward. This in turn compressed the cerebral cortex, killing the President.
Captain Grayson was in his late fifties, a trim, graying, bespectacled navy bones of pleasant, professional demeanor, the sort of man you would be thrilled to find at the foot of your hospital bed, checking your chart and issuing crisp commands to the nurses. Into the bargain, he had an impressive chestful of ribbons, including the bright yellow-and-red rectangle denoting service in Vietnam (pharmacist's mate on the aircraft carrier _Independence_ ). This was, of course, irrelevant to his testimony, but the deputy attorney general, keenly aware that the jury contained two veterans of the Vietnam War, slyly managed to insert a glancing mention of it in her direct examination of Captain Grayson. There was no point in objecting. Boyce could see that all the jurors, not just the vets, were already in love with him. They wanted him at the foot of _their_ hospital beds when the time came. Boyce knew from experience that juries tended to love doctors, except for plastic surgeons with practices in Beverly Hills and the 10021 zip code area of Manhattan. Doctors with military decorations—two credibilities in one uniform—they deemed godlike.
Deputy AG Clintick gently walked him through his testimony, reinforcing the fact that the time of death had occurred in the hour and a half following the overheard shouting match. She got him to comment that the President was in fine physical condition, extraordinary, considering his grueling ordeal during the war. Grayson had observed some evidence of coronary heart disease, but nothing serious. Perhaps in a few years, during his second term—objection, sustained—perhaps in four or five years he would have required angioplasty, or the insertion of a stent, but these were now routine procedures. He would have lived to a ripe age. A great loss. A great loss for the nation. Boyce was bursting to object, but you could have heard a pin drop, so he held back.
And then Clintick ambushed Boyce. According to her witness list, the enlarged photographs of the Paul Revere mark on the President's skull were to have been presented by a leading civilian forensic dermatologist. But seeing how mesmerized the jury was by Captain Grayson, the navy's answer to Marcus Welby, M.D., she pulled a fast one and asked that the photographs be introduced so that Grayson could explain them.
During the ensuing sidebar, Boyce protested that Captain Grayson, for all his wonderfulness, was not qualified to discuss epidermal markings that were, as he put it, "more mysterious in origin than the designs supposedly left on the Andes mountains by ancient flying saucers."
A seething Boyce and a placid deputy AG returned to their places. A clerk mounted the three-by-two-foot photographs on an easel. It was the first time they had been shown to the public.
A gasp went through the courtroom. Boyce winced. There was nothing worse for your client than the sound of an entire courtroom having its breath taken away by graphic evidence of your client's alleged handiwork.
The photograph was an enlargement of a five-by-two-centimeter rectangle of presidential forehead.
The letters were so clear that they could have been used as an eye chart. _The New Yorker_ observed, "They could have been read by Stevie Wonder."
When Boyce had first seen the depressing photographs, he had briefly contemplated an _Exorcist_ explanation: that the conspiracy against Beth included even Satan, who had malevolently stamped Paul Revere's mark on the President's skull.
He took a deep breath. Some days you earned that thousand bucks an hour.
Members of the media exchanged smirks. Let Shameless Baylor explain away _this!_
DAG Clintick, at pains to suppress her glee, gently guided Captain Grayson through his description of this damningly conclusive piece of forensic evidence. She had prepared diligently. Did such dermal embossing ever occur naturally on humans? Not like this, Captain Grayson replied. In his wide medical experience, had the captain ever seen or heard of six capital letters naturally occurring on human flesh? No, he couldn't think of any instances offhand. And what would the chances of six reversed Roman letters spelling the name of a Revolutionary War silversmith appearing naturally on human flesh be, say?
Objection. The witness was being asked to indulge in the most extravagant statistical speculation.
Sidebar.
Overruled.
The witness may answer the question.
"Approximately one in fifty-seven billion."
Murmurmurmur.
Objection.
It was a long morning.
When she finished with Captain Grayson, Sandy Clintick flashed Boyce a triumphal smirk.
Boyce rose, walked over to the witness box, and rested his arm companionably on it, as though having a conversation with an old friend.
"Your Honor," he said so casually that he might have been telling the judge that his wife had called and asked him to pick up the dry cleaning on the way home, "the defense stipulates that this mark came from the spittoon."
Sandy Clintick froze. She and scores of Justice Department lawyers had studied every one of Boyce Baylor's cases. In over two decades of aggressive lawyering, he had stipulated exactly twice. Even Judge Dutch, normally as impassive as one of the seventeenth-century Dutch burghers in his collection, raised an eyebrow.
"Only a fool or scoundrel," Boyce continued, "would stand here and waste the jury's time trying to maintain that those letters occurred naturally—or miraculously—on President MacMann's forehead." He was making a speech, but let the deputy AG object while he sounded so sweetly reasonable. "The question is _how_ they got there."
"Objection."
"Proceed with your questioning, Counsel."
First Boyce would demonstrate to the jury that he too thought Captain Grayson was the greatest thing to happen to the medical profession since Galen.
He inquired gently into the number of autopsies he had performed. My, my, that _is_ a lot. He then dismissed the notion that he thought Captain Grayson was not qualified to comment on the markings simply because he was not a forensic dermatologist. For heaven's sake, was he not the navy's top pathologist? At Bethesda Naval Hospital—custodians of presidential health?
Captain Grayson could only reply as modestly as he could that, yes, his credentials did seem to be in order.
Absolutely, absolutely.
Sandy Clintick thought, _They're having a love fest up there. Any minute he's going to give Grayson a back massage. What the hell is he up to_?
"Captain, in the course of your autopsy, I assume you would examine every square inch of the President's body?"
"Yes."
"Every part?"
"Objection. Asked and answered."
"Did you take tissue samples?"
"Yes."
"Did you take tissue samples from the bottom of his feet?"
"No."
Boyce's face assumed a look of respectful surprise. "I'd have thought that would be routine."
"Objection."
"I'll rephrase. Why didn't you take tissue samples from the soles of his feet?"
"I saw no reason to. I observed the soles of his feet."
"In observing them, did you see traces of soap residue?"
"I did not observe traces of soap residue."
"But without taking tissue samples, can you say, with one hundred percent certainty, that there was no soap residue present?"
Captain Grayson paused. "No, I could not say for certain."
Boyce did not press. Digging up a dead-and-buried U.S. president in Arlington Cemetery is a tall order, and high risk unless you were absolutely certain that his feet were slimy with soap from his last shower.
Just in case, though, DAG Clintick was ready. She had lined up a scientist from Procter & Gamble, makers of the President's favorite shower soap, as well as a physicist from the Jet Propulsion Laboratory. The P&G scientist would testify that the President's bar soap had been specifically designed to leave minimal residue—the result of an expensive lawsuit by a man who had sued the company for $100 million after slipping in the shower. The JPL physicist would demonstrate the unlikelihood—1 in 2.6 trillion—of a slip resulting in the imprint on the skull, given the spittoon's normal position in the President's bedroom. Sandy Clintick was not eager to glaze the jury's eyeballs with vectors, g-forces, square roots, and formulas with eight variables, but if it came to that, she was ready.
"Captain," Boyce said, "what effect does death have on human flesh?"
"When rigor mortis sets in, it hardens all the organs and tissue."
"Is it then accurate to say that human skin becomes more impressionable following death? In other words, if something were pressed onto it in the hours _after_ death, would an impression be made on the skin?"
"Objection. Captain Grayson is not here in the capacity of expert dermatologist."
Gotcha. Right into the trap.
"Your Honor," Boyce said in a tone of wounded reasonableness, "it was the _prosecution_ who presented the captain as expert witness in these matters. I respect and esteem his credentials and accept him as more than qualified to testify on this most basic aspect of forensic medical science."
Several of the jurors actually nodded in agreement. Such moments made Vlonko's heart leap.
"Overruled. You may answer the question, Captain Grayson."
"As rigor mortis sets in, the skin develops pallor consequent to oxygen depletion. Relevant to Mr. Baylor's question, there is a general loss of turgor and elasticity. So, yes, in theory, any indentation or impression would be more likely to persist after death than before."
Bingo.
"So, Captain, if after the President had died, the spittoon were pressed down onto his forehead—"
"Objection!"
"Your Honor, I am merely asking a medical witness if a certain set of circumstances would produce a specific result." He flashed a steely glance at the prosecution table. "And I would point out to the prosecution that I have reserved the right to recall _previous_ witnesses."
It was a long and hissy sidebar. How the public yearned to hear what was being said!
Finally Judge Dutch said, "The jury is instructed that the witness is not in a position to state what may or may not have actually taken place."
"Captain Grayson," Boyce continued, "if following the President's death, the spittoon were pressed down onto a bruised area, could that, in your considered medical opinion, have left this vivid impression of Mr. Revere's mark? Like stamping hardening clay?"
"Yes," Captain Grayson said, "that could explain it. Theoretically."
Boyce smiled. "Thank you, sir. No further questions at this time."
# Chapter 18
Boyce's cross of Captain Grayson triggered a media hurricane. The front page of the _New York Post_ screamed:
SHAMELESS TO GOVT:
_YOU_ DUNIT!
The pixel pundits said that Boyce had just "declared war" on the United States government. Mark Fuhrman, the detective in the O. J. Simpson case, was tracked down by helicopter, shooting elk in Montana, and ferried to a television studio in Bozeman where he could comment on the similarities between the accusation that he had planted the bloody glove in Simpson's backyard and Boyce's allegation that the government bogeymen had embossed the President's skull while he lay cooling.
In fact, Boyce had gotten the inspiration for this grand canard not from the Simpson case, but from the JFK assassination. The harder-element conspiracy theorists back then had asserted that President Kennedy's corpse had been altered in the ambulance on the way from Andrews Air Force Base to Bethesda Naval Hospital, to make it look as though the fatal shots had come from behind. (When any fool _knew_ that President Kennedy had been shot from the front, back, and sideways by triangulated firing squads staged by the CIA, FBI, and Mafia.) Boyce kept the source of his inspiration to himself.
Commentators on the legal shows _tsk-tsked_ that DAG Clintick should have seen it coming. Greta Van Botox, host of _Objection!_ , declared, "If you get into the ring with Shameless Baylor, you better be prepared to mud wrestle." Alan Crudman, in his now regular guest slot on _Hard Gavel_ , tried to downplay the damage to the government. Privately, he admitted that Boyce Baylor had taken a damaging piece of evidence and nicely jujitsued it to his advantage.
DAG Clintick called an unusual full-scale press conference to denounce Boyce. "In all my years as a prosecutor," she said, steam hissing from her ears, "I have never witnessed such morally and ethically disgraceful tactics as these. Today Mr. Baylor truly earned his nickname."
With that she turned and walked back into the courthouse, into the basement garage, and into her car and drove back to her office at the Justice Department to command 120 lawyers and investigators to crank up, overnight, a minute-by-minute, second-by-second timeline of the precise whereabouts of every FBI and Secret Service agent, household staff, medical personnel, and body handler on the morning of the President's death.
She knew there was no merit to Boyce's stunning allegation, and that was why it scared her. It was so outrageous, so unbelievable, that one-third of the jury would believe it. People believe unbelievable things because it's self-flattering to think that you are intellectually daring enough to accept what others find preposterous. It's why people believe in UFOs, assassination conspiracies, certain religions, and the possibility that the Boston Red Sox will someday win the World Series.
It had been a dizzying couple of weeks. The DAG called in a highly credentialed plastic surgeon who stated flatly that Boyce's allegation was not medically possible.
Under cross-examination, Boyce removed the plastic surgeon's own skin. The doctor had done a lot of work for the government—transforming the faces of turncoat mobsters and defector spies who needed new identities. By the time Boyce had finished enumerating the crimes of his patients, the doctor made Beverly Hills plastic surgeons seem saintly.
The deputy AG produced more plastic surgeons, dermatologists, skin allergists, and medical examiners to contradict Captain Grayson's assertion. Boyce destroyed them one by one. He made their testimony sound so abstruse, so ambiguous, that no one without a Ph.D. in forensic dermatology could have made sense of it. Boyce's first law of deconstructing scientific evidence was: Make it boring and make it annoying. Vlonko reported that four jurors were now grinding their teeth whenever Clintick stood up to grill another skin doctor. This was progress.
It was chess. Sandy had made a bad move by having Grayson testify about the mark, and Boyce, having seized the advantage, was pressing it, forcing her to move defensively. The media were meanwhile consumed with trying to figure out whom exactly Boyce was planning to finger as the person who'd stamped Paul Revere's name on the President's noggin.
Beth was increasingly consumed with something else.
"How was your weekend with Perri?" she asked.
"Hm?" Boyce was deep in thought. Today would be an important day in court. Babette Van Anka was taking the stand. Anticipation was high.
What was Beth saying? "Fine. Fine," he said.
"What's she like?"
"Two arms, two legs. Mammal."
"She seems bright enough. For someone who showcases her boobs."
Boyce quietly reveled in Beth's jealousy. What's more, his weekend with Perri in New York had been spent fending off oddly similar questions about Beth. In the midst of a delicious session of welcome-home fellatio, Perri had looked up in midministration and asked, "What's she really like? She seems sympathetic, for a cold-blooded murderer." It took Boyce half an hour to get back to where he had been.
"She speaks very well of you," Boyce said to Beth. "She called you 'sympathetic.' "
"Why don't you ever talk about her?"
"What do you want to know about her?"
"Are you in love with her?"
"Objection. Leading."
"I'll rephrase. How would you describe your feelings for Ms. Pettengill?"
"Amicable, certainly."
"Amorous?"
"That would depend on your definition of amorous."
"The witness is directed to answer the question." Their knees were touching.
The motorcade was pulling up in front of the courthouse.
"Court is adjourned."
"I reserve the right to recall the witness at a future time."
Nick Naylor nixed going in through the basement entrance. If Beth MacMann was going in the front door, so would Babette.
Babette was not entirely enthusiastic about the idea. Her notion of arriving somewhere consisted of pulling up in a white limousine longer than most aircraft carriers, stepping out onto a red carpet escorted by twelve steroidal immensities with ear radios, then pausing along the way in to be told by a breathless interviewer from _Entertainment Tonight_ that she looked "incredible!" and asked whether she was "excited" about her new movie or "really excited."
Shimmying her way through a media gauntlet mob baying at her with questions—"Babette! Babette! How many times did you and Ken do it that night?"—was not her preferred entrance.
Nick had arranged for bodyguards who bore at least a passing resemblance to the human species. He had also quietly arranged for twenty or so (paid) "supporters" to be on hand, waving signs expressing cheery sentiments like WE LOVE YOU, BABETTE! and YOU GO, GIRL! Some specifically hailed her commitment to peace in the Middle East. One even said, SUICIDE BOMBERS FOR BABETTE! All very welcoming and reassuring. When she took the stand, she would be basking in the warm reflected glow of fan love.
For a woman who spent enough on clothes in a year to dress the population of Liechtenstein, Babette's appearance was decidedly minimalist. She wore—what else?—a black pantsuit, the uniform of serious modern women. Her copious bosom—nearly unique among Hollywood breasts for being actually real—was not in evidence. She looked almost flat chested.
But it was the eyeglasses that prompted the most tittering among the press. They were half-glasses, the kind people wear to government hearings so that they look up from incriminating documents and say, "With all due respect, Senator, I draw quite a _different_ inference from that." Intellectual accessorizing. Babette's glasses said, "I ruined my eyes on the footnotes of _Foreign Affairs_ magazine."
Boyce studied the jury. Many of them were wide-eyed over having a certified Hollywood celebrity in their midst. If the pattern held, it would soon dawn on them that this Hollywood bigshot was here to impress _them_. Then they'd relax almost to the point of cockiness. It would be their one chance in life to have the upper hand with a celebrity.
"What is your profession, Ms. Van Anka?" the deputy attorney general began.
"I'm an activist in international affairs."
The correspondent from _Vanity Fair_ was seen pressing a balled handkerchief to his mouth.
Boyce nudged Beth beneath the table.
"What _else_ do you do, Ms. Van Anka?" the deputy AG asked.
"I am, in addition, an actor, singer, and recording artist. But at this point in my life I consider myself primarily an advocate. For peace in the Middle East. And the environment."
A collective tubercular sound came from the press section. Eyes rolled like tumblers in slot machines. Judge Dutch glanced sternly.
"You were an overnight guest in the White House on the night of September twenty-eighth and twenty-ninth of last year?"
"Yes."
Babette's attorneys, Morris, Howard, and Ben, had drilled into her the absolute necessity of answering _as briefly as possible_. They were anxious about their client's predilection for lengthy—indeed, interminable—statements. Taciturnity is as rare in Hollywood as fur.
"Had you been an overnight guest in the White House on previous occasions?"
"Yes."
Two one-word answers in a row. Observers marveled. Had Morris, Howard, and Ben installed a shock collar on her? "How many times?"
"The First Family and I were friends. I stayed there numerous occasions."
Beth scribbled on her legal pad and slid it over to Boyce: "Friend of First Family?"
"How many times, Ms. Van Anka?"
"I couldn't say."
"Couldn't say or won't say?"
"Objection."
"Sustained."
"Ms. Van Anka, this is the official log kept by the White House usher of overnight guests. It shows that you stayed overnight in the White House a total of fifty-six times."
"If that's what it says. I'm honored."
"Would you say that fifty-six times is a lot?"
"That depends on your definition of 'a lot.' "
"On how many of those occasions was your husband also an overnight guest?"
Boyce rose, Galahad defender of the sanctity of the Grab–Van Anka marriage, to express his outrage.
"Your Honor, setting aside for a moment the prosecution's sneering, harassing tone, this question is utterly without relevance."
"Overruled."
After pickaxing away at Babette, the deputy AG established that Mr. Grab "had only been able to join" Babette at the White House on four occasions. Four occasions, out of fifty-six.
"Your husband, then, was with you only seven percent of the time you were there?"
"If that's what it comes out to. Mr. Grab is a busy man. He's frequently away on business."
"What kind of business is he in?"
"The business kind of business."
"Explain that to the court, if you would."
"He finances things. He makes things happen. Wonderful things that benefit humanity."
"What sorts of things?"
"Objection. Your Honor, what possible relevance does it have that Ms. Van Anka's husband is a respected international financier and philanthropist, who has given considerable sums of money to minorities and other charities?"
"Your Honor, I object to counsel's objection."
"Approach," Judge Dutch said, glowering.
Sidebar.
"Ms. Van Anka," the deputy AG continued, finally, "are you familiar with this quote that appeared in the Los Angeles newspaper about you: 'Babette Van Anka, she's so bad you wanna spanka'?"
"Objection. Your Honor, this sacred court of law is no place for scandal-mongers and low gossip."
"Overruled."
Under his breath, Boyce muttered, "All right, then, it _is_."
"I must have missed the paper that day," Babette replied. "I'm busy, too, you know."
"You've never heard this quote before?"
"I heard it."
"And what does it refer to?"
"Not my acting or singing, hopefully."
Laughter.
Nick Naylor thought, _Yesss_. It was his line.
The deputy attorney general smiled. _Okay, toots, you wanna have fun? Let's have fun_.
"Was it your practice, Ms. Van Anka, before spending the night in the White House, to purchase erotic underwear and erotic devices, at a store in Los Angeles called QQ?"
Babette paled. Boyce leapt to his feet.
"Your Honor, this is an outrageous invasion of privacy, and an affront to women everywh—"
Judge Dutch motioned them forward to sidebar number 127.
The mention of QQ sent a palpable thrum through the courtroom. QQ was a boutique that had been started by a woman who before going to jail had operated one of Los Angeles's most elite escort services. After three and a half years of working in the prison laundry, she opened QQ with the backing of those of her former clients whom she had not publicly identified during her trial. It specialized in exotic, upscale lingerie and romantic "excess-ories," as its catalog put it. If Tiffany were taken over by _Hustler_ magazine, the result would be QQ. One popular item was the mink-lined crotchless panties ($2,500). A Victorian corset with whalebone stays went for $800. Then there was the set of four sterling silver balls connected by a string with a handle at the end ($3,200).
The TV commentators all waited for someone else to explain to the public what QQ stood for. Judge Dutch groaned inwardly.
The whole world was waiting. In Denver, a pilot delayed pulling back from the gate so that he would not miss hearing on the live radio broadcast what this QQ stood for. Moreover, the passengers approved the delay and heckled a supervisor who came aboard to order the pilot to depart.
At the defense table, Beth thought back to the time when, in an attempt to stimulate her husband's waning interest in her, she had ordered some racy silk thingees from a catalog. She'd put it all on one night, feeling foolish looking at the garter-belted, stockinged, and bustiered reflection in the mirror. Beth MacMann—in a bustier! She'd lit scented candles, dabbed perfume all over, put on romantic music, lain back on the bed, and waited for him to come through the door. And waited. When he'd finally walked in it was past three, and it had been obvious that even he had already had his fill of sex for one night.
Judge Dutch cleared his throat. "Proceed, Ms. Clintick." By now Babette was a jittery wreck. Her intellectual glasses kept sliding off her nose.
"Ms. Van Anka, do you purchase lingerie and sexual-related items at a store in Los Angeles called QQ?"
"I may have. I shop at a lot of stores in Los Angeles."
The deputy AG asked the court to enter into evidence fourteen credit card receipts in Babette's name from "QQ Enterprises, Ltd." One by one, they were projected on the screen mounted in the courtroom. The purchases totaled some $23,725. Doubtless, that would impress those on the jury supporting families of four on $30,000.
"These are yours, Ms. Van Anka?"
Babette studied them through her glasses as though they were recently unearthed Dead Sea scrolls.
"Apparently."
"Let me draw your attention to the last receipt. What is the date on that?"
"I can't tell."
"Upper left. Would you read that date for the court, please?"
"September twenty-sixth."
"The item listed, would you read it, please?"
"I can't see. These are reading glasses."
"Very well, with the court's permission, I will. 'Mink massage mitt.' Twelve hundred dollars.
"Ms. Van Anka, what does the name of the store where you purchased these items refer to?"
"I wouldn't know that."
"Does it stand for—"
"Objection. Speculation. Speculation of the most prurient—"
"Sustained."
The world would have to wait until that evening, when one unrestrained guest commentator blurted out what everyone at this point already knew. QQ stood for "Quivering Quim," the Victorian term for—never you mind.
Onto the courtroom screen was projected a series of slides. Each vertically filled one-third of the screen. The first was titled "Van Anka Purchases at QQ Boutique, by Date."
The second was titled "B. Van Anka Overnight Stays at White House, by Date."
The third, filling in the last third of the screen, was titled "Nights First Lady E. MacMann Absent from White House, by Date."
The date of each of Babette's fourteen purchases at QQ preceded by one or two days her visits to the White House. The dates of those fourteen—of fifty-six—of her stays in the White House coincided exactly with dates that Beth had been absent.
"It's a real shame," Boyce said, handing Beth a stiff vodka back in his hotel suite.
"What is?"
"That we were moneyless students when we knew each other. I'd have liked to see you in mink-lined panties. During the sidebar, Judge Dutch gave Clintick hell for going on about the silver balls."
"She seemed pretty wound up."
"She's trying the biggest case of her life. Though at this point, I think her primary objective is to hang me, not you. If that doesn't sound too conceited."
"I wouldn't want a modest lawyer defending me."
"I know we've been over this a hundred times, but you're positive he was banging Babette that night?"
"Boyce, the man was bent over from sexual exertion. I could smell her perfume on him. And when I turned on the light, he did that raccoon thing with the eyes."
"Number three did that once when she came in. I knew."
"It's charming how you've assigned numerical identities to your ex-wives. I've been in love with two men in my life, a serial adulterer and a serial divorcé. Where did I get this karma?"
"When you dumped me. Karma is as karma does. Actually, it wasn't karma at all. It was divine retribution, the wrath of an angry god, for screwing up my life."
"I love it when people of our generation feel sorry for themselves."
"I don't buy into boomer self-loathing. Our generation has accomplished many things."
"Name one."
"Disco, junk bonds, silicone implants, colorized movies, the whole concept of stress as a philosophical justification for self-indulgence. These achievements will tower above minor accomplishments like defeating Hitler, breaking the sound barrier, and inventing a vaccine for polio. Future historians will call _us_ the Greatest Generation."
"I think it's why I fell for Ken. He was real."
"What wasn't 'real' about me?"
"I didn't mean it that way."
"It all has to do with your dad. He got a medal in World War Two. Then along comes a chest full of medals from another war and—boom, you're enlisted marching in his parade."
"Maybe. I never felt like spending a hundred grand on psychoanalysis to find out. I'd so much rather spend the money on lawyers."
"Well, it's reassuring that heroes can turn out to be pricks just like the rest of us nonheroes. But that still leaves unresolved the larger question."
"What?"
"How you'd look in mink-lined panties."
After another day of making Babette Van Anka look like the Slut of the Millennium, Deputy Attorney General Clintick tossed her limp, twitching carcass at Boyce for cross-examination. As Boyce rose to begin, he felt like all the king's horses and all the king's men. Humpty-Dumpty was in a thousand pieces.
Babette warily returned his smile as he approached. By now she knew he was, for whatever reason, on her side, unlike that bitch prosecutor. This Baylor person had said, "Objection!" so many times, he must have carpal tongue syndrome by now. Morris, Howard, and Ben had explained to her why he was being nice: to take away the wife's motive for killing Ken. But they'd warned her: Be careful of him. _Short answers_.
Boyce walked Babette through some preliminaries to reinforce in the minds of jurors that, like many of them, she had grown up poor, that she had in her own, if somewhat overheated, way shown that the American dream worked. He maneuvered the subject around to the movie _They Call Me_ General _Powell!_ so that Babette could talk about how thrilling it had been to work with the actor Denzel Washington on a biographical movie about Colin Powell. He led her through a tearful account of her role in the movie _Flight 208 Is Delayed_ , in which she played an Israeli paratrooper who single-handedly rescues a jumbo jet full of Hasidic schoolchildren from fanatic Palestinian hijackers. It was this experience, she told the court, that had been the epiphany that had prompted her to become a force for peace in the Middle East.
"You and President MacMann were very close, were you not?" Boyce asked, switching topics abruptly.
Babette seemed taken aback by the question.
Boyce added, "I mean, he looked to you frequently for advice on the Middle East, did he not?"
She brightened. "Oh, you don't _know_."
"Do you mean by that, yes?"
"Yes. Yes. All the time, he was calling me."
Eyeballs careened sideways among the press.
"He called you specifically about the Middle East peace process?"
Boyce had learned from his idol Edward Bennett Williams that an ideal cross-examination elicits an unbroken string of yeses. It shows you have total control of the witness.
"Many times. Many, _many_ times."
"He relied on you heavily for your input?"
"Yes."
"Objection."
"Your Honor, I understand that the deputy attorney general might be upset because I am questioning the witness without resorting to character assassination, innuendo, and slander, but—"
_"Approach."_
Boyce endured his sidebar lecture in silence. It was worth it. Later, Vlonko reported that jurors four, five, seven, and thirteen had beamed at Boyce's outburst.
"Ms. Van Anka," he continued, "are people in your position, that is, creative artists of pronounced social conscience, sometimes ridiculed or made to suffer because they dare to speak out against injustice or on behalf of oppressed—"
"Objection!"
"I will rephrase. Ms. Van Anka, have you been attacked for your activism?"
"Constantly."
Boyce shook his head sadly, as if hearing this were all just too painful.
"It must be hard."
"Objection."
"Withdrawn. Is it hard?"
"Very hard. All the time people say terrible things."
"Ms. Van Anka, is it your understanding that powerful forces in the Middle East were aware of the fact that you had the President's ear and tried to prevent you from advising him?"
"I object. I object strenuously. Your _Honor_."
"Let me rephrase. Did powerful forces in the Middle East try to prevent you from advising the President of the United States on foreign policy?"
"I... had the impression... yes?"
Boyce nodded. "And did these forces try to accomplish that by spreading rumors that you and the President were lovers?"
Babette nodded. Her chin quivered. She looked into her lap. Then, as if on cue, she burst into tears. _"Yes,"_ she sobbed.
Boyce shook his head at the iniquity of it all. "Was this personally painful to you, Ms. Van Anka?"
"You have no idea."
"How did it make you feel, as a woman, that people would say such vile, wicked things—"
"Objection. Your Honor, is defense conducting a cross-examination or a support group session?"
"Ask your question, Counsel."
"How did it make you feel, as a woman?"
"It made me feel"—Babette sniffled—"that for all our progress as a gender, that we still have a long way to go."
"Ms. Van Anka, did you in February of two years ago personally carry a confidential message from President MacMann to the Prime Minister of Israel?"
Babette's eyes widened. Over the years, she had embroidered this nothing of a story into such a heroic tapestry as to make Bayeux blush. She had told dozens of people that the President had asked her to carry a "top secret" message to the Israeli Prime Minister. In fact, the message consisted of, "Tell that bagel-biter his new press secretary has the best honkers in the Fertile Crescent." As Babette told the story, she made it sound as though the President had entrusted her with a plea not to use atomic weapons on Syria.
"Yes"—Babette nodded—"I did. But I can't—"
"Of course I won't ask you to divulge the contents of a highly classified message having to do with national security."
_"Objection."_
"Withdrawn. Now, Ms. Van Anka, let us turn to a subject that the prosecution seemed to find so personally distasteful...."
_Okay, Babs, you're doing great. Now we're going to play connect the dots_. His investigators had found the first dot in an interview that Babette had given to one of the women's magazines years ago, just after she had married Max.
"I'm referring to the business of the personal items, the underwear and such from that store in Los Angeles. Did you not give an interview to one of the women's magazines stating"—Boyce glanced disapprovingly at the prosecution—"freely and openly and I might add, indeed proudly, that you and Mr. Grab enjoy what one might call a full and loving intimate relationship?"
Babette wasn't entirely sure where this was going, but at this point she would have followed this man up onto an exit ramp of the 405 Freeway.
"Yes. Max and I have a wonderful relationship."
The sound of choking came from the media.
"Thank you for your candor. And did you tell this magazine that you and your husband both believe that the way to sustain a healthy, intimate relationship is to"—Boyce smiled benevolently—"keep things in the bedroom _interesting_?"
"Yes." Babette blushed.
"Objection."
"Overruled." Judge Dutch, along with the billion or so people watching, was dying to see where this was going.
"And did you tell the interviewer, without embarrassment—in fact, with evident joy—that your husband enjoys it when you put on sexy underclothing?"
"I did."
"Ms. Van Anka, because of your busy schedules, you and your husband are _apart_ much of the time, is that not true?"
"Yes? Yes."
"Does it make you feel close to your absent husband to wear these articles of intimate clothing?"
"Oh, _yes_."
"Objection."
"Withdrawn. Do you wear these items only when you are with your husband, or sometimes when you are apart?"
"I bring them with me on trips. To remind me of him. When I feel them against my skin, I feel I'm... well..."
"Thank you. I know these are terribly private matters. When you are apart from your husband, do you ever put on these articles and call him on the phone?"
_Connect the dots. Come on, Babs_. Babette blushed, smiled, stared into her lap, brushed away a strand of hair. "Sometimes I put on the things so that I can _pretend_ that we're together."
Boyce phrased the next question carefully, knowing that there had been no outgoing call to Max from the Lincoln Bedroom on the night of September 28–29.
"Were you planning to make such a call to your husband that night at the White House?" _Just a few more dots and you can go back to saving the Middle East and the caribou_.
"Yes, I was. I... that was one of the reasons I retired early."
"But you fell asleep before you could?"
"Yes."
"You retired at twelve-thirty A.M. That would have been only nine-thirty P.M. on the West Coast."
"Right. Max wouldn't be home until later, so I was going to stay up and call him."
"You told the FBI that you got into bed with the television on."
"Yes. I turned on the television. I usually fall asleep."
_Now follow me very, very closely here_.
"Is there a particular show that you watch late at night?" In media interviews, Babette talked incessantly about her passion for watching public television, where you could always find something "thought-inspiring." (Like the public service announcements for Saab.) Boyce had read every one of these interviews.
"I try to watch"—Babette looked imploringly at Boyce—"you know, substantial shows."
She meant "substantive," but she was trying.
"Public television?"
"Objection! Your _Honor_."
"I'll rephrase. What sort of television?"
"Public television. Talk shows. Documentaries..."
With his back to the jury, Boyce looked directly into her eyes and said very carefully, " _Did you turn on the public television channel that night in your bed_?"
"I... yes. I'm sure I... yes."
"And you told the FBI that you fell asleep."
"Yes. I was tired. Such an evening. So many Latin dignitaries. They're exhausting. So talkative."
"So you fell asleep, _with the TV on_?"
"Yes."
"You fell asleep with the television on, set to the public TV channel. This was sometime after twelve-thirty A.M.?"
"Yes."
"Your Honor, with the court's permission, I would like to have received in evidence the _TV Guide_ listings for the evening and early morning of September twenty-eight to twenty-nine."
A copy of the _TV Guide_ from that week was duly entered after a lengthy sidebar initiated by a distinctly unhappy-looking deputy attorney general.
Boyce handed the _TV Guide_ to Babette and asked her to read the listings for the early morning of September 29 for WETA, the local public television channel.
Babette put on her intellectual glasses and read, "One A.M. to three A.M., _Who's Afraid of Virginia Woolf_?"
She looked up at Boyce.
_No, no. Do not smile at me_.
She caught herself.
A murmur went through the courtroom.
"Would you tell the court what _Who's Afraid of Virginia Woolf_? is?" Boyce asked.
"It's a film. A great film. A great American film."
"Would you tell the court what it's about, briefly?"
"It's about two couples in an unhappy marriage. I mean, _really-really_ unhappy."
"And what takes place in the movie?"
"Oh, a lot of anger. Shouting. Throwing things. Screaming."
"Shouting, screaming, throwing things?"
"For _starters_."
"The television in the Lincoln Bedroom is fifteen feet from the bed. Would you have had the volume up so that you could hear?"
"Very up."
"So a movie showing people arguing loudly was playing in the Lincoln Bedroom, not far from where Secret Service agent Birnam was stationed, between one A.M. and three A.M.?"
"Yes."
"Thank you, Ms. Van Anka. No further questions at this time, Your Honor."
# Chapter 19
They were calling it "the Timeline of the Millennium."
Normally in criminal trials, timelines—chronological orderings of events—are broken down into minutes. The one Boyce and his team beavered up was in hundredths of seconds. This prompted snide comments among the media that he must have located the White House residence cockpit flight recorder.
Boyce's timeline alleged that an incompetent, vengeful FBI agent Whepson and a resentful, hearing-impaired Secret Service agent Birnam had been alone with the President's cooling corpse for thirty-seven seconds, giving them time to emboss the President's forehead with the Revere hallmark.
Deputy Attorney General Clintick fought like a lynx to have the timeline excluded. She was under increasing pressure following Boyce's astonishing cross-examination of Babette Van Anka.
President Harold Farkley was getting more and more questions about the case, and they were being asked with even less than usual courtesy. One day, after announcing a historic engineering initiative to prevent the Missouri River from overflowing and drenching America's breadbasket, and deploring racial profiling—former secretary of state Colin Powell had again been pulled over by a Virginia State Trooper and spread-eagled across the hood of his car—he was accosted in an unseemly manner by the traveling White House press corps, demanding to know if he had played an "active" role in prosecuting his well-known nemesis, Mrs. MacMann. None, none at all, he averred. His press secretary told him that Bob Woodward, Investigative Reporter of the Previous Millennium, was "making inquiries." Harold Farkley's mouth went dry.
Having lived all his life in the shadow of his own mediocrity, he was determined to defy his karma and win the upcoming nomination. The last thing he needed was a front-page _Washington Post_ article ("First in a Series of Articles") with the headline:
SEEKING TO SETTLE OLD SCORE,
FARKLEY ENCOURAGED JUSTICE DEPT.
TO PROSECUTE FIRST LADY
He instructed his press secretary to put out the word—and not quietly—that he had been "skeptical" of the evidence against Beth "all along."
In due course, articles reflecting this new line appeared. The sources were not attributed directly, but there was enough DNA in them to alarm the attorney general. He in turn instructed his press secretary to put out the word—loudly—that he too had had "qualms" about the evidence "from the beginning" but that Deputy Attorney General Clintick had been "avid" to prosecute.
In due course those articles appeared, causing Sandy Clintick to break out in a rash. Whom the gods would destroy, they first make itch.
On _Hard Gavel_ , Alan Crudman's jealousy over Boyce's masterful handling of the case had caused him to evolve into a public second-guesser for the prosecution. It was something of a career reversal for a man who had once boasted that he could have gotten Adolf Hitler acquitted.
In the middle of last night's show, he had gotten carried away in his fever to demonstrate that Agents Whepson and Birnam could not possibly have put on latex gloves, grabbed the spittoon, embossed the presidential forehead, removed the gloves, and replaced the spittoon between 7:33:00 A.M. and 7:33:37 on September 29, while Beth was in the bathroom barfing. Crudman leapt out of his chair to reenact the scenario, lapel microphone still attached, yanking his mike out of its socket and upsetting a water glass.
Perri's attitude toward Boyce had become openly antagonistic. Boyce had stopped coming home on weekends. She couldn't even wheedle anything out of his team as to what was going on behind the scenes. When they did speak, she was unable to get anything out of him. He didn't respond to her cooey little nudgings. His conversation consisted of, "Uh-huh, uh-huh. Listen, gotta go." It was all a bit... much. Hadn't Perri nursed him back to emotional health after his disastrous fourth divorce from the socialite mountain climber—what's more, at a time when she should have been concentrating on her own career?
With her producers, she pretended that Boyce was keeping her fed with tidbits. Meanwhile, as she discussed the case on television every night for a larger and larger audience, Perri had become consumed with what, for her, was the larger question of the Trial of the Millennium: Were Boyce and Beth doing it?
The tricky part in getting Judge Dutch to allow his timeline was that the only person who could attest to Agents Whepson and Birnam's being alone with the corpse was—his client, and of course there was no way he would put her on the stand. One of the triumphs of the American justice system is that the guilty—that is, the accused—does not actually have to defend himself. He can just sit there while lawyers fire spitballs at the accusers and make them out to be the real villains.
"We go now," said evening news anchorman Peter Jennings, "to our legal correspondent. Jeff, how did it go today?"
"Peter, this was another _bad_ day for the prosecution. Mrs. MacMann's lawyer, Boyce Baylor, introduced a timeline of the morning of the President's death that is so minute, so _detailed_ , that you have to wonder if this jury, or any jury, would be able to keep it all straight. It tracks the movements of eighteen people in and out of the presidential bedroom over a period of two hours. At the heart of Baylor's argument is a critical thirty-seven-second period when, as he claims, FBI agent Jerrold Whepson and Secret Service agent Woody Birnam were alone with the President's body. He contends that, acting out of personal animosity toward Mrs. MacMann, they stamped his forehead with the Paul Revere silver hallmark on the spittoon so that it would emerge as a murder weapon. The defense contends that the President died in the night as a result of an accidental _fall_. For the past three days, Baylor has _hammered_ at Whepson and Birnam _relentlessly_. In the end, they did not categorically _deny_ that they were alone with the body while Mrs. MacMann was in the bathroom. I have to say, whatever you think of the argument that this is all some government conspiracy, these were _effective_ cross-examinations, especially coming after his _devastating_ cross-examination of Babette Van Anka. In the end, the jury may conclude not that Mrs. MacMann is _innocent_ , but that she is not, beyond a reasonable doubt, _guilty_. Peter?"
"I wondered if this was going to happen," said Boyce.
Beth's head rested on his chest. "You knew it would."
They were in bed in a tangle of sheets. Outside, beyond two sets of doors, stood silently fuming Secret Service agents. It was a Friday night, no court tomorrow, and Boyce's war room was quiet.
"How did you get them?" Boyce asked, feeling under the sheets for the silky-furry object. He held it aloft for inspection.
"Aren't they _hideous_? I called a friend in L.A. and had her buy them. Don't worry. She's discreet. It's not the sort of thing I wanted to have reported that I put on my American Express. So now the question is finally resolved."
"Not quite. We know what you look like in crotchless mink panties now. We still don't know what you would have looked like in them back in law school."
"Even more ridiculous than I look in them now."
"I can't believe you wore them in court today. What if you'd had some medical emergency and they'd taken you to the hospital and put you on the table and you're wearing _these_? What a headline!"
Beth snuggled against him. "You were really good."
"You weren't so bad yourself."
"In _court_. Don't flatter yourself."
"Oh. And how was I just now?"
"Mm, adequate."
"Your bill just went up by a million dollars."
"You were amazing. Godlike."
"Ten percent off."
"Just like old times."
It was strange, making love to a once familiar partner after a quarter century. Boyce turned over metaphors in his contented mind. Was it like drinking wine long cellared and ripened? Or was it more like entering a garden in which the vines had matured into—
"The media's saying it's over," Beth said.
"The media isn't the jury. But you heard Vlonko. I don't think I've ever seen him happier."
"Vlonko," said Beth. "Diviner of minds."
"He said jurors six, seven, ten and thirteen were actually nodding when I crossed Birnam."
"Is thirteen the—"
"Homosexual pediatric neurosurgeon of German extraction. You just don't _get_ more no-nonsense than that. And _he_ was nodding." Boyce sighed happily. "I don't want to jinx it—the gods are watching 24/7—but I _think_ we might have this thing nailed. I think they're going to acquit. You never know, but I think you're going to be okay."
Beth reached for her cigarettes and lit one. "I want my life back after this," she said. "Not the old life. My own."
"You're young."
"Ish."
"Sexy."
"Ish."
"Smart."
"How smart was I, to get into this?"
"You were smart enough to hire a good lawyer." Beth sat up on one elbow and turned to him. She was beaming with excitement. "Boyce, I want to take the stand."
"Huh?"
"I want to testify."
"Is that a cigarette or a joint you're smoking? Are you nuts?"
"No, I want to take the stand."
"I'm not even going to go into that."
"Why?"
"Because it's such an insane notion that it's not even in the realm of thinkability. This was not a particularly easy case. You may have noticed? Just because we're ahead, don't get crazy ideas."
"You've been brilliant. I'm the first to admit that. I know you've worked your heart out on this. But people still think I did it. You heard what they said on TV tonight."
"Who cares what the public thinks? Get yourself a PR guy. Whatsisname, Naylor, the one who does Babette's PR. He could make Saddam Hussein out to be Santa Claus."
"Thank you. That was truly sensitive."
_Change the subject, quick!_
"I gotta have water. These hotel rooms. My mouth is like the Mojave."
Boyce went in search of water. His mind was reeling. _Astounding. Twenty-five years in politics turns you into a—politician. She's barely off the hook on a murder rap and already she's planning her comeback_. What was she thinking? Well, she wasn't thinking, just like twenty-five years ago when she married War God. The minibar. It would have water. Cool, expensive water from some spring in Finland or Wales, so pure you could wash your contact lens in it.
Boyce got back into bed with an eight-ounce bottle of water that cost $9. He snuggled up against her. Her body was less pliant and responsive than a minute ago.
"Your cross-examinations of the staff," she said, her back to him, "made me out to be a total bitch. Lady Bethmac."
"So what, if it helps you get off a murder rap?"
She turned to face him. "What good does it do me if I get off and everyone still thinks I did it? And that I'm the Joan Crawford of First Ladies?"
"What good does it do you if you _get off_? Apart from not spending the rest of your life in federal prison? Or the death penalty? That's a hard one. I'll have to think about that."
"I need some kind of life after this."
"Boy, you're an easy one to please."
"If I take the stand, I can show them that I not only didn't do it, but that I'm not the First Bitch."
"Listen to me: A jury is probably about to acquit you of murder. Trust me, that is _the_ major goal here. It is the _only_ goal. Look, acquittal means you didn't do it."
"No, it doesn't. It just means that I got off. I'll be the O. J. Simpson of First Ladies. What am I supposed to do, hang out on public golf courses looking for the real killers?"
"It beats working in the prison laundry for the next forty years. To say nothing of lethal injection."
"What if I want to continue in public life? What if I want to run for public office myself? The Senate."
Boyce stared in the semidark, a pointless dramatic gesture.
"Beth, I know this has been very stressful for you."
"Will you please not speak to me like I'm a mental patient?"
"I won't if you don't act like one. Look, it's not true what F. Scott Fitzgerald said about American lives not having second acts. Look at Charles Manson—he's got his own Web site. You can do whatever you want after this. My God, the product endorsements alone will be enough to—pay my bill!"
"I have no intention," Beth said icily, "of becoming a product endorser. That's a line of work for overweight British royals and oversexed White House interns."
"How _are_ you planning to pay my bill? I suppose we could work something out." He nibbled her ear. "Take it out in trade."
"Stop. According to a study by the American Bar Association, three-quarters of the public thinks that a defendant who doesn't take the stand is either guilty or hiding something."
"For once I agree with three-quarters of the American public. Do you know how many times I've allowed a client to testify in a criminal case? Twice. One was a seventy-eight-year-old Mafia boss in advanced stages of emphysema. I put _him_ on the stand so the jury could listen to him wheeze. They felt so sorry for him, they let him go die at home. The second was a Catholic cardinal who'd been accused of unholy communion with an altar boy. Now it was twenty years later and the altar boy had a heroin problem, a crack cocaine problem, and a drinking problem, along with three other unpleasant diseases. So he'd decided to extort from his former parish monsignor, who was now a leading prince of the American church. In this particular case, the cardinal happened to be innocent. A rarity, I know, an actually innocent client. I should have had him stuffed. At any rate, I put him on the stand because how often is it you have an innocent client who dresses in scarlet robes and wears a cross the size of a tire iron around his neck?"
"And?"
"They found him guilty. And you want to take the stand."
Beth stubbed out her cigarette. "All those years with Ken, all those horrible years, I sucked it up, turned the other cheek, looked the other way, worked my ass off. I'm in my forties, I have no money, no visible means of support, other than endorsing antiques you can kill your husband with. I'm a widow—do not interrupt me, please—and everyone thinks I'm an assassin who offed a war hero with a spittoon. This is _not fair_. And I will _not accept it_. And I will _not_ walk out of that courtroom into a life of people pointing at me in airports as some historical freak, afraid of turning on the television because Jay Leno and Letterman might be _doing_ me in their monologues because nothing else sensational happened that day. Frankly, doing laundry in prison for forty years—or getting hooked up to the death drip—doesn't look half-bad by comparison. If I'd wanted to get off on technicalities, I'd have hired Alan Crudman or Plato Cacheris or some other hotshot. I hired _you_ because I need to _win_. Not not-lose. Win. You said I should walk in that first day and look like I'd come to accept their apology? Well, that's how I plan to walk out, like I've accepted _their_ apology and now I'm ready to accept the world's."
Boyce considered. "I think we should prepare a statement announcing that you plan to devote the rest of your life to searching for the real killers."
Beth slugged him with the pillow. Hard.
"I'm beginning," said Boyce, "to see how you got that nickname."
# Chapter 20
You might want to glance at this," Boyce's secretary said a few days later, handing him the Style section of _The Washington Post_ , folded over to Lloyd Grove's "Reliable Source" column. "Glance" was code for "Here is something that you are really not going to like."
The relationship between First Defendant **Beth MacMann** and superlawyer **Boyce "Shameless" Baylor** may be progressing beyond the normal lawyer-client stage. The two were an item in the 1970s when they were students at Georgetown Law, before Lady Bethmac unceremoniously dumped him for the future President. But she may have left the pilot light on. Or they could just be boning up on the next day's court proceedings, in Shameless's $7,500-a-night suite at the Jefferson Hotel.
Grove, the swine. Beth didn't dump him "unceremoniously." And the $7,500 was for three suites, not one.
His mind then turned to the larger issue.
Her Secret Service detail. Of course. They were the only ones who could have known, and they now despised them both. Well, who could blame them? He had accused them of monstrous conspiracy. Still, where was their quiet professionalism? Where was the "Secret" in Secret Service? "Pricks," he muttered.
The irony was that whatever "pilot light" Beth had kept burning for him had been blown out during their argument over her testifying. He probably should not have made the crack about how it was just about this time of morning that her late husband had "committed suicide by spittoon." Beth got out of bed, got dressed, and stormed out of the suite, back to her Cleveland Park Elba. So much for their reunion. It had been fun while it lasted, all three hours of it. This would present an interesting new challenge, not being on speaking terms with your client in the middle of a murder trial.
Warily, Boyce turned on the TV. Instant disaster. On came one of the morning shows. The two hosts were in the middle of a wink-wink fest over it, cracking art-imitating-life jokes about how maybe Beth would get a break on her legal bill. Ha ha ha.
Boyce flipped channels. Swell. There on the screen above the next two hosts was a photo of Beth and Boyce from law school, with the headline LOVE STORY and a caption "Love means never having to say you're guilty."
He began to dial Beth on his cell phone, then thought it better under the circumstances to use a landline. God knows who was listening in.
"I think today," he said, "we might want to use the basement garage entrance to the courthouse."
Beth was in shock, or at least as close to shock as type A personalities allowed themselves to get.
"How did this happen?" she croaked. "Who?"
"Ask your so-called Secret Service detail."
"I did. They denied it."
"Of course they denied it."
"I believe them. But I wouldn't blame them if they had. But these guys are professionals. They don't blab to the press. Even about people who've made up cockamamie stories about how they plant evidence to incriminate First Ladies."
"For the benefit of anyone listening in to this conversation, the former First Lady is obviously hysterical and not possessed of full mental faculties."
"This has to have come from your end."
Boyce thought. "The night desk clerk. When you stormed out of here in the middle of the night. He must have tipped them."
"Give me some credit. I didn't exit through the lobby. We took the elevator to the basement garage. I guess I might as well get used to basements, since I'm going to be spending the rest of my life in them."
"We'll sort it out later. Meanwhile, if anyone asks—and they will—we were working late. It's perfectly plausible."
"On a Friday night?"
"Edward Bennett Williams worked late Friday nights when he was in trial."
"Boyce," she said, "about the other night. I've been thinking."
At least she'd come to her senses about testifying. Thank God.
"I have to take the stand."
"Beth, this is not a good time to discuss this."
"Then when would be a good time? This afternoon?"
"Look, _Vanity Fair_ is going to be calling any minute asking us to pose nude in bed for next month's cover. Before we discuss whether you testify, we have some serious damage control to do."
"If you can't find a way to go along with the decision, I accept that."
"What does that mean?"
"It means that I can always add someone to the defense team who would be willing to do a direct on me."
"Team? Your _team_ consists of _me_."
"Boyce, I'm just not negotiable on this."
Time. He needed time. Time to... what?... crush Valium in her food. That's it. Keep her drugged for the rest of trial. If she nodded off at the defense table, he'd say, You _see_ what a strain this poor woman is under?
"Okay. We'll talk about it after court today. Jesus, look at the time. We're going to be late. Have the un–Secret Service drive you in through the basement garage. I'll meet you inside."
"I'm going in the front door, as I have every day."
"Don't say anything to anyone about testifying. Beth?"
She'd hung up.
Confronted with this fresh hell—as Dorothy Parker put it—Perri had only one way to proceed: full steam ahead. Somewhat nervously, she rehearsed her indignation and dialed Boyce's cell phone, his supersecret cell phone, the one whose number he gave out only to death row wardens entrusted with the temporary care and feeding of his less successful clients.
"Is this _true_?" Perri demanded, dispensing with the usual, "Hi, honey!"
"I'm on my way to court," Boyce said. "Can we talk about this later?"
So it _was_ true. As it happened, Perri had been the source of the leak to _The Washington Post_. She had called up Grove and told him that she had walked in on Beth and Boyce in the act. Call it an educated gambit.
"You bastard." Perri faked a stifled sob.
"Look, baby, I'm"—what a morning—"it's got nothing to do with you."
"Obviously not."
"I'll come up this weekend. We'll go to La Grenouille. Champagne, foie gras, that Dover sole with that sauce you like. Grand Marnier soufflé." Just what he needed in the middle of the Trial of the Millennium—a hysterical betrayed girlfriend, with her own television show, costarring Alan Crudman as the Greek chorus.
"You think I can be bought off with a _soufflé_? What do you think I am, some _flight attendant_?"
"Those happen to be the best soufflés in the world."
"Screw the soufflés!"
"Perri honey, I've been under intense pressure here. You have no idea."
_My God_ , he thought, _I sound like Babette Van Anka_.
"You could be disbarred for screwing your client, you know."
"Honey, lawyers screw their clients all the time. And make a good living."
"I'm calling the D.C. Bar Ethics Committee."
"Okay, okay. Listen, I'm pulling up at the courthouse right now. You can probably see me on your TV. Do you see me? That's me. I'm waving. That's for you." He made a kissing sound. "I'll call you the second I'm free. Okay?... Okay?... Perri?"
She'd hung up. Not yet ten o'clock and already two for two.
After court that day, they were waiting for him as his car exited the courthouse basement garage. The driver had to stop or he would have crushed a dozen photographers and cameramen. They swarmed around the car, encircling it, scanning the inside with their lenses. Boyce was no stranger to paparazzi, but this was an entirely new level of interest. Now he had insight into what it was to be a Princess Diana. Not wishing to have film footage of his car speeding away from a clamoring, shrieking press become a permanent part of the Boyce Baylor videotape archive, he rolled down the window, smiled, and said, "She's in the trunk." They liked that. They let him go after a few pointless, shouted questions.
Beth had made it abundantly clear, over a strained atmosphere and tuna-fish sandwiches in the soundproof defense room, that she now more than ever planned to take the stand. This led to their most candid discussion about the night of September 28–29.
"If you're committed to going through with this," Boyce said, "then you'd better tell me everything I need to know about what happened. Tell me everything. Including what I really _didn't_ want to know."
"All right," Beth said. "He came to bed after screwing that over-the-hill hooker—"
"Stop. Stop right there." Boyce sighed. "You'll recall I spent days doing one of the best cross-examinations of my career—and I don't say that to boast—rehabilitating that 'over-the-hill hooker,' _precisely_ so that the jury would conclude that she had _not_ been screwing the late President. Which conveniently removed your motive for killing him. And now the first thing you tell me is he was in there doing push-ups with her. Beth, do you understand why this is suicidal?"
"Okay. Ask the question again."
"Would you tell the court what happened on that night?"
"My husband came to bed about two-thirty A.M.—"
"Why so late?"
"I didn't ask."
"Oh, _that's_ convincing."
"He's the President of the United States. They get up all the time in the middle of the night to save the world."
"Weren't you curious as to what crisis had him up in the middle of the night?"
"The crisis in his crotch."
"Very good," Boyce said. "We got two questions into a cross-examination before you admitted to killing him. Here are a few more. Did you ever assault your husband? Did he require medical treatment afterwards? Was it your habit to throw heavy objects at him? Did you kill your husband? Did you assassinate the President of the United States?"
"Finished?"
"Not nearly."
"I woke up. It was after two-thirty by the clock. I heard a sound. It was the President, getting into bed. He was frequently up at night, phone calls, emergencies. I didn't think anything of it. I... no, that doesn't really work, does it? Okay, you want to know what _really_ happened that night?"
"Yes, Mrs. MacMann. Would you tell the jury what _really_ happened that night."
"Do you want to hear this?"
"Personally, I'm dying to. But if I were you, I'd stop right there."
"If I tell them what happened, they'll believe me."
"I think you spent too long in politics."
"They'll believe the truth."
"You did whack him with the Revere ware, or didn't you?"
"Not _that_ hard."
Boyce buried his face in his hands. "Oh, great."
"I didn't throw it that hard. He barely flinched. Usually he goes down when I throw something at him."
"Be sure to mention that."
"I've hit him much harder before. The time I threw the lamp at him? Four stitches. He had the press secretary say he swallowed a pretzel wrong and passed out and hit his chin on the way down."
"Well, it lets you off the hook. We'll just explain to the jury that he didn't actually die because you bashed him in the skull with a heavy metal object. He died—of something else. His war wounds..."
"But that's just it. He did. He had to. Look, he was alive and breathing when he put his head on the pillow. He was smirking."
"You didn't... get up in the middle of the night and finish him off? Beth?"
"What do you take me for?"
"I know what killed him."
"What?"
"Gamma rays from outer space."
"I don't know what happened. Maybe she screwed him to death. I don't know, except that I'm not guilty. And I won't have people thinking I am and that you got me off. If you won't help me, I'll find someone who will."
When Boyce got back to the hotel at the end of the day, the police had erected barriers outside. In addition to the mob, he counted six satellite trucks. They were all hoping for a glimpse of at least half of the Fun Couple of the Millennium.
Above he heard a helicopter.
He motioned the driver to go around. His era of basement garages had begun.
Back in his suite, he slumped into a chair and fought the temptation to pour himself a double bourbon.
He went into the bedroom.
The phone rang. Perri.
"I'm putting together tonight's show," she said in a businesslike voice. "Have you got anything for me?"
"There's a rumor." Boyce sighed. "She might be testifying."
"Really? How good a rumor?"
"That's the rumor."
"See you Friday at La Grenouille."
Boyce thrashed in his bed until around 2:30 A.M. He got up, dressed, and exited the Jefferson in an unaccustomed way—by the outside fire escape.
The ladder deposited him in a back alley that led to the street. As he turned away from the hotel, he saw that a skeleton crew was manning the cameras.
He caught a cab on Connecticut Avenue and gave the driver the cross streets a few blocks away from his ultimate destination.
"Lady Bethmac's stayin' just up the block from here," the driver ventured.
"You think she did it?"
" _Oh_ yeah. She killed him with that _bowl_. But that lawyer she got, he'll get her off. Uh-huh. Do you know what he charges?"
"No idea."
"Ten thousand dollars. For one _hour_."
"He must be good."
"Oh, he _good_. He get the devil off. He and the devil, they get along _just_ fine. Got a lot in common. Lot of lawyers in this town just like that." He chuckled. "Don't know if the devil going to have _room_ for them all."
He pulled over at the corner of Wisconsin and Newark. Boyce gave him a twenty, along with his business card. The driver inspected it in the light.
"Damn! You him! Can I get your autograph?"
The last time Boyce had snuck into somewhere had been a girl's dorm back in college days. _Plus ça change_. But that dorm had not been under the protection of the Secret Service, and as Boyce contemplated how to breach the perimeter, it crossed his mind that the director of the Secret Service would gladly pin its highest medal on any agent who shot Boyce Baylor, preferably in the balls.
At the entrance to Rosedale, the estate where Beth was encamped, was a duplication of the vigil back at his hotel. Satellite trucks, vans full of slumbering TV crews. The Secret Service had pulled a car athwart the driveway.
Boyce walked down a darkish street that bordered the six-acre property. There were houses whose backyards abutted the estate. Feeling distinctly criminal, he looked both ways and plunged into one of the backyards. He scaled a low redbrick wall, ruining some meticulously trained clematis vines in the process, and pulled himself over into the forbidden zone.
She was staying in the old yellow farmhouse where George Washington had actually slept once or twice, the home having belonged to his friend and comrade in arms, General Uriah Forrest. Forrest, a wealthy Georgetown merchant, had built the house, three miles up from the river, for his young wife, as a port town was not considered congenial to a lady. Once again the old house was serving as the domicile for a lady for whom the city had proved too much.
As Boyce walked toward the house, he decided that since he had no face mask, grappling hook, or silencer-equipped pistol, there was little point in trying to play James Bond. He would go in like the lawyer he was—threatening to sue everyone in his path.
He got his chance soon enough. He heard the scrotum-tightening sound of a German shepherd saying, in German to his handler: "Please— _bitte!_ —please let me just rip out his throat, _then_ you can arrest him." This was followed by a commanding human voice saying, " _Freeze!_ Put your hands where I can see them! Now!"
It took ten minutes of gradually escalating threats and calls to their superiors before they relented and led him to Beth's door. Three of them stood behind him, scowling, at the ready, in case he turned out to be an assassin wearing a rubber Boyce Baylor face mask.
She came to the door, wiping sleep from her eyes, amazed, clearly, to find him there, but perhaps, he thought, not altogether disappointed.
The Secret Service withdrew. He went in, closed the door.
"I have a solution to our basement garage problem," he said.
She was standing by the mantelpiece, one arm across her chest, smoking. "Oh?"
"I asked you this once before. I'm going to ask you one more time."
"Boyce, I told you—"
"Will you marry me?"
Beth's eyes widened. "At three-thirty A.M.?"
"We could wait until the morning. Judge Dutch could marry us during recess. That would give them something for tomorrow's evening news."
She sat on the couch next to him. "Remember the last time you proposed to me? In the rowboat at Fletcher's Boat House?"
"I remember you said yes. This time, if the answer is yes, I want it in writing."
"Shouldn't we get through this first?"
"We're almost there. As long as you drop this idea of testifying."
"Is that what's behind this?"
"No. I came here to ask you to marry me. I would have brought a ring, but the stores were closed. Look, we lost twenty-five years. I don't want to lose the next twenty-five. We won't have any teeth left. As for the rest of it, I'll give up practicing law, we'll go away. I'll build you a castle, in—wherever you want. I've got lots of money. It's embarrassing how much I've made. We'll be together and not give a shit what anyone thinks. We'll leave it all behind. We'll adopt Korean kids. With Russian kids, you never know what you're getting. Don't think about it. For once, go with your heart. Please. Say yes. Prove Nancy Reagan wrong. _Just say yes_."
"Yes."
"Oh baby, that's great. That's just terrific."
"After I testify."
# Chapter 21
No one believed Perri when, wearing her tightest cashmere sweater on _Hard Gavel_ , she hinted that Beth would take the stand _precisely_ because it was going so well, in order to rehabilitate herself with the public.
Alan Crudman ridiculed the idea in his most condescending tone, saying that with all due respect, that was nuts, just nuts. No defense attorney worth his salt would allow it. Forget it.
Perri smiled. Let him play her for the dumb blonde. The freckled dweeb would eat his words. Indeed, she felt confident enough to challenge him, right there and then, to a $1,000 bet. Crudman fell for it.
Next morning, while watching the day's proceedings on TV, Perri felt a warm wave of satisfaction wash through her when she saw on the screen a distinctly weary-looking Boyce—had he cut himself shaving?—approach the bench, along with an amazed, eager Deputy Attorney General Clintick. Judge Dutch leaned forward, then seemed to recoil into his black leather throne like an astronaut pinned by the G-force of blastoff. He recovered, leaned forward again, in an attitude, Perri thought, of barely controlled amazement. His face said, "Counsel, are you _quite sure_?" Boyce nodded, as if accepting reluctantly the terms of a plea bargain he had just resoundingly lost. Judge Dutch stroked his chin, waved the lawyers away from his bench like so many flies, and instructed the clerk to remove the jury. Even on television you could hear the excitement rippling through the courtroom. Clerks who had not moved in so long that some viewers were surprised to find they were actually alive and not wax effigies were suddenly all attention, heads flicking this way and that like aroused cobras.
So began the fierce deliberations by a sleepless, frustrated Boyce to try to render his client's entire prior life history inadmissible. His one consolation was that no tape recording existed of a 911 emergency call to the police from the late President, with him saying, "Help! It's my wife! She's attacking me with a spittoon!" So far as he knew.
President Harold Farkley was in the middle of a meeting with various princes of a Middle Eastern kingdom. He hoped to persuade them to increase their oil production in time to lower prices at the gas pump before the upcoming presidential election. He knew that in return they would ask him to sell them advanced U.S. fighter jets, ostensibly to protect their oil but really to annoy the Israelis. Thus the intractable American position in the Middle East: pleading with Arabs for more oil while providing their enemy with the latest weapons. Another day at the White House.
The chief of staff approached the President's chair from behind and whispered the news into his ear. President Harold Farkley's eyes went vacant in surprise.
"News about the trial?" Prince Blandar inquired.
The President wondered if he should pretend it was something else. "Just a minor development. Nothing, I assure you, as important as meeting with you and Their Royal Highnesses."
One of the other princes asked his cousin to translate.
"His Royal Highness Wazir says that he thinks it was the FBI man, not the Secret Service man, who impressioned the President's forehead."
President Farkley smiled thinly and thanked His Royal Highness for this valuable insight, then tried to steer the conversation back to fossil fuels. One of the other princes began remonstrating heatedly in his native tongue with Prince Wazir. Prince Blandar translated: "His other Royal Highness is expressing to his cousin his belief that the President _fell_ onto the silver expectoration receptacle."
He leaned in closer to the President. "What is your own belief, Mr. President?"
Harold Farkley had not been designed by nature for such critical moments. He knew this. He also knew that the right answer might open those lovely oil faucets a crack and float him to victory in November on a soft black cushion.
"Your Highness," he said, "there are times when I think that the United States could learn a thing or two about justice from some of her allies, such as your own kingdom. You certainly don't let lawyers bring your fine country to _its_ knees! Ha ha."
Prince Blandar nodded appreciatively. "It is true. A matter such as this would have been taken care of very differently. Several years ago, one of the princesses threatened her husband. It was taken care of the next day. A simple death notice was published the week following. On page seven. End of matter."
"Yes, well, it's certainly been a distraction. I'll be sure to pass along what you said to our attorney general and the secretary of state, who I know is very much looking forward to his upcoming visit with the King. Now about the oil..."
Alan Crudman was in a foul temper the next night on _Hard Gavel_ at having been aced out of $1,000 by Perri. He told the viewers Perri had obviously gotten her information from inside sources, whereas _he_ had based his judgment on "legal scholarship." She made him write out a check right then and there, to her favorite charity: a home for the emotionally troubled children of divorce lawyers.
Perri glowed that night on TV. Immediately after the show, she got a call from the head of one of the big three networks—who as it happened was himself recently separated from his second wife—telling her how impressed he was, not just by Perri's uncanny acuity, but by her show in general, and how he very much wanted to meet with her at her earliest convenience to discuss a possible relationship. With the network, of course. Was she by any chance free for dinner that night?... She was? Why, good, good... La Grenouille? Excellent. Everything was so good there. The soufflés!
He'd done his homework.
Faced with potential disaster, Boyce determined to get it all out on the table in his direct examination. He wouldn't leave anything for DAG Clintick, who was salivating to get her turn at Beth.
"Mrs. MacMann," he said, trying to look thrilled that his client was now finally getting the chance to tell her side of the story, "did you kill your husband?"
"No."
_There. See? She's innocent. No further questions_.
"Did you and your husband ever fight?"
"Yes. Often."
"Did you ever throw things at your husband?"
"Yes. On at least eight occasions that I can recall. We had a pretty spirited marriage."
"What did you throw at him?"
"Anything I could lay my hands on."
People laughed.
"Such as?"
Beth considered. "I recall... a book, a paperweight, a stapler... a carton of milk, a shoe—high-heeled—and, oh, the lamp." It sounded like a shopping list.
"Lamp?"
"A desk lamp. It was not a valuable antique, but it did make an impression on him."
More laughter. Judge Dutch glared owlishly.
"Did it require stitches by the White House physician?"
"Yes, it did. Four, I believe. Possibly more. I got him pretty good."
Judge Dutch said that if he heard one more titter, he would order the TV cameras out of the courtroom and clear it of spectators. A billion spectators trembled at the thought.
"Was this incident covered up, for the press?"
"Yes, it was. The press secretary said that Ken—my husband—had swallowed a pretzel and passed out and hit his chin on the way down, nearly killing the dog."
"You say you threw things at the President eight times?"
"While we were living in the White House. I threw things at him on other occasions, before."
"I see. Were these incidents the result of what you might call normal marital stress?"
"Objection."
"Sustained."
"Did you throw things at your husband because he made you angry for a particular reason?"
"Of course. It wasn't just target practice, Mr. Baylor."
Judge Dutch almost laughed himself.
"Did you throw things at him for a _good_ reason?"
"Objection."
"Sustained."
"Did you throw these things at him because you felt he had betrayed you?"
Beth paused. "My husband is dead, Mr. Baylor. He's not here to account for himself, so I would just as soon not go into that."
"Objection."
Judge Umin directed the witness to answer the question that had been put to her by her own defense counsel.
"I respectfully decline to answer that, Your Honor."
"I withdraw the question."
"Objection."
Judge Dutch drummed his fingers on the bench. He waved the lawyers up. Boyce was instructed to instruct his client to answer the question.
"Because," Beth said, "I was upset with him."
"Mrs. MacMann, are you a bitch?"
Murmurmurmur.
"I hope not. I've tried not to be. But in politics some people inevitably are not satisfied."
"Are you familiar with the nickname Lady Bethmac?"
"Yes. It's a pun on the awful wife of a king in Shakespeare's play _Macbeth_. She _was_ a bitch."
"How did the name Lady Bethmac get attached to you?"
"Let's see. One, I caused a number of people at the White House to be dismissed. Two, the people who were dismissed were not happy about this. Three, newspaper headline writers find it hard to resist a good pun."
"Did you cry at your husband's funeral at Arlington National Cemetery?"
"No, I did not."
"Why?"
"I did my crying that day in private. I have tried, to the extent possible, not to express personal emotions publicly."
"Is that why you never threw anything at your husband in public?"
"I suppose."
"The night he died, did you hear anything unusual?"
"I heard a noise, yes."
"Did you investigate it?"
"No."
"Why?"
"My husband was often up in the middle of the night. He was the President of the United States. It's a 24/7 job."
"The antique silver spittoon made by Paul Revere, why was that in your room?"
"It's handsome, quintessentially American. And my husband liked it. He liked to crumple pieces of paper at night and try to get them in, like a basketball hoop."
"Why were your fingerprints on it?"
"It was in my bedroom, Mr. Baylor. I moved it all the time. People do that with objects in their own bedrooms."
"Just a few more questions, Mrs. MacMann. Why didn't you and the President have children?"
Beth looked at her lap. "It was not for want of trying. I miscarried twice, earlier in our marriage."
"Did you continue to try to have children?"
"Yes. I very much wanted children. I tried very hard."
"After the President's death, you were questioned four separate times by the FBI. According to the 302 forms—that is, the FBI reports of those discussions—you did not once request to have an attorney present. Is that correct?"
"Yes."
"Were you also aware that you were under no legal obligation or compulsion even to speak with the FBI?"
"I was aware of that. I'm a lawyer myself."
"Why did you speak to the FBI?"
"I wanted to tell them what I knew."
"Why didn't you have a lawyer present when you did?"
"I had nothing to hide."
"Even when you became aware that you were a suspect?"
"I still had nothing to hide. Nor do I now. That's why I am testifying."
"Thank you. Your witness, Ms. Clintick."
# Chapter 22
Beth was in a jubilant mood. They were on their way back to the Jefferson after her testimony. The last time Boyce had seen her this happy was in the 1970s. When they'd walked out of the courthouse minutes ago, the spectators had done something they hadn't before. They'd applauded. Now she was hugging Boyce's arm in the car, nuzzling him, saying, "See?"
He smiled gamely, but his mood didn't match hers. He felt like the Japanese admiral on December 7, 1941, pacing the bridge of the _Akagi_ , thinking, _They are going to be so-oo pissed_.
When they reached the war room on Boyce's floor, everyone there applauded. Beth gave a little stage bow and a victory sign.
Boyce wasn't about to ream them out in front of her, but it made him boil. It was a violation of his most sacred rule: _Do not tempt the gods_. And most sacred rule (b): It's not over until the jury foreman says, "Not guilty."
But now here Vlonko came up to them, saying that Jeeter was "off the charts." During Beth's testimony, he reported, most of the jurors had sat there "nodding up and down like those fucking rear dashboard puppies." He reported that Judge Dutch had had his clerk go back to the jury room before the lunch break and tell them to cut it out with the nodding.
"Maybe he'll have to put fucking hoods over them!" Vlonko chortled.
It was good news. Still, this postvictory atmosphere in his war room was making Boyce very nervous. He could hear the gods murmuring, could hear the clanky sound of Vulcan's smithy as he pounded out lightning bolts.
The associate in charge of media monitoring came up, beaming like a July sunflower, to tell Beth breathlessly that the National Association of Former First Ladies had issued a statement supporting her. This _was_ significant. The NAFFL was one of the most powerful lobbies in Washington.
There was more: The television commentators were beside themselves over her performance. The associate paused in mideffusion to tell Boyce that he, too, was getting good reviews. Beth went off to watch for herself, leaving Boyce alone in the crowded room with his darkling presentiments.
Phone call for you, Boyce, line five. Judge Dutch's clerk.
The prosecution had just requested—and had been granted—a three-day recess. They needed time to serve subpoenas on several new witnesses.
A string of four-letter words went through Boyce's head.
Who were the witnesses?
The list is being faxed to you right now.
Boyce hung up. Beth emerged from the media room, all excitement. Like Marilyn Monroe returning from her world tour to tell Joe DiMaggio, "You never heard such cheering!" Someone on TV had proposed that as soon as the trial was over, Beth should announce her candidacy—for president!
Boyce, fax for you. Just came in.
He read the names. There were three. Lonetta Sue Scutt. Who was this?
He led Beth into his study and closed the door and showed her the list. She read it. Remarkable how quickly facial muscles could rearrange themselves.
Beth explained about Lonetta Sue Scutt. "Is she going to be a problem?" she asked.
"By the time I'm through with her she'll be so radioactive they won't allow her in tunnels. But tell me about Damon Blowwell. And about Dr. Mark Klatz." He sat down. "Tell me all about them."
"Damon was Ken's political director. Before that, he was his campaign mana—"
"I read the papers, Beth. I know _who_ he is. Tell me why he's suddenly a prosecution witness."
Beth considered. "I'm not sure."
"That's helpful."
"I don't know. Damon and I got along all right. I mean, he wasn't... well, he didn't really _like_ me. But I don't know why he'd be out to get me."
"Why didn't he like you?"
"He and Ken were very close. I was the pain-in-the-ass wife. You know how tribal men can be."
"Did he have any specific reason not to like you?"
"He thinks I killed Ken. That would be his main reason, I'd guess."
"Any other reasons?"
"During the primaries—this was before he stopped drinking and got religion—he was putting out word that one of the wives of the other candidates was a lesbian who was having a hot-and-heavy with her trainer. I told him to cut it out. Came down on him sort of hard. But I mean, it was no big thing."
"He's coming to testify against you. That is a big thing."
Beth considered. "He was in Vietnam."
"Not another war hero."
"Green Berets. I only heard him talk about it once. It was late, we were all stuffed into this little plane. It had been a long day. And he talked about what he did in the war. It was... kinda out there. If I'd been the Viet Cong, I wouldn't have wanted to have Damon with his knife crawling into my hootch in the middle of the night."
"Wonderful. We have a hostile Green Beret on our hands. But why is he hostile? We've got three days to find out. I can put my people on it, but it would be helpful if you could give us some direction."
"It's funny."
"What, possibly, could be funny?"
"I'm on trial for assassination. And Damon _was_ an assassin. I mean, that was his job. They had a name for it, even. Wet work. He and Ken used to joke about how it was perfect training for politics. But I don't know why he'd be testifying. Sorry."
"Well, when you're not glued to the TV listening to people talk about how you should run for president, try to come up with something."
"Are we feeling hostile?"
" _We_ are feeling that _everyone_ is being _way_ too _overconfident_. Who is Dr. Mark Klatz? Did he advise you which spot on Ken's head to aim for with the spittoon?"
"He's my gynecologist."
"Jesus. And why would your gynecologist be testifying against you?"
"It's just none of their damn business."
"What is 'none of their damn business'?... _Be-th_?"
Deputy Attorney General Clintick put Dr. J. Mark Klatz on the stand first. To Boyce this meant that he was the government's weakest witness. Damon Blowwell she'd scheduled third. That meant his testimony was the most devastating. He had six investigators poring over Damon Blowwell's military records, tax records, credit records, school records. With any luck, it would turn out that he massacred innocent civilians and was an alcoholic wife beater. Beth still swore she had no idea what he had against her.
The deputy AG spent an hour going over Dr. Klatz's impeccable credentials. Boyce already knew how impeccable they were.
He was low-key, in his early sixties, with glasses. His first name was Julius, apparently in honor of the eponymous Roman emperor whose birth gave us the term _C-section_. He had headed up the OB-GYN department at Mount Sinai Hospital. He'd advised the United Nations committee that was trying to get African countries of fundamentalist Islamic bent to outlaw the practice of cutting off the clitorises of young girls to discourage them from having sex. He wrote newspaper op-ed articles deploring this barbaric form of chastity enforcement. In short—and he was that, too, which somehow enhanced his professional aura—he was the sort of person you would want peering between your legs, going, "Hmm..."
Dr. Klatz was manifestly unhappy at being present. He looked as if he would gladly perform a clitoridectomy on the deputy attorney general, without anesthesia.
"When did the defendant first come under your care, Doctor?"
"April of 1983."
"Why did she come under your care?"
"She was recommended."
"Why was she recommended?"
"She had experienced a second miscarriage the previous month. Her physician referred her to me."
"What was your evaluation of her, medically?"
"That's none of your business," the doctor said. "It's none of anyone's business."
Judge Dutch instructed the doctor gently to answer.
Dr. Klatz shook his head. "With all respect to you and the court, you can find me in contempt and put me in jail, but I will not answer that question."
Judge Dutch drummed his fingers and contemplated the dreariness—for everyone concerned—of having the doctor dragged off in handcuffs. He waved up the lawyers.
One of the television networks had hired a lip-reader to decipher what the judge said during the sidebars. They couldn't broadcast a direct translation, of course. But their correspondent certainly seemed to have an uncanny knack for predicting just how Judge Dutch would rule.
The correspondent told his viewers, "My guess is that he will _not_ force the issue and will let the prosecution proceed along a parallel line of questioning."
"Dr. Klatz," the prosecution continued, "did you prescribe a regimen of birth control pills for Mrs. MacMann?"
Dr. Klatz looked at Beth. What was the use? They had the prescriptions.
"Yes. You already know that."
"Has Mrs. MacMann remained your patient since April 1983?"
"Yes."
"And has she continued to take birth control pills under your prescription since that time?"
"I have prescribed them. Whether she took them, I can't say."
The deputy AG asked the court to enter into evidence a thick stack of paper, prescriptions dating back two decades and continuing until recently.
"Tell me the _good_ news," Boyce said to his associate who monitored the media.
Beth had gone back to Rosedale for a long bath and, Boyce suspected, good cry.
"The women," the associate said nervously, "are furious. The message they're sending is 'Hands off her body,' 'None of your business,' echoing Klatz's line. The head of the National Organization for Women used the term _vast male conspiracy_. The National Association of Former First Ladies issued a guarded statement standing by her."
"Now the bad news."
"The word _liar_ was used twenty-three times during the evening news cycle. CBS used the term _credibility problem_. ABC is teasing tonight's _Nightline_ show with the line 'Can we believe her?' Tomorrow's _New York Post_ has a story quoting—indirectly—the archbishop of New York saying that if she and the President had had a kid, maybe all this wouldn't have happened."
Boyce groaned and went off to get Vlonko's report.
Vlonko was staring at his computer screen, scowling.
"We got problems with two, four, and eight. Maybe big-time problems."
"Two is the Catholic with four kids?"
"Five fucking kids. And the sister with the Down's baby."
"What about four and eight?" Juror number four taught Sunday school and just loved kids, according to her questionnaire. Juror eight was liable to feel betrayed at hearing that a defendant who'd told the court how desperate she was to have a baby had been popping birth control pills like breath mints for two decades. On weekends, she volunteered at an adoption agency.
"I would say, Boyce, not so fucking good," said Vlonko. "Lips very tight all day. Hardly moving. Hands on laps. This is hostile posture."
The drawback to being a trial attorney was that you couldn't, after a bad day, stun yourself into insensibility with a good stiff drink. The lovely clink of ice, the little cat's feet padding up to the cortex, the furry body rubbing up and down against it like a scratching post.
Not tonight. Tonight would be a long night, spent closely reading articles from medical journals and psychiatric journals about birth control pills as a hormone management tool and on the long-term post-traumatic effects of miscarriage.
# Chapter 23
Boyce's cross of Klatz elicited a nearly unbroken string of yeses from the eager-to-help doctor, but it felt like bailing a leaking boat with a too small bucket. He'd have kept Dr. Klatz on the stand longer if he could, just to bathe the jury in his amiable, nonjudgmental, pro-Beth aura.
The prosecution had given Lonetta Sue Scutt a good scrubbing and put her in a dress that managed to cover most of her tattoos. Her hair had been dyed so dark that it had a granular quality, like a wig made from shoe polish and fishing line. For someone who lived in the desert, she had suspiciously pale skin, and decades of two packs a day had cured her vocal cords to sandpaper. She listed her profession as "home-maker" and "exotic dancer."
Aware that her witness did not present the image of Mother Teresa, Clintick kept her direct examination brief.
Had she been in the employ of Governor and Mrs. MacMann? Oh yeah. Had she observed stress in the marriage? Ohh yeah. Had she heard Mrs. MacMann express her intention to—she was quoting here from a statement she had made to the FBI—cut off the governor's penis? Uh-huh. Is that a yes, Ms. Scutt? Oh yeah. And was she fired shortly after overhearing this? Uh-huh, and she threatened me to keep quiet or the state troopers would take care of me. Thank you. Your witness.
Boyce was courtly. He showed Lonetta Sue Scutt no less respect than he would have the Queen of England. Ms. Scutt, are you currently taking any medication? I take a few pills, uh-huh. You have a prescription for OxyContin? That's a powerful painkiller, is it not? Yeah, and I got a powerful pain. What was the cause of the pain, Ms. Scutt? There was this accident. What kind of accident? I got some battery acid powder up my—in the sinuses? Really? How does battery acid powder get into the sinuses, Ms. Scutt? It was an accident, like.
Boyce admitted into evidence the emergency room report from the Morongo Basin Hospital. Lonetta had snorted the battery acid powder. Her cocaine dealer had given it to her. She had paid him for her previous purchase with sex. Along with the sex was included a nasty dose of sexually transmitted disease. The dealer paid her back by substituting granulated battery acid powder for cocaine in her next purchase. It was a miracle she hadn't died.
"Irrelevant!" DAG Clintick cried.
Boyce fired back that she should be ashamed to have called such a witness in the first place. The photo of a foggy-glassed Judge Dutch angrily pointing his finger at them made the cover of _Time_.
"My guess," the lip-reader-assisted network correspondent told his viewers while the judge was wagging his finger and threatening to fine both Boyce and Sandy, "is that Judge Umin may be so _fed up_ at this point that he's prepared to sanction both the defense _and_ prosecution."
"Ms. Scutt," Boyce continued, "did you telephone the _National Perspirer_ and try to sell your story to them for the sum of one million dollars?"
"Why not? Everyone else connected to it is making a fortune."
Lonetta was refreshingly candid.
"Two final questions, Ms. Scutt. Did you tell the _Perspirer_ that while serving lunch to Mrs. MacMann and her friend Mrs. Hackersmith, you heard Mrs. MacMann say she was going to cut off the governor's penis?"
"That's what she said."
"And would this be the same gubernatorial organ that you—that you had had in your physical possession before serving lunch to Mrs. MacMann and her friend?"
"Objection!"
Sidebar.
"I will rephrase the question, Ms. Scutt. Were you orally acquainted with the governor?"
"I don't have to answer that. Do I, Judge?"
Before Judge Dutch could answer, Boyce said softly, "I won't keep Ms. Scutt any longer, Your Honor. No further questions. I would like to recall Mrs. MacMann."
Beth took the stand. "Mrs. MacMann, did you threaten in Ms. Scutt's presence to cut off the governor's penis?"
"No, that's inaccurate. I told Mrs. Hackersmith that I was going to cut off his balls."
It took some gaveling to quiet the court.
"Your Honor," Beth said, "I apologize for the language. I could have used a more general anatomical term, but I wanted to quote what I said verbatim."
Judge Dutch, whose glasses were now opaque with vapor, merely grunted. Boyce continued.
"Did you dismiss Ms. Scutt because she overheard you discussing your... surgical fantasies vis-à-vis the governor?"
"No," said Beth, looking directly at her accuser. "Ms. Scutt knows very well why she was dismissed."
"Objection."
"Withdrawn, Your Honor. No further questions."
The mood that night in Boyce's war room was somewhat improved. Until Beth said to him quietly, "About what Damon might say? There was this conversation I had at one point with Ken...."
Damon Jubal Early Blowwell looked as if he might still be in the military and not some K Street political consultant. He was in his midfifties, wore his hair trimmed to within a centimeter of his skull, and kept his jaw in a permanent jut. He had suspicious brown eyes, the tight lips of someone anticipating disrespect, and the physique to do something about it. When he smiled, his whole face seemed to suck inward at the center in a fierce pucker that made him look not entirely human. His normal expression was a scowl.
He answered with "Yes, sir" and "No, sir" and bit off the ends of sentences like tobacco plugs. When the clerk swore him in, Blowwell stood rigidly erect and added one last word to his answer: "So help me _God_." Vlonko told Boyce afterward that when Blowwell took the stand, all nine male jurors sat up in their seats.
Boyce had gone back over every public utterance Blowwell had made following the President's death. He had never come right out and accused Beth of murder, but for someone who had been such a faithful family retainer, his coolness toward her had been conspicuously glacial.
Blowwell had gone to work for Ken MacMann the moment that Ken announced he was running for president. He'd been a hard-partying political hack in Alabama. Getting his fellow Vietnam veteran elected president had restored a sense of purpose to his life. He quit drinking and became a born-again Christian. When a former Green Beret with two Bronze Stars finds his way back to the path of righteousness, it's prudent to get out of his way. The citation for his medals was classified. Boyce's Pentagon moles had found out that they were for assassinating eight high-level Viet Cong cadres.
Blowwell had become wealthy since leaving the White House. He now had clients all over the world. But President MacMann's death had been hard on him. Boyce's investigators had found out that he had increased visits to Alcoholics Anonymous to five a week—up from one a week when he worked at the White House. Boyce hoped he would not have to mention that in court, especially since jurors four, seven, and fourteen had relatives who attended AA. He also hoped he wouldn't have to insinuate that Damon's war experiences had left him with, as they say, "issues." Jurors one, three, six, and fifteen had friends or relatives die or be wounded in World War II, Korea, and Vietnam. Boyce really didn't want to have to blow his nose all over a military man's ribbons.
DAG Clintick took her time walking Damon through his background.
"You served _two_ tours in Vietnam?"
"Yes, ma'am."
"Was that unusual?"
Boyce knew very well she was just trying to get him to object.
"Probably was not typical."
"Why did you serve two tours in Vietnam?"
"I wanted us to win."
Jurors one, six, and fifteen were nodding.
Boyce thought, _This one is the real article, and he's coming right at us_.
"What did you do in Vietnam, Mr. Blowwell?"
Beth whispered, "Shouldn't you be objecting?"
"Shh."
"My job was to kill the enemy."
"You were Mr. MacMann's campaign manager and, when he became president, his political director at the White House. What did that job entail?"
"Killing the enemy."
The courtroom erupted in laughter. Judge Dutch himself grinned. Boyce thought, _Slick, very slick_.
"Did you and the President have frank discussions?"
"Wouldn't have been much point in having unfrank discussions."
"Of course. You were his confidant, after all." Ms. Clintick smiled. "He trusted you."
"And I trusted _him_."
"Did President MacMann ever discuss his wife with you?"
"Objection."
Sidebar.
DAG Clintick continued, "Did the President ever confide in you whether he was dissatisfied in his marriage?"
"He did. He told me that he wanted to divorce Mrs. MacMann."
The courtroom stirred.
"Did he say when he wanted to divorce her?"
"Immediately following the reelection."
Judge Dutch had to gavel the courtroom to silence.
"Did he say whether he had made this intention clear to Mrs. MacMann?"
"He told me that he had discussed it with her."
"And what was her reaction?"
"She was not pleased. He said she had called him a name."
"What name?"
"It's a pretty salty term."
Judge Dutch reluctantly gave Blowwell permission to continue.
"She called him a 'cocksucker.' "
Gasps, gaveling. Network censors scrambled, too late, to hit the bleep-out button. Throughout America, mothers cautioned their children that such language was not to be repeated in their households. In Europe, the sound of laughter could be heard through a million windows. In Asia, there was confusion as precise translations were sought. Judge Dutch finally removed his useless glasses.
"So it would be fair to say that Mrs. MacMann was displeased by the President's revelation of his intention."
"I would say yes."
"Did the President say if Mrs. MacMann had said anything further with regard to her intentions?"
"He told me that she was planning to run for public office herself—the governor's office that he had held—after the President was reelected. He said she told him that she would agree to a divorce once she had accomplished that. She told him that the only way she would leave the White House was on her terms."
Murmurmurmurmurmur.
"Thank you, Mr. Blowwell. No further questions for the witness at this time, Your Honor."
Boyce was fantasizing: His associate burst through the courtroom doors, breathless, tie askew, bruised, even missing a shoe. He was clutching a U.S. Army dossier designated "Top Secret." Inside was a report that Sergeant Damon Blowwell had been dishonorably discharged for massacring an entire elementary school of peace-loving Vietnamese children, including the school mascot water buffalo, Phong. He had decorated the bar at the noncom officers' club with their little pigtails. Not only that, but—
"Counselor?"
# Chapter 24
Well," Boyce said once they were back at the hotel behind closed doors, "your campaign to rehabilitate yourself is coming along nicely."
"Don't start."
"I think I'm doing an admirable job of not starting. You're lucky I don't have a spittoon handy."
"Damon blew that conversation way out of proportion."
"No, darling. What he blew was us. Out of the water."
"You recovered well. I thought your cross brilliant, insinuating that he was a religious fanatic and war criminal."
"We did not 'recover' today. All that was purely for the benefit of juror three."
"Which one is he?"
"She. By now you should know these people better than your own relatives. The lesbian who hated her Baptist military father."
"Oh, her."
"In case everyone else in the jury fell in love with Damon, she's our only hope. My God, what a disaster."
"Damon was spinning. It wasn't untrue, but he made it sound worse than it was."
"Did you call Ken a 'cocksucker'?"
"Yes. And he was."
"When you take the stand again you can tell the jury it was a pet name. My widdle cocksucker. You do realize that Blowwell would not have taken the stand if you hadn't testified? His lawyer told me as much. It was your testimony that finally put his needle into the red and made him come forward with all this."
Boyce took off his tie and hurled it across the room as if ridding himself of a snake that had wrapped itself around his windpipe. "What _is_ it with all these war heroes? You can't throw a stone in this case without hitting one. Didn't your husband ever hang out with _normal_ people?"
"I think Damon has a problem with female authority."
"He can discuss that on TV with Oprah when he writes his book. Meanwhile, _we_ have a problem, with _him_."
Boyce picked up the phone and buzzed.
"George? Boyce. Did you get anything on him?... No VC ears?... Are you sure? I'd lay odds there's a My Lai in that man's record somewhere. Have you spoken to _everyone_ in his platoon?... Well, track him down in goddamn Peru, George, I don't care what it takes.... Then _hire_ a goddamn helicopter. What about his AA friends? I _know_ AA types are fiercely loyal to each other, but we're not dealing with samurai warriors here, George. They're recovering alcoholics. You get 'em alone, you pull out a bottle of hundred-dollar Scotch and hold it under their noses, and I promise, within ten minutes they'll be singing 'Whaddya Do with a Drunken Sailor' and telling you everything you want to know." Boyce hung up.
"What?" he said to Beth, who was looking at him with horror.
"Remind me," she said, "was I in class the day they taught us to suborn recovering alcoholics?"
"Uncivil Procedure 101. My favorite course."
Beth suddenly paled.
"You okay?"
She bolted for the bathroom door. She emerged ten minutes later looking shaky.
"Didn't mean to upset you," Boyce said.
"I've been upset since the seventies."
"Now you're cookin', George." Boyce hung up.
"Great news," he said to Beth. They were having breakfast, Boyce tucking in heartily to his usual hot oatmeal with wheat germ and mixed berries. Beth bird-nibbled at a muffin and sipped at tea. Her color was still off.
"Guess who beat up an antiwar protester in the seventies for lipping off to him and calling him a baby murderer? Sergeant Blowhard!"
"I'd have done the exact same thing."
"This information was not easy to come by. You could be more enthusiastic about it, you know. Apparently he was tanked when he slugged the guy. That's why he brought it up in AA. This is good. We can use this."
"Oh, Boyce, you didn't pour booze down some poor alcoholic's throat to get this? I just don't think getting recovering drunks drunk is right."
"Don't get ethical me with me, Spittoon Girl."
Beth burst into tears. "I'm sorry," she said. "I just can't seem to get a grip."
"It's okay," Boyce said, helpless as any male confronted with a weeping female, "I'm not going to subpoena a recovering alcoholic."
"You're not?" Beth blew her nose into a stiff Jefferson Hotel napkin.
"Not because I've gone soft," Boyce said. "With the mood the country's in right now, the jury would award Blowwell damages for skinning his knuckles on an antiwar protester."
Beth blew her nose. "Probably right."
He patted Beth's hand. "We'll figure something out."
"Let's go now to our special legal correspondent, who is outside the courthouse. Jeff, how did it go today?"
"Peter, this was _not_ a good day for the defense. Beth MacMann's attorney, Boyce Baylor, filed a motion last week seeking to have Damon Blowwell examined by a court-appointed psychiatrist, in an effort to establish that Blowwell—as the motion put it—has a history of 'vicious sociopathic behavior characterized by extreme violence.' The basis for this is that Mr. Blowwell allegedly hit an antiwar protester back in the seventies. Baylor seized on the incident and tried to have Blowwell's court testimony, considered highly damaging to the former First Lady, thrown out by Judge Umin."
"And how did the judge rule today?"
"Just ten minutes ago, Judge Umin _denied_ the motion. Morever, he did so in unusually harsh language, indicating that he is growing rapidly _impatient_ with the defense."
"So Mr. Blowwell's testimony stands?"
"Yes. Further, we've just learned that Damon Blowwell has filed a thirty-million-dollar defamation suit against Boyce Baylor. So the atmosphere down here at the U.S. District Court is highly _charged_. Peter?"
Beth had taken some kind of downward turn that had Boyce at a loss. It was as if she'd lost interest in her own case. In court, she stared straight ahead, a terrible, guilty-looking eye posture—and twice had passed him urgent notes saying she needed a five-minute recess—"NOW!!" The moment they were granted, she flew toward the side door.
Naturally, these quick exits did not go unnoticed. It is difficult to go unnoticed when you are being seen live by over one billion viewers. Commentators remarked that she seemed to be under quite a lot of stress. Asked about this on the steps of the courthouse one day, Boyce was sorely tempted to say, "She's on trial for murdering her husband. Of course she's under 'considerable stress,' you pigeon-brained idiots." Instead he remarked that the reason for her downcast countenance was that, as former First Lady of the land, it tugged at her heartstrings to see the country she so loved torn apart by this tragedy.
But as one pundit put it, the country was not being torn apart. If anything, it was rapidly approaching unanimity on the matter of her guilt.
"Boyce?"
"What?" He was in a foul mood. Judge Dutch had denied yet another motion, his case was going down in flames, and the night before on Perri's show, Alan Crudman had declared that Boyce Baylor had made a "tragic error" in putting Beth on the stand. He knew very well Boyce had tried everything short of locking Beth in a closet to keep her from taking the stand.
"There's something I need to tell you."
He'd been afraid of this. It had happened before. And it always happened right about now: The client would break down, _just_ as Boyce was about to go in and give his closing argument, and blubber all over his legal pad that—sob, gasp—they _were_ guilty. Thank you. Thank you for sharing that with me as I prepare to go in and tell the jury that they are about to make a terrible mistake.
He said, "Beth, you could really help me right now by—"
"I'm pregnant."
They were in the car going to court. He could hear the courtroom rumbling from this atomic news, the media gasping with pleasure—a whole new layer of scandal!—the shocked, drawn faces of the jurors, spectators clamoring, Judge Dutch, eyeglasses fogged, gaveling, gaveling, ordering the bailiffs to clear the courtroom. He saw headlines, the evening news, heard the titter of his colleagues. He saw it in all its dire and awful vividness.
"That's so... great," he squeaked.
He was seized with joy. He'd never heard such good news. He'd never wanted children by any of his wives, sensing as he had that none of the marriages was likely to last. And now the only woman he had ever really loved had just announced that she was pregnant by him! Admittedly, the timing could have been better—twenty-five years later and in the middle of her trial for murdering the President of the United States. Otherwise it was wonderful news.
He detached himself from her long enough to ask, "But—you were on the pill."
"I went off them about the time we went to trial. I was getting headaches and the doctor said to stop for a while while he monitored my estrogen. I never got around to going back on. It wasn't as though I were likely to get pregnant, right? I thought you might have guessed. All those trips to the bathroom."
"A lot of clients have to use the bathroom in a hurry. Nerves. I was focused on the case."
The case!
He saw himself standing next to Beth in front of Judge Dutch. For the sentencing. Judge Dutch was wearing contacts so his glasses wouldn't fog. Beth's belly was huge with child. She was wearing maternity clothes. They held hands, not proper in court, strictly speaking, but they couldn't help it. Judge Dutch's voice kept catching in his throat. "In light of your condition, Mrs. MacMann, the United States will not avail itself of the sentence of death which would normally be imposed in such a grievous, indeed, heinous case. But because you have been found guilty of one of the most serious crimes there is—if not _the_ most serious—it is the judgment of this court that you serve the balance of your life in prison, without possibility of parole."
He heard the gasps, the sobs. He turned, saw the tears streaming down Beth's cheeks as she stood there for the last time in her life wearing nonprison clothes. Saw the marshals approaching with steel manacles and leg chains. Heard Judge Dutch straining to control his own emotions as he concluded that this terrible tragedy had claimed more than the life of the President of the United States—it had forever blemished the honor of the United States and, perhaps most tragically of all, had robbed an unborn child of its mother, who would be only a person in an orange uniform on the other side of thick glass. Case closed, and may God have mercy on us all.
Down came the gavel. Beth was led away.
"No!" he cried.
"Boyce? You okay?"
He filed motion after motion. "Loco motions," they were dubbed by _American Lawyer_ magazine. He moved to dismiss on the grounds that Beth's Secret Service detail was spying on her and passing the information along to the prosecution. Judge Dutch tossed it in the judicial wastebasket. Boyce moved for a mistrial on the grounds that one of the jurors had just dozed off for five minutes during a stretch of stultifyingly dull testimony by an expert in acoustics. Into the wastebasket. He moved for a mistrial because the second cousin of juror fourteen signed a contract for a book titled _Second Cousin of Juror 14: My Story_. Waste-basket. Three for three.
Boyce dispatched his most unethical investigator—a former U.S. intelligence agent who had had to resign after being caught selling Stinger antiaircraft missiles to the Serbs—to Vietnam with a suitcase of hundred-dollar bills with which to bribe an entire hamlet of Mekong Delta peasants into suddenly recalling that Sergeant Damon Blowwell had wantonly massacred half its population one night—just for the heck of it. The scheme fizzled when the investigator got as far as Bangkok, where he exchanged the $100,000 for heroin and caught a flight for Amsterdam, where he exchanged the heroin for $500,000 of ecstasy, which he secreted in large wheels of Gouda bound for Atlanta. It would have been awkward for Boyce to pursue him through the courts, so he let it go, charging the $100,000 to one of his corporate clients as a week's worth of "photocopying and messenger services."
Judge Dutch dismissed each of Boyce's motions with mounting choler and vexation, at one point warning him icily that if he received one more of these appalling roadblocks, he would call down upon Boyce's head "the lesser angels of my nature."
Boyce's furious motion filing was to buy a few weeks for Beth to get over her morning sickness before the deputy AG got her on the stand for cross-examination. That was going to be bad enough without Beth having to be excused every five minutes to dash to the bathroom. As a rule, juries are not impressed if you have to throw up every time a difficult question is posed.
Judge Dutch was getting suspicious of Beth's frequent calls of nature. His clerk had told Boyce that the judge was considering having her medically examined. That was to be avoided at all costs. Meanwhile, Boyce put it out to the press that Beth had been temporarily inconvenienced by a nasty "tummy bug." In moments of daydreaming, he found himself calling his child by the nickname Tummybug.
# Chapter 25
United States calls Elizabeth MacMann."
How subtle, Boyce thought, of Clintick to drop Beth's maiden middle name, Tyler, which Beth had always made such a point of using. It would start Beth's cross-exam on a note of annoyance.
He slid his legal pad across the defense table toward Beth.
_She's wearing panty hose underneath_
She gave him a smile that said, "I'll be okay."
The television commentators went into their TV golf tournament whisper.
"Elizabeth MacMann is rising... walking around the defense table... walking now toward the witness box... climbing up into the witness box... Barbara, how would you describe her outfit?"
"It's a pantsuit, of course. Black. We know that much. We do not know _who_ the designer is. It looks like a cross between Ann Taylor and Carolina Herrera...."
"Judge Umin now reminding Mrs. MacMann that she is still under oath."
"She's made a point, generally, of wearing clothes by American designers...."
"Sitting down, now..."
"You'll notice she is _not_ wearing the pearl necklace that she wore when she took the stand previously."
"What do we read into that?"
"I'm not sure. It was given to her by her late husband. So you could read all sorts of things into it. Or not."
"Deputy Attorney General Sandra Clintick, approaching the witness stand. What's she wearing, Barbara?"
"We do have that information. Saks Fifth Avenue double-breasted jacket with skirt and off-white crepe de chine blouse—"
"I have to interrupt you—here we go."
"Mrs. MacMann," DAG Clintick began, "you testified earlier that there—I'm quoting from the transcript—may have been more than eight occasions when you violently attacked your husband. Is that correct?"
"No, it's not. I said I might have thrown something at him. I did not characterize it as a violent attack."
"You do not consider throwing objects at or striking people acts of violence?"
Boyce winced. It was a textbook instance of why a defendant should never take the stand.
"I would not consider, for instance, throwing your shoe at your husband in a moment of domestic stress a quote violent attack. I would consider that pretty fairly standard husband maintenance."
There was laughter in the courtroom, though none, Vlonko pointed out later, came from the jury box.
"Would you consider throwing a heavy metal object at his skull a violent attack?"
"Objection. Conjecture."
Overruled.
"Yes," Beth said. "I certainly would. I'd consider it not only violent but unlawful and punishable at law."
"As First Lady, you spoke out against domestic violence."
"Yes, I did. On numerous occasions."
"Do you consider that hypocritical?"
"Objection."
Overruled.
"No, Ms. Clintick. I distinguish between marital spats and domestic violence."
"Even when these so-called marital spats result in contusions, lacerations, bruising, and stitches?"
"My husband was a six-foot-three former naval officer and outweighed me by more than seventy pounds, Ms. Clintick. He was perfectly capable of defending himself from the likes of me."
"Even at night, in the dark, while he slept?"
"Objection."
Sustained.
"You're asking hypothetical questions, Ms. Clintick," Beth said. "I'll answer directly: I did not hit my husband with that spittoon in the dark, as he slept."
"Was he awake when you hit him?"
"Objection. Asked and answered."
Overruled.
"I did not hit him."
"Did you throw the spittoon at him?"
"Objection, Your Honor. Asked and answered. Ms. Clintick's line of questioning constitutes harassment."
Overruled.
"I've answered that question, Ms. Clintick," Beth said tightly.
"Answer directly. Did you throw the spittoon at him?"
"I told the FBI agents that I did not."
DAG Clintick looked over toward the jury and apparently liked what she saw.
"Mrs. MacMann, as a college student, you played softball?"
Boyce knew this one was coming. The horror, the horror...
"I did."
"What position did you play on the team?"
"I was the pitcher."
"So your aim would be pretty good, wouldn't it?"
"With a softball, decades ago."
"You pitched four no-hitters in the season your senior year."
"The batters weren't that good. No disrespect to Smith College intended."
Clintick's alma mater, as it happened.
"In your testimony, after admitting that you had violently attacked your husband on numerous occasions, when you were asked why you did that, you replied that since your husband was dead, you were not going to say. Is that correct?"
"Yes, it is."
"Whom are you trying to protect, Mrs. MacMann? Your dead husband or yourself?"
"Objection."
Overruled.
"I am trying, Ms. Clintick, to defend myself against a charge of murder. But I will not do that by dragging down a man to whom I was married for twenty-five years and"—Beth sighed somewhat—"who is considered by the country a hero."
"Did your husband cheat on you?"
"That's none of your business, Ms. Clintick."
"Your Honor?"
After a sidebar, Judge Dutch instructed Beth to answer the question.
"You would have to define 'cheat' for me."
"Did he sleep with other women while he was married to you?"
"I very much doubt that."
An explosion of laughter.
"Mrs. MacMann, were you aware that your husband engaged in sex with other women?"
U.S. marshals were poised to serve subpoenas on half a dozen of Beth's friends to whom she had confided her problems over the years. If they denied that Beth had told them about it all, they would open themselves up to charges of perjury. Beth knew this. She had nowhere to go.
"I prefer not to be aware of some things," Beth said.
"Is the name Amber Swenson familiar to you, Mrs. MacMann?"
"Yes."
"Rita Ferreira?"
"Yes."
"Violet Bronson?"
"Yes."
"Jo Anne Casardo?"
_"Yes."_
"Tammy Royko?"
"Uh-hum."
"Is that a yes, Mrs. MacMann?"
"Yes."
"Cass Macklehose?"
"Yes."
"Serena Whitmore?"
"Yes."
"Objection. Your Honor, is the prosecution going to read the _entire_ phone book?"
"With the court's indulgence, there are only twelve more names on this list."
Throughout the country, phones rang. The next day, the headline ALL THE PRESIDENT'S WOMEN appeared in three hundred newspapers. DAG Clintick came under heavy fire from the women's groups. Lawsuits were threatened, none filed. Ms. Clintick's team had done their homework diligently.
As this honor roll was called, Boyce forced his features into a blank expression. Clintick moved in for the kill.
"Is the name Babette Van Anka familiar to you?"
"Of course it is," Beth snapped.
"No further questions at this time, Your Honor. Reserve the right to recall the defendant at a future time."
"How bad?" Boyce asked Vlonko as they stared at the numbers on the screen.
"Fucking bad."
"Beth, honey?" They were lying together on top of the bed in Boyce's suite, staring at the ceiling, holding hands. "There's something I need to tell you."
"Can't we just not think of anything right now?"
"Now is not a good time for not thinking of anything."
"That's so Washington," Beth said. " 'Now is not the time for partisanship.' 'Now is not the time for politics as usual.' Please."
"I have to be your lawyer for a moment."
Beth sighed.
"The case," he said, "as you may be aware, is not... is, well, it's not..." He was so unused to giving his clients bad news that he was at a loss.
"We're going down," Beth said. "In flames."
"We're not going up. We're not really even going sideways. Eliminating those directions leaves down."
"I'm so sorry, Boyce. I screwed everything up. I'm _so_ sorry."
"Now is not the time for self-recrimination."
"After I'm convicted—would that be the time?"
"Let's look at it from a purely tactical point of view. If it comes out that you're pregnant, and that's a question of when rather than if, the jury is going to feel very jerked around. You say on the stand that more than anything you wanted to have a baby. Then your OB/GYN says that you've been on the pill. Then you get pregnant in the middle of your murder trial. It's not an ideal situation."
"What _is_ it with these hormones? All I can do is burst into tears."
"Now is not the time to burst into tears. Right now I need you sharp and hard. I need you pre-sorry, pre-pregnant. I need Lady Bethmac."
Beth wiped her tears defiantly. "All right. Screw sorry."
"That's my girl. Ready?"
"Ready."
"Okay. Whatever happens, just go with it."
"What are you talking about?"
"You didn't know anything about it. Understand?"
"Boyce, I need you to defend me, not get into trouble."
"Baby, trust me. I _am_ defending you."
They were quiet awhile.
"I don't know what you have in mind. But one of us has to not be in prison to raise this child."
Boyce said nothing.
"Whatever it is, don't. I'm asking you."
# Chapter 26
The people you knew.
To get the people you knew took time. Boyce begged a day's recess under the pretext of needing time to locate a "vital" defense witness, an utterly irrelevant maid who had once done part-time cleaning for the MacManns.
Boyce used the cell phone listed to "B & B Seafood." It was his fishy phone, the one for very sensitive conversations.
He reached Felicio on _his_ cell phone, which was listed under God knew whose name, probably someone dead and the less you knew about it the better. You never knew where Felicio was at any given moment. Boyce could hear Peruvian-sounding flute music in the background. On a previous call, he had heard explosions.
"I need you," Boyce said. _"Ahora."_
The upside of knowing the people you knew was that they were grateful to you. Latino clients were grateful to the point of embarrassment. They named their children after you, offered to kill your enemies. Even inconsequential ones.
Twelve years before, Boyce had kept Felicio from spending the rest of his life in a U.S. federal prison for trying to steal one thousand pounds of C-4 explosive from a military base. The government claimed Felicio was planning to use it to blow up the local U.S. embassy, in retaliation for ending its support for Felicio's rebel group. Felicio's defense—admittedly bold—was that he was going to use it to blow up the infrastructure of the corrupt dictatorship.
Boyce took the case pro bono to show his contempt for the U.S. government's "war" on drugs that in the course of twenty years had put one-third of the black population in jail while reducing the availability of drugs by a factor of roughly zero. By the time he was finished, the jurors were ready to enlist in Felicio's rebel army and not only overthrow the corrupt dictator, but also storm the U.S. embassy. Every year since, Boyce had gotten a Christmas card from Felicio, who was now chief of security for a chain of Central and Latin American hotels.
Felicio was overcome with emotion at being asked for help by his old savior. _¡Cómo no, patrón!_ He would be in Washington on the next flight. No—he would come by private plane! He would be there before dawn! Boyce said that dawn would be early enough. He gave Felicio a general idea of the area in which his assistance was being sought, so that he could bring along whatever specialists he needed.
The breakfast was arranged at a hotel in Tysons Corner in Virginia, a half hour from Washington. Boyce booked the room himself. Felicio was waiting for him when he walked in.
Boyce's next call was to Sandy Clintick. She was surprised to get it. She didn't bother to conceal her feelings. Boyce told her he wanted to meet with her. In private.
Fine, she said. Come to my office. Boyce politely declined the opportunity to be ambushed by photographers, slipping into a side door at the Justice Department. He counterproposed the Metropolitan Club. He arranged for a private dining room. Seven o'clock? See you then.
Sandy Clintick was shown in. Boyce extended his hand. She simply nodded and sat down. Her body language said, "So?"
"I might," he said, "be able to talk my client into agreeing to an involuntary manslaughter charge. But I would insist on two and a half years. Max. House arrest."
Sandy Clintick stared at Boyce. Finally a smile appeared, not the kind that bathes you in warmth.
"When I took this case," she said, "I tried to go into it as I would any other. Do the job, don't make it personal, walk away. I was actually looking forward to going to trial against you. I've observed you over the years with interest. Sometimes with admiration. Then I finally met you. And, do you know, I've almost forgotten that you have a client. To be honest—something you may have a hard time relating to—at this point I don't care about your client. But I do care about you. And I'm going to beat you."
She paused.
"You've given me insight, Mr. Baylor. And I'm grateful to you. You've restored my belief in evil. So the answer is—go to hell. I am going to convict your client. Just for the pleasure of making you lose."
"Fair enough." Boyce grinned. The rule never failed: Forgive your enemies. It makes them madder than hell.
" 'Fair enough'?" Deputy Attorney General Clintick laughed, amazed. "It doesn't bother you to be called evil?"
"I've been called worse."
She shook her head.
"If this conversation leaks," Boyce said, "I'll know where it came from."
"I have _so_ many better things to do. Including," she said, rising, "being here."
MACMANN DEFENSE REPORTEDLY SEEKING DEAL
"What's this? What the hell is this?"
Beth held _The Washington Post_. She read aloud: " 'Sources within the Justice Department say that MacMann attorney Boyce Baylor initiated contact to explore the possibility of a reduced charge, from first-degree murder to involuntary manslaughter. According to them, Baylor was rebuffed.' " Beth looked up. "Is this _true_?"
"Do I look like the sort of lawyer who would go hat in hand to some hack prosecutor, to beg? Give me some credit."
"No no no. None of that. Answer the _question_."
"This puts us in an excellent position. PR-wise. We'll crucify her for this. Deliberate leaking. Trial by media. It's scandalous. We might even succeed in getting her thrown off the case."
Beth took his hand and placed it over her belly. "Swear. On this."
"It was just a friendly little chat."
"Dammit, Boyce!"
"Oh," Boyce said dismissively, "I just wanted to give her a chance to vent before closing arguments. It's like milking a rattlesnake. Leaves 'em with a little less venom."
"You should have _asked_ me."
"I asked you," Boyce said, "not to take the stand."
"So she wants blood?"
"The good news is it's mine she wants."
"So that's it? That's the ball game?"
Boyce sat beside her. "Look on the bright side. You have a brilliant attorney, the best in the country, and..." He sighed. "All right, it's not an ideal situation. But don't you give up. Things can change. You never know what's going to happen in a case like this."
" 'Case like this'? You've tried a 'case like this' before?"
"Oh, dozens."
"Boyce!"
"What?"
"It kicked!"
Boyce felt. "Sure it wasn't gas?"
"It wasn't _gas_. I felt it."
"Do they kick at this point? We should get a book."
"It kicked."
"Maybe it was objecting."
"Boyce, this baby is going to be born in a prison."
"It's going to be all right." He put his hand on her belly. "Swear."
Juror number fifteen emerged from his shower in his room at the Capitol Suites Hotel and moistly walked barefoot across the carpet toward the one object that had given him pleasure in the last five months, the television. The U.S. Marshals Service had installed some sort of block—one of the jurors had nicknamed it the J-chip—so that the televisions in their rooms could not receive normal programming, only documentaries, cartoons, sports, and movies. The Home Cooking Channel, Self-Discovery Channel, Celebrity History Channel, Police Chase Channel, and the Self-Abuse Channel.
He had had no news of the outside world in almost half a year. The country might be at war. The stock market might have crashed. Maybe they'd cured AIDS and landed on Mars. Maybe aliens had landed and taken over. Who knew? They didn't get newspapers or magazines, unless the marshals went through them first with scissors and cut out the references to the case.
He took off his damp towel and gave his delicates a good scratching. The only other nice thing about this incarceration, which had now lasted longer than a typical prison sentence for mass murder, was that you could stand buck naked in a room and give your balls a good fondling without the wife or kids walking in on you.
Nads firmly in hand, he picked up the remote control with his free hand and hit the power button. He looked over at the desk mirror. He'd gained fifteen pounds eating the crap they served. He was sourly contemplating the protuberance of his waistline when he heard an unfamiliar sound—a news report coming from the television. He turned and watched. There before him was one of those guys, the whatyacallem, commentators. He recognized him from the O. J. Simpson trial, the boyish-looking one with the glasses.
"Peter," the man was saying, "Boyce Baylor has _denied_ the report that he sought a plea bargain with Deputy Attorney General Sandra Clintick. But this story _refuses_ to die. Independent sources have confirmed, to me directly, that Baylor and Clintick _were_ seen entering the Metropolitan Club several days ago, just _moments_ apart...."
Juror number fifteen thought, _Hell is_ this?
Juror number seven's thoughts were similar. She had been crocheting while watching the Biography Channel but had decided fifteen minutes into it that she was just not that interested in Marie Osmond. How did these people qualify for "biographies," anyway? What was the world coming to?
She had been flicking through the channels in search of the Home Cooking Network when all of a sudden she came on the public television channel. How had that come on? It was the _news_ show, the one with that nice man from Oklahoma, Mr. MacLehrer. And heavens, here he was discussing the trial, with a young historian with a hairpiece. What on earth? How was it they were getting television news all of a sudden? She watched, fascinated. So that Boyce Baylor had tried to arrange a plea bargain, eh? Well, he certainly should, the way it was going, and if I were you, Mr. Big Shot Defense Attorney, I wouldn't drive too hard a bargain. Why he ever put that woman on the stand, she'd never understand.
Juror number nine listened to the news about the trial for half a minute before changing back to the Sports Channel. The Lakers were playing the Knicks.
# Chapter 27
Boyce was in his hotel television studio doing an interview with the _Today_ show, denouncing Deputy Attorney General Sandra Clintick for "shamelessly" leaking "false and injurious" information to the media when they moved in. They arrested him in midinterview on live television. It made for what is otherwise the most misused phrase in the English language: "must-see TV."
As he was handcuffed by the FBI agents and read his rights, with some five million viewers watching, the show's producers could only offer a silent prayer of thanksgiving to the television god for this amazing benison, the first live arrest of a lawyer on network TV.
The director of the FBI denied that the timing was intended to humiliate Boyce Baylor. The agents were simply following procedure. Of course no one believed this, much less the director of the FBI, but the media was so grateful to him for providing them with such a spectacular moment that they didn't press.
Boyce knew what was happening before the viewers did, since he knew the meaning of, "You're under arrest for violation of 18 U.S. Code 371." It took the producers a few minutes to clarify that the legalese stood for "conspiracy to tamper with a federal jury." Within a few more minutes, they rousted their legal correspondent out of bed—he apologized profusely for not having been watching—to inform them that the standard penalty for conviction of such a crime was five years in a federal prison, plus certain and permanent disbarment.
Boyce noticed as they rode down in the Jefferson's elevators that the FBI agents—there were four of them instead of the usual two, presumably in case he decided to shoot it out—were grinning. How sweet this moment must be for them.
"You're having a good day, aren't you?" he said, his tone not unfriendly.
"And it's not even coffee break yet," one replied, beaming back at him.
In the car on the way to FBI headquarters, Boyce considered clinically, _So this is what it feels like, riding in the back, handcuffed_. What empathy he would now have with his clients. Assuming he ever had any more clients.
How had they known? Had they tapped his fishy phone?
Felicio's plane? God knows whose plane it was. Felicio had some colorful friends. Maybe customs had noted the tail number and tracked it. Were Felicio and Ramon in custody?
More to the point—had the plan worked? Had the jurors been polluted? If so, Judge Dutch would have no choice but to order a mistrial. And in a new trial, Beth would not repeat the mistake of taking the stand. All she would have to do was follow the script of the first trial up to that disastrous point, and then she, and Tummybug, would be all right. Now they could visit him in jail. _Look! That's Daddy behind the glass! He can't hear us! Wave!_
They were pulling up in front of the FBI Building. There must have been two hundred of them, yelping, baying.
"I see you called ahead."
"Since you like going in front doors so much."
They were putting him through the full perp walk so that the moment could be recorded for all posterity, in all its ingloriousness.
He heard the first question shouted at him. "Boyce—who're ya gonna call?"
It was an interesting question. Whom does Boyce Baylor call when _he_ needs a lawyer?
He ran a few names through his mind. Whoever he called automatically became the top law dog of the land.
No. Boyce was not yet ready to pass the baton of greatness. He was _still_ the greatest. Though admittedly, if you were being hustled off in handcuffs, your number one status might be open to question.
When he went through the door, every agent in headquarters was there in the lobby to watch. They were all smiling. Then they all burst into applause.
"Thank you!" Boyce shouted. "Great to be here."
After the mug shot and fingerprinting, he posed for pictures with the arresting agents. Might as well be a good sport about it.
It was a mere special agent in charge who did the interrogation. Boyce was disappointed. He'd been expecting at least a deputy director. Clearly, they were determined to pretend it was "just another case."
He smiled at the SAIC and his deputy. "Oh, fellas, fellas. You don't really think I'm going to talk to you, do you?"
Judge Dutch was not a happy camper. He interviewed the jurors separately, in his chambers. In addition to being unhappy about his jury being tampered with, he was troubled by something else: the FBI's handling of it. Something was not right. Only moments after juror number seven had reported to the U.S. marshals that her television reception had undergone a miraculous improvement in reception, the FBI had swooped into the hotel in force—thirty agents, rushing into the jurors' rooms, yanking out the cords of TV sets, taking statements, isolating jurors.
In other words—they _knew_. Then why, for God's sake, hadn't they moved in sooner, to prevent it? Judge Dutch would have words with the director.
Meanwhile, he would now interview juror number fifteen. So far, only four jurors had admitted to having seen coverage on their television sets. Number nine was ambiguous. He "thought" he might have seen "something." The judge could not penetrate beyond this, other than to get a play-by-play recap of the Knicks–Lakers game. Jury of peers. God save us.
Judge Dutch had initially impaneled a jury of eighteen, giving him twelve plus six alternates. He would have to dismiss these four, leaving him with two alternates. At this point, he could only pray that of the remaining jurors to interview, no more than two had seen anything. Otherwise it would be the Mistrial of the Millennium. The prospect made Judge Dutch consider, in this order, shooting himself, giving up the law, drinking an entire bottle of gin. Option three might lead, pleasantly enough, to accomplishing options one and two.
He nodded gravely at his clerk to bring in juror fifteen.
Boyce asked simply, "What's your evidence?"
Wide grins. One of the agents walked over to a VHS and pressed play. A TV monitor anchored to the ceiling produced an image: Boyce, Felicio, and Felicio's technical man in the Tysons Corner hotel room. The quality of FBI surveillance tape had much improved since the grainy days of the DeLorean and Mayor Marion Barry busts. These days it was all so digital. Boyce listened to himself saying, "There will be a very big story on the TV news tomorrow night. That's what I want them to see. Can it be done by then?"
Felicio conferred with his guy.
" _Sí, patrón_. No problem."
The FBI man pressed stop. "Want popcorn?"
"You know," Boyce said, "if I didn't know better, I'd swear that was me."
The agent in charge chuckled. "So it was you who leaked the story about the plea bargain? That's slick, Counselor. You ask the deputy AG for a meeting just so you'd have something worth leaking."
The other agents nodded approvingly. "That's good."
"You want to make a call, or watch the second reel?"
"Have you arrested the other two people in that video?" Boyce asked.
"You mean the ones you'd swear were Felicio Andaluz and Ramon Martinez, if you didn't know better? You'll probably run into them at your arraignment. They said to say _hola_. They've been very helpful. Told us all about your phone call."
Something wrong here. Felicio had been tortured by some of the best interrogators south of the border, where forcibly extracting information was as old a profession as gold mining. This too often involved removing gold (fillings). So Felicio wouldn't have warbled for these gringos. His pain threshold _began_ at crushed knuckles. What was the worst these palefaces could do to him? Threaten to take away his cigarettes?
So if they knew about Boyce's call to Felicio, which they apparently did, they had to know _on their own_.
Had they tapped his fish phone? That was a wiretap he could fight.
Still, what a mess. He needed to call Beth. She'd have heard by now. Mongol farmers in their yurts would have heard the news by now.
"Boyce! Is this true?"
"Did I ask you that?"
"Are you all right?"
"I'll be out of here in an hour."
"Wouldn't bet on that, Counselor."
"As soon as I post bail."
"Who was that?"
"Agent Dokins, my new best friend. I'm surrounded by my fan club here." He cupped the phone. "Yo, J. Edgar, you going to claim I'm a flight risk?"
The agents grinned.
Boyce sighed. "They're being pricks about this, naturally. Make that two hours. Beth?"
"Yes?"
"You understand what this means? Don't say anything. You didn't know anything about this, so you'll be fine. You can tell them the truth about this. You didn't know. Everything's going to be fine now. Do you understand?"
"Boyce, you're calling from FBI headquarters after being arrested for jury tampering. By what definition is this 'fine'?"
"Just tell Babcock to bring a fresh shirt, the Turnbull and Asser, with the burgundy tie. Along with the bail. I'm not going to walk out of here in the same shirt."
Boyce emerged from the J. Edgar Hoover Building four and a half hours later, looking natty and upbeat for someone facing five years plus professional extinction. They tried to route him out the side door, but he insisted on going out the way he came. More or less every television camera in North America was there to record the moment.
"I have a short statement," he said above the roar of shutters. "After that I'm going to my office to check the U.S. Constitution to see where it says that the government has the right to harass and jail lawyers in the middle of trials. It would appear that the government is so afraid of losing this case—which they never should have brought in the first place—that they'll stop at nothing. Now they have stooped so low that the backbone of the law may be permanently bent. Thank you. See you all in court."
He could tell she'd spent the better part of the morning crying.
"You ought not to cry so much. It makes you puffy."
"Why did you do this?"
"This is no time for that. What it is time for, however, is a drink."
He ordered a Bloody Mary from room service. He flumped onto the sofa and loosened his burgundy tie.
"How'd I look on TV?"
"It was an improvement on the _Today_ show. At least you weren't arrested in the middle of it." Beth sat opposite him. "I won't ask if it's true."
"Thank you."
"Is it true?"
"What's true is that Judge Dutch will have no choice but to order a mistrial. Which means you get to start over. I won't be defending you this time. But whoever does will be a better attorney than I was, because they will not allow you to testify."
Beth looked pale. "You sacrificed yourself for me?"
"I can't have my child born in prison, for God's sake."
"How did they catch you?"
"Our attorney-client relationship is about to change. Maybe the less we discuss this, the better."
He looked at the phone. "I better get on the blower. Felicio's going to need a lawyer. I need a lawyer. You need a lawyer." He chuckled. "Another great day for the profession. Everyone needs a lawyer. Maybe we could get a bulk rate at one of those firms that advertise on late night TV."
"You're awfully calm about all this."
"I wonder if they give you a court-appointed attorney when you get to the Pearly Gates? I never thought of that. What a practice that would be."
# Chapter 28
Judge Dutch issued his ruling after judicious deliberation. Determined—indeed, obsessed—with not being overturned on appeal, he checked in with his old mentor on the Supreme Court, as well as with a few other solons of the bench. Then he did what judges alone can do: he ruled.
There would be no mistrial.
The result of his interrogations had produced a total of six jurors who had watched trial coverage on their hotel television sets. These would be dismissed as alternates. This left him with twelve jurors. These twelve were now isolated on a military base under twenty-four-hour guard by U.S. Army Special Forces troops. There were no television sets in their rooms. They were transported to and from U.S. District Court in a military convoy, like al-Qaeda prisoners in Cuba. Some pundits declared this slightly melodramatic, but at this point Judge Dutch no longer cared what anyone—much less the media—said. His sole goal in life was to get a verdict. If the jurors had to be housed in subterranean vaults in the New Mexico desert—fine. He began to drop dark hints that he would remove the cameras from his courtroom. Instantly the media clammed up.
He read his ruling in court. The jury was not present. He avoided looking at Boyce. Then he got to the part in his ruling stating that Boyce would not be permitted to continue as counsel. He looked him straight in the eye.
"Objection. Your Honor, may I approach?"
"No."
Judge Dutch continued reading. Defense counsel had been arraigned on charges of conspiracy to tamper with a federal jury. He also faced disbarment proceedings by the Ethics Committee of the District of Columbia Bar. Pending disposition of those cases, it would be "grossly inappropriate" and morally intolerable for Mr. Baylor to remain as counsel. Though of course he was innocent until proven guilty.
He granted Mrs. MacMann a week's delay so she could arrange for new legal representation. He gave Boyce one final, disgusted look. Boyce thought he was going to pronounce sentence of death: _And may God Almighty have mercy upon your soul_.
It was a glum little party they made back at the Jefferson. Chairs were drawn up in a circle. Boyce, Beth, and half a dozen of the best legal minds in the country chewed over how exactly to proceed. Beth's phone had started ringing minutes after the news of Boyce's arrest, lawyers salivating to represent her, no matter that chances of winning at this point were slim to nil. Beth MacMann was going down. On the bright side, it was possible that she and Boyce might end up doing their time in the same facility. Wouldn't that be cozy? Thank God no one yet knew she was pregnant.
Three television monitors were on with the sound off. As the lawyers talked, Boyce's eyes kept wandering to the screens. There was Perri on _Hard Gavel_ , looking scrumptious. Crudman was on with her. Curious as Boyce was, he couldn't bring himself to turn up the volume. Death by hanging, lethal injection, or burning at the stake he could take, but not the condescension of Alan Crudman.
Some commentators insisted that Beth had put Boyce up to the jury tampering. Others said, No no no, this has all _his_ fingerprints on it. Did he not get that terrorist Felicio Andaluz off twelve years ago for trying to blow up the U.S. embassy in Encantado? How do you think he got his nickname, anyway?
Boyce glanced at another screen.
POLL: OPINION OF BAYLOR
POSITIVE: 3%
NEGATIVE: 96%
At least 1 percent was still undecided.
The third monitor was showing old footage of a younger Boyce and Felicio during Felicio's trial. It was always strange, Boyce thought, seeing your youthful pixelgänger on TV looking back at you, thinner, hairier, inevitably wearing a wider tie. How strange and self-mocking it must be for movie stars to watch their early flicks in the winter of their lives. Felicio looked like a right-wing version of Che Guevara—revolutionary, but clean, with haircut and new shirt.
Boyce had called Linc Caplan at Skadden, Arps to ask him to take on Felicio and Ramon's defense and to send him the bills. Oh, the bills. What sweeter justice could there be for a lawyer than to find himself on the billing end of his brethren? Boyce's own income stream, that mighty green Amazon that had been flowing toward him for so many lush years, would soon be no more than a dried and cracked arroyo.
Then he saw on one of the monitors—the video of himself and Felicio and Ramon in the hotel room! As the others continued to talk, Boyce plucked the remote off the table and turned up the volume, just in time to hear himself giving Felicio the deadline for polluting the jury.
It stopped conversation in the room. As indeed, around the nation.
The FBI videotape ended and was replaced with the image of Perri. It was _she_ who had shown the tape! Perfidy, thy name is Pettengill. Boyce was stunned.
Everyone in the room was staring at him.
"Isn't someone," he said, "going to say something contemptuous about the FBI leaking its evidence to the media? Or were you waiting for me to go first?"
"It's outrageous."
"Goddamit, where do they get off?"
"A new low."
"Who the fuck—pardon me, Mrs. MacMann—do those cocksuckers think they are?"
"Thank you," Boyce said. "That was collegial of you."
"Would you excuse Boyce and me for a moment?" Beth said.
When they were alone, she said, "So that's why you went to Clintick. So there would be something newsworthy on TV the next day for your friends to beam into the jury's hotel rooms?"
"We shouldn't be discussing this."
"They would have found out it was you. Sooner or later."
"By then you'd have been in the middle of a new trial."
"So I've ruined your life twice."
" 'History repeats itself, first time as tragedy, second as farce.' "
"You realize our child is going to hate us."
"So do most lawyers' kids."
The nice thing about being a judge—aside from the big chair—was that they had to come to you.
Sitting in front of him in his dark and woody chambers hung with four-hundred-year-old still lifes and portraits of fish and merchants was the director of the Federal Bureau of Investigation. The director's evident nervousness was deeply satisfying to Judge Dutch.
We'll get to the broadcast of this video in a moment, the judge began. Though it's not my case and I have no jurisdiction, I'm very, very curious to know how a show called _Hard Gavel_ , hosted by Baylor's former girlfriend, came into possession of it.
The director began to answer evasively. The judge held up a hand, a gesture combining serenity and absolute power. Continue in that manner, the gesture said, and by this hand thou shalt know the true meaning of woe.
What I desire to hear from you first, Judge Dutch continued, running his fingertips along the edge of the blade of a letter opener that had once belonged to Supreme Court justice Potter Stewart, is the following: Your agents videotaped Boyce Baylor and those jalapeño hoodlums in a hotel conspiring to tamper with a jury. My jury. _Why, then, did you permit them to go forward and pollute the jury_? And while we're at it, why was I not informed?
The director mumbled out some boilerplate about procedure. Seeing Judge Dutch's normally placid face staring back at him like an Assyrian lion, he stopped.
We needed, he said, to catch them in the act.
_Who_... needed?
The Bureau.
Judge Dutch had indulged in violent expostulation only once before, after learning that an art dealer in Amsterdam was trying to sell him a still life of a dead flounder and two lemons, allegedly by De Grootie, but in fact by one of De Grootie's pupils, Flemm Vander Flemm.
It's no small thing to alter the metabolic rate of the director of the FBI, yet Judge Dutch succeeded. He would demand a Senate investigation. For _starters_.
Now, he continued, while we are on the subject of the Bureau's possibly criminal misconduct—Judge Dutch bit down on the final syllable as if it were a crisp breadstick— _how did that videotape leak_? Or was it that FBI evidence tapes had become the latest thing in reality TV?
We're looking into that. Be assured, Judge, we are looking into it.
Do. Look _deeply_ into it. And inform me of your findings. Before I hear about it on television.
Yes, Your Honor.
# Chapter 29
When you're hot, you're hot. When you're not, the hotel management shows up at the door with your bill. Ten months of three suites came to—after discount—$1,845,322, including room service. With a velvety clearing of his adenoids, the manager asked that Boyce vacate "at the earliest convenience." The "media situation," he explained, had become "disruptive to the other guests."
Simultaneously, Sandy Clintick moved to have Beth placed in custody as a "flight risk."
How Beth was planning to skip the country while under the "protection" of a dozen Secret Service agents was not clear. But feelings were running so high against her and Boyce that the polls showed most Americans in favor of putting her behind at least some kind of bars. For Boyce they favored hanging, preferably before he could be tried and after lengthy torture, _peine dure et forte_. It was all a glorious excuse for a national splurge of lawyer bashing.
"It's incidents such as these," Alan Crudman wrote in an op-ed article for _The Boston Globe_ , "that weaken Americans' faith in their legal system."
"I thought I'd already hit bottom," Boyce said to Beth, "but being looked down on by Alan Crudman gives new definition to bottom."
"He called me this morning," Beth said. "He wants to take the case. He wants to file a motion."
"Pass the hemlock. I'm listening."
"Mistrial for reasons of ineffective assistance of counsel. He thinks Dutch would go for it. Even if you don't get a mistrial, it's very solid grounds for a reversal after conviction."
Boyce chuckled darkly. "I'd sooner swallow leeches than spend ten minutes with Alan Crudman. But he's a helluva lawyer. You should consider it. Look at the scumbags he's gotten off."
Beth looked at him.
"Sorry. Didn't mean it that way. I meant, go for it."
"I told him to go screw himself."
"Beth, it's hard enough paying Alan Crudman a compliment. Don't make me work at it."
"I'm not going to ruin your life three times."
"I wouldn't feel it. You develop a callus after the second time. Look, it's not a bad idea he's got there. You can file an affidavit along with the motion saying that _I forced_ you to take the stand against all your better instincts. You had no idea I was conspiring to tamper with the jury. And my getting thrown off the case has completely compromised your defense. You should be given the opportunity to start over. These are all perfectly good arguments for a mistrial. Dutch will have to go for it. He's painted himself into a corner. The Supreme Court would back him up in a second. You've got to do it. You have to do it for—"
"I don't want to hear that I have to do it for the Tummybug."
"Well, you do, Beth. Even if you don't want to hear it."
"I'm not going to. So forget it. We'll find some other way."
"At what point in your life did you decide that stubbornness was one of the cardinal virtues?"
"When at age eleven I saw that the world was ruled by men."
"Fine. Condemn our child to life in the prison playground."
"Maybe it would be better not to _have_ the child."
They were silent.
"That's a terrible thing to say," Boyce said.
"I know. I didn't mean it. I was just trying to hurt you. I'm sorry." Beth considered her belly. "I'm going to start showing soon."
"What a media feast that's going to be. They ought to be paying us an entertainment fee. Think of the content we're providing."
Beth put her hands to her stomach. " _This_ is content." She smiled.
"We go now to our legal correspondent."
"Peter, _another_ tumultuous day in the MacMann case. We have just learned that Beth MacMann will take over her _own_ defense. Mrs. MacMann _is_ a lawyer, after all—it was at law school that she _met_ Boyce Baylor and the late President—though she has never _practiced_. This is a highly unusual development. It's hard, if not impossible, to think of someone taking over their own defense in a murder trial at this late stage, but then just about everything about this case has been unusual."
"What will Boyce Baylor's role be? Will he have one?"
"His _role_ , Peter, will consist of defending _himself_ on the very serious charge of jury tampering. At the moment, we understand, he is occupied with moving _out_ of the Jefferson Hotel, where he maintained his so-called war room. We've learned further that he is _suing_ the Jefferson for wrongful eviction...."
The government had seized his bank accounts, so Boyce found what are politely called alternate arrangements across the river in Rosslyn, Washington's unbohemian left bank, a suburban sprawl of glassy high-rises where most of the pedestrians work for various defense agencies and the cabdrivers come from countries badly disappointed by U.S. foreign policy.
It wasn't so bad, though living without the room service took some getting used to. The papers ran photographs of the outside of the motel juxtaposed with photos of the previous one, under the caption "How the mighty have fallen." The relative modesty of his new surroundings even gave him a bit of _nostalgie de la boue_ , reminding him of his early days of defending corrupt union officials, mobsters, and—lowest of the low—providers of illegal soft money. He became a local celebrity at the Szechuan Sizzle Chinese restaurant around from his motel. The owner paid him the highest compliment a Chinese restaurateur can bestow: he didn't charge for the soup.
Boyce listened with one ear to the machinations of his legal team, who were busy filing motions and preparing a defense in the face of dismal and overwhelming evidence. But try as he might, Boyce had no interest in his own case. He did not relish the idea of spending five years in prison, surrounded by victims of inferior lawyering. All he could think about was Beth and Tummybug.
Judge Dutch had stopped short of placing Beth in custody, but just for good measure he had ordered that she be placed under a kind of house arrest. Marshals were ordered in to "protect" the house in Cleveland Park where she was staying. So she now had two rings of federal protection around her, her Secret Service detail on the inside and bulky men in windbreakers on the outside. At night, over a phone line that they were fairly sure was tapped, Beth and Boyce joked darkly about the tunnel the two of them were digging.
In the mornings, Boyce would show up in a taxicab, with at least two vans full of television camera crews following, and ride with her in her now even longer motorcade to the courtroom. She would get out at the front entrance. He sat in the car with the contemptuous Secret Service agents, watching the proceedings on a portable television set. During breaks, Beth would return to the car, parked in the courthouse basement, to get his comments and notes. As the TV correspondent would say, it was an unusual arrangement.
Beth had called to the stand the White House curator, F. Dickerson Twumb. The idea was to show that she so revered eighteenth-century American silverware that she never would have used it to crush the skull of her husband. It somewhat left open the suggestion that she'd have been happy to use some other less precious blunt object.
"Well," Beth said, entering the car with her folders and legal pads, followed by the faithful Vlonko.
"I'm uneasy with this witness," Boyce said. "He doesn't seem to like you very much, frankly."
"You have to understand about curators. They think it's _their_ White House, and they regard all First Ladies—with the exception of Jackie Kennedy—as menopausal busybodies whose idea of decor is an Ethan Allen showroom. When we got to the White House, I had the temerity to suggest to him that I found Albert Bierstadt's landscapes boring, and he went into a snit from which he has apparently not yet recovered."
"Great witness for the defense." Boyce snorted.
"He did say I was particular about the silver."
"Vlonko?"
Vlonko shook his head. "Jury's not so fucking happy today. Not happy yesterday, either. Or the day before. Maybe they don't like army food."
"All right," Boyce said. "You go back in there and you grab this guy by his bow tie and get him the hell off the stand. Who's your next witness?"
"I want to recall Secret Service agent Birnam."
"Why?"
"I want to put it to him, 'If you think I was such a threat to the President, then why didn't you rush into the bedroom and shoot me?' "
Boyce shrugged. "Why not." He glanced furtively at Vlonko. Vlonko's look said it all.
Boyce was having a morose, solitary dinner of Chairman Mao chicken and crispy shredded beef at the Szechuan Sizzle before going back to his room. His eyes strayed to the television monitor over the bar. The sound was off, with the closed captioning on. On the screen he saw the flashing lights. The captioning scrolled:
CAPT. CARY GRAYSON, WHO PERFORMED THE AUTOPSY ON PRESIDENT KENNETH MACMANN, IS IN CRITICAL CONDITION FOLLOWING AN ACCIDENT ON THE GEORGE WASHINGTON PARKWAY. HIS CAR WENT OFF THE ROAD AND HIT A TREE. HE WAS TAKEN TO NEARBY BETHESDA NAVAL HOSPITAL AND UNDERWENT SURGERY... POLICE SAY THERE MAY BE EVIDENCE THAT HE WAS INTOXICATED...
Intoxicated? Grayson? He called Beth on his cell phone. It didn't matter if the feds listened in, though his attorney, Judd Best, had been assured by the U.S. District Attorney's Office that there were no outstanding eavesdropping warrants on him.
"Were you planning on recalling Dr. Grayson?"
"No. Why?"
"He's in surgery and may not come out of it. Car crash."
"Poor man."
"Well, the poor man may have been drunk at the wheel when he went off the road into the tree."
"Grayson?"
"Turn on the news."
On the other end, Boyce heard the sound of the channel he was watching silently at the bar.
"Jeez," Beth said.
Boyce looked around to see if any reporters were listening in. He said into his cell phone, "Not to be morbid about it, but tomorrow morning I want you to go to Dutch and say, 'Look, I was planning to call this guy again and _now_ look.' He'll ask you what you wanted to hear from Grayson. I'll think of something. Toxicology, whatever. But you'll say it's _crucial_ , and without his testimony we—you—are being deprived of _vital_ evidence. Wait."
"What?"
"I just thought of something. Pure genius!"
"Well?"
Boyce looked around again. "No, not on the phone. Tomorrow, on the way to court."
"Well?" she said next day as they were riding to court.
Boyce lowered his voice. "Wouldn't they love to hear this in the front seat up there." He whispered, "We—you—tell Dutch that you were going to recall Grayson to testify in greater detail about that suspicious postdeath imprinting of the Revere mark on Ken's forehead. _But they got to him before you could_."
"Who got to him?"
"They—the Secret Service. _They ran him off the road_."
Beth looked at him. "Boyce, honey, I know this has been an awful strain on you."
"Dutch'll hit the ceiling. But it doesn't matter. You bring it up in the closing arguments. Look—seventy-five percent of the American people _still_ think that JFK was killed by his own people. Trust me, at least one juror will think, _Hmmmm_. And all you need is one juror."
"It's nuts."
"Of course it's nuts."
"I don't know if I have it in me." Beth sighed and sat back in the seat. "What I'd really like to do is go into that courtroom today and say, 'Here, here's what happened. Here's the truth of it.' "
"Beth, how many times do I have to tell you. The truth has no _place_ in a court of law."
"I know, I know. But my belly is swelling, my tits won't fit in the bra anymore, they want to put me away. I'm tired."
Boyce spent a busy morning calling his shrinking group of media friendlies, trying to convince them that Dr. Grayson's mishap on the George Washington Parkway was no accident.
"Who," he said darkly, "had the motive? Who had the means?"
"You're saying the Secret _Service_ tried to kill him?"
"I know. Hard to believe, isn't it? But that's what they said about Vince Foster."
"Boyce, Grayson was drunk. He was tanked."
"Exactly. And who had the means to plant the booze on him?"
"Oh, come on."
"If you don't want it, fine. I gotta go, that's _Newsweek_ on the other line."
"No no no. Hold on. Do you _know_ this, or are you guessing?"
"Did Woodward know?" Boyce said. "Did Bernstein know?"
It's an axiom of journalism that if you can get one paper to print a rumor, all the others will rush in to print it on the grounds that someone else did. It was for the best that poor Dr. Grayson was still in a postoperative coma after undergoing seven hours of brain surgery. By noon the next day, it was being speculated, mostly on the Internet, that his accident was no accident.
The Secret Service was being besieged by calls wanting to know if they had run him off the road—to keep him from testifying that the Revere mark _had_ been planted by them. The day after that, the _Times, Post_ , and half a dozen other pillars of journalistic reputability were running items headlined:
SINISTER MOTIVES ALLEGED IN GRAYSON CRASH
The Secret Service directed inquiries to Bethesda Naval Hospital, where an embarrassed doctor stood before the podium at a press conference to say that Dr. Grayson's blood alcohol level upon arrival at the emergency room was—clearing of throat—"slightly elevated above the legal limit."
How elevated?
Clearing of the throat. "Point one nine."
God in heaven, the man was stinking! Plastered! Navy drunk! Oh, what do you do with a drunken sailor, so _ear-lie_ in the _mor-ning_?
Another navy doctor, this one with more ribbons than Christmas morning on his white chest, was trotted out to say that Captain Grayson was a man of impeccable reputation who was not only the navy's top pathologist, but had served his country valorously in war. Naturally, the President's death and his involvement, his testimony, had been a strain. There was no excuse for what had happened, but no one had been harmed but Dr. Grayson, and so while we all pray for his recovery, let us bear in mind that he surely is as human as the rest of us.
The media, sensing that moral outrage and opprobrium might be out of place, ceased baying for the time being and set up a death watch outside Bethesda Naval.
# Chapter 30
But _why_ was he drunk? Why was this white-gowned, medal-wearing paragon of military and medical virtue driving blotto into trees?
Boyce hid from his media pursuivants in the Szechuan Sizzle, poring over the transcript of Grayson's testimony, over the toxicology reports from the President's autopsy, over everything pertaining to the doctor. No clues to this seemingly out-of-character behavior presented themselves. At the trial, the doctor himself had been an exemplar of professional calm in every way. Reviewing the tapes, Boyce saw that his expression had been serene and unruffled. Following his testimony, the women's magazines had been full of his pictures. _People_ magazine had declared him one of its "50 Most Reassuring Men in America," an event that according to reports had given everyone at Bethesda Naval a good chuckle, no one more so than Dr. Grayson.
Meanwhile, Judge Dutch had reacted volcanically to the media reports about the Secret Service allegedly pouring whiskey into Dr. Grayson and then driving him off the road into the maple tree. He knew exactly where this canard had begun to quack. In retaliation, he gave orders that no vehicle carrying Boyce Baylor could enter even the basement garage of the courthouse. It turned out that this was, in fact, beyond even his august sovereignty. Judge Dutch's glasses now fogged at the merest mention of Boyce Baylor. The joke began to circulate around the courthouse clerks that the next car to veer drunkenly off the George Washington Parkway would be Judge Dutch's Volvo.
But at least the Trial of the Millennium was coming to a close.
For Beth, however, there was little light at the end of this long tunnel. Las Vegas bookmakers were laying thirty-to-one odds on conviction. And ominously, some commentators were remarking that her physique seemed to be changing, almost as if she were, well, pregnant.
One evening a few minutes past ten o'clock after returning from working at Rosedale with Beth on her concluding argument, Boyce sat in his usual booth at the Sizzle, nursing a glass of inferior brandy and staring halfheartedly at a motion he was filing in his own case. As he worked, a fortune cookie was placed in front of him.
He looked up to tell the waiter that he did not want a fortune cookie, only to see the back of someone disappearing briskly toward the restaurant's front door. Odd.
He looked at the fortune cookie. Its fortune protruded from the sugar clam lips.
He extracted the piece of paper cautiously, as if it might be the fuse to a bomb. In the kingdom of the tricky, the paranoid man still has all his fingers.
It was in handwriting.
_Confucius say public phone Colonial and Nash soon ring with interesting tiding. 10:15 pm. WPS_.
Boyce threw a twenty on the table and walked out of the Sizzle. There are no emptier streets at night than those of Rosslyn, Virginia. He walked the two blocks to the intersection of Colonial and Nash. It was dark and out of the way, just the place to kill someone. Not that anything that exciting ever happened in Rosslyn.
_Calm down_ , he told himself. But he was nervous.
He answered on the first ring.
The voice on the other end was cheerful, like that of someone who wanted you to try their long-distance service, at significant savings.
"I always said, if I'd stuck around for my trial, you're the man I'da hired, Counselor."
Boyce had never heard Wiley P. Sinclair's voice, so he had no way of knowing if this was really Wiley P. Sinclair, former FBI counterintelligence officer, betrayer of his country, agent of Chinese intelligence, code name Confucius. All of this he knew from the public record.
"Tell me something," Boyce replied, "that would convince me that I'm really talking to Confucius."
Chuckle. "You mean, like a PIN?"
Wiley P. Sinclair was the FBI's Most Wanted fugitive. He had made a jackass of them (not an especially daunting task). It was said that he was still working for the Chinese. His double agenting had, among other things, helped them get the Olympics and a nifty new U.S.-designed nuclear-tipped torpedo for their submarine fleet.
On visits to Beijing, U.S. presidents and secretaries of state would demand that China turn Wiley over to them. The Chinese would blink through the cigarette smoke and say that they had no knowledge of this Wiley P. Sincrair and then suggest that if there were such a person, he must be working for the imperialist lackeys in Taiwan. And that would be the end of the Wiley P. Sinclair portion of the agenda for that visit.
And now Boyce was—or might be—speaking to him from a pay phone in Dullsville.
"I figured," Wiley said, "that you might want some bona fides. They're in your room waiting for you. Bit of a step down from the Jefferson, isn't it?"
"Why are we speaking?"
"It's complicated, Counselor."
"Trust me. I can handle it."
"Combination of reasons. You've given the bastards one hell of a run for their money. I like your style. Okay, I could tell you I'm doing this just to help, but you'd figure out that's bullcrap. So I'll level with you. My current employers would be very pleased if this information came out. And you're perfectly placed to be the one to bring it out. So here's the 411."
"The what?"
"The information." Wiley P. Sinclair laughed. "Counselor, I'm surprised. I keep up better with the English slang living in Pandaland than you do."
Boyce didn't dare say it out loud to Beth in the car the next morning, just in case it was bugged. He wrote down the substance of what Wiley had told him.
Beth read, looked up sharply at Boyce. He took the paper back, tore it up, and put the pieces in his pocket.
"How can you be sure it was he?" Beth asked.
"When I got back to my hotel room, there was an envelope under the door. It was the PIN to his old ATM machine. I had someone in my office check his FBI file. It was in there. No one else would know that."
"Except for the FBI. They could be setting you up."
"I considered that," Boyce said. "But why would they bother at this point? I'm going down in flames as it is. Why pour gasoline on me now?"
"To make you burn brighter."
"Maybe. But what if it was him?"
"He."
"This is a gift, Beth."
"He's a traitor to his country."
Boyce was reminded that Beth had been First Lady of the United States. "Is this the time to be splitting ethical hairs?"
"It's not a hair, Boyce. The man is evil."
"Precisely. He and I are on the same bandwidth."
"It's wrong."
"So the Chinese got the Olympics. So they got a torpedo. Is this the end of the world?"
"He protected all their agents in California who were stealing secrets from Silicon Valley."
"No one's saying the man's a saint."
"This is like trying to explain vegetarianism to a shark."
"So why bother? This is not the time for ethical hand-wringing. Save it for your book."
"Just _why_ does this fugitive traitor want to help me? I'm asking out of curiosity, not for ethical reasons, if that makes you feel any better."
Boyce decided to leave out the part about how Wiley was doing this for Chinese intelligence.
"Because he genuinely believes you're innocent. And feels that this is a way of doing at least something good. To make up for past misdeeds. Would you deny a fellow human being the chance to atone?"
"The only person on the planet who believes I'm innocent is a former FBI agent who sold out his country to finance his gambling addiction. What a fan club."
"Now is not the time to be choosy."
Judge Dutch listened without comment, his glasses misting to opacity. Deputy Attorney General Sandra Clintick listened in silence with an expression that needled from contempt to incredulity and outrage.
Beth concluded, "I wanted to inform you both of this privately. I know this comes late. I also realize that there are ramifications, since it involves sensitive issues of national security. But there it is, and I intend to pursue it."
"This information," Judge Dutch said, "where did it come from?"
Beth cleared her throat. "From Wiley P. Sinclair."
"Objection!" Sandy Clintick snapped.
"We're not in court, Ms. Clintick," Judge Dutch observed. "You're free to express yourself in nonjudicial language." He turned back to Beth. "But I have to say, Ms. MacMann, I'm not impressed by this. Not a bit."
"I would rather myself that it had come from some other source, Your Honor."
"This is disgraceful, Your Honor," Clintick said. "Disgraceful and desperate."
The judge rocked in his chair. "Mrs. MacMann, if this turns out to be without foundation and you are ultimately found guilty, I will... weigh this at the time of sentencing. Do you understand the implications?"
Beth nodded. "Yes, Your Honor."
"What if this witness—assuming I even allow you to call him—denies it? As he well might? What then?"
"Your Honor," Beth said, "surely you don't expect me to discuss matters of legal strategy in front of the prosecution?"
Beth and Sandy Clintick left the judge's chambers together. Alone in his outer office, Clintick turned to Beth.
"When are you due?"
"June."
Clintick smiled icily. "There's nothing you two wouldn't stop at, is there?"
"It was an accident."
"Oh, right."
"You think this was part of the overall defense strategy?"
"Why not? It's actually a smart move. Makes it harder for the judge to hand down a death sentence, doesn't it?"
Beth returned the gelid smile. "Is that why you haven't mentioned it in public?"
"I'm hoping to get a verdict and sentence _before_ you show up in a maternity dress."
"I'll try not to hold you up."
Wiley P. Sinclair, being skilled in the arts of evasion, counterevasion, and even counter-counterevasion, had left Boyce with a means of contacting him, involving bright orange stickers and a stop sign on Glebe Road. A few hours later, Boyce was at a pay phone in Old Town, Alexandria. Wiley P. Sinclair laughed when Boyce told him what he wanted.
"Now doesn't _that_ take the cake," Wiley said. When he was finished being charmed by the idea, he said, "You know the three cardinal rules, right? Don't eat at a place called Mom's, don't draw to an inside straight, don't go to bed with someone who's got more problems than you do. Here's a fourth: Don't try to outfox someone named Wiley."
"I wouldn't presume to try to set a trap for you," Boyce said. "But we need this document. Otherwise they're just going to deny it, and where does that leave us? Who're we going to call then?"
"No way, Counselor."
"Do you want this to happen, or not?"
"Are you saying it's a deal breaker?"
"Yeah."
There was a long pause. Then Wiley P. Sinclair chuckled.
"Oh, are they going to be hot for my ass again. Red hot. I'm going to have to relocate so deep in Pandaland that I may end up discovering a whole new _species_ of bear. But okay. You're my kind of lawyer, Counselor. You do whatever you gotta do for your client. Say, I just gotta ask—she did it, right?"
"That's on a need-to-know basis."
"I can't wait to see their faces when she waves this thing in court. You a betting man, Counselor?"
"Not in your league."
"Do you know they had two hundred agents in Vegas looking for me? I was there, and they missed me! What a buncha numb-nuts. I went in drag!" Wiley laughed. "Lay you three to two that Judge Dutch is gonna burst a nose artery when she shows up in court with this."
Wiley insisted on making the arrangements. Boyce counted the laws that he, Boyce, was breaking in doing this. He stopped at six.
# Chapter 31
Two days later, Beth and Sandy Clintick were back in Judge Dutch's chambers.
"I will ask you a last time not to use again the word _bullshit_ , Ms. Clintick," Judge Dutch said. "We're not in court, but neither are we in a bar."
"Then I'll use the word _travesty_ ," Clintick said, fuming. "This is a travesty. And you are permitting it."
"I haven't permitted anything as of yet, Ms. Clintick. But I won't permit _that_ sort of language. Anywhere."
"She comes in here"—Clintick pointed at Beth, who was quietly enjoying her fury—"with an affidavit, by a fugitive _traitor_..."
"I've made no determination yet as to the affidavit, Ms. Clintick," he said, then cast a disdainful glance at Beth. "Other than to acknowledge that it exists."
He picked up the piece of paper imprinted with the notary public's seal. "I take it," he said, "that this notary had no idea who Wiley P. Sinclair is."
"That would be correct, Your Honor. Mr. Sinclair—or so I was informed by Mr. Baylor, who supervised the notarizing and witnessing—was asked for two forms of identification, which he provided. A driver's license, apparently still valid, and a Social Security card."
Judge Dutch sniffed. "This is appalling, Mrs. MacMann."
Beth shrugged. "I admit it's untidy, Judge. But Mr. Sinclair's legal ability to make an affidavit is unaffected by his status as a fugitive."
"We appear," Judge Dutch said with a sigh, "for the time being to have hopscotched beyond the issue of his criminality. Mr. Baylor will have to bear the burden of that matter. Since he is the one who"—he looked at her unbelievingly—"had the unlawful contact with our fugitive, Mr. Sinclair. Of course, Mr. Baylor can always claim that there was some duress, or that he was unable to effect the arrest of Mr. Sinclair."
Beth wondered—was he prompting her?
"At any rate," he continued, leaning forward, "we must now confront this Log Cabin business."
"Defense calls Roscoe Farquant."
Oh, the stir, the buzz, the rumbling, the craning of heads and shifting of glutei in Judge Umin's courtroom. Boyce, feeling more than ever like a mole in his backseat hole in the bowels of the courthouse basement garage, would have given a testicle and three Supreme Court decisions to be there in person.
It had—naturally—leaked to the media that Beth was going to call the head of the National Security Agency, the only agency left in Washington that really had any secrets worth knowing. Her subpoena had created a sensation. NSA lawyers said he would not honor the subpoena. Judge Dutch replied that in that case he would have General Farquant arrested. The NSA relented. Nothing so concentrates the mind as the prospect of handcuffs.
No one knew what exactly she planned to ask him. The pixel pundits frothed over with speculation.
Farquant was, as most NSA chiefs tended to be, a former military person. He looked it: trim, peach fuzz hair, glasses, eyes beady with the big-big secrets. He looked like a man who wouldn't tell God something on the grounds that God was not cleared to know.
"General Farquant," Beth said, addressing him in a courtly, respectful manner, "I won't waste your time or the court's establishing your credentials, which are beyond question. You are the director of the National Security Agency, and have been for the last five and a half years. That agency collects electronic information on behalf of the U.S. government. Is that an accurate description of its role?"
"It is a very general description of the agency's function," he replied.
"Does the code name Operation Log Cabin mean anything to you?"
Nothing so excites a Washington audience as introducing the term _code name_ in a public setting. Invariably, what follows is evidence that the government has, once again, been up to something disastrously ill advised, or at least very, very naughty.
General Farquant stared unblinkingly at Beth, the court, the nation, the world beyond. The only sign that a white phosphorous grenade had just gone off in his stomach was a slight lateral twitch of the eyeballs.
"I'm not in a position to comment on that."
Boyce bellowed out loud with delight, startling the Secret Service agent in the driver's seat. Yes! The crew-cut SOB hadn't denied it outright!
"Was Operation Log Cabin put into effect some eighteen months ago?"
"I'm not in a position to comment."
"Of course. General Farquant, was Operation Log Cabin a covert surveillance program, mounted by the National Security Agency, whereby electronic eavesdropping devices were placed in the Lincoln Bedroom at the White House? So that your agency could monitor the conversations?"
A giant sucking sound could be heard in the courtroom.
"I am not in a position to comment on that." For all his sangfroid, General Farquant was beginning to resemble the frog placed in the pot of water that is slowly brought to a boil.
"Objection," said Sandy Clintick. "The witness has answered the question to the best of his ability."
"He most certainly has not," said Beth.
Judge Dutch thoughtfully tapped his cheek with a finger. "Overruled."
Beth continued. "Was the purpose of Operation Log Cabin—which presumably was so named after the fact that Abraham Lincoln grew up in a log cabin—"
"Objection. Conjecture."
"Oh, honestly," Beth said.
"Sustained."
"Was the purpose of Operation Log Cabin to obtain information on persons who were guests of President MacMann and the First Lady?"
"I'm not in a position to comment."
"On the night of September twenty-eighth, year before last, Ms. Babette Van Anka, the actress and activist, was a guest in the Lincoln Bedroom. Was such a device implanted in her cell phone or other personal effects by an agent or agents working for NSA?"
"I'm not in a position to comment."
"Thank you, General Farquant," Beth said pleasantly. "You've been most forthcoming."
Boyce slammed his fist against the window and bellowed, "Yee-hah!"
"Sir," said the Secret Service agent in the front seat, "do you _mind_?"
The President summoned his chief of staff.
"This Log Cabin thing, what the hell?"
"Do you want to know, sir?"
"No, I don't. But goddammit, Henderson."
"Yes, sir, I agree completely."
"Get me distance on this. I want miles of distance between this and me."
"I—we—all understand that, sir. We are at the moment working on that."
"When did it—no, I don't want to know. Get me—who's in charge of this, this bucket of night crawlers?"
"No one at NSA seems to be stepping forward to claim credit for it, sir. We're—"
"Heads. I want heads, Henderson. Heads lined up like golf balls, in the Rose Garden."
"Understood."
"I leave for Europe tomorrow. Sweet Jesus. I'm going to be with the heads of seven countries—plus the Queen. The Queen, Henderson! Of England! Did any of them stay in the Lincoln Bedroom during the MacBeth administration?"
"Heads of state typically stay in Blair House or the Queen's Bedroom, as you know, sir." Henderson cleared his throat. "However, the Queen of England did, in fact, stay in the Lincoln Bedroom on one occasion. She had expressed interest in it. Apparently she is a fan of Lincoln's. So the MacManns put her there. I have the date here...."
The President's face drained of color. "I'm spending the night at Windsor Castle. As her guest."
"Sir, I think we can make it clear that this Operation Log Cabin was in no way sanctioned by _this_ White House. For all we know, it wasn't even sanctioned by the last White House."
"Well, let's get _that_ message out and cranking, and fast."
"Yes, sir. Right away."
"Henderson."
"Sir?"
"What were they after, for God's sake?"
It was probably just as well that Babette Van Anka was not driving herself to the studio to record her annual message of peace for the coming Easter-Passover holidays or she might have driven off the road and into a royal palm. She had given up watching the trial weeks ago, on the grounds that it was not good for her skin. Now her cell phone rang, the first eight bars of the sound track from _Fabulous, Fabulous Me_ , the movie that had cemented her status as a star. Before she managed to say hello, she heard the sound of Max, calling her a cow in every language that he and his ancestors had spoken, with a few Far Eastern languages thrown in for good measure. He then related the substance of that morning's testimony. And hung up.
Babette played the scene as she might in a movie. She rolled down the window and hurled her cell phone out of the car, in case it too was bugged. Fortunately, this being Los Angeles, there were no pedestrians to injure.
She told Massimo, her driver, to take her to LAX instead of the recording studio.
"Which airline, madame?"
" _Any_ airline! International!"
She cursed Max for swinishly not sending the plane to get her. He had brusquely informed her, between epithets, that he needed it himself to remove himself even farther from U.S. justice.
Money. She would need money. As a star, she rarely carried any, since other people paid. Her secretaries took care of the occasional pecuniary necessities, supplying her with ironed banknotes. She looked in her purse. There were a few crisp, folded one-hundred-dollar notes. More would be needed. But she could hardly present herself at the bank. She didn't even know which bank handled the Grab–Van Anka cash. Then she remembered the television commercials showing people inserting cards in machines and getting cash. Triumphantly, she produced a credit card from the purse and directed Massimo to stop at a machine.
She leapt out and inserted the card. After several minutes she raced back to the car.
"It keeps asking for a personal identification number," she shrieked at the hapless Massimo. "What the _fuck_ is a personal identification number?"
Massimo explained, earning himself a cuff on his chest. "How am _I_ supposed to know it? Call someone! No—don't! Into the car. Into the _car!_ Drive! Just _drive!_ "
At the international departures terminal, she frantically scanned the names of the airlines and ordered Massimo to stop when she saw one that seemed more foreign than the others.
She hurried in after divesting Massimo of his pocket money. Heads turned at the spectacle of America's most notorious film star. The morning's trial testimony had played over all the television monitors at the airport, so everyone was up-to-date.
She went directly to the "Emperor Class" check-in counter. A businessman, presumably an emperor, was being assisted. Babette placed a peremptory elbow on the counter. The man turned to tell the pushy broad to cool her heels. When he saw who it was, his mouth gaped.
"I require a seat," she informed the check-in agent, a lovely young woman dressed in a silk sari from her native land. "I require the _entire_ first class section. Here—" She dumped half a dozen credit cards on the counter.
"I'm afraid, Ms. Van Anka, that we are completely booked in emperor class."
"You'll have to move them to business class. I'll pay for their seats. I have a scene I have to rehearse, and I need absolute privacy."
"So charter a jet," said the man she'd elbowed aside.
"I regret very much, Ms. Van Anka, but I cannot move other passengers. But there is one seat available in business class."
Babette threw up her hands. "All right, all _right_. I was just trying to _help_ you. Give me the seat."
"May I see your passport, please?"
"I don't _carry_ a passport."
"You need a passport to enter the country."
"Don't be absurd," Babette said. She pointed to her face. " _This_ is my passport."
Glances were exchanged behind check-in. A more senior agent was summoned, a quintessence of competence and courtesy in a blazer with numerous little medallions on his lapel betokening years of competent handling of crises small and large. But on the matter of a passport, he was gently unmovable.
It was at this point that Babette, who had, to be sure, been under a strain these many past months, finally and irretrievably lost it.
She stormed off to a series of first class check-in counters of international carriers, demanding a seat, if not the entire section. Alas, the passport requirement was the deal breaker at each one. Her remonstrations drew a crowd. Her choices of words eventually caused security to be alerted.
The famous photograph of her being half carried away—she had tried to bite the Wackenhut security man—by half a dozen personnel, like some Seattle protester, was soon over the wires and onto the front pages, accompanied by the news that Max was, apparently, well on his way into somewhat deeper exile in the Far East, aboard his own plane, the first class section of which was all his own.
# Chapter 32
As Babette was being subdued by Wackenhut and the LAPD, Beth was moving to subpoena the Log Cabin tapes. This was a complicated legal maneuver, inasmuch as the National Security Agency had not acknowledged that they existed.
Allowing the Wiley P. Sinclair affidavit had become a radioactive decision. Protesters now gathered outside the courthouse carrying signs calling for Judge Dutch's impeachment. The right wing was especially in dire need of mollification.
When the director of the FBI was observed one morning entering the judge's chambers, one of the television networks promptly reported that the judge was being arrested for treason and that the highest law enforcement officer in the land was personally doing the arresting. The FBI quickly issued a statement saying that the director had merely wanted to "confer" with Judge Umin.
Boyce made himself more available to the press than a politician running in the New Hampshire primary. However, since the latest polls showed that over 80 percent of the American people now viewed him not only as "loathsome" but also as "worse than a traitor" to his country for having colluded with the fugitive Wiley P. Sinclair, he did most of the interviews à la Deep Throat, in basement garages, parking lots, and public parks. Every day he had to move from motel to motel to avoid a stakeout by the media. His face was now so recognizable that if he presented himself in public, people snarled and hurled objects at him. He wore dark glasses inside buildings. He went to a disguise shop and bought himself a mustache. During one TV interview, he forgot to take off the mustache—the producer mischievously did not point it out—which got his face replastered across the next day's front pages with snide captions.
The theme of Boyce's media drumbeat was that the President of the United States must "come clean" with the American people about the bugging of the White House. If he did not act, then Congress must surely step in.
"How do you feel," ABC News asked Boyce as they stood in a remote section of Fort Marcy Park overlooking the Potomac River, "about the fact that the majority of the American people say they despise you?"
"So do the majority of my ex-wives," Boyce said. "But that doesn't change the matter that the government is in possession of evidence that will exonerate my client."
The President of the United States, Harold Farkley, spent his week in Europe being photographed with a series of unsmiling foreign heads of state. Buckingham Palace expressed its displeasure over the putative taping of the Queen and the Duke of Edinburgh during their White House stay by refusing to be photographed shaking hands with him. The President of France declared that he would never set foot in the White House again, "in the event there is a microphone in my soup." The Prime Minister of Japan suggested that the head of the NSA should cut off his little finger by way of apology. Foreign newspapers ran cartoons showing Abraham Lincoln hiding under a bed wearing headphones, listening in on the pillow chat above. In truth, President Harold Farkley's foreign tour could not have been called a success.
Meanwhile, the 672 other people whom the MacManns had over the years invited to spend the night in the Lincoln Bedroom by way of thanking them for having donated millions in "hard" money to their campaign and political party were enduring their own individual autosda-fé at the hands of a gleeful media. They were all being tracked down and asked how they "felt" about having been bugged. The answer was generally, "Not great."
Assuming that they _had_ been bugged. The National Security Agency was in high hunker-down mode, refusing all comment. This corporate muteness, however, was rapidly exhausting the national patience. The various congressional oversight committees were being forced by public opinion and a salivating media to _tsk-tsk_ and demand—demand!—the truth. Moreover, protesters were beginning to show up at the agency's main gate in Fort Meade, Maryland, outside Washington, with furious signage saying, RELEASE THE TAPES and FARQUANT IS BIG BROTHER. The pixel pundits generally agreed that government hadn't been this much fun since the early 1970s.
"Sir?"
"What, Henderson?"
"Frigby is outside, with the latest polls."
"I don't have time. What do they say?"
"I thought you might want to hear directly from Frigby, sir," said the chief of staff, who knew from experience never to be the bearer of bad tidings when you can let someone else do it.
Frigby, reluctantly granted access, gave it to the President straight. A majority of Americans blamed him personally for Operation Log Cabin, despite the fact that he had not authorized it or been president when it was put into effect.
"How can this be, Frigby?" he pleaded.
The chief of staff looked away. Harold Farkley's pain was too much to watch.
"Sir," said Frigby, "the majority of the American people can be pretty obtuse, when you come right down to it."
"What are you saying?"
"You need to get rid of this issue, sir."
"Goddammit, Frigby. Goddammit, Henderson."
"Yes, sir."
"What I'm about to tell you must go no further than this room," Roscoe Farquant said to Judge Dutch. It was 10:30 at night. They were alone, having both entered the vacant chambers of another judge, one half hour apart, by separate doors.
"I can keep a secret, General," Judge Dutch said a bit stiffly. "The FBI agents who vetted me for this position will attest to that."
"Let's leave the FBI out of this," said General Farquant. "First you should have some background."
When he had finished, Judge Dutch's glasses had completely misted over.
"General," he finally said, "you've thrown a monkey wrench the size of the Washington Monument into my trial."
"NSA's charter is strictly collective. We are not an investigative agency. You can see why it made no sense—more to the point, why we were unable, from the standpoint of national security—to come forward with any of this."
"Whatever," said Judge Dutch, who liked every now and then to show his command of current English slang. "But this impeaches the testimony of one of the leading witnesses in the trial."
"Notwithstanding, this was a highly classified intelligence-gathering operation. Until it was compromised by Mr. Sinclair."
"How did he know about it?"
"Presumably from his employers in Beijing."
"How did _they_ know?"
"That question raises a multiplicity of modalities. Obviously, Log Cabin was compromised. At any rate, by alerting Mr. Baylor to its existence, Mr. Sinclair has effectively rendered any intelligence we gained from Log Cabin useless. That, obviously, was his objective, to protect Mr. Grab."
"But you told me you didn't get anything on Grab."
General Farquant sighed. "No, we did not. Our conclusion ultimately was that Mr. Grab does not discuss his dealings with Indonesian middlemen for Chinese intelligence with his wife."
"Grab got the Indonesian oil contracts, President MacMann got covert contributions to his reelection campaign from the Chinese military? Laundered through Grab's offshore corporations. Is that it?"
"Essentially. That being the case, you can perhaps see why electronic surveillance of Mr. Grab was warranted."
"Why didn't you just plant the bugs on him directly, at his home?"
"I would rather not go into that, Judge. It's highly sensitive."
"I'm highly insensitive. Go into it. I insist."
"The NSA is not allowed to spy on American citizens within the United States. We found a loophole."
"Go on."
"The White House is a federal facility. You do not need a court order to conduct surveillance within a federal facility. If the bug happens to leave the federal facility, we have no control over that. If you follow."
"That's some loophole. So you bugged not only the Lincoln Bedroom, but the personal effects of the people staying in it, so that when they left, you could continue to listen in."
General Farquant nodded. "Cell phones, PalmPilots, laptops."
"Why not just rely on the FBI and CIA to take care of it?"
"After the Wiley P. Sinclair incident, the Aldrich Ames incident, the Hanssen incident, our faith in the integrity of the CIA's and FBI's ability to keep secrets was, you will appreciate, minimized. And if you can't trust the CIA and FBI, who can you trust?"
"The NSA, apparently," Judge Dutch said somewhat tartly. "But if you were only after Max Grab, why did you tape six hundred and seventy-two other people?"
"Judge, I came here to convey to you privately that if a subpoena by the defendant were narrowed to one particular tape, that of September twenty-eight, then NSA might not contest the subpoena. Otherwise..."
"The NSA might just declare that no such tapes exist. And burn them all. Is that it? And that's the end of it?"
"I'm not in a position to comment on that."
"Boyce!" Beth was speaking on a cell phone given her by a Washington girlfriend, out of view of the Secret Service. She had dialed a pay phone in Arlington, just within the outer limits of the radius within which Boyce was permitted to travel pending the disposition of his own criminal case. "They just called. Dutch is going to allow the subpoena, provided we limit it to the one tape."
"That's great news. But..."
"What?"
"Don't you want to hear what's on those _other_ six hundred and seventy-two tapes?"
"Not right now."
"I want to be there in court when they play it. I'll wear a disguise. I'm getting good at them."
"Just stay in the car. For God's sake, Boyce."
"Your SS men want to shoot me. The looks they give me."
"Hold on. Oh jeez, it just came over the news."
"What?"
"Babette. Guess who's representing her?"
"Not—"
"Alan Crudman."
Alan Crudman was notorious within the legal fraternity for billing his clients not only for his legal services, but for going on television to talk about them. Typically, he would talk about them for five minutes and then devote the rest of his airtime to plugging his latest book, each of which was, "in all modesty, my best."
Within hours of being hired by Max Grab—now said to be "in seclusion" in either Macao or Kuala Lumpur—to represent Babette, Crudman's publisher announced that his book on the case, provisionally titled _Tape Rape: The Framing of Babette Van Anka_ , would be in stores two weeks after the trial ended.
The night before he accompanied Babette to court, Alan Crudman managed to appear on all three television networks, plus half a dozen cable shows, for an estimated $17,500 aggregate of billable hours.
However, despite successfully squeezing himself into the final round of the Trial of the Millennium, he faced the nightmarish fact that ultimately it was not _about_ Alan Crudman.
Eighty percent of the American public might detest Boyce, but they loved watching him. After months of listening to Alan Crudman drone endlessly on television about how he would have handled it all, no one, really, was in the mood for much more of his self-glorifying yaddayadda. Judge Dutch certainly wasn't.
"If it please the court"—Crudman rose—"we ask for a delay in order to file a motion to suppress this so-called evidence."
"Denied."
"But Your Honor—"
"Sit down, Mr. Crudman."
A ripple of laughter went through the press section.
"With all respect, I will be heard," Crudman said hotly.
"Mr. Crudman, this is not _Hard Gavel_. But I will," Judge Dutch said, raising his own hammer of authority in a distinctly minatory way, "use this if you remain standing one more second."
Crudman sat, flushing redder than a boiled ham. He whispered to a pallid Babette, "I'll crush him on appeal."
This remark was caught by the network lip-reader and duly relayed to the viewing public, along with the obligatory preface that the correspondent was just guessing at what Crudman was saying to his client.
NSA tape number 4322-LC was duly entered into evidence. A special master of evidence had been appointed by the court to take custody of it once the NSA had handed it over.
Judge Dutch warned those who were "participating" in the proceedings via television that the tape they were about to hear contained material of an _adult_ nature. This had the effect of causing every teenager in America flipping through TV channels to stop right here.
"You may start the tape," Judge Umin instructed the clerk of the court.
"Oh, baby, baby, baby, _Mr. President_..."
Heads turned toward Babette, slumped forward in her seat. There was speculation that she had, in fact, died.
The media section was in extremis. When they heard the sound of a violently creaking boxspring mattress along with repeated thumps—evidently the sound of a head, either Babette's or the President's, smacking against the venerable Lincoln headboard—several of them temporarily lost control. They saved themselves from expulsion by masking their laughter as tubercular coughing fits.
Presently a prolonged "Unh" was heard on the tape.
A groan of male exhaustion, then forced purring: "Nothing wrong with you!" Then, "Would you like me to...?" Followed by a female gasp of surprise and the sound of something suspiciously genital being plunged into water and ice cubes. Then an apparently valedictory grunt and the sound of a door opening and closing.
The judge ordered the clerk to stop the tape.
There was silence in the courtroom.
Crudman rose in his seat. "We challenge the authenticity of this tape and move to have it stricken from the record. This is obviously an attempt, _by_ the government and, if I may, _by_ this court to impugn the testimony of Ms. Van Anka."
The consensus of the pixel pundits that night on TV was that Crudman, confronted with this steaming evidence, had decided to try to shift the attention back to his own glorious self by provoking Judge Dutch to find him in contempt. However, Judge Dutch did not rise to this bait, instructing him simply, once again, to sit down and shut up—using, of course, rather more dignified language.
# Chapter 33
Crudman responded to this indignity by going on network TV that night and hinting that the judge was "anti-Semitic." It was an odd assertion, given that Judge Dutch was himself a member of the tribe. Confronted with this inconvenient fact, Crudman shot back that the judge was a "self-hating Semite." It was the general consensus that the only Semite toward whom Judge Dutch might be anti was Alan Crudman, but this was no serious charge, since it put him squarely in the mainstream of American public opinion.
It was not a quiet tableau outside Babette's hotel, the Elegant, a few blocks away from her previous home away from home, the White House. Several hundred cameramen had assembled, it was suggested, so that they would be on hand when Babette leapt out her seventh-story window. In his television interviews that night, Boyce said she really should have stayed at the Jefferson, where the management was so notably hospitable toward participants in the trial.
The scene inside Babette's suite at the Elegant was no less tumultuous. Her entourage now consisted of Crudman and his team of half a dozen lawyers and investigators doing what they could to gird Babette's loins against an almost certain criminal indictment for perjury; her nutritionist; trainer; yoga instructor; doctor armed with hypodermics and state-of-the-art beta-blockers; her publicist, Nick Naylor, now upping his own daily dosage of Prozac; and three of her most stalwart sycophants, flown in from Los Angeles to remind her how fabulous she looked and what a great movie this was all going to make—taking care not to say, "After you get out of prison, darling."
Boyce and Beth lay on the floor of her Cleveland Park aerie. They could hear the hum of the media satellite trucks parked outside the gates. Boyce's own traveling press stakeout team had followed him here, but at this point so what.
Boyce was going over court transcripts. Beth filled out legal pad after legal pad with questions for her impending cross-examination of Babette.
"I like to think that I'm a reasonably compassionate person," Beth said. "But to be honest, I'm looking forward to crucifying her."
Boyce held up the transcript of the tape recording. "Where she says, 'Nothing wrong with _you_.' Here, line twenty-five. Tell me what you know about that."
"It's the sort of thing women tell men to make them feel good. Hasn't anyone ever told you that?"
"As a matter of fact, no."
Beth patted him on the rear. "They will."
" 'They'? You mean the attractive large men with tattoos I'm going to meet in prison?"
A look of pain came over her. "Boyce. It's not _funny_."
"I'm not laughing. All right, let's take it one trial at a time. We have two problems. First, we were the ones who made Babette out to be the chaste and faithful Mrs. Grab and patroness of peace in the Middle East. We were the ones who gave her an alibi, that she was in bed with curlers watching Elizabeth Taylor on TV screaming at Richard Burton. _Now_ you're going to be telling her, You lying slut, you were in there _schtupping_ my husband. That hangs her on a hook. But it doesn't get you off yours. Because now you really have a motive for killing him."
"It's a problem."
"So let's look at this 'Nothing wrong with _you.'_ What's going on here?"
"She's flattering him on his performance."
"Beth, the man sounded like he was dying."
"All men sound like they're dying when they make love."
"And all women sound like they're pretending to die? Putting aside the sexual politics for a moment. Okay, they're banging away, someone's head is bashing against the headboard—which, by the way, you will contend was _his_ head, which _completely_ compromises the Revere bruise. Excellent. So they're screwing away... he sounds like he's about to collapse... she's going 'Oh, baby, baby, _Mr. President'_ —a little kinky, by the way—and they finish. _Unhhhh_. Now she flatters him on his performance. He doesn't respond. Then there's this gasp coming from her. Here, line thirty-four, almost like he's coming after her again for more, which isn't likely since he sounds like he's about to have a heart attack... and that sound of ice and water. Is he drinking? Did he have booze on his breath when he got back to the room?"
"I don't know. I didn't kiss him. He wasn't a boozer, though. There's always a carafe of ice water on the nightstand when guests are staying."
"Did he usually drink water after banging you?"
Beth sighed. "I can't remember. It's been a while."
"Well, _try_."
Beth thought. "Sure. Everyone does. You're thirsty after a good lay, I seem to recall."
Boyce replayed the tape in his mind. "It's not the sound of drinking. There's no gulping. There's ice tinkling and an 'ah' sound. It's the sound of something being _immersed_."
"His...?"
"What a gentleman."
"We're not dealing with a gentleman. An officer, maybe, but no gentleman."
"Back when you and War God were enjoying full marital relations, was his normal postcoital behavior to get up and immerse his hot dog in the water jug?"
"Not that I can recall."
"So why's he doing it here?"
"Maybe it was sore," Beth said with a trace of jealousy. "Maybe it was chafed. From friction."
"Well, tomorrow when she's on the stand, I want you to home in on the ice water. No matter how awkward it gets. Why did the President dip his willy in your water, Ms. Van Anka? Why? Why?"
"I can hardly wait," Beth said without relish.
"Stay on that until you get an answer. Replay that part of the tape until she crumples. That's _key_."
"Defense calls Babette Van Anka."
Babette took the stand dressed in a black pantsuit and Jackie O dark glasses, which she was asked to remove.
Judge Dutch directed that she be administered the oath. Babette said that she needed to confer with her attorney, Mr. Crudman. Judge Dutch waved Crudman forward. Crudman, a wee man, had to stand on his tippy-toes before the bench. Judge Dutch shook his head and then waved the unwanted counsel back to his corner.
The judge informed Babette that despite the fact that she was still under oath from her previous testimony, "a new oath is in order."
According to the ratings released afterward, Beth versus Babette was the most widely watched part of the Trial of the Millennium. Over one and a half billion human beings tuned in. Once again, airline pilots called in sick, elective surgeries were postponed. Even the launching of America's newest aircraft carrier, the _Tom Clancy_ , was postponed, spurious technical reasons being alleged. No one wanted to miss _this_.
Nor did Boyce, but that morning did not find him in his customary observer post, in Beth's Secret Service car in the courthouse basement, being glowered at by fuming agents in the front seat.
Bethesda Naval Hospital is a venerable, bleached-white stone presence off Wisconsin Avenue, just beyond the northern border of the District of Columbia. It is here that Marine One, the presidential helicopter, brings presidents for their annual checkups, so that their most intimate medical details can be shared with the entire world. It was after one of these visits in the late 1970s that navy doctors vouchsafed that the President of the United States was afflicted with grave hemorrhoids. The capable doctors of the United States Navy are the custodians of the health of their commanders in chief.
The people Boyce knew included those who knew about professional-quality disguise and professional-quality fake IDs. So this morning, as an estimated 178 million Americans sat glued to the television watching the two Amazons of the Trial of the Millennium have at each other, Boyce Baylor, wearing glasses, wig, and mustache and dressed in the uniform of a vice admiral in the U.S. Navy Medical Corps, strode confidently through the main gate, presented his badge to the marine guard, and proceeded on his way. He found a men's room, where in the toilet stall he removed from his briefcase a clipboard, white hospital gown, and stethoscope. He checked himself in the mirror. The most reviled man in America now looked like the most respectable. It gave him a thrill to see his new identity. Boyce saluted himself smartly, took a deep breath, and went out the door.
He'd defended enough doctors accused of gross malpractice to know his way around basic medical lingo, but just to be safe, he had crammed himself with some trauma and coma-related buzzwords like "fixed and dilated" and "Babinski reflex."
He presented himself at the nurses' station and politely but crisply asked for Dr. Grayson's room.
When the nurse, a lieutenant, looked up and saw three admiral's stars, she pointed the distinguished-looking man on his way with a respectful, "Sir." Boyce nodded pleasantly.
There was a marine guard outside his room, but marines are trained from day one to salute admirals so vigorously that they nearly concuss themselves.
"Sir!"
"As you were, marine," Boyce growled with what he deemed appropriate hierarchical condescension. He wondered if he should have barked, "Straighten that gig-line, Corporal!"
Dr. Grayson was sitting up at a forty-five-degree angle, with the usual tubes running in and out of him. Life-support machines hummed and clicked.
A nurse, red haired, pretty, was in attendance.
"Admiral?" she said, obviously surprised at seeing this unfamiliar face.
"As you were, Nurse. Admiral Quigley, from Cinclantnavmedcom." It sounded official, anyway. He added, "Norfolk."
The nurse's eyes widened at the augustness of the syllables.
"Yes, sir."
"CNO asked me to look in on Captain Grayson."
"Yes, sir."
"How's he's doing?"
Boyce knew from the item in the paper three days earlier that Captain Grayson had come out of his postcrash coma.
"Sir, his vitals have stabilized. The Medrol appears to have turned around the cerebral edema."
"Hm," Boyce grunted, apparently satisfied. "Usually does. How did he do on the Babinski?"
"No evidence of brain damage, sir."
Boyce leaned closer to her to whisper. She smelled lovely. "How is he doing _psychologically_?"
"He appears depressed, sir."
"Um." Boyce nodded knowingly. "Would you excuse us?"
"Sir." She left. Boyce approached the bedside.
"Well, good morning, Captain," Boyce greeted him heartily. "You're looking fit for sea duty."
Captain Grayson did not answer. The only sea duty he looked ready for was burial. It wasn't the damage from the car wreck. Boyce could see that. The man might be healing, but he was still broken inside. The eyes were lifeless with pain.
"You're not watching the big trial, Captain? Mrs. MacMann is cross-examining the actress Ms. Van Anka."
Captain Grayson turned and looked at Boyce. The eyelids fluttered. The eyes studied Boyce's face carefully. They moved to the admiral stars on his epaulets, then back to Boyce's face. They narrowed suspectingly.
"You know who I am, don't you, Captain?"
A placid look came across the captain's face. It was as though, standing on the bridge of a ship after a fierce engagement, he had just received the news that the damage from a torpedo had been repaired and that the ship might now not sink after all.
"Sorry about the uniform," Boyce said. "It was the only way to get past the marines. I know how much this uniform means to you. I think I know how much President MacMann's service in the navy means to you."
Pain flickered back into the eyes.
"Shall we watch, Captain? Shall we watch the trial together?"
Captain Grayson looked stricken and, for a moment, lifeless.
_Please_ , Boyce thought, _don't let the machines start beeping_.
Finally Captain Grayson nodded. Boyce rose and turned on the TV monitor.
# Chapter 34
Ms. Van Anka, I'd like to draw your attention to the testimony you have already given this court and to this jury," Beth added for good measure. "Would you at this point care to modify, or change, that testimony?"
Babette looked mournfully toward her attorney. Her entourage was seated with the spectators, beaming encouragement at her, slipping her thumbs-up gestures, but at this point it would have taken an entire Hollywood Bowl full of supporters to cheer up Babette. She asked the judge if she could confer with her counsel. Judge Dutch wearily waved Crudman, Beth, and Deputy AG Clintick forward for a sidebar.
"Ms. Van Anka," Crudman said, "is willing to modify her prior testimony, which occurred at a time of severe emotional distress, on the condition that she receives total immunity from any future prosecution for perjury."
Judge Dutch leaned back in his chair. Beth and Sandy looked at each other. Of late, the deputy AG's attitude toward Beth had softened.
The judge leaned forward and whispered to Crudman, "No way in hell, Counsel."
The network correspondent translated for his viewers, "My guess is that Judge Umin will _decline_ any petition from Van Anka's defense attorney to immunize her prior testimony."
Beth suppressed a smile at Crudman's humiliation as he went to the witness box to give Babette the unhappy news. He added that the judge would rue the day. He would be crushed on appeal! Meanwhile, go with plan B.
Beth now resumed her cross-examination. "Is that your voice on the tape, Ms. Van Anka? Along with the President's?"
"Sounds a bit like me. But I couldn't say."
This brought a gale-force expulsion of air from the lungs of the spectators. Judge Dutch did not gavel silence. He seemed too occupied trying to maintain his own composure.
Beth, too, was having a hard time. "I see. Any guesses as to _who_ it might be?"
DAG Clintick did not object. She was looking down at her table, trying to retain her composure.
"I couldn't say," Babette said. She smiled bravely. "I have many imitators."
Crudman winced. The idiot—he'd told her, Give them nothing! Keep your answers _to the minimum!_
"Imitators?" Beth said.
"I'm a well-known actress. My voice is widely known. Some people try to sound like me."
"The woman on this tape isn't you, but is trying to sound like you? Is that what you're saying?"
"I don't know _what's_ going on in that tape."
"Ms. Van Anka," Beth said sympathetically, "the authenticity of that tape has been certified by a special master of evidence appointed _by_ this court as having been recorded in the early morning hours of September twenty-ninth, during which time it is a documented fact, as recorded by the chief usher of the White House, _and_ the Secret Service, that you were a guest in the Lincoln Bedroom, where this tape _was_ recorded. Now are you telling the court, the jury, that that's _not_ you?"
It was all too much. Here Babette Van Anka's training as an actor overtook her instinct for self-preservation. If she was going down, by God, it would be a going-down worthy of Bette Davis or Joan Crawford or Gloria Swanson.
"Yes, it's me! Of course it's _me_! I loved him! Unlike you! Who murdered him!"
Crudman bolted to his feet. "Your Honor, my client is not herself. Move to strike her remarks—"
It took several minutes to restore order. "Ms. Van Anka," Judge Dutch said sternly, "another outburst like that and I will find you in contempt of court."
"Oh," Babette moaned. "You don't _know...._ "
"You will answer defense counsel's questions directly, to the best of your ability. Without commentary. Is that _understood_?"
"This is a perversion of justice," declared Alan Crudman. Indeed, that became the title of the first of his three books on the case.
"You are out of order, Mr. Crudman. And I have run out of warnings to you. The clerk of the court is instructed to remove Mr. Crudman."
Crudman was removed. Outside the courthouse, he told the media that he now knew what it was like to be a "Jew in Hitler's Germany" and vowed to "pursue justice all the way to the Supreme Court." The Supreme Court, one reporter pointed out, was really only a few blocks away.
Inside the courtroom, after a ten-minute recess, Judge Dutch told Babette that the rest of her cross-examination could be postponed until she had engaged other legal counsel.
Beth rose. "Your Honor, in that case I move that Ms. Van Anka be placed in custody as a flight risk. Her husband, Mr. Grab, is currently being sought by federal authorities and is at large abroad. The Attorney General's Office has indicated that they will seek an indictment of her for false testimony. It is therefore our contention that she may attempt to flee."
Murmuring. Sidebar. The judge leaned back in his chair, turned to Babette. "Ms. Van Anka, the court finds that given the circumstances, you present a flight risk. You may either continue your testimony here today, without legal counsel. Or you may continue it later. However, in the meantime, I will order that you be held at the federal detention center pending that testimony."
_"Jail?"_ gasped Babette.
"Detention."
"No. No, no no no no. I want to testify. Now. Right away."
"Very well. You may proceed, Mrs. MacMann."
"Ms. Van Anka," said Beth, "you admit, then, that that _is_ your voice on the tape."
"Yes. I said so, didn't I?"
"So you did. I don't wish to make this any more difficult for you than I know it must be...."
"You have _no_ idea. No one has _any_ idea how hard this is."
"I'm sure it is," Beth said, taking a breath. "It would appear, to judge from the tape, that you and the President were engaged in..."
"Pressing the flesh," whispered a reporter.
"... in sex. Is that a fair inference?"
"We made _love_. You wouldn't know about that."
"Ms. Van Anka," said Judge Dutch, "I will not warn you again."
"What? What did I say?"
"Was the President," Beth continued, "all right?"
"He was fantastic."
"On the tape he sounds... Forgive me, I'm not sure quite how to put this, he sounds very... Let me put it this way: Did you observe him..."
"I did more than observe him, honey."
"Indeed. Was he physically all right? On the tape he sounds tired."
"Of course he was tired. He'd just been to the moon and back."
"It's a long trip." Beth nodded. "So physically, he performed, um, well?"
"I said, he was great."
"Even after a long evening? At his age?"
"Maybe he was inspired."
"Let me draw your attention to the transcript...." Babette was provided with one. "Here on page seven eighty-three, line thirty-five. Your Honor, I ask that this portion of the tape be played for the court."
The sounds of Babette's gasp and the tinkling of ice cubes, followed by a short male " _Ahhh_ ," were heard.
"What was happening at this point, precisely, Ms. Van Anka?"
"He... needed... he was... he was thirsty. He was having some water."
"That doesn't sound like someone drinking. It sounds like something being immersed in water."
Babette was silent.
"Ms. Van Anka?"
_"What?"_
"Was he drinking?"
"I said that already."
"We can call in forensic acoustic experts to advise the court to reconstruct what that sound is."
"He had a hard-on, all right? He had a hard-on and he was going back to your room—where he was worried that you might kill him, which you did. He dipped his business in the ice water to make it relax. All right?"
Judge Dutch had to gavel the courtroom back to something resembling order. Babette's entourage was warned that if they did not stop making those sounds, they would be removed.
"And did his... business relax, Ms. Van Anka?"
"What do you want from me?"
"The truth. That's all."
"No, it didn't. He had to sort of... stuff it into his trousers."
"The President was in his late fifties. It had been a very long evening at this point, entertaining a head of state, many guests, then entertaining, I guess, you, in a vigorous physical manner. The time was now after two A.M. And yet even after exhausting lovemaking," she added, "if indeed it could be called that, are you telling the court that he _still_ maintained an erection?"
"A monster."
"That's unusual."
"How would you know?"
"Ms. Van Anka," said Judge Dutch, "my patience is at an end. One more comment like that and I will find you in contempt, and you will spend the weekend in detention. Is that clear?"
"Yes. _Yes_ ," Babette moaned.
"Ms. Van Anka," Beth continued, "did the President have any pharmaceutical assistance that night, to your knowledge, that would have enabled him to maintain such a... heroic erection, even after sex?"
"I..."
"Yes?"
"He had some Viagra."
Murmurmurmurmurmur.
"Viagra, the prescription medicine that enables men to achieve and maintain erections. Is that what you mean?"
" _Of course_."
"The President took Viagra?"
"Sort of. In a way."
"How do you mean, exactly?"
"Oh, God. It's..." Babette looked over imploringly at the judge. "It's _private_."
"This is a murder trial, Ms. Van Anka," said the judge. "You are legally and morally obliged to provide such evidence as you are aware of. Which you should have done the _first_ time you testified."
"All right all right. The President and I had... been intimate before. And on the last several occasions he had been unable to perform. I mean, as a man." She sighed heavily. "You know what that does to a man's ego. I wanted him to be happy and fulfilled. He was the President of the United States. If a president isn't fulfilled, then the world is at risk. I didn't want him to... I didn't want to say to him, 'Here, take this.' So I ground up a few pills into powder and mixed it with some moisturizing cream and applied it to my... self. So that it would, you know, act... topically. Like ointment."
"You created a Viagra ointment and applied it to your private parts?"
"What's _private_ anymore?"
"And this topical ointment, apparently, got into the President's system?"
"Hard as the Rock of Gibraltar."
Beth said, "No further questions for the witness at this time."
# Chapter 35
So," said Beth back at Rosedale, "was it good for you, too?"
"Not bad," Boyce said. "Not bad at all. You might just make a good trial attorney in five or six years."
"We're still not off the hook. So he was hopped up on Viagra. Funny that didn't make it into the autopsy report."
"Maybe the doctors were concerned about the dignity of the orifice."
"It still leaves me waiting for him, lurking behind the door, holding the spittoon."
"Someone else was impressed with your cross-examination. In fact, deeply moved."
"Alan Crudman? O. J. Simpson? I give up."
"Captain Cary Grayson."
"How do you know that?"
"I spent the morning with him."
"You saw Grayson?"
"Oh, I saw Grayson. Captain Grayson and I bonded today."
"How'd you get in to see him?"
"Never mind. But he's ready to be deposed. And since he looks like he's about to die any second, I suggest you get on the phone to Judge Dutch right now and set it up. Right away."
Judge Dutch gave the necessary instructions. Within an hour and a half, he, Beth, Sandy Clintick, two clerks, a stenographer, court video and sound technicians, and a notary public were roaring up Wisconsin Avenue in a U.S. marshals motorcade to Bethesda Naval Hospital. This naturally attracted the attention of the media, who joined in with their own motorcades, attaching themselves serially. By the time the procession reached Bethesda Naval, the motorcade was fifty-four vehicles long—longer even than a normal presidential motorcade. It's in the _Guinness Book of World Records_ , under "Longest Motorcade."
In the rush, no one thought to notify the main gate of Bethesda Naval that the mother of all motorcades was about to roar through. When the marines saw this imperial millipede approaching, flashing more lights than most airports, they assumed that it must be none other than the President of the United States, gravely wounded. They called inside with this information, causing such alarm that every trauma surgeon in the building—naturally wanting to succor their commander in chief—rushed to the emergency entrance. When the door of the lead limousine opened and out stepped the leading participants in the trial, the doctors stared at one another in disappointment and confusion.
The admiral in charge of Bethesda Naval didn't quite know whom to call. For a moment, it occurred to him to summon the marines in force. But Judge Dutch was the face of maximum authority in the land, and when the judge informed the admiral that he had official business, there was little the admiral could do but say, This way, sir.
Captain Grayson had to be wheeled into a larger room to accommodate the juridical crowd.
It was just as well that they had arrived when they did, for the captain died of his injuries that morning at 4:30 A.M., a few hours after his deposition was concluded, of causes not yet detectable by medical science.
The tape was played the next day, in court.
Beth: Captain Grayson, you performed the autopsy on President MacMann the morning of September 29. Your prior testimony to the court was that he died of an epidural hematoma resulting from blunt-force trauma to the head. Do you wish now, under oath, to retract that testimony?
Capt. Grayson: Yes, I do.
Would you then tell the court how the President's death came about?
There was no epidural hematoma. I did observe evidence for trauma. An apparent contusion, with modest ecchymosis, but no laceration. But this was not the cause of death.
What did the President die of, Captain?
He died of lethal cardiac arrhythmia.
In other words, his heart failed?
Yes. Most likely ventricular fibrillation due to a progressive fall in blood pressure, associated with an excessive dose of medication. His heart stopped.
Were you able to determine why his heart stopped?
The President had mild coronary heart disease. But this was not what killed him. Toxicology reported a high concentration in the blood of sildenafil citrate.
Is that the chemical name for the prescription drug Viagra? The one used to help men achieve and maintain erection?
Yes.
Are you then saying that the President died as a result of an overdose of Viagra? Is this possible?
In someone with coronary heart disease, Viagra in high concentrations can be fatal. The President received a lethal dose of it.
How much of it was there in his blood?
The equivalent of approximately 300 milligrams. The pills come in 50-milligram tablets. Six tablets' worth.
What conclusion, then, did you draw from these observations?
I concluded that the President had expired following or during an act of coitus.
Did you falsify the autopsy report, including the toxicology report?
Yes, I did.
Why, Captain?
The President was one of the most decorated veterans of the U.S. Navy. He served his country in war with distinction and with valor. I could not let history record that he had died in such a way.
So you blamed his death on the bruise?
Yes.
Did you intend, in so doing, to implicate the First Lady of the United States in a murder case?
No. No. I never intended that. I regret that truly. That was—no. No.
I understand, Captain.
At the time of the autopsy, I knew only that the President had been found in his bedroom. My intention was that it be blamed on an accident. A fall in the night, in the bathroom. An accident.
After the First Lady was subsequently charged with murder, why didn't you come forward?
I wanted to. But I could not make myself do it. I was still protecting my commander in chief. I was certain...
Certain of what, Captain?
I was certain that Mr. Baylor would get you off. He gets everyone off. I'm sorry, Mrs. MacMann. I'm so very sorry.
I understand, Captain.
Forgive me, Mrs. MacMann.
I do, Captain.
At this point in the videotape, Beth asks Deputy Attorney General Clintick if she wants to question the captain. Sandy Clintick is seen declining with a shake of her head.
# Chapter 36
The front page of the _New York Post_ showed a picture of a weepy Babette below a headline that could not have been larger had the news been that a meteor was about to crash into the earth and end human life:
_SHE_ DUNIT!
On TV, pixel pundits tripped over one another trying to respin their earlier proclamations of Beth's certain guilt.
"There was something about Van Anka's previous testimony that never sat right with me," declared _Time's_ reporter.
"I was never comfortable with the rush to convict Beth MacMann," said _The Washington Post's_ man.
Beth's phone began to ring again, now from agents and movie producers and publishers.
"Tina Brown just called. I might be able to pay your bill after all," Beth told Boyce.
Her elation was interrupted by the news that Alan Crudman had filed a motion to quash Dr. Grayson's deathbed deposition on the grounds that his medication, which included morphine, rendered it unreliable. There were precedents for such a motion, though Vlonko, still in court charting the minute-by-minute reactions of the jury, reported that Dr. Grayson's deposition had been "fucking dynamite," leaving most of the female jurors in tears. Even if Judge Dutch did throw out the Grayson deposition, the jury might still go with its emotions.
"We could still lose this thing," he said. "We're gonna have to dig him up, Beth."
"I would really, really, rather not."
The President had been buried at Arlington Cemetery as a hero, with the highest honors a nation could bestow. The caisson bearing his body had been drawn by horses across Memorial Bridge to the solemn _tum-tum_ of drums, followed by the traditional riderless horse, with reversed boots in the stirrups. At the graveside there had been a twenty-one-gun salute, an overhead flyby of a squadron of navy jet fighters in "missing man" formation, the echo of "Taps." Was it good politics for his widow and her criminally indicted lover-lawyer to send in a back-hoe to dig him up to see whether he had lethal levels of Viagra in his veins?
"On the other hand," Beth said, "I'm glad they didn't embalm him, in case we do need to go back in for another toxicology. I can't believe I'm talking this way about my husband. My whole life has turned into an out-of-body experience."
"Your whole life"—–Boyce patted her belly—–"is right here."
"Feel."
"He wants to know whether we've got his ticket on the Lisbon plane."
"Tell him Daddy's working on it."
Babette looked eerily composed as she took the witness stand. Either she was sedated or the raised stakes had concentrated her mind. She was past hysterics now and into icy defiance. She was facing criminal indictment, not only for perjury but also for assassinating the President of the United States with an overdose of erection medicine. Most legal commentators agreed, at least, that she wouldn't be charged with first-degree murder. Negligent homicide? Wrongful assassination, with an explanation? There were no precedents.
"We are," said Edgar Burton Twimm on the _Charlie Rose_ television show, "navigating in muddy water, in fog, at night, without a compass."
Beth had with her a laptop computer with a wireless Internet connection. At the other end of the connection was Boyce, still barred by an angry Judge Dutch from the courtroom. He was in a hotel room not far away, watching the proceedings on television, with his own laptop, connected to a high-speed computer line. He was able to communicate with Beth in print, on the screen, in real time.
He saw Beth on the TV, preparing to stand and begin her examination. He typed, YOU GO, GIRL.
She rose and went to the podium, bringing the unfolded laptop with her.
Boyce typed, CONTROL THE WITNESS.
"Ms. Van Anka," Beth began in a friendly way, "you are familiar with the substance of Captain Grayson's deposition?"
"The man was out of his skull on morphine," Babette said. "He didn't know what he was saying."
Alan Crudman preened by way of indicating to all that this was his ingenious line, not Babette's.
"Objection. Witness is not in a position to make a medical evaluation as to the reliability of the deposition."
Heads turned in surprise. It was Sandy Clintick. Whose side was she on, anyway? The consensus among the pixel punditariat was that with Boyce Baylor removed, the deputy AG now lacked an opponent "really worth hating."
"Sustained," said the judge. "You will confine yourself to answering the question put to you directly, Ms. Van Anka."
Beth continued, "You heard what Captain Grayson said in his deposition?"
"I _heard_."
"You told the court that you applied Viagra, mixed with moisturizer, to your... to the... to the relevant area. Is that correct?"
"Yes."
"How much Viagra did you use?"
"What do I look like, Lee Harvey Oswald? I wanted the man to be happy. Not dead."
Boyce typed, NOT FOR YOU TO DETERMINE. PS YOUR LAST MOVIE SUCKED.
"That's not for you to determine, Ms. Van Anka. That's a question that can only be resolved by a medical authority."
"A medical authority who falsifies autopsy reports and gives deposition when's he doped to the gills? Please. I wouldn't entrust an ingrown toenail to the man. May he rest in peace."
"Objection."
"Sustained. Ms. Van Anka, you are to answer the questions."
" _This_ is why my people left Europe."
"One more comment and I will find you in contempt."
Boyce typed, JUMP IN—NOW!! IT'S THEIR STRATEGY, TO FORCE A MISTRIAL. CONTEMPT→HOSTILE JURY→ DISMISSAL.→LET'S WIN **THIS** ONE NOT WAIT FOR THE NEXT.
"Babette," Beth said.
Babette started at Beth's use of her first name.
"Sorry. Ms. Van Anka. We—I—only want to find out how much Viagra you used that night. That's all. Under oath, please, just tell the court how many pills you crushed up and mixed in with the cream."
"You mean, honestly?"
The courtroom exploded with laughter. Alas, the irony was lost on Babette, who had lived for too long in a community where insincerity was the norm.
"Honestly." Beth smiled.
"Three. The fifty-milligram ones. I just wanted the man to be able to perform, not hold up the tent."
"Three pills? The blue ones?"
"Like this." Babette formed a diamond shape with her thumbs and forefingers. "You know, you can split them in two, but I figure, why?"
"I see your point."
Suddenly the two women were like old friends, chatting away knowledgeably about how much Viagra their partners required.
"They're not fatal," Babette said. "I mean, a ham sandwich can be fatal if you choke on it. I read the directions. One night I gave Max three. He was a bit flushed in the face. But he didn't die. Right now I could give him _ten_ Viagras."
"Did you administer it to your husband the same way you administered it to mine?"
"No. I—well, you know how men don't like to admit?"
"Oh, I know."
"I crushed them up and put them in his borscht."
"I see. Just one or two final questions, Ms. Van Anka. How did it occur to you to administer the Viagra to my—to the President in this way?"
"I couldn't get to his soup. The Secret Service sees you putting powder in the President's soup and they open fire. I have a friend who does it this way, with the moisturizer. She said it worked. It worked. Well, up to a point."
"Thank you, Ms. Van Anka. No further questions at this time. Reserve the right to recall the witness."
Beth looked down at her laptop.
PUT IT IN MY SOUP AND I'LL SHOOT YOU.
"Do we believe her that she ground up only three pills?" Beth said. "I wouldn't put it past her to feed the whole bottle into a blender."
"Yes," Boyce said. "I think for once she was actually telling the truth. But Grayson said he had three hundred milligrams' worth in him. That leaves three more pills unaccounted for. Did he have a prescription?"
"Are you kidding? Every time the White House doctors give a president a Tylenol, it's front-page news. He would never have gotten a prescription."
"Did you ever see any in his toilet kit?"
"I never went into his toilet kit."
"You didn't?"
"Not after I found a twelve-pack of rubbers in it."
"Twelve-pack? When did he have time to run the country? But assuming he had the pills—who gave them to him?"
Beth thought. "It would have to be someone he trusted. Trusted absolutely."
She said the name.
"We've got to be sure. If we get him up there on the stand and he says no, it'll look like we're just fishing. And we can't subpoena eighty of his best friends and ask them if they were slipping the President hard-on pills on the sly. They'd lie anyway, and who's to contradict them?"
"The advantage of this witness," Beth said, "is that he can't lie under oath."
"Defense calls Damon Blowwell."
Damon Blowwell seemed uncharacteristically subdued. Normally he looked like a pit bull who hadn't been fed in three days.
Beth asked that he be given the oath again, even though he was still technically bound by the first.
"Mr. Blowwell, you are a born-again Christian, are you not?"
"I am."
"And you have just taken an oath swearing, before God, that the evidence you give will be truthful, is that about correct?"
"I'm not a liar, if that's what you are implying."
BACK OFF, Boyce typed. GIVE HIM ROOM.
"I'm implying exactly the opposite, Mr. Blowwell. I have only one question to ask you today. Did my husband ask you to provide him with Viagra?"
Blowwell's lower lip disappeared into the upper. Every fiber in the man's mortal body wanted to say no, but the soul that he had rededicated to the Risen Lord was whispering, _The truth shall set ye free_.
"He might have." It hung there for a second or two before he added, "Yes, he did. He did."
Murmurmurmur.
"And how much did you provide him with?"
"One bottle."
"Containing approximately how many pills?"
"One hundred, I believe."
"Did you do this on one occasion, or more?"
"Yes."
"How many occasions, approximately?"
"Half a dozen. More, maybe."
Murmurmurmurmur.
"So you provided him with as many as six hundred, or more, pills?"
"That would be correct."
"And approximately when did you last fill the President's prescription, as it were?"
"It would have been about the middle of September."
"A few weeks before he died?"
"That's correct."
Blowwell's expression, for the first time that anyone could recall in public, took on a look of terrible pain. No one could remember ever before seeing Damon Blowwell look vulnerable. The man was crumpling.
"It was an accident, Damon," Beth said tenderly. "There's no need to blame yourself."
"Objection," said Sandy Clintick, almost reluctantly.
"Sustained. Mrs. MacMann," said Judge Dutch, "if you have a question for the witness, ask it."
"No further questions for the witness, Your Honor. Thank you, Mr. Blowwell."
"Yes, ma'am. I—want to add something."
Judge Dutch said, "Very well, Mr. Blowwell."
"I want to apologize to Mrs. MacMann."
# Chapter 37
For the sake of what remained of the national dignity, the exhumation of President Kenneth MacMann was carried out under wraps during the hours of two and five A.M. This did not deter the TV networks from providing live coverage of the event, consisting of telephoto nightvision lenses aimed at a dark tent with soldiers standing in front of it while commentators passed the time by speculating about what was going on inside.
"For an operation like this, Tom, they would use, probably, a back-hoe, in conjunction with—there would be a backup backhoe, in the event the primary backhoe was for whatever reason unable to dig, or malfunctioned."
"How deep is the President buried?"
"My information is that the President is between six and eight vertical feet beneath the stone _plaza_ that was erected, the one that was placed over him after the burial."
"So they have to go through that first, correct?"
"Yes, and that's tough Vermont granite, of course."
"Once they've gotten the casket to the surface, do they—what happens then?"
"We're told that the casket, which is within a bronze outer casket, to prevent—that everything, the inner and outer caskets, will be loaded onto a military transport and taken to the National Institutes of Health."
"No more Bethesda Naval autopsies."
"No. And of course it is ironic that the NIH, where this second autopsy will be performed, under the supervision of a special master of evidence appointed by the court and _six_ independent pathologists and toxicologists, none of them connected with the armed services, is practically right across the _street_ from Bethesda Naval Hospital."
"One thing I'm not clear on—why wasn't the President embalmed?"
"It's standard procedure in cases of murder or suspicious death, Tom, _not_ to embalm. In case they have to exhume the body for further medical testing. If you embalm a body, that's it as far as further toxicology testing goes."
"Talk to us for a moment about formaldehyde...."
"Did you watch?" Boyce said. _"Honestly?"_
"I was working on my concluding argument."
"You won't have to give one if the tox report comes out the way it should."
"I..."
"What is it? Did he kick?"
"No. Nothing. Just a procedural point I was going to ask you about. I've forgotten. I'll ask you about it when I see you. When will they know?"
"Possibly this afternoon. Toxicologists are a pain in the ass. They love to take forever. They know everything's hanging on them, so they get to be the center of attention. Did you see the _Times_?"
"I've given up newspapers."
"You might want to check out the front page of today's. There's a poll."
"Has the procedure"—the word Beth used for exhumation—"caused my remaining four percent of supporters to hate me?"
"Quite the opposite. Your numbers are up, as you'd put it. Seventy-five percent feel the government owes you an apology. That's quite a reversal. You should be pleased."
Beth sighed. "Yes, that's nice."
"You've gone from being Lady Bethmac to Wronged Woman. You're not happy?"
"My husband is on a metal table somewhere. You're facing five years for saving me from myself. Having created the mother of all scandals, I'm about to become a mother and haven't the slightest confidence that I won't screw that up, too. 'Happy' isn't quite the word for what I feel right now. I better get back to my concluding argument. Just in case Dr. Grayson really was gaga on morphine and hallucinating the whole thing."
"Whatever happens, you're going to be a brilliant mother. You're going to be the mother of all mothers. Do you know why?"
"No idea."
"To make up for screwing up everything else in your life. Including my life."
"That's pretty good motivation."
"Folks," CBS News anchorman Dan Rather told his viewers, looking as if he might, finally, have a fatal nosebleed on live television, "this case has got more evolutions than a species in the Galápagos. We are told that a Dr. Laftos Crogenos, chief pathologist of the team that has performed the second autopsy on the remains of President MacMann, will be making an announcement shortly. Bob, that name, Laftos Crogenos, has more vowels in it than a bowl of alphabet soup after buzzards have finished picking out all the consonants. What do we know about him?"
"Dan, Dr. Crogenos is Greek, originally. But he is a _naturalized_ American citizen—"
"So his sympathies, naturally, would be above question?"
"There's apparently a _saying_ , Dan, in the pathology community, that there _are_ no nationalities around an autopsy table."
"Good. That's what Americans at this point need to hear."
"Dr. Crogenos has been for many years chairman of the Department of Forensic Medicine at Johns Hopkins medical school. He has performed over fifteen _thousand_ autopsies and is considered to be one of the best pathologists in the _world_. In the words of one colleague, this man can open you up from stem to stern with his eyes closed."
"This is no roadkill armadillo on Route 77 north of Corpus Christi he's working on, but a former president of the United States of America."
"There he is now. Dr. Crogenos is approaching the podium, accompanied by the five other medical examiners...."
"How does he look to you, Bob? What can we say from his expression?"
"Dan, it can't be _easy_ examining the corpse of a, well, _any_ corpse. Especially one that's been in the ground for over a year. But this one in particular, with the whole world watching over your shoulder, as it were. It has to be _tremendous_ pressure."
"I'd be jumpier than a coked-up Mexican who's just found half a _cucaracha_ in his guacamole. Let's hear what he has to say."
Dr. Crogenos's statement took less than five minutes to read. As he spoke, his face was bathed in thousands of flashes. It was done with as much dignity as could be mustered. He announced that the President had been killed by a "probably accidental" overdose of sildenafil citrate. There was evidence of "moderately advanced" coronary heart disease. An estimated 300 milligrams of Viagra had put too great a strain on the heart, bringing about lethal cardiac arrhythmia. The penile epidermis showed traces of sildenafil citrate as well as ingredients commonly found in high-end brands of moisturizing cream. There was no evidence of an epidural hematoma. The bruise on his forehead, though pronounced, had not been fatal.
At this point, Dr. Crogenos looked at his colleagues and sighed. If the Republic lasted a thousand years, schoolchildren in ages hence would remember President Kenneth MacMann as vividly as, if not more vividly than, Presidents Washington, Lincoln, and Roosevelt. His place in history was assured.
# Chapter 38
Boyce's cell phone rang.
"Hello, Counselor."
It was Sandy Clintick.
"I won't bother asking you how you got this number," he said. "But it's extremely unlisted. To what do I owe this pleasure? Are you ready to move to dismiss the indictment of my client?"
"Your former client. You're no longer representing Mrs. MacMann."
"So you're calling to apologize on behalf of the federal government, for entrapping me in the jury-tampering case?"
Sandy Clintick laughed. "No, that wasn't on my agenda. I don't have any involvement in that case. But," she added, "it's one I'd frankly _love_ to try."
"I'll bet you would. For what it's worth, I'm glad you're not. You're not as good as me, but you're up there."
"When the biggest narcissist in the law tells me I'm almost in his league, I feel lavishly complimented."
"No, Alan Crudman is the biggest narcissist in the law. I'm second."
"Shall I get to the point, or shall we continue to sniff each other?"
"By all means. Fire away."
"I'm weighing whether to move to dismiss."
"Oh, come on, Sandy. I can hear the mob outside your window with torches and pitchforks, chanting, 'Justice!' "
"I have insulated windows."
"I also read that you now have a U.S. marshals bodyguard. Don't worry, after the first coupla dozen death threats, you get used to them. I get Christmas card death threats."
"I'm thrilled to be in your league, Boyce. The reason I'm weighing whether to dismiss is there's something still bothering me."
"What would that be?"
"The Grayson deposition."
"The Crogenos autopsy supports everything that he said."
"There was something else in it that bothered me. At the end, when he asks Beth to forgive him."
"Yes?" Boyce said cautiously.
"She did. Just like that."
"She's a forgiving type."
"I don't buy that. Her life became hell because of what he did. No one is that forgiving. Even Jesus Christ would have needed to think about it for five seconds before saying, Okay, let bygones be bygones."
"She's a mensch. It's why I fell in love with her back in law school."
"Some mensch. She dumped you for that asshole."
"Please, you're speaking of the dead."
"I think I know what happened the night of September twenty-eight. And I'm pretty sure you do, too."
"So move to dismiss."
"I'll weigh it."
"Look, if it's my ass you still want, don't sweat it. I'm going down. Even Alan Crudman couldn't get me off."
"It's some consolation, I admit."
"Look on the bright side. After I spend five years in prison being made love to by passionate weight lifters with AIDS, they'll disbar me. You'll never have to face me again in a courtroom."
"I'm feeling better and better."
"Let it go," Boyce said.
"I'll weigh it." She hung up.
Boyce considered whether to tell Beth about the call. He decided against it.
The next morning, shortly after 10:00 A.M., the deputy attorney general of the United States rose and went to the podium in Judge Umin's courtroom. The mood was, as Dan Rather put it, "more electric than a drenched cat with its tail stuck in a socket."
"If it please the court," she began, "the United States respectfully moves that the court dismiss the indictment in _United States_ versus _Elizabeth MacMann_ , by reason of new developments that show that justice requires that."
The courtroom exploded.
Beth sat gravely in her seat, expressionless. Boyce, watching from his hotel room, was similarly calm. His eyes bored in on Beth.
Judge Dutch said, "In view of the prosecutor's motion to dismiss, I—"
Beth rose. "If it please the court, Your Honor, may the defendant in this case make a statement at this time?"
_Oh shit_ , Boyce thought.
"I was about to make a ruling on the prosecution's motion," said Judge Dutch, as if to say, _"If you will just remain seated and quiet, Mrs. MacMann, I'll have you out of here in three minutes."_
Sandy Clintick looked at her. The entire world—some billion-plus viewers, at any rate—was looking at Beth.
"I am aware of that, Your Honor. But the defendant would like to make a statement _before_ you rule on the motion." She added, "So that the court may be fully informed."
Judge Dutch sat back wearily with the air of a reasonable man surrendering to an unreasonable world, glasses beginning to fog. "Very well, Mrs. MacMann. Proceed."
"Thank you, Your Honor. I..." She paused. "Am not sure where to begin, so I will begin with an apology. To the people of the United States. To this court. Even to my attorney, Mr. Baylor. For not telling the truth about what happened that night."
The nosebleed that had been building for decades finally burst from Dan Rather. Mercifully, it was kept from his viewers.
Judge Dutch's eyes disappeared for the last time behind the pea-soup fog of his glasses.
"Not last," Beth continued, "I apologize to the United States Secret Service and the Federal Bureau of Investigation. Terrible accusations were laid at their doorsteps, on my behalf. I—not Mr. Baylor—bear the full moral responsibility for those accusations, and I hereby retract them."
Beth looked down, swallowed, and continued. "I threw the historical... object at the President that night.
The spittoon." The courtroom stirred.
"In fact, I threw it at him hard. This happened at a moment of emotional... Oh, to hell with it... excuse me, Your Honor. I was furious with him. I knew what he'd been doing. And if I had killed him, I cannot say to you here and now that I would have regretted it at the time."
She continued, "But I was certain, somehow, that I had _not_ killed him. And it was to that certainty that I clung throughout the investigation and"—she sighed—"subsequent events. I cannot justify that certainty. I cannot excuse the accusations that were made in my defense."
Beth's hand moved abruptly to her stomach. She winced.
"Mr. Baylor, genuinely believing in my innocence, defended me to the utmost of his ability. Which, in the case of Boyce Baylor, is pretty utmost. He now faces the possibility of prison and professional ruin. If my debt to the American people is exceeded by any other, it is by my debt to him. For without Mr. Baylor's interventions, however zealous, the result of these proceedings might very well have been otherwise. Perhaps, on balance, that would have been for the best, at least for the country's sake. At any rate, now this court knows the full truth of what happened in the White House that night. And can make such disposition," Beth concluded, sitting down, "as it deems fitting."
She sat down and folded her hands on her lap.
There was no murmuring. Even normally garrulous television commentators said nothing.
At length, Judge Dutch removed his glasses and cleared his throat. He looked toward the clerk of the court, then at Beth. He hesitated for several long seconds, then spoke.
"Motion is granted."
He turned to the jury. "The jury is discharged. On behalf of the people of the United States, I would like to extend gratitude for your service in what I know have been trying circumstances."
With that he said, "Court is adjourned," and brought down his gavel on the Trial of the Millennium.
# Epilogue
Five days before _United States_ v. _Boyce Baylor et al_. was scheduled to go to trial, a trial that in the opinion of most legal observers would be a "slam dunk" for the prosecution, Boyce's co-defendants, Felicio Andaluz and Ramon Martinez, escaped from the U.S. Detention Center in Fairfax, Virginia, during a game of basketball with other inmates. This was highly embarrassing to the government and forced a delay in the start of Boyce's trial.
From Boyce's point of view, it was a welcome delay, allowing him time to be with Beth when she gave birth. Her pregnancy had been the most media-covered gestation since an actress had appeared nude and immensely gibbous on the cover of _Vanity Fair_ magazine. Their daughter, Ilsa Tyler Baylor, weighed six pounds ten ounces, exactly—a fact _The Washington Post_ pointed out—the weight of the infamous Paul Revere silver spittoon. Chatting with reporters outside the hospital, Boyce was good-humored enough to remark that he hoped his new daughter would soon be too heavy for her mother to throw at him.
The investigation into the disappearance of Felicio and his colleague proved inconclusive. They had, simply, vanished.
One night, rocking his newborn daughter to sleep, Boyce received a telephone call. After listening to what Felicio had to say, he in turn called a reporter friend. The friend had already won two Pulitzer Prizes for his investigative reporting but was by no means averse to having a third. With the information that Boyce vouchsafed him, he went to work and in six weeks, on the eve of the start of Boyce's rescheduled trial for conspiracy to commit jury tampering, brought out the first in a series of articles, thoroughly if somewhat anonymously sourced, stating that Felicio Andaluz was a longtime agent of the U.S. Central Intelligence Agency. Furthermore, it had been the CIA that had so deftly arranged his and Mr. Martinez's escape from the detention facility. The CIA had been most eager not to have one of its prize agents, one fairly teeming with sensitive information about its operations in Latin America, take the stand in a federal court case.
The CIA naturally had "no comment" on the articles. But in the aftermath of the trial, public opinion was sensitive to any suggestion that the government had been up to yet more covert mischief.
The Justice Department found itself assailed by the media, demanding to know why, if one branch of the government was secretly springing from jail a set of defendants in the same trial, another branch of the government should so assiduously pursue the remaining defendant.
High-level meetings were held.
At length it was announced that the government would nolle prosequi in the case of _U.S_. v. _Baylor et al_. In plain English, this means: _I am not going to touch this with a ten-foot pole_.
The stated justification given was that Messrs. Andaluz and Martinez had agreed to cooperate with the government and testify against Boyce Baylor. (Untrue.) In their absence, the government now felt that it had insufficient grounds to continue against him. But what about the videotape of him plotting away happily to pollute the jury? The spokesman bravely cleared his throat and said that the tape was "open to subjective interpretation."
The nolle was greeted with approval by the over 85 percent of the American people who said they were suffering from Trial of the Millennium–related exhaustion.
The Ethics Panel of the District of Columbia Bar Association convened to determine whether there had been ethical violations sufficiently grievous to warrant disbarring Boyce from continuing to practice law. The hearings were closed but were reported in the press to be "heated." There was a certain amount of harrumphing on the nation's editorial pages about the "ridiculous" spectacle of lawyers declaring each other morally unfit.
On the eve of what was said to be a "close" vote, Boyce himself, in a brief statement given on his front steps while holding his infant daughter, announced that he was retiring from the law, as he put it, "to improve humanity by reducing the number of lawyers by one."
"Won't you miss it?" asked one reporter.
"You mean, honestly?"
_The United States_ v. _Van Anka_ lasted less than two weeks. Nick Naylor, Babette's publicist, held press conferences every afternoon after the day's proceedings, to say how enormously satisfied the Van Anka camp was with how it was all going. A number of famous actors testified on Babette's behalf, as well as the Israeli defense minister and two former prime ministers. It took the jury five hours to find her guilty of perjury. The judge (not Judge Dutch) sentenced her to one and a half years in a minimum-security facility near Los Angeles, so that she could be near her agent. She'd be out in four months, pounds thinner and looking fabulous.
Her divorce from Max was complicated by the fact that Max was now a fugitive from U.S. justice living in Indonesia and Switzerland. Though not a divorce attorney per se, Alan Crudman represented her in the matter and, after ingeniously—as even Boyce admitted—managing to attach the assets of his offshore holding companies in the Netherlands, brought him to a bargaining table in Taiwan and to a settlement that was described by the _Financial Times_ as "a lulu."
Wiley P. Sinclair disappeared once again and then several years later was given Chinese citizenship. His behind-the-scenes maneuvering to help China win sponsorship of the Olympics came to public light when he was decorated by the Prime Minister and awarded the title "Hero of the Revolution." He is said to live comfortably in Beijing and to maintain a summer residence in Hangchow. Occasionally, dressed as an elderly Chinese woman, he travels to Las Vegas.
Some thirty-eight books have been written so far about the trial, sixteen of them by the jurors. Juror _Number Eighteen_ is generally considered to be the least tedious.
Judge Dutch Umin received plaudits for his handling of the Trial of the Millennium. He declined President Harold Farkley's nomination to the Supreme Court, saying that when he reached retirement age, he planned to accept an outstanding offer to be curator of the Institute for Dutch Still Life.
Harold Farkley was overwhelmingly defeated in his bid to be reelected president, confirming in everyone's mind that he was fundamentally second-rate and that he never would have achieved the number one job in government had it not been for the fact that his predecessor had died in, as one columnist put it, "pathetic" circumstances.
His opponent ran on a platform of restoring dignity to the White House. He announced that his first piece of legislation would be the Lincoln Bedroom Protection Act, barring presidents from turning the once sacred second-floor room overlooking the South Lawn into, as he put it, "a by-the-hour motel for political donors."
He also pledged to reduce the size of government.
# Acknowledgments
I am once again in the debt of Dr. David Williams, MC, USNR. Compliments and duty, _Sir!_
Steve "Dutch" Umin of Williams and Connolly was patiently and endlessly helpful and so far has yet to submit a bill. Let the record show that he finally threw up his hands over my implacable legal solecisms and should not be held accountable thereunto. Thereof? Whatever.
Before he got mad at me for something I wrote about his client, Monica Lewinsky, Plato Cacheris provided the spittoon.
C. Boyden Gray bought me a shad-roe lunch and chuckled at the idea, a reassuring sound from a tough grader and Establishmento.
The combined hourly bill of these three distinguished attorneys, a sum equivalent to the gross domestic product of the sultanate of Brunei, is worth it, so if you have killed anyone or swindled shareholders or fought with the Taliban, call them. They're in the book.
Lincoln Caplan of the Yale Law School's journal _Legal Affairs_ , in whose pages some of this first appeared, showed once again that he is a peerless editor, to say nothing of friend.
Thomas Jackson gave precociously good advice for a young man.
John Tierney was as ever generous with his wisdom and enthusiasm.
Thanks, also, to the keen eyes of Gregory Zorthian and William F. Buckley Jr.
Special thanks to Sona Vogel for her relentlessly superb copyediting and fact checking.
President George "41" Bush kindly provided certain details of life on the second floor of the White House residence.
Affectionate thanks, once again, to Amanda "Binky" Urban of International Creative Management. Again, if you have killed someone or swindled stockholders or fought with the Taliban, call her—after you call the lawyers.
I am again in deep debt to my editor, Jonathan Karp of Random House. This is our fifth collaboration. He has now said "no" 1,278 times. But this makes it sweet when he says "yes." Thank you, my very dear Mr. Karp.
Last but never leastly: wife Lucy, daughter Caitlin, and son Conor, who put up with the author. Once again, I am left wondering why anyone would marry a writer or want one for a dad.
And finally the faithful Hound Jake, who barked at everything.
_—Blue Hill_
_September 9, 2001_
# About the Author
CHRISTOPHER BUCKLEY is the author of eight previous books, including _Thank You for Smoking_ and _Little Green Men_. That would make this his, what, ninth? He is editor of _Forbes FYI_ magazine and has contributed more than fifty "Shouts and Murmurs" to _The New Yorker_. He is also credited with bringing about lasting peace in the Middle East and with alerting NASA to significant problems with its Space Shuttle Automatic Re-entry Guidance System (SSARGS), thereby sparing several square blocks of Raleigh, North Carolina, a very unpleasant surprise. He is a regular contributor to _Martha Stewart's Inside Trading_ magazine and informally advises the government of Argentina on debt rescheduling. He is the 2002 recipient of the Washington Irving Medal for Literary Excellence but has yet to actually receive it. He lives in Washington, D.C., with his saintly and long-suffering wife, Lucy, two children, and faithful Hound Jake.
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Q: Rails cached column dont get updated on destroy - Has many through I want to count how many tags that belongs to konkurrancer.
It is a has_many through Tagsmenu relation ship.
I have added the column konkurrancers_count to my Tags table.
Here is my join model:
class Tagsmenu < ActiveRecord::Base
belongs_to :konkurrancer, :counter_cache => :true
belongs_to :tag
end
But when I destroy a konkurrancer the konkurrancers_count column dont get updated.
A: if you delete a record it just runs the deletion sql. If you destroy a record, it will instantiate the model, run all the callbacks, and then run the deletion sql. I think you'll find that you need to use destroy in order to see the cache column update.
A: Take a look at this methods: update_counters
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"redpajama_set_name": "RedPajamaStackExchange"
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Q: PHP - Create a folder and put a file in it I need to put a file in a folder named after the current date (Ex : folderbefore/2017-11-15/mynewfile.sql).
I tried :
*
*change folderbefore to 777 in the server (I know it's not good, but I tried it and anyway it doesn't work)
*add www-data to the owner group of folderbefore
*set the umask to 0 before creating the folder :
$old = umask(0);
$directory_was_created = mkdir($path, 0777);
umask($old);
*I tried adding chmod after to change the permission
$old = umask(0);
$directory_was_created = mkdir($path, 0777);
chmod($path, 0777);
umask($old);
I'm always getting a folder with 775 permissions and no right to write in this newly created folder (tried with file_put_content who gives me FALSE and fopen who's throwing an error).
What am I missing ?
A: Ok I found the problem right after I posted the question.
I'm working on a test server on a vagrant machine. In order to give my computer access to folderbefore, I needed to add this in my Vagrantfile and restart the machine :
config.vm.synced_folder "path/to/folderbefore", "path/to/folderbefore/on/my/computer", id: "unique-id-for-this-folder",
owner: "vagrant",
group: "vagrant",
mount_options: ["dmode=777,fmode=777"]
A: $directory_was_created = mkdir($path, 0777);
Needs to be
$directory_was_created = mkdir($path, 0777, true);
If you want to create nested directories within the newly created directory
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2.4GHz RF Modules Manufacturer - AIRWAVE Technologies Inc.
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Supercarrier – Die Welt der amerikanischen Flugzeugträger ist der Titel eines von Tom Clancy geschriebenen, 1999 in den USA erschienenen Buches. Die deutsche Erstveröffentlichung ist aus dem Jahr 2001. Das Buch befasst sich mit Flugzeugträgern, besonders mit den Supercarriern der Nimitz-Klasse und auf ihnen verwendetem Gerät.
Inhalt
Dieses Buch ist das sechste von Clancy, das sich mit einer speziellen Waffengattung bzw. mit verwendeten Einheiten beschäftigt. Zum zweiten Mal nach Atom U-Boot schreibt Clancy über einen Schiffstyp der United States Navy. Das Buch hat einen Umfang von ca. 470 Seiten.
Nach einer Einleitung über die Marinefliegerei führt Clancy ein ausführliches Interview mit Admiral Jay Johnson, dem damaligen Chief of Naval Operations. Die folgenden Kapitel bilden das Herzstück des Buches. Zuerst wird ausführlich die Ausbildung zum Marineflieger beschrieben, bevor sich Clancy dem Bau und Aufbau der Flugzeugträger zuwendet. Er beschreibt sehr ausführlich den Bau in der Werft Newport News Shipbuilding in Newport News, Virginia. Den Aufbau erfährt der Leser durch den "Rundgang", den Clancy mit ihm auf der USS George Washington (CVN-73) durchführt. Im dritten Kapitel des Herzstückes des Buches werden die Flugzeuge und Waffen beschrieben, die derzeit an Bord amerikanischer Flugzeugträger verwendet werden. Dazu zählen die F-14, die F/A-18, die Sidewinder, die Sparrow und die Phoenix.
Im Anschluss daran beschreibt Clancy den Aufbau von Carrier Vessel Battle Groups (CVBG) und eine Übung eines Flugzeugträgerverbandes. Abgeschlossen wird das Buch durch ein fiktives Szenario im Jahre 2016 vor der Küste Sri Lankas.
Literatur
ISBN 3-453-19074-2
Literarisches Werk
Literatur (Englisch)
Literatur (20. Jahrhundert)
Sachliteratur
Werk von Tom Clancy
Literatur (Vereinigte Staaten)
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La batalla de Huaylacucho fue una acción de armas librada el 17 de abril de 1834, en las cercanías de Huancavelica, Perú. Enfrentó a las fuerzas orbegosistas al mando del general Guillermo Miller y las bermudistas dirigidas por el general José Rufino Echenique, en el marco de la guerra civil que se desató en el Perú tras la elección del general Luis José de Orbegoso como presidente provisorio del Perú en 1833. Triunfaron los bermudistas, pero una semana después ambos bandos se reconciliaron en el llamado abrazo de Maquinhuayo.
Contexto
El día 20 de diciembre de 1833, tras finalizar el período constitucional del presidente Agustín Gamarra, la Convención Nacional (congreso constituyente) procedió a elegir a un presidente provisorio, tras haberse frustrado las elecciones presidenciales convocadas meses atrás. Gamarra apoyó la candidatura del general Pedro Bermúdez, adicto suyo, pero los liberales que dominaban la asamblea optaron por elegir al general Luis José de Orbegoso, un militar menos autoritario. No bien asumió éste el poder, cuando los partidarios de Gamarra hostilizaron al nuevo régimen con la intención de imponer a la fuerza en la presidencia a Bermúdez. Los gamarristas o bermudistas alegaron que la presidencia de Orbegoso era ilegal pues no le correspondía a la Convención Nacional elegir al Presidente.
Temiendo un golpe de Estado, Orbegoso decidió refugiarse en la Fortaleza del Real Felipe, en el Callao, el 3 de enero de 1834. Allí instaló la sede de su gobierno y comenzó a relevar a los gamarristas de los altos mandos del Ejército. En respuesta a esta acción, la guarnición de Lima se sublevó al día siguiente y proclamó Jefe Supremo a Bermúdez. Las tropas bermudistas sitiaron la fortaleza del Callao. A nivel nacional la autoridad de Bermúdez fue acatada por algunas guarniciones. Pero en Lima la civilidad se mostró contraria al golpe y se alzó en armas; los bandos se enfrentaron en las calles de la ciudad, siendo la primera vez en la historia peruana que el pueblo de Lima se enfrentaba con éxito al ejército (28 de enero de 1834). Ante tal situación Bermúdez y sus partidarios tuvieron que retirarse a la sierra.
Desatada la guerra civil, esta tuvo tres escenarios:
El frente sur, localizado en Arequipa, donde el general Domingo Nieto intentó defender el orden constitucional representado por Orbegoso, fracasando finalmente.
El frente norte, donde Felipe Salaverry, sumado al bando orbegosista, apresó en Trujillo al general Juan Francisco de Vidal La Hoz.
El frente centro, hacia donde marchó Orbegoso en persona, subiendo a la sierra en busca de Bermúdez y su ejército. Aquí fue donde se decidió el resultado de la lucha.
Los ejércitos
Orbegoso, al frente de un reducido ejército, marchó hacia Jauja en persecución de Bermúdez. Si bien tenía bajo su mando a oficiales competentes como los generales José de la Riva Agüero, Mariano Necochea, Guillermo Miller, Antonio Gutiérrez de la Fuente, Blas Cerdeña, Francisco de Paula Otero y Felipe Santiago Salaverry, sus fuerzas eran muy débiles y heterogéneas.
Por su parte, Bermúdez, también con un pequeño ejército pero formado por veteranos disciplinados, abandonó el valle de Jauja en dirección de Ayacucho para unirse con el general Frías, prefecto de ese departamento. Muy cerca le seguía el general orbegosista Guillermo Miller, quien recibió en el trayecto refuerzos del general Salaverry.
Bermúdez no gozaba del apoyo popular; tampoco sus tropas sentían apego hacia él, ya que no se preocupaba por alentarlos ni de satisfacerles en sus necesidades. Uno de sus principales oficiales, el general José Rufino Echenique, cuenta en sus memorias que acordó con Frías deponer a Bermúdez una vez que terminaran con Orbegoso; los acontecimientos posteriores modificarían su plan inicial pero no su idea primordial.
Movimientos preliminares
Tras abandonar Huancayo, Bermúdez continuó su marcha hacia Ayacucho. Por su parte, Miller pasaba a Huancavelica, extendiendo sus avanzadas hasta la quebrada de los Molinos, donde libró un combate con las avanzadas de los bermudistas. Como resultado de este encuentro, Miller se vio obligado a replegarse al pueblo de Huaylacucho (al oeste de Huancavelica). Mientras tanto, Orbegoso llegó a Jauja donde pasó revista a sus tropas. Sabiendo que Frías y Bermúdez se habían concentrado en Acobamba (al este de de Huancavelica), envió a Miller dos batallones, los cuales llegaron en la noche del 16 de abril.
Escenario
Huaylacucho es un pueblo pequeño, situado a unos 5 kilómetros al Este de Huancavelica. Ocupa una quebrada surcada por tres profundos barrancos, abiertos al oriente; las de los costados son escarpados y de difícil acceso; la del centro la recorre un pequeño río. Fue en este escenario donde las fuerzas bermudistas (cuyos jefes, después de Bermúdez, eran Echenique y Frías) se encontraron con las fuerzas orbegosistas comandadas por Guillermo Miller.
La batalla
El escenario era desventajoso para los orbegosistas, pero aun así Miller dispuso su línea, colocando a la derecha al batallón Pichincha, al centro al batallón Lima y a la izquierda al batallón Zepita (bajo mando de Salaverry) con los escuadrones de caballería que mandaba Loyola. En total sumaban unos 1.350 hombres.
Al amanecer del día 17 de abril una columna de bermudistas a las órdenes del general Frías avanzó hacia la vanguardia del ala derecha de Miller; este, para frenar el ataque, mandó primero al comandante Solar con una compañía y luego al batallón Pichincha como apoyo. Pero los bermudistas lograron repeler el contraataque y tomaron la barranca.
Los orbegosistas, sometidos a fuego convergente e incesante, vieron perdida la contienda y optaron por retirarse. Casi en desorden, procedieron a cruzar el río y muchos perecieron ahogados. Al contemplar el desastre, Salaverry avanzó por la izquierda con el batallón Zepita y consiguió detener a los bermudistas, facilitando la retirada de los suyos y salvándolos así de una destrucción completa. En esta última acción tuvo una destacada participación el Sargento Mayor de Caballería Cosme Pacheco, quien al mando de un grupo de lanceros, cubrió la retaguardia del ejército orbegosista en retirada, salvaguardando asimismo el parque de municiones que luego serviría para reforzar las líneas de defensa contra otra posible embestida de los bermudistas.
No intervino la caballería orbegosista, pero el general Frías, creyendo que podía ganársela (pues anteriormente había sido jefe de ella), se acercó con un oficial y cinco soldados, pero Loyola cargó sobre ellos y los destrozó. Frías fue muerto de un lanzazo. Loyola reunió y organizó a los dispersos.
En el bando orbegosista hubo 50 muertos y unos 32 heridos, así como 200 dispersos.
Si bien en este encuentro de armas hubo movimientos tácticos, cargas de caballería y unos cuantos disparos, no puede decirse que esta acción fuera una batalla en el sentido cabal del término, pero la historiografía peruana tradicionalmente la ha denominado así.
Consecuencias
Bermúdez no persiguió a los orbegosistas, quienes aprovecharon para reagruparse y se replegaron a Izcuchaca (norte de Huancavelica).
La acción de Huaylacucho no decidió nada. Se esperaba un encuentro definitivo, pero fue entonces cuando Echenique convenció al resto de oficiales bermudistas para llegar a un acuerdo pacífico con Orbegoso, prescindiendo de Bermúdez. Todos ellos eran conscientes de estar al servicio de una causa perdida, pues se veían repudiados por todas partes y sin recursos para continuar la lucha. Procedieron pues, a deponer a Bermúdez y de inmediato enviaron emisarios al campamento de Orbegoso. El 24 de abril llegaron al llano de Maquinguayo, a 24 km al norte de Jauja, donde encontraron a los orbegosistas en formación de batalla. Luego de colocar sus armas en pabellones, ambos ejércitos avanzaron hasta encontrarse y se estrecharon en fraterno abrazo. A este episodio singular de la historia peruana se conoce como el abrazo de Maquinhuayo.
Los bermudistas o gamarristas reconocieron así la autoridad de Orbegoso. Gamarra huyó del país.
Véase también
Abrazo de Maquinhuayo
Referencias
Basadre Grohmann, Jorge: Historia de la República del Perú (1822 - 1933), Tomo 2. Editada por la Empresa Editora El Comercio S. A. Lima, 2005. ISBN 9972-205-64-9 (V.2)
Tauro del Pino, Alberto: Enciclopedia Ilustrada del Perú. Tercera Edición. Tomo 8. HAB/IZQ. Lima, PEISA, 2001. ISBN 9972-40-157-X
Vargas Ugarte, Rubén: Historia General del Perú. La República (1833-1843). Octavo Tomo. Primera Edición. Editor Carlos Milla Batres. Lima, Perú, 1971.
Varios autores: Historia general de los peruanos. Tomo 3. Primera Independencia Nacional y Revolución Peruana. Con el auspicio del Gobierno Revolucionario de las Fuerzas Armadas. Impreso en Talleres Gráficos de Iberia S.A. Lima, agosto de 1973.
Batallas de Perú
Perú en 1834
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This Dock at the Sea Pier Photography was take on a hot summer day at the Salton Sea, California. The film was damaged and turned out to be a great "defect" with light leak and numbers impressed on it from the 120 roll's paper. A great scene for any nature lover or for one who enjoys post apocalyptic scenery.
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\section{Introduction}
The aim of this article is to investigate the pairs of ternary positive-definite quadratic forms $f$ and $g$ with perfectly identical representations over $\IntegerRing$.
For any $N$-ary quadratic forms $f$ and $g$ with real coefficients,
we will use the notation $f \sim g$ when
$f$ and $g$ are \textit{equivalent over $\IntegerRing$}, \IE $f(\mathbf{x} w) = g(\mathbf{x})$ for some $w \in GL_3(\IntegerRing)$.
The set of the representations of $f$, \IE $\{ f(x) : 0 \neq x \in \IntegerRing^N \}$,
is denoted by $q_\IntegerRing(f)$.
When $N = 2$, the corresponding result was proved by a number of mathematicians \cite{Watson80}.
If two binary positive-definite quadratic forms $f \not\sim g$ satisfy $q_\IntegerRing(f) = q_\IntegerRing(g)$, then
there exists $d > 0$ such that $f$ and $g$ are equivalent to either of $d (x_1^2 - x_1 x_2 + x_2^2)$ and $d (x_1^2 + 3 x_2^2)$ over $\IntegerRing$.
All the indefinite cases were provided in \cite{Delang82}, \cite{Delang87}.
An immediate consequence of the result in the binary case is that
there are infinitely many pairs of equivalence classes
of ternary positive definite forms $f, g$ such that $q_\IntegerRing(f) = q_\IntegerRing(g)$.
In fact, it is not difficult to verify that $f$ and $g$ in each of the following families, satisfy $q_\IntegerRing(f) = q_\IntegerRing(g)$, regardless of the values of $c, d$ (see Section \ref{Case of Disc(A_i, B_i) = 0 (proofs of Propositions 1, 2)}):
\begin{enumerate}[(i)]
\item $\{ f, g \} = \{ c(x_1^2 - x_1 x_2 + x_2^2) + d x_3^2, c(x_1^2 + 3 x_2^2) + d x_3^2 \}$,
\item $\{ f, g \} = \{ c(x_1^2 - x_1 x_2 + x_2^2) + d (x_1 + x_2 + 3 x_3)^2, c(x_1^2 + 3 x_2^2) + d (x_1 + 3 x_3)^2 \}$.
\end{enumerate}
It was recently conjectured by Kaplansky in his letter to Schiemann that
except for the cases identical to either of the above (i), (ii),
$f \not\sim g$ with $q_\IntegerRing(f) = q_\IntegerRing(g)$ will be constant multiples of regular quadratic forms; an integral-valued quadratic form $f$ is said to be \textit{regular}, if
$f$ can represent all the genus representations, \IE $m \in \IntegerRing$ represented by $f$ over $\IntegerRing_v$ for any primes $v$ including $v = \infty$.
It was proved in \cite{Do2012} that the conjecture holds if only diagonal quadratic forms are considered.
(There are only two cases if both of $f \not\sim g$ with $q_\IntegerRing(f) = q_\IntegerRing(g)$ are diagonal; see No.34 and No.50 in Tables \ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}--\ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(3/3)}.)
In order to obtain more detailed information about this problem,
an exhaustive search for such $f, g$ with integral quadratic coefficients were carried out.
It can be proved that if $f, g$ over $\RealField$ satisfy $q_\IntegerRing(f) = q_\IntegerRing(g)$,
infinitely many $f_2, g_2$ over $\IntegerRing$ with $q_\IntegerRing(f_2) = q_\IntegerRing(g_2)$ are generated from these $f, g$ (Lemma 2.1). Hence, the search also provides information about the case of real forms.
The result is presented in Tables \ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}--\ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(3/3)}
in Section \ref{A table of quadratic forms with the same representations over Z},
which indicates that
the existence of such pairs is rather limited as conjectured by Kaplansky, although the current list includes some non-regular cases.
If the quadratic forms contained in the above (i), (ii) are excluded,
our exhaustive search finds only 151 equivalence classes of quadratic forms
that have perfectly identical representations over $\IntegerRing$ as another class.
Among the 151 classes, 36 are not provided by regular quadratic forms.
In addition, the list includes a case that has been proved to be regular only under the Generalized Riemann hypothesis \cite{Oliver2014}.
In what follows, $\{ f, g \} \sim \{ f_2, g_2 \}$ means that either of $f \sim f_2$, $g \sim g_2$ or $f \sim g_2$, $g \sim f_2$ holds.
The following is suggested from the computational result:
\begin{description}
\item[Kaplansky conjecture (modified version):]
If two ternary positive-definite quadratic forms $f \not\sim g$ over $\RealField$ satisfy $q_\IntegerRing(f) = q_\IntegerRing(g)$,
either of the following holds:
\begin{enumerate}[(i)]
\item $\{ f, g \} \sim \left\{ c ( x_1^2 - x_1 x_2 + x_2^2 ) + d x_3^2, c ( x_1^2 + 3 x_2^2) + d x_3^2 \right\}$ for some
$c, d \in \RealField$,
\item $\{ f, g \} \sim \left\{ c ( x_1^2 - x_1 x_2 + x_2^2 ) + d (x_1 + x_2 + 3 x_3)^2, c ( x_1^2 + 3 x_2^2) + d (x_1 + 3 x_3)^2 \right\}$ for some
$c, d \in \RealField$,
\item $\{ f, g \} \sim \{ c f_2, c g_2 \}$ for some $c \in \RealField$ and $f_2, g_2$ contained in either of the No.1--53 in Tables \ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}--\ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(3/3)}.
\end{enumerate}
\end{description}
In the above, the non-regular cases newly found in our search are also included.
However, it should be noted that
they were just confirmed to have the identical set of representations up to 3,000,000, by computation.
The same thing can be proved up to $\infty$, with regard to regular quadratic forms.
Before proceeding to our theoretical results motivated by the Kaplansky conjecture,
first we provide the following proposition; for any commutative ring $R$,
the set of all the $n$-ary quadratic forms over $R$
is denoted by ${\rm Sym}^2 (R^n)^*$,
and the set of all the pairs of such forms
is denoted by ${\rm Sym}^2 (R^n)^* \otimes_{R} R^2$.
For any subring $R_2 \subset R$,
the elements of $q_{R_2}(A, B) := \{ (A(x), B(x)) : 0 \neq x \in R_2^n \}$
are called \textit{simultaneous representations of
$(A, B) \in {\rm Sym}^2 (R^n)^* \otimes_{R} R^2$ over $R_2$}.
\begin{prop}\label{thm:proposition 1}
The Kaplansky conjecture is true for any positive-definite $f \in {\rm Sym}^2 (\RealField^3)^*$
with $c f \notin {\rm Sym}^2 (\RationalField^3)^*$ for any $c \in \RealField^\times$,
if and only if the following (*) is true:
\begin{description}
\item[(*)]
If both of $(A_i, B_i) \in {\rm Sym}^2 (\RationalField^3)^* \otimes_{\RationalField} \RationalField^2$ ($i = 1, 2$) satisfy
\begin{enumerate}[(a)]
\item $A_i$ and $B_i$ are linearly independent over $\RationalField$,
\item $c A_i + d B_i$ is positive-definite for some $c, d \in \RationalField$ (\IE d-pencil),
\item $q_\IntegerRing(A_1, B_1) = q_\IntegerRing(A_2, B_2)$, then
\end{enumerate}
$(A_1, B_1) = (w, 1) \cdot (A_2, B_2)$ holds for some $w \in GL_3(\IntegerRing)$,
or otherwise, $\{ (A_1, B_1), (A_2, B_2) \}$ equals either of the following as a set, for some $(w_i, v) \in GL_3(\IntegerRing) \times GL_2(\RationalField)$ ($i = 1, 2$):
\begin{enumerate}[(i)]
\item $\left\{
(w_1, v) \cdot ( x_1^2 - x_1 x_2 + x_2^2, x_3^2 ),
(w_2, v) \cdot ( x_1^2 + 3 x_2^2, x_3^2 )
\right\}$,
\item $\left\{
(w_1, v) \cdot ( x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2 ),
(w_2, v) \cdot ( x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2 )
\right\}$,
\end{enumerate}
where $GL_3(\RationalField) \times GL_2(\RationalField)$ acts on ${\rm Sym}^2 (\RationalField^3)^* \otimes_{\RationalField} \RationalField^2$ by
\begin{eqnarray*}
\left(w,
\begin{pmatrix}
r & s \\
t & u
\end{pmatrix}
\right) \cdot (A, B) = (r A({\mathbf x} w) + s B({\mathbf x} w), t A({\mathbf x} w) + u B({\mathbf x} w)).
\end{eqnarray*}
\end{description}
\end{prop}
For any field $k$,
we shall say that a pair $(A, B) \in {\rm Sym}^2 (k^n)^* \otimes_{k} k^2$ is \textit{singular},
if $\det(A x - B y) = 0$ as a polynomial in $k[x, y]$, and \textit{non-singular} otherwise.
A non-singular $(A, B)$ is said to be \textit{anisotropic} over $k$ if $A(x) = B(x) = 0$ does not hold for any $0 \neq x \in k^n$.
According to \cite{Uhlig79},
it was first proved in \cite{Finsler36} that
any pair $(A, B) \in {\rm Sym}^2 (k^n)^* \otimes_{k} k^2$ with $n \geq 3$
is a d-pencil if and only if $(A, B)$ is non-singular and anisotropic over $\RealField$.
If only $(A_i, B_i)$ with ${\rm Disc}(A_i, B_i) = 0$ are considered,
(*) holds true, which can be proved without difficulty.
\begin{prop}\label{thm:theorem 1}
If the discriminants ${\rm Disc}(A_i, B_i) := {\rm Disc}(4 \det(A_i x - B_i)) = 0$,
the above (*) holds true.
\qed
\end{prop}
Motivated by the Kaplansky conjecture and the computational result, the following are proved in this article.
\begin{thm}\label{thm:main result over RationalField}
We assume that $(A_1, B_1), (A_2, B_2) \in {\rm Sym}^2 (\RationalField^3)^* \otimes_{\RationalField} \RationalField^2$ satisfy
\begin{enumerate}[(a)]
\item \label{item: assumption (a)} $A_i$ and $B_i$ are linearly independent over $\RationalField$.
\item[(b')] \label{item: assumption (b)} $(A_i, B_i)$ is non-singular and anisotropic over $\RationalField$.
\item[(c')] \label{item: assumption (c)} $q_\RationalField(A_1, B_1) = q_\RationalField(A_2, B_2)$.
\end{enumerate}
In this case,
$(r_1 A_1, r_1 B_1)$ is equivalent to $(r_2 A_2, r_2 B_2)$ by the action of $GL_3(\RationalField) \times \{ 1 \}$
for any integers $r_1, r_2$ that satisfy $r_1^{-1} \det(A_1 x - B_1) = r_2^{-1} \det(A_2 x - B_2)$.
\qed
\end{thm}
Considering that (b) is equivalent to the condition obtained by replacing $\RationalField$ in (b') with $\RealField$,
the situation of Theorem \ref{thm:main result over RationalField} is more general than that in the conjecture.
In the proof, the result by Bhargava on the one-to-one correspondence between the set of pairs of quadratic forms and the set of quartic rings and its resolvent cubic rings\cite{Bhargava2004}, is used.
From Theorem \ref{thm:main result over RationalField}, it is seen that $q_\RationalField(A_1, B_1) = q_\RationalField(A_2, B_2)$
leads to $\det(A_1 x - B_1 y) = c \det(A_2 x - B_2 y)$ for some $c \in \RationalField^\times$,
as proved in Proposition \ref{prop:same det(Ax+By)}.
In what follows, we shall mention an application of the above result to experimental science;
on account of the need to determine the crystal lattice (\IE the equivalence class of a real ternary $f$ over $\IntegerRing$) from information about $q_\IntegerRing(f)$ that is extracted from experimental data,
it has been recognized in crystallography that
some $f \not\sim g$ have the perfectly identical representations over $\IntegerRing$ (\cite{Mighell75}, \CF \cite{Tomiyasu2016}).
A three-dimensional lattice is \textit{hexagonal} if and only if
it has a basis $v_1, v_2, v_3$ satisfying
\begin{eqnarray*}
( v_i \cdot v_j )_{1 \leq i, j \leq 3} =
\begin{pmatrix}
c & -c /2 & 0 \\
-c/2 & c & 0 \\
0 & 0 & d \\
\end{pmatrix},
\end{eqnarray*}
for some $c, d \in \RealField$. A three-dimensional lattice is \textit{rhombohedral} if and only if
it has a basis satisfying
\begin{eqnarray*}
( v_i \cdot v_j )_{1 \leq i, j \leq 3} &=&
\begin{pmatrix}
c+d & -c/2+d & -c/2+d \\
-c/2+d & c+d & -c/2+d \\
-c/2+d & -c/2+d & c+d
\end{pmatrix} \nonumber \\
&=&
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-1 &-1& 1
\end{pmatrix}
\begin{pmatrix}
c+d & -c/2+d & 3 d \\
-c/2+d & c+d & 3 d \\
3 d & 3 d & 9 d
\end{pmatrix}
\begin{pmatrix}
1 & 0 & -1 \\
0 & 1 & -1 \\
0 & 0 & 1
\end{pmatrix},
\end{eqnarray*}
for some $c, d \in \RealField$. Hence, (i) $c (x_1^2 - x_1 x_2 + x_2^2) + d x_3^2$
and (ii) $c (x_1^2 - x_1 x_2 + x_2^2) + d (x_1 + x_2 + 3 x_3 )^2$ ($c, d \in \RealField$)
parametrize all hexagonal and rhombohedral lattices, respectively.
\section*{Notation and symbols}
Throughout this paper,
a quadratic form $\sum_{1 \leq i \leq j \leq n} s_{ij} x_i x_j$ is always identified with the symmetric matrix
with $s_{ii}$ in the $(i, i)$-entry and $s_{ij}/2$ in the $(i, j)$-entry.
For any quadratic forms $f(x_1, \ldots, x_m)$, $g(x_1, \ldots, x_n)$,
their \textit{direct sum} is the $(n+m)$-ary quadratic form $f(x_1, \ldots, x_m) + g(x_{m+1}, \ldots, x_{m+n})$,
and denoted by $f \perp g$.
A quadratic form $\sum_{i=1}^n c_i x_i^2$ is represented as a diagonal matrix or $[c_1, \ldots, c_n]$.
In particular, $[c]$ means the unary quadratic form $c x^2$.
For any $f$, $g \in {\rm Sym}^2 (R^n)^*$,
if there exists $w \in GL_3(R)$ such that $f(\mathbf{x} w) = g(\mathbf{x})$,
it is said that $f$ and $g$ are \textit{equivalent over $R$}, and denoted by $f \sim_R g$.
For any $f \in {\rm Sym}^2 (R^n)^*$,
the \textit{representations over $R$}
are the elements of $q_R(f) := \{ f(v) : 0 \neq v \in R^N \}$.
For any $f_1, \ldots, f_s \in {\rm Sym}^2 (R^n)^*$,
their \textit{simultaneous representations over $R$}
are the elements of $q_R(f_1, \ldots, f_s) := \{ (f_1(v), \ldots, f_s(v)) : 0 \neq v \in R^N \}$.
\section{A table of quadratic forms with the same representations over $\IntegerRing$}
\label{A table of quadratic forms with the same representations over Z}
The algorithm used to exhaustively searched for
sets of positive-definite ternary quadratic forms with the identical representations is presented in Table 1 of Section \ref{Algorithm},
with all the discussions required to prove that the algorithm works well.
The following $P$1--3 describe the searched region; the representations of each quadratic form $\sum_{1 \leq i \leq j \leq 3} s_{ij} x_i x_j$ that satisfies the following conditions are passed to the algorithm as an argument $\Lambda = \langle q_1, \ldots, q_t \rangle$.
All quadratic forms over $\RationalField$ (or their scalar multiples)
can be contained in the searched region by increasing the number 115 in (P3) sufficiently.
\begin{enumerate}[($P$1)]
\item all of $s_{ij}$ are integral and do not have a common divisor more than 1.
\item The form is reduced, \IE satisfies Eq.\ (\ref{eq:condition3}) and the following:
\begin{enumerate}[(i)]
\item $s_{12}, s_{13}, s_{23} > 0$ or $s_{12}, s_{13}, s_{23} \leq 0$,
\item $s_{11} = s_{22} \Longrightarrow \abs{s_{23}} \leq \abs{s_{13}}$,
\item $s_{22} = s_{33} \Longrightarrow \abs{s_{13}} \leq \abs{s_{12}}$.
\item case of positive Eisenstein forms (\IE $s_{11}, s_{22}, s_{33}> 0$):
under the assumption that $s_{ij} = s_{ji}$, for any distinct $1 \leq i, j, k \leq 3$,
$s_{ii} = 2 \abs{s_{ij}} \Rightarrow \abs{s_{i k} } \leq 2 \abs{ s_{j k} }$.
\item case of non-positive Eisenstein forms (\IE $s_{11}, s_{22}, s_{33} \leq 0$):
for any distinct $1 \leq i, j, k \leq 3$, $s_{ii} = 2 \abs{s_{ij}} \Rightarrow s_{i k} = 0$.
In addition,
\begin{eqnarray*}
s_{11} + s_{22} = 2 \abs{ s_{12} + s_{13} + s_{23} } \Rightarrow s_{11} \leq \abs{ s_{12} + 2 s_{13} }. \label{eq:condition2}
\end{eqnarray*}
\end{enumerate}
The above inequalities were provided by Eisenstein to give the representatives of the
classes of positive-definite ternary quadratic forms \cite{Eisenstein1851}.
\item $s_{33} \leq 115$.
\end{enumerate}
The set contained in either of the hexagonal and rhombohedral families were removed from the output.
The algorithm
was also applied to all (possibly) regular quadratic forms in the tables of \cite{Jagy97},
in order to check that all the pairs of regular forms are contained in the output.
The results are presented in Tables \ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}--\ref{Forty-nine groups of ternary positive
definite quadratic forms representing the same numbers(3/3)}.
By using a computer,
the quadratic forms in each set have been confirmed to have the identical representations over $\IntegerRing$ up to 3,000,000.
\begin{table}[htbp]
\caption{Ternary positive-definite quadratic forms with the same representations over $\IntegerRing$ (Continued)}
\label{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}
\begin{minipage}{\textwidth}
\begin{scriptsize}
\begin{tabular}{p{1mm}p{19mm}p{4mm}p{4mm}p{4mm} p{4mm}p{4mm}p{4mm}
lr}
\hline
No.
& Determinant (Ratio) & $s_{11}$ & $s_{22}$ & $s_{33}$ & $s_{12}$ & $s_{13}$ & $s_{23}$
& Integers not represented by the genus
& Bravais Type\\
\hline
1 &
$^{**}2^9 3^3$ (8) & 11 & 32 & 44 & -8 & -4 & -16
& $4 n + 1$, $4 n + 2$, $8 n + 7$, $2^2 (4 n + 1)$,
$2^2 (4 n + 2)$,
& Triclinic \\
&
$^{**}2^6 3^3 11$ (11) & 11 & 32 & 59 & 8 & 10 & 8
& $3n + 1$, $3^{2k+1}(3n + 1)$
& Triclinic \\
\hline
2 &
$^{**}2^9 3^2$ (8) & 5 & 20 & 48 & -4 & 0 & 0
& $4 n + 2$, $4 n + 3$, $8 n + 1$, $2^2 (4 n + 2)$, $2^2 (4 n + 3)$,
& Monoclinic(P) \\
&
$^{*}2^6 3^2 11$ (11) & 5 & 20 & 68 & -4 & -4 & -8
& $3^{2k}(3n + 1)$
& Triclinic \\
\hline
3 &
$^{**}2^9 3^2$ (8) & 17 & 17 & 20 & -14 & -4 & -4
& $4 n + 2$, $4 n + 3$, $8 n + 5$, $2^2 (4 n + 2)$, $2^2 (4 n + 3)$,
& Monoclinic(C) \\
&
$^{**}2^6 3^2 11$ (11) & 17 & 20 & 20 & -4 & -4 & -8
& $3^{2k}(3n + 1)$
& Monoclinic(C) \\
\hline
4 &
$^{**}2^9 3$ (8) & 7 & 15 & 16 & -6 & 0 & 0
& $4 n + 1$, $4 n + 2$, $8 n + 3$, $2^2 (4 n + 1)$, $2^2 (4 n + 2)$,
& Monoclinic(P) \\
&
$^{*}2^6 3 \cdot 11$ (11) & 7 & 15 & 23 & -6 & -2 & -6
& $3^{2k + 1}(3n + 1)$
& Triclinic \\
\hline
5 &
$^{**}2^9 3$ (8) & 11 & 11 & 16 & 6 & 8 & 8
& $4 n + 1$, $4 n + 2$, $8 n + 7$, $2^2 (4 n + 1)$, $2^2 (4 n + 2)$,
& Monoclinic(C) \\
&
$^{**}2^6 3 \cdot 11$ (11) & 11 & 11 & 19 & 6 & 2 & 2
& $3^{2k+1}(3n + 1)$
& Monoclinic(C) \\
\hline
6 &
$^{**}2^5 3^3$ (8) & 8 & 11 & 11 & -4 & -4 & -2
& \multirow{2}{*}{ $4 n + 1$, $4 n + 2$, $3n + 1$, $3^{2k+1}(3n + 1)$ }
& Monoclinic(C) \\
&
$^{**}2^2 3^3 \cdot 11$ (11) & 8 & 11 & 15 & -4 & 0 & -6
&
& Triclinic \\
\hline
7 &
$^{**}2^5 3^2$ (8) & 5 & 5 & 12 & -2 & 0 & 0
& \multirow{2}{*}{ $4 n + 2$, $4 n + 3$, $3^{2k}(3n + 1)$ }
& Orthorhombic(C) \\
&
$^{**}2^2 3^2 11$ (11) & 5 & 5 & 17 & -2 & -2 & -2
&
& Monoclinic(C) \\
\hline
8 &
$^{**}2^5 3$ (8) & 4 & 4 & 7 & 0 & -4 & 0
& \multirow{2}{*}{ $4 n + 1$, $4 n + 2$, $3^{2k+1}(3n + 1)$ }
& Orthorhombic(C) \\
&
$^{**}2^2 3\ 11$ (11) & 4 & 7 & 7 & -4 & 0 & -6
&
& Monoclinic(C) \\
\hline
9 &
$^{**}2^2 3^3 5$ (5) & 5 & 8 & 17 & 4 & 2 & 8
& \multirow{2}{*}{ $4 n + 2$, $4 n + 3$, $3n + 1$, $3^{2k+1}(3n + 1)$ }
& Monoclinic(C) \\
&
$^{**}2^5 3^3$ (8) & 5 & 8 & 24 & -4 & 0 & 0
&
& Monoclinic(P) \\
\hline
10 &
$^{**}2^2 3^2 5$ (5) & 3 & 8 & 8 & 0 & 0 & -4
& \multirow{2}{*}{ $4 n + 1$, $4 n + 2$, $3^{2n} (3 n + 1)$ }
& Orthorhombic(C) \\
&
$^{**}2^5 3^2$ (8) & 3 & 8 & 12 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
11 &
$^{**}2^2 3\ 5$ (5) & 1 & 4 & 16 & 0 & 0 & -4
& \multirow{2}{*}{ $4 n + 3$, $4 n + 2$, $3^{2k + 1}(3n + 1)$ }
& Orthorhombic(C) \\
&
$^{**}2^5 3$ (8) & 1 & 4 & 24 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
12 &
$2^{-2} 3^3 5^2$ (25) & 5 & 5 & 8 & -2 & -4 & -1
& \multirow{2}{*}{ $3 n + 1$, $3^{2k+1}(3 n - 1)$ }
& Triclinic
\\
&
$2^{-2} 3^3 37$ (37) & 5 & 8 & 8 & -4 & -1 & -5
&
& Triclinic
\\
\hline
13 &
$2^{-2} 3^2 5^2$ (25) & 3 & 4 & 7 & 3 & 3 & 4
& \multirow{2}{*}{ $3^{2k}(3 n - 1)$ }
& Monoclinic(C)
\\
&
$2^{-2} 3^2 37$ (37) & 3 & 4 & 7 & 0 & 0 & -1
&
& Monoclinic(P)
\\
\hline
14 &
$2^{-2} 3\ 5^2$ (25) & 1 & 4 & 5 & 0 & -1 & -1
& \multirow{2}{*}{ $3^{2k+1}(3 n - 1)$ }
& Monoclinic(C)
\\
&
$2^{-2} 3\ 37$ (37) & 1 & 4 & 7 & 0 & 0 & -1
&
& Monoclinic(P)
\\
\hline
15 &
$2 \cdot 3^4$ (1) & 4 & 7 & 7 & 2 & 2 & 5
& \multirow{2}{*}{ $4 n + 2$, $3 n - 1$, $3 (3 n \pm 1)$, $2^{2k+1}(8n + 7)$ }
& Monoclinic(C)
\\
&
$2^3 3^4$ (4) & 4 & 7 & 25 & -2 & -2 & -4
&
& Triclinic
\\
\hline
16 &
$2^{-1} 3^5$ (1) & 2 & 2 & 41 & 2 & 2 & 1
& \multirow{2}{*}{ $3 n + 1$, $3^2(3 n + 1)$ $3^{2k+1}(3n + 1)$ }
& Orthorhombic(F)
\\
&
$2 \cdot 3^5$ (4) & 2 & 6 & 41 & 0 & -1 & -3
&
& Triclinic
\\
\hline
17 &
$2^{-1} 3^4$ (1) & 2 & 2 & 14 & 2 & 2 & 1
& \multirow{2}{*}{ $3 (3 n + 1)$, $3^{2k}(3n + 1)$ }
& Orthorhombic(F)
\\
&
$2 \cdot 3^4$ (4) & 2 & 6 & 14 & 0 & -1 & -3
&
& Triclinic
\\
\hline
18 &
$2^{-1} 3^3$ (1) & 2 & 2 & 4 & 0 & -2 & -1
& \multirow{2}{*}{ $3 (3 n - 1)$, $3^{2k+1}(3n + 1)$ }
& Monoclinic(C)
\\
&
$2 \cdot 3^3$ (4) & 2 & 4 & 8 & 1 & 1 & 4
&
& Triclinic
\\
\hline
19 &
$2 \cdot 3^4$ (1) & 5 & 7 & 7 & 5 & 1 & 6
& \multirow{2}{*}{ $4 n + 2$, $3 (3 n \pm 1)$, $3^3 (3 n \pm 1)$, $2^{2k+1}(8n + 7)$ }
& Triclinic
\\
&
$2 \cdot 3^4$ (1) & 5 & 5 & 8 & -3 & -4 & 0
&
& Triclinic
\\
\hline
20 &
$2 \cdot 29$ (1) & 3 & 5 & 5 & 3 & 1 & 3
& \multirow{2}{*}{ $4 n + 2$, $2^{2k+1}(8n + 3)$ }
& Triclinic
\\
&
$2 \cdot 29$ (1) & 3 & 3 & 7 & 1 & 2 & 1
&
& Triclinic
\\
\hline
21 &
$^{*}3^3$ (1) & 1 & 4 & 7 & 0 & -1 & 0
& \multirow{2}{*}{ $4 n + 2$, $3 (3 n + 1)$, $3^{2k+1}(3 n - 1)$ }
& Orthorhombic(C)
\\
&
$^{*}3^3$ (1) & 1 & 5 & 7 & 1 & 1 & 5
&
& Monoclinic(C)
\\
\hline
22 &
$^{*}2^{-2} 3^3$ (1) & 1 & 1 & 7 & 0 & -1 & 0
& \multirow{2}{*}{ $3 (3 n + 1)$, $3^{2k+1}(3 n - 1)$ }
& Orthorhombic(C) \\
&
$^{*}2^{-2} 3^3$ (1) & 1 & 2 & 4 & -1 & 0 & -1
&
& Monoclinic(C) \\
\hline
23 &
$2^{-2} 3^2 59$ (59) & 5 & 5 & 6 & -2 & -3 & 0
& \multirow{2}{*}{ $3^{2k}(3n + 1)$ }
& Triclinic
\\
&
$2^{-2} 3^2 71$ (71) & 5 & 5 & 8 & -4 & -2 & -1
&
& Triclinic
\\
\hline
24 &
$2^{-2} 3 \cdot 59$ (59) & 2 & 4 & 7 & 1 & 2 & 4
& \multirow{2}{*}{ $3^{2k+1}(3n + 1)$ }
& Triclinic
\\
&
$2^{-2} 3 \cdot 71$ (71) & 2 & 4 & 7 & -1 & -1 & 0
&
& Triclinic
\\
\hline
25 &
$3^4$ (4) & 4 & 4 & 6 & -2 & -3 & 0
& \multirow{2}{*}{ $3(3 n + 1)$, $3^{2k}(3 n - 1)$ }
& Triclinic
\\
&
$2^{-2} 3^4 7$ (7) & 4 & 6 & 7 & 3 & 2 & 3
&
& Triclinic
\end{tabular}
\end{scriptsize}
\footnotetext[1]{
The mark $**$ indicates that the form is the only one in its genus. Therefore, it is regular.
}
\footnotetext[2]{
The mark $*$ indicates that the quadratic form is regular.
}
\footnotetext[3]{
The notation $*!$ indicates that the form is one of the 14 quadratic forms that may be regular. The regularity has been proved only under the generalized Riemann Hypothesis \cite{Oliver2014}.
}
\end{minipage}
\end{table}
\begin{table}[htbp]
\caption{Ternary positive-definite quadratic forms with the same representations over $\IntegerRing$ (Continued)}
\label{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(2/3)}
\begin{scriptsize}
\begin{tabular}{p{1mm}p{20mm}p{4mm}p{4mm}p{4mm} p{4mm}p{4mm}p{4mm
lr}
\hline
No.
& Determinant (Ratio) & $s_{11}$ & $s_{22}$ & $s_{33}$ & $s_{12}$ & $s_{13}$ & $s_{23}$
& Integers not represented by the genus
& Bravais Type\\
\hline
26 &
${}^{**}3^3 5$ (5) & 5 & 5 & 8 & 5 & 4 & 2
& \multirow{3}{*}{ $4 n + 2$, $3 n + 1$, $3^{2k+1}(3n + 1)$ }
& Monoclinic(C)
\\
&
${}^{**}2^3 3^3$ (8) & 5 & 8 & 8 & 4 & 2 & 8
&
& Monoclinic(C)
\\
&
${}^{**}3^3 11$ (11) & 5 & 8 & 9 & 2 & 3 & 6
&
& Triclinic \\
\hline
27 &
${}^{**}3^2 5$ (5) & 3 & 5 & 5 & 3 & 3 & 5
& \multirow{3}{*}{ $4 n + 2$, $3^{2k}(3n + 1)$ }
& Orthorhombic(I) \\
&
${}^{**}2^3 3^2$ (8) & 3 & 5 & 5 & 0 & 0 & -2
&
& Orthorhombic(C) \\
&
${}^{**}3^2 11$ (11) & 3 & 5 & 8 & -3 & 0 & -2
&
& Monoclinic(C) \\
\hline
28 &
${}^{**}3 \cdot 5$ (5) & 1 & 4 & 5 & 0 & -1 & -4
& \multirow{3}{*}{ $4 n + 2$, $3^{2k+1}(3n + 1)$ }
& Orthorhombic(I) \\
&
${}^{**}2^3 3$ (8) & 1 & 4 & 7 & 0 & 0 & -4
&
& Orthorhombic(C) \\
&
${}^{**}3 \cdot 11$ (11) & 1 & 5 & 7 & -1 & 0 & -1
&
& Monoclinic(C) \\
\hline
29 &
${}^{**}2^4 3$ (3) & 1 & 8 & 8 & 0 & 0 & -8
& \multirow{3}{*}{ $4 n + 2$, $4 n + 3$, $2^{2k} (8 n + 5)$ }
& Hexagonal \\
&
${}^{**}2^4 11$ (11) & 1 & 8 & 24 & 0 & 0 & -8
&
& Orthorhombic(C) \\
&
${}^{*}2^6 3$ (12) & 1 & 8 & 24 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
30 &
${}^{**}3$ (3) & 1 & 2 & 2 & 0 & 0 & -2
& \multirow{3}{*}{ $2^{2k} (8 n + 5)$ }
& Hexagonal \\
&
${}^{**}11$ (11) & 1 & 2 & 6 & 0 & 0 & -2
&
& Orthorhombic(C) \\
&
${}^{**}2^2 3$ (12) & 1 & 2 & 6 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
31 &
${}^{**}2^{-1} 3$ (3) & 1 & 1 & 2 & -1 & 0 & 0
& \multirow{3}{*}{ $2^{2k + 1} (8 n + 5)$ }
& Hexagonal \\
&
${}^{**}2^{-1} 11$ (11) & 1 & 2 & 3 & 0 & -1 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 3$ (12) & 1 & 2 & 3 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
32 &
${}^{**}2^4$ (1) & 3 & 3 & 3 & -2 & -2 & -2
& \multirow{3}{*}{ $4 n + 1$, $4 n + 2$, $2^{2k}(8n + 7)$ }
& Cubic(I) \\
&
${}^{**}2^6$ (4) & 3 & 3 & 8 & -2 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2^4 3^2$ (9) & 3 & 3 & 19 & -2 & -2 & -2
&
& Orthorhombic(F) \\
\hline
33 &
${}^{**}2$ (1) & 1 & 1 & 3 & 1 & 1 & 1
& \multirow{3}{*}{ $4 n + 2$, $2^{2k+1}(8n + 7)$ }
& Rhombohedral \\
&
${}^{**}2^3$ (4) & 1 & 3 & 3 & 0 & 0 & -2
&
& Orthorhombic(C) \\
&
${}^{*}2 \cdot 3^2$ (9) & 1 & 3 & 7 & 1 & 1 & 2
&
& Monoclinic(C) \\
\hline
34 &
${}^{**}1$ (1) & 1 & 1 & 1 & 0 & 0 & 0
& \multirow{3}{*}{ $2^{2 k}(8 n + 7)$ }
& Cubic(P) \\
&
${}^{**}2^2$ (4) & 1 & 2 & 2 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}3^2$ (9) & 1 & 2 & 5 & 0 & 0 & -2
&
& Orthorhombic(C) \\
\hline
35 &
${}^{**}2^{-1} 3^3$ (2) & 1 & 4 & 4 & 1 & 1 & 2
& \multirow{3}{*}{ $3 n - 1$, $3^{2k+1}(3 n + 1)$ }
& Orthorhombic(I) \\
&
${}^{**}2 \cdot 3^3$ (8) & 1 & 6 & 9 & 0 & 0 & 0
&
& Orthorhombic(P) \\
&
${}^{*!}2^{-2} 3^3 11$ (11) & 1 & 6 & 13 & 0 & -1 & -3
&
& Monoclinic(C) \\
\hline
36 &
${}^{**}2^{-1} 5^2$ (2) & 1 & 4 & 4 & 1 & 1 & 3
& \multirow{3}{*}{ $5^{2 k} (5 n \pm 2)$ }
& Orthorhombic(I) \\
&
${}^{*}2^{-2} 3 \cdot 5^2$ (3) & 1 & 4 & 5 & -1 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 5^2$ (8) & 1 & 5 & 10 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
37 &
${}^{**}2^{-1} 5$ (2) & 1 & 2 & 2 & 1 & 1 & 2
& \multirow{3}{*}{ $5^{2k+1}(5 n \pm 2)$ }
& Orthorhombic(I) \\
&
${}^{*}2^{-2} 3 \cdot 5$ (3) & 1 & 2 & 2 & 0 & 0 & -1
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 5$ (8) & 1 & 2 & 5 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
38 &
${}^{**}2^{-2} 5^2$ (1) & 2 & 2 & 2 & -1 & -1 & -1
& \multirow{3}{*}{ $5^{2k}(5n \pm 1)$ }
& Rhombohedral \\
&
${}^{**}5^2$ (4) & 2 & 3 & 5 & -2 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}5^2$ (4) & 2 & 2 & 7 & -1 & -1 & -1
&
& Monoclinic(C) \\
\hline
39 &
${}^{**}2^{-2} 5$ (1) & 1 & 1 & 2 & 1 & 1 & 1
& \multirow{3}{*}{ $5^{2k+1}(5n \pm 1)$ }
& Rhombohedral \\
&
${}^{**}5$ (4) & 1 & 2 & 3 & 0 & 0 & -2
&
& Orthorhombic(C) \\
&
${}^{**}5$ (4) & 1 & 2 & 3 & -1 & 0 & -1
&
& Monoclinic(C) \\
\hline
40 &
$2^4 3 \cdot 13$ (39) & 5 & 12 & 12 & -4 & -4 & 0
& \multirow{4}{*}{ $4 n + 2$, $4 n + 3$, $2^{2k}(8n + 1)$ }
& Monoclinic(C)
\\
&
$2^4 71$ (71) & 5 & 12 & 21 & -4 & -2 & -4
&
& Triclinic
\\
&
$2^4 79$ (79) & 5 & 12 & 24 & -4 & 0 & -8
&
& Triclinic
\\
&
$2^4 5 \cdot 19$ (95) & 5 & 12 & 28 & -4 & -4 & 0
&
& Triclinic
\\
\hline
41 &
$3 \cdot 13$ (39) & 3 & 3 & 5 & 0 & -2 & -2
& \multirow{4}{*}{ $2^{2k}(8n + 1)$ }
& Monoclinic(C)
\\
&
$71$ (71) & 3 & 5 & 6 & 2 & 2 & 4
&
& Triclinic
\\
&
$79$ (79) & 3 & 5 & 6 & -2 & -2 & 0
&
& Triclinic
\\
&
$5 \cdot 19$ (95) & 3 & 5 & 7 & -2 & 0 & -2
&
& Triclinic
\\
\hline
42 &
$2^{-1} 3 \cdot 13$ (39) & 3 & 3 & 3 & 3 & 1 & 1
& \multirow{4}{*}{ $2^{2k+1}(8n + 1)$ }
& Monoclinic(C)
\\
&
$2^{-1} 71$ (71) & 3 & 3 & 5 & 1 & 3 & 2
&
& Triclinic
\\
&
$2^{-1} 79$ (79) & 3 & 3 & 5 & -1 & -2 & -1
&
& Triclinic
\\
&
$2^{-1} 5 \cdot 19$ (95) & 3 & 5 & 5 & 3 & 2 & 5
&
& Triclinic
\\
\hline
\end{tabular}
\end{scriptsize}
\end{table}
\begin{table}[htbp]
\caption{Ternary positive-definite quadratic forms with the same representations over $\IntegerRing$ (Continued)}
\label{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(3/3)}
\begin{scriptsize}
\begin{tabular}{p{1mm}p{20mm}p{4mm}p{4mm}p{4mm} p{4mm}p{4mm}p{4mm
lr}
\hline
No.
& Determinant (Ratio) & $s_{11}$ & $s_{22}$ & $s_{33}$ & $s_{12}$ & $s_{13}$ & $s_{23}$
& Integers not represented by the genus
& Bravais Type\\
\hline
43 &
${}^{**}2^4 7$ (7) & 5 & 5 & 5 & 2 & 2 & 2
& \multirow{4}{*}{ $4 n + 2$, $4 n + 3$, $2^{2k}(8n + 1)$ }
& Rhombohedral \\
&
${}^{**}2^4 3 \cdot 5$ (15) & 5 & 5 & 12 & -2 & -4 & -4
&
& Orthorhombic(F) \\
&
${}^{**}2^4 23$ (23) & 5 & 8 & 12 & 0 & -4 & -8
&
& Monoclinic(C) \\
&
${}^{*}2^6 7$ (28) & 5 & 8 & 12 & 0 & -4 & 0
&
& Monoclinic(P) \\
\hline
44 &
${}^{**}7$ (7) & 2 & 2 & 3 & 2 & 2 & 2
& \multirow{4}{*}{ $2^{2k}(8n + 1)$ }
& Rhombohedral \\
&
${}^{**}3 \cdot 5$ (15) & 2 & 3 & 3 & -2 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}23 $ (23) & 2 & 3 & 5 & -2 & 0 & -2
&
& Monoclinic(C) \\
&
${}^{**}2^2 7$ (28) & 2 & 3 & 5 & 0 & 0 & -2
&
& Monoclinic(P) \\
\hline
45 &
${}^{**}2^{-1} 7$ (7) & 1 & 1 & 5 & 1 & 1 & 1
& \multirow{4}{*}{ $2^{2k+1}(8n + 1)$ }
& Rhombohedral \\
&
${}^{**}2^{-1} 3 \cdot 5$ (15) & 1 & 3 & 3 & 1 & 1 & 1
&
& Orthorhombic(I) \\
&
${}^{**}2^{-1} 23$ (23) & 1 & 3 & 5 & 1 & 1 & 3
&
& Monoclinic(C) \\
&
${}^{**}2 \cdot 7$ (28) & 1 & 3 & 5 & 0 & 0 & -2
&
& Monoclinic(P) \\
\hline
46 &
${}^{**}2^{-2} 3^3$ (1) & 2 & 2 & 2 & 1 & 1 & 1
& \multirow{4}{*}{ $3n + 1$, $3^{2k+1}(3 n - 1)$ }
& Rhombohedral \\
&
${}^{**}3^3$ (4) & 2 & 3 & 5 & 0 & -2 & 0
&
& Orthorhombic(C) \\
&
${}^{**}3^3$ (4) & 2 & 2 & 8 & 1 & 2 & 2
&
& Monoclinic(C) \\
&
${}^{*}2^{-2} 3^3 7$ (7) & 2 & 3 & 8 & 0 & -1 & 0
&
& Monoclinic(P) \\
\hline
47 &
${}^{**}2^{-2} 3^2$ (1) & 1 & 1 & 3 & -1 & 0 & 0
& \multirow{4}{*}{ $3^{2k} (3 n - 1)$ }
& Hexagonal \\
&
${}^{**}3^2$ (4) & 1 & 3 & 3 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}3^2$ (4) & 1 & 3 & 4 & 0 & -1 & -3
&
& Orthorhombic(I) \\
&
${}^{*}2^{-2} 3^2 7$ (7) & 1 & 3 & 6 & 0 & 0 & -3
&
& Orthorhombic(C) \\
\hline
48 &
${}^{**}2^{-2} 3$ (1) & 1 & 1 & 1 & -1 & 0 & 0
& \multirow{4}{*}{ $3^{2 k + 1}(3 n - 1)$ }
& Hexagonal \\
&
${}^{**}3$ (4) & 1 & 1 & 3 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}3$ (4) & 1 & 2 & 2 & 1 & 1 & 1
&
& Orthorhombic(I) \\
&
${}^{*}2^{-2} 3 \cdot 7$ (7) & 1 & 2 & 3 & -1 & 0 & 0
&
& Orthorhombic(C) \\
\hline
49 &
${}^{*}2^{-2} 3^3$ (1) & 1 & 1 & 9 & -1 & 0 & 0
& \multirow{4}{*}{ $3 n - 1$, $3^{2k + 1}(3 n - 1)$ }
& Hexagonal \\
&
${}^{*}2^{-2} 3^3$ (1) & 1 & 3 & 3 & 0 & 0 & -3
&
& Hexagonal \\
&
${}^{*}3^3$ (4) & 1 & 3 & 10 & 0 & -1 & -3
&
& Orthorhombic(I) \\
&
${}^{**}3^3$ (4) & 1 & 3 & 9 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
50 &
${}^{**}2^{-1}$ (1) & 1 & 1 & 1 & 1 & 1 & 1
& \multirow{4}{*}{ $2^{2k + 1} (8 n + 7)$ }
& Cubic(F) \\
&
${}^{**}2$ (4) & 1 & 1 & 2 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}2^{-1} 3^2$ (9) & 1 & 2 & 3 & 0 & -1 & -2
&
& Orthorhombic(I) \\
&
${}^{**}2^3$ (16) & 1 & 2 & 4 & 0 & 0 & 0
&
& Orthorhombic(P) \\
\hline
51 &
${}^{**}2^{-1} 3^3$ (2) & 2 & 2 & 5 & 2 & 2 & 1
& \multirow{5}{*}{ $3 n + 1$, $3^{2k+1}(3n + 1)$ }
& Orthorhombic(F) \\
&
${}^{**}2^{-2} 3^3 5$ (5) & 2 & 5 & 5 & 2 & 1 & 5
&
& Monoclinic(C) \\
&
${}^{**}2 \cdot 3^3$ (8) & 2 & 5 & 6 & -2 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 3^3$ (8) & 2 & 5 & 6 & -1 & 0 & -3
&
& Triclinic \\
&
${}^{**}2^{-2} 3^3 11$ (11) & 2 & 5 & 8 & -1 & -1 & -2
&
& Triclinic \\
\hline
52 &
${}^{**}2^{-1} 3^2$ (2) & 2 & 2 & 2 & 2 & 2 & 1
& \multirow{5}{*}{ $3^{2k} (3 n + 1)$ }
& Tetragonal(I) \\
&
${}^{**}2^{-2} 3^2 5$ (5) & 2 & 2 & 3 & -1 & 0 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 3^2$ (8) & 2 & 3 & 3 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}2 \cdot 3^2$ (8) & 2 & 2 & 5 & 1 & 1 & 1
&
& Monoclinic(C) \\
&
${}^{**}2^{-2} 3^2 11$ (11) & 2 & 3 & 5 & 0 & -1 & -3
&
& Monoclinic(C) \\
\hline
53 &
${}^{**}2^{-1} 3$ (2) & 1 & 1 & 2 & 0 & -1 & -1
& \multirow{5}{*}{ $3^{2k + 1} (3 n + 1)$ }
& Tetragonal(I) \\
&
${}^{**}2^{-2} 3 \cdot 5$ (5) & 1 & 1 & 4 & 0 & -1 & 0
&
& Orthorhombic(C) \\
&
${}^{**}2 \cdot 3$ (8) & 1 & 1 & 6 & 0 & 0 & 0
&
& Tetragonal(P) \\
&
${}^{**}2 \cdot 3$ (8) & 1 & 2 & 4 & 1 & 1 & 2
&
& Monoclinic(C) \\
&
${}^{**}2^{-2} 3 \cdot 11$ (11) & 1 & 2 & 5 & 1 & 1 & 1
&
& Monoclinic(C) \\
\hline
\end{tabular}
\end{scriptsize}
\end{table}
Overall, 53 sets consisting of 151 quadratic forms were obtained as the candidates that may have the identical representations up to $\infty$.
It should be noted that all of them
can be obtained if the algorithm in Section \ref{Algorithm}
is carried out under the constraint $s_{33} \leq 48$.
Since the number is rather small compared with the upper bound $115$ of the searched region,
it is expected that the 53 cases, in addition to those in the hexagonal and rhombohedral families, provide all the searched quadratic forms, up to the action of $GL_3(\IntegerRing)$ and constant multiple.
The tables also have information about the regularity of each quadratic form. This is based on the tables of Jagy \textit{et\ al.\ }\cite{Jagy97} and \cite{Oh2011}.
If a form in Tables \ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(1/3)}--\ref{Forty-nine groups of ternary positive definite quadratic forms representing the same numbers(3/3)}
is regular, it is marked with $**$ or $*$.
In the tables, 38 out of the 53 cases consist of regular (or possibly regular) quadratic forms.
The others are neither regular nor spinor-regular from the result in \cite{Benham90}.
In general, it is difficult to exactly determine the set of integral representations for a ternary quadratic form (\CF \cite{Ono97}).
For example, it has not been proved that the quadratic forms in No. 35 really have the same representations,
up to $\infty$.
If the quadratic forms are equivalent over $\RationalField$,
it may be possible to prove that they have the same representations, without providing the exact set of their representations,
as done for the hexagonal and the rhombohedral families,
although the author could not do this for the non-regular forms in the tables.
When two positive-definite quadratic forms $f$, $g \in {\rm Sym}^2 (\RationalField^3)^*$ satisfy $q_\IntegerRing(f) = q_\IntegerRing(g)$,
it can be proved that they are equivalent over $\RationalField$ if and only if $\det f / \det g$ is a square in $\RationalField^\times$.
\section{Preliminaries for main theorem}
Some basic properties on simultaneous representations that will be used in the proof of Theorem \ref{thm:main result over RationalField} are presented herein.
The following lemmas and Corollary 1 are repeatedly used to prove the theorem.
In what follows, $k$ is an arbitrary field, and $\bar{k}$ is the algebraic closure of $k$.
${\rm Sym}^2 (k^3)^* \otimes_{k} k^2$ and $GL_3(k) \times GL_2(k)$
are denoted by $V_k$ and $G_k$, respectively,
as in the theory of prehomogeneous vector spaces.
The following can be seen as a generalization of the well-known fact that any pairs of positive-definite quadratic forms are simultaneously diagonalized over $\RealField$:
\begin{lem}\label{lem: diagonalization over F}
For any $(A, B) \in {\rm Sym}^2 (k^n)^* \otimes_{k} k^2$ with $\det A \neq 0$,
let $K \subset \bar{k}$ (\textit{resp.}, $K_2 \subset \bar{k}$) be the field generated by all the roots (\textit{resp.}, all the roots of multiplicity $> 1$) of $\det(Ax - B) = 0$ in $\bar{k}$ over $k$.
If $(A, B)$ is anisotropic over $K_2$, then
$(A, B)$ is simultaneously diagonalized by the action of $GL_n(K)$.
Consequently, for any $\alpha \in K$,
the rank of $A \alpha - B$ equals $n$ minus the multiplicity of $\alpha$ as a root of $\det(A x - B) = 0$.
\end{lem}
\begin{proof}
Let $w \in GL_n(K)$ be the matrix that provides a Jordan decomposition of $S := A^{-1} B$.
Since $B$ is symmetric, we have $\tr{w} B w = (\tr{w} A w) (w^{-1} S w) = \tr{(w^{-1} S w)} (\tr{w} A w)$.
Hence, if $w^{-1} S w$ is diagonal,
$(\tr{w} A w$, $\tr{w} B w)$ is a simultaneous block-diagonalization of $(A, B)$
and each block corresponds to an eigenspace of $S$.
If $[A_1, \cdots, A_m]$ and $[B_1, \cdots, B_m]$ are the blocks of $A$ and $B$, respectively, then
one of $A_i$, $B_i$ is a constant multiple of the other.
Hence, $(A, B)$ can be simultaneously diagonalized.
We next assume that $S$ has a Jordan block of size $m > 1$
that corresponds to a multiple root $\alpha \in K_2$ of $\det(A x - B) = \det A \det(I x - S) = 0$.
Let $S_2$, $A_2$, $B_2$ be the $m \times m$ blocks of $w^{-1} S w$, $\tr{w} A w$, $\tr{w} B w$, respectively, corresponding to the eigenspace of $\alpha$.
Since we have $B_2 = A_2 S_2 = \tr{S}_2 A_2$,
the $(1, 1)$-entries of $A_2$ and $B_2$ equal $0$.
This is impossible since $(A, B)$ is anisotropic over $K_2$.
Hence, all the Jordan blocks in $S$ must have size $1$. The lemma is proved.
\end{proof}
In particular, $K_2 = k$ always holds for $n=3$.
Therefore, any $(A, B) \in V_k$ anisotropic over $k$, is simultaneously diagonalized by the action of $GL_n(K)$.
\begin{lem}\label{lem: equivalence conditions on isotropy}
Let $k$ be a field with ${\rm char} k \neq 2$.
We assume that $(A, B) \in V_k$ is linearly independent,
non-singular and anisotropic over $k$, and satisfies $\det A \neq 0$, ${\rm Disc}(A, B) \neq 0$.
Let $\alpha_i \in \bar{k}$ ($i = 1, 2, 3$) be the roots of $\det(A x - B) = 0$
and $K_i$ be the Galois closure of $k_i = k(\alpha_i)$ over $k$.
We choose $C_i \in {\rm Sym}^2 (k_i^2)^*$ satisfying $A \alpha_i - B \sim_{k_i} C_i \perp [ 0 ]$.
Then,
$C_i$ is anisotropic over $K_i$ for at least two of $i = 1, 2, 3$ and all of $i$ with $k_i \neq k$.
\end{lem}
\begin{proof}
Since ${\rm Disc}(A, B) \neq 0$,
the vector $0 \neq v_i \in k_i^3$ with $(A \alpha_i - B) v_i = 0$ is uniquely determined, up to constant multiple of $k_i^\times$.
From $(A \alpha_i - B) v_i = 0$ and $(A \alpha_j - B) v_j = 0$,
we have $\tr{v}_i A v_j = \tr{v}_i B v_j = 0$ for any distinct $1 \leq i, j \leq 3$.
Hence, if $w$ equals the matrix $\tr{(v_1\ v_2\ v_3)}$, then
$w A \tr{w}$ and $w B \tr{w}$ are diagonal.
It may be assumed that $[d_1, d_2, d_3]$ and $[d_1 \alpha_1, d_2 \alpha_2, d_3 \alpha_3]$ are diagonal entries of $A$ and $B$.
For any $1 \leq i, j \leq 3$,
if $k_i, k_j \neq k$ are conjugate over $k$,
it may be assumed that
the isomorphism $\sigma_{ij} : k_i \rightarrow k_j$ induced by $\alpha_i \mapsto \alpha_j$ over $k$ maps $v_i$ to $v_j$.
Then, $\sigma_{ij}$ maps $d_i$ to $d_j$.
We now have the following:
\begin{small}
\begin{eqnarray}\label{eq: transform of (A,B)}
\left( w,
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}
\right)
\cdot
(A, B)
=
\left(
\begin{pmatrix}
0 & 0 & 0 \\
0 & d_2 (\alpha_1 - \alpha_2) & 0 \\
0 & 0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix},
\begin{pmatrix}
d_1 (\alpha_2 - \alpha_1) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right). \nonumber \\
\end{eqnarray}
\end{small}
In what follows, the pair in the right-hand side is denoted by $(\tilde{A}, \tilde{B})$.
We may assume one of the following:
\begin{enumerate}[(i)]
\item $k_1 = k_2 = k_3 = k$,
\item $k_1, k_2$ are quadratic over $k$,
\item $k_1, k_2, k_3$ are cubic over $k$.
\end{enumerate}
We shall show that the assumption that the diagonal matrices $D_1 = [d_2 (\alpha_1 - \alpha_2), d_3 (\alpha_1 - \alpha_3)]$
and $D_2 = [ d_2 (\alpha_1 - \alpha_2), d_3 (\alpha_1 - \alpha_3) ]$ are isotropic over $K_1$ and $K_2$, respectively,
leads to a contradiction.
The assumption holds if and only if
some $\beta_1 \in K_1$, $\beta_2 \in K_2$ satisfy
$d_2 d_3 (\alpha_1 - \alpha_2)(\alpha_1 - \alpha_3) = -\beta_1^2$ and
$d_1 d_3 (\alpha_2 - \alpha_1)(\alpha_2 - \alpha_3) = -\beta_2^2$.
Then, $(\tilde{A}, \tilde{B})({\mathbf x}) = 0$ holds,
if we put ${\mathbf x} := (-\beta_2 / d_1 (\alpha_1 - \alpha_2), \beta_1 / d_2(\alpha_1 - \alpha_2), \pm 1)$.
Hence,
$(A, B)$ is isotropic over $k$ in case (i).
The same is true in case (ii),
since
$(A, B)({\mathbf x} w) = 0$ and ${\mathbf x} w \in k^3$,
if we choose $\beta_2$ such that $\beta_2 = \sigma_{12}(\beta_1)$.
In case (iii),
$A \alpha_i - B$ is isotropic over $K_1 = K_2 = K_3$ for all $i = 1, 2, 3$
due to conjugacy.
There exist $\beta_i \in k_i$ ($i = 1, 2, 3$) such that either of the following holds for every distinct $1 \leq h, i, j \leq 3$:
\begin{itemize}
\item $d_i d_j (\alpha_h - \alpha_i)(\alpha_h - \alpha_j) = -\beta_h^2$,
\item $d_i d_j (\alpha_h - \alpha_i)(\alpha_h - \alpha_j) = -\beta_h^2 (\alpha_1 - \alpha_2)^2 (\alpha_2 - \alpha_3)^2 (\alpha_3 - \alpha_1)^2$.
\end{itemize}
It is possible to choose $\beta_i$ so that $\beta_1, \beta_2, \beta_3$ are conjugate.
If we put ${\mathbf x}_2 := (1/\beta_1, 1/\beta_2, 1/\beta_3)$,
we have $(\tilde{A}, \tilde{B})({\mathbf x}_2) = (A, B)({\mathbf x}_2 \tr{w}) = 0$ and ${\mathbf x}_2 w \in k^3$.
Thus, the lemma is proved.
\end{proof}
As a fundamental fact, $q_\IntegerRing(A_1, B_1) = q_\IntegerRing(A_2, B_2)$ leads to $q_\RationalField(A_1, B_1) = q_\RationalField(A_2, B_2)$,
and $\{ (0, 0) \} \cup q_{\IntegerRing_p}(A_1, B_1) = \{ (0, 0) \} \cup q_{\IntegerRing_p}(A_2, B_2)$,
by taking the topological closure of $q_\IntegerRing(A_i, B_i)$ in $\RationalField_p^2$.
For any global field $k$, if we start from $q_k(A_1, B_1) = q_k(A_2, B_2)$,
the following are immediately obtained:
\begin{enumerate}[($C$1)]
\item if both $(A_1, B_1)$ and $(A_2, B_2)$ are anisotropic over $k_v$,
$q_{k_v} (A_1, B_1) = q_{k_v}(A_2, B_2)$, since the topological closure of $q_k(A_i, B_i)$ in $k_v^2$ equals
$\{ (0, 0) \} \cup q_{k_v}(A_i, B_i)$.
\item if both $(A_1, B_1)$ and $(A_2, B_2)$ are isotropic over $k_v$,
$q_{k_v} (A_1 \pi - B_1) = q_{k_v}(A_2 \pi - B_2) = k_v$ for any $\pi \in k_v$.
\item if $(A_1, B_1)$ is anisotropic and $(A_2, B_2)$ is isotropic over $k_v$,
$A_1 \pi - B_1$ is isotropic over $k_v$ for any $\pi \in k_v$.
\end{enumerate}
Thus, the following is obtained as a corollary of Lemma \ref{lem: equivalence conditions on isotropy}.
\begin{cor}\label{cor: anisotropic iff anisotropic}
We assume that $(A_1, B_1), (A_2, B_2) \in V_k$
satisfy $q_k(A_1, B_1) = q_k(A_2, B_2)$.
For any finite prime $v$ of $k$, if $\det (A_1 x - B_1 y)$ completely splits in $k_v$,
the above ($C$3) cannot happen, hence,
$q_{k_v} (A_1 \pi - B_1) = q_{k_v}(A_2 \pi - B_2)$ for any $\pi \in k_v$.
$(A_1, B_1)$ is isotropic over $k_v$ if and only if $(A_2, B_2)$ is.
\end{cor}
\section{A canonical form for elements of $V_k$}
\label{A canonical form for elements of $V_k$}
Herein, the method to obtain a \textit{canonical form} of $(A, B) \in V_k$ over a field $k$ is discussed, assuming that ${\rm char} k \neq 2, 3$.
Any $(A_1, B_1)$, $(A_2, B_2) \in V_k$ can be simultaneously transformed
to the canonical form, if $\det (A_1 x - B_1 y) = c \det (A_2 x - B_2 y)$ for some $c \in k^\times$,
and some $(q_A, q_B) \in q_k(A_1, B_1) \cap q_k(A_2, B_2)$ satisfies $\det (q_B A_i - q_A B_i) \neq 0$.
We first recall that
a one-to-one correspondence between the elements of $V_\IntegerRing := {\rm Sym}^2 (\IntegerRing^3)^* \otimes \IntegerRing^2$
and all pairs of a quartic ring and its resolvent ring was recently proved in \cite{Bhargava2004}.
The result was generalized to the case of any base scheme $S$ \cite{Wood2011}.
Herein, the version for Dedekind domains \cite{O'Dorney2016} is adopted for the study of quadratic forms.
A \textit{Dedekind domain} is a Noetherian, integrally closed integral domain that has the property that every nonzero prime ideal is maximal.
Under the definition, a field is also a Dedekind domain.
A {\textit lattice} over a Dedekind domain $R$ is a finitely generated, torsion-free $R$-module.
If $M$ is a lattice over $R$ and $k$ is the field of fractions of $R$,
the {\textit rank} of $M$ is defined as the dimension of $M \otimes_R k$ over $k$.
A unitary commutative associative $R$-algebra is called a \textit{quartic ring} (\textit{resp. cubic ring}), if it has rank 4 (\textit{resp. 3}).
When we put $L := Q/R$, $M := C/R$ and $L^* := {\rm Hom}(L, R)$,
a \textit{quadratic map} means an element of ${\rm Sym}^2 L^* \otimes_R M$.
\begin{defin}
Let $R$ be a Dedekind domain.
For any quartic ring $Q$,
its \textit{cubic resolvent ring} $C$ (also called numerical resolvent in \cite{O'Dorney2016}) is defined as the $R$-algebra with the following properties:
\begin{itemize}
\item It is equipped with an $R$-module isomorphism $\theta: \Lambda^2 (C/R) \rightarrow \Lambda^3 (Q/R)$ and a quadratic map $\phi: Q/R \rightarrow C/R$ such that
\begin{eqnarray}\label{eq: eq for Q}
x \wedge y \wedge x y = \theta(\phi(x) \wedge \phi(y)).
\end{eqnarray}
\item The multiplicative structure of $C$ is determined by
\begin{eqnarray}\label{eq: eq for C}
x \wedge x^2 \wedge x^3 = \theta(\phi(x) \wedge \phi(x)^2) \text{ for any } x \in Q/R,
\end{eqnarray}
where $\phi(x)^2$ is the square $\tilde{\phi}(x)^2$ of any lift $\tilde{\phi} : Q \rightarrow C$ of $\phi$.
\end{itemize}
The quadratic map $\phi$ is called the resolvent map of $(Q, C)$.
\end{defin}
If $R$ is a field, and $Q$ is the quartic extension $R[\alpha]$, the \textit{classical resolvent map} from $Q$ to $R$ is defined by $\alpha \mapsto \alpha \alpha^\prime + \alpha^{\prime\prime} \alpha^{\prime\prime\prime}$,
by using the conjugates $\alpha^\prime$, $\alpha^{\prime\prime}$, $\alpha^{\prime\prime\prime}$ of $\alpha$ over $R$.
The following explains a generalization of the Bhargava correspondence for quartic rings
to the case over a Dedekind domain:
\begin{thm}[Theorem 1.4 \cite{O'Dorney2016}]
Let $R$ be a Dedekind domain. There is a canonical bijection between
\begin{enumerate}[(i)]
\item Isomorphism classes of pairs $(Q, C)$, where $Q$ is a quartic ring
and $C$ is a cubic resolvent ring of $Q$.
\item Quadruples $(L, M, \theta, \phi)$ where $L$ and $M$ are lattices of ranks $3$ and $2$ over $R$, respectively,
$\theta: \Lambda^2 M \rightarrow \Lambda^3 L$ is an isomorphism and $\phi: L \rightarrow M$ is a quadratic map.
\end{enumerate}
Under this bijection, the identifications $Q/R \cong L$ and $C/R \cong M$ are obtained.
Any quartic ring $Q$ has a cubic resolvent, and if $Q$ is Dedekind, the resolvent is unique.
\end{thm}
In what follows, we explain how the pair of $Q$ and $R$ as in (i), is associated with $(L, M, \theta, \phi)$ in (ii).
Since $R$ is a Dedekind domain,
the lattice $Q/R$ is isomorphic to ${\mathfrak a}_1 \bar{\xi}_1 \oplus {\mathfrak a}_2 \bar{\xi}_2
\oplus {\mathfrak a}_3 \bar{\xi}_3$
for some $\bar{\xi}_1, \bar{\xi}_2, \bar{\xi}_3 \in Q/R$ and ideals ${\mathfrak a}_1, {\mathfrak a}_2, {\mathfrak a}_3$ of $R$.
Similarly, $C/R$ is isomorphic to ${\mathfrak b}_1 \bar{\omega}_1 \oplus {\mathfrak b}_2 \bar{\omega}_2$
for some $\bar{\omega}_1, \bar{\omega}_2 \in C/R$ and ideals ${\mathfrak b}_1, {\mathfrak b}_2$ of $R$.
If these ${\mathfrak a}_i$, $\bar{\xi}_i$, ${\mathfrak b}_j$, $\bar{\omega}_j$ are fixed,
the quadratic map $\phi : Q/R \rightarrow C/R$
is uniquely associated with a pair of ternary quadratic forms $(A, B)$ over the fraction field $k$,
by
\begin{eqnarray}\label{eq: relation between phi and (A, B)}
\phi(x_1 \bar{\xi}_1 + x_2 \bar{\xi}_2 + x_3 \bar{\xi}_3) = B(x_1, x_2, x_3) \bar{\omega}_1 + A(x_1, x_2, x_3) \bar{\omega}_2.
\end{eqnarray}
If we put $L := Q/R$, $Q$ is isomorphic to $R \oplus L$ as an $R$-module.
Let $\xi_i \in Q$ ($1 \leq i \leq 3$) be the element corresponding to $\bar{\xi}_i$ in $Q / R$.
We first note that (\ref{eq: eq for Q}) implies
\begin{eqnarray*}
\left( \sum_{i=1}^3 x_i \bar{\xi}_i \right) \wedge \left( \sum_{i=1}^3 y_i \bar{\xi}_i \right)
\wedge \left( \sum_{i=1}^3 \sum_{j=1}^3 x_i y_j \bar{\xi}_i \bar{\xi}_j \right)
&=& \theta\left( \phi \left( \sum_{i=1}^3 x_i \bar{\xi}_i \right) \wedge \phi \left( \sum_{i=1}^3 y_i \bar{\xi}_i \right) \right).
\end{eqnarray*}
The multiplicative structure of $Q$, \IE all $c_{ij}^k \in R$ of the equalities $\xi_i \xi_j = c_{ij}^0 + \sum_{i=1}^3 c_{ij}^k \xi_k$, is determined from (\ref{eq: eq for Q}) as follows;
we denote the coefficients of $A$ and $B$ by $A(x_1, x_2, x_3) = \sum_{1 \leq i \leq j \leq 3} a_{ij} x_i x_j$,
$B(x_1, x_2, x_3) = \sum_{1 \leq i \leq j \leq 3} b_{ij} x_i x_j$.
We then have
\begin{eqnarray*}
(\xi_1 \wedge \xi_2 \wedge \xi_3)
\begin{vmatrix}
x_1 & y_1 & \sum_{i, j=1}^3 c_{ij}^1 x_i y_j \\
x_2 & y_2 & \sum_{i, j=1}^3 c_{ij}^2 x_i y_j \\
x_3 & y_3 & \sum_{i, j=1}^3 c_{ij}^3 x_i y_j \\
\end{vmatrix}
=
-
\theta\left( \bar{\omega}_1 \wedge \bar{\omega}_2 \right)
\sum_{1 \leq i \leq j \leq 3}
\sum_{1 \leq k \leq l \leq 3} x_i x_j y_k y_l
\begin{vmatrix}
a_{ij} & b_{ij} \\
a_{kl} & b_{kl} \\
\end{vmatrix}.
\end{eqnarray*}
Replacing each $\xi_i$ by $-\xi_i$ if necessary,
we can fix the sign as follows:
\begin{eqnarray*}
\begin{vmatrix}
x_1 & y_1 & \sum_{i, j=1}^3 c_{ij}^1 x_i y_i \\
x_2 & y_2 & \sum_{i, j=1}^3 c_{ij}^2 x_i y_i \\
x_3 & y_3 & \sum_{i, j=1}^3 c_{ij}^3 x_i y_i \\
\end{vmatrix}
= -
\sum_{1 \leq i \leq j \leq 3}
\sum_{1 \leq k \leq l \leq 3} x_i x_j y_k y_l
\begin{vmatrix}
a_{ij} & b_{ij} \\
a_{kl} & b_{kl} \\
\end{vmatrix}.
\end{eqnarray*}
Comparing the coefficients of each term, the following equations are obtained:
\begin{eqnarray}\label{eq: multiplicative structure of Q}
c^j_{ii} &=& \epsilon \lambda^{ii}_{ik}, \nonumber \\
c^k_{ij} &=& \epsilon \lambda^{jj}_{ii}, \nonumber \\
c^j_{ij} - c^k_{ik} &=& \epsilon \lambda^{jk}_{ii}, \\
c^i_{ii} - c^j_{ij} - c^k_{ik} &=& \epsilon \lambda^{ij}_{ik},\ \nonumber
\lambda^{ij}_{kl} = \begin{vmatrix} a_{ij} & b_{ij} \\ a_{kl} & b_{kl} \end{vmatrix},
\end{eqnarray}
where $(i, j, k)$ denotes any permutation of $(1, 2, 3)$ and $\epsilon = \pm 1$ is its sign.
The $c^0_{ij}$ are determined from the associative law of $Q$.
Consequently, all $c_{ij}^k \in R$, are uniquely determined, up to the transformations given by
$c^j_{ij} \mapsto c^j_{ij} + a$, $c^k_{ij} \mapsto c^k_{ij} + a$, $c^i_{ii} \mapsto c^i_{ii} + 2a$ ($a \in Q$)
which corresponds to the replacement of $\xi_i$ by $\xi_i + a$.
In order to uniquely determine $c_{ij}^k$, the following constraints are applied in \cite{Bhargava2004}:
\begin{eqnarray}\label{eq: normalization}
c^1_{12} = c^2_{12} = c^1_{13} = 0.
\end{eqnarray}
If we put $f_{det}(x, y) := 4 \det(A x - B y)$,
the following equation holds for any $x = \sum_{i=1}^3 x_i \bar{\xi}_i \in Q/R$.
\begin{eqnarray}\label{eq: formula for 4 det(B(x) A - A(x) B}
x \wedge x^2 \wedge x^3 = f_{det}(B(x_1, x_2, x_3), A(x_1, x_2, x_3)) \bar{\xi}_1 \wedge \bar{\xi}_2 \wedge \bar{\xi}_3.
\end{eqnarray}
Hence, the ring structure of $C$ determined by (\ref{eq: eq for C}), is same as that provided by $f_{det}(x, y) := 4 \det(A x - B y)$ under the Delone-Faddeev-Gan-Gross-Savin correspondence (\cite{Delone64}, \cite{Gan2002}).
If the coefficients are denoted by $4 \det(A x - B y) = a x^3 + b x^2 y + c x y^2 + d y^3$,
the basis $\langle \bar{\omega}_1, \bar{\omega}_2 \rangle$ of $C/R$ as an $R$-algebra is lifted to a basis $\langle 1, \omega_1, \omega_2 \rangle$ of $C$ that satisfies:
\begin{eqnarray}\label{eq: multiplicative structure of R}
\omega_1^2 &=& -ac + b \omega_1 - a \omega_2, \nonumber \\
\omega_1 \omega_2 &=& -ad, \\
\omega_2^2 &=& -bd + d \omega_1 - c \omega_2. \nonumber
\end{eqnarray}
In what follows, for any fixed field $k$ and $(A, B) \in V_k$ with entries $A = (a_{ij})$ and $B = (b_{ij})$,
$(\QA{k}{A}{B}, \langle 1, \xi_{1}, \xi_{2}, \xi_{3} \rangle)$
and $(\RA{k}{A}{B}, \langle 1, \omega_{1}, \omega_{2} \rangle)$ denote the quartic $k$-algebra
and its resolvent cubic algebra with their bases, assigned to $(A, B)$ by the equations (\ref{eq: multiplicative structure of Q}), (\ref{eq: normalization}) and (\ref{eq: multiplicative structure of R}).
\begin{lem}\label{lem:generator of Q otimes RationalField}
For any $h_0 \in k$ and $h := (h_{1}, h_{2}, h_{3}) \in k^3$,
if we put $\alpha := h_0 + \sum_{j=1}^3 h_j \xi_{j} \in \QA{k}{A}{B}$, the following holds:
\begin{eqnarray*}
1, \alpha, \alpha^2, \alpha^3 \textit{ is a basis of } \QA{k}{A}{B} \text{ over } k
\Longleftrightarrow
\det(B(h) A - A(h) B) \neq 0.
\end{eqnarray*}
When $(A, B)$ is non-singular and anisotropic over $k$,
$\QA{k}{A}{B}$ contains such an $\alpha$
if and only if $A, B$ are linearly independent over $k$.
\end{lem}
\begin{proof}
For any element $\alpha \in \QA{k}{A}{B}$,
we denote the image of $\alpha$ by the natural epimorphism $\QA{k}{A}{B} \longrightarrow \QA{k}{A}{B} / k \cdot 1$ by $\overline{\alpha}$.
By (\ref{eq: formula for 4 det(B(x) A - A(x) B}), the following matrix $M$ satisfies $\det M = 4 \det(B(h) A - A(h) B)$:
\begin{eqnarray}
M
\begin{pmatrix}
\overline{\xi_1} \\ \overline{\xi_2} \\ \overline{\xi_3}
\end{pmatrix}
=
\begin{pmatrix}
\overline{\alpha} \\ \overline{\alpha^2} \\ \overline{\alpha^3}
\end{pmatrix}.
\end{eqnarray}
The first statement is obtained from this.
With regard to the second one,
the only-if part immediately follows from the fact that $\QA{k}{A}{B}$
is isomorphic to $k[x, y, z] / (x^2, y^2, z^2, xy, yz, xz)$, if $A, B$ are linearly dependent.
We now prove the if part.
If $\det(A x - B y) = 0$ does not have roots $[x : y] \in {\mathbb P}^1(k)$, then
$\det(B(h) A - A(h) B) = 0 \Leftrightarrow A(h) = B(h) = 0$.
None of $0 \neq h \in k^3$ satisfy this, owing to the anisotropy of $(A, B)$ over $k$.
If $\Pi \subset {\mathbb P}^1(k)$ is the set of all the roots of $\det(A x - B y) = 0$ in ${\mathbb P}^1(k)$,
then,
\begin{eqnarray*}\label{eq:condition det(B(h) A - A(h) B) = 0}
\det(B(h) A - A(h) B) = 0 \Longleftrightarrow u A (h) = v B (h) \text{ for some } [u : v] \in \Pi.
\end{eqnarray*}
Owing to $u A - v B \neq 0$ for any $[u : v] \in {\mathbb P}^1(k)$,
if the cardinality of $\Pi$ equals 1,
some $h \in k^3$ does not satisfy $u A (h) = v B (h)$.
Otherwise, all roots of $\det(A x - B y) = 0$ belong to ${\mathbb P}^1(k)$,
hence, $(A, B)$ is simultaneously diagonalized over $k$ by Lemma \ref{lem: diagonalization over F}.
Thus, it is easily seen that this case is eliminated as well.
\end{proof}
In the following, for fixed $(A, B) \in V_k$, we will take $\alpha$ as in Lemma \ref{lem:generator of Q otimes RationalField}
and put $4 \det (A x - B y) := a x^3 + b x^2 y + c x y^2 + d y^3 = 0$.
Let ${\rm ch}_\alpha(x) := x^4 + a_{3} x^3 + a_{2} x^2 + a_{1} x + a_{0}$ be the characteristic polynomial of $\alpha \in \QA{k}{A}{B}$.
From (\ref{eq: multiplicative structure of Q}), these $a_i$ ($0 \leq i \leq 3$) are represented as a polynomial of $h_i$ ($0 \leq i \leq 3$) and the coefficients of the quadratic forms $A$, $B$ over $\IntegerRing$.
The resolvent polynomial of $f(x) := x^4 + a_{3} x^3 + a_{2} x^2 + a_{1} x + a_{0}$ is defined as
$f^{res}(x) := x^3 - a_{2} x^2 + (a_{1} a_{3} - 4 a_{0}) x + (4 a_{0} a_{2} - a_{1}^2 - a_{0} a_{3}^2)$.
It was pointed out in \cite{Bhargava2004} that if $Q_k(A, B)$ is a field over $k$, the image of $\alpha$ by the classical resolvent map equals $z + B(h) \omega_1 + A(h) \omega_2$ for some $z \in k$.
In general, it can be verified by direct calculation that the following equality holds as a polynomial of the components of $A$, $B$ and $h_i$.
\begin{eqnarray*}
{\rm ch}_\alpha^{res}(x)
&=& \prod_{i=1}^3 \left( x - z + a B(h) \frac{u^{(i)}}{v^{(i)}} - d A(h) \frac{v^{(i)}}{u^{(i)}} \right) \nonumber \\
&=& (x-z)^3 + ( c A(h) - b B(h) ) (x-z)^2 + \{ b d A(h)^2 + (3 a d - b c) A(h) B(h) + a c B(h)^2 \} (x-z) \nonumber \\
& & + a d^2 A(h)^3 - (b^2 d - 2 a c d) A(h)^2 B(h) + (a c^2 - 2 a b d) A(h) B(h)^2 - a^2 d B(h)^3, \label{eq: ch_beta^{res}(x)}
\end{eqnarray*}
where $[u^{(i)} : v^{(i)}] \in {\mathbb P}^1(\bar{k})$ ($1 \leq i \leq 3$)
are the roots of $4 \det (A x - B y) = 0$.
The $z$ equals $(a_2 + c A(h) - b B(h)) / 3$, which is obtained by comparing the coefficients of $x^2$.
\begin{prop}\label{prop: monogenic ring}
We assume that $(A, B) \in V_k$ is linearly independent, non-singular and anisotropic over $k$.
As in Lemma \ref{lem:generator of Q otimes RationalField}, $h \in k^3$ with $\det( B(h) A - A(h) B) \neq 0$ and $\alpha \in \QA{k}{A}{B}$ are fixed.
With regard to this $\alpha$ with the characteristic polynomial ${\rm ch}_\alpha(x) = x^4 + a_{3} x^3 + a_{2} x^2 + a_{1} x + a_{0}$, we
$\Lambda := \langle 1, \alpha, \alpha^2 + a_{3} \alpha + a_{2}, \alpha^3 + a_{3} \alpha^2 + a_{2} \alpha + a_{1} \rangle$
gives a basis of $k[\alpha]$.
Let $(R, \Phi)$ be the three-dimensional $k$-algebra corresponding to the cubic form ${\rm ch}_\alpha^{res}(x + a_{2}/3)$.
Then, the following $(\tilde{A}, \tilde{B})$
corresponds to the pair of $(k[\alpha], \Lambda)$ and $(R, \Phi)$ under the Bhargava correspondence:
\begin{eqnarray}\label{eq: definition of tilde{A} and tilde{B}}
\tilde{A} =
\begin{pmatrix}
0 & 0 & 1/2 \\
0 & -1 & 0 \\
1/2 & 0 & 0
\end{pmatrix},\
\tilde{B} = -
\begin{pmatrix}
1 & a_{3}/2 & a_{2}/6 \\
a_{3}/2 & 2a_{2}/3 & a_{1}/2 \\
a_{2}/6 & a_{1}/2 & a_{0}
\end{pmatrix}.
\end{eqnarray}
\end{prop}
\begin{proof}
It is straightforward to verify that $\Lambda$ satisfies (\ref{eq: normalization}), and $(\tilde{A}, \tilde{B})$ corresponds to the pair of $(k[\alpha], \Lambda)$ and $(R, \Phi)$ by direct computation.
\end{proof}
In particular, we have $4 \det \left( \tilde{A} x - \tilde{B} \right) = {\rm ch}_\alpha^{res}(x + a_2/3)$.
We shall construct $(W, V) \in G_k$
such that
$(\tilde{A}, \tilde{B}) = (W, V) \cdot (A, B)$ holds;
let $W$ be the matrix uniquely determined by
\begin{eqnarray*}\label{eq: definition of W}
W
\begin{pmatrix}
\overline{\xi_1} \\ \overline{\xi_2} \\ \overline{\xi_3}
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 & 0 \\
a_{3} & 1 & 0\\
a_{2} & a_{3} & 1
\end{pmatrix}
\begin{pmatrix}
\overline{\alpha} \\ \overline{\alpha^2} \\ \overline{\alpha^3}
\end{pmatrix}.
\end{eqnarray*}
The determinant is given by $\det W = 4 \det(B(h) A - A(h) B) \neq 0$.
Using $q_A := A(h)$ and $q_B := B(h)$,
we define $V \in GL_2(k)$ by
\begin{eqnarray*}\label{eq: definition of V}
V
&:=& \det W^{-1}
\begin{pmatrix}
1 & 0 \\
(c q_A - b q_B)/3 & 1
\end{pmatrix}
\begin{pmatrix}
q_B & -q_A \\
-c q_A q_B - d q_A^2 & -a q_B^2 - b q_A q_B
\end{pmatrix}.
\end{eqnarray*}
This $V$ is derived from the following equality:
$$a q_B\frac{u^{(i)}}{v^{(i)}} - d q_A \frac{v^{(i)}}{u^{(i)}} = \frac{ ((u^{(i)} / v^{(i)}) (a q_B^2 + b q_A q_B) + c q_A q_B + d q_A^2 }{ -(u^{(i)}/v^{(i)}) q_A + q_B }.$$
$(\tilde{A}, \tilde{B}) = (W, V) \cdot (A, B)$ is proved by direct computation.
As a result, the following corollary is immediately obtained:
\begin{cor}\label{cor: (A, B) is isotropic over k}
We assume that $(A, B)$ and $\alpha$ are taken as in Proposition \ref{prop: monogenic ring}.
For any field $k \subset K \subset \bar{k}$,
$(A, B)$ is isotropic over $K$
if and only if ${\rm ch}_\alpha(x) = 0$ has a root in $K$.
\end{cor}
\begin{proof}
Using the action of $G_k$,
we may replace $(A, B)$ with $(\tilde{A}, \tilde{B})$ in (\ref{eq: definition of tilde{A} and tilde{B}}).
This is proved as follows:
\begin{eqnarray*}
(\tilde{A}, \tilde{B})({\mathbf x}) = 0 \text{ for some } 0 \neq {\mathbf x} \in K^3
&\Leftrightarrow& (\tilde{A}, \tilde{B})(s^2, st, t^2) = 0 \text{ for some } [s : t] \in {\mathbb P}^1(K) \\
&\Leftrightarrow& {\rm ch}_\alpha(u) = 0 \text{ for some } u \in K.
\end{eqnarray*}
\end{proof}
\section{Automorphisms of cubic polynomials}
\label{Automorphisms of cubic polynomial}
The purpose of this section is to prove the lemma \ref{lem:automorphism of (A, B) with V}
which deals with the case of $q_k(A, B) = q_k((I, V) \cdot (A, B))$.
The lemma is used in the proof of Theorem \ref{thm:main result over RationalField}.
The assumption ${\rm char} k \neq 2, 3$ is also used herein so as to simplify the discussion.
\begin{lem}\label{lem:automorphism of cubic polynomial}
If a cubic polynomial $f (x, y) = a x^3 + b x^2 y + c x y^2 + d y^3 \in k[x, y]$ has
no multiple roots and satisfies $(\det V)^{-1} f ( (x, y) V ) = u f(x, y)$ for some $V \in GL_2(k)$ and $u \in k$, then
$V^n = u^n I$ for some $n = 1, 2, 3$.
\end{lem}
\begin{proof}
Let $[p_i : q_i] \in {\mathbb P}^1(\bar{k})$ ($i = 1, 2, 3$) be the roots of $f(x, y) = 0$.
$V \in GL_2(k)$ exchanges $[p_i : q_i]$. Hence, $V^n = v I$ for some $n = 1, 2, 3$ and $v \in k$.
From
$(\det V^n)^{-1} f((x, y)V^n) = v f(x, y) = u^n f(x, y)$,
$v=u^n$ is obtained.
\end{proof}
In what follows, $p_i, q_i$ that satisfy $f(x, y) = \prod_{i=1}^3 (q_i x - p_i y)$ are fixed.
The following examples list all the cases in which $V \neq I$.
\begin{exa}[Case $V^2 = u^2 I$]
It may be assumed that $V$ swaps $[p_1 : q_1]$ and $[p_2 : q_2]$, and fixes $[p_3 : q_3]$.
For some $0 \neq r_1, r_2 \in \bar{k}$,
\begin{eqnarray}
V
&=& u
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2
\end{pmatrix}^{-1}
\begin{pmatrix}
0 & r_2 \\
r_1 & 0
\end{pmatrix}
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2
\end{pmatrix}.
\end{eqnarray}
From $V^2 = u^2 I$, $r_1 r_2 = 1$ is obtained.
Since
we have $(\det V)^{-1} f((x, y)V) = - u^{-2} f((x, y)V) = u f(x, y)$,
$V \begin{pmatrix} q_3 \\ -p_3 \end{pmatrix} = -u \begin{pmatrix} q_3 \\ -p_3 \end{pmatrix}$ is obtained.
Consequently, $r_1 = r_2^{-1} = -(q_2 p_3 - p_2 q_3) / (q_1 p_3 - p_1 q_3)$.
Thus,
\begin{eqnarray*}
& & \hspace{-10mm}
-(q_2 p_3 - p_2 q_3) (q_1 p_3 - p_1 q_3) V \nonumber \\
&=& \frac{u}{ p_1 q_2 - p_2 q_1 }
\begin{pmatrix}
q_2 & -q_1 \\
-p_2 & p_1
\end{pmatrix}
\begin{pmatrix}
0 & (q_1 p_3 - p_1 q_3)^2 \\
(q_2 p_3 - p_2 q_3)^2 & 0
\end{pmatrix}
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2
\end{pmatrix}
\nonumber \\
&=& u
\begin{pmatrix}
p_1 p_2 q_3^2 - p_3^2 q_1 q_2
& (p_1 q_2 + p_2 q_1) q_3^2 - 2 p_3 q_1 q_2 q_3 \\
(p_1 q_2 + p_2 q_1) p_3^2 - 2 p_1 p_2 p_3 q_3
& -p_1 p_2 q_3^2 + p_3^2 q_1 q_2
\end{pmatrix}.
\end{eqnarray*}
Therefore, if we put
$C :=
p_3 / \frac{\partial f}{ \partial y }(p_3, q_3) =
-q_3 / \frac{\partial f}{ \partial x }(p_3, q_3)$,
\begin{eqnarray}\label{eq:definition of V}
V
&=&
u C
\begin{pmatrix}
b p_3 + c q_3
& -3 a p_3 - b q_3 \\
c p_3 + 3 d q_3
& -b p_3 - c q_3
\end{pmatrix}.
\end{eqnarray}
If $[p_3 : q_3] \notin {\mathbb P}^1(k)$,
$V \in GL_2(k)$ implies that
$3 a c = b^2$ and $3 b d = c^2$, then $f(x, y) = (3b c)^{-1} (b x + c y)^3$.
Since $f$ is assumed to have no multiple roots, we obtain $[p_3 : q_3] \in {\mathbb P}^1(k)$.
\end{exa}
\begin{exa}[Case $V^3 = u^3 I$]
It may be assumed that
$V$ maps $[p_i : q_i]$ to $[p_j : q_j]$ for any $(i, j) = (1, 2), (2, 3), (3, 1)$.
Since $V^3 = u^3 I$, for some $c_i \in \bar{k}$ with $c_1 c_2 c_3 = u^3$, we have
\begin{eqnarray}
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2 \\
p_3 & q_3 \\
\end{pmatrix}
V
=
\begin{pmatrix}
0 & c_2 & 0 \\
0 & 0 & c_3 \\
c_1 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2 \\
p_3 & q_3 \\
\end{pmatrix}.
\end{eqnarray}
We have $\det V = c_{i_2} c_{i_3} (p_{i_2} q_{i_3} - p_{i_3} q_{i_2}) / (p_{i_1} q_{i_2} - p_{i_2} q_{i_1})$
for any $(i_1, i_2, i_3) = (1, 2, 3), (2, 3, 1), (3, 1, 2)$. Hence,
\begin{eqnarray}
\begin{pmatrix}
c_1 \\
c_2
\end{pmatrix}
=
c_3 \frac{p_3 q_1 - p_1 q_3}{p_1 q_2 - p_2 q_1}
\begin{pmatrix}
\frac{p_2 q_3 - p_3 q_2}{p_1 q_2 - p_2 q_1} \\
\frac{p_3 q_1 - p_1 q_3}{p_2 q_3 - p_3 q_2}
\end{pmatrix}.
\end{eqnarray}
From $c_1 c_2 c_3 = u^3$,
some $\zeta \in \bar{k}$ with $\zeta^3 = 1$
satisfies $c_3 = u \zeta (p_1 q_2 - p_2 q_1)/(p_3 q_1 - p_1 q_3)$.
Thus,
\begin{eqnarray}
V
&=& u \zeta
\begin{pmatrix}
p_1 & q_1 \\
p_2 & q_2
\end{pmatrix}^{-1}
\begin{pmatrix}
\frac{p_3 q_1 - p_1 q_3}{p_2 q_3 - p_3 q_2} & 0 \\
0 & \frac{p_1 q_2 - p_2 q_1}{p_3 q_1 - p_1 q_3}
\end{pmatrix}
\begin{pmatrix}
p_2 & q_2 \\
p_3 & q_3
\end{pmatrix}.
\end{eqnarray}
From $\det V = u^2 \zeta^2 \in k$
$\zeta \in k$ is obtained.
Hence, $u$ may be replaced with $u \zeta^{-1}$.
If we put $\Delta = (p_1 q_2 - p_2 q_1)(p_2 q_3 - p_3 q_2)(p_3 q_1 - p_1 q_3)$,
we have ${\rm Disc}(f(x, 1)) = \Delta^2$ and
\begin{eqnarray*}
\Delta &=& -2 (p_1^2 p_2 q_2 q_3^2 + p_2^2 p_3 q_1^2 q_3 + p_1 p_3^2 q_1 q_2^2) - b c + 3 a d \\
&=& 2 (p_1 p_2^2 q_3^2 q_1 + p_3 p_1^2 q_2^2 q_3 + p_2 p_3^2 q_1^2 q_2) + b c - 3 a d, \\
V
&=&
\frac{u}{\Delta}
\begin{pmatrix}
(9 a d - b c - \Delta)/2
&
b^2 - 3 a c \\
-c^2 + 3 b d
&
-(9 a d - b c + \Delta)/2
\end{pmatrix}.
\end{eqnarray*}
Since $V \in GL_2(k)$, we have
$\Delta \in k$.
Therefore, $f(x, y) = 0$ completely splits over $k$, or
$k[x]/(f(x, 1))$ is a Galois cubic field over $k$.
\end{exa}
In the following lemma, $4 \det (A x - B y)$ is denoted by $f_{det}(x, y)$.
\begin{lem}\label{lem:automorphism of (A, B) with V}
We assume that $(A, B) \in V_k$ with ${\rm Disc}(A, B) \neq 0$, is linearly independent, non-singular and anisotropic over $k$.
We further assume that there is a matrix $\pm I \neq V \in GL_2(k)$ and $u \in k$
such that $({\det V})^{-1} f_{det}((x, -y) V) = u f_{det}(x, -y)$ and
$q_k(A, B) = q_k((I, V) \cdot (A, B))$.
Then,
there exists
$W \in GL_3(k)$ such that $(W, -u^{-1} V) \cdot (A, B) = (A, B)$.
\end{lem}
\begin{proof}
Using the action of $GL_2(k)$,
we may assume $\det A \neq 0$.
Let $\alpha_1, \alpha_2, \alpha_3 \in \bar{k}$ be the roots of $\det(A x - B) = 0$, and $k_i$ be the field $k(\alpha_i)$.
We fix $0 \neq v_i \in k_i^3$, $d_i \in k_i$ ($1 \leq i \leq 3$)
and $w = \tr{(v_1\ v_2\ v_3)}$
as in the first paragraph in the proof of Lemma \ref{lem: equivalence conditions on isotropy}.
$(A, B)$ is transformed as in (\ref{eq: transform of (A,B)}).
There are the following two cases:
\begin{itemize}
\item ($V^2 = u^2 I, V \neq I$)
It may be assumed that $V$ exhanges $[\alpha_1 : -1]$ and $[\alpha_2 : -1]$.
We then have $k_1 = k_2$ and $\alpha_3 \in k$. Furthermore,
\begin{eqnarray}
V
&=& u
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}^{-1}
\begin{pmatrix}
0 & -\frac{\alpha_1 - \alpha_3}{\alpha_2 - \alpha_3} \\
-\frac{\alpha_2 - \alpha_3}{\alpha_1 - \alpha_3} & 0
\end{pmatrix}
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}.
\end{eqnarray}
Hence,
\begin{footnotesize}
\begin{eqnarray*}
& & \hspace{-10mm}
\left( w,
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix} V
\right)
\cdot
(A, B) \nonumber \\
&=&
\left( I, u
\begin{pmatrix}
0 & -\frac{\alpha_1 - \alpha_3}{\alpha_2 - \alpha_3} \\
-\frac{\alpha_2 - \alpha_3}{\alpha_1 - \alpha_3} & 0
\end{pmatrix}
\right)
\cdot
\left(
\begin{pmatrix}
0 & 0 & 0 \\
0 & d_2 (\alpha_1 - \alpha_2) & 0 \\
0 & 0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix},
\begin{pmatrix}
d_1 (\alpha_2 - \alpha_1) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right) \\
&=&
\left(
-u
\begin{pmatrix}
-\frac{d_1(\alpha_1 - \alpha_3)}{d_2(\alpha_2 - \alpha_3)} \cdot d_2 (\alpha_1 - \alpha_2) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix},
-u
\begin{pmatrix}
0 & 0 & 0 \\
0 & -\frac{d_2(\alpha_2 - \alpha_3)}{d_1(\alpha_1 - \alpha_3)} \cdot d_1 (\alpha_2 - \alpha_1) & 0 \\
0 & 0 & d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right).
\end{eqnarray*}
\end{footnotesize}
Owing to $q_k(A, B) = q_k((I, V) \cdot (A, B))$,
for any primes ${\mathfrak p}$ that completely splits in $k_1 = k_2$, we must have
\begin{eqnarray*}
q_{k_{\mathfrak p}}\left(
\begin{pmatrix}
d_2 (\alpha_1 - \alpha_2) & 0 \\
0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix}
\right)
&=& q_{k_{\mathfrak p}} \left( u
\begin{pmatrix}
\frac{d_1(\alpha_1 - \alpha_3)}{d_2(\alpha_2 - \alpha_3)} \cdot d_2 (\alpha_1 - \alpha_2) & 0 \\
0 & -d_3 (\alpha_1 - \alpha_3)
\end{pmatrix}
\right), \\
q_{k_{\mathfrak p}}\left(
\begin{pmatrix}
d_1 (\alpha_2 - \alpha_1) & 0 \\
0 & d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right)
&=& q_{k_{\mathfrak p}}\left(
u
\begin{pmatrix}
\frac{d_2(\alpha_2 - \alpha_3)}{d_1(\alpha_1 - \alpha_3)} \cdot d_1 (\alpha_2 - \alpha_1) & 0 \\
0 & -d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right).
\end{eqnarray*}
This implies that there exists $\beta \in k_i$
such that $- d_1(\alpha_1 - \alpha_3) / d_2(\alpha_2 - \alpha_3) = \beta^2$
and $\left( W,
-u^{-1} V
\right)
\cdot
(A, B)
=
(A, B)$ if we put:
\begin{eqnarray*}
W = w^{-1}
\begin{pmatrix}
0 & \beta & 0 \\
\pm 1/\beta & 0 & 0 \\
0 & 0 & 1
\end{pmatrix} w.
\end{eqnarray*}
If $k_1 = k_2 = k$, then $W \in GL_3(k)$ has the required property.
Otherwise, $k_1 = k_2$ is quadratic over $k$.
If the signature of $\pm 1/\beta$ is chosen so that $\beta$ and $\pm 1/\beta$ are conjugate over $k$,
$W \in GL_3(k)$ is obtained.
\item ($V^3 = u^3 I, V \neq I$)
In this case, $k_1 = k_2 = k_3 = k$ or a Galois cubic field over $k$. As shown in the above example,
\begin{eqnarray*}
V
&=& u
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}^{-1}
\begin{pmatrix}
\frac{\alpha_3 - \alpha_1}{\alpha_2 - \alpha_3} & 0 \\
0 & \frac{\alpha_1 - \alpha_2}{\alpha_3 - \alpha_1}
\end{pmatrix}
\begin{pmatrix}
\alpha_2 & -1 \\
\alpha_3 & -1
\end{pmatrix}, \\
&=& u
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}^{-1}
\begin{pmatrix}
0 & \frac{\alpha_3 - \alpha_1}{\alpha_2 - \alpha_3} \\
-\frac{\alpha_2 - \alpha_3}{\alpha_3 - \alpha_1} & -1
\end{pmatrix}
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix}.
\end{eqnarray*}
Hence,
\begin{small}
\begin{eqnarray*}
& & \hspace{-10mm}
\left( w,
\begin{pmatrix}
\alpha_1 & -1 \\
\alpha_2 & -1
\end{pmatrix} V
\right)
\cdot
(A, B) \nonumber \\
&=&
\left( I, u
\begin{pmatrix}
0 & \frac{\alpha_3 - \alpha_1}{\alpha_2 - \alpha_3} \\
-\frac{\alpha_2 - \alpha_3}{\alpha_3 - \alpha_1} & -1
\end{pmatrix}
\right)
\cdot
\left(
\begin{pmatrix}
0 & 0 & 0 \\
0 & d_2 (\alpha_1 - \alpha_2) & 0 \\
0 & 0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix},
\begin{pmatrix}
d_1 (\alpha_2 - \alpha_1) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & d_3 (\alpha_2 - \alpha_3)
\end{pmatrix}
\right) \\
&=&
\left( -u
\begin{pmatrix}
\frac{d_1(\alpha_3 - \alpha_1)}{d_2(\alpha_2 - \alpha_3)} \cdot d_2 (\alpha_1 - \alpha_2) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & d_3 (\alpha_1 - \alpha_3)
\end{pmatrix},
-u
\begin{pmatrix}
d_1 (\alpha_2 - \alpha_1) & 0 & 0 \\
0 & \frac{d_2(\alpha_1 - \alpha_2)}{d_3(\alpha_3 - \alpha_1)} \cdot d_3 (\alpha_2 - \alpha_3) & 0 \\
0 & 0 & 0
\end{pmatrix}
\right).
\end{eqnarray*}
\end{small}
Then, there exits $\beta_{i_1} \in k_1 = k_2 = k_3$
such that $d_{i_2} (\alpha_{i_1} - \alpha_{i_2}) / d_{i_3} (\alpha_{i_1} - \alpha_{i_3}) = \beta_{i_1}^2$
for every $(i_1, i_2, i_3) = (1, 2, 3), (2, 3, 1), (3, 1, 2)$.
$\left( W,
-u^{-1} V
\right)
\cdot
(A, B)
=
(A, B)$ and $W \in GL_3(k)$, if we put:
\begin{eqnarray*}
W = w^{-1}
\begin{pmatrix}
0 & \beta_3 & 0 \\
0 & 0 & \beta_1 \\
\beta_2 & 0 & 0
\end{pmatrix}
w.
\end{eqnarray*}
Consequently, the lemma is proved.
\end{itemize}
\end{proof}
\section{Proof of Theorem \ref{thm:main result over RationalField}}
In what follows, we assume that $k = \RationalField$.
In order to prove the theorem, we will first show that $q_\RationalField(A_1, B_1) = q_\RationalField(A_2, B_2)$
implies that either of the determinants is a constant multiple of the other.
Hence, $(A_i, B_i)$ ($i = 1, 2$) are simultaneously transformed into their canonical forms,
defined in Section \ref{A canonical form for elements of $V_k$}.
\begin{prop}\label{prop:same det(Ax+By)}
If $(A_i, B_i) \in V_k$ ($i = 1, 2$) satisfy the conditions (\ref{item: assumption (a)}), (b') and (c') of Theorem \ref{thm:main result over RationalField},
there are coprime integers $r_1, r_2$ such that $r_1^{-1} \det (A_1 x - B_1 y) = r_2^{-1} \det (A_2 x - B_2 y)$.
\end{prop}
\begin{proof}
Using the action of $GL_2(k)$ on $V_k$, we may assume that $\det A_1 \neq 0$.
If ${\rm Disc}(A_1, B_1) = 0$,
let $\alpha \in k$ be the multiple root of
$\det(A_1 x - B_1) = 0$.
In this case, $A_1 \alpha - B_1$ has rank $\leq 1$.
Owing to $q_k(A_1 \alpha - B_1) = q_k(A_2 \alpha - B_2)$, $A_2 \alpha - B_2$
has the same rank. Therefore, $\alpha$ is a multiple root of $\det(A_2 x - B_2) = 0$.
Let $\beta \neq \alpha$ be another root of $\det(A_1 x - B_1) = 0$.
Since $(A_1, B_1)$ is anisotropic over $k$,
$C \in {\rm Sym}^2 (k^2)^*$ with $A_1 \beta - B_1 \sim_{k} C \perp [ 0 ]$ must be anisotropic over $k$.
Hence, $A_2 \beta - B_2$ has rank 2, and $\det(A_2 \beta - B_2) = 0$.
Thus, the proposition is proved if ${\rm Disc}(A_1, B_1)=0$.
The same holds if ${\rm Disc}(A_2, B_2)=0$.
We next assume that ${\rm Disc}(A_i, B_i) \neq 0$ ($i = 1, 2$).
Let $\alpha \in \bar{k}$ be the roots of $\det(A_1 x - B_1) = 0$
and $K$ be the Galois closure of $k(\alpha)$ over $k$.
If $C \in {\rm Sym}^2 (K^2)^*$ with $A_1 \alpha - B_1 \sim_{K} C \perp [ 0 ]$ is anisotropic over $K$, then
there are a finite prime of $k_{\mathfrak p}$ and an embedding $\iota: K \hookrightarrow k_{\mathfrak p}$ such that
$\iota(C)$ is anisotropic over $k_{\mathfrak p}$.
In this case, ($C$3) immediately above the Corollary \ref{cor: anisotropic iff anisotropic} cannot happen,
hence $\det(A_2 \alpha - B_2) = 0$ is proved
by $q_{k_{\mathfrak p}}(A_2 \iota(\alpha) - B_2) = q_{k_{\mathfrak p}}(A_1 \iota(\alpha) - B_1)
= \{ 0 \} \cup q_{k_{\mathfrak p}}(\iota(C))$.
Thus, by Lemma \ref{lem: equivalence conditions on isotropy},
the equations $\det(A_i x - B_i) = 0$ ($i = 1, 2$)
have at least two common roots $\alpha \neq \beta$.
In addition, if there exist distinct $\gamma_1 \neq \gamma_2$ such that $\det(A_i \gamma_i - B_i) = 0$ ($i = 1, 2$),
both $\gamma_i$ belong to $k$ and both $A_i \gamma_i - B_i$ are isotropic over $k$.
Thus, by using the action of $G_k$,
we may assume the following:
\begin{eqnarray}
(A_1, B_1) &=&
\left(
\begin{pmatrix}
a_{11} & a_{12} & 0 \\
a_{12} & a_{22} & 0 \\
0 & 0 & a_{33}
\end{pmatrix},
\begin{pmatrix}
0 & c_1 & 0 \\
c_1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\right), \\
(A_2, B_2) &=&
\left(
\begin{pmatrix}
0 & c_2 & 0 \\
c_2 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
b_{11} & b_{12} & 0 \\
b_{12} & b_{22} & 0 \\
0 & 0 & b_{33}
\end{pmatrix}
\right).
\end{eqnarray}
Furthermore, by using the action of $G_k$,
we may assume that $a_{11} = b_{11} = 1$ and $c_1 = c_2 = -1/2$.
For the anisotropy of $(A_i, B_i)$ over $k$,
$-a_{33}, -b_{33}, -a_{22} a_{33}, -b_{22} b_{33} \notin (k^\times)^2$ is required.
Under all the assumptions, if we put $g(x) := (x - \alpha)(x - \beta) \in k[x]$, then
\begin{eqnarray*}
4 \det ( A_1 x - B_1 ) &=& -\{ 4 (a_{12}^2 - a_{22}) x^2 + 4 x a_{12} + 1 \} a_{33} x = -4 (a_{12}^2 - a_{22}) a_{33} x g(x), \\
4 \det ( A_2 x - B_2 ) &=& \{ x^2 + 4 x b_{12} + 4 (b_{12}^2 - b_{22}) \} b_{33} = b_{33} g(x).
\end{eqnarray*}
In particular, we have $4 (a_{12}^2 - a_{22}) b_{12} = a_{12}$ and $4 (b_{12}^2 - b_{22}) a_{12} = b_{12}$.
From this, we obtain
$(a_{22} - a_{12}^2)/(b_{22} - b_{12}^2) = a_{22} / b_{22} = a_{12}^2 / b_{12}^2 \in (k^\times)^2$.
Furthermore, for any $0 \neq x \in k$ we have
\begin{eqnarray}
A_1 x - B_1
&\sim_k& \begin{pmatrix}
x & 0 & 0 \\
0 & -4 (a_{12}^2 - a_{22}) g(x) / x & 0 \\
0 & 0 & a_{33} x
\end{pmatrix}, \\
A_2 x - B_2
&\sim_k&
\begin{pmatrix}
-1 & 0 & 0 \\
0 & g(x) & 0 \\
0 & 0 & -b_{33}
\end{pmatrix}.
\end{eqnarray}
Let $P_0$ be the set of odd primes $p$ of $k = \RationalField$
such that $p$ completely splits in $k[x] / (g(x))$ (\IE
$a_{22}, b_{22} \in (k_{p}^\times)^2$).
For a fixed ${p} \in P_0$,
we will denote the roots of $g(x) = 0$ in $k_{p}$ by $\alpha_{p}$ and $\beta_{p}$.
Let $P \subset P_0$ be the subset consisting of all $p \in P$ with
$4 b_{12} \in \IntegerRing_p$ and
$a_{33}$, $b_{33}$, $4 (a_{12}^2 - a_{22}), 4(b_{12}^2 - b_{22}) \in \IntegerRing_p^\times$.
For any $p \in P$, $\alpha_p$ and $\beta_p$ belong to $\IntegerRing_p^\times$.
By setting $x$ to an element of $\RationalField_p$ close to $\alpha_{p}$ (\textit{resp.} $\beta_{p}$),
$g(x) \in p \IntegerRing^\times$ and $4 (a_{12}^2 - a_{22}) / x \in \alpha_{p}^{-2} \beta_p^{-1} \in (\IntegerRing_p^\times)^2$
(\textit{resp.} $\alpha_{p}^{-1} \beta_p^{-2} (\IntegerRing_p^\times)^2$) can be assumed.
$\left( \frac{-a_{33} }{p} \right) = \left( \frac{-b_{33}}{p} \right)$ is then obtained from $q_{k_p}(A_1 x - B_1) = q_{k_p}(A_2 x - B_2)$. In addition, either of the following holds:
\begin{enumerate}[(i)]
\item $\left( \frac{-a_{33} }{p} \right) = \left( \frac{-b_{33}}{p} \right) = 1$.
\item $\left( \frac{ -\alpha_p }{p} \right) = \left( \frac{-\beta_p }{p} \right) = 1$.
\end{enumerate}
Let $K \subset \bar{k}$ be the extension of $k$ obtained by attaching $\sqrt{b_{22}}$ to
$\RationalField$, \IE the splitting field of $g(x)$ over $\RationalField$.
If $K(\sqrt{-b_{33}})$ is denoted by $F_1$, we have $F_1 \supsetneq K$, since $-b_{33}, -b_{22} b_{33} \notin (k^\times)^2$.
Let $F_2$ be the extensions of $K$ that are obtained by attaching the roots of $g(-x^2) = 0$ to $K$.
$F_2 = K$ is proved as follows;
if $F_2 \supsetneq K$, let $F_3$ be the composition $F_1 F_2$.
Let $Q_i$ ($i = 1, 2, 3$) be the set of all primes of $K$ that completely splits in $F_i$.
Since the extension $F_i / K$ is Galois,
the Kronecker density
$d_{i} := \lim_{s \downarrow 1} \frac{ \sum_{p \in Q_i} p^{-s} }{ \log (1/(s - 1)) }$
equals $1 / [F_i : K]$ (Theorem 8.41 (2), \cite{Kato2011}).
However, from $p \in P \Rightarrow$ (i) or (ii), we obtain $d_1 + d_2 \geq 1$.
Owing to $d_1 = 1/2$, in order that $d_3 > 0$, $d_2 = 1$, \IE $K = F_2$ is required.
This implies $-\alpha, -\beta \in (K^\times)^2$, and $b_{12}^2 - b_{22} = \alpha \beta/4 \in (k^\times)^2$.
Because $(a_{12}^2 - a_{22})/(b_{12}^2 - b_{22}) \in (k^\times)^2$, $a_{12}^2 - a_{22}$ is also a square in $k^\times$.
Let $d \in k^\times$ be the element satisfying $d^2 = a_{12}^2 - a_{22}$.
Then, $A_1({\mathbf x}) = 0$
if and only if ${\mathbf x} = (x_1, x_2, x_3) \in k$
satisfies $(x_1 + (a_{12} + d) x_2)(x_1 + (a_{12} - d) x_2) = -a_{33} x_3^2$.
From $\left( \frac{-a_{33} }{p} \right) = \left( \frac{-b_{33}}{p} \right)$ for any $p \in P$,
we also have $a_{33} / b_{33} \in (K^\times)^2$, hence,
$a_{33} / b_{33} \in (k^\times)^2$ or $a_{33} / b_{22} b_{33} \in (k^\times)^2$.
Consequently, if $T_{1, F}, T_{2, F}$ are defined as follows,
$T_{1, k} = T_{2, k}$ is obtained by considering the representations $(0, *)$:
\begin{itemize}
\item $T_{1, F} := \left\{ (x^2 + a_{33} y^2) ( (a_{12} - d) x^2 + (a_{12} + d) a_{33} y^2 ) z^2 :
\begin{matrix}
x, y, z \in F,\\ (x, y) \neq 0, z \neq 0
\end{matrix}
\right\}$,
\item $T_{2, F} := q_{F}( x^2 + b_{33} y^2 ) \cup q_{F}(b_{22} x^2 + b_{33} y^2)$.
\end{itemize}
If the binary quadratic forms are defined by $g_1(x, y) = x^2 + a_{33} y^2, g_2(x, y) = (a_{12} - d) x^2 + (a_{12} + d) a_{33} y^2$,
\begin{eqnarray*}
Q_1 \in q_{k_v}(g_1) & \Longleftrightarrow & (Q_1, -a_{33})_v = 1, \\
Q_2 \in q_{k_v}(g_2) & \Longleftrightarrow & (Q_2, -a_{22}a_{33})_v = (a_{12} - d, (a_{12} + d) a_{33})_v,
\end{eqnarray*}
where $v$ is any primes of $k$, and $(, )_v: k_v^\times \times k_v^\times \mapsto \{ \pm 1 \}$ is the Hilbert symbol.
Thus, for any $Q_1 \in q_{k_v}(g_1)$, $Q_2 \in q_{k_v}(g_2)$,
we have:
\begin{eqnarray*}
(Q_1 Q_2, -b_{33})_v
&=& (Q_1, -b_{33})_v (Q_2, -b_{33})_v \\
&=&
\begin{cases}
(Q_2, -a_{33})_v & \text{if } a_{33} / b_{33} \in (k_v^\times)^2, \\
(Q_1, -a_{22} a_{33})_v (a_{12} - d, (a_{12} + d) a_{33})_v & \text{if } a_{33} / b_{22} b_{33} \in (k_v^\times)^2,
\end{cases} \\
(Q_1 Q_2, -b_{22} b_{33})_v
&=& (Q_1, -b_{22} b_{33})_v (Q_2, -b_{22} b_{33})_v \\
&=&
\begin{cases}
(Q_1, -a_{22} a_{33})_v (a_{12} - d, (a_{12} + d) a_{33})_v & \text{if } a_{33} / b_{33} \in (k_v^\times)^2,\\
(Q_2, -a_{33})_v & \text{if } a_{33} / b_{22} b_{33} \in (k_v^\times)^2.
\end{cases}
\end{eqnarray*}
This implies
\begin{eqnarray*}
(Q_1 Q_2, -b_{33})_v = 1
&\Longleftrightarrow&
\begin{cases}
Q_2 \in q_{k_v}(g_1) & \text{if } a_{33} / b_{33} \in (k_v^\times)^2, \\
Q_1 \in q_{k_v}(g_2) & \text{if } a_{33} / b_{22} b_{33} \in (k_v^\times)^2,
\end{cases} \\
(Q_1 Q_2, -b_{22} b_{33})_v = (b_{22}, b_{33})_v
&\Longleftrightarrow&
\begin{cases}
Q_1 / a_{22} \in q_{k_v}(g_2) & \text{if } a_{33} / b_{33} \in (k_v^\times)^2,\\
Q_2 / a_{22} \in q_{k_v}(g_1) & \text{if } a_{33} / b_{22} b_{33} \in (k_v^\times)^2.
\end{cases}
\end{eqnarray*}
Considering that $q \in q_{k_v}(g_1) \Leftrightarrow a_{33} q \in q_{k_v}(g_1)$ and $q \in q_{k_v}(g_2) \Leftrightarrow a_{22} a_{33} q \in q_{k_v}(g_2)$, for any $Q_1 \in q_{k_v}(g_1)$, $Q_2 \in q_{k_v}(g_2)$,
$Q_1 Q_2$ belongs to $T_{2, k_v}$ if and only if the following holds:
\begin{enumerate}[(i)]
\item Case of $a_{33} / b_{33} \in (k_v^\times)^2$: $Q_1/a_{33}$ or $Q_2$ belongs to $q_{k_v}(g_1) \cap q_{k_v}(g_2)$.
\item Case of $a_{33} / b_{22} b_{33} \in (k_v^\times)^2$: $Q_1$ or $Q_2/a_{22}a_{33}$ belongs to $q_{k_v}(g_1) \cap q_{k_v}(g_2)$.
\end{enumerate}
If $a_{33} / b_{33} \in (k^\times)^2$, owing to $T_{1, k} = T_{2, k}$,
$q_{k_{v_1}}(g_1) \not\subset a_{33} q_{k_{v_1}}(g_2)$
and
$q_{k_{v_2}}(g_1) \not\supset q_{k_{v_2}}(g_2)$ cannot simultaneously hold for any primes $v_1 \neq v_2$,
because otherwise, by using the Chinese remainder theorem,
we may choose $(s, t) \in k^2$ with
$Q_1 := g_1(s, t) \in q_{k_{v_1}}(g_1) \setminus a_{33} q_{k_{v_1}}(g_2)$ and
$Q_2 := g_2(s, t) \in q_{k_{v_2}}(g_2) \setminus q_{k_{v_2}}(g_1)$.
This $Q_1 Q_2 \in T_{1, k}$ does not belong to $T_{2, k}$, hence a contradiction is obtained.
However, if $a_{22} \notin (k^\times)^2$,
there exist infinitely many primes $v$ such that
$g_i$ is isotropic over $k_v$ and $g_j$ is not for both $(i, j) = (1, 2), (2, 1)$.
Hence, $a_{22}, b_{22} \in (k^\times)^2$ is required, which can be proved similarly in case of $a_{33} / b_{22} b_{33} \in (k^\times)^2$.
Let $m \in \RationalField$ be the smallest positive number such that
$a_{12} = m \tilde{a}_{12}$ and $a_{22} = m^2 \tilde{a}_{22}$
holds for some $\tilde{a}_{12}, \tilde{a}_{22} \in \IntegerRing$.
Then,
$\tilde{d} = d / m$ satisfies $\tilde{d}^2 = \tilde{a}_{12}^2 - \tilde{a}_{22}$.
If an odd prime $p$ divides $\tilde{d}$, $p$ does not divide $\tilde{a}_{12}$ owing to the choice of $m$.
With regard to this $p$, $(a_{12}-d)/(a_{12}+d) = (\tilde{a}_{12}-\tilde{d})/(\tilde{a}_{12}+\tilde{d}) = 1 \mod p$, hence, any
elements of $T_{1, k_p}$
belong to the class $(a_{12} - d) (k_p^\times)^2$ in $k_p^\times / (k_p^\times)^2$.
However, this is impossible owing to $T_{1, k} = T_{2, k}$.
Similarly, it can be proved that $4$ cannot divide $\tilde{d}$.
Consequently, $\tilde{d}^2 = 1$ or $4$, which can happen only when $\tilde{a}_{22} = a_{22} = 0$, which is impossible because $(A_1, B_1)$ is anisotropic over $k$. Thus, the proposition is proved.
\end{proof}
We can now proceed to the proof of Theorem \ref{thm:main result over RationalField}.
As in the previous section, we denote the quartic and cubic $\RationalField$-algebras assigned to $(A_i, B_i)$ by
$(Q_\RationalField(A_i, B_i), \langle 1, \xi_{i,1}, \xi_{i,2}, \xi_{i,3} \rangle)$ and
$(R_\RationalField(A_i, B_i), \langle 1, \omega_{i,1}, \omega_{i,2} \rangle)$.
\begin{proof}[Proof of Theorem \ref{thm:main result over RationalField}]
We fix $(A_i, B_i) \in V_\RationalField$ ($i =1, 2$) as stated in Theorem \ref{thm:main result over RationalField}.
From Proposition \ref{prop:same det(Ax+By)},
we have $r_1^{-1} \det (A_1 x + B_1 y) = r_2^{-1} \det (A_2 x + B_2 y)$ for some coprime integers $r_1, r_2$.
In what follows, we denote $4 \det (A_i x - B_i y) / r_i$ by $f_{det}(x, y) := a x^3 + b x^2 y + c x y^2 + d y^3$ ($a, b, c, d \in \RationalField$).
From Lemma \ref{lem:generator of Q otimes RationalField},
some $h_1 := (h_{1,1}, h_{1,2}, h_{1,3}) \in \RationalField^3$ satisfies $f_{det}( B_1(h_1), A_1(h_1) ) \neq 0$.
From the assumption (c'),
there exists $h_2 := (h_{2,1}, h_{2,2}, h_{2,3}) \in \RationalField^3$ such that $(A_1, B_1)(h_1) = (A_2, B_2)(h_2)$, hence,
$f_{det}( B_2(h_2), A_2(h_2) ) \neq 0$.
Using
these $h_1$, $h_2$ and arbitrarily chosen $h_{1,0}, h_{2,0} \in \RationalField$,
we define $\alpha_i := h_{i,0} + \sum_{j=1}^3 h_{i,j} \xi_{i,j} \in Q_\RationalField(A_i, B_i)$.
If we put $q_A := A_1(h_1) = A_2(h_2)$ and $q_B := B_1(h_1) = B_2(h_2)$, then $\alpha_i$
has the characteristic polynomial ${\rm ch}_{\alpha_i}(x) := x^4 + a_{i,3} x^3 + a_{i,2} x^2 + a_{i,1} x + a_{i,0}$ as follows:
\begin{eqnarray*}
{\rm ch}_{\alpha_i}^{res} \left(x + \frac{ a_{i,2} + r_i (c q_A - b q_B) }{3} \right)
&=& x^3 + r_i (c q_A - b q_B) x^2 + r_i^2 \{ b d q_A^2 + (3 a d - b c) q_A q_B + a c q_B^2 \} x \nonumber \\
& & + r_i^3 \{ a d^2 q_A^3 - (b^2 d - 2 a c d) q_A^2 q_B + (a c^2 - 2 a b d) q_A q_B^2 - a^2 d q_B^3 \}.
\end{eqnarray*}
Hence,
\begin{eqnarray}\label{eq:relation between ch(beta_i)^{res}(x)}
r_1^{-3} {\rm ch}_{\alpha_1}^{res} \left( r_1 x + \frac{ a_{1,2} }{3} \right) = r_2^{-3} {\rm ch}_{\alpha_2}^{res} \left( r_2 x + \frac{ a_{2,2} }{3} \right).
\end{eqnarray}
Let $W_1$, $W_2$ and $V$ be the rational matrix determined by
\begin{eqnarray*}
W_i
\begin{pmatrix}
\overline{\xi_1} \\ \overline{\xi_2} \\ \overline{\xi_3}
\end{pmatrix}
&=&
\begin{pmatrix}
1 & 0 & 0 \\
a_{i,3} & 1 & 0 \\
a_{i,2} & a_{i,3} & 1
\end{pmatrix}
\begin{pmatrix}
\overline{\alpha_i} \\ \overline{\alpha_i^2} \\ \overline{\alpha_i^3}
\end{pmatrix}, \\
V_i
&:=& \frac{1}{ r_i f_{det}(q_B, -q_A)}
\begin{pmatrix}
1 & 0 \\
r_i (c A(h) - b B(h)) /3 & 1
\end{pmatrix}
\begin{pmatrix}
q_B & -q_A \\
-r_i (c q_A q_B + d q_A^2) & - r_i (a q_B^2 + b q_A q_B)
\end{pmatrix}.
\end{eqnarray*}
Since
$\begin{pmatrix}
r_1 & 0 \\
0 & 1
\end{pmatrix} V_1
=
\begin{pmatrix}
r_2 & 0 \\
0 & 1
\end{pmatrix} V_2
$
the following $(\tilde{A}_i, \tilde{B}_i)$ ($i = 1, 2$)
have identical sets of simultaneous representations over $\RationalField$ by ths assumption (c'):
\begin{eqnarray}
(\tilde{A}_i, \tilde{B}_i)
:=
\left( W_i,
\begin{pmatrix}
r_i & 0 \\
0 & 1
\end{pmatrix} V_i \right)
(A_i, B_i).
\end{eqnarray}
From $4 \det (\tilde{A}_i x - \tilde{B}_i) = {\rm ch}_{\alpha_i}^{res} \left( r_i x + a_{i,2} / 3 \right)$,
$\det (\tilde{A}_1 x - \tilde{B}_1) = (r_1/r_2)^{3} \det (\tilde{A}_2 x - \tilde{B}_2)$ is obtained.
Even if $\alpha_i$ is replaced by $\alpha_i - \trace{\alpha_i}/4$ (${\rm Tr}$ is the trace function), all the above hold.
The proof of the theorem is completed by the following Proposition \ref{prop:case of monogenic}.
\end{proof}
\begin{prop}\label{prop:case of monogenic}
We assume that $r_1, r_2$ are coprime integers
and the following $(A_i, B_i) \in V_\RationalField$
satisfy
the assumptions (b') and (c') of Theorem \ref{thm:main result over RationalField} and $\det (A_1 x - B_1) = (r_1/r_2)^{3} \det (A_2 x - B_2)$:
\begin{small}
\begin{eqnarray}\label{eq:definition of (A_i, B_i)}
(A_i, B_i) :=
\left(
\begin{pmatrix}
0 & 0 & r_i/2 \\
0 & - r_i & 0 \\
r_i/2 & 0 & 0
\end{pmatrix},
-
\begin{pmatrix}
1 & 0 & a_{i,2} / 6 \\
0 & 2 a_{i,2} / 3 & a_{i,1}/2 \\
a_{i,2} / 6 & a_{i,1}/2 & a_{i,0}
\end{pmatrix}
\right).
\end{eqnarray}
\end{small}
Then, either of the following holds:
\begin{enumerate}[1.]
\item
there exist coprime integers $u_1, u_2$ such that $r_i = u_i^2$ ($i = 1, 2$),
and
$(A_1, B_1) = (w, I) \cdot (A_2, B_2)$
for the following $w \in GL_3(\RationalField)$:
\begin{eqnarray}
w =
\begin{pmatrix}
1 & 0 & 0 \\
0 & u_1/u_2 & 0 \\
0 & 0 & (u_1/u_2)^2
\end{pmatrix}.
\end{eqnarray}
\item
There exist $s_1, s_2 \in \RationalField$ such that $(r_2 / r_1) (s_1^4 + a_{1,2} s_1^2 + a_{1,1} s_1 + a_{1,0}) = (r_1 / r_2) (s_2^4 + a_{2,2} s_2^2 + a_{2,1} s_2 + a_{2,0}) = c^2$ for some $c \in \RationalField^\times$,
and $(w, (r_2/r_1) I) \cdot (A_1, B_1) = (A_2, B_2)$
for the following $w \in GL_3(\RationalField)$:
\begin{eqnarray}\label{eq:case of t neq 0}
w =
\begin{pmatrix}
1 & 0 & 0 \\
2 s_2 & 1 & 0 \\
s_2^2 & s_2 & 1
\end{pmatrix}^{-1}
\begin{pmatrix}
0 & 0 & 1/c \\
0 & 1 & 0 \\
c & 0 & 0
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
2 s_1 & 1 & 0 \\
s_1^2 & s_1 & 1
\end{pmatrix}.
\end{eqnarray}
\end{enumerate}
The 1.\ always holds, if ${\rm Disc}(A_i, B_i) = 0$ for (either of) $i = 1, 2$.
\end{prop}
For the proof, the following lemma is used.
\begin{lem}\label{lem:isomorphic k-algebras}
Let $k$ be a global field with ${\rm char} k \neq 2$.
Assume that $f_1(x)$, $f_2(x) \in k[x]$ are monomial quartic polynomials
with no roots in $k$, no multiple roots in $\bar{k}$,
and $k[x] / (f_1^{res}(x))$ and $k[x] / (f_2^{res}(x))$ are isomorphic as $k$-algebras.
Furthermore, assume that $f_1(x)$, $f_2(x)$ have a root in $k_{\mathfrak p}$ with respect to the same ${\mathfrak p} \in P$,
where $P$ is the set of all the primes of $k$ that completely splits in $k[x] / (f_i^{res}(x))$.
In this case,
$k[x] / (f_1(x))$ and $k[x] / (f_2(x))$
are isomorphic as $k$-algebras.
\end{lem}
\begin{proof}
It may be assumed that the coefficient of $x^3$ in $f_i(x)$ equals $0$.
Fix a prime ${\mathfrak p} \in P$ so that both of $f_1(x)$ and $f_2(x)$ have a root in $k_{\mathfrak p}$.
Let $\alpha_{i, {\mathfrak p}} \in k_{\mathfrak p}$ ($i = 1, 2$) be the root.
If the roots of $g_{i, {\mathfrak p}}(x) := f_i(x) / (x - \alpha_{i, {\mathfrak p}}) \in k_{\mathfrak p}[x]$ are denoted by
$\beta_{i,1}, \beta_{i,2}, \beta_{i,3} \in k_{\mathfrak p}$, then
\begin{eqnarray*}
\alpha_{i, {\mathfrak p}} \beta_{i,1} + \beta_{i,2} \beta_{i,3}
&=& -\alpha_{i, {\mathfrak p}} (\alpha_{i, {\mathfrak p}} + \beta_{i,2} + \beta_{i,3}) + \beta_{i,2} \beta_{i,3} \\
&=& -2 \alpha_{i, {\mathfrak p}}^2 + (\alpha_{i, {\mathfrak p}} - \beta_{i, 2}) (\alpha_{i, {\mathfrak p}} - \beta_{i, 3}) \\
&=& -2 \alpha_{i, {\mathfrak p}}^2 - f_i^\prime(\alpha_{i, {\mathfrak p}}) /(\beta_{i, 1} - \alpha_{i, {\mathfrak p}}),
\end{eqnarray*}
where $f_i^\prime(x)$ is the first derivative of $f_i(x)$ with respect to $x$.
Thus, there exists $0 \neq C \in k_{\mathfrak p}$ such that
\begin{eqnarray}
g_{i, {\mathfrak p}}(x)
= C (x - \alpha_{i, {\mathfrak p}})^3
f_i^{res} \left( - \frac{ f_i^\prime(\alpha_{i, {\mathfrak p}}) }{ x - \alpha_{i, {\mathfrak p}} } - 2 \alpha_{i, {\mathfrak p}}^2 \right).
\end{eqnarray}
From the assumption about $f_1^{res}(x)$ and $f_2^{res}(x)$,
${\mathfrak p}$ completely splits in $k[x] / (f_1(x))$ if and only if it does in $k[x] / (f_2(x))$.
Let $f_i(x) = \prod_{j=1}^m g_{ij}(x)$ ($g_{ij} \in k[x]$) be a factorization in $k$.
By assumption, $k[x] / (f_i(x))$ is a direct sum of $k[x]/(g_{ij}(x))$ with degree $2$ or 4 over $k$.
For each $g_{ij}(x)$, we
fix an embedding $\iota_{ij} : k[x]/(g_{ij}(x)) \hookrightarrow \bar{k}$
and let $K_i$ be the composite field of $\iota_{ij} (k[x]/(g_{ij}(x)))$ ($j = 1, \cdots, m$).
$K_1$, $K_2$ are quadratic or quartic fields over $k$,
and any prime ${\mathfrak p}$ of $k$ completely splits in $K_1$ if and only if it does over $K_2$.
Hence, $K_1$, $K_2$ have identical Galois closure over $k$ (Theorem 8.8, \cite{Kato2011}).
If both of $K_i$ are Galois over $k$, then $K_1 = K_2$.
If either of $K_1$, $K_2$ is not Galois over $k$, both must be a quartic field not Galois over $k$.
Even in this case, $K_1$ and $K_2$ are isomorphic over $k$. This can be seen as follows;
first suppose that $f_i^{res}(x) = 0$ has a root in $k$.
In this case,
$f_i(x) := x^4 + a_{i, 2} x^2 + a_{i, 1} x + a_{i, 0}$ is decomposed
as follows:
\begin{eqnarray}
f_i(x)
=
(x^2 + u_i/2)^2 - (u_i^2 - 4 a_{i,0}) \left( \frac{ a_{i, 1} }{ u_i^2 - 4 a_{i,0} } x - 1/2 \right)^2.
\end{eqnarray}
Even if $u^2 = 4 a_{i,0}$, $a_{i,1}=0$ follows from $f_i^{res}(x) = (x - a_{i, 2})(x^2 - 4 a_{i, 0}) - a_{i, 1}^2$.
Therefore $K_1$, $K_2$ are quadratic extensions of a quadratic field over $k$.
Since they have the same Galois closure, the quadratic field is common,
and $K_1, K_2$ are conjugate over $k$.
Next, suppose that $f_i^{res}(x)$ does not have a root in $k$.
In this case, the Galois closure $F$ of $K_1$, $K_2$ contains a cubic field isomorphic to $k[x]/(f_i^{res}(x))$.
Since ${\rm Gal}(F/k)$ is isomorphic to a subgroup of $S_4$, this happens only when ${\rm Gal}(F / k) \cong S_4$ or $A_4$.
Since all the subgroups of $S_4$ (\textit{resp.} $A_4$) of index $4$
are conjugate to $S_3 = \langle (1, 2, 3), (1, 2) \rangle$ (\textit{resp.} $A_3 = \langle (1, 2, 3) \rangle$),
$K_1, K_2$ are conjugate over $k$.
Consequently, if $k[x] / (g_{ij}(x)) = K_i$ for all $1 \leq j \leq m$, $k[x] / (f_{1}(x))$, $k[x] / (f_{2}(x))$
are isomorphic as $k$-algebras.
It remains to verify the case of
$K_i = F_1 \cdot F_2$ and $K_2 = F_1 \oplus F_2$,
where $F_1, F_2$ are distinct quadratic fields.
They correspond to the $G_k$-orbits
containing
\begin{itemize}
\item $k(\sqrt{d_1}, \sqrt{d_2})$,
\begin{eqnarray}
(A, B) :=
\left(
\begin{pmatrix}
-d_1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
\begin{pmatrix}
d_2 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatrix} \right).
\end{eqnarray}
\item $k(\sqrt{d_1}) \oplus k(\sqrt{d_2})$,
\begin{eqnarray}
(A, B) :=
\left(
\begin{pmatrix}
-d_1 & 0 & 0 \\
0 & -d_2 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
0 & -1/2 & 0 \\
-1/2 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix} \right).
\end{eqnarray}
\end{itemize}
However, $\det(A x - B y)$ of the former has only rational roots,
whereas that of the latter has irrational roots.
Therefore, $f_1^{res}(x) = f_2^{res}(x)$ cannot happen. Hence, this case can be eliminated.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:case of monogenic}]
We first put $f_i(x) := x^4 + a_{i, 2} x^2 + a_{i, 1} x + a_{i, 0}$.
It follows from $\det(A_1 x - B_1y) = (r_1/r_2)^3 \det(A_2 x - B_2 y)$ that
\begin{eqnarray*}
(1/r_1)^2 (4 a_{1,0} + a_{1,2}^2/3) &=& (1/r_2)^2 (4 a_{2,0} + a_{2,2}^2/3), \label{eq: equation 1 between a_{i,0} + a_{i,2}} \\
(1/r_1)^3 (a_{1,1}^2 - 8 a_{1,2} a_{1,0}/3 + 2 a_{1,2}^3 / 27) &=& (1/r_2)^3 (a_{2,1}^2 - 8 a_{2,2} a_{2,0}/3 + 2 a_{2,2}^3 / 27).
\end{eqnarray*}
Hence, $r_1^{-1} a_{1,2} = r_2^{-1} a_{2,2}$ implies that
$r_1^{-2} a_{1,0} = r_2^{-2} a_{2,0}$ and
$r_1^{-3} a_{1,1}^2 = r_2^{-3} a_{2,1}^2$. Thus, in this case,
case 1.\ occurs. Therefore, in what follows we assume that $r_1^{-1} a_{1,2} \neq r_2^{-1} a_{2,2}$.
In case of ${\rm Disc}(f_1^{res}) = 0 = {\rm Disc}(f_2^{res}) = 0$,
the multiple root of $f_i^{res}(x) = \det (A_i x - B_i)$ for $i = 1, 2$,
equals $x = 2 a_{1, 2} / 3 r_1 = 2 a_{2, 2} / 3 r_2$, which is seen by considering when $A_i x - B_i$ in (\ref{eq:definition of (A_i, B_i)}) is rank $1$. In particular, $r_1^{-1} a_{1,2} = r_2^{-1} a_{2,2}$ follows in this case.
Since ${\rm Disc}(f_1) = 0$ implies ${\rm Disc}(f_1^{res}) = 0$,
we may now assume that both $f_i$ and $f_i^{res}$ have no multiple roots.
By Corollary \ref{cor: (A, B) is isotropic over k},
$\RationalField[x] / (f_i(x))$ is a direct sum of number fields of degree greater than $1$ over $\RationalField$.
Let $p$ be a finite prime that completely splits in $\RationalField[x] / (f_i^{res}(x))$.
$(A_i, B_i)$ is isotropic over $\RationalField_p$
if and only if $\RationalField_p$ contains a root of $f_i(x) := x^4 + a_{i, 2} x^2 + a_{i, 1} x + a_{i, 0} = 0$.
Hence,
$f_1(x) = 0$ has a root in $\RationalField_p$ if and only if $f_2(x) = 0$ does.
By Lemma \ref{lem:isomorphic k-algebras},
$\RationalField[x] / (f_1(x))$ and $\RationalField[x] / (f_2(x))$ are isomorphic as $\RationalField$-algebras.
Thus, there exists $x_2 \in \RationalField[x] / (f_1(x))$
such that $x \mapsto x_2$ provides an isomorphism $\RationalField[x] / (f_2(x)) \rightarrow \RationalField[x] / (f_1(x))$.
Now $\QA{\RationalField}{A_1}{B_1}$ is a $\RationalField$-algebra isomorphic to $\RationalField[x] / (f_1(x))$
with the basis $\langle 1, r_1 x, r_1 x^2, r_1 (x^3 + a_{1,2} x) \rangle$.
We assume that $x_2$ is represented as $h_0 + r_1 \{ h_1 x + h_2 x^2 + h_3 (x^3 + a_{1,2} x) \}$ in $\QA{\RationalField}{A_1}{B_1}$,
using some $h_0 \in \RationalField$ and $h := (h_{1}, h_{2}, h_{3}) \in \RationalField^3$.
If $b_0$, $b_1 \in \RationalField$ are the coefficients of $x^3 + b_1 x + b_0 = 4 r_1^{-3} \det (A_1 x - B_1)$, then
by using the formulas given in the proof of Proposition \ref{prop: monogenic ring},
$(W, V) \in G_\RationalField$ satisfying $(W, V) \cdot (A_1, B_1) = (A_2, B_2)$ is obtained as follows
(herein, $\bar{x}$, $\bar{x_2}$ are the classes of $x$, $x_2$ in $\QA{k}{A_1}{B_1} / \RationalField$):
\begin{eqnarray*}
W
\begin{pmatrix}
\overline{r_1 x} \\ \overline{r_1 x^2} \\ \overline{r_1(x^3 + a_{1,2} x)}
\end{pmatrix}
&=&
\begin{pmatrix}
\overline{x_2} \\ \overline{x_2^2} \\ \overline{x_2^3 + a_{2,2} x_2}
\end{pmatrix},
\end{eqnarray*}
\begin{eqnarray*}
V
&:=& (\det W)^{-1}
\begin{pmatrix}
r_2 & 0 \\
\frac{ r_1^3 b_1 A_1(h) }{3} & 1
\end{pmatrix}
\begin{pmatrix}
B_1(h) & -A_1(h) \\
-r_1^3 ( b_1 A_1(h) B_1(h) + b_0 A_1(h)^2 ) & -r_1^3 B_1(h)^2
\end{pmatrix} \\
&=& (\det W)^{-1}
\begin{pmatrix}
r_2 B_1(h) & - r_2 A_1(h) \\
-r_1^3 ( \frac{2}{3} b_1 A_1(h) B_1(h) + b_0 A_1(h)^2 ) & - r_1^3 ( \frac{1}{3} b_1 A_1(h)^2 + B_1(h)^2)
\end{pmatrix}.
\end{eqnarray*}
These matrices have the determinants $\det W = r_1^3 ( B_1(h)^3 + b_1 A_1(h)^2 B_1(h) + b_0 A_1(h)^3 )$
and $\det V = -r_2 \det (W)^{-1}$.
From $q_\RationalField(A_1, B_1) = q_\RationalField(A_2, B_2)$,
$q_\RationalField(A_1, B_1) = q_\RationalField((I, V) \cdot (A_1, B_1))$ is obtained.
If $V \neq I$, by Lemma \ref{lem:automorphism of (A, B) with V},
there exist $W_2 \in GL_3(\RationalField)$ and $u \in \RationalField$ such that
$(W_2, u^{-1} V) \cdot (A_1, B_1) = (A_1, B_1)$ and $\det V = -u^2$.
Hence, replacing $(W, V)$ with $(W, V) (W_2^{-1}, u V^{-1})$,
we may assume that $V$ is a scalar multiple of $I$.
Thus,
$A_1(h) = 0$, $B_1(h) = - r_1^{-3} r_2$, $\det W = - r_1^{-6} r_2^3$,
and $V = r_1^3 r_2^{-1} I$.
Therefore,
$(r_1^{-1} W, r_1/ r_2)$ maps $(A_1, B_1)$ to $(A_2, B_2)$.
Since $A_1(h) = 0$,
there are $0 \neq C \in \RationalField$ and $0 \neq (s, t) \in \RationalField$ such that $h = C (s^2, s t, t^2)$.
If we put $f_i(X, Y) := Y^4 f_i(X/Y)$ ($i = 1, 2$), then
$C^2 f_1(s, t) = r_1^{-3} r_2$ follows from $B_1(h) = - r_1^{-3} r_2$.
Using $\trace{x^2} = - 2 a_{1,2}$, $\trace{x^3} = - 3 a_{1,1}$ and $\trace{x_2} = 0$, we obtain
$x_2 = C r_1 ( s^2 x + s t (x^2 + 2 a_{1, 2}) + t^2 (x^3 + a_{1,2} x + 3 a_{1,1}))$.
Hence,
\begin{eqnarray*}
& & \hspace{-10mm}
(t x - s)\{ t x_2 + C r_1 (s^3 - a_{1,2} s t^2 - 2 a_{1,1} t^3) \} \\
&=& C r_1 (t x - s) \left\{ s^3 + s^2 t x + s t^2 (x^2 + a_{1, 2}) + t^3 (x^3 + a_{1,2} x + a_{1,1}) \right\} \\
&=& -C r_1 \left\{ s^4 - t^4 x^4 + a_{1, 2} t^2 (s^2 - t^2 x^2) + a_{1,1} t^3 (s - t x) \right\} \\
&=& -C r_1 f_1(s, t).
\end{eqnarray*}
If we put $\tilde{C} := -C r_1 f_1(s, t)$,
$\tilde{C}^2 = (r_2 / r_1) f_1(s, t)$ follows from $C^2 f_1(s, t) = r_1^{-3} r_2$.
Hence, if $t = 0$, then $r_1 / r_2 \in (\RationalField^\times)^2$, and $x_2 = u x $ holds for $u := C r_1 s^2$,
which satisfies $u^2 = C^2 r_1^2 s^4 = r_2/r_1$.
As a result, $a_{2, 2} = u^2 a_{1, 2}$, $a_{2, 1} = u^3 a_{1, 1}$ and $a_{2, 0} = u^4 a_{1, 0}$ are obtained.
If $t \neq 0$,
the characteristic polynomials of $\tilde{x} := t x - s$, $\tilde{x}_2 := \tilde{C} (t x - s)^{-1}$
are as follows:
\begin{eqnarray*}
{\rm ch}_{\tilde{x}}(X)
&=& f_1(X + s, t)
= X^4 + 4 s X^3 + (6 s^2 + a_{1, 2} t^2) X^2 + (4 s^3 + 2 a_{1, 2} s t^2 + a_{1, 1} t^3) X + f_1(s, t), \\
{\rm ch}_{\tilde{x}_2}(X)
&=& f_1(s, t)^{-1} X^4 {\rm ch}_{\tilde{x}}(\tilde{C} / X) \\
&=& f_1(s, t)^{-1} \{ \tilde{C}^4 + 4 \tilde{C}^3 s X + \tilde{C}^2 (6 s^2 + a_{1, 2} t^2) X^2 + \tilde{C} (4 s^3 + 2 a_{1, 2} s t^2 + a_{1, 1} t^3) X^3 + f_1(s, t) X^4 \}. \nonumber
\end{eqnarray*}
Therefore, in this case,
the equation (\ref{eq:case of t neq 0}) is obtained
by putting $s_1 = s$, $s_2 = \tilde{C} (s^3 + a_{1,2} s t^2 / 2 + a_{1,1} t^3 / 4) / f_1(s, t)$ and $c = \tilde{C}$.
Thus, the proposition is proved.
\end{proof}
\section{Case of ${\rm Disc}(A_i, B_i) = 0$ (proofs of Propositions \ref{thm:proposition 1}, \ref{thm:theorem 1})}
\label{Case of Disc(A_i, B_i) = 0 (proofs of Propositions 1, 2)}
From the known result in the binary case, it is immediately obtained that
$q_\IntegerRing(x_1^2 - x_1 x_2 + x_2^2, x_3^2) = q_\IntegerRing(x_1^2 + 3x_2^2, x_3^2)$.
In the case of (ii), \IE when $(A_1, B_1) = (x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2)$
and $(A_2, B_2) = (x_1^2 + 3x_2^2, (x_1 + 3 x_3)^2)$, we have
\begin{eqnarray*}
(A_2, B_2)(x_1, x_2, x_3) &=& (A_1, B_1)(x_1 + x_2, 2 x_2, -x_2 + x_3) \\
&=& (A_1, B_1)(2 x_1, x_1 + x_2, -x_1 + x_3) \\
&=& (A_1, B_1)(x_1 + x_2, x_1-x_2, -x_1 - x_3).
\end{eqnarray*}
Consequently, $q_\IntegerRing(A_2, B_2) \subset q_\IntegerRing(A_1, B_1)$,
and the converse is also true, since any $(y_1, y_2) \in \IntegerRing^2$
can be represented in either of the ways $(x_1 + x_2, 2 x_2)$, $(2 x_1, x_1 + x_2)$ or $(x_1 + x_2, x_1-x_2)$
for some $x_1, x_2 \in \IntegerRing$.
Proposition \ref{thm:theorem 1} can be also proved in an elementary way.
\begin{proof}[Proof of Proposition \ref{thm:theorem 1}]
As proved in Proposition \ref{prop:same det(Ax+By)},
$q_\IntegerRing(A_1, B_1) = q_\IntegerRing(A_2, B_2)$
implies that all the roots of $\det (A_1 x - B_1 y)$
and $\det (A_2 x - B_2 y)$ are common.
In particular, ${\rm Disc}(A_1, B_1) = 0$ leads to ${\rm Disc}(A_2, B_2) = 0$.
By the action of $G_\IntegerRing$, $(A_1, B_1)$ can be transformed into:
\begin{eqnarray*}
(A_1, B_1) = (M, 1) \cdot
\left(
\begin{pmatrix}
a_{11} & a_{12} & 0 \\
a_{12} & a_{22} & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\right),\
M
:=
\begin{pmatrix}
1 & 0 & m_1 \\
0 & 1 & m_2 \\
0 & 0 & m_3
\end{pmatrix},
\end{eqnarray*}
where $m_1, m_2$ and $0 \neq m_3 \in \IntegerRing$ may be assumed to have the greatest divisor 1.
By comparing the representations $(0, *)$ of $(A_1, B_1)$ and $(A_2, B_2)$, it is seen that $(A_2,B_2)$ can be simultaneously transformed into:
\begin{eqnarray*}
(A_2, B_2) = (\tilde{M}, 1) \cdot
\left(
\begin{pmatrix}
\tilde{a}_{11} & \tilde{a}_{12} & 0 \\
\tilde{a}_{12} & \tilde{a}_{22} & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\right),\
\tilde{M}
:=
\begin{pmatrix}
1 & 0 & \tilde{m}_1 \\
0 & 1 & \tilde{m}_2 \\
0 & 0 & m_3
\end{pmatrix}.
\end{eqnarray*}
The greatest common divisor of $\tilde{m}_1, \tilde{m}_2$, $0 \neq m_3 \in \IntegerRing$ is $1$, owing to $q_{\IntegerRing}(B_1) = q_{\IntegerRing}(B_2)$.
Furthermore, since $q_{\IntegerRing}(A_1) = q_{\IntegerRing}(A_2)$,
either of the following may be assumed
:
\begin{enumerate}[(I)]
\item
$\begin{pmatrix}
a_{11} & a_{12} \\
a_{12} & a_{22}
\end{pmatrix}
=
\begin{pmatrix}
1 & -1/2 \\
-1/2 & 1
\end{pmatrix},\
\begin{pmatrix}
\tilde{a}_{11} & \tilde{a}_{12} \\
\tilde{a}_{12} & \tilde{a}_{22}
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 3
\end{pmatrix}$.
\item
$\tilde{A} := \begin{pmatrix}
a_{11} & a_{12} \\
a_{12} & a_{22}
\end{pmatrix}
=
\begin{pmatrix}
\tilde{a}_{11} & \tilde{a}_{12} \\
\tilde{a}_{12} & \tilde{a}_{22}
\end{pmatrix},\
0 < a_{11} \leq a_{22},
0 \leq - 2 a_{12} \leq a_{11}$, \IE $\tilde{A}$ is reduced.
\end{enumerate}
For any $m \neq 0$ and $n \in \IntegerRing$,
the class of $n$ in $\IntegerRing/ m \IntegerRing$ is denoted by $\Mod{n}{m}$.
In case (I),
the following is proved by
considering the representations $(A_i(h), B_i(h)) = (1, *)$, $(3, *)$:
\begin{eqnarray*}
\left\{ \pm \Mod{m_1}{m_3}, \pm \Mod{m_2}{m_3}, \pm \Mod{m_1+m_2}{m_3} \right\}
&=&
\left\{ \pm \Mod{\tilde{m}_1}{m_3} \right\}, \\
\left\{ \pm \Mod{m_1 - m_2}{m_3}, \pm \Mod{m_1 + 2 m_2}{m_3}, \pm \Mod{2 m_1 + m_2}{m_3} \right\}
&=&
\left\{ \pm \Mod{\tilde{m}_2}{m_3} \right\}.
\end{eqnarray*}
The above can hold only when either of the following holds:
\begin{itemize}
\item $\Mod{m_1}{m_3} = \Mod{m_2}{m_3} = \Mod{\tilde{m_1}}{m_3} = \Mod{\tilde{m_2}}{m_3} = \Mod{0}{m_3}$ or
\item $m_3 = 3$, $\Mod{m_1}{3} = \Mod{m_2}{3} = \pm \Mod{\tilde{m_1}}{3}$, $\Mod{\tilde{m_2}}{3} = \Mod{0}{3}$.
\end{itemize}
We note that each corresponds to the cases (i), (ii), respectively. Thus, the theorem is proved in case (I).
In case (II), by changing the basis of $\IntegerRing^3$ to another reduced one,
$\Mod{m_1}{m_3} = \Mod{\tilde{m}_1}{m_3}$, $\Mod{m_2}{m_3} = \pm \Mod{\tilde{m}_2}{m_3}$ may be assumed.
If $\Mod{m_1}{m_3} = -\Mod{m_1}{m_3}$ or $\Mod{m_2}{m_3} = -\Mod{m_2}{m_3}$, then
$(A_1, B_1)$ and $(A_2, B_2)$ are equivalent by the action of $GL_3(\IntegerRing)$.
In what follows, we assume that
$\Mod{m_1}{m_3} \neq -\Mod{m_1}{m_3}$ and $\Mod{m_2}{m_3} = -\Mod{\tilde{m}_2}{m_3} \neq -\Mod{m_2}{m_3}$.
The multiplicity of the representation $a_{11} + a_{22} + 2a_{12}$ of
$\tilde{A}$ must greater than $1$, otherwise we would have $\Mod{m_1 + m_2}{m_3} = \Mod{\tilde{m}_1 + \tilde{m}_2}{m_3}$
(hence $\Mod{m_2}{m_3} = \Mod{\tilde{m}_2}{m_3}$).
This can happen only if $a_{12} = 0$ or $a_{12} = -a_{11}/2$.
If $a_{12} = 0$, then
$(A_1, B_1)$ and $(A_2, B_2)$ are equivalent by the action of $GL_3(\IntegerRing)$.
If $a_{12} = -a_{11}/2$,
then we see from the representations $(A_i(h), B_i(h)) = (a_{11} + a_{22} + 2a_{12}, *)$ that either of the following is required:
\begin{itemize}
\item
$\left\{ \pm \Mod{m_2}{m_3}, \pm \Mod{m_1 + m_2}{m_3} \right\}
=
\left\{ \pm \Mod{m_2}{m_3}, \pm \Mod{m_1 - m_2}{m_3} \right\}$.
\item
$a_{11} = a_{22} = - 2 a_{12}$ and \\
$\left\{ \pm \Mod{m_1}{m_3}, \pm \Mod{m_2}{m_3}, \pm \Mod{m_1 + m_2}{m_3} \right\}
=
\left\{ \pm \Mod{m_1}{m_3}, \pm \Mod{m_2}{m_3}, \pm \Mod{m_1 - m_2}{m_3} \right\}$.
\end{itemize}
If $\Mod{m_1 + m_2}{m_3} = \pm \Mod{m_1 - m_2}{m_3}$, then
$\Mod{m_1}{m_3} = -\Mod{m_1}{m_3}$ or $\Mod{m_2}{m_3} = -\Mod{m_2}{m_3}$ must hold, which contradicts with the above assumption.
In the former case, $\Mod{m_1 + m_2}{m_3} = \pm \Mod{m_1 - m_2}{m_3}$ always holds .
In the latter case, if $\Mod{m_1 + m_2}{m_3} \neq \pm \Mod{m_1 - m_2}{m_3}$ is assumed, then
$\Mod{m_1 + m_2}{m_3} = \pm \Mod{m_i}{m_3}$ and $\Mod{m_1 - m_2}{m_3} = \pm \Mod{m_j}{m_3}$ must hold for either $(i, j) = (1, 2)$ or $(i,j)=(2, 1)$.
Consequently, $\Mod{m_1}{m_3} \neq -\Mod{m_1}{m_3}$ and $\Mod{m_2}{m_3} \neq -\Mod{m_2}{m_3}$ implies
$(\Mod{m_1 + m_2}{m_3}, \Mod{m_1 - m_2}{m_3}) = (\Mod{-m_1}{m_3}, \Mod{m_2}{m_3})$ or $(\Mod{-m_2}{m_3}, \Mod{-m_1}{m_3})$.
Comparing the representation $(a_{11} + a_{22} - 2 a_{12}, *)$, we have
\begin{eqnarray*}
& & \hspace{-20mm}
\{ \pm \Mod{m_1 - m_2}{m_3}, \pm \Mod{m_1 + 2 m_2}{m_3}, \pm \Mod{2 m_1 + m_2}{m_3} \} \\
&=&
\{ \pm \Mod{m_1 + m_2}{m_3}, \pm \Mod{m_1 - 2 m_2}{m_3}, \pm \Mod{2 m_1 - m_2}{m_3} \}.
\end{eqnarray*}
If $(\Mod{m_1 + m_2}{m_3}, \Mod{m_1 - m_2}{m_3}) = (\Mod{-m_1}{m_3}, \Mod{m_2}{m_3})$, by using
$\Mod{m_1 + 2 m_2}{m_3} = \Mod{2 m_1}{m_3} = -\Mod{m_2}{m_3}$ and
$\Mod{2m_1 - m_2}{m_3} = -\Mod{2 m_2}{m_3} = -\Mod{m_1}{m_3}$,
\begin{eqnarray}
\{ \pm \Mod{m_2}{m_3}, \Mod{0}{m_3} \}
&=&
\{ \pm \Mod{m_1}{m_3}, \Mod{0}{m_3} \}.
\end{eqnarray}
If $(\Mod{m_1 + m_2}{m_3}, \Mod{m_1 - m_2}{m_3}) = (\Mod{-m_2}{m_3}, \Mod{-m_1}{m_3})$, by using
$\Mod{2 m_1 + m_2}{m_3} = \Mod{2 m_2}{m_3} = -\Mod{m_1}{m_3}$ and
$\Mod{m_1 - 2 m_2}{m_3} = \Mod{2 m_1}{m_3} = \Mod{m_2}{m_3}$,
\begin{eqnarray}
\{ \pm \Mod{m_1}{m_3}, \Mod{0}{m_3} \}
&=&
\{ \pm \Mod{m_2}{m_3}, \Mod{0}{m_3} \}.
\end{eqnarray}
In the both cases, $\Mod{m_1}{m_3} = \pm \Mod{m_2}{m_3}$ is obtained, which leads to $\Mod{m_1 + m_2}{m_3} = \pm \Mod{m_1 - m_2}{m_3}$, hence is impossible.
\end{proof}
Proposition \ref{thm:proposition 1} is proved in the remaining part.
\begin{proof}[Proof of Proposition \ref{thm:proposition 1}]
Since the ``only if'' part is clear, we shall prove the ``if'' part;
for any positive-definite $f_1, f_2 \in {\rm Sym}^2 (\RealField^3)^*$ with $q_\IntegerRing(f_1) = q_\IntegerRing(f_2)$,
take $\lambda_{1}, \ldots, \lambda_{s} \in \RealField$ linearly independent over $\RationalField$ and
positive-definite $A_{1j}$, $A_{2j} \in {\rm Sym}^2 (\RationalField^3)^*$ ($1 \leq j \leq s$) satisfying $f_1 = \sum_{j=1}^s \lambda_j A_{1j}$, $f_2 = \sum_{j=1}^s \lambda_j A_{2j}$
as in Lemma \ref{lem:decomposition of positive definite symmety matrices}.
In this case, $q_\IntegerRing(A_{11}, \ldots, A_{1s}) = q_\IntegerRing(A_{21}, \ldots, A_{2s})$ holds.
Therefore, the proposition is obtained by proving the following (**):
\begin{description}
\item[(**)]
If both of $(A_{i1}, A_{i2}, \ldots, A_{is}) \in {\rm Sym}^2 (\RationalField^3)^* \otimes_{\RationalField} \RationalField^s$ ($i = 1, 2$) satisfy all of
\begin{enumerate}[(a)]
\item $A_{i1}, \ldots, A_{is}$ span a linear space of dimension more than 1 over $\RationalField$,
\item $\sum_{j = 1}^s c_j A_{ij}$ is positive definite for some $c_1, \ldots, c_s \in \RationalField$,
\item $q_\IntegerRing(A_{11}, \ldots, A_{1s}) = q_\IntegerRing(A_{21}, \ldots, A_{2s})$,
\end{enumerate}
there exists $w \in GL_3(\IntegerRing)$ such that $(A_{11}, \ldots, A_{1s}) = (w, 1) \cdot (A_{21}, \ldots, A_{2s})$,
or there exist $w_1, w_2 \in GL_3(\IntegerRing)$ and $v \in GL_s(\RationalField)$ such that $\{ (A_{11}, \ldots, A_{1s}), (A_{21}, \ldots, A_{2s}) \} $ equals either of the following:
\begin{enumerate}[(i)]
\item
$\left\{
(w_1, v) \cdot ( 0, \cdots, 0, x_1^2 - x_1 x_2 + x_2^2, x_3^2 ),
(w_2, v) \cdot ( 0, \cdots, 0, x_1^2 + 3 x_2^2, x_3^2 )
\right\}$,
\item $\left\{
\begin{matrix} (w_1, v) \cdot ( 0, \cdots, 0, x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2 ), \\
(w_2, v) \cdot ( 0, \cdots, 0, x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2 ) \end{matrix}
\right\}$,
\end{enumerate}
where $0$ is the ternary quadratic form that maps all the ${\mathbf x} \in \IntegerRing^3$ to 0, and
$GL_3(\IntegerRing) \times GL_s(\RationalField) \ni (w, v)$
acts on ${\rm Sym}^2 (\RationalField^3)^* \otimes_{\RationalField} \RationalField^s$
by
\begin{eqnarray*}
\left(w, (v_{ij})
\right) \cdot (A_1 \ldots A_s) =
\left( \sum_{j=1}^s v_{1j} A_j({\mathbf x} w), \sum_{j=1}^s v_{2j} A_j({\mathbf x} w), \ldots, \sum_{j=1}^s v_{sj} A_j({\mathbf x} w) \right).
\end{eqnarray*}
\end{description}
The case of $s = 2$ is equivalent to the assumption (*).
In order to prove (**) by induction, we assume (**) is true for $s \leq n$
for some $n \geq 2$.
If $A_{11}, \ldots, A_{1 n+1}$ span a linear space of dimension $m < n+1$, by using the action of $\{ 1 \} \times GL_{n + 1}(\RationalField)$,
$(A_{11}, \ldots, A_{1 n+1})$ is mapped to $(\tilde{A}_1, \ldots, \tilde{A}_{m}, 0 \ldots, 0)$ for some $\tilde{A}_j \in {\rm Sym}^2 (\RationalField^3)^*$.
From $q_\IntegerRing(A_{11}, \ldots, A_{1s}) = q_\IntegerRing(A_{21}, \ldots, A_{2s})$,
$(A_{21}, \ldots, A_{2s})$ is simultaneously mapped to $(\tilde{B}_1, \ldots, \tilde{B}_{m}, 0 \ldots, 0)$ for some $\tilde{B}_j \in {\rm Sym}^2 (\RationalField^3)^*$.
The proposition is proved by induction in this case.
Thus we assume the dimension of the space spanned by $A_{11}, \ldots, A_{1 n+1}$ is exactly $n+1$.
By using the action of $GL_3(\IntegerRing) \times GL_{n+1}(\RationalField)$,
either of the following may be assumed:
\begin{enumerate}[(I)]
\item $A_{1j} = A_{2j}$ ($1 \leq j \leq n$) and $\sum_{j=1}^n c_j A_{1j} \succ 0$ for some $c_j \in \RealField$,
\item $n = 2$ and $(A_{11}, A_{12}) = (x_1^2 - x_1 x_2 + x_2^2, x_3^2)$, $(A_{21}, A_{22}) = (x_1^2 + 3 x_2^2, x_3^2)$,
\item $n = 2$ and $(A_{11}, A_{12}) = (x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2)$, $(A_{21}, A_{22}) = (x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2)$.
\end{enumerate}
In the case (I),
define $H := \{ g \in GL_3(\IntegerRing) : A_{ij} = g \cdot A_{ij} \text{ for all } 1 \leq j \leq n \}$.
For any $c \in \RationalField$,
owing to
$q_\IntegerRing(A_{11}, \ldots, A_{1 n-1}, A_{1 n} + c A_{1 n+1}) = q_\IntegerRing(A_{21}, \ldots, A_{2 n-1}, A_{2 n} + c A_{2 n+1})$
and the assumption of induction,
there exists
$h \in H$
such that $(h, 1) \cdot (A_{11}, \ldots, A_{1 n-1}, A_{1 n} + c A_{1 n+1}) = (A_{21}, \ldots, A_{2 n-1}, A_{2 n} + c A_{2 n+1})$.
Since $H$ is a finite group, some $h \in H$ satisfies $h \cdot (A_{1 n} + c_i A_{1 n+1})= A_{2 n} + c_i A_{2 n+1}$ for some distinct $c_1 \neq c_2$.
Hence $h \cdot A_{1j} = A_{2j}$ holds for all $1 \leq j \leq n + 1$.
In the cases (II) and (III), owing to $A_{11} \not\sim A_{21}$,
${\rm Disc}(A_{11}, A_{12} + d A_{13}) = {\rm Disc}(A_{21}, A_{22} + d A_{23}) = 0$
is required for any $d \in \RationalField$.
Therefore $A_{i1}, A_{i2}, A_{i3}$ are linearly dependent over $\RationalField$ for each $i = 1, 2$.
Now it is straightforward to see that (**) is true in this case.
\end{proof}
\paragraph{Acknowledgments}
The author would like to extend her gratitude to Professor T. Kamiyama of KEK.
He gave her an opportunity to apply the theory of quadratic forms to a real scientific problem.
She also appreciates Professor T. Oda of the university of Tokyo, Dr. S. Harashita of Yokohama National University and Dr. K. Gunji of
Chiba Institute of Technology for their valuable comments during a regular seminar about geometry and modular forms.
For various checks of our program and computation of genera and spinor genera, we used the algebra system Magma \cite{MR1484478}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 108
|
#
Published 2018 by Prometheus Books
_World War Trump: The Risks of America's New Nationalism_. Copyright © 2018 by Hall Gardner. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, digital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via the Internet or a website without prior written permission of the publisher, except in the case of brief quotations embodied in critical articles and reviews.
Trademarked names appear throughout this book. Prometheus Books recognizes all registered trademarks, trademarks, and service marks mentioned in the text.
Cover design by Jacqueline Nasso Cooke
Cover design © Prometheus Books
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Library of Congress Cataloging-in-Publication Data
Names: Gardner, Hall, author.
Title: World war Trump : the risks of America's new nationalism / by Hall Gardner.
Description: Amherst, New York : Prometheus Books, 2018. | Includes index.
Identifiers: LCCN 2017042513 (print) | LCCN 2018004448 (ebook) | ISBN 9781633883963 (ebook) | ISBN 9781633883956 (hardcover)
Subjects: LCSH: United States--Foreign relations--2017- | Nationalism--United States--History--21st century. | World politics--1989-
Classification: LCC E895 (ebook) | LCC E895 .G36 2018 (print) | DDC 320.540973/0905--dc23
LC record available at <https://lccn.loc.gov/2017042513>
Printed in the United States of America
Acknowledgments
Introduction: A Self-Fulfilling Prophecy
Chapter 1: The Perils of the New "America First" Nationalism
Chapter 2: Inauguration Tremors: Rifles, Tanks, and Nuclear Weapons
Chapter 3: The New Bogeyman: Russians, Immigrants, Muslims—and the Question of Impeachment
Chapter 4: Risks of the New American Nationalism for the European Union
Chapter 5: The Risk of War over Crimea, the Black Sea, and Eastern Europe
Chapter 6: The Global Impact of the China-Russia Eurasian Alliance
Chapter 7: China, North Korea, and the Risk of War in the Indo-Pacific
Chapter 8: Syria and Widening Wars in the "Wider Middle East"
Chapter 9: Peace through Strength? Or World War Trump?
Chapter 10: Defusing the Global Crisis
Postscript: It Can Happen Here
Notes
Index
Writing this book, has, like many of my previous works, been like shooting at a moving target. But in this case, the Trump administration has been zigzagging through ever-changing policies like a vampire in flight. I would like to thank the American University of Paris library staff for their assistance with my book projects over the years, and Nina Bechmann and Mohammad Abdalhaleem for their valuable assistance in volunteering to check over my endnotes, as well as Soyoung Park and Anita Maksymchuk for helping build my website. And once again I would like to thank Isabel, who had to put up with me working on this project from early morning to late at night for several weeks. And my daughters, Celine and Francesca, whom I have generally neglected in the process. I would also like to thank my editor, Steven L. Mitchell, Jade Zora Scibilia, Hanna Etu, Cheryl Quimba, and Jackie Nasso Cooke for their support and help in working with me on this project, and for those at Prometheus Books who originally proposed the title, _World War Trump_.
Let us hope that this project is not in vain—and that the Trump administration policies will not generate a global war. But, even then, it is crucial to begin to turn around the new arms race and concentrate on the real need for negotiated peace, development, and human fulfillment in the United States and abroad—and on a truly healthy and inhabitable planet.
"Disarmament, with mutual honor and confidence, is a continuing imperative."
—President Dwight D. Eisenhower, 1961
"At this moment, for example, in 1984 (if it was 1984), Oceania was at war with Eurasia and in alliance with Eastasia. In no public or private utterance was it ever admitted that the three powers had at any time been grouped along different lines.... The enemy of the moment always represented absolute evil, and it followed that any past or future agreement with him was impossible."
—George Orwell, _1984_ (1948)
_"For the first time in all history, a great nation must go on arming itself more and more, not for conquest—not for jealousy—not for war—but for peace!"_
—Sinclair Lewis, _It Can't Happen Here_ , 1935
Toward the end of the Cold War, it became a cliché to cite the adage that the North Atlantic Treaty Organization (NATO) was created "to keep the Americans in, the Russians out, and the Germans down." Someone else later added as a joke: "to keep the French happy." But this original rationale for expanding NATO (as a collective defense organization in such a way as to keep the new Russia out of Eastern Europe and former Soviet states, the Germans/Europeans restrained, and the French happy, with the Americans fully in control) should have been thrown out the window once the Warsaw Pact dissolved in 1989. The Clinton administration should have put the whole NATO enlargement process on hold in the late 1990s and begun a full reassessment—just like one of the founders of the anti-Communist containment policy, Paul Nitze, among other officials and experts, had urged at the time.
The Clinton administration was not entirely oblivious to the possibility that NATO enlargement could eventually provoke Moscow. And it did at least superficially consider a range of options for European security that could have provided an alternative to NATO as the primary supplier of European security. But in the process of expanding a large and complex political-military bureaucracy, President Clinton decided to hedge his bets and opt for what could be called the NATO "self-limitation approach."
In the NATO self-limitation approach, NATO would not deploy foreign troops and nuclear weapons on the territory of new NATO member states. This approach was then confirmed by both NATO and Russia with the signing of the 1997 NATO-Russia Founding Act. The latter was intended to represent the basis for a closer NATO-Russia relationship, but Moscow saw it as almost immediately breaking down—given the fact that NATO's so-called exceptional war 'over' Kosovo, which was fought by NATO in 1999 against the interests of one of Russia's historical allies, Serbia, was not granted approval by the UN Security Council. Not only was that war in technical violation of the North Atlantic Treaty that had founded NATO, and against the spirit of the 1997 NATO-Russia Founding Act that Moscow had just signed, but the Clinton administration had opted to reject Russian proposals to deal jointly with the ongoing sociopolitical conflict in Kosovo—without even permitting Moscow to discuss those proposals thoroughly with either the United States or the other NATO members. And combined with the open NATO enlargement, this war represented one of the major factors that helped bring to power Vladimir Putin.
For more than twenty years, I have been warning that the uncoordinated NATO and European Union enlargements into former Soviet space would result in a Russian revanchist backlash—and that the major focal point of dispute would be Crimea. My argument was the following: If the Russian Federation was not fully included and engaged with both the United States and the Europeans in the formulation of the new post–Cold War security architecture, then the world could eventually expect a Russian backlash and the militarization of Russian behavior.
The militarization of Russian behavior would, in large part, result from the ambiguous nature of the 1997 NATO-Russia Founding Act. On the one hand, the expansion of NATO's integrated military capabilities and infrastructure to new members in eastern Europe, particularly once NATO proposed expansion to the three Baltic states, Georgia, and Ukraine in particular, risked a counter-military reaction by Moscow. On the other hand, the open NATO enlargement also made it more difficult to defend NATO's new members—that is, without the deployment of conventional forces coupled with direct threat to use of nuclear weapons the closer that NATO moved to the Russian border without a geostrategic "buffer" of neutral states.
That easily predictable Russian backlash against both NATO and European Union enlargement has now taken place. Much as NATO was seen by Moscow as containing Russia in geostrategic and military terms, Moscow also saw the European Union as seeking to isolate Russia in political-economic terms. And it is now the 1997 NATO-Russia Founding Act, among other international accords, that have been put in question in the aftermath of the early 2014 Russian annexation of Crimea, Moscow's political military interference in eastern Ukraine, and the buildup of Russian forces in the Baltic region. These Russian counteractions have led to the subsequent deployment of NATO forces in Poland and the Baltic states on a rotating, yet possibly permanently rotating, basis. In effect, rather than working together with Moscow to forge a new conjoint system of post–Cold War European security since the Gorbachev and Yeltsin administrations, the United States, NATO, and the European Union have achieved what can truly be considered a self-fulfilling prophecy by provoking a Russian backlash.
But that is not all. I also argued that along with a Russian backlash, a general militarization of interstate behavior would concurrently take place. Such a general militarization—which would include states such as the People's Republic of China in closer alliance with the Russian Federation, Iran, and other countries—would develop over time as the new powers that would emerge after Soviet collapse would soon resist US efforts to expand its global hegemony in eastern Europe, in the Indo-Pacific, and throughout the "wider Middle East" and much of the world.
Not only that, but these major and regional power rivalries are beginning to merge with the ever-expanding Global War on Terrorism that was initiated by George W. Bush against both anti-state organizations such as al-Qaeda and so-called rogue states including Afghanistan under the Taliban, Iraq under Saddam Hussein, and Libya under Muammar Gaddafi. The concern raised here is that the human and political costs of the US retaliation for the September 11, 2001, attacks—a retaliation which should have focused on al-Qaeda alone—have far exceeded the actual damage caused by those attacks, and that these wars on both "terrorism" and "rogue states" cannot be judged to be "successful" by any standard. Since 2001, approximately 370,000 people have been killed by violence in Iraq, Afghanistan, and Pakistan. At least 200,000 civilians have died in this fighting. Moreover, at least 10.1 million Afghans, Pakistanis, and Iraqis have been surviving as war refugees in other countries, or have been forcibly displaced from their homes.
In terms of costs, the United States alone has spent or committed at least $4.8 trillion on the wars in Afghanistan, Pakistan, and Iraq—a sum paid for almost entirely by heavy borrowing. Depending on the costs of the ongoing wars against the Islamic State, future interest payments could total over $7.9 trillion by 2053. Despite these huge costs, and despite the highly unlikely possibility that Washington can bring many of these conflicts to "successful" conclusions, the Global War on Terrorism is now being extended by President Donald Trump in Syria, Iraq, and Afghanistan, and countries such as Niger, among others. Trump has proclaimed that he would "eradicate" radical Islamist terrorism "from the face of the Earth"—but without necessarily pointing to feasible diplomatic solutions to establish peace in the long term in the aftermath of those military interventions.
On the domestic side, the intensification of major power rivalries and sociopolitical struggles within states, coupled with the 2008 global financial crisis, has indirectly resulted in the rise of a number of authoritarian or "illiberal" democracies. Recall, President Bill Clinton had originally justified NATO enlargement as least in part on the basis that NATO would help to stabilize fledgling eastern European democracies—even if NATO was not exclusively democratic at the time of its conception. And yet, in contemporary circumstances, NATO members Hungary and Poland—and particularly Turkey—can now be considered authoritarian states, or in the new formula, "illiberal democracies." And NATO members Bulgaria, Hungary, the Czech Republic, and Turkey have all begun to flirt with Russia, for better or for worse.
On the international side, the ongoing conflict between major and regional powers and differing anti-state "terrorist" groups has increasingly taken place both outside and within domestic societies through new forms of hybrid warfare, cyber-sabotage, and acts of partisan "terrorism." The modernization of nuclear and conventional weaponry, combined with the deployment of advanced US missile defense systems, not to overlook the deceptive tactics of hybrid and cyber-warfare, have largely rendered the Cold War concept of mutual assured destruction (MAD) obsolete. As indicated by the tremendous risks involved in President Trump's nuclear brinksmanship with North Korean leader Kim Jung Un, it has become increasingly evident that any number of direct, or even indirect, conflicts in differing regions of the world could draw major and regional powers into a direct confrontation. The possibility of major power war—most likely with the use of nuclear weaponry—is real.
The danger that is now confronting the world is that these twenty-first-century hybrid wars against both "terrorists" and "rogue states," combined with major power rivalries, are now leading to the formation of two contending systems of alliances. The United States, NATO, Ukraine, Japan, Israel, and Saudi Arabia have all begun to take steps to align more closely with each other against Russia, China, or Iran. States such as Belarus, Bulgaria, Hungary, Turkey, the Czech Republic, Serbia, Qatar, the Philippines, Pakistan, and India, among others, have all been caught up in the cross fire and may either strengthen their present ties with either the United States/NATO or with Russia and China—or else switch to the other side, if they do not opt for neutrality. In any case, the very threat to switch sides further exacerbates regional and global tensions.
_World War Trump_ argues that the new "America First" nationalism—coupled with Trump's largely unexpected and erratic foreign-policy flip-flops and willingness to use force—will provoke even greater regional sociopolitical-economic instability and interstate disputes than those that already exist. In essence, Trump's threatening actions and the general militarization of American policy could soon polarize much of the world into two rival alliances.
Trump's impatience; his Nixonian "madman" behavior; and his wild, unstatesmanlike foreign-policy flip-flops make both rival states and present allies automatically assume worst-case scenarios—as leaderships fear that the United States will not keep its promises or that Washington will radically alter its policies. Trump claims that he wants to bargain from a "position of strength," yet the United States is already seen as the predominant global power by far. America is already great and does not need to be "made great again"—at least not in Trump's militaristic manner of thinking. The risk is that Trump's "Peace through Strength" approach could soon spark a number of potential military confrontations—if his foreign policies are not accompanied by a sincere effort to seek out compromises and even make concessions through intense bilateral and multilateral negotiations. In this respect, Trump missed a major opportunity in his address to the UN General Assembly on September 19, 2017, to formulate a concerted path toward global peace that would involve the United States, the Europeans, Russia, and China, among other concerned states.
In order to defuse a truly critical state of affairs, _World War Trump_ proposes that Washington find ways to work with both Moscow and Beijing, in addition to other major and regional powers, through bilateral meetings and through multilateral Contact Groups, backed by the United Nations or the Organization for Security and Co-operation in Europe (OSCE), in the diplomatic effort to ameliorate political-economic tensions and disputes in key regional "hot spots" throughout the world. These multilateral Contact Groups need to prevent the global geopolitical system from polarizing into essentially two hostile systems of alliances, by seeking to better manage, if not resolve, key regional disputes that could potentially draw major and regional powers into direct conflict.
**CHAPTER OUTLINE**
Before we move forward, I want to present to you the general structure of _World War Trump_. As you have seen, this introduction, "A Self-Fulfilling Prophecy," argues that the Russian backlash to the uncoordinated NATO and the European Union "double enlargement" represents a self-fulfilling prophecy.
Chapter 1, "The Perils of the New 'American First' Nationalism," outlines the general arguments of the book and argues that Trump's "America First" nationalism and his often-contradictory foreign policies and policy flip-flops will prove destabilizing and provocative. The chapter discusses multiple tactics of the "hybrid warfare," including cyber-sabotage, "democracy engineering," and Russian "nationalist engineering." It argues that Trump-Pence policies, geopolitical rivalries between major and regional powers, and the new US-Russian arms race could lead the global system to polarize into two contending alliance systems. And that new techniques of hybrid warfare and acts of "terrorism"—which impact both the domestic and the international relations of differing states—could help spark a major power war.
Chapter 2, "Inauguration Tremors: Rifles, Tanks, and Nuclear Weapons," critically examines Trump's "Make America Great Again" budget and discusses the real social and political-economic costs of his proposed military buildup and the perverse impact of what President Eisenhower called the "military-industrial" (and congressional) complex on the American political economy and society.
Chapter 3, "The New Bogeymen: Russians, Immigrants, Muslims—and the Question of Impeachment," examines the domestic political impact on the United States of both the alleged Russian cyber-tampering and the accusations of Trump complicity with Moscow on the US presidential elections. The chapter critically discusses Trump-Pence policies and the apparently growing popular sense of alienation from the American system of democratic governance. Issues include the US electoral college system; the growing gap in wealth; Trump's opposition to the Affordable Care Act ("ObamaCare"); domestic violence, terrorism, drugs, and gun control; and Trump's controversial policies toward Mexican immigration and toward immigrants from Muslim-majority countries. The chapter concludes with a discussion of the prospects for Trump's impeachment.
Chapter 4, "Risks of the New American Nationalism for the European Union," discusses the impact of Trump's strong support for the British exit from the European Union ("Brexit") and for nationalist movements in general. The chapter examines Trump's approach to Germany; the impact of sanctions placed on Russia in the aftermath of its annexation of Crimea in 2014; the rise of anti-EU and anti-NATO movements in France and throughout Europe after Brexit and plans for strengthening the European Union; Russian and US attempts to influence elections in Europe; and Moscow's negative reaction to EU efforts to bring former Soviet bloc states into a closer political-economic partnership with Europe.
Chapter 5, "The Risk of War over Crimea, the Black Sea, and Eastern Europe," examines why Trump suddenly flipped from opposing Ukraine's efforts to regain Crimea after the Russian annexation in early 2014 to supporting Kiev's efforts to regain Crimea, and why Trump no longer calls NATO "obsolete." The chapter discusses the sociopolitical ramifications of Trump-Pence efforts to press all NATO members to boost their defense expenditures in the effort to counter Russian military pressures in the Black Sea region and in eastern Europe in general. Given the rise of the authoritarian Erdoğan regime in NATO member Turkey, plus an authoritarian leadership in Hungary, NATO has begun to lose its "democratic" credentials. While Washington has begun to fear the potential breakup of NATO due to a potential Turkish defection, Moscow has begun to fear the potential defection of Belarus from the Russian-led Collective Security Treaty Organization (CSTO). These fears are leading both Washington and Moscow to attempt to tighten their alliance relationships against one another.
Chapter 6, "The Global Impact of the China-Russia Eurasian Alliance," explains how US defense and alliance policies have been pushing Russia and China into a closer alignment, not only in Eurasia but also throughout much of the world, including Mexico, Venezuela, and much of Latin America, where they intersect with the ongoing War on Drugs, thus impacting regional US and domestic interests. In analyzing the growing influence of China's Belt and Road Initiative (BRI) and its Regional Comprehensive Economic Partnership (RCEP) on Pakistan and on other states throughout the Indo-Pacific region, the chapter argues that India represents the key "pivot" state that could either move closer to a Russian-Chinese-Iranian Eurasian alliance or else toward a US-European-Japanese alliance—if New Delhi cannot remain neutral and become a potential mediator.
Chapter 7, "China, North Korea, and the Risk of War in the Into-Pacific," analyzes the regional implications of Chinese-Taiwanese-Japanese disputes and conflicts over the South China and East China Seas for the United States and Russia. The chapter then focuses on the real threat of nuclear war with North Korea, which could engulf the entire region—if the Trump administration does not soon engage in real negotiations involving the six powers most concerned, the United States, China, Russia, Japan, South Korea, as well as North Korea.
Chapter 8, "Syria and Widening Wars in the 'Wider Middle East,'" critically examines Trump's decision to bomb a Syrian airfield with 59 Tomahawk cruise missiles in April 2017 after the Syrian regime of Bashar al-Assad allegedly used poison gas against its own population. The chapter argues that Trump's strong backing for Saudi Arabia and his opposition to the Iran nuclear accord signed by the Obama administration will not only antagonize Iran but also divide the Europeans, and press Tehran closer to Russia and China—given the ongoing proxy war between Iran and Saudi Arabia that has enveloped most of the wider Middle East and that is spreading into new regions throughout the world.
Chapter 9, "Peace through Strength? Or World War Trump?" critically examines Trump's "America First" policies and argues that they could lead to polarization of the world into two rival alliances. To prevent the latter, the United States, along with the Europeans and Japan, must engage with both Russia and China in the effort to resolve disputes and conflicts over a number of regional hot spots. In effect, it is argued that geopolitical tensions will not be abated until the issues of Crimea, Kaliningrad, Kashmir, Taiwan, and North Korea are fully addressed by the major powers themselves through UN- or OSCE-backed Contact Group diplomacy—in a geostrategic context in which countries such as India and Turkey could play major diplomatic roles. Concurrently, the major powers need to bring Saudi Arabia and Iran into a rapprochement over the Israeli-Palestinian conflict and the ongoing wars in Syria, Iraq, and Yemen.
Chapter 10, "Defusing the Global Crisis," outlines ways to reduce, if not eliminate, nuclear weaponry. It argues for engaging in multilateral Contact Group diplomacy, backed by the United Nations and the OSCE, to help resolve a number of key regional disputes and conflicts. It emphasizes the need for NATO and the European Union to build effective peacekeeping forces that can work with Russia, China, and other major and regional powers under UN or OSCE mandates. In addition to arguing for implementing international legal norms to establish joint-sovereignty arrangements for territorial disputes, Chapter 10 also critiques Trump's decision to drop out of the 2015 United Nations Climate Change Conference (COP 21) and argues that Trump's antediluvian emphasis on fossil fuels will amplify the global environmental crisis and exacerbate the very sociopolitical-ecological problems that could lead to wider wars—while also isolating the United States in the world community.
And, finally, America cannot truly help to resolve many of the world's problems unless it also engages in major social and political reforms at home—by reforming the electoral college system, by better controlling and reducing spending on the federally subsidized military-industrial-congressional complex, and by engaging in new approaches to the War on Drugs, gun control, and immigration reforms. If federal, state, and local debts (roughly $23.2 trillion in 2017) continue to skyrocket, more radical constitutional reforms of the US bicameral system of democratic governance and restructuration of the fifty-state system could be considered in order to reduce costs and provide fairer and more effective national, regional, and local governance that brings American leadership much closer to the needs and interests of the population. Most crucially, and in priority, the tremendous gaps in income need to be reduced through the implementation of practical and non-ideological systems of shared capitalism and workplace democracy in different kinds of enterprises.
_World War Trump_ is primarily focused on reorienting American foreign and defense policy away from the pursuit of global America First hegemony and toward an omnidirectional peace-oriented diplomacy of interstate conflict resolution and inter-societal reconciliation intended to prevent a new arms race and the subsequent polarization of the world into two rival alliances. Nonetheless, the final chapter sketches a number of possible domestic US reforms that represent a practical alternative to those proposed by the Trump administration and that can hopefully be developed in greater detail in a sequel to this book. Given the depths of the global geopolitical and financial crisis now confronting the United States and the world, the point is that the prevention of major power war will also require radical reforms of the military-industrial-congressional complex, as urged by President Dwight D. Eisenhower in his January 1961 farewell address. The United States will not only need to reformulate its foreign and defense policy but also radically reform its system of governance and its domestic political-economy—if it is to both achieve peace abroad and work to mitigate tendencies toward even deeper social, economic, and political polarization within the United States itself.
Even before Donald Trump's first one hundred days in office had finished, his administration was already confronted with a number of domestic and international crises. Trump's policy proposals—and efforts to implement those policies without full consultation with the parties involved—have been met with significant domestic and international political opposition.
Trump's hastily conceived and executed foreign- and domestic-policy decrees; his often-incoherent statements, tweets, and actions with respect to Russia, Ukraine, Crimea, China, and Taiwan, and the ongoing wars in Syria and Iraq; his strong criticism of President Barack Obama's nuclear accord with Iran; his attempts to impose a ban on immigration to the United States from six or seven Muslim-majority "countries of concern"; his policies toward "illegal" immigration and Mexico and Venezuela; his efforts to extend the Global War on Terrorism to Afghanistan (again); his rejection of the COP 21 United Nations Climate Change treaty on global climate change; his failure to strongly condemn the white supremacist and neo-Nazi "Unite the Right" demonstration in Charlottesville, Virginia; his threats to "totally destroy" the country of North Korea in response to its nuclear weapons and missile programs; coupled with many other issues, have all generated considerable domestic and international controversy, protest, and dissent.
**TRUMP'S MAJOR POLICY FLIP-FLOPS ON RUSSIA AND CHINA**
It was not a very long time after he had become president that Trump had begun to alter many of his presidential campaign pledges, at least in respect to US foreign and security policy. Trump, who had depicted himself in simplified terms as essentially "pro-Russian" and "anti-Chinese" during the US presidential campaign, soon began to flip-flop on both positions, thus creating confusion as to what US global strategy should be toward its two major rivals.
In August 2016, with respect to Russia, presidential candidate Donald Trump had warned that US efforts to regain Crimea on behalf of Ukraine against Russia could result in World War III. Yet just two weeks after he became president, the Trump-Pence administration dramatically reversed course and took a much tougher approach toward the Russian annexation of Crimea and its political-military interference in eastern Ukraine. A year later, in August 2017, Trump reluctantly signed into law H.R. 3364, "Countering America's Adversaries through Sanctions Act," which strengthens sanctions placed on Iran, North Korea, and Russia.
Ironically, Trump's own contradictory foreign-policy proposals, plus congressional investigation into his alleged collusion and business deals with Moscow, and those of his associates, could potentially undermine his promised campaign efforts to achieve a deal with Russian President Vladimir Putin. In addition to Trump's newfound support for Ukrainian claims to Crimea, which Moscow had rapidly annexed in 2014, other policies that could alienate Moscow include Trump's massive nuclear and conventional arms buildup, his opposition to the Iran nuclear accord, and his strong support for Saudi Arabia against Iran. And, in the long term, his push for US shale energy development, coupled with his support for the 2016 Polish-Croatian "Three Seas Initiative," could potentially put the United States into direct rivalry with Russian energy exports to Ukraine and both eastern and western Europe.
Each of Trump's foreign-policy flip-flops are very problematic. Contrary to Trump's frequent declarations that NATO was "obsolete," the Trump-Pence administration soon claimed that it will strongly support NATO, although it still expected allies to spend up to 2 percent of their GDP on defense. And, contrary to Trump's statements that he was "indifferent" to the European Union (although this was dubiously the case), Trump spokespersons began to claim that they strongly supported the European Union. (See chapter 4.)
Prior to becoming president, Trump had stated that he did not care whether or not Ukraine joined NATO. Although Trump no longer appears to propose, as he did in November 2015, that Germany and the Europeans should play the major role in defending Ukraine, he has not yet stated whether he would seek to formalize Ukrainian neutrality or else bring Kiev into NATO. Instead of seeking a formal recognition of Ukrainian neutrality, the Trump administration, in part under congressional pressure, could decide to provide even greater US military assistance to Kiev in its struggle against Russian-supported "autonomists" in eastern Ukraine, thus further antagonizing Moscow—if no diplomatic solution can soon be found. (See chapters and .)
Both NATO and the European Union are in dire need of major reforms. Trump's policy flip-flops and unstatesmanlike emotional outbursts are not very helpful when concrete proposals are needed to solve complex problems. Most important, as discussed in this book, Trump needs to address the key issues of the proposed enlargement of NATO to Ukraine and Georgia, as they impact vital Russian security concerns. The United States and Europeans need to explore with Moscow the question as to whether alternative security systems for the Black Sea/Caucasus region and Ukraine can be implemented. And the world still needs to find ways to reduce, if not eliminate, step-by-step, nuclear weaponry where possible—in the process of de-escalating nuclear tensions with North Korea. (See chapters , , , and .)
With respect to China, Trump had initially planned to take a very confrontational approach toward Beijing. But he suddenly backed off. Prior to becoming president, Trump had threatened to play the Taiwan independence "card" in an attempt to obtain political, military, and economic concessions from Beijing. Then, just a few weeks after becoming president, Trump suddenly engaged in an about-face in a phone call with the president of China, Xi Jinping, in early February 2017. This reversal of policy was taken after the Chinese president stated he would not agree to speak with President Trump until after Trump had publicly acknowledged the "One-China" policy. Trump may have also suddenly switched positions due to his realization that Beijing was needed to help quell North Korea's nuclear weapons and missile programs. (See chapters and .)
**OTHER POLICY FLIP-FLOPS**
Trump likewise flipped on his Syrian policy. In April 2017, in the midst of his dinner with Chinese President Xi, Trump opted for "cruise missile diplomacy" by firing 59 Tomahawk cruise missiles at a Syrian airbase. This action was ostensibly taken to punish the regime of Syrian leader Bashar al-Assad for the purported use of chemical weaponry against the Syrian people in the ongoing civil war. The irony is that Trump had previously opposed similar missile strikes against Syria when Obama declared that Syria had crossed the "red line" in August 2013 after Damascus had previously been accused of using chemical weaponry. While Obama had opted not to strike, Trump decided to act: For Trump, Syria had now crossed too many "red lines." Under domestic pressure to act, Trump felt he needed to show what he believed to be strength and decisiveness. But his cruise missile diplomacy did not result in any major changes in Syrian or Russian policies in that brutal war.
Trump has also threatened Iran if the latter does not live up to its promises not to develop a nuclear weapons capability in accord with the Joint Comprehensive Plan of Action (JCPOA). That accord has been strongly backed by Russia and China, as well as by US allies France, Germany, and the United Kingdom. Trump's strong criticism of the JCPOA, and US and Israeli threats to engage in missile strikes against presumed Iranian nuclear sites, not only threaten to undermine Trump's promises to achieve a positive relationship with Moscow but also could alienate Iran, particularly if Trump or the US Congress eventually decides to decertify the JCPOA without clear evidence of Iranian cheating. This could then encourage Teheran and other countries in the region to develop nuclear weaponry; Saudi Arabia, for example, could look toward nuclear-capable Pakistan for assistance. And Trump's major $110 billion arms sale to Saudi Arabia in May 2017, which was designed in part to counter Iran's missile testing and conventional weapons buildup, will only exacerbate Iranian-Saudi proxy wars throughout the wider Middle East, from Syria and Iraq to Yemen and up into Afghanistan. (See chapter 8.)
Not only that, but Trump's strong criticism of the Iranian nuclear accord could undermine the possibility that North Korea would accept a somewhat-similar future accord. The failure to press North Korea toward a nuclear freeze, and then hopefully toward denuclearization, could then lead to the further proliferation of nuclear weaponry in the Indo-Pacific—if not to a nuclear war that devastates the Korean Peninsula and much of the region. Here, Pyongyang heard the opposite message than that which Trump had intended in Syria: Pyongyang has continued to test a range of weapons systems in the aftermath of Trump's Tomahawk cruise missile attacks on the Syrian airbase—in preparation for a possible war with the United States. Likewise, in the aftermath of Trump's September 2017 speech to the United Nations, in which Trump called Kim a "Rocket Man...on a suicide mission," Pyongyang boosted the rhetoric by threatening to detonate a hydrogen bomb over the Pacific Ocean, and by calling Trump "a mentally deranged US dotard." Put crassly, Trump and Kim have entered into a radioactive pissing match with potentially lethal consequences. (See chapters and .)
Trump's anti-immigrant and America First protectionist stance also impacts countries closer to the US homeland and could further destabilize Mexico. In effect, given the ongoing social conflict in Venezuela, Trump-Pence policies toward Latino immigrants could impact US regional security by opening up Central America and the Caribbean to even greater Chinese and Russian political-economic influences—while exacerbating the drug wars and terrorist activities. The need for stability in the region—and for counterbalancing Chinese and Russian influence—is obtaining global attention. NATO has been considering making Venezuela's neighbor, Colombia, NATO's first Latin American partner. In his September 2017 UN speech, rather than urging regional diplomacy, Trump threatened the possibility of US military intervention in Venezuela ostensibly in the effort to help "them...regain their democracy." An alternative is concerted mediation with Cuba, but Trump has downgraded US ties to Havana. (See chapters and .)
Trump has, however, maintained at least one presidential campaign promise. He has sustained his hardline stance against the Islamic State (IS). The proof is the Pentagon's decision (most likely approved by Trump, although denied by the White House) to pulverize a network of Afghan caves that were occupied by IS fighters, with the use of the massive "Mother of all Bombs." Trump has thus demonstrated his potential to deploy massive force in the Global War on Terrorism by dropping the most powerful non-nuclear bomb in American arsenal. Not only will the use of such a weapon make it more likely that innocent people will be killed as so-called collateral damage, but it also lowers the threshold for the possible use of nuclear weaponry.
**"AMERICA FIRST" NATIONALISM AND PROTECTIONISM**
Trump has called his right-wing revolution "America First." But it is not certain what this means in a highly interdependent world—even if that interdependence is highly uneven. The United States may still remain the predominant or hegemonic power for a decade or more, but its global interests are being challenged in different regions of the world. It is a situation in which Washington cannot manage all complex political, economic, security, and ecological issues singlehandedly. Truly global problems cannot be managed or resolved without the full cooperation of other states.
By claiming that the whole world has been "ripping off" the United States over the years, Trump has begun to criticize not only US rivals but US allies as well. He has threatened to place high tariffs not only on China but also on the US allies Germany, Japan, South Korea, Mexico, and other countries that have large trade deficits with Washington. These protectionist policies, if implemented, could result in new global and inter-allied trade and currency wars—and could potentially lead to an even deeper long-term recession/depression.
Trump's major economic concern has been the competition from China, with which Washington has a $310 billion trade deficit. Prior to becoming president, Trump attacked what he saw as unfair trade competition due to China's low wages, lack of environmental restrictions, and purported currency manipulation. These, he believed, were the major cause of the loss of an estimated 5 million US manufacturing jobs since 2001. The Trump-Pence administration consequently threatened to counter China's (formerly) cheap labor advantage by raising tariffs.
The key issue, however, is how much of the US job loss is actually due to manufacturing import competition from China or from NAFTA or other trade pacts—and how much of the job loss is actually due to technological innovation and automation that reduce the need for manpower. It is not at all certain that an America First protectionist policy will help regain the considerable amount of manufacturing jobs lost since 2001; this is due to the fact that job loss has also been caused by technological modernization and automation. Another issue is that while import substitution does impact manufacturing and other jobs, those jobs cannot easily be shifted into other sectors. Assuming that US firms cannot fully enter the China market, they could then seek out low-cost trade with other countries. Putting up protectionist barriers on China or other countries could then cause dangerous trade and monetary wars as a result of globally interconnected industries and technologies.
On the one hand, the US turn toward protectionism will press other states to also search for ways to reduce the types of interdependence that leaves them most vulnerable to economic coercion and financial sanctions. In this way, the search for new markets could be positive, in that it will provide states with greater freedom to pursue new markets other than the huge American market. On the other hand, it could also be negative, in that states might be less willing to abide by international laws, technical standards, and coordinated regulations—assuming that they can break somewhat free from corporate or technological interconnections. And by not forging multilateral trade agreements, states and their major corporations may have great difficulties in finding new markets and guaranteed access to increasingly scarce or expensive resources—hence opening the doors to both domestic and international conflict.
Trump's complaints about excessive Chinese imports, lack of US access to Chinese markets, and Chinese currency manipulation, not to overlook China's significant holdings of the US debt, represent a sign of major tensions between the two countries. With respect to Chinese imports, many goods are produced in China by US firms, to the greater profit of the latter. Both Washington and Beijing can be seen as manipulating their currency ratios, but these rates tend to fluctuate in differing cycles anyway, so that one side tends to criticize the other only when the terms of the exchange rate are not in that side's favor. So in April 2017, when the dollar was at a relatively high value versus other currencies, Trump no longer accused China of currency manipulation. Yet the issue has not suddenly disappeared; either country could still opt take strong measures by devaluing their currencies against the other if there is no formal monetary cooperation between the two sides.
The possibility of a trade and monetary war with China is further augmented by the significant growth of the American national debt (over 100 percent of GDP since 2012) that has made the United States dependent, at least in part, on Chinese finance, along with that of Japan. Beijing has been the major foreign purchaser of US government bills, notes, and bonds, and it holds even more debt than the amount owned by American households. As such, Beijing owned $1.24 trillion of the US debt as of September 2016, but has begun to slowly sell its holdings. The rest of the $19.5 trillion US gross federal debt (not including the fifty states and localities) is owned by either the American people or by the US government itself, in part in the form of trust funds for Social Security and for other programs such as retirement accounts. Combined with high levels of personal debt, the massive US debt could eventually impact the future well-being of the American population if the US economy goes into yet another tailspin after that of 2008. And a very large percentage of the US gross national debt is due to US investments in nuclear weaponry and infrastructure since the beginning of the Cold War, plus borrowing for the US-led military interventions in Afghanistan and Iraq since the September 11, 2001, attacks. (See discussion, chapter 2.)
The general danger is that the significant dependence of the United States upon foreign capital (primarily Chinese and Japanese), coupled with a substantial US current account and trade deficit, has historically resulted in an increase in domestic protectionist pressures. And this dependence on foreign capital could mean that the United States will be impacted by the vagaries of Chinese governmental policies—since China is not an US ally. Beijing's control over a significant portion of the US debt accordingly makes it difficult for Washington to challenge Chinese policies and provides Beijing with considerable political-economic leverage over the United States, given its occasional threats to sell its dollar holdings.
It is true that, at the present time, Beijing's threat to sell its US dollar Treasury holdings would lead the value of the US dollar to drop and the price of Chinese exports to the United States to rise. So it is dubious that Beijing would act on those threats any time soon. But in the not-so-long term, Beijing could opt to sell its US Treasury holdings—likely only once it had sufficiently expanded China's domestic consumption for its own products by augmenting incomes, while also expanding its exports to the markets of countries other than the United States.
In this regard, China is, in fact, expanding its regional and global markets though its Belt and Road Initiative (BRI) and through the funds invested by the Chinese Investment Corporation and the Asian Infrastructure Investment Bank (which Beijing sees as a means to counter to the US-dominated World Bank and International Monetary Fund, or IMF). And, given Trump's own hastily conceived decision to dump the Trans-Pacific Partnership (TPP), China will be able to more easily pursue its major new trade accord, the Regional Comprehensive Economic Partnership (RCEP), which includes Russia and possibly India—and which could soon become the largest trading bloc in the world.
The danger is that this US political-economic battle with China could soon play itself out in real battle zones in the Indo-Pacific region. Such a scenario could prove plausible once China develops sufficient military capabilities, with Russian backing, to protect its political-economic interests throughout the Indo-Pacific and overseas, including in Latin America and the Caribbean. It also depends on how Japan will react to the Chinese quest for regional, if not global, hegemony. Already there are signs of increasing Japanese militarization in response to both China's burgeoning military capabilities and North Korea's missile and nuclear weapons provocations. And Tokyo could, somewhat like Beijing, try to use its own holdings of US debt as leverage to draw the United States to defend Japanese interests. (See chapter 6.)
**FOCAL POINTS OF CONFLICT AND ALLIANCES**
Trump believes that he will be able to preserve American hegemony by engaging in one of the greatest military buildups in US, if not in world, history. Such a major US military buildup, combined with a NATO and Japanese military buildup, Trump believes, will preserve peace with Russia, China, Iran, North Korea, and other states and anti-state movements that might attempt to challenge US hegemony in their specific regions. As he put it, "nobody will dare question our military might again. We believe in peace through strength, and that's what we'll have."
Yet the situation is not quite so simple. Trump's foreign-policy flip flops over Crimea and Taiwan are extremely significant in that the primary geo-strategic, military, and political-economic tensions between the United States/NATO and Russia and between the United States, Japan, and China, revolve to a large extent around these two focal points respectively. In terms of geo-strategic and political-economic interests, both Crimea and Taiwan appear as crucial as the island of Gibraltar or the Falkland Islands for the United Kingdom, or the Panama Canal for the United States. In this perspective, Russian claims to Crimea appear to parallel Chinese claims to both Taiwan and islands in the South and East China Seas—as stated by US Secretary of State Rex Tillerson in his testimony before Congress.
Focal points of conflict represent areas where differing powers struggle for access to raw materials and markets, seek to secure spheres of influence and security, and attempt to establish geostrategic positions for defense or attack. In addition to Taiwan and Crimea, other focal points include Russian-controlled Kaliningrad, which most immediately impacts Poland, the Baltic states, Sweden, the European Union, and NATO. Disputes in the Black Sea region and Caucasus (Russia vs. Georgia; Azerbaijan vs. Armenia; pan-Islamist movements in Dagestan and elsewhere in the Russian-controlled Caucasus) similarly impact directly or indirectly the global geo-economic interests of the United States, the European Union, NATO, Turkey, and Russia. Disputes over differing islands in the South China and East China Seas impact the United States, Japan, and China, as well as the neighboring countries.
This is not to overlook how the burgeoning tensions over North Korea's nuclear program impact South Korea, Japan, and the United States, as well as China and Russia. The ongoing proxy wars in Syria, Iraq, and Yemen, and elsewhere throughout the wider Middle East between Saudi Arabia and Iran, represent focal points that have already drawn Russia, Iran, Turkey, Israel, Egypt, France, and the United States into the conflict in Syria, with many of the anti-Assad militias financed by Saudi Arabia and other Arab Gulf states or Turkey. US and NATO intervention in Afghanistan, and Indian-Pakistani rivalry over Kashmir, is beginning to draw in China and Russia. There is furthermore a real possibility that sociopolitical conflicts in Mexico and Venezuela, and other countries in Central America and the Caribbean, in part related to drug wars, could begin to draw the United States—if not NATO for the first time as well—into different forms of police or military intervention, as occurred during (and before) the Cold War.
The above areas all represent focal points of conflict that could either spark a major power war or else become theaters of conflict once a major power war breaks out. New regional or major power wars could soon be sparked if the geo-strategic, military, and political-economic disputes that surround these territories are not managed carefully and prudently. And in the background behind each of these conflicts there are deeper geostrategic and political-economic concerns that are often combined with politico-economic instability and domestic crisis. One of these major issues that could generate major power war is the potential breakup of the Russian-led Collective Security Pact (CSTO), as feared by Moscow, and the potential breakup of NATO, as feared by Washington.
**RECIPROCAL FEARS OF THE POTENTIAL BREAKUP OF NATO, THE EUROPEAN UNION, AND THE RUSSIAN-LED CSTO**
After seeking to check the NATO and EU double enlargement into Ukraine by annexing Crimea and by supporting "autonomist" forces in the ongoing conflict in eastern Ukraine, Moscow now fears the "loss" of Belarus and the potential breakup of its Collective Security Treaty Organization (CSTO). Moscow likewise opposes the further encroachment of NATO influence into Russian-proclaimed spheres of influence and security in the Black Sea region. At the same time, Moscow fears the possibility that Russian Federation itself will begin to disaggregate in the face of the NATO and EU double enlargement, combined with the rise of pan-Islamist secessionist movements within Russia itself. This is not to overlook the pressure of so-called democracy movements inside the Russian Federation or else regional secessionist movements in Kaliningrad and in Siberia. (If the latter, for example, were to secede from the Russian Federation, Moscow would not be able to benefit from Siberia's significant oil and gas wealth.)
The possible disaggregation of the Russian Federation itself is exacerbated by the fact that certain regions are nearly bankrupt, a factor that caused protest in 2017 throughout the country, and not just in Moscow. And unlike the disaggregation of the Soviet Union, the feared disaggregation of the Russian Federation could lead Moscow to engage in a full-scale Russian political-military buildup, which could involve more vengeful Russian actions that are intended to undermine US interests. Given the rise of a new Russian nationalism, it appears dubious that a new Russian reformer, somewhat similar to former Soviet president Mikhail Gorbachev, will be able come to power once Vladimir Putin eventually steps down.
While Moscow fears the eventual breakup of the CSTO, Washington somewhat similarly fears the potential breakup of NATO but for differing reasons. As Turkey becomes an illiberal democracy, it could potentially break away from NATO, for example. Some NATO members, such as Hungary, the Czech Republic, and Bulgaria, are beginning to seek closer relations with Moscow, while NATO member Poland appears to be sliding toward authoritarianism—even if it is a strong supporter of NATO. Most problematic, both the French far right and far left strongly oppose NATO membership, as do most far-right and far-left groups in Europe.
On the one hand, Trump-Pence administration pressures on NATO members to raise defense spending to 2 percent of their GDP could lead some states—specifically those that are in a dire financial situation—to drop out of the alliance. On the other, even an economically powerful state, such as Germany, could drop out of NATO if Berlin enters into a trade war with the United States over sanctions on Russia, among other disputes, and in the effort to forge an all-European system of security and defense. A relatively more independent Europe could then seek a separate accord with Russia or, more likely, with China. In fact, Trump's June 2017 decision to drop out of the 2015 United Nations Climate Change Conference (COP 21), coupled with major disputes with Russia, could help press the Europeans into closer political-economic and technological relations with China. Ironically, Trump's decision against the COP 21 appears to contradict his own concept of America First—that is, if that concept can be defined to mean that the United States should take leadership in innovation. (See chapters and .)
Added to the US fear of a breakup of NATO is the potential breakup of the European Union after the United Kingdom's exit from the European Union (Brexit). Trump's proclaimed indifference to the European Union after Brexit—coupled with his ideological support for anti-EU, anti-NATO nationalist movements, combined with very strong criticism of Germany as the political-economic leader of the European Union—have risked undermining the European Union, even if Trump has attempted to backtrack from his previous positions.
One of the major reasons for the Trump-Pence administration to so suddenly reverse course and no longer call NATO "obsolete"—and to speak so highly of both NATO and the "wonderful" EU—is precisely to whip US allies into line. In effect, US efforts to boost NATO defense spending is intended to tighten the defense relations of NATO allies against their common foes. But high levels of defense spending will not resolve the financial crisis impacting European economies. Instead, high defense spending could exacerbate that crisis through increased government borrowing to pay for military capabilities. As the Trump-Pence administration seeks to press US allies (in Europe, in Asia, and in the wider Middle East) to spend more on defense (generally expecting those allies to "buy American"), Trump's America appears to be acting somewhat like ancient Athens as the latter tried to force its allies to pay their dues in the struggle against Sparta during the Peloponnesian War.
In effect, both Washington and Moscow have been attempting to whip their respective NATO and CSTO allies in line—while concurrently probing the weaknesses of the rival alliance. For its part, Moscow has sought to strengthen its ties to China and Iran while also boosting its own defense capabilities in an effort to strengthen the defense relations of its CSTO allies against their common foes. Moscow has also hoped to counter US nuclear and conventional force superiority by asymmetrical military means—leading to a new arms race.
**THE NEW ARMS RACE**
Initially, during his presidential campaign, and in the early days of his presidency, Trump had proposed to reduce US and Russian nuclear weaponry—in the optimistic assumption that he would be able to make an arms deal with President Putin that would then eliminate economic sanctions placed on Moscow after its annexation of Crimea in 2014, for example. Yet once becoming president, Trump has planned to surpass Obama's own nuclear and conventional force military buildup—in order to pressure Moscow and other recalcitrant states to make deals on US terms in accord with the maxim: Peace through Strength. Trump wants the Pentagon to engage in a much larger and costlier conventional and nuclear weapons buildup than that which Obama had reluctantly initiated in the midst of his second term.
Obama had initiated a new nuclear and conventional arms race after his failure to "reset" US-Russian relations in 2009 in the aftermath of the five-day August 2008 Georgia-Russia war. This war was, in fact, initiated by Georgia after South Ossetian provocations, with Moscow waiting to pounce on the sidelines. And once Russian troops entered into Georgia itself, the Russian military then threatened Tbilisi, the Georgian capital. After Georgia fought very effectively for its size against Russia in the 2008 Georgia-Russia war, Putin opted for a major military modernization of its armed forces. This, in turn, combined with Beijing's push to militarize islands in the South China Sea and North Korea's nuclear weapons testing, led Obama to engage in a new nuclear modernization program. All of these factors led Obama to engage in a further buildup of US naval forces in what was called "rebalancing" to Asia.
Ironically, however, before announcing the new US nuclear modernization program, Obama had promised to abolish genetically genocidal nuclear weaponry altogether in his speech in Prague in April 2009. At that time, Obama had promised that the United States "would take concrete steps towards a world without nuclear weapons...and urge others to do the same." (See chapter 10.)
Obama's efforts to "reset" US-European-Russian relations in 2009 had failed to pursue two proposals that might have opened the door to peace. The first was then Russian President Dmitri Medvedev's June 2008 call for a new European Security Treaty. The second was Turkish President Recep Tayyip Erdoğan's August 2008 call for a new Caucasus Stability and Cooperation Platform. If both of these proposals had been taken more seriously, and more thoroughly discussed and negotiated by the United States, NATO, and the European Union, with Turkey and Russia, then the present crisis might not have reached such a dangerous impasse. (See chapter 9.)
In sum, Trump has advocated building up US military power beyond Obama's military buildup—by means of advocating a Nixon-Reaganite policy of Peace through Strength. The goal of this military buildup is intended to put North Korea and Iran, as well as Russia and China, among other states as well, on their guard. By threat to use force, if not nuclear weapons, Trump has hoped to press these countries into making concessions on US terms, where possible. At the same time, in pushing for an American military buildup, the Trump-Pence administration has also planned to press the Europeans, Japan, and South Korea, Saudi Arabia, and other states to boost their defense spending as well—while concurrently waging the War on Terrorism against the Islamic State (IS) and affiliates of al-Qaeda, among others. The risk, however, is that this tremendous US and allied military buildup appears to be bringing Russia, China, and Iran, among other states, into an even closer alignment. (See chapters , , and .)
**HYBRID WARFARE AND THE NEW GLOBAL RIVALRIES**
The global rivalries between the United States, the Europeans, Japan, India, Russia, and China, among other regional powers and anti-state organizations, are becoming even more dangerous—precisely because a number of countries are trying to intervene in the domestic affairs of their rivals through techniques of "hybrid warfare." This new form of warfare includes both cyber- sabotage and the cyber-manipulation of popular opinion. Moscow, among other states and anti-state groups, has been accused of engaging in cyber-tampering in different countries. Russian military intelligence, the GRU, has, for example, been accused of intruding into both the US and French presidential elections, through its cyber-espionage group, APT 28 (also known as Fancy Bear). In addition to propagandizing against US policies through RT and Sputnik broadcasts, Moscow has likewise been accused of paying trolls to manipulate US and European public opinion through Facebook and Twitter accounts.
Evidently, this is not the first time that foreign governments have tried to manipulate the views of domestic populations or steal secrets or destroy assets. Throughout history, states and anti-state sociopolitical political movements have used techniques of propaganda, industrial warfare, sabotage, theft, and assassination. Both Washington and Moscow have interfered in the domestic politics of differing countries throughout the world—as well as each other's. Yet it is the United States that has sought to intervene in the elections of other countries roughly twice as much as Russia or the Soviet Union have, at least since 1945.
From the Russian perspective, Moscow has opposed what it sees as US- and EU-inspired democracy engineering in countries with close ties to Russia or within the Russian Federation itself. Moscow considers this a form of "hybrid" or what it calls "nonlinear" warfare. From the Russian perspective, democracy engineering can be traced to US-supported democratic revolutions in Warsaw Pact countries (1980s), Serbia's Bulldozer Revolution (2000), Georgia's Rose Revolution (2003), Ukraine's Orange Revolution (2004), and Ukraine's Euromaidan Movements (2013–2014). This is not to ignore the Cedar Revolution in Lebanon (against Syrian occupation in 2005) and the Arab Spring movements in 2011–2013. The latter prodemocracy movements appeared, at least in part, to be aimed at overthrowing Russian allies, such as Libyan leader Muammar Gaddafi and Syrian leader Bashar al-Assad.
At the same time, however, these sociopolitical movements were not controlled or perfectly manipulated by Washington. The Russian critique of the Arab Spring movement, for example, ignores the fact that US and Europeans allies, such as Ben Ali and Hosni Mubarak, were also overthrown by democracy movements in Tunisia and Egypt, respectively. This indicates that the United States is not entirely a puppet master that can manipulate the entire theater of events through democracy engineering.
Nevertheless, after what Moscow considers US and EU interference in the Ukrainian election process in 2013–2014, President Putin stills fears the possibility that US- and EU-supported democracy movements could eventually attempt to overthrow his own rule—and, perhaps more likely, that of his unstable ally in Belarus, Alyaksandr Lukashenka. In November 2014, Putin observed: "In the modern world extremism is being used as a geopolitical instrument and for remaking spheres of influence. We see what tragic consequences the wave of so-called color revolutions led to.... For us this is a lesson and a warning. We should do everything necessary so that nothing similar ever happens in Russia." In effect, this statement implies that Moscow will try to repress such movements inside Russia, while also playing its own game of "nationalist engineering"—by supporting differing pro-Russian sociopolitical movements throughout the world and by interfering in the elections of other countries, if necessary. (See chapters , , and .)
Yet Russia is not the only country that has been accused of cyber-intrusions in American affairs. Chinese hacking into US military and corporate websites has been a major concern. Beijing has purportedly obtained access to US government, aerospace, military-technological, and corporate secrets, from Westinghouse and US Steel, for example. The latter included downloading 4.2 million government personnel files in 2015. The Chinese military purportedly possesses a 100,000-man cyber-espionage division. Beijing has also been an effective political lobbyist in the US Congress, more so than Moscow. More recently, North Korea has been accused of devastating hacking into Sony Pictures, banks, and corporate accounts, ostensibly in order to raise funds for its military programs. North Korean hacking against Sony represented a protest against the film _The Interview_ , a satire about a plot to assassinate Kim Jong Un.
And while the United States has pointed the finger at Russia, China, and North Korea for allegedly engaging in differing forms of cyber-tampering and cyber-sabotage in the United States and other countries, Russia and Iran have pointed the finger at the United States and Israel for allegedly engaging in Stuxnet malware attacks against Iran's Natanz nuclear facility—where Iran had been suspected of enriching uranium for military purposes. Both Russian and Iranian officials denounced the Stuxnet cyber-attack as an "act of war." The key issue raised by the Stuxnet attacks is not so much that the computer virus could spread out of control but that state and anti-state actors that possess the appropriate know-how could soon develop similar malware that can be used for cyber-sabotage. These prototypes could then proliferate much easier than does nuclear weaponry—with potentially devastating results.
The point here is that social-political manipulation of domestic societies by external powers is expanding beyond mere media propaganda and classical techniques like assassination and industrial sabotage. This manipulation is becoming part of a larger geopolitical struggle involving techniques of "hybrid" warfare and high-tech processes of manipulation and PSYOPS (psychological operations) that are intended, successfully or not, to transform domestic and elite opinion and mass social behavior. These manipulations are intended to influence the totality of a rival domestic society and its foreign policies and can thus impact a rival state's technological and industrial infrastructure, economies and banking systems, stock markets, corporations, state bureaucracy, governance processes, and international policies. Due to the deep interpenetration of each other's societies and political, economic, and financial processes, these new tactics make the prospects of new kinds of war even more likely. This means that the quest for American military superiority—short of being able to totally annihilate the enemy's military and its population through nuclear weaponry, as threatened by President Trump against North Korea—is impossible to obtain, given the new tactics of hybrid warfare that can engage in malevolent actions inside or outside the territorial perimeters of rival countries.
The danger is that these new forms of hybrid warfare coupled with the general rise of nationalism throughout the world—both of which are being exacerbated by the Trump-Pence administration's military buildup and America First doctrine—tend to press state leadership to demand presumed unilateral "solutions." Yet no conflict can be fully resolved unilaterally—all disputes and conflicts need concerted attention of the major and regional powers and representatives of the populations most concerned.
**TOWARD A WORLD OF POLARIZED ALLIANCES?**
If Washington—as the still globally hegemonic power—cannot, in the next few years, work to achieve a concerted and cooperative relationship with both Russia and China, then the post–Cold War constellation of major and regional powers could soon polarize into the formation of two rival alliance systems. On the one hand, Washington has already attempted to tighten its alliances with NATO, EU members, Japan, ASEAN states, Israel, Saudi Arabia, and the other Gulf states. On the other, Moscow and Beijing have begun to forge a new Eurasian alliance by means of linking the Russian-led Collective Security Treaty Organization (CSTO) with the Shanghai Cooperation Organization (SCO)—in addition to backing both Iran and Syria, while seeking out other potential allies, such as Brazil, South Africa, Qatar, and Turkey, if not both India and Pakistan, if possible.
As a burgeoning major power, India represents the key global pivot state that could join either alliance—as the United States, Europe, and Japan compete with Russia and China for New Delhi's political-economic and military allegiance. Should India join either alliance, that could represent a decisive shift in the global balance of power, norm, and strategic intent—which, in turn, could destabilize the global system, possibly leading to wider regional, if not major power, wars.
Without sustained steps toward a US diplomatic engagement with both Russia and China, a potential clash between two rival encircling and counter-encircling alliances appears to be in the making. These two essentially insular free trade versus continental protectionist alliances appear to be acting somewhat like the two opposing alliance systems, the British and French–led Triple Entente versus the German-led Triple Alliance, which unexpectedly exploded into major power war in August 1914.
In this perspective, the possession of nuclear weaponry by major and regional powers will not necessarily deter major power war. Despite the lingering Cold War myth of mutual assured destruction (MAD)—that a rough equality of nuclear weaponry will prevent two states from engaging in major power warfare—war between major powers in the new polycentric post–Cold War global system, in which differing states and anti-state actors possess highly uneven power capabilities and influence, becomes even more plausible given the rise of new high-tech forms of combat, advanced stealth systems, renovated tactical nuclear weaponry, non-nuclear Prompt Global Strike missiles, and hypersonic and thermobaric weaponry, combined with cyber-sabotage that can destroy military and civilian infrastructure. It is theoretically possible for major powers to involve themselves in a direct but intermittent conflict, keeping nuclear weapons on reserve or using tactical nuclear weaponry with "low" explosive yields. By contrast, lesser powers might be tempted to use nuclear weapons first—in order to offset their relative military weakness vis-à-vis major and regional powers. The real possibility of major power war is furthermore being augmented by the fact the Global War on Terrorism since September 11, 2001, is beginning to merge with rivalries between major and regional powers that could lead to the polarization of the global system into two rival alliances.
In such circumstances that appear to be forging a proto-Sino-Russian alliance, Washington will not be able to play the "China Card" against Moscow as it did during the Cold War. Nor will the United States be able to play Russia against China, given their close financial, political-economic, energy, and burgeoning military relationship. In order to prevent a Sino-Russian alliance from more strongly backing a number of countries, which could possibly include North Korea, it is absolutely essential to better coordinate US and European policy toward North Korea and Iran, among other states, with both Beijing and Moscow. This means that the United States will need to engage in intensive diplomacy with both Moscow and Beijing so as to prevent an even tighter Sino-Russian military alliance, while preventing the outbreak of war on the Korean Peninsula, and in the wider Middle East.
Given the above, Washington needs to engage in a strategy intended to prevent two possible scenarios. The first is the polarization of the world into two rival alliance systems. The second is the feared breakup of the US systems of alliances that could take place—if the United States cannot work with the Europeans, Japan, India, Russia, and China, as well as key regional actors, to establish new systems of regional and global security. Either scenario could result in widening regional wars—if not major power war.
To prevent either of these scenarios, the Trump-Pence administration (or a future US leadership) needs to pursue a full-fledged and concerted diplomatic engagement with both friends and foes alike. In other words, the United States, the Europeans, and Japan must begin to engage in _a truly peace-oriented diplomacy_ with _both_ Moscow _and_ Beijing. The concern raised in this book is that it is not at all clear that Washington can reach international agreements and forge solid alliances—a hope that President Trump himself has expressed—if all countries continue to assert their presumed national interests _above_ the interests of other states.
As Trump put it in his inauguration speech: "It is the right of all nations to put their own interests first." But here, Trump makes no real distinction between vital and secondary interests. And he does not indicate whether so-called vital interests can be modified through diplomatic compromises and concessions. Trump's threats and actions have additionally raised questions as to whether his policies are truly of the general American interest or those of his personal interests (or those of his associates). (See chapters and .)
Trump's repeated statement that NATO is "obsolete" does not address the major issue and, in fact, makes the security situation even more precarious. The main issue should be how to reform NATO and permit it to retract gracefully from its promises of an open enlargement, while concurrently pressing for a rapprochement between NATO, the European Union, and Russia. The key problem is to put an end to Russian fears of encirclement and regime change, while also mitigating NATO fears of Russian efforts to probe NATO political and military weaknesses. More specifically, the fundamental question is whether NATO should continue to expand its "open" membership policy to Ukraine and Georgia or other countries (as promised in the 2008 Bucharest, 2010 Lisbon, 2012 Chicago, 2014 Wales, and 2016 Warsaw NATO summits) and how to bring NATO and the European Union into more positive relations with Moscow.
As to be argued, if bargained cautiously, a resource- and industrial-rich "neutral" and "decentralized" Ukraine with adequate self-defense capabilities, and with Crimea as an "free-trade zone" under Russian sovereignty, could begin to defuse tensions between the United States, the European Union, NATO, and Russia. It is essential that Russia and Ukraine learn to live side by side—as these two large and contiguous countries will remain in uneven political-economic, energy, and financial interdependence upon each other. Concurrently, it is crucial to find ways to bring China into a closer rapprochement with Japan—while all concerned states need to work to de-escalate the North Korea nuclear weapons and missile threat as soon as possible. (See chapters and .)
Trump has rightfully hoped that NATO, the Europeans, and Russia can look for ways to cooperate in the Global War on Terrorism, yet such actions do not appear sufficient to prevent a new global arms race. Nor will such cooperation prevent the real potential for widening conflict—unless diplomatic cooperation between the major powers is soon expanded. Here, the United States, Europeans, Russia, China, and Japan will need to find ways to reach a rapprochement between Saudi Arabia and Iran, India and Pakistan, and Israel and the Palestinians—if the Global War on Terrorism is ever to come to an end.
The concern raised here is that nationalist America First Trump-Pence administration ideology tends to undermine the crucial need for concerted multilateral diplomatic efforts to achieve global peace through international organizations such as the United Nations, the OSCE, and even more practically through multilateral Contact Groups, among other intergovernmental and nongovernmental inter-social and inter-religious forums that can help find solutions to disputes and conflicts between differing states and societies. The option of joint sovereignty arrangements, for example, backed by the United Nations, the OSCE, or a proposed Asian OSCE-like forum, may represent options that can help resolve interstate disputes over islands in the Indo-Pacific, if not over Crimea or elsewhere.
If the Trump-Pence administration continues to spout America First nationalism— _in such a way that Washington might not envision where and when it is absolutely necessary to compromise or concede on presumed vital interests_ —World War Trump could well be the result.
Most Americans—if not most of the world—were not expecting Donald Trump to be elected as the forty-fifth president of the United States. Trump nevertheless obtained a solid victory in the electoral college by gaining the majority of Midwestern and Southern states. His rival, former Secretary of State Hillary Clinton, actually won the popular vote by a colossal 2.8 million votes. Yet Clinton's electoral college votes, primarily from coastal states, were not sufficient to win the presidency. Had she obtained roughly 80,000 votes to win a few key states, Michigan, Pennsylvania, and Wisconsin, she might have won the election.
Much as was the case for the victory of George W. Bush, who had won his second presidential term in 2000 against his Democratic opponent, Al Gore, after winning the electoral college votes by a slim majority, Trump's victory once again raised significant questions as to whether the American system of democracy needs major reforms. Even though it was only the fifth time in US history that a president had won the election without winning the popular vote, some form of electoral reforms appear necessary in the short run. In the longer term, a restructuration of the fifty-state system may prove necessary—given significant federal, state, and local debts and huge imbalances in the population sizes and industrial/rural areas across the fifty American states. (See chapter 10.)
**PROTEST AT THE PRESIDENTIAL INAUGURATION**
Trump's inauguration festivities were confronted by a predominantly peaceful protest of an estimated 500,000 demonstrators (with some estimates as high as 2 million) in Washington, DC—out of an estimated 4.5 million people also protesting throughout the country and much of the world. There were some acts of violence: in Washington, DC, some 209 of the 230 individuals arrested, including journalists, medics, and legal advisers, were charged with felony rioting. This raised fears that Trump could engage in a future crackdown on journalistic freedom.
These inauguration protests attracted many more people than initially expected—and many more than those who came to hail Trump on the day of his inauguration—although this latter fact was denied by the Trump administration as "fake news." The demonstrators had marched on Washington to protest Trump's proposed policies with respect to women, ethnic minority groups, the LGBTQ communities, and in opposition to his campaign pledges to eliminate key environmental programs, develop highly polluting shale energy, and to make deep budget cuts in governmental programs that deal with social issues. (This is not to overlook opposition to his plans for funding a major military buildup.) The protests put Trump on warning that he does need to address the concerns of all Americans—even those who proclaim, "Trump is not my president!"
**TRUMP'S DECREES**
Once in power, Trump almost immediately launched an attack on the "progressive" or "egalitarian" domestic American policies that are generally demanded by both radical and liberal Democrats. In opposition to President Obama's essentially liberal policies, Trump's nationalist America First policy has generally opposed multilateralism, international trade pacts, arms control, high taxes on corporations and wealthy individuals, and strong federal-government regulation with respect to environmental protection, healthcare, gun control, and US-government backing for the political and legal rights and entitlements of women, minority groups, and immigrants.
Assuming Trump can eventually push his agenda through a Republican-dominated Congress—which will not be so easy, given the fact that not all Republicans necessarily support his policies—some of his proposed reforms appear to promise a boon for a number of corporations (depending on the sector), the banks, and the military-industrial complex in particular. In essence, Trump seeks to dismantle many federal government programs—in seeking budget cuts of up to $3.6 trillion over the next decade—while seeking to boost military spending considerably.
Trump's decrees are generally—but not necessarily—seen as welcome by a number of private enterprises and corporations, particularly given his promises to reduce corporate taxes significantly. Trump initially promised major corporate tax cuts (down to 15-20 percent from 35 percent); cuts in federal regulations (including the banking/finance sector) by 75 percent, if not by more; and a controversial protectionist import or border tax of 20 percent to 35 percent. Yet Trump's tax cut proposals could increase the national debt by $3 to 7 trillion over the next decade.
In order to reduce competition in the name of "economic nationalism," Trump furthermore promised to place a high tax on products from countries such as Germany, Japan, Mexico, China, and others. Trump likewise criticized American automobile firms that build cars in lower-cost foreign factories. Trump's proposals appeared to ignore the fact that many of these countries are highly dependent on the US market, and that many of these products are manufactured abroad by US-based multinational corporations. Moreover, his initial plans to tighten immigration controls were not necessarily seen as positive for agro-industrial firms, food services, restaurants, and retail concerns, as well as short-term construction projects or agricultural needs that often seek out both authorized and unauthorized immigrant labor. Here it has been argued that US firms need the influx of migrant labor, given the fact that the US harvest requires between 1.5 million and 2.2 million workers annually and at least 50–70 percent of farm laborers in the United States are unauthorized. If the United States did not actually face a shortage of labor in agricultural production, the US GDP would have grown by almost $12.4 billion in 2012 and would have produced almost $4.9 billion more in annual farm revenues.
In his first hundred days of office, Trump issued more executive orders than any president since Harry Truman—despite previously criticizing Barack Obama for engaging in "major power grabs of authority."
**THE GROWING GAP IN WEALTH**
One of the issues that is beginning to delegitimize the American system of governance both at home and abroad is the growing gap between the very wealthy in the United States and the rest of population. In a word, as it takes either tremendous wealth—or else access to finance—just to run for office, the United States is beginning to look like it is being run as a plutocracy. In the United States, the median top 5 percent of households possess more than ninety times the wealth of the median US family. In his presidential campaign, despite his own considerable wealth, Trump himself had denounced outrageously high CEO pay. The chief executive officers (CEOs) of America's biggest companies can earn at least three times more than they did twenty years ago and at least ten times more than thirty years ago. Some estimates put CEO pay at 373 times the average worker's pay. Once he became president, however, Trump's proclaimed concerns with income inequities were quickly forgotten.
Ironically, the independent political machine of Donald Trump, the billionaire, actually spent less money to win the election than did the well-oiled Democratic machine of Hillary Clinton. The total amount spent for the election process is outrageous. Clinton spent almost $1.2 billion, and lost—because of the electoral college. Trump won the electoral college but spent only a little over half of what Clinton spent, roughly $650 million. Trump also needed less money because he was able to use some $66 million of his own money, his own private jet, and other Trump facilities.
Trump has furthermore surrounded himself with wealthy advisers—showing a nepotistic penchant for members of his own family. Trump has packed his Cabinet and administration with a number of individuals who are highly successful in financial terms but who do not necessarily possess significant (or any) governmental and political experience. Trump prefers to support those whom he believes represent real world "success"—as opposed to the input from academic and policy experts. And if Trump's appointees are not wealthy individuals, then they often possess considerable military experience—as has been the case with his National Security Council. The careers of many of the latter individuals (including John Kelly as chief of staff, Jim Mattis as head of the Pentagon, and H. R. McMaster as National Security Advisor) appear heavy on military experience yet lighter on expertise in the areas of diplomacy and diplomatic engagement. Trump's dual emphasis on leadership from the corporate world and from military is taking place at the same time as he seeks to cut funding for the State Department and foreign service—all in a global situation in which effective diplomacy is most needed. At the time of Trump's first one hundred days of office, there were still about two hundred positions at the State Department that required Senate confirmation. The United States lacked ambassadors to NATO, the European Union, France, Germany, and Russia. For some very important diplomatic positions, Trump has been very slow to name nominees, while some of his appointments have questionable or politically biased qualifications.
Yet the very fact that Trump has put the new captains of industry, high tech, and finance, and high-ranking military, in charge of governmental affairs, symbolically puts the capitalist system and the military-industrial complex on trial for the American population and the world. In other words, Trump's success or failure could prove symbolic for the success or failure of capitalism and the American democracy as well. (See chapter 10.)
From this perspective, the very nature of the Trump administration serves as evidence that the United States is ruled by a wealthy plutocracy—and not by the neo-liberal myth of meritocracy. Moreover, Trump's policies of high defense spending—what is called military Keynesianism—appear to advocate a new form of supply-side trickle-down economics for the greater benefit of the defense-sector industries that are spread across most of the fifty US states. Trump has promised that the tremendous benefits accrued by the wealthy through tax reductions and for the military-industrial complex will eventually be passed down to the middle class and poor—but without any political guarantees whatsoever that that will prove to be the case.
**POST-OBAMACARE AND THE PROSPECTS FOR DOMESTIC SOCIAL PROTEST AND VIOLENCE**
In many ways, the growing gap in wealth in the United States is clearly illustrated by the lack of healthcare for many Americans. Trump's policies include the abolition of "ObamaCare" (the Affordable Health Care Act, or ACA), which is generally seen as very costly for business. And while Trump has promised to remake the Affordable Care Act, given its costs, he also, at least initially, promised not to cut Social Security, Medicare, and other entitlements. Because Medicare and Medicaid provide coverage for around 70 million poor, disabled, and elderly people, it is politically risky to touch; but this has not stopped some Republicans from so doing. Yet the potential failure of Trump to deliver on his promises to achieve a more effective healthcare system could substantially augment social discontent in the United States—at the same time that federal, state, and local debts continue to skyrocket to over $23 trillion—which is now over 100 percent of the US GDP.
By the end of September 2017, the predominantly Republican Congress was not able to find a way to replace the ACA. Influential Republicans such as Susan Collins, John McCain, and Lisa Murkowski, and libertarian Rand Paul, all opposed the Graham-Cassidy repeal measure to replace it. Saner individuals realized that Republicans had not put together a viable option. Following the publication of the first version of Republican (GOP) healthcare plan in March 2017, it had been projected by the Congressional Budget Office that 14 million more people would be uninsured under the GOP legislation than under current law by 2018. Then, following the additional proposed changes to subsidies for insurance, the increase in the number of uninsured people relative to the number under current law was expected to rise to 21 million in 2020 and then to 24 million in 2026 under the March 2017 GOP proposal. If it had passed, the GOP plan could have meant that millions of people with preexisting conditions could find themselves priced out of the market.
Those who supported the proposals argued that the first GOP plan would reduce the federal deficit by $337 billion in first decade—but most likely at the price of considerable domestic American discontentment and protest. The failure to replace Obama's ACA divided the Republican Party and represented a major defeat for Trump. But this defeat will not prevent future efforts to reform ObamaCare, hopefully more prudently, with the advice of experts from both nongovernmental advocacy groups and healthcare providers, and with greater consideration for all political perspectives. Job insecurity and lack of health insurance are both factors that can undermine ideological support for the American system of democratic governance, resulting in social protest, if not increasing drug abuse, criminality, and violence. Washington must eventually work out a more fair and equitable deal on these issues.
**US GOVERNMENTAL RESPONSE TO VIOLENCE**
The Obama administration had to manage a number of mass urban protests and violence, somewhat reminiscent of urban protests of the 1960s and early 1970s. These protests were often in response to police killings of unarmed individuals—one factor that helped to generate the sociopolitical movement Black Lives Matter, among others. In the current situation, race relations appear to be deteriorating even further. And depending in part on the general level of employment, unemployment, and underemployment, the situation could deteriorate even further.
It is consequently not surprising that a significant April 2017 opinion poll indicated that 36 percent of those interviewed believed that racism and bigotry are an imminent threat to the country (up from 29 percent from two years ago), while 23 percent thought it was serious (up from 22 percent). In other words, almost 60 percent of those interviewed believed racism was on the rise. The poll also expressed concern that Americans believed that they were losing their rights to freedom of speech. If accurate, this poll forewarns of greater domestic American violence ahead. Such violence could be mixed with acts of differing forms of race-related "terrorism" for and against minority groups—combined with both pro- and anti-Islamist "terrorist" actions. And the poll had been taken prior to the violent clashes in which white supremacists, neo-Nazis, and the Ku Klux Klan took the offensive against peaceful counterprotesters in Charlottesville, Virginia, in a counterprotest that took place in August 2017 during a "Unite the Right" demonstration.
The issue of dealing with protest and social discontent is complicated by the some of the completely understandable difficulties that the American police force will confront in dealing with new spates of urban protest. The problem is that even initially peaceful urban protests will tend to be tainted by fears of acts of terrorism. These fears could in turn generate excessive reactions by law enforcement officers, thereby resulting in unnecessary police violence and acts of repression. Trump's often crude, blatantly chauvinistic, and vociferous style of public speech has not done much to calm the situation, and a number of far-right groups and white supremacist leaders, such as former imperial wizard of the KKK David Duke and Richard B. Spencer, the president of the National Policy Institute (a white supremacist think tank), believe that they possess Trump's tacit support, given his refusal to strongly condemn their movements. This belief is also due to Trump's ongoing relationship with his right-wing former National Security Advisor, Steve Bannon.
In the effort to prevent police violence from reoccurring, Trump has promised to provide local police with sufficient resources to be able to build better relationships in their immediate community. At the same time, Trump has also threatened to engage in tougher measures than did the Obama administration in order to control urban violence, drug use, and crime. One of his proposals is to adopt the controversial and dubiously effective "stop-and-frisk" practice nationwide. This, however, implies an expansion of police powers that could be deemed unconstitutional, depending on the way the stop-and-frisk procedure is carried out.
Backed by the National Rifle Association (NRA), one of the most powerful lobbies in the country, Trump has been a strong opponent of gun control, although he has publicly opposed the sale of assault weapons. Yet if accepted into law, his proposed policies could make it easier for both "good" and "bad" guys to obtain weaponry and then carry that weaponry across the country. Permission to take weapons across state lines is thus being proposed in an era in which mass shootings for different personal and ideological motives—and not only Islamist—appear to be becoming more frequent and more lethal in terms of the number of people killed, as illustrated, for example, by the October 1, 2017, mass shooting in Las Vegas. (See chapter 3.)
At the same time, however, handguns and rifles are not the only problem. In the new age of hybrid warfare, everyday technologies and chemicals/substances can become weapons. Airplanes (as on September 11, 2001), trucks (as in Nice, France, on July 14, 2016 and in New York City on October 31, 2017), and fertilizer bombs soaked in diesel fuel and other chemicals (as in the left-wing bombing of the University of Wisconsin-Madison Army Mathematics Research Center in protest of the Vietnam War in 1970 and the right-wing Oklahoma City bombing of the Alfred P. Murrah Federal Building in 1995), can also be used for purposes of mass killing—whatever the social or political cause might be.
Stricter gun control laws might not prevent all kinds of violence and criminality, but blocking access to assault weapons, among other forms of weaponry, including controlling cop-killer bullet sales, can help limit some mass killings. The 1994 assault weapons ban had expired on September 13, 2004—and subsequent efforts to renew the ban or propose alternative legislation have thus far failed.
Given the significant social problems related to unemployment and underemployment, job discrimination, high levels of drug-related crime, and other forms of criminality, which generally involve the spread of guns, will Trump be able to obtain the trust of urban communities? Or will Trump be confronted by the outbreak of mass urban protests against police violence of the kind that confronted the Obama administration in a number of major US cities? Or will there be more protests like those that took place in Charlottesville, North Carolina, in August 2017—in which a man drove a truck into a group that was counterprotesting the "Unite the Right" demonstration, killing one woman and injuring many others?
Trump's so-called solution to the problems of criminality and "terrorism" inside American society has largely been to justify the right of people to bear arms—rather than to seek ways to control weapons that do not necessarily infringe on personal freedoms. Trump's rhetoric does not appear to recognize the dangers that an even wider spread of heavy weaponry and their bullets will cause—if these weapons are not properly managed and controlled. Trump's so-called solution to both domestic and external threats has been to produce more rifles, tanks, and nuclear missiles, and the like.
**TRUMP AND THE MILITARY BUILDUP**
One can only speculate how the populations of the world might have reacted to the Trump team's decision to deploy tanks and missile launchers for his inauguration parade—if that had happened. This is actually what Trump's transition team initially proposed in a symbolic effort to show off American military power—as is the case for democratic countries such as France, which is a major arms producer, but which is also endemic for authoritarian and militaristic countries such as Russia, China, and North Korea.
Trump's team evidently did not think about the destructive impact that such heavy military equipment would have had on already heavily traveled Washington, DC, roads and infrastructure. Military flyovers, involving the latest US fighter jets, were approved instead, but they were ruled out at the last minute, ostensibly because of poor weather conditions. Yet what message did Trump's team hope to convey by such a show of force? And how would those protestors, the American people, and the rest of the world, both US allies and rivals, have interpreted that message—if such a display of massive US military power—symbolic of the new America First nationalism—had taken place for the first time on the streets of the US capital, Washington, DC? And will Trump initiate a costly military parade for the future July 4th celebration of the Declaration of Independence, as he has indicated after his visit to France's July 14, 2017, Bastille Day celebrations?
It is clear that Trump's team believes that the United States must reassert itself on the global stage—if US interests and the "civilized world" (in Trump's words) are to be defended and if America's global hegemony is to be sustained. Trump's choice of Mike Pence as his vice presidential running mate had already revealed his alignment with Christian conservatives who generally oppose domestic "political correctness" and "international cosmopolitanism." In terms of foreign policy, Christian conservatism advocates Peace through Strength—and the ostensible need for the United States to show leadership and intervene militarily, even unilaterally, when deemed necessary, but not necessarily, "legitimate."
Pence's formula for confronting Moscow, for example, can be summed up in his own words: "The provocations by Russia need to be met with American strength.... We are going to rebuild our military. This whole Putin thing, look, America is stronger than Russia. Our economy is 16 times larger than the Russian economy. Our political system is superior to the crony corrupt capitalist system in Russia it every way." Yet given the fact that the Russian GDP is roughly the size of New York State's, building up the US defense budget—ostensibly in order to negotiate with Moscow (among other states) from a position of strength—is not the way to deal with Moscow, which will continue to react with countermeasures.
As furthermore indicated by its "America First: A Budget Blueprint to Make America Great Again," the Trump-Pence administration believes that the United States must reassert itself on the global stage through greater defense spending. In mid-February 2017, Trump proposed a major arms buildup that, he believed, would be able to more effectively assert US interests versus its adversaries—while engaging more deeply in the Global War on Terrorism. Trump accordingly proposed to augment defense spending above the already major increase sought by President Obama, to more than $639 billion. (See discussion, this chapter.)
The Obama administration had already planned to spend around $1 trillion on modernizing the US nuclear triad (land, sea, and air forces) over the next three decades against Moscow, China, North Korea, Iran, and other potential threats. This military buildup involves the deployment of five new nuclear weapons systems and the deployment of new naval forces so as to better deter the possibilities of war with North Korea and China in the Indo-Pacific. Obama had initially intended to improve US force capabilities in the Indo-Pacific primarily. Yet Moscow's not-entirely-unexpected military intervention in Crimea and buildup of forces in eastern Europe in early 2014 forced the United States and NATO to look toward ways to build up military capabilities in eastern Europe as well as in the Indo-Pacific. (See chapters and .)
In his telephone call to Russian President Putin in February 2017, President Trump denounced the 2010 New START (Strategic Arms Reduction Treaty), a treaty that had been negotiated by the Obama administration—as a "bad deal," just after Putin had raised the possibility of extending that same treaty. New START permits both countries to possess no more than 700 deployed and 100 non-deployed land-based intercontinental and submarine-launched ballistic missile launchers and heavy bombers equipped to carry nuclear weapons. The treaty also limits each side to no more than 1,550 deployed warheads.
Putin has asserted that Moscow has no intention of reneging on New START and on other arms control obligations. And yet Washington has accused Moscow of deploying a new intermediate-range ground-based cruise missile in violation of the 1987 INF treaty that bans the deployment of US and Russian land-based intermediate-range missiles. This, from a strategic perspective, would provide Moscow with an advantage in a military confrontation, as Russian intermediate-range missiles could be fired against targets in Europe and Japan, while Moscow would hope to protect itself from US retaliation with the threat to strike the US continent with its new intercontinental Topol'-MR (or RS-24 Yars) missile, which Moscow claims possesses multiple hypersonic warheads that can avoid missile defense systems. In 2016, Washington had previously accused Moscow of adding more warheads and missiles, thereby surpassing the 1,550 warhead limit set by New START. But Moscow has until February 2018 to comply. (See chapter 10.)
This new buildup of American military capabilities has accordingly been aimed to cover a number of possible contingencies. Such capabilities would, it is believed, be sufficient to handle two major wars or "major regional contingencies" (MRCs) nearly simultaneously. Trump also claims that he intends to develop a state-of-the-art missile defense system that includes the modernization of US naval cruisers with such antimissile capabilities. As part of this new military buildup, the Pentagon has been urging the Trump administration to consider a review of the US nuclear arsenal and force posture. The Pentagon also hopes to make the United States more capable of prosecuting a "limited" nuclear war—whether against North Korea, the Islamic State, Iran, Russia, China, or others—even if it is dubious such a war would remain "limited."
For these scenarios, the US Army would need to increase in size to about 540,000 active-duty soldiers. This would represent an actual increase above the 460,000 active soldiers that the Pentagon itself called for in its $583 billion budget proposal for FY2017.
In May 2017, Trump sent Congress a finalized proposed budget request for FY2018 of $639.1 billion. It would spend $574.5 billion for the base budget and allocate $64.6 billion for the Overseas Contingency Operations (OCO) budget, which is not included in the official defense base budget. The latter account pays for operations in Syria, Iraq, Northern Pakistan, and other regions. This overall budget request was $52 billion above the defense budget cap demanded by the Budget and Control Act (BCA) of 2011, and it does not fully include other defense-related costs to be discussed in this chapter.
**REAL DEFENSE COSTS**
In 2016, official US defense spending for the base budget was already an astronomical $597 billion—almost as much as the next fourteen countries put together. China's budget of $145.8 billion, for example, was less than a third of the US budget. Official US defense spending was thus roughly $385 billion more than China, $500 billion more than Saudi Arabia, and $530 billion more than Russia.
But these official figures do not tell the whole story. To obtain total defense spending, one needs to add onto the Department of Defense budget those items related to defense spending in other government agencies. These include: veteran affairs (in the Veteran's Administration); military retiree payments and interest payments on money borrowed to fund previous military programs (in the Treasury budget); military aspects of the space program (in the NASA budget); energy programs that go for secret defense and nuclear weapons research, testing, and storage purposes (in the Energy budget); and foreign military aid in the form of weapons grants for allies (included in the State Department budget). Other defense costs include sales and property taxes at military bases (in local government budgets), plus the hidden expenses of tax-free food, housing, and combat pay allowances. One also needs to add on the costs of maintaining seventeen intelligence agencies, including Homeland Security, that are involved in international intelligence gathering and operations. If one adds all of these, the figure grows much higher, way above 20 percent of the total federal budget. And it averages per fiscal year about double what the Pentagon officially reports.
But even then, the yearly DOD base budget does not account for the fact that the wars in Afghanistan, Pakistan, Iraq, and Syria represent supplemental spending, which is placed in the separate Overseas Contingency Operations (OCO) account, as mentioned above. The overall costs of these wars (plus other expenses that are not included the official DOD budget) from 2001 to 2016 added up to some $4.8 trillion in 2016 and are still rising. And the costs of these wars would have been even greater if US interest rates had been higher since 2001.
**QUESTION OF INTEREST RATES**
The US Federal Reserve's controversial decision to keep federal interest rates artificially low in the period from 2002 to 2004 during the George W. Bush administration made the costs of borrowing for the Global War on Terrorism relatively much less expensive than what might have been the case otherwise. The Federal Reserve Bank is supposed to be independent of the US executive branch of government, yet the low interest rates at that time (whether or not they were made low "accidentally on purpose") just happened to make borrowing for the Global War on Terrorism in 2001 and the subsequent US-led intervention in Iraq in 2003 much cheaper. This is an issue that should be investigated. In the view of many economists who argued that interest rates should have been raised at the time of the dot-com crisis of 2000–2002, these low interest rates eventually had a severe impact on the economy. They represent one of the major causes of the 2008 global recession, as low interest rates provided cheap money to both China and US consumers, which in turn fueled the housing market bubble and its collapse, as illustrated in the subprime and "liars' loan" crisis, while assisting China's financial boom in the longer term.
**SOCIAL COSTS OF HIGH DEFENSE SPENDING**
In order to help cover for the increase in defense spending, Trump has promised to cut governmental nondefense spending—instead of cutting the fat off military spending. Trump's proposed budget cuts include cuts in the State Department and foreign aid by some 30–37 percent, in addition to cuts in environmental protection programs, among other areas. At present, roughly 60 percent of foreign aid goes to economic and development assistance, and 40 percent goes to security concerns. These cuts could impact US relations with a number of strategically important countries. The highest aid recipients in 2016 include Afghanistan, Israel, Egypt, Iraq, Jordan, and Pakistan, then Kenya, Nigeria, Tanzania, and Ethiopia.
Through such cuts, to a certain extent resisted by Congress, Trump has hoped to achieve a total reduction in federal spending by $10.5 trillion over ten years. In the past, increases in the Pentagon's budget have generally been financed by cuts in nonmilitary public spending, by borrowing from the Social Security Trust Fund, and by debt and deficit spending—and at a tremendous cost for the overall economy. At the same time, this excessive government expenditure is justified by sustaining US superiority in arms and in arms sales to US allies so as to partially amortize the costs of public expenditure for the US military-industrial complex.
In 2015, the United States led in arms transfer agreements worldwide, making agreements valued at $40.2 billion (50.29 percent of all such agreements), up from $36.1 billion in 2014. The United States also led in the actual transfer or delivery of arms: In 2015, the United States ranked first in the value of all arms deliveries worldwide, making nearly $16.9 billion in such deliveries, or 36.62 percent of arms transfers. US arms sales have outpaced Russia's for the past eight years, with France in third place. Yet even these sales do not amount to the total overall public costs of maintaining the military-industrial complex.
Coupled with the fact that defense spending is roughly double what is officially reported, assuming all defense budget categories are added up in their entirety, it is clear that the high costs of the defense program cut significantly into social needs. Trump's proposed cuts in nonmilitary affairs appear all the more absurd given the fact that defense spending dwarfs the mere 2 percent spent on the State Department (which also includes foreign military aid in the form of weapons grants for allies) and on international affairs—at a time when the United States and the world needs more diplomacy, with quality leadership at the head of the desks in key State Department positions. And defense spending furthermore dwarfs the amount the United States provides each year to the United Nations for peacekeeping, environmental protection, refugees, public health, and so on. The State Department should have been taking the lead in foreign affairs, but over the years, the military and intelligence agencies have gained predominance within the US governmental bureaucracy—much as President Eisenhower warned. (See chapter 10.)
**THE FAILURE TO MAKE SAVINGS**
The Pentagon purportedly squashed a major study to restructure its approach to business operations. That 2016 study proposed that the United States could save $125 billion over five years and even more in the future. This could be done by streamlining bureaucracy, cutting back on the amount of high-priced contractors, and by making better use of information technology without necessarily reducing military forces or firing administrators.
The Congressional Budget Office (CBO) also reported that the Pentagon could save tens of billions of dollars over the next ten years alone by delaying, reducing, or canceling deployments of a number of weapons systems. These weapons systems include _Columbia_ -class nuclear-armed submarines; a new intercontinental missile, the Ground-Based Strategic Deterrent; the new nuclear cruise missile, the Long-Range Standoff Cruise Missile (LRSO); and the Long-Range Strike Bomber. It is also possible to reduce or eliminate the numbers of B-61-12 "tactical" nuclear warheads (with 480 now scheduled for production by 2020). These weapons can be carried by the F-35 and the Long-Range Strike Bomber—in addition to already-existing fighter jets and bombers. All of these systems can be questioned on both strategic (unnecessary or provocative) and financial grounds (excessively costly). A number of these powerful weapons systems could be unilaterally reduced or eliminated, or else bargained down, as part of a general arms accord with Moscow and possibly with China. But to do that, the United States will need to engage in truly peace-oriented diplomacy.
During his presidential campaign, Trump had stated that he opposed the congressionally mandated sequestration process of the 2011 Budget Control Act that was aimed at cutting defense spending. From Trump's perspective, sequestration cut both wasteful and necessary military spending by equal percentages. This meant that absolutely necessary budget cuts due to excessive military spending might not necessarily make strategic or even practical sense. At that time, before he became president, Trump claimed that he would not seek an across-the-board military buildup, but instead would build only those weapons systems that are truly "needed" by the military—and that are not pushed on the military by demagogic congressional demands of what should be called the "military-industrial-congressional complex."
Once he became president, however, in seeking to abolish the sequestration process, Trump consequently tried to claim personal credit for reducing excessive expenditures on the F-35, for example. The F-35 advanced fighter is one of the most expensive defense boondoggles in US history, and is expected to cost upward of $1.45 trillion over its fifty-year life span. But contrary to Trump's boasts, the Government Accountability Office (GAO) stated that the fifth-generation fighter still faces over $1 billion in cost overruns in FY2017 and is officially estimated to cost $10.3 billion in FY2018. And there is some dispute about F-35 capabilities versus the Russian SU-35 and integrated air defenses. In fact, the United States, Russia, India, and China are all having both technological and cost problems with stealth fighter jets. And F-35s could also prove vulnerable to Russian-made anti-aircraft missiles, such as the S-400.
**TRUMP'S PLAN TO BOOST INFRASTRUCTURE INVESTMENTS**
It is true that Trump's plans to boost defense expenditure will most likely generate high-skill high-tech work in the military-industrial complex throughout many of the fifty states, with some trickle-down effect. But increased defense spending risks reducing government funding available for other much-needed non-defense alternatives and alternative-energy projects, while augmenting the national debt. The irony raised here is that the roots of the US infrastructure crisis stems from mistakes that Ronald Reagan had already made with high military spending hikes and supply-side tax cuts. Reagan's policies did help expand the economy in the 1980s, but they also resulted in "insufficient investment in physical public capital such as highways, bridges, mass transit, waste water facilities, hazardous waste sites, and the like."
To improve American infrastructure, Trump has proposed using $200 billion in public money as a means to leverage $1 trillion in private funding. This amount, for example, is expected to include some $40 billion in Saudi funding that was promised after the $110 billion arms deal with Saudi Arabia that was announced in May 2017. Trump thus hopes to raise money by leveraging public funding, while replacing publicly supplied services and infrastructure with private-supplied goods and services. This approach will nevertheless continue to diminish and weaken the public sector—with uncertain consequences.
Perhaps more ironically, the Chinese Investment Corporation (CIC) has been looking to invest in Trump's infrastructure projects ostensibly in an effort to build trust between the two countries. Beijing's CIC claims that the United States will need as much as $8 trillion in infrastructure development, not just $1 trillion. Contrary to Trump's anti-Saudi and anti-Chinese presidential campaign stances, it looks like both Saudi and Chinese finance are needed to save Trump's infrastructure program—and will help indirectly pay for his major military buildup while also exposing American politics to greater Chinese and Saudi political-economic influence.
It is furthermore not yet clear what kind of infrastructure projects would be funded—as Democrats oppose the use of healthcare as an infrastructure project, for example. Trump's ideology of "economic nationalism" emphasizes some areas of US "infrastructure development." These areas include the military-industrial complex and fossil fuels, for example, and roads, bridges, navigable waterways, and dams. But the Trump administration might not consider the infrastructure needed for mass transit, healthy drinking water, waste recycling, housing, alternative energy, and ecologically safe technologies. The dilemma is that much of the money spent on the defense buildup could be better spent on college scholarships for increasingly expensive education, for example, among other crucial nondefense needs. Trump's emphasis on physical infrastructure development also overlooks the fact that a solid education in both practical studies and liberal arts provides the basis for all infrastructure development and for future innovations.
**CONGRESS AND THE MILITARY-INDUSTRIAL COMPLEX**
Not only does Trump's leadership and the US democratic process appear more closely linked with capitalist elites, but also it appears even more perilously linked with the military-industrial complex that President Eisenhower warned about in his 1961 farewell address: "we have been compelled to create a permanent armaments industry of vast proportions."
The problem is that the American system of spoils leads congresspeople to push for news arms projects. Arms lobbyists often try to press the US Congress to accept their policy, development, and even military-technology proposals upon promises of investment and additional jobs for the state—or else upon their particular definition of "national security." On at least two occasions since the sequestering laws that seek to automatically cut defense costs went into effect in 2011, it was Congress (and not the Pentagon) that had decided to lift the spending caps so as to increase defense spending. The concern raised here is that the military-industrial complex operates and provides jobs in a vast majority of US states, so that the more weapons systems or parts of weapons systems that a state receives to construct, the more jobs (combined with the trickle-down effects of military Keynesianism) the state will support and more votes Congress can expect. The system is difficult to break, particularly because the House of Representatives is elected on two-year terms and congresspeople need show positive results as soon as possible. (This represents a significant reason to restructure the bicameral legislative system. See chapter 10.)
The fact that Vice President Mike Pence has supported the building of the extremely costly F-22 Raptor, among other excessively expensive and strategically unnecessary weapons systems, indicates the influence of the military-industrial-congressional complex. Here, contrary to his pretended opposition to wasteful and excessive military spending, Trump himself has stated: "Accomplishing this military rebuild will be a fifty-state effort—every state in the union will be able to take part in rebuilding our military and developing the technologies of tomorrow." Yet this linkage between the federally financed military-industrial complex and the fifty states is the root of the overspending!
**POLITICIANS AND THE MILITARY-INDUSTRIAL COMPLEX**
Perhaps more ironically, of the top five presidential candidates, it was the Democrat Hillary Clinton who obtained more money during the 2015–2016 election cycle from defense contractor employees than the next four presidential candidates combined. And the proclaimed democratic socialist Bernie Sanders actually obtained $100,000 more from defense contractors than did Donald Trump. And among Senators, Bernie Sanders also obtained more funding from defense contractor employees than Republican Senators Ted Cruz and John McCain—who are strong supporters of Peace through Strength.
To his credit, Senator McCain has criticized excessive defense spending on some projects, such as the extremely costly F-35 stealth fighter. Yet McCain also proposed a $640 billion base defense budget for 2017, much larger than that of Trump. For his part, Sanders has nevertheless continued to argue for much deeper defense cuts than McCain—despite his continued support for the controversial F-35 fighter jet, which has three production plants based in Sanders's home state of Vermont.
**THE DOMESTIC AND INTERNATIONAL IMPACT OF A NEW ARMS RACE**
How will Trump's America First military buildup impact American society? Will it really lead to a form of growth that more equitably distributes wealth, as Trump promised during his election campaign? Will excessive US governmental spending in the highly capital-intensive military-industrial complex really trickle down to benefit the general population, even if military industries are spread throughout most of the fifty states? Or will Trump's military buildup simply exacerbate the growing gap in the United States between rich and poor and indirectly press young people into state-financed military careers? And will Trump's display of a big stick necessarily result in peace—if the tools of diplomacy are undermined by excessive emphasis on military power?
The point is that government funding for the military-industrial complex—ostensibly in order to sustain US military superiority against all potential rivals alone—is not necessarily sufficient to keep the peace. On the international side, the dilemma is that the United States as a global hegemonic power cannot cover all regional contingencies where rival major and lesser powers and anti-state movements may possess a tactical advantage—or believe that they can seize one. On the domestic side, should Trump's new military Keynesian version of trickle-down economics fail to lift the economy, the result will be a major political-economic crisis and the augmentation of social and political conflict at home.
The danger is that Trump's massive military program will create more pork-barrel projects, significantly boost the national debt, and antagonize firms and workers who do not partake directly or indirectly in the military-industrial complex—while fomenting a global arms race among those state and anti-state partisan movements most opposed to America First policies. And because the United States still represents the predominant global power, a failure to lead the country at home will lead to even greater social and geopolitical strife abroad.
What is needed is a _greater_ , not lesser, emphasis on omnidirectional peace-oriented diplomacy in an effort to prevent a new arms race—so as to thoroughly reduce excessive government spending on armaments and to foster positive conditions for national and international socioeconomic development and environmental protection. (See chapters and .)
The fact that Donald Trump did not obtain a clear mandate from the American people—in an election in which only roughly 58 percent of eligible voters participated—has appeared to delegitimize the very nature of the American democratic system. Yet what has additionally worked to discredit the American elections have been accusations by both Trump and his opponents that foreign influence—both "illegal" immigrants and the "Kremlin"—had tampered with this important dimension of American society. This is not to overlook Trump's earlier allegations that the elections were supposedly rigged in advance—but he nevertheless won unexpectedly and without a majority of the popular vote (See also chapter 10.)
**THE ALLEGED ROLE OF MOSCOW**
Just after the presidential vote count was in, President Obama rapidly accused Moscow of tampering with the US election process through cyber-intrusions. Obama ordered the FBI and five other law enforcement and intelligence agencies to start an investigation as to whether the Kremlin had attempted to influence the presidential election. Republican Senator John McCain declared that alleged Russian hacking into American domestic affairs represented an "act of war." Senator McCain, with Senator Lindsey Graham, then pressed for a full-fledged Senate Select Committee investigation. In response, Trump replied in a tweet that urged McCain and Graham to halt their investigation into Russia and to "focus their energies on ISIS, illegal immigration and border security instead of always looking to start World War III." Nevertheless, both the House Committee and a Senate Select Committee continued to investigate Russian activities in the US elections and Trump's relations with Moscow, as have two other committees. For his part, Senator McCain also pushed for an independent inquiry.
McCain not only sought to inflame the American public opinion against Russia, but also sought to condemn Trump for his ostensibly pro-Putin political stance. McCain did state that he would not, however, actually go to "war" with Russia over the issue. From a legal standpoint, such actions involving interference in the election process, assuming they could be proven, could not be considered a rationale for an initiation of armed conflict. At the same time, such intrusions would represent an "illegal" external intrusion in American internal affairs and could therefore invite some form of retorsion.
By mid-May 2017, as events unfolded, and just after Trump fired FBI Director James Comey, who had begun to investigate Trump's alleged connections to Russia, former FBI Director Robert Mueller was appointed as a special counsel with the power to investigate whether Moscow had in any way covertly aided President-elect Donald Trump, and whether or not Trump himself, or any of his associates, were hoping to profit, for example, from its contacts with Russian officials. One accusation was that the Trump administration could put an end to the sanctions placed on Moscow in 2014 in the aftermath of the Russian annexation of Crimea and its political-military interference in eastern Ukraine, in exchange for business favors. Specific areas of investigation have included the alleged Trump compliance with the hacking and leaks of emails written by key figures in the Democratic Party; Donald Trump Jr.'s meeting with a lawyer with purported Russian government connections, in an effort to obtain "dirt" on Hillary Clinton; whether or not the Trump campaign was involved in Russian efforts to spread fake news targeted at voters in key states; and whether or not Moscow did attempt to hack into US election-related computer systems. Unlike the investigations of the congressional committees, Mueller has the power to file federal charges. By late October, Mueller indicted three people, Trump's former campaign chief, a former Trump business associate, and Trump's ex-foreign policy adviser, with alleged crimes that included money laundering, lying to the FBI, and conspiracy against the United States.
**DOMESTIC ATTACKS AGAINST TRUMP**
The key domestic dilemma for the Trump administration is that President Trump's campaign promises to engage in a rapprochement with President Putin immediately evoked strong congressional opposition. Both Republicans and Democrats have strongly opposed the lifting of economic sanctions on Russia without also seeking a guarantee that Moscow will withdraw its clandestine forces from eastern Ukraine and then return Crimea, which Moscow annexed in early 2014, back to Ukraine. This was seen in the passing of the August 2016 H.R. 3364 Act, "Countering America's Adversaries through Sanctions Act." Given its strong bipartisan congressional backing, Trump was pressed to sign the bill into law despite the fact that he considered it to be "flawed"; Trump argued that the bill "encroaches on the executive branch's authority to negotiate" and "makes it harder for the United States to strike good deals for the American people and will drive China, Russia and North Korea much closer together." The bill could also "hinder our important work with European allies to resolve the conflict in Ukraine," he argued.
In American domestic debates, the crisis in Ukraine has generally been blamed on Moscow _alone_ due to the Russian annexation of Crimea and its political-military interference in eastern Ukraine. American observers tend to downplay issues concerning the highly centralized nature of the Ukrainian domestic power structure and Kiev's discrimination against the Russophone Ukrainian minority. There is thus a tendency for Moscow _alone_ to be blamed for the continuation of the conflict in eastern Ukraine by arming the autonomist movements—as opposed to Kiev's failure to follow up on the Minsk II accords and its refusal to negotiate directly with the "autonomist" factions in the Donbas region due to Kiev's strong opposition to federalism and decentralization of power. (See chapters and .)
That Trump has realized that his efforts to seek a rapprochement with Russia are not "good" for him in political terms is seen in his following statement: "If we could get along with Russia, that's a positive thing. We have a very talented man, Rex Tillerson, who's going to be meeting with them shortly, and I told him, I said, 'I know politically it's probably not good for me.... I would love to be able to get along with Russia. Now, you've had a lot of presidents that haven't taken that tack. Look where we are now."
Trump has largely based his hopes on greater cooperation with President Putin through concerted cooperation in the Global War on Terrorism—in the hope that such cooperation will lead to cooperation in other areas. Trump accordingly said: "I respect a lot of people, but that doesn't mean I'm going to get along with him. He's a leader of his country. I say it's better to get along with Russia than not. And if Russia helps us in the fight against ISIS, which is a major fight, and Islamic terrorism all over the world—that's a good thing. Will I get along with him? I have no idea." In his own eccentric way, Trump has argued that his efforts to deal with Putin are not personal—but a national security issue, an affair of the state. Nevertheless, Trump has been suspected of seeking personal profit from his dealings with Moscow.
Trump's off-the-wall and off-key statements regularly bring him political trouble. When a journalist insisted that Putin was a killer, Trump replied: "There are a lot of killers. We have a lot of killers.... Well, you think our country is so innocent?" It has been alleged without absolute proof that that Putin had ordered the death of Russian journalists (including Anna Politkovskaya, who reported on human rights abuses during the brutal Russian military intervention in Chechnya) and other opponents of the regime (such as Boris Nemtsov, an opposition politician who strongly denounced Putin's corruption and Russian intervention in Crimea and in eastern Ukraine). Putin has also been accused of killing Alexander Litvinenko, former FSB and KGB agent, who denounced Putin as coming to power with the help of the Russian secret police, the FSB. Litvinenko was murdered by the use of the radioactive substance polonium in a London restaurant in 2006—an affair that soured Anglo-Russian diplomatic relations for years. Litvinenko had openly accused Putin of killing the Russian investigative journalist Anna Politkovskaya and engaging in other criminal actions.
Although it does not justify murder, of course, the fact of the matter is that Putin is far less repressive, in comparative historical terms, than Lenin or Stalin, or even previous tsars. And Washington has had to negotiate with the leadership of far more repulsive regimes (including Joseph Stalin and Mao Zedong, among others, who have engaged in mass murder) as part of geopolitical realities. And although Trump could have replied more tactfully, the Central Intelligence Agency (CIA) has, in fact, been accused of assassination, both before and after the Cold War, and has engaged in so-called targeted killing, with the use of drones in Pakistan and Afghanistan. The US government has also been accused of torturing or attacking journalists critical of US policy. NATO, for example, as the US-led defense organization, bombed Radio Television Serbia in downtown Belgrade, Serbia (a Russian ally) in 1999 during the war over Kosovo, for example, killing sixteen people. Washington has likewise been accused of killing and torturing Al Jazeera journalists, among others. In addition to the use of so-called enhanced interrogation techniques (which included waterboard torture) at Guantanamo Bay, the Abu Ghraib prison scandal—which involved torture, rape, and murder—has represented a national disgrace. Since the scandals at both prisons have worked to undermine fundamental American values, the United States cannot claim the higher moral ground versus Putin's Russia. Moreover, Putin's preclusive interventions in Crimea and eastern Ukraine in 2014 without a UN mandate have not caused even close to the same degree of death and destruction as has George W. Bush's so-called preemptive decision to engage in a major military intervention in Iraq in 2003, for example, also without a UN mandate. In this sense, one cannot point the finger at Putin alone.
**ALLEGATIONS OF RUSSIAN TAMPERING IN US ELECTIONS AND IN US DOMESTIC AFFAIRS**
Trump has consequently faced severe criticism at home and abroad for his hope to achieve a more positive relationship with Russia—and with Russian leader Putin, in particular. The dilemma is this: Even if Trump is finally able to reach out to Putin and Russia—even in such a way that appears consistent with American national security interests—there will nevertheless be a cloud of suspicion that he did so with his own personal interests, or those of his business associates, in mind.
As the so-called Russia-gate has evolved, the Kremlin has been accused of cyber-tampering in an effort to embarrass Hillary Clinton and the Democrat National Committee, allegedly to help Trump be elected president. These alleged actions resulted in the releasing of private emails of key Democratic leaders that exposed the Democrat's campaign tactics and political manipulations. Moscow (or someone) had also allegedly tapped into Republican Party communications, but these materials were not released.
Alleged Russian meddling in US elections is a major issue that Trump himself made much worse when he urged Russian hackers to target Hillary Clinton's nonsecure personal email server, saying: "Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing." It has been alleged that Moscow used WikiLeaks and other critical media to divulge information that was intended to embarrass Clinton and the Democrats. This accusation has, however, been denied by WikiLeaks founder Julian Assange—even though he himself was not absolutely certain who was ultimately responsible for distribution of the information. But it does appear plausible that disgruntled Democrats who opposed Hillary Clinton may have leaked some materials. In short, Moscow's Fancy Bear espionage group may possibly have engaged in cyber-intrusions, but so did others. In May 2017, Russian President Putin claimed that the Kremlin was not responsible, but that "patriotic-minded" Russians could have been responsible for the alleged cyber-attacks.
At the same time, on the US side, some leaks of highly classified materials at the highest level of government appear to be ignored by Washington, while other leaks have been investigated and prosecuted with the fullest intensity of the law, such as the significant information leaked by Chelsea Manning. In general, the dilemma is that some of these leaks may represent a legitimate reaction to perceived negative government policies that seek to perpetrate conflict or that augment governmental powers. But other leaks may be a result of the opposition by special interest groups to positive state efforts to mitigate conflicts or solve disputes or deal with controversial issues, but which do not serve the interests of those particular groups or individuals. Some leaks may thus be self-serving and done to settle scores. Other leaks may be mis- or dis-information intended to divert attention and shift the focus of journalistic investigations—what Trump himself has called "fake news"—but which is an area in which he himself excels by exaggeration, distortion, or invention of so-called information.
It has been argued that Moscow's alleged cyber hacking was one of the causes in the decline of popular support for Hillary Clinton's bid for the presidency. For her part, Clinton herself did blame Russian President Putin, at least in part, for undermining her presidential campaign, but she also blamed FBI Director James Comey for the way he handled the federal investigation into her use of a non-secure email account for US government purposes. Comey's accusations against Clinton were accordingly seen as impacting voters just days before the election—more so than Russian influence.
But it has not yet been proved whether the alleged Russian hacking or the FBI investigation into Clinton's use of a non-secure email account—if either—was truly responsible for impacting the votes in three key states in Michigan, Pennsylvania, and Wisconsin. Had Clinton obtained roughly 80,000 votes in the aforementioned states, then she might have obtained a large enough number of electoral college votes to win the presidency. Clinton did not even campaign in Wisconsin, because she believed it was solidly blue-collar pro-Democrat state—even though many blue-collar voters were shifting toward Trump. It will be very difficult to prove whether Russian influence and/or the FBI investigation were really the primary reasons for some 80,000 people not to vote for Clinton. Some people did not vote at all or backed third-party candidates simply because they disliked both Trump and Clinton.
**MOSCOW LOSES FAITH IN TRUMP**
Whether or not Moscow had any real influence in helping Trump get elected appears dubious. It is certain that Moscow did attempt to influence popular opinion in the United States and elsewhere, but to what extent is not clear. Moscow did propagandize against US policies through Russian Television (RT) and Sputnik broadcasts. Moscow most likely paid trolls (as can any group or state) in an effort to secretly manipulate US and European public opinion through Facebook and Twitter accounts. Yet whether and to what extent Russian military intelligence, the GRU, intruded into the US presidential elections, through its cyber-espionage group, APT 28 (also known as Fancy Bear), is not certain.
If, and to what extent, Trump himself might have colluded with Moscow is a major dimension of the FBI investigation. The fact that Trump fired FBI Director Comey in May 2017—ostensibly for the poor way he had handled the Clinton email investigation—even if Comey's actions may have helped Trump and cost Clinton some votes—raised real questions as to the possibility that Trump obstructed justice. Comey was fired just a few days after he had purportedly demanded more resources for the FBI investigation into the alleged ties of Trump and his associates to Moscow. (Comey's alleged demands were denied by the Justice Department but affirmed by a number of congresspeople.) Moreover, Trump had previously urged Comey to kill the investigation of the alleged ties of former National Security Advisor Michael Flynn to Russia (an allegation first denied by the White House). Another possible reason for Comey's firing was that the FBI had begun to investigate Jared Kushner, Trump's son-in-law. Kushner has been accused of opening a secret back channel to Putin, ostensibly to discuss global strategy—an action on the surface that appears appropriate, given the dangerous state of the US-Russian relationship. Yet Kushner; Trump's son Donald Trump Jr.; and Trump's campaign chairman at the time, Paul J. Manafort, all attended a meeting with a lawyer who claimed to possess close connections to the Russian government, in the hope they could obtain damaging information on Hilary Clinton.
Interestingly, while Trump's team was attempting to find dirt on Clinton, Clinton was obtaining dirt on Trump. It is now known that the controversial Fusion GPS report on Russian influence on Trump, which was completed by former British MI6 intelligence agent Christopher Steele, was paid for in part by Hillary Clinton's campaign and the Democratic National Committee. While the Fusion GPS report does not absolutely prove collusion between Trump and the Russian government, and argues that Moscow possessed information that could blackmail Trump, and while fact can be mixed with fiction, the report did point the way to engage in even deeper investigations, given Trump's extensive and long-term connections to Russia.
What gives at least some credence to the Fusion GPS report, and what is not generally stated by the media, is its analysis of the differing factions inside the Russian government. It sounds credible that Prime Minister Dmitri Medvedev, for example, was purportedly furious over the Russian hacking of the Democratic National Council and the subsequent anti-Russian backlash in the United States. The Russian Ambassador to the United States Sergei Kislyak; the Ministry of Foreign Affairs; and independent foreign-policy adviser Yuri Ushakov, had purportedly urged caution. As evidence of tampering came out, Putin's chief of staff, Sergei Ivanov, purportedly argued that the only thing to do was sit back and deny everything—and the blowback would amount to nothing. But then Ivanov was sacked as chief of staff in August 2016, with no public explanation, and was replaced by Anton Vainov, who had no role in the operation. This indicates that Putin may have realized his mistake, although Russian foreign minister Sergei Lavrov, who may also have played a role, remains in office. In effect, some Russian elites, in opposition to Ivanov and Putin, had feared Trump's erratic character and preferred Clinton as the "devil you know" rather than the "devil you don't know." Now, after Trump's election, US-Russian relations are at their lowest level since the end of the Cold War.
Despite alleged clandestine efforts of the Kremlin to support Trump and despite Trump's initially strong support in the Russian government-controlled media, Trump's popular and official backing in Russia began to diminish almost immediately once he became president—in part due to some of his Cabinet choices. By early February, the Trump-Putin dating game was at an end. The tone of the new Trump administration changed dramatically after the new US ambassador to the United Nations, Nikki Haley, issued her first salvo against Russia. Haley condemned Moscow for the upsurge in violence in eastern Ukraine at that time and demanded that Russia return Crimea to Ukraine. (See chapters and .) Concurrently, accusations of Russian cyber-tampering in the US presidential elections hit the news headlines and Moscow continued to deny governmental involvement in the US election process.
**THE RUSSIAN PERSPECTIVE**
In Russian eyes, it was the Obama administration who had acted first by trying to influence the outcome of both the Russian and Ukrainian national elections. In December 2011, Hillary Clinton had publicly supported mass protests against the results of the Russian elections after Putin's party, United Russia, had suffered significant losses. At that time, Putin claimed that hundreds of millions of US dollars were being distributed to influence those elections. Putin accused Clinton of publicly declaring that the elections "were not honest and not fair" before she had even "received the material from the observers"; according to Putin, Clinton "set the tone for some actors in our country and gave them a signal" and "with the support of the US State Department, [they] began active work." Putin further resented Clinton's public remarks and US and news media accusations that Putin, his family, and his associates, had become billionaires. Clinton's criticisms, and those of others, made Putin believe that Washington was seeking "regime change" in Russia.
Putin likewise criticized the influence of NGOs (nongovernmental organizations) backed by the US State Department. Moscow passed legislation to make the paperwork much more difficult for all NGOs to register in Russia. In 2012, Putin signed the so-called foreign agent law. This law required NGOs that received funding from outside Russia to register as foreign agents. NGOs would then be subject to mandatory audits. Moreover, Putin raised the penalties for those caught protesting at unauthorized rallies.
Then, during the Euromaidan protest movement of late 2013–2014 in Ukraine, Moscow intercepted and broadcast comments made by then Assistant Secretary of State Victoria Nuland. In what has been called "democracy engineering," which backs ostensibly democratic sociopolitical movements that hope to change authoritarian governments into democracies, Nuland's comments indicated US support for specific Ukrainian leaders and expressed her disdain for the less assertive policies of the European Union, for example, in a four-letter word. The Euromaidan movement, backed by the United States, then succeeded in ousting the kleptocratic regime of Ukrainian President Viktor Yanukovych, who was seen as backed by Moscow—even if Yanukovych did always not support Putin on all issues. Yanukovych, for example, refused to join the Russian-led Collective Security Treaty Organization (CSTO), Russia's much weaker version of NATO.
Nevertheless, angered by perceived American efforts to undermine Yanukovich through democracy engineering, Moscow retaliated by using Washington's interference in Ukrainian politics to rationalize its own clandestine military intervention into Crimea and into eastern Ukraine in early 2014—in the effort to check NATO and EU enlargement into its self-defined "near abroad." At that time, Clinton, who was no longer secretary of state, compared Putin's annexation of Crimea with Hitler's annexation of the Sudetenland—but qualified that analogy by stating that she did not think Putin himself was like Hitler. It was nevertheless a remark that must have infuriated Putin—and represented one of the factors that could possibly have led him to engage in alleged Russian cyber-tampering and social-media propaganda in the US elections against Clinton.
**THE TRUMP ADMINISTRATION, RUSSIA, AND ENERGY INTERESTS**
As the congressional investigations into the Russia-Trump relationship continued, a number of members of Trump's team have been shown to have had some form of close ties to Russian officials or businessmen. And if allegations can be proved to possess some veracity, such a major scandal could easily be used by both Democrats and Republicans in Congress to block any potential Trump efforts to "appease" Moscow. In fact, the publication of the "Paradise Papers" in November 2017 shows the extent to which a number of Trump administration officials are linked financially to offshore financial paradises and to Russian business interests. These disclosures warrant open congressional hearings.
Initially, it was believed that former ExxonMobil CEO Rex Tillerson, whom Trump nominated as Secretary of State, would attempt put an end to the sanctions on Moscow once he took the post. This belief was due to the fact that ExxonMobil had lost more than $1 billion in 2015 on account of US sanctions placed on Russia in 2014, and that ExxonMobil could lose hundreds of billions in the future if it cannot sustain many of its large-scale energy projects in Russia. Yet Tillerson, who had received the Russian Order of Friendship Award from Putin for his role in setting up ExxonMobil's extensive investments in Russian energy reserves, took a very tough stance against Moscow's actions in Crimea and eastern Ukraine during his Senate confirmation hearings for US Secretary of State. Tillerson strongly criticized Obama's policy as being too weak from the outset. Obama, Tillerson asserted, should have advised Ukraine to move all available military assets to its eastern border and provide those assets with defensive weapons, US or NATO air surveillance, and intelligence. Yet had such an approach been taken by Obama, it could have drawn the United States, NATO, and Russia directly into the conflict in eastern Ukraine, as Moscow would seek to fend off Ukrainian forces and foreign surveillance too close to its borders.
Tillerson's tough stance on both Moscow and Beijing at his Senate confirmation hearing helped him to obtain the post as secretary of state. Tillerson was also able to survive congressional scrutiny, in large part by arguing in favor of maintaining sanctions on Moscow—a position that he has generally sustained since becoming secretary of state. Nevertheless, this did not prevent ExxonMobil from applying to the Treasury Department in April 2017 for a waiver from US sanctions on Russia in a bid to resume its joint venture with state oil giant Rosneft. And Tillerson's stance on sanctions has not prevented ExxonMobil from eventually suing the Treasury Department in July 2017 over a $2 million fine for purportedly violating US sanctions against Russia in 2014 at the time when Tillerson was CEO. In general, Trump and Tillerson appear less supportive of sanctions on Moscow than does Vice President Pence.
Yet the first person to fall as a part of this so-called Russia-gate scandal was not Tillerson but Trump's pro-Russian former National Security Advisor Michael Flynn. The key issue here is that Flynn appeared to be setting up a political network inside Ukraine that would attempt to undermine Ukrainian President Petro Poroshenko's rule and his policies toward eastern Ukraine and Crimea and toward Russia—and then forge a new Ukrainian-Russian accord and put an end to sanctions. This is significant, since Washington will eventually need to find a path for both Ukraine and Russia to compromise over eastern Ukraine and Crimea in the near future—even if Flynn's plans have been thrown to the garbage heap.
In addition to his paid role to represent Turkish interests for a company with close ties to the Turkish government, and his work for RT, Flynn may have been in trouble for another reason. It was Flynn, as director of the Defense Intelligence Agency at the time, who claimed that Washington had made a willful "decision" set up a "Salafist principality" in eastern Syria to counter the Assad regime. This frank statement implied that Washington, along with its Arab Gulf allies, supported radical Islamist groups, including al-Qaeda, and may have helped to create the Islamic State. Such an affirmation by a former high-ranking intelligence officer may have enraged a number of invisible faces in US governmental agencies in Washington against him. (See chapters and .)
**SANCTIONS ON MOSCOW**
A number of US senators, both Republican and Democrat, have accordingly wanted to make it extremely difficult for Trump to reduce or eliminate sanctions on Russia—even if those sanctions have ostensibly hurt the profits of major US and European businesses and oil or agricultural interests. Senator John McCain has warned that if President Trump did not soon put an end to speculation that he is still willing to ease sanctions on Russia, "for the sake of America's national security and that of our allies," McCain would work with his colleagues "to codify sanctions against Russia into law."
Senator McCain, and Democratic Senate Minority Leader Chuck Schumer, accordingly initiated the bipartisan effort to require congressional approval before Trump lifts any sanctions against Moscow. "Sectoral sanctions" on Russia that impact major energy companies and banks were due to expire in December 2017—unless extended by Congress. (See chapter 4.)
**COUNTER-ALLEGATIONS OF ILLEGAL IMMIGRANTS AND TERRORIST INFILTRATION**
Trump and his spokespersons denied the general consensus among US intelligence agencies that Moscow had, with high probability, tampered in the US presidential elections. Instead, Trump repeatedly countered with claims that "3–5 million" unauthorized immigrants had voted "illegally" for Clinton. Trump then stated he would begin an investigation of purported voter fraud once he became president. Such an investigation raised the prospects that the fifty US states would try to implement more restrictive voting regulations (such as voter ID laws) that could further limit the ability of the poor and minorities to vote.
Trump's claim that unauthorized immigrants were permitted to vote for Clinton in the millions is evidently false. But what is true is that when states count all residents for purposes of redistricting, whether or not those residents are eligible to vote, that can change the social and political balance of the election districts at the time of the census. This fact can lead the party who is in the majority in each state's district to try to fix the size and shape of those districts in that party's favor, using the process of gerrymandering.
Yet Trump's negative approach to illegal and unauthorized immigration overlooks the fact that many immigrants fill seasonal and other low-paying jobs that most Americans generally do not want. Trump has nevertheless threatened to expel illegal and unauthorized Mexican and other immigrants. He has furthermore demanded that Mexico pay for the border wall/fence that he plans to extend between the two countries. And he has also threatened to block the remittances that are sent back to the families of undocumented workers living in the United States—an action that would be probably be ruled illegal.
Nevertheless, if Trump is able to force large numbers of aliens to leave the United States, this act could destabilize Mexico and other Central American countries whose populations depend on those remittances. As some 50 percent of Mexicans live in poverty, the remittances of undocumented workers accounted for as much as 2.1 percent of the Mexican GDP in 2010, and they considerably exceeded earnings of Mexico's oil exports in 2016. In fact, immediately after Trump threatened to block bank transfers and helped to erode confidence in the peso during his campaign, Mexicans abroad sent nearly $2.4 billion in transfers in November, 24.7 percent higher than in 2015—the fastest expansion since March 2006.
Trump's threats have generally hurt the Mexican economy. A weak Mexican economy will augment the power of the mafia and the fees of coyotes who take immigrants across the US border, and who will not necessarily be stopped by a wall or fence. A weakened Mexican economy would concurrently strengthen the hands of drug lords in Mexican society, and create greater socioeconomic instability that could impact US urban areas as well. This is true given the degree of violence, political corruption, money laundering, and arms trafficking that takes place as drug traffickers attack each other's gangs for control over territories and populations, and engage in extortion from industries, such as avocado production. This appears plausible given the link between higher remittances and lesser crime rates in Mexico.
A politically unstable failed state on or near US borders could potentially require US police, if not military, interventions. Trump purportedly warned the Mexican president that Mexico was not doing enough to stop those "bad hombres down there"; he said, "You aren't doing enough to stop them. I think your military is scared. Our military isn't, so I just might send them down to take care of it." Jest or not, in December 2010, Mexican drug smugglers were involved in a controversial gun battle with US Border Patrol agents that killed a US officer in a remote canyon in southern Arizona. Incidents like the latter, which involved US government efforts to trace US manufactured guns sent to Mexican drug gangs, could drag the United States into more overt intervention along the border. Such a jest could become a self-fulfilling prophecy—if relations between the United States and Mexico continue to deteriorate—along with the relations between the United States and Venezuela, among other Latin American countries—in part due the perverse political-economic regional impact of the illegal drug trade and the War on Drugs, which results in drug addiction and the spread of guns and criminality in many US cities. (On Venezuela, see chapter 6.)
By contrast, a major change in US policy that would legalize some drugs, and treat the drug epidemic as a social and health issue, could help wind down the War on Drugs, improve US relations with Mexico and Latin America, reduce the number of prisoners in US jails, permit law enforcement authorities to focus on other forms of criminality (including white-collar financial crime and terrorist groups), while concurrently helping to reduce the spread of guns throughout the United States. (See chapter 10.)
**FEARS OF DOMESTIC ISLAMIST TERROR**
To add to this heightened sense of paranoia, Trump's accusations against millions of "illegal" immigrant voters, who purportedly voted in favor of Hillary Clinton, have been combined with Trump's claims that Islamist terrorists have been trying to infiltrate into American society—by means of both legal and illegal immigration.
In January 2017, Trump activated a ninety-day immigration ban on individuals from seven "countries of concern" and which happened to be Muslim-majority countries. Five of these were predominantly Sunni: Sudan, Libya, Somalia, Yemen, and Syria. And two were predominantly Shi'a: Iran and Iraq. Yet after his first immigration ban was blocked in the courts, Trump issued a new order in March 2017 that dropped Iraq from the list—after reassurances from the Iraqi government that it would provide increased information sharing with the United States. But this shift also appears to have resulted from the fact that US businesses and the US military have major interests in Iraq. The new order would not include American green card holders or previous visas, for example, and it would no longer permanently ban Syrian refugees. At the same time, Trump authorized new decrees that would make it easier to deport any illegal or unauthorized aliens.
Ironically, however, Trump's efforts to ban immigrants from six or seven Muslim-majority countries do not impact the nationalities of the individuals who were directly involved in the September 11, 2001, terrorist attacks: Saudi Arabia had nineteen citizens involved; the United Arab Emirates, two; Egypt, one; and Lebanon, one. In addition, the September 11 attacks were not even orchestrated from Saudi Arabia, or even from Afghanistan—which the United States, under a UN mandate, then attacked in December 2001; the attacks were coordinated by an al-Qaeda cell operating out of Hamburg, Germany—a NATO ally.
Trump has denied that the immigration ban is aimed at Muslims alone. He has claimed that his administration has only been pinpointing those individuals from "countries of concern" who could engage in extreme violence. Nevertheless, the fact that Trump had demanded a "total and complete shutdown of Muslims entering the United States" in December 2015 during his election campaign after the San Bernardino killings by Islamic State sympathizers—and that he plans to "eradicate" radical Islam from the face of the earth, as stated in his January 2017 inauguration address—project the public image that Trump intends to engage in a crusade against the whole Muslim world.
**THE QUESTION OF DOMESTIC TERRORISM**
Even if some form of immigration ban is eventually upheld by the courts, it is not at all clear that such a ban would necessarily address the real security problem involving acts of terrorism inside the United States. This is because acts of extreme violence are not always influenced by radical Islamist theology, and many of these acts have been carried out by US citizens, not by immigrants or Muslims.
Such acts of terrorism, among many others, include the 2012 mass shooting in Aurora, Colorado, in which the killer was not an Islamist. They also include the 2013 attack in Orlando, Florida, in which the killer purportedly pledged allegiance to the Islamic State. What the two cases had in common was that both killers were mentally disturbed "lone wolves"—with very different personal and social grievances. And both had legally purchased semiautomatic weaponry as American citizens. While al-Qaeda and Islamic State represent organized groups that can provide some technical know-how and information to individuals, lone wolves often act in the hope that their violent actions will be disseminated by the media and then inspire others.
The issue raised here is that given Trump's near-obsessive focus on Islamist militants, the real dangers of "right-wing," anti-Semitic, Islamophobic, and other forms of violent fanaticism and xenophobia must not be ignored. The Norwegian Anders Behring Breivik, for example, murdered seventy-seven young Norwegians on July 22, 2011, at a Labour Party Youth summer camp in the hope that the media attention he would obtain would help his fascist cause gain support in Europe and in Russia. Breivik had hoped to spark a war against both politically correct "cultural Marxism" and against Islamist movements. Breivik's attack and his right-wing ideology then influenced the American Adam Lanza, who shot and killed twenty-six people (including twenty children) at Sandy Hook Elementary School in Newtown, Connecticut. Lanza then shot himself.
In August 2017, white supremacists, neo-Nazis, and the Ku Klux Klan organized a "Unite the Right" demonstration in Charlottesville, Virginia, to protest a decision to remove a statue of Confederate General Robert E. Lee; the violence at that demonstration, which resulted in the death of one counterprotester and the injuring of nineteen others, shows that these right-wing movements are coming out of the closet in the belief that Trump has appeared to sympathize with their cause. Trump did not strongly condemn the white supremacist movement and its racist and neo-Nazi slogans, and instead affirmed that the violence was caused by "hatred, bigotry and violence on many sides." Former imperial wizard of the Ku Klux Klan David Duke stated that protesters were "going to fulfill the promises of Donald Trump" to "take our country back." (Further, other acts of terrorism have been influenced by ideologies other than political Islamism. The killings of police officers in Dallas and Baton Rouge in 2016 by two disturbed US veterans, Micah Johnson and Gavin Long, respectively, were facilitated by the access to assault weapons. These acts appeared to be more inspired by the Black Lives Matter movement and black nationalism than by Islamist ideology.)
And finally, Trump was absolutely silent on the issue of gun control when a deranged individual, Stephen C. Paddock, engaged in a mass shooting in Las Vegas, using a device that turned semiautomatic weapons into automatic. Paddock killed himself, and no one seems to know the motivation for the shooting, except perhaps that he had lost a significant amount of wealth and feared losing his casino "high-roller status." Armed with at least twenty guns, Paddock was able to bring four thousand rounds of ammunition into his hotel room before killing at least fifty-eight people and wounding 546.
**TRUMP'S CONTINUING FOCUS ON MUSLIMS**
In his February 2017 address to Congress, Trump became the first president to use the term "radical Islamic terrorism." He purportedly overruled his newly appointed National Security Advisor, Lt. Gen. H. R. McMaster, who had argued that these groups are better described as "un-Islamic"—even if they claim to represent Islamist beliefs. Neither George W. Bush nor Barack Obama officially referred to "Islamic terrorism." This is because the term conflates Islam as a religion with terrorist organizations that manipulate Islamic beliefs for their own political purposes. In addition, anti-state partisan groups that engage in acts of terrorism and violent extremism might or might not possess "Islamist" ideologies. Trump reiterated the same offensive term in his September 2017 speech to the UN General Assembly, when he proclaimed that he would "stop radical Islamic terrorism."
The issue raised here is that because Trump is focusing on what he has called "radical Islamic terrorism" primarily, US policy might avoid focusing its attention on far-right-wing or far-left-wing or other individuals/groups that are also plotting the use of extreme violence—but that do not possess Islamist ideologies (e.g., the Charlottesville attack and the Las Vegas mass shooting). Yet the October 31, 2017, attack in a Lower Manhattan bike path by Sayfullo Saipov, an Uzbek citizen whom Trump claims had won the green card—but who had ostensibly converted to Islamist extremism only since living in the United States, and who was then praised by the Islamic State for his actions—has permitted Trump to once again go on the attack against violent Islamist movements and immigrants in general, while trying to avoid crucial issues raised by right-wing violence in Charlottesville and by the mass shooting in Las Vegas.
The problem is that the official US government use of the term "radical Islamic terrorism" can be manipulated by militant groups with Islamist ideologies to augment recruitment. This is very problematic because one the major sources of pan-Islamist recruitment is the very high un- and underemployment rate of youth throughout much of the Arab/Islamic world, plus police repression. In addition, these youths are often attracted to militant Islamist movements because Islamists argue that the tremendous Arab oil wealth, which is controlled by the royal families of these countries, which are in turn backed by the United States, European (or Russian) military-industrial complexes, could be better invested in job creation and development of the poorer Arab states. Groups like al-Qaeda and IS oppose not only the United States, Europeans, Russia, China, and Israel but also the Arab Gulf monarchies. (See chapter 8.)
In sum, Trump is doing much too little to unite the country against "hate and evil" in all forms. On the contrary, his anti-Muslim propaganda and perceived sympathy for extreme right-wing movements appear to be helping to spread "hate and evil" and to polarize American society.
**THE QUESTION OF IMPEACHMENT**
Many of the allegations against Trump discussed in this chapter could be used to impeach him, but the process could prove to be very long and make it even more difficult to pursue US diplomatic initiatives that are truly intended to foster regional and global peace. All accusations against him have, of course, been denied by Trump as "fake news - a total political witch hunt."
Nevertheless, there appear to be viable grounds for the alleged complicity of a number of Trump's political advisers with Russian officials, but it really depends on what was the precise purpose of those contacts between the Trump team and Moscow. The Trump team has been accused of complicity with Moscow in the effort to find damaging information on Hilary Clinton, with information allegedly provided by Russian intelligence services. There have been further allegations that Trump and/or his associates might have obtained significant stock shares in the Russian government's Rosneft Gas Company (close to Putin) in exchange for the lifting of sanctions against Moscow and downplaying the Ukrainian question. As previously mentioned, the disclosure of the "Paradise Papers" could open the door to further congressional investigations of alleged Trump administration corruption.
While a number of the previously mentioned charges have not yet been proved, Trump and his associates have engaged in major business deals with Chinese firms—which has also raised ethical and constitutional questions. In March 2017, for example, Chinese officials rapidly approved thirty-eight new Trump trademarks, including branded businesses from hotels to insurance to bodyguard and escort services.
In May 2017, Trump was additionally accused of allegedly sharing US secrets with Russian Foreign Minister Sergey Lavrov. In this meeting, Trump was said to have revealed the city in the Islamic State's territory where the US intelligence partner, Israel, had detected the purported IS threat to laptops. This sharing of information, which is the president's prerogative, apparently did not go through the appropriate intelligence channels and was not approved by Israel, which feared that it could expose the source of the information. This action, by itself, did not present a case for Trump's impeachment, but it could be considered a violation of his oath of office, in addition to representing an example of presidential incompetence. If it is proven that Trump's firing of FBI Director James Comey did represent obstruction of justice, then this could provide at least one basis for impeachment.
If Trump were impeached, or forced to step down by his own cabinet under Article 25 of the US Constitution, Christian conservative Vice President Mike Pence—who is at least partly responsible for Trump's turn toward a more hardline anti-Russian position—could take his place. But these scenarios will take time to carry out and it is not clear that the Republican-controlled Congress wants Trump out yet—at least not until its radical tax-reduction proposals are implemented into law, after the Republicans failed to defeat ObamaCare and implement a new plan.
More likely, Trump will continue to use both external and internal "threats" such as illegal immigrants, Islamist terrorists, and North Korean nuclear weapons testing to deflect criticism away from the investigation into his affairs with Russia for as long as possible. At the same time, critics of Trump will point to Russia as the predominant "threat." Not all of these "threats" are imagined, and Russia, along with China, could soon represent a real danger—but only because of the failure of US diplomacy to deal with Russia, China, and other perceived threats more effectively.
**THE IMPACT OF DOMESTIC THREATS ON FOREIGN POLICY**
In sum, allegations of the Kremlin's influence in the US election process, combined with accusations of "illegal" immigrant votes taking part in the US elections, plus fears of Islamist "terrorist" infiltration into American society, not to overlook Beijing's and Pyongyang's alleged cyber-intrusions, all appear designed to illustrate purported foreign threats to the sacrosanct American democratic process and to the safety and security of American society as a whole. On the one hand, Senator John McCain was not alone in decrying alleged Russian cyber-tampering as an "act of war." On the other, President Trump himself has made both "illegal" immigration and fears of "Islamic terrorist" infiltration into American society major issues of his presidency—while Trump's public pronouncements have worked to escalate nuclear tensions with Pyongyang.
If steps are taken to impeach Trump, the danger in such a situation is that Trump's presidential decisions that impact both domestic and foreign policy could be made for domestic tactical reasons, to protect Trump himself—and not for any greater national or international purpose that could serve the greater cause of global peace. As events continue to play out under the looming possibility of Trump's impeachment process, any of these foreign "threats" could be manipulated by the president, his cabinet, or Congress in such a way as to start a war, either accidentally or accidentally on purpose, in the effort to divert American popular attention away from domestic controversies that involve the ongoing struggle for power within Washington and with rival states abroad. A major war could break out—particularly if a rival or anti-state organization engages in something, correctly or incorrectly, interpreted by Washington to represent an action hostile to US interests.
Much as Aristotle pointed out more than two thousand years ago, it is when the "distant threat" can be brought home, and into one's daily life, that the possibility of war with a distant power or powers becomes much more viable. And once the distant threat is brought home, then the burdens of militarization, if not the sacrifices of war, can be met with greater toleration by the society once the general population can actually see how that otherwise "distant threat" actually challenges their safety and way of life. And the possibility of war abroad is then further exacerbated by the perceived threat of "enemies" at home—who are seen as opposing, whether peacefully or violently, the politics and goals of those in power—thus undermining the legitimacy and effective ability of the powerful to rule.
There is a real risk that a number of Trump administration policies that tend to encourage US protectionism and "economic nationalism" will likewise help to legitimize both far-right-wing and far-left-wing anti-EU and anti-NATO political parties throughout Europe. And if Trump continues to push for increased military expenditure for an ongoing NATO enlargement, coupled with a policy of strong US and EU sanctions on Moscow that are seen as harming European political-economic interests, such a policy could backfire. This could cause some states to abandon EU and/or NATO membership—without necessarily implementing constructive and viable alternatives. Wolfgang Ischinger, head of the Munich Security Conference, and a former German senior diplomat, warned: "Is President Trump going to continue a tradition of half a century of being supportive of the project of European integration, or is he going to continue to advocate EU member countries to follow the Brexit example?...If he did that, it would amount to a kind of nonmilitary declaration of war. It would mean conflict between Europe and the United States. Is that what the US wants? Is that how he wishes to make America great again?"
Likewise, the president of the European Commission, Jean-Claude Juncker, warned that the collapse of the European Union could, for example, lead to war in the western Balkans. Juncker further stated that Trump's America First doctrine was scaring Europeans into thinking that the United States no longer cared for Europe. Soon after, Trump suddenly changed tune and claimed that the European Union was a "wonderful" organization.
Prior to becoming president, Trump initially claimed to be "indifferent" to the European Union—in addition to asserting that NATO was "obsolete." But he nevertheless appeared to be actively seeking to undermine the European Union by encouraging states to leave that international regime, given his strong support for the United Kingdom's exit from the European Union (Brexit). Trump also engaged in strong attacks on Germany, which he sees as the major global economic competitor to the United States, given Germany's roughly $65 billion trade surplus with the United States. The risk is that Trump's America First policies could eventually open the door for Beijing and Moscow to play their own games of offering finance and trade (China) and energy and resources (Russia) to needy countries—in their effort to obtain greater political-economic influence inside Europe. And in destabilizing Europe—coupled with Russia's overt and covert backing for anti-EU and anti-NATO movements—Trump's policies could destabilize much of the world. (See chapter 1.)
With respect to NATO, Trump had initially argued that NATO was obsolete due to the fact that it had been created during the Cold War, and because of its ostensible lack of relevance for the Global War on Terrorism, combined with the fact that not all of its membership spends the expected 2 percent of its GDP on defense. From Trump's perspective, this has forced Washington to remain the predominant provider of European and global security; the United States contributes some 70–75 percent of NATO's military capabilities—a fact that implies that the Europeans were free riders, paying for only 25–30 percent of total defense costs. Trump therefore made the provocative argument that the United States might not defend allies who did not pay their fair share of the NATO defense burden. Trump's statements accordingly appeared to downplay NATO's Article V defense commitment.
Trump's negative positions toward NATO and the European Union appeared, at least on the surface, to reverse themselves by February 2017, once Vice President Pence spoke up for both NATO and the European Union. Countering European fears that Trump would abandon NATO and the European Union, Vice President Mike Pence rushed to Brussels in late February 2017 to declare "unwavering" US support for NATO and "unequivocal" support for the European Union—in a speech before an audience of skeptical European elites. The Belgium capital, Brussels, ironically, houses the offices of both the European Union and NATO. But the two rarely speak together, and when they do, they generally cannot find a common understanding!
Vice President Pence accordingly gave a resounding "yes" to three crucial questions from EU Council-Consilium President Donald Tusk. The first question was whether the Trump administration was committed to maintaining an international order based on rules and laws. Tusk's second question was whether Trump was committed to NATO and to "the closest possible trans-Atlantic cooperation." The third question was whether Europe could count, "as always in the past, on the United States' wholehearted and unequivocal, let me repeat, unequivocal support for the idea of a united Europe." By mid-February 2017, Trump himself suddenly changed tune, stating his strong support for the European Union and claiming that it was a "wonderful" organization.
Nevertheless, the Trump-Pence administration continued to warn that the alliance might cease to function altogether if allies did not contribute their share of the defense burden. This possibility was due to the fact that underspending on NATO eroded "the very foundation of our alliance," in Mike Pence's words. Secretary of Defense Gen. James Mattis, put it this way: "America will meet its responsibilities, but if your nations do not want to see America moderate its commitment to the alliance, each of your capitals needs to show its support for our common defense." At the May 2017 NATO summit in Brussels, Trump argued that even 2 percent of GDP (roughly $119 billion) would be "insufficient to close the gaps in modernizing, readiness, and the size of forces. Two percent is the bare minimum for confronting today's very real and very vicious threats."
Vice President Pence likewise stated that some of the largest US allies still lacked "a clear and credible path" in order to build up their military capabilities to 2 percent of GDP. Here Germany, the main political-economic power in Europe, and major arms exporter to Qatar and Saudi Arabia (along with the United States) is still below the mark and is considered "short of everything" from manpower to military equipment. And even though France spends more for defense than other NATO members, it has nevertheless reduced its actual capacities by roughly 50 percent under Presidents Nicolas Sarkozy and François Hollande.
Yet the problem is that even an increase in European defense spending will only prove helpful if is better coordinated by a European defense organization. Of the five countries that spend more than 2 percent of GDP on defense—the United Kingdom, Estonia, Poland, Greece, and the United States—Greece is more concerned with NATO member Turkey than it is with Russia. For its part, the United Kingdom still retains a global empire, as does the United States—so that European NATO is not its only defense and security concern. Romania has declared that it will reach 2 percent of GDP in 2017, while Latvia and Lithuania will also reach 2 percent. All three of the latter are primarily concerned with a potential Russian "threat," as are Estonia and Poland. At the same time, it is not clear whether all NATO allies, including Germany, can or will fall in line with US demands to increase defense spending.
The real problem is that the two sides, the United States and Europeans, continue to divide on a number of key political and strategic questions. These questions involve the nature and costs of a NATO/European defense buildup, the Global War on Terrorism, as well as policies toward Russia, Ukraine, Iran, and China, and others. Here there are many issues where the United States and Europeans disagree and where the Europeans themselves disagree, as eastern European states generally want NATO and the European Union to focus on Russia, while states such as France want NATO and the European Union to focus on the issues confronting the Mediterranean and the Global War on Terrorism. The question as to whether or not Ukraine and Georgia should become NATO members, for example, remained the unspoken "elephant in the room" in the NATO summit of May 2017. And, as to be argued, another option to forcing individual NATO countries to augment their defense spending would be to implement a post-Brexit European defense entity that would better coordinate defense spending. (See chapter 9.)
**TRUMP AND THE EUROPEAN UNION**
The new American nationalism risks playing with fire if Trump continues to engage in policies that would even indirectly support "economic nationalism" in Europe. The demise of the European Union risks the return of rabid nationalist rivalries inside Europe. This is because the rise of the European Union has played a historical role in helping to bring rivals France and Germany into cooperation with other the European powers—following the formation of the European Coal and Steel Community in 1957 and the Franco-German decision to forge a common currency, the Euro.
Trump had fully supported the British exit from the European Union (Brexit), and he stated his belief that other countries could soon leave the European Union as well. The so-called PIIGS states of Portugal, Ireland, Italy, Greece, and Spain—whose political economies need significant reforms in order remain in the Euro monetary system—could possibly drop out of the European Union. Greece is still suffering after reaching a crisis point at which it almost left the European Union. European austerity measures have resulted in cutting public expenditures, high unemployment, loss of pensions, the sale of public assets, plus efforts to raise taxes—in a situation in which the Greek economy has lost 25 percent of its former value. International efforts to bail Greece out have been promised only after Athens engages deep structural reforms and further austerity measures. Having already accumulated over €300 billion in bad loans, the possible collapse of Italian banks could impact much of Europe, including Austria, Greece, Spain, and Portugal. And if Italy fails, France may be next.
Despite the election of the liberal-centrist and pro-EU Emmanuel Macron as president of France in April 2017, there is still a real possibility of a French exit from the European Union in the next five to ten years. If so, this would risk tearing apart the remaining Franco-German-EU political-economic and defense relationship. In dividing Europe, such an option is even more dangerous than the British exit—as the United Kingdom never shared a common currency with the European Union. Brexit and Frexit are not the same thing!
While the United Kingdom will most likely continue to work with the Europeans through bilateral national security accords and through NATO, even if it is no longer a member of the European Union, it is highly unlikely that other states that leave the European Union will necessarily remain in NATO. Unlike the British conservatives, like Theresa May who backed Brexit, but who underscored the importance of NATO in her meeting with President Trump in January 2017, many continental Europeans, on the left, right, and even center, are anti-NATO.
**BREXIT AND THE EUROPEAN UNION**
Brexit nevertheless opens up a number of complex questions with respect to the United Kingdom's security and defense relationship with Ireland and Scotland and with the European Union as a whole; questions that indirectly impact NATO. As of September 2017, London plans to leave the European Union by March 2021—at a cost of about £40 billion and a slight loss of Britain's credit status thus far.
First, the United Kingdom, which accounted for roughly 20 percent of European defense, will no longer be able to block all-European defense plans. This theoretically permits EU to press ahead with an all-European defense system, if France and Germany, the next two major EU actors, can gradually coordinate policies. At the same time, however, bilateral UK-French strategic nuclear accords reached at the 1998 Saint Malo agreement indicate the continuing need for bilateral UK-French nuclear-strategic and defense cooperation. In 2016, London and Paris agreed to deepen their joint efforts to develop missile technology. How Brexit will politically impact these bilateral UK-French accords, which are outside the NATO and EU context, remains to be seen.
Brexit also opens up a number of questions for Ireland and Scotland as members of the United Kingdom. Brexit raises border questions with Ireland, and once again raises the question of Northern Ireland's independence, given the fact that hundreds of thousands of Irish live in the United Kingdom and hundreds of thousands of UK-born and eastern European citizens reside in Ireland. Brexit also raises the issue as to how to regulate trade and traffic along the border of UK-controlled Northern Ireland and EU member Ireland. This concern led to UK Prime Minister Theresa May and Irish Prime Minister (the taoiseach) Enda Kenny reconfirming the 1998 Good Friday accords in Northern Ireland, which has led to peace through power-sharing between the Unionists (who want to remain in the United Kingdom) and the Irish nationalists.
The issue of border controls has brought back memories of the Troubles—as British military outposts and customs posts were often attacked by the Irish Republican Army. If new border controls are implemented, and if the borders are not perceived to be free and "fluid," they could once again become targets of new violent anti-UK movements by Irish militants who still hope to leave the United Kingdom—particularly if the aforementioned power-sharing arrangement reached in the 1998 Good Friday agreement continues to break down periodically, as it did in January 2017, for example.
Brexit could also lead to Scottish independence from the United Kingdom. This was proposed by Scottish Prime Minister Nicola Sturgeon, who has planned a second independence referendum for the fall 2018 to the spring 2019. Scottish nationalists have generally wanted to remain in the European Union, but not necessarily in the NATO "nuclear alliance"—given the Scottish National Party's (SNP's) opposition to nuclear weaponry. The Scottish independence issue impacts the future of the United Kingdom's Trident nuclear submarine system, which is based on the Clyde. Many Scots oppose Trident, as well as NATO/UK airbases in Scotland.
An independent Scotland may also need to reapply to join the European Union, which could open monetary questions concerning the Scottish use of the British pound after Brexit. And if Scotland wants to remain in NATO, it might need the approval of the all NATO members. This could possibly be opposed by some states that do not want to encourage independence movements. NATO members Spain, Greece, Romania, and Slovakia, for example, did not want to recognize Kosovo independence, which was backed by the United States. Some sort of intermediary arrangement could possibly be worked out with both NATO and the European Union—but not without time-consuming negotiations.
Brexit has also opened questions as to whether the United Kingdom or EU member Spain will control Gibraltar, which guards the gates of the Mediterranean. Like the people in Northern Ireland, the population in Gibraltar fears the possibilities of economic instability once they leave the European Union's external border. In addition, UK disputes with Argentina over the Falkland Islands appear to be returning. This is because Argentina believes that the European Union may no longer back British control over the islands after the 1982 UK-Argentine war. Argentina may seek to bargain post-Brexit bilateral accords with Britain in exchange for a return of the islands to Argentina. The region around the Falklands is rich in fishing, resource deposits, and oil. These concerns were responsible, at least in part, for the 1982 sea battle over the Falklands between Argentina and the United Kingdom. Just prior to that conflict, the option of joint sovereignty had been proposed as a way to forge a compromise and prevent war—but it failed to be implemented. (See chapter 10.)
**GERMANY**
Trump has argued that the European Union represents an "instrument of German domination designed with the purpose of beating the United States in international trade." And largely because he sees the European Union as a mere instrument of German political-economic power, Trump claims that he is basically "indifferent" as to whether or not the European Union will break up. Germany does represent a major economic competitor of the United States (in automobiles, for example, and given Germany's huge $65 billion trade surplus with the United States). But while it is true that Germany may dominate the economy of the European Union, that by itself is more reason to engage in political and economic reforms of the European Union, rather than lending support to those who want to break up it entirely. (For EU reform options, see chapter 9.) After discussions with Chancellor Angela Merkel, Trump stated: "Nevertheless, Germany owes vast sums of money to NATO & the United States must be paid more for the powerful, and very expensive, defense it provides to Germany!"
The issue raised here is that Trump's harsh criticism of Germany's immigration policy, plus his demands that Germany spend at least 2 percent on defense (Germany spent roughly 1.19 percent of GDP on defense in 2016), combined with threats to impose up to 35 percent border taxes on German products, could unleash a wave of thus far hidden anti-American nationalist forces in Germany and elsewhere, particularly after the right party, Alternative for Germany party (AfD), entered the Bundestag with roughly 13 percent of the vote in September 2017. (See discussion on right-wing parties, this chapter.)
Although Germany hopes to diversify its energy sources, it has opposed strong US sanctions on Moscow that might interfere with the construction of Nord Stream 2, an energy pipeline that would run through the Baltic Sea, circumventing Ukraine. US sanctions could also hurt the financing of European firms doing business in Russia. Berlin has called the sanctions imposed by the "Countering America's Adversaries through Sanctions Act" in August 2017 as illegal and has urged the European Union to take countermeasures—a step that could enter into a trade war if the United States and the European Union do not coordinate sanctions policy and negotiate the eventual lifting of sanctions together.
American threats not to support NATO and Brexit, alongside Russian threats to Ukraine and eastern Europe, could then press Germany and the new European Union (without the United Kingdom) to forge tighter political-economic, if not military, ties with China, with which Germany has already established a "special relationship," in which China believes that Berlin will strengthen its influence in the European Union to China's political and economic advantage. Germany has, for example, been lobbying the European Union to put an end to the EU arms embargo placed on China since the June 1989 repression of the Chinese democracy movement. In exchange for the European Union ending its arms ban, Beijing has promised greater finance to European countries, and greater trade and investment opportunities. Beijing has been investing heavily, for example, in nuclear energy plants in the Czech Republic, Hungary, and Poland, in rivalry with both the United States and Russia, and Beijing is looking to the Bulgarian nuclear energy market as well. Because the United Kingdom had been one of the strongest supporters of the arms embargo on China, the lack of a British presence in the European Union could then open the door to closer European-Chinese defense relations. Such an approach would ostensibly be intended to attempt to draw China away from closer defense ties to Russia, but to the chagrin of the Japanese, while it could be interpreted as an act of "encirclement" by Moscow. Such an EU-China defense linkage could occur if NATO does start to fall apart, and assuming that Germany and the new European Union (without the United Kingdom) cannot reach a separate accord with Russia over Ukraine, among other EU-Russia disputes.
Prior to Trump, the Obama administration had been encouraging Germany, France, and the Europeans to look toward Japan and India for closer political-economic ties, including arms sales—despite Germany's already close ties to China. Closer German-Japanese political-economic relations in Europe and the Indo-Pacific would then represent a counterpoint to China's Belt and Road Initiative (BRI), which seeks to expand Chinese trade and investment between Asia and Europe. Both Moscow and Beijing, however, see this approach as seeking to forge a new "encircling" alliance in Europe and in the Indo-Pacific. (See chapters and .)
**IMMIGRATION INTO EUROPE**
The general financial crisis in the European Union since 2008, coupled with the immigrant crisis, and US and EU sanctions on Russia, has been making it even more difficult for the European Union to function—as a number of states have decided to strengthen their borders. In mid-2016, among EU countries, Hungary possessed the most asylum applicants per capita, followed by Sweden and Germany. In terms of numbers, Germany has accepted the most migrants, then Sweden, Italy, France, the Netherlands, and the United Kingdom. Ironically, Germany's decision to take in at least 1.1 million immigrants, due, in large part, to Germany's growing need for labor, has been criticized by Trump, even though this is a German, and not an American, affair, and should be of no concern to Trump.
One of the main issues that has caused an anti-EU backlash has accordingly been the failure of the European Union to deal effectively with the post-2015 immigration crisis from Syria, Afghanistan, Iraq, and other North African countries. Waves of immigrants followed in the wake of the Arab Spring movements in 2011, the French- and UK- led military intervention in Libya (backed by NATO) in March 2011, and then the Syrian civil war, which began in 2011. More than a million refugees arrived in Europe in 2015 alone. The failure of the European Union to check the wave of refugees at the edges of the European Union in Greece and Italy, as well as into Hungary through the Balkans, then led many European states to close their national borders. This led to the building of a whole series of national walls/fences/barriers in Europe, as generally demanded by rising right-wing nationalist groups.
Ironically, refugees were blocked in Calais, France, unable to travel to the more liberal United Kingdom, where they believe they will find jobs—as the United Kingdom was not part of Schengen group that oversaw immigration policy inside the European Union. The large immigrant camps in France were then broken up by force by the government of François Hollande, and most immigrants were forced to move to different locations or flee the country. France was in the ironic situation of trying to keep in the country refugees who did not want to stay! Here, although fears of immigration were a factor in Brexit, it was largely opposition to immigration from eastern Europe that was believed to have impacted the jobs of English citizens—and not the fear of non-European refugees who had entered the Schengen zone of continental Europe.
**SANCTIONS ON MOSCOW**
By contrast with east European states (with the exception of Serbia, Hungary, Slovakia, and perhaps Bulgaria and the Czech Republic), many western European states (Germany, Italy, and France, among others) have generally wanted to maintain positive relations with Russia. These states were initially reluctant to place strong sanctions on Moscow's banking, oil, and defense sectors in 2014.
For its part, the United Kingdom has generally taken the toughest stance on Russia in part due to the poisoning of former Russian agent Alexander Litvinenko. But British legal investigations into that affair, plus sanctions placed on Russia in the energy sector, have not, for example, prevented British Petroleum (BP) from making profitable deals with the Russian energy firm Rosneft by 2016. Likewise, German Chancellor Merkel has nevertheless insisted that the United States and the European Union sustain sanctions on Russia in response to its annexation of Crimea and political-military interference in eastern Ukraine, while likewise denouncing Putin for his support of Syrian leader Bashar al-Assad.
Europeans who argue against sanctions generally argue that sanctions have no impact on Russian policy in Ukraine. Contrary to the Americans, who trade less with Moscow, many businesses did not want to lose access to Russian markets. And when Moscow unexpectedly placed counter-sanctions on European agricultural exports to Russia, those counter-sanctions generally hurt Europeans more than Americans. European farmers have subsequently been hurt significantly by counter-sanctions placed by Moscow on European farm products. Trade between Russia and the European Union consequently dropped by over €180 billion between 2013 and 2015 while EU farmers and agricultural cooperatives claim that they have lost their main export market worth €5.5 billion. EU agriculture was additionally hurt by a general drop in Chinese demand due to a downswing in the Chinese economy.
While some US auto companies and banks have been hurt by sanctions on Moscow, a number of American food companies, including MacDonald's, Yum! and Burger King, and the agro-industrial firm, Cargill, have actually been benefiting from them—as Moscow began to engage in import substitution (which it claimed to be beneficial) while also seeking to import food products from Brazil, Argentina, and Asia. Israeli agriculture has also hoped to benefit—a factor drawing Israel and Russia closer together. The enterprises of states that do not agree to placing sanctions on Russia, most importantly China, but also Japan, Turkey, Brazil, Argentina, Qatar, and South Africa, have begun to benefit.
In June 2016, the German and Austrian foreign ministers began to have second thoughts. They stated that EU sanctions on Russia should be gradually phased out as the peace process progresses. This represented an effort to reverse previous positions that sanctions could be lifted only once the Minsk peace plan is fully implemented. Both France and Greece have likewise sought a change in policy toward sanctions as well. EU sanctions have reduced Russian GDP by 1 to 1.5 percent, and the EU's own GDP by 0.1 percent. The EU has lost an additional 0.3 percent of its GDP as a consequence of the Russian counter-embargo on EU agricultural products. The Baltic states, Finland, and Poland are paradoxically the most supportive of sanctions but also the ones to suffer most from them. This is while Italy, for example, has been less harmed.
For its part, the United States, which most strongly supports sanctions, has lost a mere 0.005 percent of its GDP because of its own sanctions on Russia and the counter-sanctions imposed by Russia on the United States. Yet despite the harm done to European agriculture, EU ambassadors nevertheless agreed to first extend their economic sanctions against Russia to January 2017, and then later to July 2017, and again to January 2018. These renewed sanctions were due to Moscow's perceived lack progress on Minsk II accords that has been intended to establish peace in eastern Ukraine—and even if not all the problems can be blamed on Moscow for not following through on the Minsk accords.
**THE RISE OF ANTI-NATO ANTI-EU MOVEMENTS**
The US and European financial crisis since 2008 has led a number of left-wing and right-wing political parties to demand that their countries drop the Euro as a currency, dump their creditors, and then exit the European Union—rather than attempting to further reform the European Union itself. The Austrian, Dutch, and French elections have, however, appeared to have stemmed the tide of nationalist-populist movements for the moment, but they might not be able to hold out for long—if political-economic conditions do not eventually improve in the European Union as a whole. In this regard, the rise of the German far right, which seeks to break monetary ties with France and other countries, in the September 2017 elections is very worrying. In general, both far-right and far-left movements have gained in strength as a result of the decline of traditional parties, due in large part to the rise in social inequity and the decline of the middle classes.
In 2016, in the Austrian presidential elections, the Green Party and European Federalist candidate Alexander Van der Bellen just barely defeated far-right party Freedom Party candidate Norbert Hofer. In the Netherlands, in the March 2017 elections, three centrist parties (People's Party for Freedom and Democracy, VVD; Labour Party, PvdA; and Democrats 66, D66)—which are all pro-European and pro-NATO, but with some differences on immigration—ran against the nationalist anti-EU Party for Freedom (PVV) of Geert Wilders. The PVV only won 13.1 percent of the votes, but due to the pluralist party system, the PVV still became the second largest party.
In April 2017, Emmanuel Macron of the new, liberal Federalist party En Marche! (Onwards!) won the French presidential elections by 67 percent. But Macron will nevertheless face an uphill battle to obtain popular support for his proposed liberal market-oriented reforms. The far-left presidential candidate, Jean-Luc Mélenchon, refused, for example, to publicly endorse Macron against Marine Le Pen in the second round of the French elections and called for an international conference to adjudicate border conflicts in eastern Europe. Along with the far right, Mélenchon has continued to oppose Macron's liberal labor policies, calling for strikes in September 2017. (See discussion on France, this chapter.)
Chancellor Angela Merkel, whose pro-immigration policies were also denounced by Donald Trump, won her fourth term in September 2017. The Social Democrats (SPD) had surged briefly in popularity since choosing left-wing Martin Schulz to run against Angela Merkel, but he nevertheless lost the elections. The SPD then announced that it would no longer sustain an alliance with Merkel's CDU (Christian Democratic Union of Germany). This will make it difficult for Merkel to forge a coalition in the Bundestag particularly as the extreme right-wing party, Alternative for Germany party (AfD), gained roughly 13 percent of the Bundestag. The CDU will hold 246 MPs in the Bundestag, and the SPD, which scored poorly, has 153. The right-wing AfD took 94 seats; the FDP, 80; the Left, 69; and the Greens, 67.
The rise of the far right in Germany negates what were otherwise more or less positive signs. In addition, there is a real concern that Trump's policies of economic nationalism, his support for Brexit, his strong criticisms of Germany, and his pretended support for the European Union, plus the general financial crisis, combined with sanctions on Moscow—and US pressure on NATO members to spend more on defense—are all encouraging a number of anti-EU, anti-NATO left-wing, populist, and nationalist parties to rise to power in Europe. These movements have been given ideological support by Steve Bannon, one of Trump's advisers, who has remained behind the scenes even after he was pressed out of the US National Security Council in April 2017. And behind a number of these populist movements, there are a number of even more overtly fascist/ racist movements that could follow in their footsteps.
After winning the presidency, Trump had met with Nigel Farage of the right-wing UK Independence Party, even before meeting with Theresa May, the British Prime Minister. Trump himself gave far-right-wing candidate Marine Le Pen a tacit endorsement just before the April 2017 French presidential elections, by saying that she was "strongest on borders, and she's the strongest on what's been going on in France.... Whoever is the toughest on radical Islamic terrorism, and whoever is the toughest at the borders, will do well in the election." Trump's statement proved completely wrong, since the liberal-centrist Emmanuel Macron won the election.
Some of these right-wing and left-wing movements are backed by Moscow; others oppose Moscow. But both left-wing and right-wing movements have stated their opposition to NATO and the European Union. On the right, these include the Front National (France); FPÖ (Austria); Golden Dawn (Greece); KSCM (Czech Republic); and Jobbik (Hungary). On the left: Front de Gauche (France); AKEL (Cyprus); and Die Linke (Germany).
In 2014, in Hungary, for example, the pro-Russian Prime Minister Viktor Orbán's nationalist Fidesz-KNDP party had won 44.5 percent of the votes, but the far-right Movement for a Better Hungary, Jobbik, won 20.54 of the vote, meaning far-right-wing parties possess some 65 percent of the vote in that country. In 2015, in Poland, the right-wing Law and Justice Party has begun to challenge the separation of powers between the state and legal system. While anti-Russian and pro-NATO, Poland's Prime Minister Beata Szydło may also oppose the European Union's new project—a European Pillar of Social Rights.
Many left-wing movements tend to take a pro-Russian stance on many issues, even if Putin's politics are far from being "left-wing." European left-wing movements have gained strength following the 2008 financial crisis, but they tend to lose support on account of the fear of a massive influx of foreign immigrants entering Europe. Among left-wing movements, European United Left/Nordic Green Left (GUE/NGL) in the European Parliament generally supports the Russian position in the Council of Europe, and in OSCE general assemblies—especially on issues related to Ukraine and Syria. Greece's Syriza party, in coalition with the right-wing populist ANEL, often backs Russia on energy, foreign affairs, and defense. The left-wing Podemos rapidly became the third largest party in Spain since 2014, while Die Linke is strong in eastern Germany. The Cyprus Communist AKEL party has obtained as much as 30 percent of the vote in parliamentary elections. Sinn Fein, the political wing of the IRA, is one of the top three political parties in Ireland. In September 2017, as previously mentioned, the populist Alternative for Germany (AfG) party entered the Bundestag for the first time, with ninety-four seats.
Moreover, given the fact that the far-right nationalist-populist parties possess very different goals, such movements will not be able to forge a unified policy—except for their general opposition to the European Union and NATO. There are, however, a few nationalist parties (primarily from eastern Europe, but also the Scottish, who want to remain in the European Union once the United Kingdom leaves) who want to not abolish the European Union but reform it. Most of the Euro-nationalist parties thus oppose EU bureaucracy but disagree on other issues. The Alternative for Germany (AfG) party, for example, regards Le Pen's National Front (NF) as a national socialist movement in that the NF does not believe in the free-market, pro-business, and national-libertarian policies that the German AfG party supports. The AfG tends to be anti-immigrant, anti-Muslim, anti-EU federalism, anti-NATO, anti-Ukraine, and pro-Russian.
In general, far-right parties in countries in close proximity to Russia tend to oppose Russian influence, while far-right parties farther away from Russia are generally more supportive of Russian interests. Far-right parties in Finland, Latvia, and Romania tend to be hostile to Russia, while those in Germany, Croatia, Denmark, Hungary, Poland, and Sweden tend to be more open or neutral. Pro-Russian parties in Estonia are influential due to strong ethnic Russian influence, but these groups are not necessarily pro-Putin. Hungary's Prime Minister Viktor Orbán and Slovakia's Robert Fico have taken pro-Russian stances, and so has the Czech Republic's new prime minister, Andrej Babis, the billionaire "Czech Trump."
Other right-wing parties tend to be strongly pro-Russian in the rest of Europe—most crucially, the National Front in France and the UK Independence Party (UKIP) and the British National Party (BNP) in the United Kingdom; in Germany, the Alternative for Germany (AfG) and the Nationalist Democratic Party (NPD); and in Italy, the Northern League. Even the populist Italian Five Star Movement (M5S), which claims to go beyond left-wing and right-wing schisms, has shifted in a pro-Russian direction. A M5S foreign-policy spokesperson stated that M5S was neither pro-Russian nor pro-American, but it has opposed NATO "aggression" and called for the end of EU sanctions against Russia. It has also called for the strengthening of intelligence ties between the European Union and Russia.
Like left-wing movements, right-wing nationalist movements have been gaining political capital from high unemployment. But these nationalist movements have also been gaining support due to the failure of the European Union and the Schengen system of border controls inside Europe to deal with the post-2015 immigration crisis.
**FRANCE AS THE KEYSTONE OF THE EUROPEAN UNION**
The fate of the European Union could well depend on the nature of French politics in the coming five to ten years, after liberal-centrist pro-EU Emmanuelle Macron was elected president in May 2017. The United Kingdom's decision to leave the European Union has put the focus on French politics. It is now France that could eventually determine the future of Europe, as the outcome of the September 2017 German elections means that Merkel will probably have to concentrate on domestic issues, with the rise of the far-right AfG party and the alienation of the SPD. Macron and Merkel appeared to have established a good working relationship in support of a more effective European Union, but may profoundly disagree on the details. Berlin has thus far opposed French proposals to integrate European budgets, to create a European Monetary Fund and a minister of finance and the economy to surpass the economic crisis, coupled with a common European defense. Instead Germany, contrary to French counsel, has sought to impose austerity on Greece, for example, to the benefit of German banks, for example, with Greek airports and ports under German administration.
Both the far right and the far left in France have highly criticized, if not opposed, French membership in the European Union and NATO. Out of nineteen French presidential candidates, only Emmanuel Macron was supportive of NATO and the European Union; the rest were critical of both international regimes. But even if Emmanuel Macron won the French presidential elections by 66.1 percent, an estimated 43 percent voted for him only in an effort to block Le Pen—with 8.6 percent of the voters voting blank or nul, and with high abstention at one-quarter (25.4 percent) of the voting population.
After the May 2017 French presidential elections, all the political parties were shaken up by Macron's victory and are in the process of reconstituting themselves. Macron possesses a strong federalist vision of the European Union. He is calling for a banking union and an integrated EU budget that all EU states must follow, overseen by an EU finance minister. And yet, given the social and political divergence of the EU membership, only a limited degree of political and social integration has thus far taken place that would fit into this mold. It seems a more decentralized and interstate model of cooperation would be more appropriate, one that addresses the EU "democracy deficit" at both the local and national level. Whether Macron will try to push through a more centralized model or one that is more decentralized remains to be seen. He says he is open to discussion. And much depends on Merkel and Germany. (See chapters and .)
**RUSSIAN (AND AMERICAN) EFFORTS TO INFLUENCE EUROPEAN ELECTIONS**
Just like it had been claimed that Moscow was supporting Trump against Clinton, it is believed that Moscow had been supporting the French Republican candidate François Fillon, whose political career as prime minister had brought him in close contact with Vladimir Putin, and who has been critical of NATO and US policy toward Russia. Moscow was also said to support the anti-EU, anti-NATO National Front candidate Marine Le Pen over any other candidates. Le Pen met with Putin in March 2017. In addition to being accused of interfering in the American elections, Moscow has also been accused of both overtly and covertly (through cyber-attacks) interfering in the elections and the domestic politics in Holland, Montenegro, Germany, and France in 2016–2017, in addition to in Estonia and in Georgia in the past.
Marine Le Pen was able to borrow funds from a Czech-Russian bank after being unable to borrow from banks in the European Union, but she has had problems paying her debts after failing to win the presidential election. In addition to demanding a national referendum of France's membership in the European Union and NATO, the National Front has recognized Russia's annexation of Crimea and sent observers to the Crimean referendum, which was intended to legitimize the Russian annexation. But what was perhaps even more disturbing than Russian support for Le Pen was the fact that Marine Le Pen was seen at Trump Towers prior to the presidential inauguration in January 2017 in effort to gain political support and financing from some of Trump's associates. As Trump tacitly endorsed Le Pen just prior to the April–May 2017 French presidential elections, it was consequently feared that Trump associates (along with Putin) were still supporting a French exit (Frexit) from the European Union for nationalist-ideological reasons. In France, Le Pen was seen as an all-American "Trumpette." But Le Pen then claimed that it is Trump who has been following French National Front policies of "economic patriotism." Yet Trump's steps away from a positive relationship with Russia since February 2017 have greatly deceived Le Pen.
In France, Richard Ferrand, the secretary-general of Macron's En Marche! (Onwards!) stated that the Macron campaign had been hit by "hundreds, if not thousands" of attacks that were attempting to probe the campaign's computer systems from locations inside Russia. If true, this sounds much like what has been called a "cyber riot" involving angry individual hackers, as opposed to a direct Kremlin-sponsored "cyber-attack." Moscow was said to strongly oppose the Liberal-Centrist Emmanuel Macron, who is pro-EU, even though he had opposed further NATO enlargement, but so might be "patriotically minded" Russian hackers, as Putin himself has claimed, in reference to attacks on the United States. (See chapter 2.)
Then, just two days before the French presidential elections, Macron's offices were hacked once again, purportedly by APT 28 (also known as Fancy Bear)—a cyber-espionage group tied to Russian military intelligence, the GRU, which may have also been involved with hacking during the US elections. The involvement of Russian military intelligence, of course, was vehemently denied by the Kremlin. After the French elections, Putin met with Macron at Versailles in late May 2017—but without making any major changes in French-Russian policy—in part as the G-7 opted to sustain sanctions on Moscow. There was, however, despite Macron's not-very-warm encounter with Putin at Versailles, a general recognition by Macron that many problems, including that of Syria and the battle against the Islamic State, could not be resolved unless Moscow was involved.
**MOSCOW AND THE EU PARTNERSHIP WITH EASTERN EUROPEAN STATES**
In 2008–2009, Moscow had begun to oppose stronger EU efforts to expand its political-economic interests into former Soviet bloc states by means of the EU Eastern Partnership and neighborhood program. Before that time, even though the European Union began to implement a visa regime that blocked the entry of Russian citizens, Moscow generally did not consider the European Union a potential "threat." Yet the European Union concurrently developed a common security and defense policy and mutual defense clause based on Article 42 (7) of the Treaty of the European Union, introduced in 2009. This treaty, in effect, links the defense of NATO members with the defense of EU members—and could potentially mean that both NATO and EU members could be drawn into support of both non-NATO members and partners of the European Union.
Moscow soon began to interpret the 2008–2009 EU Eastern Partnership as being aimed at bringing its six Eastern European, yet former Soviet bloc, neighbors away from Russian spheres of influence and security. The EU partnership thus limits Russian political-economic influence over its six post-Soviet Eastern European neighbours—Armenia, Azerbaijan, Belarus, Georgia, Republic of Moldova, and Ukraine. In such a way, the European Union was seen as redirecting the political-economic orientation of Ukraine and other former Soviet republics toward Europe and away from Moscow—as the latter's political-economic and energy interests were not taken into account.
This leads to questions as to how, and if, the European Union will balance its relations with Russia and with the new EU "Eastern Partners." Although EU supporters do not like to admit it, it had been EU efforts since 2008 to 2014 to bring Ukraine—along with other post-Soviet states into EU Associate Agreements—that represented one of the major factors that provoked the Russian annexation of Crimea in early 2014. It is possible that the Russian response would have been very different if there had been greater political-economic coordination between the European Union, Ukraine, and Russia.
Kiev did, however, sign the EU Association Agreement after the Russian annexation of Crimea in 2014 despite the ongoing conflict in the Donbas region—only to see that association accord rejected by a referendum in a European state, Holland, in April 2016. Georgia and Moldova already possess association agreements with the European Union. Moldovans, Georgians, and Ukrainians obtained visa-free access to the European Union in the spring of 2017, while Armenia and Azerbaijan should complete negotiations on somewhat-similar partnership deals.
Even Belarus could soon participate in Eastern Partnership summits—as the European Union hopes to distance the country from Moscow, despite its authoritarian leadership under President Alyaksandr Lukashenka. In February 2016, the European Union had lifted sanctions against Lukashenka and other Belarus defense-sector officials for human rights abuses. This was true even though UN observers have seen no substantial improvement in the treatment of journalists and others who have criticized the government. The fact that Belarus has been attempting to facilitate the Minsk accords between Moscow, Kiev, Paris, and Berlin has represented a sign that Belarus wants to move closer to Europe, in part due to fear of Russian irredentist claims to Belarus. At the same time, Minsk needs to closely balance its relations with Moscow and Brussels, as Moscow opposes the defection of Belarus from the CSTO and the Eurasian Economic Union.
**A BREAKUP OF THE CSTO?**
One of Moscow's key concerns is the fear that Belarus could break away from the Russian-led Collective Security Treaty Organization (CSTO) and Eurasian Union and shift toward the European Union. Part of Russia's militarization has been intended to prevent the breakup of its system of alliances as the bitter conflict in Ukraine also indirectly impacts Russian ally and trading partner, Belarus, given Russian irredentist claims to Belarus and Ukrainian territory. Belarus could well be the next former Soviet state to enter into a political succession crisis similar to that which took place in Ukraine.
A political succession crisis in Belarus appears highly likely given strong social opposition to President Alyaksandr Lukashenka, his difficulties in finding ways to balance political-economic relations between the German-backed EU association promises and Polish influence, and the Russian difficulties in subsidizing the Belarusian economy. The drop in world energy prices, and the imposition of US and European sanctions since 2014, has led Moscow to fear the breakup of its CSTO—which represents a mini-version of the Warsaw Pact.
Moscow is also in competition with the European Union in Serbia, which is both a candidate for the EU Stabilization and Association Agreement (SAA) and a candidate for membership in the Russian Eurasian Union. Interestingly, this presents the same dilemma for Serbia as that which faced Ukraine in the choice between the EU Associate Partnership and the Russian-led Eurasian Union. In theory, Belgrade could join both. In practice, however, it might not work because once it joins the EU Associated Partnership, Serbia would have to implement the exact same tariffs on trade with the Russian-led EEU, which the European Union, as a single trade bloc, does. Perhaps much like EU efforts to bring Ukraine into a closer partnership in 2013–2014, steps to bring Serbia into a closer partnership with the EU also forewarns of potential political-economic instability in Serbia as the latter splits between those who want closer ties to Russia and those who want closer ties with Europe, which dominates Serbian imports and exports—that is, if the European Union and Russia cannot begin to forge some form of compromise deal. (See chapter 9.)
The general problem is that it is not certain the European Union can offer by means of these Association Accords anything that is truly better than what Russia or China or other countries can offer. In addition, Russia could react by force once again with respect to Belarus, for example, if it sees its interests threatened by an expanding European Union, like it did in Ukraine in 2014. Will the European Union renew economic sanctions against Russia, which are set to run out by the end of January 2018, thus risking the further alienation of Moscow? Will all EU countries continue to abide by the sanctions regime? Will the United States and the European Union be able to continue to align their policies toward Moscow? Or will US and EU policies diverge? Could the European Union then look to closer security and defense relations with China? Or will it seek an accommodation with Moscow without US input?
**RISKS OF EU COLLAPSE**
In sum, the Trump administration's encouragement of economic nationalism could facilitate the breakup of the European Union (after Brexit)—if not the disintegration of NATO as well—particularly if Italy enters into a financial crisis and/or if France veers to the left-wing or right-wing and opts to break out of the European Union in the next in five to ten years. A collapsed European Union would then be preyed upon by US, German, Russian, and Chinese political, economic, and military pressures. Both Russia and China have been attempting to draw a number of states closer to the Eurasian orbit—hoping to further splinter Europe and weaken its global influence.
European Council President Donald Tusk, the former premier of Poland, has warned that acts of Russian imperialism, an assertive China, anarchy in the Middle East and Africa, plus the threat posed by radical Islamist groups—combined with the economic nationalism of Donald Trump—all represent threats to European unity. For his part, despite very strong policy differences toward global warming, for example, and how to handle the Iranian and North Korean disputes, President Macron has thus far sought a close working relationship with Trump, while seeking to convince Trump that alternative multilateral strategies, as outlined in Macron's address to the UN General Assembly, are plausible if France, the European Union, and the United States can work together. For Macron, it is urgent to "rebuild multilateralism" with regard to the conflicts in Syria, Ukraine, Iran, and North Korea, and to sustain the COP 21 process to reduce global warming.
Both Macron and Tusk mentioned above realize that a breakup of the European Union and NATO could lead individual states, including Germany, potentially under the pressure of the far right, to "re-nationalize" their defenses while renewing old territorial and nationalist/ethnic rivalries within Europe. Already, after Brexit, the United Kingdom appears willing to rebuild its special relationship with the United States, if the United States is willing to do so as well. Yet the complex process of the United Kingdom's withdrawal from the European Union could potentially leave Brussels relatively impotent and unable to work toward greater unity. The United States, Germany (after breaking out of the European Union), Russia, and China would then begin to compete to obtain political, economic, and financial, if not military, influence over a divided Europe in turmoil.
If this latter scenario is to be avoided, the United States, NATO, and a more effective European Union (with close French-German cooperation) need to begin to find ways to work with, and not against, Russia. The United States/NATO, Europeans, and Russians also need to begin to defuse political-military tensions and reinitiate efforts to reduce the ongoing buildup of nuclear and conventional arms in Europe, while trying to find as many political-economic and ecological areas as possible where they can work in common both in Europe and abroad. (See chapters and .)
In August 2016, presidential candidate Donald Trump had warned that US efforts to regain Crimea on behalf of Ukraine against Russia in the aftermath of Moscow's annexation of the Ukrainian-controlled Crimea in early 2014 could result in World War III.
But in February 2017, the Trump administration appeared to have completely reversed position. The new US ambassador to the United Nations, Nikki Haley, condemned Moscow's military support for the autonomists in eastern Ukraine: "We do want to better our relations with Russia, however the dire situation in eastern Ukraine is one that demands clear and strong condemnation of Russian actions. The United States—calls for an immediate end to the Russian occupation of Crimea. Crimea is a part of Ukraine. Our Crimea-related sanctions will remain in place until Russia returns control over the peninsula to Ukraine."
Ambassador Haley's statement caused resentment in Moscow since the fighting had only begun to escalate once again after Kiev's forces engaged in a "creeping offensive" since mid-December 2016 into the buffer zone closer to the positions of the eastern Ukrainian autonomists in the Donbas region. (See further discussion on eastern Ukraine, this chapter.) These forward actions, in which Russia was then blamed for the outbreak of the fighting, took place at a time when Kiev feared that Trump would make a separate deal with Moscow over US sanctions on Russia, Crimea, and eastern Ukraine, but without Kiev's participation.
Haley's statement thus appeared to have reversed Trump's previous position, that, in order to improve US-Russian relations, and prevent a possible major power war, the United States would seek out a new "deal" with Moscow. In the aftermath of the US ambassador's address to the United Nations, Trump did, however, promise to "work with Ukraine, Russia, and all other parties involved to help them restore peace along the [Russian-Ukrainian] border." But it was not clear how this would be achieved. Nor was it clear what how the Trump administration would approach the issue of Moscow's annexation of Crimea.
At this point, Trump had been proposing the possibility that the United States would lift the fairly tough economic and political sanctions that had been placed on Russia in the aftermath of Russia's annexation of Crimea—if Moscow would, in turn, begin the process of reducing the size of its strategic nuclear forces. The issue was that some of the major sanctions against Russia were due to expire in December 2017—if Congress did not renew them. The problem with this approach, from the perspective of Trump's critics, was that it appeared to delink the sanctions issue away from Russia's illegal annexation of Crimea and its political-military interference in eastern Ukraine. Moreover, Trump's proposal to remove sanctions on Russia appeared to be tied to the "Russia-gate" controversy in which Trump and his associates have been accused of collusion with Moscow. (See chapter 3.)
**SANCTIONS AND ENERGY QUESTIONS**
Yet a deeper factor that underlies the debate as to whether or not to sustain or lift sanctions on Russia is another debate raised by Trump's America First nationalist ideology. Is it better to sustain US-based multinational corporate energy investments in Russia for the long term? Or would it be better to invest in the development of US shale energy industry? (The real debate, however, should be on how to fully develop alternative sources of energy that produce jobs and that are more ecologically sustainable, but this does not appear to be on the Trump-Pence administration agenda. See chapter 10.)
On the one hand, ExxonMobil, for example, has claimed that it has been taking considerable losses due to US sanctions placed on Moscow. Some elements in the Trump administration want to eliminate, or at least minimize, sanctions and safeguard ExxonMobil's considerable joint investment deals in the Arctic Kara Sea, western Siberia, Sakhalin island, and the Black Sea, which had been reached with Russian-government energy company, Rosneft, in 2012–2013, given the massive size of Russian reserves. In April 2017, ExxonMobil applied to the Treasury Department in April 2017 for a waiver from US sanctions on Russia in a bid to resume its joint venture with state oil company Rosneft. In July 2017, ExxonMobil sued the Treasury Department over a $2 million fine for purportedly violating US sanctions against Russia in 2014. In fact, a number of energy deals sought by ExxonMobil, Chevron, Royal Dutch Shell, Repsol, and Petrochina, with Kiev and with its state-owned Chornomornaftogaz, have been placed in jeopardy and in legal limbo due to the Russian annexation of Crimea. This makes it nearly impossible for Moscow to legally make deals with these same companies, among others, over formerly Ukrainian owned assets—until there is a political and legal settlement between Kiev and Moscow.
Specifically, both the American-based multinational Chevron and Royal Dutch Shell pulled out of their promised investments in Ukraine. Shell pulled out in part due to the annexation of Crimea and the outbreak of war in eastern Ukraine and in part due to the fact shale oil and gas reserves in both Poland and Ukraine were not as large as previously believed. ExxonMobil, among other oil companies, will need to renegotiate its Black Sea energy exploration deals with Moscow, after it annexed Crimea, if possible—as the area is still disputed with Kiev. But Chevron had actually pulled out in December 2014 due to, in large part, Ukraine's complex tax laws, and not the conflict in eastern Ukraine.
For its part, Moscow has nevertheless hoped that it can ride out US and EU sanctions. Moscow still believes that it can eventually develop Black Sea energy and other resources seized from Ukrainian jurisdiction that are potentially worth trillions of dollars. Moscow hopes to obtain finance and investment from Swiss, Qatari, and Chinese, if not Japanese and South Korean, banks and corporations that are not subject to US and EU sanctions. Moscow also hopes to engage in trade with countries in Latin America, in the Middle East (including Israel and Turkey), and in the Indo-Pacific (including China and Japan) that have not imposed sanctions. Moscow wants to believe that the United States and/or the European Union will eventually abandon sanctions altogether.
The sanctions have initially tended to cut off Russia's access to Western capital markets and know-how and scare off foreign investors, even though the Russian economy began to strengthen in 2016. Coupled with generally low global oil prices, the general economic crisis has cut real incomes, fueled inflation, and caused significant capital flight. Since 2012, consumer prices in Russia have risen by 50 percent. There had been a drop in the value of the ruble against the dollar, and average salaries fell by 36 percent from 2012 to 2016 in dollar terms. Inflation has been officially described at 5.4 percent but is probably much higher.
Even though it does open some opportunities for non-US and non-EU firms, the Russian annexation of Crimea comes at a major political-economic cost to Moscow. It also undermines US and EU trust in the Russian leadership—which it makes it even more difficult to build a positive relationship.
**IDEOLOGICAL DIVISIONS IN THE TRUMP ADMINISTRATION**
The Trump administration appears divided into two ideological camps. Against the globalists, there is a general move among other America First economic nationalists to reduce, or at least minimize, multinational energy investments in Russia in favor of the development of US shale oil and gas. Trump's nationalist America First ideology stands against multinational corporate investments, such as those of ExxonMobil in Russia, and wants ExxonMobil to invest back in the United States.
This group generally wants to further develop the US domestic energy market through strong support for the Keystone XL pipeline, through cutbacks on strict Environmental Protection Agency regulations, and by opening public lands for energy development purposes. Shale energy supporters see US and European sanctions on Russia as an opportunity to export shale gas to Europe, including Poland and Ukraine, and thus reduce eastern European dependency on Russian energy. In addition, shale energy supporters want to support the Baltic states to reduce their dependence on Russian energy, by helping to supply the technology to build regasification plants. Despite the ongoing Ukrainian-Russian conflict, Kiev is still dependent on Moscow for about 50 percent of its natural gas. Expanded exports of highly polluting US shale gas and oil to Europe are furthermore seen as a means to potentially undercut one of Moscow's major sources of revenue, which represents some 68 percent of Russia's total exports. American energy companies also hope that they can eventually break the Organization of the Petroleum Exporting Countries (OPEC), which seeks to control energy pricing. The US effort to export shale energy to Europe (despite the heavy debts involved in shale oil and gas production) could soon put the United States and Russia into direct competition for former Russian energy markets.
In effect, Trump's "energy dominance" plan was written into the August 2017 H.R. 3364 act, "Countering America's Adversaries through Sanctions Act." As mentioned in an earlier chapter, Trump appeared to be forced to sign this bill into law, given its strong bipartisan congressional backing, despite the fact that he publicly stated it to be "flawed." H.R. 3364, Trump argued, "encroaches on the executive branch's authority to negotiate" and "makes it harder for the United States to strike good deals for the American people and will drive China, Russia and North Korea much closer together" and could additionally "hinder our important work with European allies to resolve the conflict in Ukraine."
Trump claimed to have opposed the "Countering America's Adversaries through Sanctions Act," but he nevertheless more strongly backed America First energy interests against those of Moscow by supporting the 2016 Polish-Croatian Three Seas Initiative. The latter initiative is intended to foster trade, infrastructure, energy, and political cooperation among the ex-Communist countries bordering the Adriatic, the Baltic, and the Black Seas. Nearly all the countries involved—Poland, Hungary, the Czech Republic, Slovakia, Romania, Bulgaria, Lithuania, Estonia, Latvia, Croatia, Slovenia, and Austria—are heavily reliant on Russian gas and oil imports. The Three Seas Initiative seeks to minimize these states' dependence on Russian energy imports, and thus could clash with Russian interests.
It is predicted that by 2040, given significant and risky investment, US shale oil production will increase 45 percent from 2015 levels to 7.1 million barrels a day—and it most like grow rapidly with the assistance of Trump's pro–fossil fuels policies. Over the past decade, shale gas has risen from 2 percent to 37 percent of US natural gas production. After surpassing Russia, the United States is now the world's largest producer of natural gas and is beginning to develop its export capabilities. On January 24, 2017, President Trump signed presidential memoranda to revive both Keystone XL and Dakota Access pipelines in order to speed the process after President Barack Obama had rejected the fourth phase of the Keystone XL pipeline in November 6, 2015, in an effort to reduce greenhouse gases.
The Keystone XL pipeline has been opposed by both ecology groups and Native Americans. In November 2014, the Rosebud Sioux Tribe had called the US decision to build the pipeline "an act of war" and had vowed to block the project from crossing its lands. The debate has, in part, revolved around the social, health, and economic impacts of potential pipeline leakage and whether or not the pipeline will generate a significant number of long-term jobs, while also raising the question of whether these risky investments are creating new debt bubbles. Perhaps most important, the Trump administration support for the project raises real questions as to whether the government will be able to move energy producers away from carbon emissions that exacerbate the dangerous trends toward global warming, and toward nonpolluting and healthier energy options. (See chapter 10.)
**TRUMP TAKES AN ANTI-MOSCOW STANCE**
It was only once Trump became president that he began to stress the importance of NATO as a collective defense organization. Throughout his presidential campaign, and in the early days of his presidency, Trump repeatedly called NATO "obsolete." On the one hand, this raised Moscow's hopes that Trump would begin to reform NATO, a collective defense organization that dates from the Cold War. On the other hand, Trump's critical statement raised the fears of eastern European countries that the United States and NATO might no longer back them against ongoing and future Russian military pressures and threats in accord with NATO's Article V security guarantees.
Prior to becoming president, Trump had appeared to ignore NATO's role as a collective defense organization against potential threats from Russia or other states. And even though NATO has thus far played only a limited role in the fight against the Islamic State, Trump also appeared to ignore NATO's role in Afghanistan and Libya as part of the Global War on Terrorism. At that time, Trump appeared unaware of the debate inside NATO as to how many resources to concentrate on defenses to the north and east against Russia and how much to concentrate on defending Europe to the south in relationship to immigration, refugees, and terrorist movements.
At the same time, however, Trump did agree to accept the Balkan state of Montenegro into NATO membership. This action was taken in order to check Russian influence inside the country and for NATO to secure the coast of the Adriatic—angering Moscow, which purportedly tried to stage a coup, assuming the government did not pretend to stage a coup itself, as has been alleged in this age of disinformation. But it nevertheless remains unclear how NATO should or could deal with the high level of corruption in Montenegro, including counterfeit Euros in the country, plus the strong pro-Russian sentiment among the population.
**THE QUESTION OF RUSSIAN "AGGRESSION" AND THE THREAT OF WAR**
For understandable historical reasons, most eastern European states are afraid of Russia and believe that Moscow is inherently "aggressive." Yet the term "aggression" does not fully explain _why_ Russia has acted in the way it has—and at that particular moment. As Moscow had made what it believed to be reasonable deals with the previous Ukrainian Yanukovych leadership, a better explanation is that Russian military intervention in Crimea and political-military interference in eastern Ukraine was intended as a _preclusive action_ that has sought to check further enlargement by both NATO and the European Union into the Russian-defined "near abroad." Here, Moscow believes the EU expansion since 2008 has sought to undermine Russian political-economic influence in the former Soviet states of Belarus, Ukraine, Moldova, Georgia, Armenia, and Azerbaijan.
Since at least March 2014, if not earlier, Moscow has been engaging in provocative overflights and submarine incursions into the territories of NATO and EU members, including EU member states Sweden and Finland. As a consequence, the latter countries have been considering closer ties to NATO. Likewise, in mid-March 2014, Moscow engaged the Russian Baltic Fleet in exercises along the Baltic coast, while also placing infantry, air force, and Spetsnaz troops on alert throughout the Russian Federation and along the borders of NATO member states from the Arctic region to the Baltic and Black Seas. A Russian military buildup has accordingly been taking place in Kaliningrad, where Moscow's warships are not as vulnerable as they are when they sail from Saint Petersburg and pass between Estonia and Finland. In addition to threatening to deploy nuclear weapons in Kaliningrad, Moscow has also threatened a nuclear and conventional weapons buildup in the newly annexed Crimea.
These are just a few of many incidents that have taken place since the Russian annexation of Crimea in 2014 that could spark conflict. Somewhat-similar incidents involving both Russian and Chinese overflights have taken place in the Asia-Pacific region as well, while both Russia and China point to the buildup of NATO and US (and Japanese) military capabilities in Europe and Asia. From the Russian perspective, such repeated overflights into European, Japanese, and US airspace have been justified by the fact that the number of nuclear-capable fighters in the NATO Baltic air-policing mission had been significantly increased in early March 2014. In Moscow's view, advanced NATO fighter jets with potential nuclear weapons capabilities now patrol regularly along the sensitive Baltic state border only a few minutes by supersonic flight to Saint Petersburg and Moscow.
Moscow has also opposed NATO membership for EU members Sweden and Finland, who are no longer technically neutral, having joined the European Union. Should Finland join NATO, Moscow would fear the rebirth of Finnish irredentist claims, backed by NATO, to territories taken by the Soviet Union after the Winter War and the Continuation War. Finland has claimed that it has no irredentist claims, yet Moscow nevertheless fears that if Finland joins NATO, then NATO might back Finnish claims to Karelia, Salla, and Petsamo. Claims to Petsamo, if pursued, could give Finland access to the Barents Sea—which will become a major sea line of communication in the coming years as the polar ice caps, unfortunately, melt. (This issue might be better addressed by bilateral Finnish-Russian negotiations than by drawing NATO into the picture.)
Russia already feels "encircled" by NATO in the Arctic region. Out of the eight Arctic Circle states, in addition to Russia itself, five are NATO members: Norway, the United States, Canada, Denmark (Greenland), and Iceland (where it passes through the small offshore island of Grímsey). As previously mentioned, two are EU members considering NATO membership, Sweden and Finland.
There are a number of major legal and defense issues here. The close overlapping links between NATO and EU members raises a delicate political-legal-security question. If an EU member is attacked, the United States and NATO could automatically become involved. This is because NATO and the European Union possess close overlapping memberships, which themselves possess mutual defense clauses. The EU defense clause, however, is much tighter than the NATO Article V clause.
**BUDAPEST ACCORDS AND NATO-RUSSIA FOUNDING ACT**
Russian actions raised the question as to whether the United States, France, and the United Kingdom were necessarily obliged to provide military assistance to Ukraine in accord with the Budapest Memorandum of 1994, but which was not a formal treaty. The United States and the European Union did agree to apply economic sanctions on Russia in 2014, but they are not legally mandatory for all states. Nevertheless, Moscow's ostensible violation of the Budapest Memorandum, among the other international agreements, including the 1997 NATO-Russia Founding Act, which had established the NATO-Russian relationship, led the United States and the European Union to argue that Russian actions in Crimea and eastern Ukraine could not be permitted without some form of punishment. (See the introduction.)
The 1994 Budapest Memorandum had provided security assurances (not security guarantees) to Ukraine that obliged its signers, the United States, Russia, and the United Kingdom (plus France and China later and separately), "to respect the independence and sovereignty and the existing borders of Ukraine" once Kiev gave up its nuclear weaponry left over from the ex–Soviet Union. But "respect for borders" was not intended as an absolute guarantee of military assistance to Ukraine.
The NATO-Russia Founding Act of 1997 stated, "Any actions undertaken by NATO or Russia, together or separately, must be consistent with the United Nations Charter and the OSCE's governing principles." From the Russian perspective, this principle was already broken by NATO itself after it declared war on Serbia over Kosovo in 1999 without a UN Security Council mandate.
The 1997 NATO-Russia Founding Act had also promised that NATO had no plans to deploy troops or nuclear weapons on the territories of the new members—at least at that time. In terms of conventional forces, the Founding Act stated that "the Alliance will carry out its collective defense and other missions by ensuring the necessary interoperability, integration, and capability for reinforcement rather than by additional permanent stationing of substantial combat forces." Both sides were to "prevent any potentially threatening build-up of conventional forces in agreed regions of Europe, to include Central and Eastern Europe."
The Russian military intervention into Crimea and eastern Ukraine has accordingly threatened to tear apart the 1997 NATO-Russia Founding Act. For this reason, the option to support Kiev through direct military assistance to counter eastern Ukrainian secessionists secretly backed by Moscow is a decision of individual states—not of NATO or the European Union as a whole. Under the Obama administration, Washington decided to provide only limited US training and support for Kiev's military units—instead of providing significant arms shipments.
Since 2014, Russian military pressures have led NATO to consider a rotating deployment of troops on the territory of the new NATO members, so as to not technically violate the NATO-Russian Founding Act, with the questionable rationalization that the "rotation" of forces did not represent a "permanent" deployment of forces. By January 2016, the US Congress mandated the European Reassurance Initiative, which promised $985 million for the Pentagon to augment a "rotational" US troop presence in Europe for NATO activities and to preposition US military infrastructure and assets in Europe. President Obama then requested $3.4 billion for 2017 for the deployment of an additional "rotational" Armored Brigade Combat Team (ABCT) to central and eastern Europe, plus the prepositioning of combat equipment, and additional training and exercises in Europe.
US aid has also been allocated to help build the defense capacity of new NATO members in addition to assisting potential NATO members, which include Georgia, Macedonia, Moldova, and Ukraine. Yet proposed NATO membership for the latter states continues to fuel tensions with Moscow. The latter interprets US calls for greater NATO defense spending as an anti-Russian gesture that is intended to force Moscow to spend much more on defense against the combined forces of NATO in return.
Nevertheless, members of the US Congress, the Pentagon, and European hardliners have continued to argue for a permanent NATO deployment, while Russia already considers the decision to be a "permanently rotating" deployment—and thus a violation of the NATO-Russia Founding Act. Hardliners have also argued for NATO to display a nuclear capability in military maneuvers in order to symbolically counter Russian threats to deploy tactical nuclear weapons in Kaliningrad, for example. But these measures would most likely be met by Russian counter-threats. And the risk is that Ukraine's 1,300-mile-long and porous border with Russia can only be defended by NATO's use of nuclear weapons if relations between Ukraine and Russia remain acrimonious. A NATO defense of Georgia, Moldova, Ukraine, and the Baltic states could easily escalate into a nuclear conflict given the Russian tactical advantage in each area.
In effect, President Obama had argued that Ukraine was part of Russia's vital interests, but not those of the United States—and left the situation dangerously ambiguous. Obama did not seek to formally renounce NATO enlargement. Nor did he seek to formally establish Ukraine as a neutral country. Despite the fact that Ukraine is not a NATO member, NATO has become involved in defending Ukraine with what it considers defensive assistance. This is in part because NATO and Ukraine formed the NATO-Ukraine Commission just after NATO first formed the NATO-Russia Council in 1997. In 1997 the idea was to approach both sides and suggest cooperative measures but give Russia priority. Now NATO appears to be granting Ukraine priority in its conflict with Russia. Members of the US Congress, the Pentagon, and some members of the Trump administration, have been considering greater military supports for Kiev, including lethal aid, which Russia could easily counter. NATO Secretary General Jens Stoltenberg and other European leaders have proposed a policy of "defense and dialogue" vis-a-vis Russia. But the question remains: How much defense? And how much _real_ dialogue?
**WAR OVER CRIMEA AND EASTERN UKRAINE**
From the Russian perspective, the events that resulted in the Russian annexation of Crimea and Russian political-military interference in eastern Ukraine in 2014 stem from US and European attempts to expand their spheres of influence into the Russian "near abroad." In effect, Moscow feared that its naval base at Sevastopol would fall into the hands of NATO, while its political-economic interests in eastern Ukraine would be undermined by the more competitive European economy.
Moscow not only opposed US and European efforts to engage in "democracy engineering" against former Ukrainian Prime Minister Viktor Yanukovych in 2013–2014, but also opposed any political-economic deal between the European Union and Ukraine that did not also incorporate Russian gas, Ukrainian debts to Moscow, and other political-economic interests. Moscow then pressured Yanukovych to refuse to sign the EU Association Accord—which helped spark the Euromaidan protests in Kiev in 2013–2014. Moscow then took advantage of the chaos once the kleptocratic, but not always pro-Russian, Yanukovych leadership collapsed.
The general chaos then taking place in Ukraine permitted Moscow to engage in preclusive actions intended to rapidly annex Crimea by means of deploying "little green men" without insignias in strategic locations throughout the isthmus. Moscow also began to engage in clandestine political-military intervention in eastern Ukraine in support of Ukrainian "autonomists." This also meant that Moscow seized waters surrounding Crimea in which Ukraine had just offered US and European multinational energy companies, such as ExxonMobil, to explore.
By March 2014, Moscow had formally annexed Crimea after staging a public referendum that ostensibly legitimized Russian actions. This put an end to Kiev's controls over the peninsula and thus safeguarded the Russian Black Sea fleet from possible eviction by the new government in Kiev. Moscow, of course, denied any wrongdoing in that it saw Yanukovych as being overthrown by an "illegal" coup (even if Ukrainian lawmakers backed that "coup" by opposing Yanukovich's corruption and kleptocracy).
In effect, Moscow claimed that it was fighting Ukrainian "fascists" by supporting the right of self-determination for the ethnic Russian majority of the Crimean populations, according to its own national security interests defined during the Yeltsin administration. At the same time, whether Moscow has been able to improve the living conditions, quality of life, and sociopolitical freedoms of both Russian and non-Russian minorities, such as Tatars and Ukrainians, living in Crimea after the annexation is another question. The potential failure to do so could undermine Moscow's rationale for the annexation.
As fighting intensified between pro–Ukrainian government supporters and Russian-backed autonomists in the Donbas region, the Minsk I accords were signed in September 2014 talks between representatives from Russia and the opposing two Ukrainian factions under the auspices of the Organization for Security and Co-operation in Europe (OSCE). The Minsk I agreement, which had followed previous attempts to stop fighting, tried, but failed, to implement a cease-fire. These accords were then followed by the February 2015 Minsk II accords between Ukraine, Russia, France, and Germany, once again under the auspices of the OSCE.
The February Minsk II accords urged greater "decentralization" by means of a reform of Ukraine's Constitution. By mid-July 2015, the new Ukrainian Prime Minister Petro Poroshenko introduced a bill to the parliament that would ostensibly devolve powers to localities. Poroshenko insisted that these constitutional changes would not turn Ukraine into a "federation" or "special status" as demanded by Moscow. Kiev has opposed greater "autonomy" or "federation"—a position opposed by many in the Ukrainian parliament and violently opposed by right-wing centralists—in the fear that greater autonomy for the Donbas could eventually lead to political secession and independence. On March 16, 2017, three of Ukraine's major far-right groups—Svoboda, Right Sector, and National Corps—signed a manifesto that called for "establishing and developing a great national state."
Nevertheless, Poroshenko has claimed that he would grant local authorities more power throughout the country. But this is to be done by the strengthening of presidential control over local self-governments by means of "centrally assigned 'prefects' with broad powers." Kiev's efforts to find an in-between position that will somehow satisfy both centralists and "autonomists" who demand a special status (while actually asserting presidential powers over localities in the process), appears to have failed miserably with the resumption of fighting in mid-2016. At the same time, despite pressures from the World Bank and NGOs pressing for greater governmental transparency, corruption runs high. Ukraine is tied with Russia as two of the most corrupt states in the world, with a rating of 131 out of 176 countries.
In May 2015, Moscow's own propaganda in favor of the Novorossiya movement for a potential union with the Donbas region and other southern Ukrainian regions suddenly ceased. Not only was such an option opposed by France and Germany in the Minsk II accords, but the costs of such a venture, coupled with strong Ukrainian resistance, the probable need for a long-term Russian occupation force, the costs of long-term Russian political-economic isolation from the United States and Europe, and the general collapse of global energy prices, appeared to put a damper on such imperialist expansion. In an effort to show that it does not possess an imperial design, Moscow had permitted a series of Ukrainian overflights under the Open Skies Treaty in March 11, 2014, and it likewise granted Ukraine's request to conduct an inspection of a "non-declared military activity" in a border region. (NATO, however, was not impressed by what it called Moscow's "selective" implementation of the Open Skies Treaty.)
Moreover, the very fact that Moscow has been unwilling to admit to its own population the role of Russian special forces in Ukraine appears to indicate that Moscow does not want to take over the burden and responsibility for the entire region, as has been the case for Crimea. Moscow does not want an unpopular war in which it must enlist the general population. Whether or not the killing was orchestrated by Putin, as alleged, one of the purported rationales for the assassination of Boris Nemtsov, an opposition politician, in February 2015 was that he was attempting to make public proof that Russian forces were involved in the intervention in eastern Ukraine. (See discussion, chapter 3.)
Ukrainian autonomists have still not given up their struggle, despite the fact that they are not obtaining full backing from Moscow. In August 2016, Russian-backed forces engaged in a major military buildup around Ukraine (to north in Bryansk, to the east near Rostov, to the south in Crimea, and to the west in the Transnistria area of Moldova) after it claimed that Kiev had engaged in a military incursion and terrorist sabotage in Crimea. This incident followed a number of sabotage attempts—which may or may not be backed by Ukrainian authorities. These include efforts to disrupt the supply of electricity to Crimea and to blockade transportation routes and water supplies.
Kiev does not appear willing to accept the loss of Crimea and has sought US and NATO support to regain it. The Trump administration has now supported Kiev's position since February 2017—particularly after conflict flared up again after Kiev engaged in a "creeping offensive" in mid-December 2016, which has nonetheless stepped deeper into the grey zone between the two sides. This offensive was ostensibly intended to check supplies going to the Russian-backed autonomists, while trying to preempt militias from the Donetsk and Luhansk "people's republics" from seizing more territory. The autonomists have, in turn, begun to expropriate Ukrainian businesses in the Donetsk and Lugansk regions. For its part, Kiev has claimed that Moscow was building up its forces in Crimea to turn it into an "isolated military base" and was attempting to justify "aggressive actions of [Russian] military units...on the territory of the currently occupied peninsula."
Kiev's strategy has been intended to further divide and then defeat the "autonomist" Russophone forces that have generally split between those seeking independence (the self-proclaimed yet unrecognized "republics" of Donetsk and Lugansk) and those seeking greater autonomy from Kiev's centralized controls, but who are not necessarily pro-Putin. The dilemma is that Ukrainian President Petro Poroshenko appears incapable of implementing the February 2015 Minsk decentralization proposals, which would involve changes to the constitution and would permit local elections.
Concurrently, Moscow appears reluctant to make good on its security commitments because of its commitment to eastern Ukrainian autonomists. The latter see the open border with Russia as key to their survival. At the same time, Kiev has, particularly in 2015–2016, insisted on being able to control the Ukrainian-Russian border first before implementing the Minsk accords. This is a major factor that has led to a breakdown in discussions—and which, in addition, led to Kiev's December–February 2016 "creeping offensive" into the grey zone between the two sides that is in or near the war-ravaged cities of Avdiivka, Debaltseve, Dokuchaievsk, Horlivka, and Mariupol, closer to the positions of the eastern Ukrainian autonomists. Control of the Mariupol region for the autonomists appears key in geo-economic terms—as it opens up the possibility of trade and transportation links with Russian-held Crimea.
The United States and European states have appeared to have granted Kiev sufficient financial and military assistance in order to counter autonomist movements that are not-so-secretly backed by Russia. Yet the fighting in the Donbas has moreover proved very costly for all sides, and rebuilding the region will prove very difficult. A collapsed Donbas region that is potentially separated from a partitioned Ukraine could soon become a much larger and unstable version of Russian-backed Transnistria, South Ossetia, and Abkhazia combined. Such political-economic instability will continue to pollute the whole area with black marketeering, weapons smuggling, and other forms of criminality.
Moreover, a failed "state" in eastern Ukraine would prove very troublesome not only for an essentially bankrupt Kiev and the rest of the region, but for Moscow as well—as the latter, for example, will need to deal with refugees fleeing to Russia. Some 1,554,497 people have already fled the country, with the vast majority (1,226,104) moving to the Russian Federation—which has not necessarily accepted them with open arms. Roughly 148,867 have gone to Belarus. The costs of reconstruction and development in the aftermath of the conflict will be considerable.
Putin's annexation of Crimea has been proving unexpectedly costly for Moscow to achieve rapidly in the short term—and even more so with US and European sanctions in place. Moscow has needed to augment salaries and pensions of the Crimean population to Russian standards, while tourism will remain much lower than normal until the situation stabilizes. The Kerch bridge that is needed to supply Crimea from Russia will probably cost much more than the officially estimated $4.5 billion and may not prove long-lasting due to the harsh nature of the surrounding climate. In addition to the need for Moscow to supply Crimea with gas and electricity, Kiev's blockade of the North Crimean Canal has negatively impacted Crimean agriculture, as well as the overall Crimean economy, ecology, and population.
By blocking the North Crimean Canal, for example, Kiev has prevented as much as 85 percent of Crimea's water supply from entering Crimea. In this new form of hybrid and environmental warfare, these actions have already provoked a crisis in agricultural production and could force migration back to Russia—if Moscow cannot soon find ways to provide water for the isthmus.
**THE FAILURE OF US POLICY**
The NATO-Russia Founding Act—which was intended to bring NATO and Russia into closer post–Cold War cooperation—is now being challenged. The breakdown of the NATO Founding Act could potentially result in the permanent deployment of troops and nuclear weaponry in eastern Europe. This could not only result in a new partition of Europe but also provoke an even more dangerous Russian backlash—if the crisis cannot soon be abated. It could also lead to a major arms race in which Moscow will seek to counter US military superiority by asymmetrical and "hybrid" methods. Not only do "rotating" deployments risk undermining that fundamental NATO-Russia Founding Act, but so does the deployment of the F-35 stealth fighter, which is capable of carrying the renovated B-61-12 tactical nuclear bomb, in NATO military exercises along the Estonian-NATO border, for example. Although this is not the official explanation, these exercises are designed to demonstrate capabilities that could counter a potential Russian advance into the region.
The essential dilemma is this: As long as NATO, Ukraine, and other countries "will not" recognize Russian sovereignty over Crimea, and as long as Moscow claims it "will not" give up that sovereignty, there will be no lasting peace. Other diplomatic options to a military buildup and arms race must be forthcoming, and yet not only is the State Department divided, but the White House and Congress appear totally at odds on this issue—as the passage of the "Countering America's Adversaries through Sanctions Act" in August 2017 has indicated. There is a real risk that an eventual partition of Ukraine, coupled with the permanently rotating deployment of NATO forces in the Baltic region, could in turn lead to a new partition of Europe though Ukraine—followed by the polarization of the world into two rival alliances. (See chapters and .)
US and NATO policies have thus far been pushing Russia and China closer together. While Trump was initially right to seek a rapprochement with Russia, despite the controversial way he has gone about it, the dilemma is that the Trump administration's policy flip-flops could push Beijing and Moscow even closer together. This is true given the Trump-Pence administration's newfound support for Kiev's claims to Crimea combined with Trump's initial threat to support Taiwanese independence (even though he backed off to support the One-China policy). Washington has also been seeking to check China's access to islands in the South and East China Seas, while concurrently threatening trade sanctions against both North Korea and possibly China itself—if Beijing cannot convince North Korea to give up, or at least freeze, its nuclear weapons program.
On a geostrategic level, Washington has been raising China's suspicions of a US-inspired "encirclement" by calling on India to join the United States, Japan, and Australia to deal with common security challenges in the Indo-Pacific region, through the Quadrilateral Security Dialogue (or Quad). For its part, Japan, China's historical rival, has called for the formation of a "democratic security diamond" that would include Japan, the United States, Australia, and India to counterbalance China. Washington's threat to build up naval forces and alliances in the Indo-Pacific region, plus the deployment of US missile defense systems in Europe, the Middle East, and Asia since the Obama administration, has accordingly begun to press Russia and China into an even closer defense relationship against US conventional and nuclear weapons superiority. China has furthermore opposed US-South Korean, THAAD missile defense deployments. A close Sino-Russian defense relationship could then lead to tighter Sino-Russian defense relations with Iran—and possibly with India.
For its part, Beijing has hoped to overcome its century of humiliation since its Opium wars with Great Britain and its subsequent political-economic exploitation by the Europeans, by the United States, and particularly by Japan. Now China wants to establish itself as a major power in the twenty-first century. Beijing first seeks to make itself a major political-economic and financial actor by expanding its global political-economic hegemony through its Belt and Road Initiative (BRI) and by working to develop a massive trading bloc, the Regional Comprehensive Economic Partnership (RCEP). China's huge financial, economic, and technological capabilities will then permit it to develop significant military capabilities.
China's Belt and Road Initiative (BRI), which was initially said called "One Belt, One Road," comprises the Silk Road Economic Belt and the Twenty-First-Century Maritime Silk Road. Beijing's objective has been to develop a trade and infrastructure network that connects Asia with Europe and Northern Africa along the ancient Silk Road routes. These routes cover more than 60 countries and regions from the Far East to Europe and the Middle East (with Egypt and the Suez a key geo-economic focal point). The BRI countries currently account for some 30 percent of global GDP and more than 35 percent of the world's merchandise trade.
By 2050, Beijing hopes that the BRI will contribute up to 80 percent of global GDP growth and bring as many as three billion more people into the middle class. In many ways, the legitimacy of the Chinese Communist Party ironically depends to a large extent on its ability to expand the share of its economic pie to as many Chinese citizens as possible. By 2020, Beijing aims to double the yearly average personal disposable income of roughly $3,000 in 2010 (20,000 yuan) to roughly $6,000 (40,000 Yuan) for at least one billion people. And in the process, China is straining regional, financial, and global resources (as well as the health of its population through excessive pollution)—in rivalry with US, Japanese, and European incomes and mass consumerism, while concurrently risking conflicts with its neighbors and with the United States itself.
Concurrently, Beijing has been finalizing the RCEP negotiations with the ASEAN states (Indonesia, Malaysia, Philippines, Singapore, Thailand, Brunei Darussalam, Cambodia, Laos, Myanmar [Burma], and Vietnam), plus India, Japan, South Korea, Australia, and New Zealand. If the RCEP is finalized by the end of 2017 as expected, and assuming India, Japan, South Korea, and Indonesia do not stall progress, the RCEP would create the largest trading bloc in the world, covering nearly one third of the global economy. But it would exclude the United States and Europeans.
**TOWARD CLOSER RUSSIAN AND CHINESE POLITICAL-ECONOMIC AND MILITARY INTEGRATION**
To achieve their long-term goals, China and Russia appear to be moving into a closer political-economic and strategic Eurasian alliance. At the roots of this Sino-Russian rapprochement was the final border delimitation of 2004. This had been the fruit of negotiations since 1986, when Gorbachev initiated the Soviet rapprochement with China. Since at least since 2005, Russia and China have been engaging in major military maneuvers in the framework of the Shanghai Cooperation Organization (SCO). By 2012, they announced their Strategic Partnership, followed by the comprehensive 2014 Strategic Partnership. The latter has been symbolized by closer Sino-Russian defense collaboration involving joint naval maneuvers in the eastern Mediterranean in May 2015—in which China appeared to backing Russian military actions in Syria. Russia and China likewise engaged in joint naval maneuvers in September 2016 in the contested waters of the South China Sea.
It was in Moscow on May 8, 2015, that Moscow and Beijing stated their combined intent to integrate the Chinese-led Silk Road Economic Belt and the Twenty-First-Century Maritime Silk Road with Russia's Eurasian Economic Union (EEU). This accord took place at the ceremony that commemorated the end of World War II—but which was boycotted by most US and European leaders, due to Moscow's annexation of Crimea in 2014. The boycotting of this event, at least in the Russian perspective, appeared to denigrate Moscow's crucial role in defeating Hitler.
For its part, Moscow has additionally tried to widen its markets and build a stronger economic infrastructure by forging its new Eurasian Economic Union. The EEU is to include Armenia, Belarus, Kazakhstan, Kyrgyzstan, and Russia. Moscow has also been attempting to press Moldova, Kyrgyzstan, Tajikistan, and Serbia into joining. Moscow, however, has failed to incorporate Ukraine, which joined the EU Association Agreement in late 2014 after the Euromaidan movement overturned the kleptocratic government of President Viktor Yanukovych, but after Russia had annexed Crimea and supported the autonomist movements in eastern Ukraine.
Moscow has also hoped to enlarge the Collective Security Treaty Organization (CSTO), which represents a Russian effort to mimic NATO. The CSTO thus far includes Russia, Belarus, Kazakhstan, Kyrgyzstan, and Tajikistan, but Moscow has invited China, India, Iran, Mongolia, and Pakistan to be CSTO observers. Afghanistan and Serbia are already observers. Pakistani observer status was, however, surprisingly rejected by Armenia due to Pakistan's support for Azerbaijan and its refusal to recognize Armenia. (Even before 2014, Ukraine had remained out of both the CSTO and the Eurasian Union—giving preference to possibility of joining NATO and the EU Association Agreement.)
**REASONS FOR A CLOSER SINO-RUSSIAN ALLIANCE**
The first major rationale for closer Russian ties to China is a consequence of Russian fears of isolation in Europe in response to the double expansion of NATO and the European Union into former Soviet spheres of influence and security. In the Indo-Pacific, the US-Japanese alliance has been tightening relations with South Korea and Australia, while seeking to strengthen relations with India, if possible. In effect both Moscow and Beijing fear that Washington will reactivate its policy of containment or "encirclement."
Moreover, both Moscow and China tend to see US and European ideological appeals to "democracy" and "human rights" as a threat both to one-party Communist leadership in the case of China and to the dynamic authoritarian duo of Vladimir Putin and Dmitri Medvedev in the case of Russia—a situation in which Putin's political party, United Russia, predominates over all other Russian political parties, with more than 3/4 of the 450 seats in the Russian Duma. From Beijing's perspective, the student-led Chinese democracy movement in Tiananmen Square and throughout the country in April–June 1989, and the more recent Hong Kong democracy movements (the Umbrella Revolution, since 2014), have been seen as backed by US and European (and Taiwanese) democratic ideology, media, and secret financial supports.
Rather than engage in necessary reforms designed to check corruption and open the decision-making processes to greater civil-society inputs, both Moscow and Beijing see these democracy movements as an essentially US-directed tool of hybrid warfare primarily aimed at undermining the Russian and Chinese power structure and ultimately aimed at breaking up both countries. (On Russian views, see chapter 3.)
**THE QUESTION OF RUSSIAN ARMS SALES TO CHINA**
Moscow has, for the first time in years, begun to sell China advanced weaponry. In 2016, Moscow promised to sell four advanced Russian Su-35 fighter jets to Beijing—with a total of twenty-four jets to be completed in three years. And China will probably obtain more than just the initial twenty-four Su-35s. This significant arms sale has generated tensions with India, Taiwan, and Japan. In addition, Moscow has also been considering the sale of _Amur_ -class submarines to China, plus advanced aircraft engines and radars. Moreover, Russia is expected to deliver its S-400 surface-to-air missile to China by 2018. Overall, Beijing and Moscow had signed roughly $8 billion in defense contracts as of November 2016.
In effect, Russian technology could boost China's air defense capability significantly—even though Moscow still fears Beijing could clone Russian military technology. In 2014, Beijing was seen as copying the Su-27/30 fighter—which then became the J-11. The Chinese have also been accused of cloning US military technology. The Chinese J-31, for example, looks like a US F-35—as the Chinese were believed to have hacked into the files of F-35 sub-contractors. Yet Russian fears of Chinese high-tech copying appear less worrisome due to the fact that Moscow expects to deploy the even more advanced S-500 surface-to-air missile and T-50 fighter in the near future. Nevertheless, these major arms sales to China are significant in that they appear to put to an end Moscow's own informal ban on selling advanced weapons systems to Beijing, which had in place since roughly 2004.
Moscow elites do fear that Russia could, in the not so-distant future, become a junior partner of China in military terms. Already, China possesses world's largest land army, and it is rapidly developing its blue-water navy, plus ICBM missile capabilities. For this reason, many Russian elites have called for restraining the sale of advanced military technologies to China. These elites worry that Russian arms sales to China could backfire, if Beijing eventually turns against Russia. By contrast with this view, other Russian elites have hoped to co-opt China by means of working closely with the leadership. They consider it futile to confront Beijing.
The dilemma is that the long-term ability of Russia (and of the United States) to stay ahead of China in the game of military innovation is not guaranteed. There are still some items Russia will not sell China—such as technology that allows the Iskander cruise missile to maneuver at extremely high speed, making it difficult to intercept. Moscow will also not supply Beijing with satellite systems that could detect ballistic missile launches. At the same time, Russian arms sales to China are helpful in permitting Moscow to obtain a better conception of real Chinese military capabilities. The nature of Russia and Chinese defense relations indicates that the two countries could deepen military-to-military trade through the joint development of new space technology, airplanes, and helicopters. Advanced Russian arms sales appear designed to support China militarily as Beijing has begun to confront the US-Japanese alliance—while both Moscow and Beijing have hoped to splinter ASEAN where possible. Cambodia, Laos, and Myanmar, but more importantly Malaysia, even Vietnam, and now the Philippines, appear to be moving closer to both Russia and China.
**ENERGY COOPERATION**
Russian arms deals with China were bargained just when Russia and China were finalizing their "historic" gas deal worth $400 billion in May 2014, just after the Russian annexation of Crimea. Even though there have been ongoing price disputes over the gas deal, China and Russia also worked out a series of energy agreements involving the doubling of oil supplies and the construction of a natural gas pipeline to China from Russia. Additionally, the two agreed to develop Russian coal resources for China's benefit.
In this case, Russian-Chinese rivalry could be seen in the fact that China had previously sought to block Russia's attempts to establish a Russia-based energy infrastructure in Central Asia, by establishing contracts with a number of Central Asian states to build gas and oil pipelines directly to China. The fact that this deal came just after the United States and the European Union imposed sanctions on Moscow after its annexation of Crimea, however, indicated that Moscow could find financing for its major projects from other sources. From the Russian perspective, defense and security issues are now overriding political-economic concerns and previous disputes with China. Likewise, major Chinese banking deals with Russia to develop the Yamal liquefied natural (LNG) gas venture in the Arctic permit Russia to reduce its reliance on gas-export sales to Europe, while supplying China and opening up LNG shipping exports to Asia. And these deals once again show that Moscow can still find financing for a major project despite US and EU sanctions, but at the risk of too great a dependence on Beijing.
In late September 2017, CEFC China Energy purchased a 14.16 percent stake ($9.1 billion) in the Russian oil producer Rosneft, a firm closely linked to Putin, which will provide China access to eastern Russian oilfields near the Chinese border. It is the first time China has acquired a significant stake in a major Kremlin-controlled corporation.
Moreover, these deals will boost not only China's economy but also its energy security. This is because the energy supply chains, particularly over land but also on the sea, tend to avoid the maritime choke points dominated by the United States and its allies. This latter strategic economic fact provides a major rationale for China to sustain its close Eurasian relationship with Russia in the future.
**COMMON VALUES**
What appears very significant is that Russia and China have begun to enhance cooperation not only by means of cultural exchanges but also by cooperation in the actual making of domestic policies. In early 2016, Russian President Vladimir Putin and Chinese leader Xi Jinping urged the legislative bodies of their respective countries to enhance their exchanges and mutual experience so as to further strengthen China-Russia ties. This resulted in the agreement of the Russia's Federation Council and Russia's State Duma, and China's National People's Congress to strengthen their cooperation on legislative initiatives and supervision, so as "to enhance coordination on regional, municipal and industrial development policies and plans."
While it is generally argued that Russia and China are not compatible in ideological terms, as China is still run by the Chinese Communist Party, the above factors have tended to bring China and Russia even closer together rather than to tear them apart. While potential disputes and a "quiet rivalry" does exist between the two Eurasian powers, the two countries appear to be moving beyond a marriage of convenience and toward a proto-alliance in opposition to the United States and its allies—as indicated by their June 2016 joint statement on strengthening global strategic stability.
**RUSSIA AS A JUNIOR ECONOMIC PARTNER TO CHINA**
By reaching out to China, Russia is not so gradually becoming to China a junior partner in political-economic and financial terms. And Moscow could become even more financially dependent on China if Russia remains isolated from US and European sources of capital for too long. It has been argued that a close Chinese-Russian relationship is probably more important to Russia than to China, but Beijing also hopes to avoid the possibility that the United States will also impose sanctions on China in the future.
In March 2013, China, along with the other BRICS countries of Brazil, Russia, India, and South Africa, set up the New Development Bank. In October 2014, China set up the Asian Infrastructure Investment Bank (AIIB). Founded in September 2007, the China Investment Corporation (CIC), which, with $813.5 billion in assets in June 2017, is the third largest soverign wealth fund, after that of Norway and the United Arab Emirates, and possesses a capital funding that dwarfs the Japanese-led Asian Development Bank (ADB) and the World Bank. China's AIIB and the CIC also possess assets that Russia cannot match—particularly with world energy prices down from their previous heights before 2008. Since 2014, and particularly in 2015, Russia has become one of the five largest recipients of Chinese outbound direct investment, in relation to the Chinese government's BRI that connects Asia with Europe. Meanwhile, China was Russia's largest bilateral trade partner in 2015.
In September 2016, the China-Russia dialogue emphasized principles such as the "rule of law." The two countries hoped to promote new tax and legal concepts for enhancing investments, investment protection, privatization, and providing state guarantees on finance for projects. There was also dialogue on how to deal with differing interpretations of legislative concepts, such as public-private partnerships and concession agreements. While Washington has often reiterated that Russia and China do not accept the "rule of law," Russia and China appear to be developing their own laws and rules!
**RCEP VS. TPP**
Russia has augmented its support for China to build the BRI. China is also implementing its major trade accord, RCEP, which would tighten its political-economic relationship with Russia and other Eurasian states. These trade pacts were initially forged to counter the US-led Transatlantic Trade and Investment Partnership (TTIP) and Trans-Pacific Partnership (TTP). But Trump unilaterally abandoned the TPP, while the TTIP has been in limbo—although the Trump administration talked about reviving it in May 2017 through a joint US-EU task force.
In terms of the international political economy, the Trump decision not to implement the TPP could actually permit Beijing to play a more significant political-economic role in the Indo-Pacific relative to Japan and the United States, particularly if it can implement the RCEP. Moreover, Trump has been seeking Chinese investment in the United States, with China's CIC and Goldman Sachs promising to co-partner in a five-billion-dollar fund, which is also intended to expand US exports to China. Contrary to his own campaign rhetoric, Trump has looked to both Saudi Arabia and China to finance his infrastructure projects, which will augment Saudi and Chinese political-economic influence in the United States.
At the same time, however, Beijing could nevertheless threaten to back away from its significant financial supports for the over-indebted US economy and its promised investments in the United States—if Trump administration continues to threaten sanctions against China due to Beijing's efforts to expand its regional sphere of influence and security through the BRI—coupled with Beijing's efforts to implement a Chinese version of the US Monroe Doctrine. (See discussion on China's holding of US debt and the possibility of a trade and monetary war, chapter 1.)
As China expands its political-economic influence in the Indo-Pacific, one dilemma is that ASEAN countries find themselves caught between the promises of Chinese trade and finance versus the uncertain American promises of security and defense assistance. ASEAN countries need strong US political-economic backing if they are to also engage in military commitments. Given Trump's decision to dump the TPP, ASEAN states do not feel confident that Trump will continue to back the Obama administration's policy of "rebalancing to Asia" that sought to strengthen the US political-economic and defense role in the Indo-Pacific.
Both Singapore and Japan had essentially argued that the TPP would provide economic "substance" through trade to the US military efforts to rebalance the region. TPP had thus been seen as a means to counterbalance China's massive political-economic influence in the region that impacts all states, from Cambodia to the Philippines to Australia. It had also been hoped that the formation of TPP would have pressed China to reform its generally poor labor standards and upgrade its environmental standards, among other concerns. Seen in this way, the TPP was intended to press China to reform its own economy so that Beijing could have eventually joined.
Ironically, now that Trump threw the TPP out the window, it is the "liberal democratic" United States that appears to be advocating economic protectionism under Trump's new America First nationalism. And it is Communist China that appears to be arguing for liberal exchange and international trade. Chinese President Xi Jinping affirmed at the Davos Forum in 2017: "We must remain committed to developing global free trade and investment, promote trade and investment liberalization and facilitation through opening-up and say no to protectionism. Pursuing protectionism is like locking oneself in a dark room. While wind and rain may be kept outside, that dark room will also block light and air. No one will emerge as a winner in a trade war." After this speech, the motto of the Chinese President Xi Jinping could be called "People First." For Trump it is "America First"—or even "Trump First." (For French President Macron it is "Planet First." See chapter 10.)
Yet China's approach is not altogether altruistic. While China claims that its investment deals do not come with political strings attached, countries that want to trade with China must drop their recognition of Taiwan. China can drive hard bargains on interest rates for loans. And Chinese investment projects, which Beijing insists on controlling, often bring in Chinese labor via "gated communities" of workers who live and eat on the premises—instead of hiring local workers.
When he had signed the TPP, Obama stated,
Today, these countries signed the Trans-Pacific Partnership—a new type of trade deal that puts American workers first. Right now, the rules of global trade too often undermine our values and put our workers and businesses at a disadvantage. TPP will change that. It eliminates more than 18,000 taxes that various countries put on Made in America products. It promotes a free and open Internet and prevents unfair laws that restrict the free flow of data and information. It includes the strongest labor standards and environmental commitments in history—and, unlike in past agreements, these standards are fully enforceable. TPP allows America—and not countries like China—to write the rules of the road in the 21st century, which is especially important in a region as dynamic as the Asia-Pacific.
In effect, despite Obama's warnings and Trump's anti-Chinese rhetoric, it looks like Trump's anti-TPP policy will actually permit China to write the rules of the future as it develops the BRI and RCEP step-by-step in an effort to dominate world trade by 2050. This appears true unless India, Japan, South Korea, and Indonesia stall the RCEP, and if Japan and New Zealand can put together a revised TPP to counterbalance China. But here, the proposed participation of Taiwan in a revised TPP could spark tensions with China, as Taiwan could escape from Beijing's efforts to force countries not to recognize Taipei and thus isolate Taiwan in the global economy. Taiwan has formal state-to-state relations with only twenty countries, after Panama shifted to recognize Beijing in June 2017.
The question remains: if Trump does convince Beijing to invest heavily in the United States in the near future, and augment US exports to China, what political strings will be attached?
**MEXICO, LATIN AMERICA, AND CHINA AND RUSSIA**
From a global standpoint, America First protectionism also opens the door for Mexico to sell its products to China, and for China, India, and Russia to enter the Mexican market—in addition to obtaining greater political-economic influence in the rest of the Caribbean and Latin America—as is already the case in the highly unstable country of Venezuela.
Trump's threats to expel Mexican immigrants, for example, and make Mexico pay for a US border wall/fence, in addition to threatening to place a "border tax" on imports from Mexico, could further destabilize a highly inequitable Mexican society. Such threats could exacerbate the drug and mafia wars inside Mexico and on the streets in urban America. They could furthermore force Mexico to look toward China, India, and Russia for trade, aid, and assistance. A Chinese geo-strategic foothold in Mexico, Venezuela, or elsewhere in the region could eventually be regarded as a challenge to the US Monroe Doctrine—particularly if China, backed by Russia, is eventually seen by Washington as trying to go too far.
Beijing has rapidly extended its influence in Latin America. In 2000, the Chinese share of Latin American trade was merely 2 percent, while the US share was 53 percent. By 2010, the Chinese share had grown to 11 percent of the total; the United States share dropped to 39 percent. By 2016, the United States was still Latin America's overall top trade partner, but China became the top trade partner of Brazil, Chile, Peru, and Venezuela. China has also been expected to overtake the European Union as the second largest trade partner of Latin America and the Caribbean in 2016. China has, for example, already initiated a new multilateral forum in 2014 with the CELAC (the Community of Latin American and Caribbean States)—a forum that excludes the United States. And in addition to looking to China for trade, investment, and finance, Latin American states such as Brazil, Venezuela, Peru, Bolivia, Nicaragua, and Argentina have begun to import Chinese arms.
**CHINA'S EFFORTS TO CONTROL GLOBAL SHIPPING**
In addition to its port investments in Asia (see discussion in this chapter), Beijing has sought to expand its interests in the geo-strategically sensitive region of the Panama Canal zone, acquiring Panama's largest seaport, Margarita Island Port, in May 2016. Margarita Port is part of the Colón Free Trade Zone, the largest free-trade zone in the Western Hemisphere and one of the world's major cargo distribution centers with sea-land-air-rail multimodal transport. Chinese elites are also looking to Nicaragua, given the proposed canal through the latter despite difficulties involving domestic opposition and environmental concerns. Chinese shipping companies have likewise been investing in Darwin, Australia; Athens, Greece (the Port of Piraeus in Greece is a gateway to Asia, eastern Europe, and North Africa); Istanbul, Turkey; and Venice, Italy—all as part of its BRI.
The concern raised here is that Chinese shipping and port investment ventures raise questions in a global situation in which geostrategic/security issues and political-economic issues are becoming intertwined. The close ties between the Chinese government and Chinese shipping and investment firms raises questions as to whether these firms are operating in Beijing's military interests or strictly in China's economic interests.
**CHINA AND RUSSIA ARE STILL IN CUBA**
With respect to China and Russian influence in Cuba, President Putin signed ten bilateral economic and commercial agreements in Cuba in July 2014 and then promised to forgive more than $35 billion in Cuban debt to the former Soviet Union, during a time when President Obama was seeking to formally recognize the country (the latter was achieved one year later, on July 1, 2015). Russia announced that it would invest in Cuban offshore oil exploration, while hoping to help other Caribbean countries develop energy sources as well. It has been reported that Moscow could also be seeking basing rights to refuel its long-range bombers in Venezuela, Cuba, and Nicaragua. Moscow and Havana have purportedly been seeking a deal to permit Russian intelligence-gathering vessels to operate off of the US East Coast and in the Gulf of Mexico through proposed logistical outposts for resupply, shore leave, and repairs.
Russian and Chinese backing for the Cuban government represents one factor in leading Trump to reverse some of Obama's policies that were intended to open US relations with the country. As of November 2017, Trump intends to restrict the ability of American citizens to engage with those Cuban businesses, restaurants, and hotels that possess close ties to Cuban government officials and the security services, as determined by the State Department. Yet such policies are likely to strengthen, not weaken, Cuban ties to China and Russia.
**THE CRISIS IN VENEZUELA**
Since 2007, China has loaned as much as $60 billion to Venezuela alone, out of more than $120 billion in Chinese loans and investments over Latin America as a whole. Access to Venezuelan oil has been the focal point of Chinese investment. Without Chinese finance, it is unlikely that the governments of Hugo Chavez and Nicolás Madder could have paid for their welfare and social programs that helped them to stay in power. Beijing has also been able to obtain many of the legal rights to Venezuelan oil and other natural resources throughout Latin America. In such a way, China has become the new social imperialist—once critiqued by Maoists.
The burgeoning civil war in Venezuela, has begun to impact the entire region. President Donald Trump labeled Venezuela's ongoing political and economic turmoil "a very very horrible problem" and placed sanctions on a number of corrupt government officials. Criminal human- and drug-trafficking organizations have begun to take advantage of black-market activities and the flight of refugees from the country. In a presumed effort to help provide stability in the region, NATO has been considering making Venezuela's neighbor, Colombia, NATO's first Latin American partner. NATO's rapprochement comes just after the historic peace deal between the Colombian government and the Revolutionary Armed Forces of Colombia FARC that ended the five-decade war that had killed more than 220,000 people and had left nearly eight million people as internal refugees. In addition to seeking to shield Colombia from sociopolitical instability from Venezuela, and securing the peace, NATO could seek to counterbalance Chinese and Russian influence in the near future.
Now that Venezuela is embroiled in civil conflict, it is not certain how either China, Russia, or the United States will react to the possibility of growing instability there and throughout the region. As the state collapses, Venezuelan finance from its oil exploitation will no longer provide funding for mismanaged countries in the region—thus exacerbating social tensions throughout Central and Latin America and the Caribbean. For its part, Moscow sees Venezuela as a strategic partner in the area, while China seeks Venezuelan oil and holds significant amounts of Venezuelan debt, which are generally repayable in oil and other forms of investment. At the same time, one can expect more problems in the Caribbean, which represents a major entry point for terrorist and drug financing.
In his September 2017 speech at the United Nations, President Trump threatened to intervene militarily, ostensibly in the effort to help Venezuelans "regain their democracy." Yet a much better option would be for Washington to negotiate a domestic peace settlement in Venezuela by working with Havana, even if Trump may have spoiled relations with Cuba by placing sanctions on the country, as previously discussed. If the United States and Cuba can work together to resolve the domestic crisis in Venezuela, that approach may help the United States and Cuba to make amends as well. But if Trump does not attempt, or cannot obtain, a negotiated domestic settlement in Venezuela, his threats to intervene militarily could further polarize the world into rival alliances, with Latin America destabilized and Venezuela looking to Russia and China for arms.
**INDIA: THE KEY PIVOT STATE**
In addition to Latin America, Moscow and China appear to be working somewhat in tandem particularly in Eurasia. Moscow had initially hoped to work with China in the Chinese-inspired Shanghai Cooperation Organization (SCO), founded in 2001, which seeks to expand security cooperation in Eurasia. The SCO is expected to include India and Pakistan as new members in 2017. This prospect has been symbolized by the annual trilateral summit between India, China, and Russia.
At the same time, however, Chinese-Russian strategic and energy ties today have become far more extensive than Russian-Indian ties as Russia and China increasingly coordinate their global policies. This has raised a question that will impact the global strategic balance: Will a presently neutral India reach out to a possible Eurasian alliance with China and Russia? Or will India move closer to the United States, Europeans, and Japan in opposition to China and Pakistan in particular? Or can India remain neutral and a potential mediator between the two blocs?
Both China and Russia have been attempting to counter the tightening US-French-Japanese-Australian alliance, but China has been checking Russian efforts to draw India into a closer three-way defense collaboration despite Russian, Chinese, and Indian summitry; for example, Russia itself has moved closer to China than to India. China has vetoed India's application to join the Nuclear Suppliers Group (NSG), despite Russia's request. As membership in the NSG would help legitimize New Delhi as a nuclear power, India sees China's attitude as representing an effort to keep India down as a major global actor. China also refused to permit India from participating in an April 2017 defense meeting with Russia. And a highly protectionist India has been hesitant to join the Chinese-backed RCEP, making it the potentially largest trade bloc in the world.
In response to Russian efforts to forge a closer trilateral defense relationship with China and India, the United States and France have hoped to draw India away from Russian political-economic influence through arms sales. In addition, the United States, Japan, and Australia hope to work with India to deal with common security challenges in the Indo-Pacific region, through the Quadrilateral Security Dialogue (or Quad). For its part, India has only slowly begun to expand its defense ties with Japan and southeast Asian states. Indian Prime Minister Narendra Modi renamed India's policy toward the region from "Look East" to "Act East" in 2014. India is now the largest recipient of Japan's $35 billion overseas assistance for infrastructure development and for its "Make-in-India" manufacturing programs. Overall, India is seeking high technology to boost its defense-industrial base.
In March 2016, India held one of its largest military exercises on its territory, involving all ten ASEAN member states. ASEAN possesses close economic relationships with both India and China, yet the ASEAN countries generally remain wary of what they see as a Chinese policy of counter-encirclement through its "string of pearls" strategy. And China is the largest arms supplier for most of India's neighbors.
The aforementioned "string of pearls" expression refers to port construction along the Indian Ocean, from Chittagong in Bangladesh, to Colombo Port City and Hambantota in Sri Lanka, to Gwadar in the Baluchistan province of Pakistan, and to the Maldives. Fears of China's military usage of these ports were raised in 2014 when a Chinese military submarine docked at the Chinese-owned facility at Colombo Port City, Sri Lanka—which represents a key trading hub between East Asia, the Middle East, and Africa, and thus one of the most prominent financial and commercial centers on China's Twenty-First-Century Maritime Silk Road (or BRI). New Delhi has additionally been highly critical of Beijing's efforts to check freedom of the seas and has protested Chinese nuclear submarine patrols in the Bay of Bengal.
India is additionally reaching out to Mongolia, Vietnam, the Philippines, Australia, Indonesia, Singapore, Malaysia, Thailand, Cambodia, Laos, and Brunei—in part in an effort to establish maritime patrols to counter terrorism, piracy, and smuggling, and in part to counter Chinese influence. The Thai and Indian leadership have, for example, stated their intent to "enhance the ASEAN-India Strategic Partnership." Thailand had reached out to China in 2015, yet relations with India appear to be taking priority. The two governments intend to prioritize the completion of the India-Myanmar-Thailand trilateral highway. This would establish a land route to Southeast Asia through India's northeast and help to counter Chinese political-economic influence in Pakistan.
To counter China's influence in Southeast Asia and to affirm India's status as a rising global power, New Delhi needs to secure sea lines of communication, enhance security in the Malacca Strait trade route, and strengthen its trade relations with ASEAN. The problem, however, is that India, as compared to China, lacks the infrastructure that is required to support more trade with ASEAN. Until India enacts land and labor reforms to encourage manufacturing investments, China's trade with ASEAN will continue to surpass that of India. India likewise possesses difficulties in competing with China in Nepal, Bhutan, and other regional countries. China is the top source of imports for Pakistan, Sri Lanka, and Bangladesh, for example, as well as for India; between 2010 and 2015, China's exports to Pakistan doubled.
In the summer of 2017, the Chinese and Indian militaries engaged in a face-off over the Siliguri "Chicken Neck" corridor in western Bhutan where the Chinese were constructing a mountain road seen as a potential strategic threat to India. New Delhi fears that if China would bring tanks and troops to the corridor, it could cut off India from its northeastern states in case of war. Tensions, however, calmed by the time of the Brazil, Russia, India, China, and South Africa (BRICS) summit in September, but the issue of burgeoning Chinese political, economic, and military influence in Bhutan, Pakistan, and other regions remains.
**RUSSIAN AND CHINESE RELATIONS WITH INDIA**
Throughout the Cold War and immediately after, Moscow has largely sought to counterbalance the interests of India with those of China. In the past, Moscow wanted to make certain that India had a favorable military-technological balance vis-a-vis China. Now, however, Russia appears to be switching toward support of China, if not Pakistan as well. In terms of joint exercises, for example, Russia's military exercises with India take place in the context of international UN peacekeeping operations. By contrast, Chinese-Russian military exercises have included maneuvers in which China and Russia have engaged in potentially offensive exercises.
Moscow thus appears to be tilting somewhat away from India as its preferred partner in defense sales and military cooperation. As previously discussed, Russia has been selling relatively more advanced weaponry systems to China—a fact that aggravates Indian security concerns. Yet as the United States and France and other arms suppliers have begun to enter the India defense market, Russia risks losing not only its arms market in India but also contracts for high technology. India has recently sought American economic and technological assistance to expand its own military and naval position in the Indian Ocean region. And it has purchased some $14 billion worth weapons and technology from the United States over the past decade, in seeking to augment its capabilities vis-à-vis China on the Himalayan border and in the Indian Ocean.
Given India's importance in Russian arms sales and foreign policy, Moscow accordingly fears that the Americans, French, and Japanese are beginning to cut Russia out of arms markets and other deals in India. This appears true even if the French Rafale may not hold up against China's (Russian-made) Su-35s, according to some experts. In short, both Moscow and Beijing have begun to support Pakistan more strongly, with Russia beginning to step back from its previous strong support for India.
**KASHMIR**
Indian-Pakistani conflict over Kashmir and other regions is concurrently beginning to interlink with the ongoing war in Afghanistan. Given the poor nature of US-Pakistani relations, at least since the US assassination of Osama bin Laden in May 2011 on Pakistani territory, Russian President Putin has begun to open discussions with Pakistan and even with the Taliban—to the chagrin of India and other regional powers that fear the potential formation of an Islamist state in Afghanistan or in other places in southwest Asia. For its part, China has opposed a number of differing Uighur Islamist movements in Xinjiang province—which China hopes to make the hub of its BRI in Central and South Asia.
The key issue is that Pakistan has been supporting differing pan-Islamist movements in Afghanistan and in Indian-controlled Kashmir. In Afghanistan, Pakistan's goal has been to somehow achieve "strategic depth" versus India by aligning itself with pan-Islamist forces. In Kashmir, Pakistan hopes to press Indian troops out of the Indian occupied zones. Pakistan has likewise supported various major terrorist strikes on Indian territory. Indian-Pakistani conflict came close to going nuclear in the dispute over Kargil in 1998—a conflict that was largely mitigated by US diplomacy.
Moscow's rapprochement with Pakistan appears to be in accord with Moscow's close alliance with China and Beijing's plans to extend the BRI into Pakistan—but in areas that India finds strategically sensitive. Relations between Putin and Indian Prime Minister Modi thus far appear to be positive in the areas of general economic cooperation, defense and arms sales, and nuclear energy. Moscow claims it wants to sustain its "special" relationship with India, and that its relations with Pakistan are strictly economic. Yet the Indian concern is that Russian and Pakistani relations appear to be warming. This was illustrated by joint Russian-Pakistani military exercises in Pakistan in September 2016—at a time when relations between New Delhi and Islamabad were tense. Moscow has also begun to discuss with Islamabad a wide range of regional issues and key areas of mutual interest—but which may or may not coincide with Chinese interests as well.
Beijing has additionally been expanding its sphere of influence and security into Pakistan, raising Indian concerns. As part of China's BRI, China is implementing the China-Pakistan Economic Corridor (CPEC). This means the development of transport and energy infrastructure roads, from China's western Xinjiang province to the Chinese-built port of Gwadar, Pakistan, on the Indian Ocean. This consequently creates an alternative direct transport link between western China and the Indian Ocean. It permits China to bypass the South China Sea by land, if strategically necessary, while developing areas of western China along the historic Silk Road from China's Xinjiang province.
While China's plans in Pakistan have not been entirely finalized, there is a possibility that some elements of the CPEC project will cross into the highly insecure and militarized territories of Pakistan Occupied Kashmir (PoK) and Gilgit-Baltistan—areas that are still claimed by India. Given the uncertain nature of security in this entire region, Beijing has been urging the Pakistan Army to take the leading role in CPEC, as opposed to civilian authorities. This could make China "a major target of Pakistani extremists and separatists," in which some secessionist groups could possibly be backed by India. In addition to perceived manipulation of the Pakistani military, China's tendency to use its own engineers and a large Chinese labor force while only hiring a minimum number of Pakistani workers could cause a popular backlash among local impoverished populations.
On the one hand, if the CPEC project succeeds, and the Chinese economy does not face its own financial meltdown in its global efforts to finance such a large number of gargantuan overseas projects, while Beijing also attempts to boost domestic incomes, repress domestic protest, and put its severe problems of pollution under control, then the CPEC could benefit Pakistan's infrastructure development and economy in the long run. But the CPEC project could also entangle Pakistan in China's global network. This could link the Pakistani military and police to Chinese political-economic and military interests. In effect, Chinese BRI and CPEC ambition raises the Kashmir dispute to the forefront of Chinese and Indian relations.
**KASHMIR: FOCAL POINT OF CONFLICT**
Initially, when the Hindu nationalist Bharatiya Janata Party (BJP) government came to power in 2014, Prime Minister Narendra Modi appeared to offer an olive branch to Pakistan. But the quest for peace did not last very long. In August 2016, terrorist attacks by Pakistan-based militant groups such as Lashkar-e-Taiba and Jaish e-Mohammed, among many previous attacks, led India to engage in a crackdown on the Indian side of Kashmir. At that time, India crossed the "line of control" that divides Kashmir and purportedly engaged in "surgical strikes" on the Pakistani side. Concurrently New Delhi threatened to block Pakistan's water supply by means of speeding up the building of new hydropower plants along the three rivers that flow into Pakistan. If so, this would have been in violation of the 1960 Indus Waters Treaty. And it could cause a negative counteraction by Pakistan's ally, China, which controls much of the water that flows into India before it enters Pakistan.
On Pakistani Independence Day, August 14, 2016, Prime Minister Nawaz Sharif called directly for the freedom of Kashmir. In effect, this statement overtly admitted Pakistani support for Kashmiri insurgence against India. In response, on Indian Independence Day, August 15, Prime Minister Modi then sent his greetings to "people of Baluchistan, Gilgit [and] Pakistan-occupied Kashmir." This statement in effect raised the prospects that India could more overtly support anti-Pakistan movements, such as the Baluchistan Republican Army and the Sindhudesh Liberation Army, in the Pakistani province of Baluchistan. If India does strongly support these movements, such actions would in turn raise tensions in the Pakistani-controlled regions of Azad Jammu and Kashmir and Gilgit-Baltistan, which are in turn claimed by India.
By contrast with his 2016 India Independence Day speech, in August 2017, Prime Minister Modi pledged that he would focus on domestic development and social issues that are confronting Indian society, so that India could be "free of terrorism, corruption, dynasty politics, communalism and casteism." But on Kashmir, he stated: "Neither gaali (abuse), nor goli (bullet) will bring a change. The change will take place when we embrace every Kashmiri." For his part, on Pakistani Independence Day, Prime Minister Sharif criticized the "expansionist designs of India" and urged the "international community to play its role in the resolution of the regional conflicts, particularly the Kashmir dispute in conformity with the UN Resolutions on the subject with a view to ensuring durable peace in the region." Here it appears essential for the Trump administration to address the Kashmir question despite Indian reluctance, and before it is too late to prevent the conflict from escalating. (See chapter 9.)
**WAR? OR CONCERTED DIPLOMACY?**
Russian and Chinese ties to Pakistan and to the Taliban consequently raise major concerns for India—despite the invitation of both India and Pakistan into the Chinese-led Shanghai Cooperation Organization (SCO) in 2017. But while the SCO will provide a forum for bilateral discussions, and could possibly help build relations between Indian and Pakistani military forces and intelligence, it appears dubious that the SCO will help India and Pakistan to find a solution to the question of terrorism and counterterrorism, and to the Kashmir problem. Nor will the SCO help find a solution of their nuclear disputes.
In addition, any Russian deal with Pakistan tightens Russian relations with China as well. Russia may hope to play intermediary between China, Pakistan, and India, but this will not prove easy, as the United States and its allies will seek to draw India closer to their geostrategic and political-economic interests through security cooperation and arms sales. These include the major sale of the thirty-six advanced French Rafale fighter jets to India in 2016—at the same time that the United States has cut assistance to Pakistan. France has also been limiting arms sales to Islamabad, while selling to New Delhi. The US Senate had blocked a deal with Pakistan for eight F-16 fighter aircraft in May 2016 and refused $300 million in defense aid to Islamabad in August 2016. These steps have led Pakistan to buy Chinese JF-17 fighter jets.
In accord with US and Japanese policy, India has strongly opposed China's maritime and naval buildup. New Delhi has moreover rejected China's claims to islands in the South China Sea, in the demand for freedom of navigation for all states in the region—which corresponds with US concerns. Despite Russian efforts to bring India into closer relationship with China, the above factors could press India closer to the new global alliance that is forming between the United States, France, Australia, and Japan.
In sum, India is hesitant about joining the Chinese-backed RCEP, and while New Delhi has hoped to maintain strong ties to Russia, Moscow appears to moving closer to China. Moreover, India's skepticism of the goals of China's CPEC and BRI combined with Modi's provocative statements with respect to Baluchistan and Gilgit-Baltistan against Pakistani claims to support Jammu and Kashmiri independence raise the risk of Indian conflict with Pakistan. And since India additionally claims Aksai Chin and Shaksgam Valley, which are presently controlled by China and because China claims Arunachal Pradesh (calling it South Tibet), which is controlled by New Delhi, Indian-Pakistani conflict could potentially set off a wider confrontation.
Trump's Indo-Pacific policies have flipped like a fish in a frying pan since he came to power in January 2017 up until his visit to South Korea, China, Vietnam, the Philippines, and Japan in November 2017. Whether he will obtain a better grasp of the region after his visit remains to be seen, but Trump has so far flipped from supporting Taiwan, to fawning upon China, to threatening to "totally destroy" North Korea. Kim Jung Un upped the ante by repeating threats that he would soon explode a hydrogen bomb over the Pacific Ocean in the process of developing nuclear missiles capable of striking the continental United States.
But, perhaps even more incredibly, Trump actually told Secretary of State Rex Tillerson not to engage in even a half-hearted attempt at diplomacy. As he put it in one of his (in)famous tweets: "I told Rex Tillerson, our wonderful secretary of state, that he is wasting his time trying to negotiate with Little Rocket Man." In effect, Trump's tweet signals to North Korea that the United States is interested not in a diplomatic solution but in regime change.
In addition, even if Trump does eventually opt for a diplomatic path, his tweet has already helped to undercut the credibility of his secretary of state, at the same time that Tillerson is attempting to restructure the State Department in accord with Trump's directives, by shutting down more than thirty special envoys offices and trying to eliminate at least 2,300 employees. This situation creates a tense working atmosphere that will make it even more difficult for the State Department to effectively deal with a number of critical issues discussed in this book, including the truly existential threat posed by North Korea.
The argument of this chapter is that there is absolutely no alternative to diplomacy with North Korean leader Kim Jung Un except for a war that could kill more than a million people. It is crucial for the United States to engage in effective diplomatic leadership that would seek a settlement between South Korea and North Korea through multilateral negotiations, and that would involve Japan, Russia, and China; and that would soon lead to direct US-North Korean talks.
**TRUMP AND HIS TAIWAN-CHINA FLIP-FLOPS**
As presidential candidate, Trump had questioned the One-China policy that has governed US, Taiwanese, and Chinese relations since the Shanghai Communiqué of 1972 and that is fundamental to sustaining positive US-Chinese relationship. In an effort to obtain political, military, and economic concessions from Beijing, which Trump has seen as the primary threat to the US economy and US interests abroad, Trump threatened to play the Taiwan independence "card." By backing Taiwan, Trump had hoped to obtain the support of anti-Communist Asian American voters. Trump also hoped to alter China's expansionist political-economic and military policies in the Indo-Pacific region.
In opposition to the One-China accord, Trump had stated that he did not see why he must be "bound by a 'one China' policy" unless the United States could make a deal with China "having to do with other things," including trade, devaluation, border taxes, their military buildup in the South China Sea. He continued, "and, frankly, they're not helping us at all with North Korea...You have North Korea, you have nuclear weapons, and China could solve that problem. And they're not helping us at all." But Trump also added as part of his bargaining strategy: "I wouldn't want them to know what my real thinking is."
Trump then offended Beijing by accepting a phone call from the Taiwanese president, Tsai Ing-wen, whom Beijing has refused to recognize as a leader. This is because Beijing sees Taiwan as a secessionist province that must eventually be reunified with the mainland—and by the use of force if necessary. Trump's efforts to play the Taiwanese card against Beijing consequently caused great popular resentment and protest in China—and continues to raise China's suspicions.
In mid-December 2016, much as it has done in the past at the beginning of the administrations of both George W. Bush and Barack Obama, Beijing flashed its dragon claws. Just after Trump's telephone conversation with Taiwanese president, Beijing sailed its (former Ukrainian) aircraft carrier, the Liaoning, accompanied by three guided missile destroyers and two frigates, into the South China Sea. This resulted in a symbolic crossing—for the first time—through the waterway between Okinawa and Miyakojima Island near Japan. The aircraft carrier thus sailed only twenty nautical miles outside Taiwan's air defense identification zone (ADIZ) in the Bashi Channel between Taiwan and the Philippines. This action forced Taiwan to deploy an unspecified number of F-16 fighter jets.
China was also purported to have flown a nuclear-capable Xian H-6 bomber along the so-called Nine-Dash Line which outlines China's territorial claims to roughly 90 percent of the South China Sea. Chinese claims clash with the claims of Vietnam, Malaysia, Indonesia, Brunei, and the Philippines to their own exclusive economic zones as determined by the international law of the sea. China's Nine-Dash Map, which is ironically based on Chinese nationalist Kuomintang party claims, sets China's boundaries in permanent dispute with the claims of almost all of China's neighbors, whose claims in the same area are not as extensive. Even if the Chinese bomber did not possess nuclear weaponry, these actions nonetheless represented symbolic statements that were intended to demonstrate that Beijing would be willing to use nuclear weaponry in order to defend its territorial claims. The incident also involved Japan, which reportedly sent out two of its fighter jets in response to the flight of the Chinese bomber.
Trump's Secretary of State, Rex Tillerson, had already warned China during his congressional confirmation hearings: "We're going to have to send China's leaders a clear signal: that, first, the island building stops and, second, your access to those islands is not going to be allowed." And according the US reports, China has appeared to be upping the ante, at least since Trump came into office, by installing antiaircraft and antimissile systems, among other weaponry, on all seven of the artificial islands it has built in the South China Sea. With Beijing ramping up the pressure, Trump suddenly backed off and was forced to capitulate just a few weeks after becoming president. This reversal of policy was taken in large part as the Chinese President Xi Jinping would not agree to speak with Trump until after he had publicly acknowledged the One-China policy. Both Tillerson and Trump then radically softened their statements toward China.
**THE ONE-CHINA POLICY**
Trump's policy reversal was largely due to the fact that the One-China policy has been at the heart of a more or less stable US relationship with the People's Republic of China since at least the 1972 Shanghai Communiqué, when President Richard Nixon and Secretary of State Henry Kissinger began to reach out to China. At that time, Washington brought the People's Republic of China into the UN Security Council (UNSC) by way of dismissing the Republic of China (Taiwan) from UNSC membership. The One-China policy is deliberately ambiguous in that it permits a Sino-US _modus vivendi._ In other words, the United States has publicly agreed to accept Chinese claims to unify with Taiwan, but Washington is nevertheless permitted to sell arms to Taiwan, without formal recognition of Taiwan as a sovereign state. Yet there is another essential issue: as Washington did not press Beijing to renounce the use of force against Taiwan when agreeing to the One-China policy, US defense support for Taiwan gives China more reason to look to Russia for arms. And it gives Moscow more reason to play the China card against the United States—so as to obtain greater concessions. Pressing China and Taiwan to mutually renounce the use of force will be key to peace. (See chapter 10.)
Beijing fears that Taiwanese independence would undermine the Communist Party's legitimacy to control "all" of China, and that US support for Taiwanese independence would impact China's claims to sovereignty and territorial integrity. Beijing furthermore fears that Taiwanese demands for independence could spark other independence movements inside the mainland (e.g., Tibet, Xinjiang province, and Inner Mongolia). For Beijing, Taiwan is nonnegotiable: Taiwan cannot be part of a Trump bargaining strategy that swaps trade and financial questions.
Trump has nevertheless appeared to be offering China a positive trade deal if Beijing will help "contain" North Korean nuclear ambitions. On the one hand, Trump's threats to engage in economic sanctions and military actions against China give Beijing very little incentive to pressure North Korea, even if Beijing does fear that the North Korean nuclear program could lead both Japan and South Korea to develop nuclear weaponry that would in turn threaten both North Korea and China itself. On the other, Trump's efforts to forge a new trade deal with China, as a possible reward for its role in dealing with North Korea, could be opposed by the US Congress as well as by Trump's domestic Republican and Independent base of support, who oppose Chinese economic competition.
Another dilemma is that Beijing represents the second most powerful political-economic actor in the world, if not the very first in some categories. In 2015, for example, China possessed the world's four biggest banks. And among the world's one hundred largest banks, China possessed the most at the top, with thirteen banks in total, while the United States placed second, with eleven banks. By some measures, China will surpass the United States in terms of gross domestic product by 2018, after already surpassing the country in terms of purchasing power parity in 2014–2015. And given its burgeoning financial, economic, and technological prowess, Beijing could, in just another decade, challenge the United States and its remaining allies in military terms, particularly in its own region—if Washington does not play its cards right. (See chapter 6.)
**THE REGIONAL CHALLENGE**
For Beijing, the state that possesses hegemony over Taiwan in turn possesses control over what the US Navy calls the "sea lines of communication" from Japan to the Arab-Persian Gulf and to the Suez Canal. Japan, South Korea, and Australia all rely upon shipping routes that pass close to Taiwan, through which an estimated $5–$7 trillion worth of goods are transported each year. Some two-thirds of South Korea's energy supplies, nearly 60 percent of Japan's and Taiwan's, and 80 percent of China's crude oil imports, flow through the South China Sea.
In addition to the geo-economic importance of who controls the sea routes through the South China Sea, there are significant amounts of resources in the region. Optimistic estimates claim that the South China Sea may contain 17.7 billion tons of crude oil. Pessimists claim that the proven reserves of oil may be only about 1.1 billion tons. For China's energy needs, cheaper alternatives are readily available, particularly given the currently low oil prices. But for the Philippines and Vietnam, losing access to the South China Sea's potential oil and gas wealth would be far more significant in economic terms than it would be for China. Another major concern is that fish stocks in the South China Sea area have fallen 70 to 95 percent from their levels in the 1950s, and in the next twenty years they could decline an additional 59 percent from their 2015 levels. The decline of fish stocks has already led China to look in waters beyond the South China Sea and as far as West Africa.
According to the Pentagon, Beijing has built up more than 3,200 acres of islands by dumping rocks in at least seven areas it occupies in the waters, in strategic positions near the disputed Spratly Islands. This provides China with long-term "civil-military" outposts from which it can project power. In addition to expanding its claims to fishing areas, resources, and energy deposits in the South China Sea and elsewhere, one of the main reasons for China to construct artificial islands in the South China Sea, which are far more extensive than those of Vietnam, for example, is to provide a barricade to protect its fleet of advanced nuclear submarines, which is based on the southern tip of Hainan Island.
China has not yet formally declared an Air Defense Identification Zone (ADIZ) in the South China Sea, as it did in the East China Sea in 2013. (See further discussion, below.) An ADIZ would identify, monitor, and control all civilian and military aircraft in the region—in a form of anticipatory self-defense. Nevertheless, Beijing has repeatedly threatened to declare an ADIZ, particularly after the Hague Permanent Court of Arbitration (PCA) ruled unanimously in July 2016 in a suit filed by the Philippines against the legal basis for China's Nine-Dash Line territorial claims to the South China Sea. China sees the PCA's decision against its irredentist claims as illegitimate: In effect, China is seeking to justify its demands to establish a Chinese version of the Monroe Doctrine for the East and South China Seas.
Beijing furthermore sees itself as potentially "encircled" by the United States, Japan, and India; this is because China considers all three to be strengthening their military presence in the Indo-Pacific. China's fears of encirclement have led Beijing to develop the concept of "forward edge defense." This is China's new strategic doctrine that calls for the projection of its strategic capabilities from land toward the ocean. This new defense concept appears designed to establish a Chinese "arc-shaped strategic zone that covers the Western Pacific Ocean and Northern Indian Ocean." Beijing has accordingly sought to develop a defense umbrella to counter US military strategies involving anti-area denial and air-sea battle concepts that could bring a potential conflict to the Chinese mainland. (See chapter 6.)
The major focus of China's defense spending remains building its capabilities to seize Taiwan—particularly in the assumption that the latter country will be defended by Washington. The problem is that, in Beijing's view, Taiwanese independence continues to represent a real "threat" that has been given greater credence by Trump's conversation with the Taiwanese president. In addition, Moscow's rapid yet relatively peaceful annexation of Crimea may have given Chinese elites the idea that they might be able to capture Taiwan in a somewhat-similar way—but only if the United States is distracted by other events or if Washington appears willing to appease Beijing.
**JAPAN AND CHINA**
Tokyo has sought to bolster its alliance contributions through the revision of the US-Japan Defense Cooperation Guidelines in April 2015. Tokyo has also insisted that the United States extend its nuclear deterrent to cover Japanese claims, including the Senkaku/Diaoyu Islands, rather than simply relying on the US nuclear umbrella. In addition to developing its own missile defense system involving the deployment of US Patriot and Aegis missile defense systems (but not yet THAAD, Terminal High Altitude Area Defense), Tokyo has increased its military spending for the past five years since 2012. At the same time, the Abe government has urged the revision of its "pacifist" constitution. So far, Tokyo has extended the ability of its self-defense forces to engage in overseas operations under the United Nations—or alongside allied forces.
To this goal, Tokyo has acquired surveillance drones, fighter jets, naval destroyers, and amphibious vehicles to counteract China's military activity in the region. In 2016, the Abe government sought to purchase an additional submarine and new fighter aircraft. The government proposed deploying roughly 1,300 soldiers from Japan's Self-Defense Force on the southern islands of Kagoshima and Okinawa—which are close to the Senkaku/Diaoyu Islands also claimed by China. Tokyo has deployed a new amphibious unit modeled on the US Marines. The latter could be called on to respond to Chinese naval and aerial activity near the disputed Senkaku/Diaoyu Islands.
One of the main concerns that could provoke war between China and Japan has been the Japanese "nationalization" of the Senkaku/Diaoyu Islands in 2013. This nationalization (the purchase of three islands from their private owner by the Japanese government) then inadvertently led to a confrontation between contending Chinese and Japanese claims. Japan has claimed the islands since the 1895 Sino-Japanese War, at a time when they were said to be uninhabited. Chinese claims go back to the Ming period, when they were said to be linked to Taiwan. China has considered Taiwan to be an inalienable part of its territory—even if the Taiwanese do not see it that way.
China then proclaimed an Air Defense Identification Zone (ADIZ) for the East China Sea in late 2013. Here it can be argued that international law has ironically begun to force a number of confrontations. Even though international law (the United Nations Convention on the Law of the Sea, UNCLOS) is intended to help states resolve disputes, international organizations have also begun to insist that states define their territorial boundaries instead of emphasizing joint sovereignty. Due to the complex situation and emphasis of national sovereignty over territorial control, both Japan and China would probably have preferred to let the Senkaku/Diaoyu issue rest in its ambiguous format.
China's claims to an ADIZ not only clash with Japanese claims over the Senkaku/Diaoyu Islands but also, to a limited extent, South Korean and Taiwanese ADIZ claims. Foreign aircraft must now identify themselves before entering the Chinese air-defense zone. The Pentagon responded with what it calls Freedom of Navigation Operations, which are intended to demonstrate that China cannot try to maintain these areas as an exclusive defense zone. As a provocative demonstration that the United States does not accept China's ADIZ, the Pentagon flew a nuclear-capable B-52 bomber over the East China Sea. With respect to Sino-Japanese conflict over the Senkaku/Diaoyu Islands, Rex Tillerson said the United States had "made a commitment to Japan in terms of a guarantee of their defense." Tillerson stated that Beijing's unilateral declaration in 2013 of an air defense identification zone overlapping Japanese airspace over the Senkaku/Diaoyu Islands was "illegal"—even if it was Japan that had first declared its own ADIZ. Tillerson nevertheless affirmed that the United States needed to defend Japan in accord with Article 5 of the US-Japanese security treaty. This Article 5 is similar to the Article V of the North Atlantic Treaty.
Japan has consequently accused China of violating international law, while Tokyo has also tightened its defense relations with the United States. Washington has clearly backed the Japanese position and has not attempted to play honest broker. At the same time, a Japanese conventional military buildup is not sufficient to defend Japanese interests unless the United States is also pressed into playing the role of nuclear gendarme—a role that Trump appeared _not_ to want to play as presidential candidate, but has threatened to play after becoming president.
**RUSSIA, CHINA, AND ASIA-PACIFIC COUNTRIES**
Joint Russian-Chinese military exercises have in part been intended to press regional states to work more closely with both China and Russia. Throughout the Cold War and up until recently, Moscow has largely sought to counterbalance the interests of Vietnam with those of China—but now it seems to be somewhat tilting toward China. For its part, China has used Russian military backing to support its own claims to expansion in the South China Sea and East China Sea (Senkaku/Diaoyu Islands), against Japan and against Taiwan, in addition to seeking to divide the ASEAN alliance. In many ways, due to Chinese "sweet and sour" policies (promises of trade and investment, coupled with military threats), Cambodia, Malaysia, Vietnam, and now the Philippines appear to be moving closer to both China and Russia—in part because of China's promises of financial assistance through the Asian Infrastructure Investment Bank (AIIB). (See chapter 6.)
Moscow has only recently begun to open political-economic relations with Southeast Asian countries. At the same time, while seeking trade with Japan, Russia continues to pressure Japan through a military buildup on the Kurile Islands/Northern Territories, largely in response to the Japanese military buildup. Concurrently, Moscow has also been keeping a rising China on guard. Moscow has, for example, doubled the number of S-400 air defense systems that it deploys in its Far East region, while it has also deployed the S-400 system in its southern coasts near Japan, while its S-500 system is in development. These systems could potentially be used against either Japan or China.
Russia has likewise been building up its Pacific Fleet. The main goal of its Pacific Fleet is to control the Northern Sea Route in the Arctic—which creates tensions with the other Arctic powers, which are primarily NATO members. This includes the stationing of new offensive missile complexes on the Kurile Islands/Northern Territories. (Because the United States is to head the Arctic Council until 2017, US-Russian cooperation will be key to avoiding conflict.)
While Moscow has not overtly backed China's claims in the Indo-Pacific, Moscow has wanted to demonstrate to the United States that Moscow and Beijing are now working together militarily in both the Mediterranean and the Indo-Pacific. This cooperation is intended to provide a possible Sino-Russian joint defense of China's Belt and Road Initiative (BRI). For its part, Beijing may want to engage in joint Sino-Russian maneuvers in order to demonstrate, to countries such as Vietnam, that Russia, despite its promises, might not protect them—in case of potential conflict with China.
Nevertheless, despite Japanese-Russian disputes over the Kurile Islands or Northern Territories, Japan's primary security concern is with North Korea and China—and not with Russia. This is due to North Korea's close political-economic ties to China, plus China's not-so-long-term threat to control sea lines of communication to the Arab-Persian Gulf. In August 2016, for example, after China had sent a warship close to the Senkaku/Diaoyu Islands in June, North Korea fired a submarine-launched ballistic missile toward Japan. Throughout 2017, North Korea launched a number of missiles over Japanese airspace, while concurrently testing its nuclear weaponry.
The dual North Korean and Chinese "threats" have consequently led Tokyo to try to prevent Russia from more strongly backing China. These factors accordingly led Japanese Prime Minister Shinzo Abe in May 2016 to advocate an eight-point plan for Vladivostok that was designed to ameliorate Russia-Japanese tensions. In effect, due to its assessment of a truly threatening situation, Japan began to both break the US and EU sanctions regime placed on Moscow after its annexation of Crimea in 2014 and engage in a rapprochement with President Vladimir Putin (See chapter 9.)
**PHILIPPINES**
By May 2016, the election of Rodrigo Duterte, as Philippine president, has led to a new Filipino-style of rebalancing to China and Russia in response to Obama's policy of "rebalance" or "rebalancing" to Asia.
In the case of the US ally, the Philippines, Manila has looked to China's Asian Infrastructure Investment Bank (AIIB) to provide it with $24 billion in economic assistance—even if it is not certain whether China can fulfill its promises given its present economic downturn. While Washington has hoped that it could help Manila to stand up to China in the South China Sea, backed by the ruling of the Hague Permanent Court of Arbitration, Washington has not concurrently been able to provide sufficient aid and assistance to help improve Philippine infrastructure. Nor has Manila perceived the United States (and the World Bank and the International Monetary Fund, IMF) as helping to improve living conditions in that highly impoverished country—in part because US and World Bank loans generally require countries to engage in significant market-oriented domestic-economic reforms.
In addition, President Barack Obama's criticism of Philippine President Duterte's repressive actions against individuals accused as being involved in the underground drug trade has led the Philippines to try to better balance its relations (including arms purchases) between the United States and the authoritarian states of China and Russia. Duterte hopes Beijing can put pressure on China's drug mafias that deal in the Philippines so as to better control drug-related criminality. How long Manila's balancing act can continue remains to be seen.
Given Obama's strong criticism of Duterte's human rights abuses in his war on drugs (which include an estimated seven thousand extrajudicial killings by March 2017), Trump purportedly told Duterte, "I just wanted to congratulate you because I am hearing of...what a great job you are doing and I just wanted to call and tell you that." When Duterte replied by calling drugs the "scourge of my nation," Trump responded: "I...fully understand that and I think we had a previous president who did not understand that." Trump's approach appears to represent a vulgar realist effort to draw Duterte closer to the United States, but will dubiously succeed.
**VIETNAM**
In the case of the former US rival Vietnam, Moscow and Hanoi became strategic partners in 2001. That relationship was then upgraded to a comprehensive strategic partnership in 2012. A Russia-Vietnam military cooperation pact then formalized the two governments' defense cooperation. Moscow promised arms to help to protect Vietnam's offshore energy interests and defend its claims in the South China Sea. Yet, in order to hedge its bets, Vietnam has also sought defense cooperation with India, the Philippines, and the United States—in the aftermath of then President Bill Clinton's diplomatic recognition of Vietnam in 1995—largely to counter Beijing's claims to portions of the Spratly Islands, which are also claimed by Hanoi.
**RISK OF ANOTHER KOREAN WAR**
In one 2016 interview, Trump had stated in reference to the 28,500 US troops deployed to South Korea: "We get practically nothing compared to the cost of this. Why are we doing this?" Yet he would subsequently reverse this position by February 2017, after he became president. When North Korea threatened to test an ICBM (intercontinental ballistic missile), Trump tweeted what appeared to be a "red line"—suggesting that if North Korea does test an ICBM, the United States will somehow be able stop it: "it won't happen." But then Pyongyang tested a number of nuclear weapons systems anyway.
One of the rationales for Trump's decision to bomb Syria in April 2017, after the latter purportedly used chemical weaponry against its own population, was to send a message to North Korea, China, and Russia that Washington was prepared to use force to assert its interests and values. As a show of force, Trump made the decision to bomb Syria at the very time that he was dining with Chinese leader Xi Jinping in Florida. Issues discussed with Xi included the $310 billion US trade deficit with China; Taiwan; Tibet; and China's military expansion in the South China Sea. Before Xi arrived in Florida, Trump had warned that he was prepared to go it alone to remove the threat presented by North Korean nuclear and missile programs—if China fails to control its closest ally.
In April–May 2017, Trump's Peace through Strength policy was blatantly challenged by North Korea in a show of conventional forces. Yet Trump's display of force in Syria did not halt North Korean missile and nuclear testing. Although Pyongyang's missile launches failed go off on schedule in April, Pyongyang successfully tried again in May with a long-range missile that could potentially strike the US military base in Guam. It has been estimated that North Korea will be capable of deploying as many as one hundred warheads by 2020 (up from twelve in 2017)—if its nuclear weapons and long-range missile program cannot be frozen in the near future by diplomatic agreement. In addition to developing a long-range missile that could strike the US continent, North Korea has purportedly been able to miniaturize its nuclear warheads.
In Trump's September 2017 UN speech, Trump threatened to "totally destroy" the state of North Korea and its population of 26 million, in rhetoric that went way beyond the George W. Bush's threats to Pyongyang in his "Axis of Evil" speech (in which Bush aimed to engage in regime change in the name of democracy). In addition to effectively blaming the North Korean people for not standing up to overthrow Kim Jung Un ("If the righteous many don't confront the wicked few, then evil will triumph"), Trump criticized the North Korean leader in an apparently improvised attack—against the advice of his advisers, who argued that Kim would take such comments personally. Trump dubbed Kim a "rocket man...on a suicide mission." Not surprisingly, Pyongyang then boosted rhetoric against the "mentally deranged US dotard" by threatening to detonate a hydrogen bomb over the Pacific Ocean. However, it appears that Trump cannot handle personal insults very maturely either: In addition to signing an executive order to introduce new sanctions against those who trade with or finance Pyongyang, Trump upped the ante by flying B-1 Stealth bombers and F-15s to the farthest point north of the Korean demilitarized zone (DMZ) that any US fighter or bomber aircraft has flown since the George W. Bush administration, according to the Pentagon. (See chapter 9.)
**MISSILE DEFENSES**
Both Beijing and Moscow have opposed the deployment of the US Terminal High Altitude Air Defense (THAAD) missile-defense system on Japanese and South Korean territory. THAAD is viewed as representing a means to counter China's strategic missile deterrent though a potential preemptive strike. And to the south and west, China also sees as a major threat India's deployment of sophisticated BrahMos cruise missiles in areas of Arunachal Pradesh that are close to disputed borders with China. At the time of this writing, Japan has purchased the Patriot and Aegis antimissile systems, among others, but has not yet decided to purchase the THAAD system.
In denouncing North Korean provocations, which are in large part intended to break US alliances with Japan and South Korea, Trump proposed accelerating the deployments of missile defense systems for Japan and South Korea. The United States had already begun to deploy advanced Terminal High Altitude Area Defense (THAAD) missile defense systems in South Korea, with the backing of the interim South Korean government, prior to the election of President Moon Jae-in. When the decision to deploy THAAD was announced, Chinese relations with South Korea plummeted—despite the US caveat that the MD systems would be designed only to counter North Korean missiles. Nevertheless, Beijing began an unofficial boycott of South Korean products. Both Beijing and Pyongyang have purportedly augmented cyber-attacks against South Korea.
From Beijing's perspective, the THAAD deployments symbolized a closer defense relationship among the United States, South Korea, and Japan. The United States additionally appears to be forging an encircling alliance against China in the Indo-Pacific by also reaching out for defense accords with Australia, New Zealand, Vietnam, the Philippines, and China's major rival, India. Both Beijing and Moscow fear that US advances in missile defenses and radar systems, coupled with the speed and accuracy of US nuclear missiles, could permit Washington to launch preemptive strikes. Both Moscow and Beijing have accordingly engaged in overflights designed to test US-Japanese defense capabilities, but such actions appear to be further militarizing Japan.
In addition to the political-corruption scandals that confronted the government of South Korean President Park Geun-hye, who was impeached and forced to step down, the THAAD deployment is generally not popular among the South Korean population—even though Beijing's meddling in the affair may possibly have led more people to accept it. Trump himself further exacerbated domestic tensions in South Korea by stating that South Korea must pay $1.1 billion for the THAAD—an issue that US officials tried to downplay. In effect, South Korea would have to pay for a system that protects Seoul, but the THAAD system that would protect US forces from attack would be "free," as the United States would pay for it One can imagine that Trump might not have been happy with that prospect either.
The winner of the South Korean presidential elections in May 2017, Moon Jae-in, initially stated that he would reconsider Seoul's previous agreement to host the THAAD, if possible—given the fact that the corrupt Park government and the United States had rushed to sign the contract. But by September 2017, he reversed his previous stance and agreed to limited THAAD deployments, despite public opposition. President Moon Jae-in has likewise hoped to bring back former president Kim Dae-jung's Sunshine Policy, which sought greater diplomatic and political-economic engagement with Pyongyang. He has also sought to revive the Kaesong industrial project that involved South Korean investment in North Korea. If possible, he would like to engage in direct talks.
To help improve relations with North Korea, Moon hopes to encourage greater South Korean investment in Russia, in the hope that by reaching out to Moscow, the latter would help facilitate a new dialogue between Seoul and Pyongyang. Japan and now South Korea both appear to have adopted a more cooperative approach to dealing with Moscow. This position opposes US and European sanctions on Russia. At the same time, Moon Jae-in wants to press for greater South Korean input into Trump's America First policy orientation as it relates to Seoul's interests.
Here it cannot be ruled out that Moscow might want to play the game of spoiler, in taking advantage of disputes between China and North Korea. Moscow could then seek to expand its economic and financial cooperation with North Korea, which has included support for transportation networks, fuel supplies, and employment. But it appears unlikely that Moscow would support North Korea to the extent that it would alienate Beijing. More likely, the two powers would pursue a more coordinated approach, much as they did with the Iranian nuclear question. (See chapter 9.)
Before the April 2017 Chinese-American summit, Trump appeared convinced that China could do much more than not just purchase North Korean coal exports—in order to change North Korea's nuclear policy. Yet Chinese President Xi was said to give Trump a ten-minute lecture on relations between China and North Korea. The lecture appeared to convince Trump how difficult it was for China to pressure North Korea: "I felt pretty strongly that they had a tremendous power over North Korea...But it's not what you would think."
Washington has accused Chinese companies of supplying North Korea with European- and Chinese-made dual-use and military technology and hardware. Washington has demanded that these sales be stopped, given estimations that show that trade between China and North Korea has increased and that secondary banks continue to provide North Koreans with loans. At the same time, Beijing and Washington have reportedly begun to share intelligence, which could permit the interdiction of arms and other illicit trade to North Korea.
As previously discussed, soon after the US missile strikes on Syria, on April 15, 2017, North Korea displayed new weaponry—which included what appeared to be a new ICBM and a new submarine-launched ballistic missile—during the Day of the Sun military parade. The Day of the Sun parade celebrated the birthday of North Korea's founder, Kim Il-Sung, who had launched the Korean War in 1950, with Soviet and Chinese backing.
Just after Pyongyang displayed its new weaponry, Washington tested an unarmed Minuteman III intercontinental ballistic missile (ICBM) from Vandenberg Air Force Base off the coast of California as part of a series of testing. Then, in May 2017, the Pentagon deployed two supersonic B-1B Lancer bombers to conduct military operations with Japan and South Korea. The United States has furthermore engaged in major military exercises with South Korea that North Korea had ritually denounced as provocative.
In regard to US military actions, China has repeatedly asked that the Pentagon stop such exercises, in order to reach a peace accord. In April 2017, when the United States appeared to be threatening an attack on North Korea, Beijing warned Trump not to engage in a Syrian-like strike against North Korean nuclear facilities—as Pyongyang would most likely strike back against South Korea, Japan, and/or US bases in the region. At the same time, despite their 1961 mutual assistance clause, it was not clear whether Beijing would be obliged to assist North Korea in the case of a US attack. This is true given Beijing's view that Pyongyang's development of nuclear weapons appears to represent a breach in their bilateral pact. Nevertheless, there is still a danger that Pyongyang will try to draw Beijing to support North Korea against Washington.
One way for North Korea to draw in China for support in case of war would be to expand the conflict to as many countries as possible. North Korea has accordingly hoped to build missiles as rapidly as possible to penetrate US, South Korean, and Japanese missile defenses. This can be accomplished by firing multiple missiles in rapid succession or simultaneously—in a tactic called "salvo fire." Enough simultaneous launches could then overwhelm the THAAD missile defense system, which requires that several antimissiles be fired in order to be assured that a missile will be destroyed, and hence increase the possibility that a nuclear missile could reach its target in South Korea. Patterns in North Korea's missile testing behavior since 2014 indicate the regime's possible intention of deploying nuclear weapons to missile units throughout the country.
At what point North Korea might use nuclear weapons is not certain, but Pyongyang appears to want to make a war as destructive as possible, in the hope that this will force its opponents to seek peace on North Korean terms. And attacking South Korea's twenty-four nuclear power plants with conventional weapons, for example, could prove an option. Even given the heavy damages that the United States inflicted upon North Korea during its invasion of the south in 1950, through the use of Napalm, for example, Pyongyang continued its struggle until 1953, until Stalin's death. Because there has never been a peace treaty between the North and South, Pyongyang has been preparing for the next round of conflict ever since. Yet a war could result in 1 million casualties, at the cost of $1 trillion. And it could go nuclear.
Given the fact that North Korea wants formal recognition as a "nuclear state," the major question for the Trump administration is whether the era of "strategic patience" is truly "over" as Vice President Mike Pence declared. Should the Trump administration attempt preemptive strikes now, at the risk of a very bloody war? Or should it engage in diplomacy that is intended to freeze Pyongyang's nuclear program where it is in 2017—as it appears unlikely that North Korea will totally denuclearize, at least in the near future?
The problem is that multilateral talks have yet not gone anywhere. The efforts of the six-party talks (between the United States, Japan, Russia, China, South Korea, and North Korea) to achieve peace ended in 2009. Tensions between North and South Korea since the end of the Cold War appear to be surging—and particularly once North Korea repeatedly tested nuclear weapons in 2006, 2009, 2013, and 2016. North Korean nuclear and missile threats, China's reluctance to drop North Korea as an ally, China's coercive tactics in the South China Sea, plus its military buildup aimed primarily at Taiwan, are all factors that are seen by Trump as threatening war in the Indo-Pacific.
**WAR WITH NORTH KOREA? AND WITH CHINA?**
Trump appears ready to extend Obama's "rebalancing to Asia" policy (originally called by Obama the "pivot to Asia") with his Peace through Strength buildup of US military power in the region, as illustrated by his November 2017 visit to the region. But here, Trump appears to be taking a more forward-deployed stance than did Obama. Trump appears to be directly threatening North Korea with unilateral preemptive strikes instead of engaging in Obama's neo-realist strategy of "offshore balancing." This latter, more defensive, approach uses regional allies to counter China and North Korea, while relying on a large offshore US naval presence in the Asia-Pacific region to deter a threatened attack. By contrast, as part of a forked strategy, while pursuing a forward military approach that threatens Pyongyang, Trump has hoped to influence China to engage in quiet diplomacy with North Korea.
The dilemma is that Trump's efforts to engage a reluctant China in diplomacy with North Korea could lead Beijing to demand considerable concessions from Washington in exchange. Moreover, China sees US policies at fault and has urged Washington to engage in direct bilateral diplomacy with Pyongyang, which will in effect legitimize the regime. Washington has been afraid that direct diplomacy will not lead North Korea to give up its nuclear weapons program.
Furthermore, given Trump's ideology of "economic nationalism" and his threats of a trade war with China, there is a real danger that Washington will not respond to Chinese political-economic demands due to Beijing's ostensible lack of assistance on the North Korean question, and that a Sino-American economic war could break out. China then could decide not to buy American products, and instead choose European products, such as Airbus over Boeing, for example. China has already been slowly selling American treasuries and can continue to do so if it begins to establish new markets in the Asia-Pacific through the Chinese-led Regional Comprehensive Economic Partnership (RCEP), which could soon represent the largest market in the world without US participation, while Beijing concurrently tightens its political-economic and military alliance with Russia. (See chapter 1.)
If tensions grow even greater, Beijing could begin to threaten Taiwan by cutting off joint economic agreements. It could further increase the number of intermediate-range nuclear missiles aimed at Taiwan, while militarizing islands and strategic positions in the South China Sea by deploying air-defense systems and advanced fighter jets on a number of China's artificially constructed islands. And if China threatens to attack Taiwan, Taipei has threatened to attack the massive Three Gorges Dam in return so as to flood vast areas.
These actions and threats could be combined with a formal declaration of an Air Defense Identification Zone in the South China Sea, much like how Beijing had established an ADIZ in the East China Sea in late 2013. Beijing, along with North Korea and Russia, could augment its cyber-attacks against private corporations, financial institutions, government agencies, and military and intelligence networks in the United States.
And, finally, if tensions really escalate, Beijing could decide not to sanction North Korea, while increasing Chinese economic aid (and possibly covert military aid) to Pyongyang. In such a way, an economic war could lead to a military conflict with both North Korea and China—as China has not given up its goal to unify with Taiwan, by force, if necessary.
The above sketch of the possible outbreak of war in Asia is of course hypothetical and hopefully will not take place. Nevertheless, after Trump's visits Asia in November 2017, a number of key questions remain: How might US frictions with China over Taiwan and North Korea, among other political-economic issues, impact the Sino-Russian relationship? Will Trump's policy toward China bring Russia and China even closer together? Or will Trump be able to draw Beijing and Moscow away from closer defense ties, if not an alliance? Will all three powers cooperate with Japan and South Korea as well, to prevent North Korea from wreaking havoc?
In sum, it is crucial for the United States to find ways to work with South Korea to open the doors to discussions with the North. Concurrently, Washington must engage with both China and Russia in such a way that not only seeks a peaceful resolution to the disputes with North Korea, but also concurrently sets the stage for a more fundamental rapprochement between the United States, Japan, China, and Russia as well. And as multilateral diplomacy between the United States, South Korea, Japan, Russia, and China with North Korea progresses, Trump will eventually need to meet directly with Kim Jung Un in order to mend relations and provide security assurances that the US will not attempt to destabilize or overthrow the regime.
But is the flip-flop Trump administration and presently demoralized State Department capable of such a sophisticated diplomatic engagement in the quest for peace? (See chapter 9.)
It has been estimated by different sources that the ongoing war in Syria has killed between 321,358 and 470,000 people and has turned millions into internal or external refugees, as of July 2017. Some 13.5 million people require humanitarian assistance, including 4.6 million who are trapped in besieged and hard-to-reach areas, as of September 2017.
The possibility that the conflict could provoke a major power conflagration is illustrated by the shooting down of a Russian military aircraft by NATO member Turkey in November 2015 as a Russian jet passed out of Turkish airspace. US efforts to coordinate effective military actions among the partners in the Global Coalition against Daesh (Islamic State), which was formed in September 2014, and which possesses 73 partners, have proved difficult, to say the least. This is true given the fact that many coalition partners oppose Jabhat al-Nusra (or al-Nusra Front), an al-Qaeda affiliate, and the Islamic State, as well as the Assad regime, while some countries, such as Saudi Arabia, Qatar, and Turkey, have been accused of secretly supporting either the al-Nusra Front or the Islamic State, or ignoring private funding that supports or trades with those terrorist organizations. In addition, Israel, for example, is not a coalition member, yet it periodically intervenes militarily in Syria, primarily against Hezbollah, when Tel Aviv sees its interests being threatened, while Iran, which is not a member of the coalition, supports Hezbollah in Lebanon and other Shi'a militias in Syria and Iraq. Concurrently, Turkey, which is a member of the coalition, has opposed US military support for Kurdish militias that are fighting against IS in Syria and Iraq. While most NATO member countries have participated in the Global Coalition against Daesh, it was not until the May 2017 NATO summit that NATO countries agreed to engage in that Global Coalition as a collective NATO operation. This represented a political statement designed, in large part, to obtain greater political support for NATO from Donald Trump, who had been highly critical of what he saw as NATO's lack of participation in the Global War on Terrorism. (See chapters and .)
For its part, Moscow, since its entry into the war in support of Bashar al-Assad's regime in September 2015, but which is not a member of the Global Coalition against Daesh, targeted not only the Islamic State but also most of the Syrian opposition forces. The latter have been backed by Saudi Arabia, Turkey, the United States, and other states, plus Syrian Kurds, as they fight the regime of Assad. Yet the Trump-Pence administration upped the ante by bombing a Syrian airbase in April 2017 after the Assad regime had allegedly used chemical weapons in a region largely controlled by Jabhat al-Nusra (or al-Nusra Front).
President Donald Trump's April 2017 decision to engage in so-called limited Tomahawk cruise missile attacks represents yet another flip-flop in Trump foreign policy. In September 2013, before Obama ruled out the use of airstrikes against Syria in the aftermath of the Assad regime's alleged use of chemical weaponry in favor of diplomacy with Russia, Trump had tweeted: "Again, to our very foolish leader, do not attack Syria - If you do many very bad things will happen & from that fight the US gets nothing!" Then, two days later, Trump tweeted: "President Obama, do not attack Syria.... There is no upside and tremendous downside. Save your 'powder' for another (and more important) day!"
On April 7, after months of domestic American criticism for being "soft" on Putin and on Russian allies, including Syrian leader Bashar al-Assad, Trump opted to strike a Syrian airbase with fifty-nine Tomahawk cruise missiles—after ostensibly weighing all of the options in his new National Security Council cabinet. Trump, as he later admitted, reversed his previous opinions about Assad and Syria—to the chagrin of some of his strongest supporters and to the praise of his political opponents. Trump's adviser, Steve Bannon, had purportedly opposed the strikes, but was apparently overruled by Jared Kushner.
But in firing fifty-nine Tomahawk cruise missiles (at roughly one million dollars each) at an airbase used by Russian military aircraft, Trump took the tremendous risk that he would not provoke some form of a Syrian or Russian retaliation at some point in the future. Putin's style is not to react immediately but to wait until conditions are in Russia's favor before he strikes back. And without UN Security Council backing, or even a legal justification on the grounds of self-defense, the Tomahawk strike nevertheless represents a unilateral decision that could open the door to additional unilateral and illegal actions—with potentially unexpected and dangerous consequences.
The Pentagon is now engaged in at least seven very different, yet increasingly interrelated, wars in the "wider Middle East" in Afghanistan/Northern Pakistan, Iraq, Syria, Yemen, Libya, and Somalia, while also imposing sanctions on Iran. Trump's actions (and particularly his backing for Israel and Saudi Arabia against Iran, as to be argued) risks a further widening of these conflicts. The problem is that even the use of more massive airstrikes will not address the root causes of these wars. A real strategy involving concerted diplomacy is needed. (See chapter 9.)
In many ways, Trump's cruise missile attacks were intended to send a message that the United States is back in the game of global geopolitics, and that Washington would now be able to negotiate with Syria and Russia and other states from a "position of strength." Washington hoped it would help press Moscow to reconsider its support for the Assad regime. (See discussion of US policy, this chapter.)
Yet this ostensibly "limited" US military action has showed no sign of drawing Syria and Russia to the bargaining table: Moscow has not shown itself to be less willing to back the Assad regime. And the Tomahawk cruise missile strikes have definitely not changed North Korean, or even Iranian, policies. The question remains: why did Trump not ask for congressional or UN authorization? How will this action help to achieve peace—if it was intended to do so?
One of the major reasons that President Obama had pulled backed from military intervention in Syria in 2013, after the Syrian regime allegedly used chemical weaponry at that time, was the fear that the intervention could result in the collapse of the Assad regime and rise of radical pan-Islamist forces without an effective government to take Assad's place. This issue has been made more complex by the fact that a number of so-called moderate forces (generally backed by Saudi Arabia or other Arab Gulf states) nevertheless possess ties with the militant Jabhat al-Nusra (which is linked to al-Qaeda), among other militant partisan movements that may oppose the United States, Europeans, Russians, Israel, and Saudi Arabia. Because al-Nusra, among other militant Islamist groups, often possesses the best trained and effective fighting forces against the Assad regime, both Qatar and Saudi Arabia have often been seen as turning a blind eye to the private networks that support these groups financially. This was true at least in the past, before Riyadh and Doha joined the Global Coalition against Daesh in 2014. At the same time, suspicions between Qatar and Saudi Arabia still remain. (See discussion on the rupture of Saudi-Qatar relations, this chapter.)
Not so ironically, Trump's Tomahawk cruise missile attacks in April 2017 were accordingly supported by the Islamic State and al-Nusra. One member of the Jaysh al-Islam faction argued that a single strike was "not enough"—as there were twenty-six airbases that strike civilians. In addition, IS appeared to take advantage of the strikes in a failed effort to retake the ancient city Palmyra in Syria's Homs Province. The Shayrat Airbase, located southeast of Homs city, which was attacked by the United States, was ironically the airbase that was being used by the Syrian Army to protect Palmyra against its destruction by the Islamic State, which had razed at least three temples and destroyed or tried to sell ancient artifacts from the World Heritage Site protected by UNESCO (the United Nations Educational, Scientific and Cultural Organization).
Other non–al-Qaeda, non-IS groups, such as the Southern Front, saw the attacks as "political" and aimed primarily at Russia. A member of a Turkish-backed Sultan Murad group stated that it would "welcome any action that will put an end to the regime that is committing the worst crimes in history." The National Coalition of Syrian Revolution and Opposition Forces welcomed the strike and urged Washington to neutralize Syria's ability to carry out air raids. In general, the strikes were seen by opposition forces as much too limited to really engage in regime change—but regime change was evidently not the goal.
Yet now that Trump (and not Obama) is in command, will there be "no upside and tremendous downside," as Trump himself had previously warned when he had urged Obama not to engage in a military response to the alleged Syrian use of chemical weaponry? What would happen if the Syrian regime is accused of using chemical weapons again or else engages in a major massacre? Would this represent another "red line"? What would Trump do then?
The major issue raised here is that even the use of force by the Trump-Pence administration will not necessarily bring about diplomatic compromise with either Syria, Russia, or other countries. In fact, after the cruise missile strikes, the Syrian regime purportedly offered the United States access to the airfield that the Pentagon had bombed; this was done only on the condition that the rebels likewise offer access to the site where the poisonous sarin gas was deployed, but which lies in territory held by the rebels. In addition, the Syrian regime was believed (although it was unclear which planes did the actual bombing) to have attacked the same opposition positions in Khan Shaykhun the very next day, from the same airbase that had just been hit by US cruise missiles. These attacks were aimed at the very same positions where the Syrian regime had allegedly used chemical weapons less than a day before.
The first issue is that Trump's attack blocked the possibility that the United Nations could rapidly begin to investigate who was actually behind the use of deadly chemicals. Some experts argue the evidence at the site of the attack tends to show that the regime did do it. The October 2017 Seventh Report of the Organization for the Prohibition of Chemical Weapons–United Nations Joint Investigative Mechanism was "confident that the Syrian Arab Republic is responsible for the release of sarin at Khan Shaykhun on 4 April 2017." Yet the evidence provided by the latter UN report is not absolutely verified.
Even though the Trump-Pence administration has claimed that it had intercepted Syrian messages that stated the plans to use chemical weaponry, Trump still could have waited and exposed the horror to the world. And Trump's action has not stopped the Syrian regime from continuing to kill by other means (including chemicals used in barrel bombs) in opposition to UN Resolution 2139, which had ordered all parties to the conflict in Syria to end the indiscriminate use of barrel bombs and other weapons in populated areas. The point is that a more concerted international approach to the use of chemical weaponry could have put even more diplomatic pressure on the Syrian regime.
The second issue is the question, Why didn't Moscow use its advanced S-400 Triumph antimissile systems, which it had installed in 2015 on the Syrian coast in Latakia and that have raised Israeli security concerns? Could Putin have actually been complacent with Trump's decision to use force, after Trump gave Moscow notice of the cruise missile attacks? Or, more likely, did Moscow not want to demonstrate the S-400 system's capabilities— _or_ its lack of capabilities? This issue raised a debate in Iran as to whether Russia's S-300 and S-400 air defense systems were even capable of thwarting the US Tomahawk missiles. And it also raises questions whether the US attack could now pressure Moscow to deliver more advanced weapons systems to Iran and other countries if US-Russian relations sour even further. Since 2014–2017, Moscow has considered sales of its advanced S-400 Triumph antimissile system to Iran, to China and India, as well as to close US allies Turkey and Saudi Arabia, as the latter hope to diversify their arms suppliers and not rely only on US weapons systems. With regard to both Iran and Saudi Arabia, the eventual deployment of more advanced Russian antimissile systems and other weaponry to these two countries could intensify Iranian proxy wars with Saudi Arabia in Syria, Iraq, and Yemen.
US unilateral actions certainly made Moscow look weak. Moscow then cut off the US-Russian "hotline"—but then restored it after Secretary of State Rex Tillerson's discussions with Putin in April 2017. Yet despite the reinitiation of US-Russian talks in April, Moscow has nevertheless promised to bolster defenses with deployment of its most advanced Black Sea "carrier-killer" frigate into the eastern Mediterranean, which is to be deployed at its Tartus naval base. Moscow has increased the size of its naval and air force in the eastern Mediterranean—but these deployments are still not sufficient to counter US and NATO deployments.
The other issue is whether this attack represented an opening salvo for more attacks. Some congressional spokespersons have claimed that this was "one-time attack"—a symbolic attack. But the US ambassador to the United Nations contradicted that statement. Ambassador Nikki Haley threatened Moscow with the following: "[The United States] is not going to have you cover for this regime anymore. And we're not going to allow things like this to happen to innocent people." Haley further stated that the United States took a "moderate approach" but is still capable of doing much more. The question remains whether this action represents a first step toward "regime change," perhaps more like Libya in 2011 than like Iraq in 2003. Or will Moscow be able to hold Assad in power?
National Security Advisor H. R. McMaster has called for Moscow to rethink its support for the Syrian president, to reevaluate its actions, and to see that Russia could actually be part of the "solution" instead of part of the "problem." But if the United States does not soon show how it can lead Moscow out of its Syrian debacle by means of a general settlement over all of the issues that divide the two sides—from Crimea to tactical nuclear weapons and missile defenses—then the confrontation between the United States, Europe, and Russia will only continue to augment, particularly if the two sides do not effectively coordinate strategy and end up "accidentally" striking each other's forces.
**INTERNATIONAL REACTIONS**
The US attack on Syria was met with strong public support from the United Kingdom, France, Germany, Canada, Poland, Saudi Arabia, Israel, Turkey, and much of the Sunni world, plus Japan, Australia, NATO, and the European Union. It was condemned by Russia, Belarus, Venezuela, Bolivia, Iran, and North Korea, while Brazil, Ireland, Switzerland, and China urged UN diplomacy.
German Chancellor Angela Merkel and French President François Hollande, for example, had issued a joint statement that declared that Bashar al-Assad "bears sole responsibility" for the US strike following the suspected chemical attack. But by contrast to the French and German positions, Washington pointed to Russian responsibility for not enforcing a 2013 agreement it brokered with Syria to eliminate its chemical weapons—in an effort to shame Moscow. As Rex Tillerson put it bluntly: "Clearly, Russia has failed in its responsibility to deliver on that commitment from 2013.... So either Russia has been complicit or Russia has been simply incompetent in its ability to deliver on its end of that agreement."
The issue raised here is that the American position appears to put Putin in a bad light. Blaming Russia could make it more difficult for Putin to save face. The Trump-Pence administration may hope that Putin would put the blame on Assad for the use of such weaponry, in which case the United States and Russia could put joint pressure on Assad to step down. Yet so far this does not appear to be the case.
For its part, Beijing warned against the "further deterioration of the situation" while also opposing the use of chemical weaponry under any circumstances. Nevertheless, Beijing stated that it would have preferred that the Trump administration conduct a complete investigation under UN auspices before engaging in the use of force. Beijing insisted that all parties continue to search for a "political solution." Yet President Trump probably made too much over the fact that China abstained on the UN Security Council vote that dealt with the alleged use of chemical weaponry by the Syrian regime in April 2017. China has vetoed six UN resolutions related to Syria since the civil war began in 2013. But even if China did turn away from joining Russia in vetoing UN resolutions, it did not oppose Moscow's position either. At the same time, Beijing was worried that Trump might engage in cruise missile strikes against North Korea—given the fact that Bashar al-Assad's Syria is not the same as Kim Jong Un's North Korea. (See chapters and .)
UN Secretary-General António Guterres called for a diplomatic settlement. Guterres warned that there was "no other way to solve the conflict than through a political solution." He continued, "Mindful of the risk of escalation, I appeal for restraint to avoid any acts that could deepen the suffering of the Syrian people." The diplomatic dilemma has been how to get Moscow to withdraw support from Assad—but also to find a way to ensure a stable transition to an effective and more inclusive and legitimate new government.
**THE ISRAELI RESPONSE TO THE CRUISE MISSILE ATTACKS**
Israel's prime minister, Benjamin Netanyahu, stated that Trump had "sent a strong and clear message" that "the use and spread of chemical weapons will not be tolerated"; Israel has feared that such "game-changing" weapons, and more than 100,000 short-range missiles, armaments, could reach Hezbollah based in Lebanon. Israel has also engaged in military strikes in Syria when it believed that Syria or Iran was backing Hezbollah. Yet Israel is also aware of the fact that the US focus on the conflict in Syria tends to deflect criticism away from Israel—which has not yet made any progress toward resolving its conflict with the Palestinians. Yet lack of progress in peace negotiations between Israel and the Palestinians, coupled with a Saudi-Israeli rapprochement, helps to fuel the propaganda machines of differing Islamist movements. (See chapter 9.)
**THE SAUDI RESPONSE**
A Saudi foreign ministry official praised Trump as "courageous" for taking action when "the international community has failed to put a halt to the regime's actions." Riyadh not only has supported Syrian opposition forces but also has been involved in a proxy war with Iran throughout the wider Middle East, in Iraq and Yemen, while secretly seeking closer strategic cooperation with Israel.
In Yemen, Riyadh has been accused of severe human rights violations in its battle against Iranian-backed Houthis. As of 2016, more than ten thousand civilians have died in the conflict in Yemen, which has been confronted with a major cholera epidemic that has infected over 600,000 people and mass starvation as a result of the Saudi embargo.
In effect, Riyadh fears Iranian efforts to infiltrate Yemen through its support for Houthis. Riyadh's brutal military intervention is intended both to cut off the Iranian arms supply and to make the Houthis think twice about aligning with Iran. The Saudis fear that Iran could ultimately gain a geostrategic foothold in the Gulf of Aden. The UN Special Envoy to Yemen, Ismail Ould Cheikh Ahmed, has hoped to convince Abd-Rabbu Mansour Hadi, president of Yemen, that a UN-sponsored peace agreement, "including a well-articulated security plan and the formation of an inclusive government, is the only way to end the war that has fuelled the development of terrorism in Yemen and the region." The UN plan would strengthen the powers of the vice president, who would oversee elections that would lead to a coalition government that would provide Houthi representation on the basis of power-sharing. A domestic political settlement could then help to end Saudi-Iranian rivalry over which state controls Yemen and the narrow Bab el-Mandeb maritime chokepoint, which in turn oversees naval access and trade to and from the Red Sea. Riyadh fears an Iranian or an al-Qaeda foothold in the region that could potentially block the passage of commercial shipping through the Red Sea and the Suez Canal, in which approximately 3.3 million barrels a day of oil cargoes pass from the Gulf to Western ports.
If all sides can cooperate, a concerted internationalized control over the Bab el-Mandeb chokepoint by means of US, French, Indian, Japanese, Saudi, and Chinese naval deployments in Djibouti appears in the making. Yet as Djibouti is China's first overseas "support facility," the United States believes that China is engaging in military expansionism to protect its overseas trade and investments.
**IRAN AND SAUDI RIVALRY**
For its part, Iran, as the strongest ally of Syria, condemned the Tomahawk attack as "dangerous, destructive and a violation of international law." Iran condemned the US attack even if it should likewise be concerned with Syria's alleged use of chemical weaponry after Saddam Hussein used such weaponry against Iranian, Iraqi Kurdish, and Iraqi Shi'ite forces during the Iran-Iraq War.
And, angering Iran, Trump has additionally begun to escalate the war in Yemen in support of the Saudi position against the Houthis. While the United States has ostensibly focused its airstrikes and Special Forces operations on al-Qaeda affiliates in Yemen, it has also provided considerable military support to Saudi Arabia. Obama transferred more than $100 billion in over forty different arms sales to Saudi Arabia. And President Obama made a major military sale to Saudi Arabia just as the US Congress overrode his veto and adopted the Justice Against Sponsors of Terrorism Act (JASTA). The latter permits US citizens to sue Saudi Arabia for the damages and death caused by the September 11, 2001, attacks. This bill was passed on the still-unproven presumption that Saudi officials may have supported al-Qaeda or that Riyadh may have known something about the attacks. Saudi Arabia is now facing a $6 billion lawsuit. US attorneys are also investigating whether Iran may have assisted al-Qaeda members by letting them pass through Iranian territory, for example.
Obama had tried to veto the JASTA bill in the belief that it could further alienate Saudi Arabia from US policy and thus intensify the conflict in the wider Middle East. In addition, Riyadh had threatened to pull hundreds of millions of dollars in assets and investments out of the United States in response to the lawsuits. Perhaps most crucially, the law opens the United States itself to lawsuits from people in other countries who have been impacted by US military interventions.
For his part, Trump had strongly opposed Obama's veto of JASTA. Yet Trump then opted for a major arms sale to both Saudi Arabia and Bahrein, which had initially been delayed by the Obama administration on account of human rights concerns resulting from the unconventional way Riyadh is fighting in Yemen. In 2015, Trump had tweeted, "Saudi Arabia should be paying the United States many billions of dollars for our defense of them. Without us, gone!" On his trip to Riyadh in May 2017, however, Trump opted to sell Saudi Arabia seven THAAD missile defense batteries, over 100,000 air-to-ground munitions, and billions of dollars' worth of new aircraft, among other weapons and satellite systems, in addition to assistance in border security and counterterrorism, maritime and coastal security, air force modernization, and cyber-security and communications upgrades. This deal, initiated by Obama, is potentially worth a total of $110 billion, assuming the US Congress does accept all proposed sales to the kingdom. Yet Riyadh, seen as hedging its bets on long-term US support with the particularly fickle Trump presidency, also raised eyebrows in Washington when it sought to purchase the Russian-made S-400 antimissile system.
**THE RUSSIAN RESPONSE**
Putin's immediate response was to call the April 2017 Tomahawk strikes on Syria a violation of international law and a "significant blow" to the Russian-American relationship. The strikes were not necessarily interpreted by Moscow as a legitimate means to punish the Assad regime for its alleged use of chemical weaponry against its own population. Russian diplomats furthermore condemned the United States for being a "partner of Daesh [that is, IS] and al-Nusra Front terrorist groups operating in Syria." Former US Defense Intelligence Agency director Michael Flynn, who lost his post as one of Trump's National Security Advisors, had affirmed that Washington, along with its Arab Gulf allies, had made a willful "decision" set up a "Salafist principality" in eastern Syria to counter the Assad regime.
One Russian diplomat described the US Tomahawk strike as a ploy to distract attention from the "tragedy" that the so-called US-led Coalition Against Daesh has created by targeting Iraqi civilians in the Iraqi city of Mosul. Russian Prime Minister Dmitry Medvedev stated the strikes had been only "one step away from military clashes with Russia"; Russian Foreign Minister Sergei Lavrov stated: "It reminds me of the situation in 2003 when the United States and Britain, along with some of their allies, attacked Iraq." Yet Lavrov hoped that "this provocation will not lead to irreparable damage" to the ties between Washington and Moscow.
One of the reasons for Moscow's direct military intervention in Syria in 2015 was to compel the United States and Europeans to recognize Moscow as a strong equal partner and to better respect Russia's perceived "vital" national security interests. Moscow hoped to signal that it does not want to lose its spheres of influence and security either in the Black Sea region or in the eastern Mediterranean and Levant region. In essence, Moscow has feared that the collapse of the Assad regime could permit pan-Islamist movements to undermine Russian controls not only in the Northern Caucasus but also in Muslim areas of the Russian Federation and in Central Asia.
The Russia intervention in Syria also represented an effort to help deflect US and European attention away from its _fait accompli_ in Crimea—while also hoping to use Syria as a form of bargaining chip to press the United States and Europeans to put an end to the political and economic sanctions that were placed on Moscow after the annexation of Crimea.
**NATO MEMBER TURKEY AND RUSSIA**
With respect to the states closest to the conflict, Turkey's Foreign Minister Mevlut Cavusoglu affirmed that Ankara saw the US intervention in Syria as appropriate but that if the intervention was limited only to a missile attack on a Syrian air base, then it represented only a "cosmetic intervention" unless it eventually removed President Bashar al-Assad from power. NATO ally Turkey welcomed the missile strikes as "positive." Syrian relations with Turkey and the Arab Gulf states had begun to deteriorate in 2011 when the Assad government engaged in a brutal repression of Arab Spring protesters, who were seen by Damascus as supported by the Muslim Brotherhood, among other pan-Islamist groups. At that time, Ankara abandoned its ties to Assad when it began to implement its idealist policy of "zero problems with its neighbors." In November 2015, after Moscow entered the Syrian conflict, Ankara shot down a Russian military aircraft that was flying out of Turkish airspace, after it had ended its mission over Syrian territory near the Turkish border. The Russian jet had struck Turkmen villages that Moscow believed were engaged in the war against Assad. It was the kind of incident that could spark World War III—if NATO had backed Turkey.
Nevertheless, NATO member Turkey, despite previous Turkish opposition to Moscow's support for Assad, began to look toward Moscow for several reasons. First, Turkey began to turn against the United States in the belief that Washington was behind the failed Gülen coup attempt in the summer of 2016. Fethullah Gülen, who lives in the United States, is a self-exiled Islamist who used to be Turkish President Recep Tayyip Erdoğan's ally when the two were engaged in curbing the political power of the military in Turkey's secular "deep state." But Erdoğan broke with Gülen in 2013—who was accused by Ankara of masterminding the coup. Erdoğan has subsequently initiated a major purge of an estimated 150,000 civil servants, teachers, prosecutors, judges, journalists, army officers, and police who have been suspended or dismissed. At least 52,000 have been put in prison. In April 2017, Erdoğan was able to pass a close and disputed referendum that augmented the powers of the presidency by revising the Turkish constitution.
**TURKEY, RUSSIA, AND THE KURDS**
Despite his unexpected rapprochement with Moscow since 2016, Erdoğan stated that he did not see Trump's strikes as going far enough. Turkey's Foreign Minister Mevlut Cavusoglu said Assad's government "must be removed from leading Syria as soon as possible, and the best way to do that is by starting the transitional process." In addition, Ankara called for the United States and its allies in the Global Coalition against Daesh to set up a no-fly zone in Syria in the wake of the April 2017 US Tomahawk cruise missile strikes. Yet this proposal has not been realized.
Second, US support for Kurdish factions in Syria to fight against IS also has also enraged Turkey. Ankara believed correctly or incorrectly that Kurdish political parties and militias in Syria, such as the Democratic Union Party (PYD), are linked to the radical Kurdish PKK in Turkey. These Kurdish groups have been seen by Turkey as demanding independence and not "autonomy" as they claim. When US airstrikes helped the Syrian Kurdish People's Protection Units (YPG)—whom Ankara saw as aligned with the Kurdish PKK inside Turkey itself—to defend the town of Kobani from the Islamic State, Erdoğan suddenly flipped sides. Turkey then began to align more closely with Russia in the fear that US support for Kurdish opponents of the Syrian Assad regime could ultimately result in the formation of independent Kurdish state that could break away from Syria and then support the PKK inside Turkey.
US support for Syrian Kurds then led Turkey to bomb US-backed Kurdish fighters in northern Syria and in Iraq, in April 25, 2017, for example, as Ankara also did in October 2016. These actions were denounced by Washington, which stated that these strikes were not approved by the Global Coalition against Daesh. And despite strong Turkish objections, the Trump administration decided in early May 2017 to provide weaponry for the Syrian Kurds of the YPG to take on fortified Islamic State fighters in Raqqa, Syria.
Third, Turkey was also impacted by Russian blackmail: Moscow could also threaten support the PKK and Assad against Turkey much as Moscow did in the past—if the two sides are not able to reach a compromise. Accordingly, Moscow has wanted Ankara to begin to work with Assad again, as Ankara had done prior to the 2011 Arab Spring movement. Moscow would then reopen mutually beneficial trade, tourism, and energy deals that had been cut off when the two sides broke off relations in November 2015 after Turkey shot down a Russian military aircraft. Russia had also threatened to cut off the South Stream energy pipeline that would provide rents for Turkey.
This situation has been made even further complex by the fact that the Turkish government considers as "terrorists" the Syrian and other Kurdish factions that are supported by the United States to fight IS. At present, the main focus of the United States is not Assad but the Islamic State (IS). In the past two and a half years, as of mid-2017, Washington has spent some $12 billion in fighting IS alone. The United States has hoped to defeat IS but by putting the fewest boots on the ground as possible. This means that not only has the Pentagon bombed IS positions since 2014, but it has also worked with both Shi'a groups in Iraq and Kurdish groups in Syria and Iraq.
The dilemma is that US support for Syrian and Iraqi Kurds has raised Ankara's fears that the United States could purposely or inadvertently be supporting Kurdish independence movements inside Turkey—even if the Pentagon has promised to closely monitor its supplies of weapons and ammunition to the Kurds. The Turks have accordingly moved across the border into Syria to control Syrian Kurdish forces and prevent them from linking to the PKK. They have periodically bombed PKK forces in northern Iraq. Turkish attacks against the Kurds come at a time when the Turkish PKK ostensibly no longer claims "independence" from Turkey. Instead, it is the Kurdistan Regional Government (KRG) leadership that has begun to demand independence. In mid-September 2017, Massoud Barzani, the president of the KRG, held a nonbinding referendum on independence of Iraqi Kurdistan; this was done against the advice of the US government, and it is strongly opposed by Iran, Syria, Turkey, and Iraq itself—but it has been backed by Israel. Also in September, the US Senate blocked the sale of $1.2 million in small arms to Erdoğan's personal security guards after accusing them of using excessive force against primarily Kurdish protesters in May. This was seen as another pro-Kurdish step by Washington against Ankara's interests.
The "threat" of an Iraqi Kurdish independence movement accordingly opens a new can of worms that could further splinter Iraq, Syria, Iran, if not Turkey itself, and can exacerbate ongoing regional conflicts, since Turkey has threatened an energy blockade and Iran has purportedly mobilized troops along the Iraqi-Kurdish border. The more the United States is seen as supporting the Kurds, the more Turkey will threaten to turn to Moscow. In addition, given growing sociopolitical tensions between Germany and Turkey in part due to immigration and human rights issues, plus the fact that the European parliament threatened to suspend EU accession talks with Ankara in mid-2017 after President Erdoğan cracked down severely on the alleged Gülen coup attempt, likewise presses Turkey closer to Russia.
Close ties to Moscow are moreover indicated by Turkish interest in purchasing Russian-made S-400 antiaircraft missiles. These weapons, which would require much closer Turkish-Russian military training and cooperation, could permit Ankara to diversify its military capabilities so that Turkey is not entirely under NATO oversight. Such a sale could also open Russian arms sales to the Arab Gulf countries or to other US allies. The sale would have proved much more acceptable to Washington and NATO, that is, if the United States, NATO, and Russia were on better terms, but it now threatens the relationship between NATO and Turkey.
Because the United States and Europeans, with their close ties to Saudi Arabia, were unable to reach an accord with Russia and Iran over Syria, Turkey joined with Russia and Iran for peace talks in Astana, Kazakhstan. The three countries then brokered a cease-fire for Syria in the effort to create "de-escalation" zones, which took effect (more or less) on December 30, 2016. But these talks took place without US, European, or Saudi participation, raising questions as to whether they can be successfully implemented. By not working with both Russia and Turkey, Washington risks letting Moscow control the show. Washington also risks the rupture of relations with Turkey—in exchange for the not entirely certain benefits of working with Syrian Kurds to try to defeat the Islamic State. Washington will eventually need to work with both Russia and Turkey, and the Kurdish factions, if not Iran as well, in the effort to find mutual accords—if a Syrian settlement is to be found.
Despite the above concerns, Trump nevertheless claimed that the Turkish president "is becoming a friend of mine" and that "he is running a very difficult part of the world."
**INDIA: A MORE NEUTRAL RESPONSE**
India made no strong comments about the US cruise missile attacks, so as not to offend the United States. India also did not want to offend Russia, its traditional ally. But New Delhi additionally does not want to make statements that could possibly enrage Muslim populations inside India itself.
New Delhi has generally seen Assad as an ally; for example, when in 2016 there was a UN vote on a Syrian cease-fire, New Delhi abstained from voting. Despite the ongoing conflict, Syria has sought contracts with Indian companies in the effort to build electrical plants, iron and steel mills, and oil and gas refineries. Syria has also sought Indian financial support to help reconstruct the country—which will prove a daunting task. India appears to be trying to balance itself between the US and Russian positions. But for how long it can do so remains to be seen. (See chapter 6.)
**THE IRAN NUCLEAR ACCORD AND THE THREAT OF US INTERVENTION**
In addition to intervening militarily in Syria and striking an IS target in Afghanistan with the Mother of all Bombs, Trump has likewise threatened to strike Iran, which is directly involved in defending the Syrian regime and in supporting Hezbollah—if the latter does not fully comply with the Iran nuclear accord, or the Joint Comprehensive Plan of Action (JCPOA), which seeks to prevent Tehran from acquiring a nuclear weapons capability.
In effect, the JCPOA took at least a decade for the UN Security Council plus Germany to negotiate through concerted Contact Group diplomacy. The Obama administration argued that the JCPOA was absolutely crucial in that it would limit the chances of a regional nuclear arms race. And it would also limit the possibility that Iran would develop a covert weapons-grade enrichment program. Trump, however, has claimed that the Iran JCPOA nuclear deal puts "limits on [Iran's] military nuclear program for only a certain number of years, but when those restrictions expire, Iran will have an industrial-sized, military nuclear capability ready to go and with zero provision for delay, no matter how bad Iran's behavior is." The truth, however, is that the Iran nuclear deal promised fifteen years of Iranian compliance, plus international inspections. Trump's concerns appear to be less about the Iran nuclear accord itself and more about his opposition to Iran's actual foreign policy toward Israel and Saudi Arabia, and its support for Syria.
Iran further angered Trump with the testing of a new intermediate-range missile in January 2017. For its part, however, Iran claimed that its missile test did not violate UN Resolution 2231. The missile was ostensibly not designed to carry nuclear weaponry and was only to be used for purposes of conventional "deterrence." The JCPOA nuclear deal is strongly backed by the Russians, Chinese, and Europeans. The problem is that Iran's missile test violated the spirit, but not the actual letter, of the JCPOA accord. Neither Russia, China, nor the Europeans would permit Washington to apply language that would prohibit all kinds of missile tests. Only tests for those missiles capable of carrying nuclear warheads would be prohibited.
A possible Trump-Pence administration or a US congressional decision to reject the JCPOA accord could (1) undermine US credibility; (2) start a new nuclear arms race in wider Middle East with Saudi Arabia, Turkey, and possibly with Egypt; and (3) make it more difficult, if not impossible, to achieve a nuclear arms accord with North Korea. Alienating Iran so that it does decide to engage in a nuclear weapons program could furthermore make it more difficult to find ways for Teheran and Riyadh to establish a _modus vivendi_ that would seek to dampen the "terrorist" proxy wars between the two rivals and thus attempt to achieve regional political settlements. A rejection of the JCPOA not only would threaten to undermine Trump's promises to achieve a positive relationship with Moscow but also could further alienate Beijing, if not the Europeans—who are just as close as Israel to a possible Iranian missile attack. And a possible US or Israeli attack on the suspected Iranian nuclear program could work to mobilize the Iranian population against the United States, as it did during the 1980–1988 Iran-Iraq War.
By criticizing the nuclear accord, the Trump-Pence administration is ironically putting pressure on Iran just when Iran rejected a fundamentalist Shi'a leader and elected an ostensibly reformist leader, Hassan Rouhani, and just when Boeing has signed two aircraft deals worth $22 billion that could supply an estimated eighteen thousand American jobs. Given the fact that part of the Boeing deal was signed when Trump was president, it appears highly unlikely that Trump would strike Iran. Nevertheless, the proxy war between Saudi Arabia and Iran could continue to escalate. And in response to Saudi military purchases from the United States and to the threat of potential new US economic sanctions on Iran, or if Washington should brand the Iran's Revolutionary Guards as a terrorist organization, then their commander, General Mohammad Ali Jafari, warned that US military bases in Bahrain, Iraq, Oman, and Afghanistan would be at risk of an Iranian missile attack.
Prior to Trump's arrival to power, the Obama administration had hoped that the JCPOA nuclear accord would eventually open the door to better US-Iranian relations, trade, and a settlement of regional conflicts. Yet the JCPOA nuclear accord was also signed at a time in which there was no apparent progress toward a resolution of the Israeli-Palestinian conflict, nor a resolution of regional disputes that involve a surrogate war between Iran and Saudi Arabia. The risk is that Trump's major $110 billion arms sale to Saudi Arabia in May 2017, as previously discussed, may have jeopardized any possibility of a Saudi-Iranian rapprochement in the near future and could turn Iran even closer to Moscow and Beijing for arms.
Here a new dimension of the global rivalry manifests itself. The fact that Iran has been moving closer to both Russia and China in the post–Cold War era raises questions as to whether the three countries could forge a new Eurasian Alliance. This appears plausible, given the fact that Iran has been considered for membership in both the Russian-led CSTO military alliance and also the more security and cooperation–oriented Chinese-led Shanghai Cooperation Organization (SCO)—even though it has not yet joined either. Teheran has thus far closely aligned with Moscow and Damascus in the conflict raging in Syria and Iraq.
**THE RUPTURE BETWEEN SAUDI ARABIA AND QATAR**
One of the most recent signs of the polarization of the world was the decision in June 2017 by Saudi Arabia, the United Arab Emirates, Bahrain, Yemen, and Egypt to isolate Qatar for its ostensible support for Iran, al-Qaeda, the Islamic State, and the Muslim Brotherhood. This effort to isolate the super-wealthy country, which is not much bigger than the size of the state Delaware, could represent a prelude to a much larger conflict for control over finance, oil, and gas resources throughout the wider Middle East—if diplomacy cannot eventually settle the dispute. Here, Turkish troops have been deployed in Qatar to protect it from a potential Saudi invasion. But more likely, the country could be threatened by an internal pro-Saudi coup d'état intended to replace the present emir of Qatar, Tamim bin Hamad al Thani.
The dispute with Qatar is threatening to splinter the Gulf Cooperation Council (GCC) of Saudi Arabia, Kuwait, the United Arab Emirates, Bahrein, and Oman, in which the latter finds itself trying to resist Saudi pressures. The dispute is also beginning to polarize countries in Africa and the wider Middle East. Already, Djibouti, Somaliland, Chad, Senegal, Maldives, and Mauritania have tended to side with Saudi Arabia and the United Arab Emirates. States such as Sudan, Ethiopia, Somalia, Eritrea, Guinea, and the Seychelles have tried to remain neutral, while Turkey is Qatar's strongest supporter. The United Arab Emirates is building a naval base in Eritrea (a country that also has close ties to Qatar), which worries Ethiopia. The United Arab Emirates has also obtained the backing of three semiautonomous regions inside Somalia, which is strategically crucial for providing airspace for Qatar. The United Arab Emirates removed its ambassador from Somalia when it declared neutrality. Riyadh is building a naval base in Djibouti, along with the United States and China. Israel likewise opposes Qatari foreign policy in the region, given Qatar's close relations with Hamas, Hezbollah, and Iran. In spreading its portfolio, Qatar has also invested in Russia's Rosneft energy company, as well as other strategic investments in Russia, so that Qatar is hedging its bets by maintaining closer ties not only to Iran, but to Russia as well—despite the fact that the US maintains its major Al Udeid military base in Qatar. Yet according to Trump, the United States "would have 10 countries willing to build us another one, believe me, and they will pay for it."
**THE WAR IN IRAQ: THE MOSUL OFFENSIVE**
In October 2016, the United States and forces of the Global Coalition against Daesh—involving Iraqi forces, Kurdish Peshmerga fighters, Sunni tribesmen, and Shi'a militiamen backed by Iraq and Iran—began a major offensive against IS in Mosul, Iraq. This is the location of Great Mosque of al-Nuri, where IS leader Abu Bakr al-Baghdadi had initially proclaimed the creation of a "caliphate" in July 2014. This major battle resulted in 420,000 refugees. Mosul was ostensibly "liberated" in January 2017; yet fierce fighting continued until July, raising questions as to how to reconstruct the devastated city.
Although IS is being defeated step-by-step in battles with the Global Coalition against Daesh, its fighters have begun to spread out in small numbers throughout the wider Middle East—from Libya to Egypt and Afghanistan. In mid-January 2017, US B-2 bombers struck IS positions in Libya, which is divided into at least two major warring factions, plus splinter groups. Concurrently, Egyptian forces have tried to eradicate IS in the Sinai. Trump bombed proclaimed IS fighters with the Mother of All Bombs in Afghanistan. The military dilemma is that air strikes do not control the ground, thus airpower does not necessarily prevent the Islamic State or pro-al-Qaeda forces from dispersing to new regions and regrouping to engage in a nomadic style of hybrid warfare.
**THE ONGOING WAR IN AFGHANISTAN**
The US effort to reconstruct a corrupt and essentially insolvent Afghanistan has represented the largest expenditure to rebuild a single country in US history. Despite a $70 billion US investment in the Afghan security forces, only 63 percent of the country's districts are under Afghan government control or influence. And since 2001, 2,247 US military personnel have died and more than 20,000 have been wounded in Afghanistan alone. And Afghanistan still leads the world in opium production—despite $8.5 billion in US counter-narcotics investment.
The September 11 attacks had actually been masterminded in Hamburg, Germany, and in not Tora Bora. Osama bin Laden—who was the leader of these terrorist attacks—was killed a decade later in 2011 by US Navy SEALs in his hiding place in Pakistan, which is ostensibly a major non-NATO ally but which has not given up its secret supports for differing radical Islamist factions in Afghanistan or in Kashmir. (See chapter 6.) Although al-Qaeda has lost some influence since bin Laden's assassination, affiliated groups are still influential in Yemen and in Syria, for example.
Further, if the actual goal of the US intervention in Afghanistan since 2001 was to set up energy pipelines and gain access to an estimated $1 trillion worth of strategic raw materials, as some analysts have argued, US firms are not necessarily in the forefront for gaining those contracts. Chinese and Indian firms appear willing to take the risks. One could argue that the 2001 intervention in Afghanistan, followed by the 2003 intervention in Iraq, which reduced US resources for Afghanistan, has done nothing but spread pan-Islamist movements—while also doing very little to resolve the domestic sociopolitical problems for either Iraq or Afghanistan, despite the billions invested.
Moreover, the US struggle against the Taliban, al-Qaeda, and now the Islamic State in Afghanistan, ironically, has served Russian interests more than American interests—while US intervention did not achieve its initial goal to destroy al-Qaeda. On the one hand, Moscow has sought to force NATO out of the Black Sea region; on the other, Putin has urged NATO to stay on in Afghanistan since 2014. This is because NATO has helped stabilize some key regions of the country while preventing the Taliban from returning to power. Yet the Russian position toward the Taliban has been shifting.
**FACTORS LEADING TO POTENTIAL CONFLICT**
The situation in Central and Southwest Asia, in Afghanistan and Kashmir, has accordingly begun to heat up again due to a number of factors. First, Obama promised that the United States and NATO would to leave Afghanistan during his administration. Yet Trump has been considering a significant increase in NATO and US troops in the country. The main purpose of a renewed surge would be to shore up Afghan troop morale—but it would also provide the Afghan government with greater firepower. Trump would, in the process, put an end to Obama's restrictions that had limited the ability of the US military to act on the battlefield. And he would give the Pentagon greater authority to use air strikes, such as the April 2017 MOAB (Massive Ordnance Air Blast, or Mother of All Bombs) attack against an IS tunnel complex. Trump would purportedly authorize the Pentagon, not the White House, to set troop numbers in Afghanistan.
Trump's National Security Advisor, H. R. McMaster, who had led anticorruption efforts in Afghanistan, is said to be one of the main backers of the new Afghan strategy. McMaster was also one of the architects of President George W. Bush's generally failed troop surge in Iraq. Donald Trump had initially claimed that he opposed—and would continue to oppose—unnecessary, if not disastrous, US military interventions, such as those in Iraq and Libya, but these initial assertions appear completely false. And it was only after the fact, that Trump argued that the George W. Bush administration's decision to invade Iraq may have been the worst foreign-policy decision in US history. Trump has also stated that he would attempt to limit the exposure of US servicemen and servicewomen to combat situations, but will he?
Despite his campaign statements, Trump appears to be engaging in yet another futile military intervention after the Bush administration had severely maltreated American service members during the wars in both Afghanistan and Iraq. Ironically, in 2013 Trump had tweeted that the United States should leave Afghanistan immediately, or, "if we have to go back in, we go in hard & quick." But it does not look like the United States will be going in hard and quick. And, at the same time, by unleashing the Pentagon rather than pressing for renewed diplomacy, Trump may actually end up extending this seemingly endless war.
Trump's call to deploy more US and NATO forces (as many as five thousand above the already eight thousand present) is a due to the fact that the Taliban appear to be making advances concurrent with the arrival of the Islamic State in the region. In effect, this will complicate US calculations as it attempts to fight both movements, which in turn will fight each other—unless they join forces. (It is possible that the new IS groups could be disgruntled Taliban elements who have formed their own groups and have declared themselves to be IS.)
Moreover, Moscow appears to be seeking a rapprochement with the Taliban—given the Obama administration's promises to withdraw from Afghanistan, which Moscow believes might then lead to a Taliban takeover. In 2014, the Taliban almost took over Kunduz, after NATO began to phase out of Afghanistan. At that time, Pakistan had concurrently sought to clear Islamist militants out of the North Waziristan tribal area; this forced those groups into Afghanistan, even though there are still Taliban sanctuaries on Pakistani territory, which make it almost impossible to achieve peace.
The dilemma for Russia is that if the United States and NATO do eventually pull out of Afghanistan, as Obama had promised, it appears unlikely that the Afghan government will survive for long. On the other hand, if the United States and NATO do engage in a new troop surge, as Trump has indicated, Moscow could play the role of a spoiler that attempts to sabotage US foreign policy wherever possible, given thus far unproven US accusations that Russia has begun to provide the Taliban with weaponry—a charge vehemently denied by Moscow.
Moscow may have decided to make a deal with the Taliban "devil." It is accordingly possible that both Beijing and Moscow might try to make some deal with the Taliban so that the latter (which has begun to struggle with the Islamic State in Afghanistan) will not support the Muslim Uighurs in Xinjiang province in China (former East Turkestan). As they build BRI together, neither Beijing nor Moscow want the Taliban to support those Islamists who want to destabilize Muslim regions in southwest Asia or in the Russian Federation itself. The problem then for Moscow and Beijing is to buy off the Taliban and to play the Taliban against the Islamic State—while also countering the United States and NATO in Afghanistan.
Yet the new Russian policy could also lead the United States to negotiate with the Taliban, which the State Department does not formally designate as a "terror organization." Proposed negotiations with the Taliban, which Secretary of State Rex Tillerson appeared to support at a NATO conference in March 2017, could have two different results. Either the United States could return to the Obama policy of supporting an "Afghan-owned, Afghan-led" negotiation, or the United States could take the lead in negotiations with the Taliban, which the Taliban have claimed they would prefer. Such negotiations could take place in US-led multilateral framework, with countries such as Russia, China, India, Pakistan, Afghanistan, and Qatar involved. But unfortunately it will take much more time, death, and destruction before such negotiations begin, because there is no trust between the opposing sides after years of warfare. And these talks could be further delayed if Trump does unleash the Pentagon so that it is no longer under State Department controls.
**THE WIDENING OF THE GLOBAL WAR ON TERRORISM**
In his first public speech in Saudi Arabia in May 2017, Trump argued that the Global War on Terrorism was "not a battle between different faiths, different sects, or different civilizations.... This is a battle between barbaric criminals who seek to obliterate human life, and decent people of all religions who seek to protect it."
While Trump did not then use the term "radical Islamic terrorism" in this speech (which he did use in his address to the United Nations in September 2017), he also did not define what he meant by "terrorists and extremists." (See chapter 3.) Nevertheless, Trump urged that "they" (whoever "they" are) be driven "out of this earth" by all the Sunni Arab states and societies that are involved in the Global War on Terrorism. It is not clear how the terms "terrorists and extremists" will be interpreted in the Sunni Arab/Islamic cultural context—as these terms could be interpreted to mean those who believe in Shi'a Islam or other "unbelievers" and atheists.
In his May 2017 speech in Saudi Arabia, Trump had focused on Shi'a Iran, as if Tehran were the source of all the problems: "From Lebanon to Iraq to Yemen, Iran funds, arms, and trains terrorists, militias, and other extremist groups that spread destruction and chaos across the region. For decades, Iran has fueled the fires of sectarian conflict and terror." So instead of seeking a way to end the regional arms race and urge Saudi Arabia and Iran to settle their differences—which is absolutely fundamental if it will ever prove possible to put an end to the Global War on Terrorism—Trump appears to have sided fully with Riyadh. In effect, rather than attempting to play honest broker, Trump has greased the fire.
On the one hand, Trump's speech (accompanied by promises of $110 billion in arm sales, plus $300 to $400 billion in mutual investments) could lead some pro-Saudi holy warriors, among others, to shift their focus from the United States, Europeans, Russia, and Israel to fight against Shi'a Iran and the Syrian regime. Saudi Crown Prince Mohammed bin Salman, who has been designated by the eighty-two-year-old King Salman to run the country, has initiated major political, social, and economic reforms so that Saudi Arabia can attempt to sustain its regional hegemony in the long term by diversifying the economy away from oil production by 2030. At the same time, Salman has begun to purge potential domestic rivals (some two hundred individuals arrested on corruption charges, including members of his own ruling al-Saud family). In addition to seeking to isolate Qatar, due in part to its ties to Iran, Salman has militantly opposed Iranian efforts to increase its political-economic influence in Lebanon, Iraq, Syria, Bahrain, and Yemen, and along the Red Sea in the Gulf of Aden. Trump appears to be falling into Salman's game plan.
On the other hand, Trump's speech will not stop IS or similar groups from recruiting young Muslims and converts—particularly those individuals who oppose what they see as the corrupt Saudi Kingdom, which controls Mecca and Medina and whose regime they believe is backed by US and European weaponry. This creates a tacit alignment between these groups and Tehran, which opposes Saudi Arabia, which is seen as the major supporter of pan-Sunni movements that could destabilize the northern Caucasus, Central Asia, and other areas in the wider Middle East against Iranian and Russian interests. At the same time, Tehran also opposes a nuclear-capable Israel, which is threatening to preempt Iran's "peaceful" nuclear program in large part due to Iran's support for the Shi'a militias of Hezbollah in Lebanon.
And despite US/NATO efforts to fight the Taliban, al-Qaeda-affiliated movements, and IS, both Moscow and Beijing see the United States as indirectly backing a number of pan-Sunni Islamist movements throughout the wider Middle East via bilateral US alliances with Saudi Arabia and the other Sunni Arab Gulf countries. US policy has led Russia, and increasingly China, to support Shi'a Iran and Syria. At the same time, Pakistan may be shifting sides, looking to both China and Russia.
Trump's propaganda risks once again falling into the trap first set by bin Laden's attack on the World Trade Center and Pentagon and that sought to draw the United States and other countries into wider wars within the Islamic world. But in this case, Trump's speech represents a call to arms for a war with Iran and those pan-Islamist groups that oppose Saudi Arabia as well. As long as Trump's pro-Saudi, anti-Iranian policies and America First, anti-"radical Islamic" ideology prevails, the Global War on Terrorism appears doomed to last a very long time.
In his book _Trump: The Art of the Deal_ , Trump stated: "My style of deal-making is quite simple and straightforward. I aim very high, and then I just keep pushing and pushing and pushing to get what I'm after." But he also said, "I never get too attached to one deal or one approach.... I keep a lot of balls in the air, because most deals fall out, no matter how promising they seem at first."
For all of his pro-Russian and anti-China rhetoric expressed during his presidential campaign, the Trump-Pence administration has totally reassessed his negotiation strategies with both Russia and China, among other states—much like his book _The Art of the Deal_ suggested he might. In the case of Russia, is Trump now taking a much tougher line in which he fully expects Moscow to hand over Crimea to Kiev? Or is he taking a hardline stance on Crimea in order to make a compromise deal? And if so, what kind of compromise? In the case of China, can Trump be tough after initially backing down on the One-China policy? Or will Trump need to make concessions—and if so, which ones? And how will North Korea impact Trump's calculus? Will China agree to play a key role in mediation with North Korea? Should he renew the six-party talks? Or should Trump meet directly with Kim Jung Un? Or both? What will be the impact of China's Belt and Road Initiative (BRI) and Regional Comprehensive Economic Partnership (RCEP), which could become the major trading bloc in the world, without US involvement? And will Moscow and Beijing be able to forge a full-fledged Eurasian military alliance?
Will his potential deals succeed or "fall out," in Trump's language? And if it is true that "most deals fall out, no matter how promising they seem at first," as he himself had affirmed—then the world is in real trouble. Gambling with war and nuclear weapons is not quite the same thing as gambling in Trump's casinos; although one can lose a fortune in a casino, _everyone_ loses in a nuclear war.
**INITIAL DISCUSSIONS WITH MOSCOW**
Despite his policy flip-flops and feigned efforts to engage in Nixonian madness in order to keep all sides guessing as to his actual policies and tactics, Trump has been more or less correct that the United States needs to begin to engage in a rapprochement with Russia, otherwise the situation could spiral out of control. But he initially moved much too fast to reach out to Moscow and raised suspicions that he sought to profit personally from his Russian contacts, while Putin's alleged efforts, whether effective or not, to interfere in the US elections caused a significant, and hopefully not irreconcilable, American backlash. (See chapter 3.) Given these circumstances, how is a more concerted US-European-Russian relationship to be established?
Based on Russian reports of the January 2017 Putin-Trump conversation, the issues the two presidents began to consider included: The creation of an anti-IS coalition; the establishment of a US-Russia partnership on an equal basis; and the possible "restoration" of trade and economic relations. This implied that sanctions imposed on Moscow after its annexation of Crimea could be reconsidered—but coupled with a _quid pro quo_ with respect to a reduction of nuclear weapons. Putin and Trump accordingly reviewed possible cooperation on a wide range of issues, including Syria, Ukraine, Iran, the Korean Peninsula, and nuclear nonproliferation.
At the same time, it was clear even then that the conditions were still not ripe enough to reach out for a full-fledged US-Russian accord. On the one hand, Trump's approach was opposed by US and European critics who argued that it was too soon to consider lifting sanctions, given the perceived Russian failure to implement the Minsk II agreements, and without pressing Russia to give Crimea back to Ukraine. On the other hand, Moscow did not completely accept Trump's proposals, either. Moscow was not prepared to give up elements of its nuclear weapons capability without a _quid pro quo_ on the part of the Americans. Nor was Putin willing to give up Crimea.
Despite the backlash caused by Moscow's efforts to influence the American elections, the dilemma now is how to put US-Russia relations back on course so that they do not spin totally out of control. A first step to ameliorate tensions would be a US-Russia summit that would address all the disputes between the United States and Russia (ranging from Ukraine, to the wider Middle East, to China and North Korea), while seeking measures to reduce, if not eliminate, nuclear weaponry, and finding ways move away from putting nuclear missiles on high levels of alert and launch-on-warning. Such a summit would lead both sides to pledge to engage in concerted cooperation to resolve these disputes and other issues of concerns where possible. (See discussion, chapter 10.) Perhaps most crucially, from an American domestic standpoint, a Trump-Putin summit could also issue a sincere pledge of noninterference in each other's domestic politics, given allegations that both sides have interfered in each other's elections. (See discussion, chapter 3.)
A US-Russia summit should then be followed by a revival of talks between the NATO-Russia Council. The concern raised here is that the buildup US and NATO power capabilities through the deployment of "permanently rotating" forces in the Baltic region, as has been pushed by the Trump-Pence administration since February 2017, risks undermining the 1997 NATO-Russia Founding Act and will only result in a counter Russia military buildup. This buildup will in turn further escalate US-European-Russian (and Chinese) tensions—if such an approach is not simultaneously accompanied by a detailed plan for an alternative system of European security and a full US, NATO, and EU willingness to compromise with Moscow over what some NATO members might consider to be "vital" interests.
The point is that the United States/NATO and Europeans need to define clear objectives so that they can convince Moscow that they are sincerely seeking to forge a grand compromise on Euro-Atlantic security. The United States and Europeans thus need to provide positive incentives and enticements and not just threats and sanctions in order to convince Moscow that a general militarization is not in Russian interests and that a closer Russian military alliance with China is not in Russian interests either. The promise to lift economic sanctions, for example, must be only one of the levers of a much larger and coordinated bargaining strategy that is intended to reduce overall NATO-EU-Russian tensions. The dilemma is how to implement a systemic approach to these issues of contention and show how they relate to each other, while seeking to find compromises or concessions where possible. (Nuclear-weapons issues are discussed in chapters and .)
**EASTERN UKRAINE**
In many ways, it does not seem that Germany and France possess enough political leverage to press both Ukraine and Russia into a compromise in the Minsk discussions that have been facilitated by Belarus. For this reason, the United States, and possibly Turkey, should soon join the Contact Group discussions between Germany, France, Ukraine, and Russia—since this conflict cannot be completely resolved unless the issues of both the NATO and EU enlargements are also on the table.
One major goal (backed by Moscow but thus far opposed by Kiev) is to work toward the "decentralization" of the country. Kiev has opposed what it calls federalism, or what is really "asymmetrical federalism," in the not-entirely-unjustified fear that federalism could eventually lead regions to demand independence. The dilemma, however, is that no one can really force _from the outside_ the two sides to cooperate—so the situation in the Ukraine needs to move toward one of a "mutually hurting stalemate," in which each side realizes on its own volition that it needs to agree to a mutual accord. Once again, Washington should join the process to help press both sides into an agreement.
From the Russian perspective, Ukraine has not yet fulfilled its part of the February 2015 Minsk II agreements. The latter involves the parliamentary vote on decentralization—a vote which was postponed in 2016, but which is a crucial part of the Minsk process. But more constitutional reforms may be needed in order to ensure that Ukraine develops a true separation of powers and a true "decentralization." German proposals for a special status law, a broad amnesty law, and a special election law for the Donbas appear possible to implement—but only if the Ukrainian coalition government under President Poroshenko can press for a resolution to the conflict in eastern Ukraine. But these already-difficult-to-implement measures are taking place in the midst of a Ukrainian financial crisis and need for deep structural and anticorruption reforms. Once again, a more engaged US diplomatic presence is needed.
To achieve peace, Kiev (backed by the United States and Europeans) needs to take official steps to meet with autonomist leaderships in order to negotiate peace settlement. Efforts to meet with the leadership of Russia-backed eastern-Ukrainian autonomists, as proposed by Ukrainian lawmaker Nadia Savchenko (and former prisoner of war in Russia), and to engage in prisoner exchange or other issues, need to be officially backed by Kiev, rather than denounced as "negotiating with terrorists."
If a political settlement can be found, one possibility is the deployment of a "multinational brigade of neutral but armed peacekeepers with unrestricted access throughout the Donbas" in order to enforce Minsk II—otherwise Minsk II will fail. This proposal has merit, and Russia and Ukraine have considered the possibility of deploying peacekeepers. Thus far, Moscow has agreed only to a temporary and limited presence, but which does not control the territory linking Russia with the autonomist movements. For its part, Kiev wants to set up a more permanent and expanded peacekeeping force.
But if these steps, as proposed by the Minsk II accords, fail, another more pessimistic scenario might result in a highly instable partition of the country, which will not benefit either side and which will create permanent tensions beyond Ukraine. This possibility becomes more probable if the United States does provide lethal assistance to Kiev. (See chapter 5.) A partition of the country could, ironically, force Moscow to financially subsidize the secessionist region—which would prove to be a more daunting task than holding up the Russian-supported breakaway republics of Transnistria (claimed by Moldova), South Ossetia, and Abkhazia (claimed by Georgia), and it could further militarize Russian actions. This represents yet another reason for a concerted international solution.
There is a real risk is that an eventual partition of Ukraine, coupled with the permanently rotating deployment of NATO forces in the Baltic region, could in turn lead to a new partition of Europe—followed by the polarization of the world into two rival alliances. (See chapter 5.)
**TOWARD UKRAINIAN NEUTRALITY**
Once, and if, the Minsk II accords can be implemented (assuming Kiev can eventually implement the necessary constitutional changes and negotiate directly with the Donbas "autonomists"), the United States, Europe, and Russia then need to establish Ukraine as an internationally recognized neutral and "decentralized" country—with Crimea as a free-trade zone, yet under Russian sovereignty, while providing aid and assistance for Ukraine's development.
Both former US Secretaries of State Henry Kissinger and Zbigniew Brzezinski have called for a "neutral" Ukraine. Yet the Kissinger proposal for the formal "neutralization" of Ukraine, coupled with the non-recognition of Crimea, will not put an end to US-Russian-European geostrategic and political-economic tensions. And the crisis will continue unabated if Kiev does not finally grant some form of decentralization or autonomy arrangement for eastern Ukraine, as demanded by the Minsk II accord. There will also be no settlement if Kiev, backed by the United States and NATO, continues to demand that Crimea be returned to Ukraine as well, while Kiev concurrently develops a strong independent military capability. In short, a settlement of the Crimean question is essential if the conflict is to be resolved in the long term. If bargained cautiously, a resource-rich yet "neutral" and decentralized Ukraine—with sufficient self-defense forces and that does not harbor irredentist claims to Crimea—could begin to defuse tensions between NATO, the European Union, and Russia.
To arrive at a full accord with Moscow, there needs to be formal agreement over NATO expansion, European security, and Ukrainian neutrality. To accomplish this, NATO should formally announce a full suspension of NATO enlargement—but as part of a larger negotiation process that is intended to reach a deal with Moscow over Crimea, eastern Ukraine, and the Caucasus. While NATO has claimed that Article X of the Washington Treaty is "one of the Alliance's great successes," there is nothing in Article X that supports the contention that NATO must expand its membership. NATO members "may" invite other states by unanimous agreement—but there is no necessity to offer an invitation to other states.
Moreover, membership in both the US-led NATO and the Russian-led CSTO are against Ukraine's initial statement of sovereignty. The statement of Ukrainian sovereignty (adopted on August 24, 1991) affirmed that Ukrainian SSR "declares its intention of becoming a permanently neutral state that does not participate in military blocs and adheres to three nuclear free principles: to accept, to produce and to purchase no nuclear weapons." It is furthermore possible that Ukraine could define itself as a permanently neutral country in that its constitution contains principles that state that it will not participate in coalitions.
The United States and NATO would accordingly need to modify NATO's "open enlargement" policy—at least for the Black Sea/Caucasus region. Such a modification of NATO's open-door policy would take place in exchange for the implementation of a new regional, yet internationalized, system of cooperative/collective security for the entire Black Sea and Caucasus regions—much as was envisioned by Turkish proposals for a Caucasus Stability and Cooperation Platform in 2008. Here, Ankara could enter the Minsk talks in order to deal with security issues concerning Crimea and the Caucasus that impact NATO-member Turkey.
In other words, instead of extending full NATO membership to Ukraine, Georgia, or other states, and then attempt to integrate these countries back into NATO's command structure, the United States, Europeans, and Russians would extend overlapping US, European, and Russian security guarantees for states in the entire Black Sea and Caucasus region in the formation of a neutral "peace and development community"—in working with Turkey and other regional powers. The purpose would be to implement a cooperative/collective security approach to the region under the auspices of the Organization for the Security and Co-operation in Europe (OSCE) that seeks to protect "vital" Russian and Ukrainian interests, while at the same time looking for new forms of regional and international cooperation.
A full suspension of NATO enlargement would accordingly be part of a larger negotiation process that is intended to reach a deal with Moscow over Crimea, eastern Ukraine, and the Caucasus—among other issues that are dividing the United States, the European Union, and Russia. The promise of NATO enlargement has done nothing but send mixed signals to Russia, Ukraine, and Georgia. NATO enlargement has antagonized Moscow, while it has been concurrently disingenuous with Kiev and Tbilisi—as full NATO membership for Ukraine and Georgia remains a highly unlikely prospect. Guarantees of Ukrainian neutrality, coupled with a gradual reduction of NATO and Russian forces in the Baltic region, could also minimize the ostensible need to expand NATO membership to EU members Sweden and Finland, given the Russian military buildup in the Baltic region. (See chapters and .)
**CRIMEA AS A FREE-TRADE ZONE**
There are many who argue that the United States and Europe should simply accept the Russian annexation as a _fait accompli_. Yet this position does not address the question of competing claims and the political, economic, energy and legal disputes over Crimea, energy resources, and other issues in the vicinity—and the need to develop and eventually demilitarize the whole Black Sea-Caucasus region that will serve Russian interests as well. To fully develop the region will require international assistance. (See chapter 5.)
The establishment of Crimea as an international free-trade zone with relative autonomy—but under Russian sovereignty—and the opening of Sevastopol to regional security and development cooperation, could open the doors for Russia to cooperate with Ukraine, as well as with the United States and Europeans. This is despite the evident friction over the annexation of Crimea, which shows no signs of dissipating at the time of this writing. The eventual opening of Crimea could, in turn, lead both Russia and Ukraine to forge new forms of memberships with a reformed (and renamed) NATO and the European Union, given deeper security and political-economic cooperation.
One option is to call for a new Crimean referendum. Such a plan was proposed by Andrii Artemenko, who was one of the contacts of President Trump's former National Security Advisor Michael Flynn in Ukraine, before Flynn was forced to resign. Essentially, Artemenko's plan would have required the withdrawal of all Russian forces from eastern Ukraine. Ukrainian voters would then decide in a new referendum whether Crimea would be leased to Russia for a term of fifty or one hundred years. The plan also outlined a way to lift sanctions on Russia.
Yet, if such a plan could be implemented, it appears unlikely that Moscow would accept the results of a new referendum in Ukraine that could potentially lead Moscow to give up sovereignty over Crimea after having already gone to such great lengths to annex it. On the other hand, it is not entirely impossible that Moscow could accept a way to pay for a lease of the isthmus in a negotiated deal, given the need for a legal accord that would attract foreign investment to Russia in the face of sanctions. As proposed above, a more feasible option might be the establishment of a free-trade zone under Russian sovereignty, or even a system of joint Russian-Ukrainian sovereignty. (See chapter 10.)
Artemenko claimed that he had "compromising" evidence on the Poroshenko government, yet his plan was opposed by a number of Ukrainian politicians, who saw it as capitulating to Russian interests, and who accused Artemenko of treason. Further, it appears highly unlikely that Moscow would have accepted that plan either. Whether the United States and NATO can propose a more feasible plan that Moscow might accept will depend on the evolution not only of relations between the United States, the European Union, and Russia, but also between the Trump administration and Congress. (See chapters and .)
**PEACEKEEPING IN THE CAUCASUS?**
In addition to the potential deployment of peacekeepers under a general UN or OSCE mandate so as to assure security and development prospects in the Donbas region, peacekeepers could also be considered for the states of the southern Caucasus. Instead of demanding that Russia "evacuate Transnistria, Abkhazia, South Ossetia, as well as Crimea and Donbass"—as the hardline position that seeks to isolate Russia advocates and which Moscow would definitely consider as a _casus belli_ —joint NATO, Russian, EU, and international peacekeepers under an OSCE mandate could be deployed in the so-called frozen conflicts in the southern Caucasus. The problem here will be how to engage in joint US, European, and Russian arrangements in an effort to share or internationalize (but not monopolize) those nonvital Russian spheres of influence and security where mutual agreement is possible.
Joint NATO-Russia/CSTO-EU or multinational Partnership for Peace (PfP) peacekeeping deployments (under an OSCE or UN general mandate) could accordingly be deployed in the Donbas region as well as in the frozen conflicts in the Transnistria, Abkhazia, South Ossetia, and Nagorno-Karabakh. These deployments could take place under a joint NATO-EU-Russian command structure in which Russia plays a positive role. These peacekeeping operations would be somewhat similar to joint deployments in the former Yugoslavia in 1995, but Moscow would be truly represented as an equal. Given the fact that the tense situation in Nagorno-Karabakh between Armenia and Azerbaijan appears to be heating up, it seems that joint US/NATO, EU and Russia/CSTO peacekeeping efforts could prove essential to sustaining peace in this region.
The above proposals presume the absolute need to strengthen NATO's Partnership for Peace (PfP) and an EU peacekeeping force that could be deployed under an OSCE or UN mandate and work with UN and Russian/CSTO peacekeepers as well. NATO's PfP and EU peacekeeping is an underused tool that can succeed on the ground in helping to build trust between warring parties—if given the proper resources. And these proposals furthermore provide a real mission for EU foreign and defense policy to cooperate with the local states, Russia, and the United States under a general OSCE mandate. Peacekeepers are real heroes, and they should be considered so by Trump.
**WHY SHOULD RUSSIA ACCEPT SUCH A PROPOSAL?**
Washington and Moscow need to revisit the Obama administration's failure to "reset" US-European-Russian relations after the 2008 Russo-Georgian War. Both countries need to reexamine both Dmitry Medvedev's June 2008 call for a new European Security Treaty and Turkish President Erdoğan's call for a Caucasus Stability and Cooperation Platform.
With respect to Russia and Turkey, both Medvedev's and Erdoğan's proposals raised critical questions as to whether it was truly necessary to enlarge NATO membership to Ukraine and Georgia, among other states. And both proposals raised questions as to whether there might be other viable options that could provide security for the Black Sea region. The joint US/NATO and EU failure to address these Russian and Turkish diplomatic initiatives—and thus to reconsider the process of enlarging both NATO and EU memberships into Russian-defined spheres of influence and security—represented a major reason for the Russian backlash in early 2014. Moscow consequently took matters into its own hands and decided to annex Crimea and support autonomous movements in eastern Ukraine.
There is a further danger that renewed tensions in Georgia and between Armenia and Azerbaijan, as well as the rise of pan-Islamist movements in Dagestan and elsewhere in the Russian Caucasus, could set off a new series of terrorist acts. Moreover, conflicts in the general Black Sea region and in both the southern and northern Caucasus could begin to interlink with the conflict in Syria and in the wider Middle East—given the return of Islamic State fighters to these regions and given Trump's strong support for Saudi Arabia against Iran. Both Riyadh and Tehran possess interests throughout the region in support of differing Islamist factions that could spark significant conflict. (See chapter 8.)
The establishment of a formally neutral Ukraine, coupled with a negotiated settlement over Crimea and eastern Ukraine, would provide the United States and Europe some bargaining leverage to at least seek a reduction of the ongoing buildup of military forces in the Black Sea region, while likewise seeking to restrain Kiev's demands for a return of the Crimean isthmus. At the same time, it could permit Crimea greater autonomy within the Russian Federation, and help protect non-Russian minorities, including Tatars and Ukrainians.
The Russian Federation is divided into republics, krais, oblasts, cities of federal importance, an autonomous oblast, and autonomous okrugs. In 2014, Sevastopol and the Crimean Republic became the eighty-fourth and eighty-fifth federal subjects of Russia. But could they, along with Kaliningrad, become free-trade and development zones under Russian sovereignty? Or could some form of joint system of sovereignty between Ukraine and Russia be established?
Unless absolutely strapped for cash, it is highly unlikely that Moscow would ever give up total sovereignty over Crimea, much like it is unlikely that it would give the Kuril Islands/Northern Territories to Japan. But for Russia to hold onto Crimea and try to develop it without some form of international investments and backing is already proving quite costly and difficult. (See chapter 5.) Putin needs an economic boost to help him rule, as domestic opposition movements have begun to strengthen. Putin has accordingly pressed for an end of sanctions in order to attract foreign investment and lift the economy out of recession. Russian presidential elections will take place in 2018, and the Trump-Pence administration could possibly try to push for a grand US-Russian compromise just after those elections—that is, if it is possible, given strong domestic US opposition to any accords with Russia. Given the fact that Putin is still seen by the general population as willing to assert presumed Russian interests and to stand up against the pressures of the United States in particular, he will most likely win the 2018 Russian elections—but without as much popular domestic support as in the past given the generally faltering economy. Yet this should not stop the United States and Europeans from engaging in negotiations with Moscow simply in the belief that Putin's rule might soon collapse. (See chapter 4.)
**ROLE OF EUROPE**
The Ukrainian conflict now holds the key as to whether Russia can ever move closer to the Euro-Atlantic community. On the one hand, if the brutal war in eastern Ukraine continues to rage, it will stimulate a general arms race and destabilize US-EU-Russian and global relations. On the other hand, an overarching geopolitical settlement could help defuse tensions throughout the region and the world.
The dilemma is that even closer political-economic ties to the European Union will not prove to be a panacea for Kiev, given its deep financial crisis. It is accordingly essential that the European Union work out mutual political-economic and financial accords with both Ukraine and Russia as soon as possible once the disputes over eastern Ukraine can be resolved. This can be accomplished in the process of rescinding sanctions against Russia (and vice versa for Russia to rescind sanctions on the European Union and the United States). In addition, the United States, Europeans, and Japanese need to look for ways to reduce the prospects of a US-Russia energy rivalry that could lead to war. One step would be to bring Moscow back into the G8 discussions after Russian membership was suspended in March 2014. The global energy (and environment) question could be one of the main concerns of future G8 talks—even if the May 2017 G7 conference had failed to reach a significant compromise over the issue of global warming with Donald Trump. Both G8 and EU-Russian discussions could likewise lead the European Union to work out a political-economic association accord that better balances both Russian and Ukrainian financial and political-economic interests, given the European Union's failure/refusal to do so in 2013–2014.
Such efforts must be taken to achieve reconciliation between Kiev and eastern Ukraine within Ukraine itself at the national level and at the international level so that a neutral (and non-nuclear) Ukraine could truly serve as a gateway between the United States, Europe, and Russia, much as former Soviet President Mikhail Gorbachev, among others, has proposed. It is in US, European, and Russian interests to prevent Ukrainian state collapse, bankruptcy, and sociopolitical instability from degenerating into a wider sociopolitical conflict. Evidently such an approach can take place only if all sides realize that compromise over presumed "vital" issues is in their mutual interests.
**TOWARD A NEW EU DEFENSE AND SECURITY CAPABILITY**
Trump's insistence that NATO was "obsolete" has ironically led the Europeans to consider other options—if they can coordinate policies. The possibility that Washington might no longer back NATO has led the French and Germans to consider forging a new post-Brexit European defense entity that would be more autonomous than NATO.
In September 2016, the French and Germans called for a joint and permanent EU command headquarters for its civilian and military missions and a strengthening of the Eurocorps. The initiative would not create a European army as such. But it would seek to advance the European Union's own joint land, air, and sea transport; relief/medical capabilities; peacekeeping; plus its ability to participate in the Global War on Terrorism. It would also initiate a European defense research program that would be funded by the common EU budget by 2021–2027, with France and Germany initiating the program in 2017.
A more unified European defense and security capability, which would pool defense resources, could theoretically engage in peacekeeping, anti-piracy, and anti-terrorism operations, while more effectively handling the waves of immigrants trying to cross the Mediterranean. A new European defense force would represent a coalition of the willing and should be directed only toward defensive and peacekeeping measures. It would be more autonomous and could theoretically work with both NATO and Russia. But to do this, the European Union first needs to implement a Common Foreign and Defense Policy step-by-step. The European Union has outlined five possible scenarios for its future, but it needs US diplomatic backing if it is to succeed in creating a truly common defense.
Over the years, the Europeans have demanded greater defense autonomy from NATO. Now, following the exit of the United Kingdom from the European Union, the Europeans need to put their money where their mouths are. The United Kingdom, the most powerful military actor in NATO after the United States, had previously blocked a more integrated European defense structure, due to its close defense ties with Washington. At the same time, the political-military dilemma is that Germany has been more supportive of a more federal system of European integration, while France's approach has been more intergovernmental.
If the United States, Europe, and Russia can eventually reach a common accord over defense and security concerns in Europe, NATO can begin to wither way—but it must not totally collapse without first working to put in its place a new Euro-Atlantic security architecture that leads the Europeans to cooperate with both the United States and Russia. If, however, NATO cannot soon find ways to cooperate with Moscow, the defection of Turkey and France, among other states, from NATO is a real possibility.
**TOWARD GREATER EU POLITICAL-ECONOMIC COORDINATION**
It is also crucial that the Europeans begin to coordinate both its geostrategic and political-economic strategy toward Moscow by looking for ways to incorporate both Russia and the other post-Soviet states, including Ukraine, into new forms of European Associate Partnership agreements. Eventually post-Soviet states could enter the post-Brexit EU as "Associate Members," as to be argued. The European Union likewise needs to work with the United States, Russia, and Turkey to establish a new system of Euro-Atlantic security based on cooperation over the Black Sea region. Such a practical project—the effort to bring peace and development to the Black Sea and Caucasus regions—could in turn provide a means for the Europeans to unify their divergent foreign and security policies, particularly if the conflict between Greece and Turkey over Cyprus can soon be resolved.
Given the real possibilities of the further breakup of the European Union after Brexit, Washington needs to back strong partners, and these actors need to be politically and economically stable and possess governments that are perceived as legitimate. The rise of a number of anti-EU movements that seek to drop the Euro as a currency, dump their creditors, and then exit the European Union, means that the European Union needs to engage in substantial internal reforms and changes in its external policies as well, in order to surmount the financial crisis since 2008. The latter crisis largely stems from low US interest rates since 2001 the US mortgage and toxic-loans crisis, which in turn exacerbated the crises of banking and finance, public debt, and underinvestment in Europe.
The European Union needs to tighten the Franco-German relationship with the rest of its core members, step-by-step, but it also needs forge new kinds of loose linkages with the United Kingdom and other states, such as Russia and Turkey, that do not meet full membership requirements. Here, depending on how the separation between the European Union and the United Kingdom is negotiated (a process that could prove very lengthy and costly), Brexit could possibly provide a new model for forging new EU political-economic relationships with other states that could then develop differing forms of partnership and associated membership with a new, more decentralized, European Union, with a hard Franco-German core and looser appendages.
In effect, a confederal approach to European unity should be pursued that addresses both intergovernmental relations and relations between the different European societies and their values—as opposed to a strong federal approach. The European Union, for example, needs to eliminate its "democratic deficit" and to decentralize some of its decision-making in accord with its own doctrine of subsidiarity, in attempting to better balance local, national, and European interests and concerns—so as to permit greater local and national input into the EU decision-making processes.
One possibility is, thus, for the full or "core" members of the European Union to create a new category of "Associate Membership." European or even non-European states with important financial, trade, or even military interests in Europe—and which are willing to contribute to EU activities and responsibilities—could, if accepted, become Associate Members that possess limited membership rights. This Associate Membership category could possibly include limited voting rights in specific areas of mutual interests (thus going beyond the status of partnership). In such a way, Russia, Ukraine, and Turkey could become Associate Members—particularly if they work with the Europeans and the Americans to establish a new Black Sea regional peace and development community, for example.
Even though Turkey remains a NATO member, it is highly unlikely that it will become a full EU member under present EU rules. But Turkey could become an Associate Member if the European Union begins to reform itself significantly after Brexit. Many politicians on all sides of the political spectrum have argued that it is time to end European hypocrisy and tell Ankara that it will never join the European Union. But this statement implies that the European Union itself will not reform and that it will not eventually morph into a new international regime after Brexit, with core and associate members. Despite President Erdoğan's significant turn toward authoritarianism and repression in Turkey, Berlin, for example, has thus far wanted to keep the door open to Turkish membership in the European Union.
From this perspective, the European Union will need to readjust its EU Partnership program so as to include Russian, Ukrainian, and Turkish political-economic interests. EU political-economic deals with Belarus and other states will also need to find ways to accommodate Moscow's interests as well. The Ukrainian financial crisis will prove almost impossible to resolve without Russian (and Chinese) finance along with European and American funding. This all implies the need for the Trump-Pence administration to work with the Europeans to reform, not abolish, the European Union. Otherwise, there is a real possibility of a new partition of Europe, running through eastern Ukraine.
**THE QUESTION OF TURKEY**
As argued above, it is highly unlikely that Turkey will become an EU member—unless both the European Union and Turkey begin to reform themselves significantly after Brexit along the lines proposed in this chapter—given the extremely poor relations between the United States, the European Union, and Turkey in recent years. The fact is that Turkey, which is a NATO member but not an EU member, has made it very difficult for NATO and the European Union to fully cooperate. In particular, the need to resolve the intra-NATO dispute between Greece and Turkey over Cyprus, among other issues, is one of the keys to improve Turkish-EU relations.
Turkish-EU relations have been in a bad state for years, given the EU refusal to accept Turkey as a member due to its large population, its predominantly Muslim culture, and its threats to bring back the death penalty. These issues have been made even worse in the aftermath of the immigrant crisis, which began in 2011, following the Arab Spring movements. This crisis led Ankara to threaten that it might unleash thousands of non-European immigrants into the European Union. In addition, since the July 2016 Gülen coup attempt, Turkey's Prime Minister Erdoğan has taken a significant turn toward authoritarianism, or "illiberal democracy," given his still high level of popular support.
Nevertheless, it is crucial that Turkey plays a balancing role between the United States/NATO, Russia, and the Europeans. But here, even if Ankara continues to veer toward authoritarianism and "illiberal democracy," there may still be ways to mitigate its authoritarian tendencies and reconcile Europe and Turkey. One way would be to negotiate a federated accord between Greek and Turkish Cypriots, guaranteed by Greece, Turkey, and the United Kingdom under UN auspices. A UN-brokered deal between Greek and Turkish Cyprus that could resolve property disputes and population displacement could then permit Cyprus to become the first major outlet of maritime trade outside the Suez Canal. And the European Investment Bank/European Bank for Reconstruction and other funds could help develop massive energy reserves in the eastern Mediterranean. If proportioned carefully and fairly, the vast reserves of the eastern Mediterranean could then benefit littoral states, including Greece, Turkey, Israel, Egypt, Palestine, Lebanon, and Syria.
Such an accord could open up Turkey to tremendous trade opportunities with Greece and the European Union, while likewise more closely linking the European Union with NATO member Turkey, in order to limit Turkey from shifting too close to Moscow. Nevertheless, Turkey could still play a key role in bringing NATO, the European Union, and Russia into a closer confederated relationship. After resolving the Cyprus issue, it might prove easier to resolve somewhat-similar disputes in the Caucasus, if not in Crimea as well. Once again, this appears plausible if, after Brexit, the European Union could forge differing membership criteria that would permit Turkey, Russia, and other states to enter the European Union in a new form of associate partnership. As previously argued, Associate Members of the European Union could possess limited voting rights on certain issues that would not be related to their size of population and would possess clearly defined responsibilities related to those issues.
**COOPERATION IN THE GLOBAL WAR ON TERRORISM**
Donald Trump has made US-European-Russian cooperation in the Global War on Terrorism one of the major themes of his presidency. This is a legitimate goal, if carried out appropriately, but one that has proved very difficult to implement, particularly given the rise of strong anti-Russian sentiment in the United States and the difficulties involved in finding ways for Russia and the United States to agree about what to do with the Assad regime in Syria. Despite the difficulties, a truly concerted approach involving Russia and the other major and regional powers is crucial for preventing the further spread of the Islamic State, al-Nusra, and other anti-state, partisan "terrorist" groups.
This approach seeks ways to put an end to what is essentially a proxy war between Saudi Arabia and Iran throughout the wider Middle East. What is needed is joint US-European-Russian cooperation to press for a cease-fire in Syria and Yemen and to eventually press Saudi Arabia and Iran into cooperation, thus putting an end to their proxy wars and the regional arms race. These steps will require finding ways to balance conflicting domestic socio-political groups, while co-opting some Islamist parties and isolating or destroying others. The key dilemma will be finding an accord with Iran and Hezbollah. If it is given some encouragement and incentives by the European Union, among other incentives from the United States, as argued above, then Turkey could play a key role in helping to mediate between Saudi Arabia and Iran, as well as between Russia and the United States.
One option may be for Syrian leader Bashar al-Assad to take asylum in either Russia or Iran in order to permit the formation of a new Syrian governmental coalition backed by the United States, Europeans, and Russia. This is because most of the Syrian opposition has thus far refused to participate in peace talks that permit the Assad leadership a future role in the governance of the country. On the other hand, a resolution to the Syrian conflict that could result in a loose confederation, or else a possible partition, might mitigate the need for al-Assad to step down, as he would hold power in the Alawite regions along the coast.
Yet the only way Syria could be held together in a confederal arrangement would probably be if some form of joint Arab-Turkish or international peacekeeping force could be deployed under a general UN mandate. The dilemma is that Turkish forces have threatened to move into Syria to check the formation of separate Kurdish enclaves that could, Turkey fears, support Kurdish independence movements inside Turkey. This means the United States, Europeans, and Russia need to find some mutual accords between Syrian Kurds and Kurds in Turkey itself. In this view, Ankara and differing Kurdish movements in Syria, Turkey, and Iraq need to forge Kurdish autonomy agreements, with US, EU, and Russian backing.
Here, instead of demanding full independence, which is nearly impossible to sustain in a highly interconnected world, and particularly as a landlocked society, Kurdish parties in Syria, Iran, and Turkey could opt for autonomy arrangements, much as the Kurds have achieved in Iraq after the 2003 US-led military intervention. Yet the regional situation will prove even more complicated and dangerous if Iraqi Kurds continue to demand independence from the Iraqi federal government due to disputes with Baghdad over the issues of autonomy and taxation in the aftermath of the nonbinding September 2017 referendum on the independence of Iraqi Kurdistan. Iraqi Kurd leader Marwan Barzani claimed victory for independence demands even before the ballots were counted, while Ankara and Baghdad threatened strong sanctions. Baghdad, backed by Iran, then began to seize Kirkuk and other oil-rich territories formerly controlled by the Kurds, potentially opening the door to a new regional conflict. (See chapter 8.)
**ISRAEL AND PALESTINE**
Trump was furious when the Obama administration had refused to veto the UN Security Council's resolution that condemned Israeli settlements in the West Bank and East Jerusalem. Obama's ambassador to the United Nations, Samantha Power, abstained from voting, on the basis that Israeli irredentist claims and plans to expand settlements in Palestinian territory could undermine any prospect of reaching a possible "two-state" solution to the ongoing conflict. At that time, a neophyte Trump supported not only expanding Israeli settlements but also Israeli claims to Jerusalem as the Israeli capital.
To obtain Jewish American votes, Trump, like many presidents before him, had suggested moving the US embassy to a new Israeli capital in Jerusalem. Trump has, however, appeared to back off on such a position after becoming president, just as he has on so many other questions. King Abdullah II of Jordan purportedly warned Trump that moving the embassy to Jerusalem could threaten the two-state solution, exacerbate the Israeli-Palestinian conflict, and augment recruits for the Islamic State. At the same time, Jordan is concerned that the radicalization of the Palestinian movement might lead to secessionist or Islamist Palestinian movements inside Jordan, due to the large number of Palestinian refugees (more than two million) who have settled there. The president of the Palestinian Authority, Mahmoud Abbas, also warned Trump against supporting Israeli claims to all of Jerusalem.
And while Trump appears to have backtracked on his previous support for the renewed expansion of Israeli settlements in the West Bank, the Trump administration has not yet developed a full-fledged policy—except that it could be secretly pressing for a rapprochement between Saudi Arabia, the United Arab Emirates, Egypt, and Israel. Trump has met with both Israeli Prime Minister Benjamin Netanyahu and President Mahmoud Abbas, the Palestinian leader of the West Bank, but it is still not clear what kind of accord can be achieved. To achieve peace, Trump will need to eventually press both sides into a settlement. Here, however, Trump may be pressing Israel and Saudi Arabia into a deal that would take Palestinian interests only into limited consideration in the effort to forge a united front against Iran. If so, such an unholy alliance could enrage many Islamist groups throughout the Arab/Islamic world.
Yet as an alternative strategy to achieve peace in the long term that would not rely primarily on Saudi Arabia, the United States could reach out to both Saudi Arabia and Qatar for diplomatic support and financing through bilateral and multilateral diplomacy, in an effort to press Israel to accept a modified Arab peace plan with further negotiations. This could open the door to greater Arab investment not only in Palestine (which would also expand Israeli markets) but also in Israel itself. The Arab Gulf states have agreed to set up telecommunication lines with Israel, open trade negotiations, and allow Israeli planes to fly over their airspace. In exchange, Israel would need to freeze settlement construction in the West Bank and relax trade restrictions with the Gaza Strip.
The two-state solution is the Arab and the UN-sponsored approach. There have been reports that the president of the Palestinian Authority, Mahmoud Abbas, might present a plan to the Israelis in which the Palestinians would give up 6.5 percent of their lands to Israel—three times as much as previously offered. The proposal would not include Jerusalem. In exchange, one option is for Israel to return to the Palestinians land equivalent to 5.8 percent of the West Bank, along with lands that would connect the West Bank to the Gaza Strip.
The political dilemma is that Benjamin Netanyahu appears determined to implement a "one-state" solution with Saudi support. President Trump, after his meetings with the Israeli leader, publicly voiced the option of a one-state solution. While it appears feasible that Saudi Arabia, the United Arab Emirates, Egypt, Bahrain, and Jordan might support such an approach, it appears unlikely that the rest of the Arab world, Iran, and the divided Palestinians would acquiesce.
A third option is a confederal solution, but as a slight modification of the two-state concept. This is feasible given the fact that Fatah in the West Bank and Hamas in Gaza have largely separate leaderships and will need to forge power-sharing accords, perhaps like those in Northern Ireland. At the same time, Hamas has been trying to change its image in part due to its own mismanagement of Gaza. Israeli sanctions are not the only factor undermining the full development of the Gaza enclave. The fact that Hamas also opposes groups like the Islamic State also leads it to seek new backers. Concurrently, the rivalry between Saudi Arabia and Qatar impacts Hamas, but also represents a major issue over which Doha and Riyadh could find compromise, particularly if assisted by US and multilateral diplomacy. (See chapter 8.)
An Israeli-Palestinian peace settlement is not a panacea that will suddenly put an end to the Global War on Terrorism, and it will not put a sudden end to the conflicts that have been raging between Shi'a and Sunni, between Kurds and Turks, and among differing Sunni factions. But it would help bring the majority of "moderate" and undecided individuals and groups of Arab/Islamic background (plus others who may support Palestinian rights but not Islamist movements) closer to the positions of the United States, Europeans, and Russia. Concrete steps toward a full-fledged Israeli-Palestinian peace settlement would begin to socially and politically isolate those who continue support violent pan-Islamist movements.
Without concrete progress toward resolving the Israeli-Palestinian conflict, Trump will find it very difficult to forge a concerted approach to the Global War on Terrorism that involves both major and regional powers—including those states most impacted by differing Islamist movements. Trump's rhetoric has only tended to inflame the already-grave crisis.
**THE IMPORTANCE OF THE IRAN NUCLEAR ACCORD**
What is needed to resolve the crisis with Iran is a new regionally based diplomacy, backed by Washington, that brings Iran into differing multilateral discussions with Saudi Arabia, Turkey, Iraq, the Kurds, the Palestinians, and Israel, if possible, among other states in the region. Placing heavy political and economic sanctions and military pressures on Tehran in the effort to force it to stop developing its ballistic and cruise missile capabilities and to stop it from supporting Hamas, Hezbollah, and the Houthis in Yemen, as Trump has proposed in opposing the Iran nuclear accord or Joint Comprehensive Plan of Action (JCPOA), will not succeed without regional negotiations. Nor will such pressures help deal with the Kurdish independence question. In addition, the possibility that the United States, Saudi Arabia, and Israel will focus primarily on the Iranian "threat" will tend to sidetrack the Global Coalition against Daesh from fully focusing on efforts to destroy the Islamic State and al-Qaeda—while undermining the possibility of a nuclear accord with North Korea.
First, Iran, Turkey, and Iraq need to find ways to compromise with Iraqi Kurdish demands for independence through mediating the formation of loose confederal arrangements among Turkish, Iraqi, and Iranian Kurds, for example, but without engaging in a major alteration of borders. With respect to Iranian support for Hamas, Washington needs to encourage Israel, Saudi Arabia, Qatar, Iran, and other states, to settle their differences with respect to the Palestinian "two-state" solution and toward Hamas in particular. By the same token, the questions of Hezbollah and Houthis need to be addressed in the process of settling the ongoing wars in Syria and Yemen, by means of finding a domestic power-sharing settlement in Yemen, as proposed by the United Nations. (See chapter 8.) Evidently none of these proposals represent an easy process, but they can be dealt with only by diplomacy, not by force.
With respect to the Iranian nuclear issue, it is highly unlikely that Israel would give up its purported "existential" nuclear deterrent, as demanded by most Arab states and Iran. Turkey, Saudi Arabia, and Egypt have all threatened to develop a nuclear weapons capability—particularly if Iran acquires a nuclear weapons capability in addition to Israel. But given Israel's intransigence to even discuss its nuclear weapons capabilities, another possible option is to pursue an agreement involving a "no-first-use" of weapons of mass destruction (WMDs) for all states in the wider Middle East region. Such a "no-first-use" of WMDs accord would permit states in the region to engage in a new strategic dialogue that could eventually result in the control and reduction of conventional weaponry, including ballistic and cruise missiles. Washington could help initiate multilateral negotiations intended to limit missile capabilities through the Missile Technology Control Regime (MTCR), for example, while seeking compromise on a number of geopolitical disputes. And instead of seeking an Israeli-Saudi alliance against Iran, the United States should not drop out of the Iran nuclear accord; it should try to improve the JCPOA verification procedures and its implementation—but gradually, as confidence is restored over time.
A Trump administration and US congressional effort to suddenly try to renegotiate perceived weaknesses in the JCPOA nuclear accord will not succeed—as it will prove very difficult, if not impossible, to restore mutual confidence between the United States and Iran, whose revolutionary guards have threatened war if Washington labels them as a terrorist organization. (See chapter 8.) It is thus up to Washington to build upon the JCPOA, not by undermining that treaty, but by negotiating a new series of multilateral treaties. New multilateral treaties would seek to establish an agreement over the "no-first-use" of WMDs; the control and limitation of ballistic and cruise missile capabilities; joint development of alternative energies, such as solar and geothermal, in an effort to reduce Iran's demand for nuclear power; plus diplomatic agreements that seek to ameliorate regional rivalries (between Israel, Saudi Arabia, Turkey, Iraq, the Palestinians, the Kurds, and Iran, among other states and populations) that have resulted in horrific acts of state-supported and anti-state terrorism.
The JCPOA can significantly reduce the threat of further nuclear proliferation throughout the wider Middle East, but the treaty will only work in the long term if the United States, Israel, Saudi Arabia, and Iran can sustain full confidence. Trump's demands that the JCPOA Iran nuclear accord be renegotiated is doomed to failure and will only exacerbate the conflicts that are now confronting the wider Middle East and the world—and could lead other regional states to develop a nuclear deterrent, in addition to North Korea.
**INDIA, PAKISTAN, AND KASHMIR**
Areas of mutual US-Russian-Indian interest need to be explored for the development of joint policies: A Taliban victory in Afghanistan, for example, does not appear to be in Russian, Chinese, or US/NATO interests. This represents a major area in which these three sides could potentially find ways to cooperate. India could also engage more effectively in a Contact Group to help resolve disputes in Central and Southwest Asia, if not the Indo-Pacific as well. At the same time, however, Moscow has opened up talks with the Taliban in expectation of NATO withdrawal and an eventual Taliban victory. But India would most likely oppose recognition of the Taliban, given the latter's links to Pakistan. These issues need to be thoroughly discussed in a new Contact Group format.
The fact that India joined the Chinese-led Shanghai Cooperation Organization in 2017, and that India has sought to speed up the signing of a free-trade agreement (FTA) with the Russian-led Eurasian Economic Union, which includes Belarus, Kazakhstan, Russia, Armenia, and Kyrgyzstan in 2017, has raised questions as to whether India will move out of relative neutrality and move closer toward a Eurasian alliance—despite its ongoing disputes with Pakistan and China. Given the development of closer defense ties with the United States and Japan, how India's relations with both Russia and China will develop remains to be seen. (See chapter 7.)
Yet instead of siding with one alliance or the other, India could play a mediating and balancing role. But this will also require India-Pakistan reconciliation over Kashmir and other issues. Here New Delhi has been reticent to engage in such discussions, whether mediated by the United States or more recently by China. As the situation in Kashmir appears to be getting worse, one option is to propose joint Pakistani-Indian sovereignty agreements. But this will prove difficult to implement, given the rise of militant groups that seek total independence—as if total independence would be possible in today's highly interconnected world.
Could India shift toward Russia and China despite the apparent development of even closer Russian-Chinese ties to India's rival Pakistan and to the Taliban? Or could China, as it seeks to develop areas in Pakistani-controlled Kashmir and in Pakistan itself, mediate the Indo-Pakistan dispute over Kashmir, for example, as has been suggested by the Chinese media, but opposed by New Delhi? Or will India continue to tilt toward the United States, Europe, and Japan? Or could New Delhi eventually play a mediating role between the United States, Europeans, and Japan on the one hand, and China and Russia on the other?
**TOWARD PEACE IN ASIA**
In May 2016, Japanese Prime Minister Shinzo Abe advocated an eight-point plan for Vladivostok that was designed to ameliorate Russia-Japanese tensions. His goal was to "make Vladivostok a city that links Eurasia and the Pacific Ocean.... Let's diversify Russia's industry and make it more efficient. Let's turn the Russian Far East into an export base for the entire Asia-Pacific region." As this action seemed to break US- and EU-led sanctions on Russia, it was purportedly opposed by President Obama.
These accords include a joint $1 billion Russian-Japanese investment fund between Russian Direct Investment Bank and Japan's Bank for International Cooperation; a special system for joint economic activity for Kurile Islands/Northern Territories; and an easing of travel restrictions to the islands. The ultimate goal is to achieve a peace treaty—but there was no accord signed that would resolve the territorial dispute. Nevertheless, Putin did admit, "that the absence of a peace treaty between Russia and Japan is, of course, an 'anachronism.'"
To prevent the real possibility of conflict in the Indo-Pacific, particularly given the North Korean threat, Japanese efforts to seek out a rapprochement with Russia eventually need to be backed (at least tacitly) by the Trump administration with a focus on joint US-Russian-Japanese energy projects on Sakhalin island, among other bilateral Russian-Japanese accords. Given the fact that Tokyo took the first step in engaging in a rapprochement with Russia after the sanctions placed on Moscow in 2014 (after Russia's annexation of Crimea and political-military interference in eastern Ukraine), the first steps toward a US-Russian entente could possibly take place in Asia, and in particular by working cooperatively to resolve the North Korean nuclear crisis. (See chapter 7.) But steps toward a full US and Japanese rapprochement with Russia will prove very difficult without a political and legal settlement with respect to Ukraine and Crimea.
Concurrently, the United States, ASEAN states, Japan, Russia, and China will need to work out new joint development projects over disputed islands in the South China Sea and East China Sea. New multilateral agreements could involve joint naval patrols and combined task forces in combating terrorism, piracy, and trafficking of illicit materials (to North Korea, for example). Trump's offer, on his trip to Asia in November 2017, to negotiate island disputes in the South China Sea between China and its neighbors appears extremely positive—assuming Trump will actually provide the State Department with sufficient resources to do the job correctly.
But the possibility of a concerted approach has thus far been undermined by the buildup of tensions with North Korea and the inability of the United States and China to coordinate strategy, while also involving Moscow. Here, Japanese and South Korean efforts to engage in positive relations with Moscow, through trade and investments in Russia, for example, are crucial to obtain Russian assistance in helping to settle the North Korean crisis. These steps in Asia could eventually open the door to much stronger US-EU-Russian relations in Europe—that is, if Trump does not also inadvertently destabilize relations with South Korea by renegotiating or terminating the US-South Korean free-trade agreement (KORUS FTA) because of a $17 billion US trade deficit with Seoul in 2016. (See chapter 7.)
**THE CRISIS OF NORTH KOREA**
Finding a common policy among the United States, China, Russia, South Korea, and Japan toward North Korea is absolutely crucial to preventing war on the Korean Peninsula. (See chapter 7.) The United States, Russia, China, Japan, North Korea, and South Korea had already taken part in six-party talks from 2003 to 2009 in an effort to denuclearize North Korea. The problem now is how to reinitiate those Contact Group talks that had stalled in the aftermath of renewed nuclear and missile testing by North Korea, and in the aftermath of the US refusal in September 2005 to provide North Korea with light-water nuclear power plants as Pyongyang requested.
Although North Korea and China are said to possess a closely interrelated relationship of "lips to teeth," Beijing does not support North Korea's nuclear provocations, as it would prefer to preserve its own nuclear hegemony over Pyongyang. In effect, Beijing would prefer the "denuclearization of the peninsula," But this demand might not prove realistic. Beijing has thus far tried to take a balanced position between North Korea and the United States. Beijing fears that an effort to engage in a complete embargo on coal and food supplies or to engage in regime change, polices proposed by the United States, will eventually destabilize the country and thus undermine its sphere of security and influence against Japanese, South Korean, and American political-economic and military influence, while also undercutting Chinese economic interests within the country.
For its part, the United States is prepared to impose sanctions not only on North Korea but also on other countries where companies or individuals are known to have helped North Korea's military programs. This is in accord with UN Security Council Resolution 2270, which Beijing had supported in the UN Security Council, and which gives a mandate to all countries to search every ship and aircraft coming in or out of North Korea to make sure there are no illicit goods and to prevent nuclear proliferation. The latter UN sanctions are important in that they impact the Strategic Rocket Force of the Korean People's Army and systems of illicit financing arranged by the Koryo Bank, in addition to sanctions placed on certain individuals. Both China and Russia did agree to consider even tougher UN sanctions, and they could possibly consider an oil embargo. But so far, sanctions do not seem to be stopping North Korea from testing its missiles.
Beijing signed the UN Security Council Resolution 2270 on sanctions, yet it does not believe sanctions alone are sufficient. China would thus prefer to press Pyongyang to freeze its missile and nuclear weapons programs in exchange for a halt to US-South Korean annual drills and freezing the deployment of missile defense systems—positions thus far opposed by Washington. China does not want war, nor does it want a destabilization of the Korean Peninsula, which could result in a massive refugee crisis. Nor does Beijing want a South Korean buyout of the North in which South Korea, backed by the United States, would dominate North Korea in a way similar to how West Germany, backed by the United States, bought out East Germany. At the same time, Beijing fears that a nuclear North Korea with effective delivery capabilities could provoke Japan and South Korea to obtain nuclear weaponry—as is being proposed by both Japanese and South Korean nationalist groups. Despite Chinese objections, Kim Jung Un appears determined to keep North Korea's nuclear weaponry as a deterrent against any rival, Japan, South Korea, the United States—and potentially China itself.
China's foreign minister, Wang Yi, has been worried that the region is faced with a "precarious situation" in which "one has the feeling that a conflict could break out at any moment.... If a war occurs, the result is a situation in which everybody loses and there can be no winner." In addition to urging both North Korea and the United States to calm down, Wang Yi has also urged Moscow to play a diplomatic role as well. Both Beijing and Moscow hope to revive six-party talks over North Korea—if the Trump administration will agree.
Yet the Trump position, as thus far formulated by Rex Tillerson, is that the United States would not negotiate with Pyongyang unless it first gave up both its nuclear and missile programs. This is a nonstarter, as North Korea has already declared itself to be a "nuclear state" much as NATO has declared itself to be a "nuclear alliance." President Trump and his Secretary of State Rex Tillerson have accordingly tried to put the onus on China to resolve the Korean issue through diplomatic pressure and economic sanctions. Here, Trump has hoped to attract China by offering a good trade deal if Beijing helps out with North Korea. At the same time, Trump has threatened to act alone militarily if Beijing does not put sufficient pressure on the North Korean regime to prevent it from expanding its nuclear and missile-delivery programs. In this regard, South Korean government officials and the Pentagon have discussed the option of deploying US tactical nuclear weapons in South Korea to counter the North Korean nuclear threat—an option that could cause a dangerous counter-reaction by Pyongyang.
As previously argued, Trump and Kim have entered into a personal radioactive pissing match without any end in sight. This was indicated by Trump's visit to South Korea in November 2017, during which Trump once again warned North Korea (but in less apocalyptic language) not to test the resolve of _his_ administration, which he claimed was tougher than past US administrations. But for Trump, the only way for the United States to begin negotiations with North Korea is for Kim Jung Un to take the first step, by stopping the development of his nuclear missiles and by accepting a "complete, verifiable, and total denuclearization."
Once again, it is dubious North Korea will accept Trump's formula. Nevertheless, from the perspective of China, and possibly for North Korea as well, the option of denuclearization could prove feasible, but only if the United States denuclearizes its own armed forces in the area or if the two sides can engage in step-by-step downsizing of their conventional and nuclear capabilities with international verification. Chinese Foreign Minister Wang Yi has accordingly called for the denuclearization of the entire peninsula and for an end to US-South Korea military exercises: "The use of force does not solve differences and will only lead to bigger disasters.... [China should not be] a focal point of the problem on the peninsula.... The key to solving the nuclear issue on the peninsula does not lie in the hands of the Chinese."
While Beijing has wanted Washington to engage in quiet talks with Pyongyang, Rex Tillerson has replied that it was up to North Korea to take the first concrete steps, not Washington: "We will not negotiate our way back to the negotiating table.... We will not reward their bad behavior with talks." As Tillerson indicated on Trump's tour of the Asia-Pacific in November 2017, the United States has accordingly been reluctant to initiate talks with Pyongyang—as it has not wanted to be seen as caving to North Korea, which could reinitiate nuclear weapons and missile tests unexpectedly, even if peace talks were announced. Tillerson had previously affirmed that the United States is not seeking "regime change" in North Korea, even though it is not absolutely certain that Trump is in full agreement. (See chapter 7.) In his November 2017 trip to Asia, Tillerson then argued that he could envision the possibility that the United States and North Korea could agree to hold talks at some point, but only as a precursor to formal negotiations. Tillerson stated that he would seek a signal from Kim Jong Un himself indicating that he would like to have some type of a meeting, communicating through one of the two or three communication channels that the United States possesses with North Korea. But Tillerson also forewarned that the United States would continue to threaten North Korea as long as Kim did not give up his nuclear weapons program.
US policy is faced with a real conundrum. On the one hand, Washington wants to halt North Korea's nuclear and missile program while also "containing" the ambitions of both China and Russia. On the other hand, Washington also wants Russia to resolve its dispute with Ukraine on Washington's terms and argues that China should be the prime agent in resolving the dispute with North Korea. This evidently makes positive support from both Moscow and Beijing difficult to obtain. Nevertheless, China appears to possess greater incentive to act in the North Korea case because it feels more threatened by a potential war on the Korean Peninsula than Moscow feels threatened by a potentially wider war over Ukraine that could encompass the Black Sea and eastern European regions.
Another dilemma is that the Trump-Pence administration needs to figure out how to demonstrate a minimum show of American force as a deterrent in support of South Korea and Japan, while also engaging in an informal dialogue with Pyongyang—an option that has been urged by Beijing for the last couple of years, but had been refused by the Obama administration, and thus far by the Trump administration. Without real dialogue with Pyongyang, the Trump administration could fail to prevent a nuclear war. Such an initial dialogue, which need not be interpreted as capitulation, could then help strengthen the Chinese position and permit Beijing apply more pressure, including tighter sanctions, on North Korea, if necessary. The problem remains: How much should Washington rely on Beijing to do the negotiating? And how much of the negotiating with Pyongyang should be led by Washington? Beijing has purportedly told Washington that it had informed North Korea "that if they did conduct further nuclear tests, China would be taking sanctions actions on their own." And in November 2017, Chinese President Xi Jinping did say he was unequivocally against North Korea sustaining a nuclear capability.
At some point in the near future, the United States should begin to offer Pyongyang some practical proposals through quiet diplomacy. Washington should promise to normalize US-North Korean relations and no longer threaten preemptive strikes against Pyongyang's nuclear program, if North Korea would agree to a "nuclear freeze," rejoin the Non-Proliferation Treaty, and permit comprehensive inspections by the International Atomic Energy Agency (IAEA), along the lines of Iran JCPOA nuclear accord. (See chapter 8.) Hopefully, North Korea could then promise to eventually eliminate all of its nuclear weapons, but most likely only in exchange for firm joint security guarantees from the six powers. A step-by-step process of gradual demilitarization by both the North and South Koreans, in which both sides would then begin to withdraw their forces from the Demilitarized Zone, could then be matched by a step-by-step withdrawal of US forces and elimination of economic sanctions on North Korea. The United States would be prepared to return to the assistance of South Korea if necessary but would work with all the permanent members of the UN Security Council to provide joint security assurances for North Korea. This would ensure that North Korea remains a "buffer" between China and South Korea and that South Korea (whose economy is fifty times the size of North Korea's) would not buy up the North as West Germany bought up East Germany. Both North and South Korea could then enter into a very loose confederation until other political arrangements could be made. In the meantime, the United States, South Korea, China, Japan, and Russia would fund North Korean energy needs and development, while seeking ways to decommission North Korea's heavy investments in the military (Pyongyang may spend as much as 22 percent of its GDP on the military, at the expense of the population) and convert its economy to more useful and productive consumer activities. Hopefully this approach will prove feasible, as even a very limited and inspected North Korea nuclear deterrent, as perhaps demanded by Pyongyang, appears unacceptable to Washington, Seoul, Tokyo, and Beijing.
Trump could eventually propose something like this in meeting with Kim Jung Un. The realist side of Trump has stated that he would be "honored," to meet directly with Kim Jung Un, but "under the right circumstances." Yet Trump also stated: "There is a chance that we could end up having a major, major conflict with North Korea. Absolutely, We'd love to solve things diplomatically but it's very difficult."
The United States needs to take a less self-righteousness approach and engage in a real and intensive dialogue—which is the only way to end the impasse and prevent the real risks of either a collapse of the North Korean regime or a possible regional war. It seems informal US-North Korean dialogue leading to a direct meeting between Trump and Kim Jung Un could begin to provide North Korea with the international respect and attention that it has been craving in the aftermath of the 1950–1953 Korean War, in which the United States devastated the country with conventional bombs and Napalm while threatening the use of nuclear weaponry. International aid and assistance could then help to develop the country in exchange for step-by-step demilitarization and informal accords between North and South Korea. North Korea would then be backed by mutual security assurances by the United States, Russia, China, France, and the United Kingdom (as outlined above).
The question remains, Will Trump eventually meet with the North Korean leader in direct bilateral diplomacy if North Korea does eventually agree to a freeze on its nuclear and missile tests? Could this result in US capitulation to North Korean demands? Or could it result in a mutual compromise, which could then prevent the real possibilities of a more general conflagration? No one has yet accused Trump of seeking business deals with North Korea or seeking to construct a Trump Tower in the midst of Pyongyang, but would anyone really complain if he is able to make peace with Kim Jung Un by way of eliminating North Korea's nuclear threat? But can he eliminate that threat without provoking war?
Trump's proposed military buildup and American First nationalism appear intended to press from a position of strength Moscow, China, Iran, Syria, and North Korea, among other countries, into a settlement of a number of disputes. But is Peace through Strength to be achieved on American terms, in accord with a posture of power-based bargaining? Or will such power-based bargaining bring about mutual compromise? Will one side or the other capitulate? Or will major power war be the consequence?
Former Soviet President Mikhail Gorbachev has warned that "no problem is more urgent today than the militarization of politics and the new arms race. Stopping and reversing this ruinous race must be our top priority." Gorbachev has proposed "that a Security Council meeting at the level of heads of state adopt a resolution stating that nuclear war is unacceptable and must never be fought. I think the initiative to adopt such a resolution should come from Donald Trump and Vladimir Putin—the Presidents of two nations that hold over 90 percent of the world's nuclear arsenals and therefore bear a special responsibility."
**TOWARD NUCLEAR AND CONVENTIONAL ARMS REDUCTIONS/ELIMINATIONS**
Because the United States and Russia have entered into a new nuclear and conventional arms race since at least 2002–2008, the dilemma is that the more the Ukrainian crisis blocks Russia from cooperating fully with the United States and the European Union, the more it will prove difficult to cooperate on other issues of common concern, including the need to reduce arms expenditures, concentrate on social and environmental concerns, and deal with the dangerous crisis unfolding on the Korean Peninsula. (See chapters and .)
Nevertheless, as soon as is politically possible, in addition to the need for further discussions on reductions of long-range ballistic missiles going beyond New START (Strategic Arms Reduction Treaty), which was implemented in 2011 and which is hoped to last at least until 2021, there should be full-fledged discussions to reduce, if not eliminate, all tactical nuclear weaponry as soon as politically possible. These arms reduction/elimination talks could take place in the aftermath of a US-Russia summit.
On the one hand, Moscow has been enhancing its A2/AD (Anti-Access/Area Denial) tactics in Kaliningrad by deploying S-400 air defense missiles and tactical, nuclear-capable, Iskander surface-to-surface missiles, plus shore-based cruise missiles, so as to block NATO from potentially resupplying the Baltic states in case of war. Russia has developed a new Satan missile, while Washington has accused Moscow of violating the 1987 Intermediate-Range Nuclear Forces Treaty (INF Treaty) by testing a new intermediate-range missile, which, if true, could set US/NATO-Russian relations back to the 1980s, when NATO opted to counter Soviet intermediate-range SS-20s with the deployment of cruise and Pershing missiles before the 1987 INF accord was signed.
For its part, the Pentagon has been modernizing the B61-12 tactical nuclear weapon and extending its rage, even when the utility of such weaponry has been questioned, and when its deployment could be considered in violation of nuclear non-proliferation treaty. The United States has also been modernizing its launch capabilities for missiles with both conventional and nuclear warheads, such as the Ground-Based Strategic Deterrent and the Long-Range Standoff Cruise Missile. In addition to deployments in Guam, the United Arab Emirates, Turkey, and South Korea, and possibly Japan, the Pentagon has been considering deploying THAAD missile defense systems and penetrating radar in Europe and Saudi Arabia.
A first step would be to take the roughly two thousand US and Russian warheads off "high-alert" status. Both Moscow and Beijing, in a sign of strategic-nuclear insecurity, have put a large number of their nuclear missiles on hair-trigger alert in order to maintain the survivability of their nuclear weapons against superior US delivery nuclear systems. So as to reduce tensions, Washington and Moscow could furthermore agree to a mutual "no-first-use" of nuclear weapons. These steps could be strengthened if Washington considers eliminating land-based elements of the nuclear triad so as to reduce some pressure on Moscow to feel that it must launch its missiles "on warning." This approach needs to be taken, but without undermining a strong US air- and sea-based deterrent that should remain in the background. Similar confidence measures will prove necessary with Beijing, which opposes the deployment of THAAD defense systems in South Korea and potentially Japan. Both China and Russia fear that the United States can use its missile defense (MD) systems and advanced missiles to launch a preemptive strike.
A thorough US-Russian discussion of tactical and intercontinental nuclear systems could be accompanied by potential compromises on the deployment of US missile defense systems in Europe. The United States could agree to removing all provocative forward-deployed tactical nuclear weapons from Europe, such as the B61-12, in exchange for significant reductions in Russian nuclear weaponry, while also reducing conventional capabilities on both sides. This approach would likewise mean bargaining within NATO—as NATO's consensus-based decision-making process means that small powers most interested in keeping nuclear weapons in Europe, such as the Baltic states, can oppose a decision to remove those weapons.
With respect to MD, the fact that Washington had initially justified MD deployments in eastern Europe based on the fear that Iran would eventually obtain a nuclear weapons and long-range missile capability should open the door to discussions in the aftermath of the Joint Comprehensive Plan of Action (JCPOA) nuclear deal with Iran. (See chapters and .) To reach a compromise, the United States and Russia should revisit some of the previous proposals for joint missile defense systems that were proposed before UN-backed Contact Group negotiations pressed Iran to give up on its nuclear program.
There has been almost no strategic nuclear confidence between Washington and Moscow since the United States unilaterally dropped out of the Anti-Ballistic Missile Treaty (ABM Treaty) in 2002 without seeking a substitute treaty on missile defenses with Moscow. Here, the Bush Jr. administration argued that the spread of nuclear missiles made the treaty obsolete but did not discuss the matter with Moscow to see if the treaty could be revised in a new format. This is the major strategic issue that helped set off the new arms race, and it will still take some time to build up trust. And Moscow will not give up unilaterally or even compromise without very tough power-based bargaining. Nevertheless, these proposals are not impossible to implement.
**FULLY ABIDING BY THE NUCLEAR NON-PROLIFERATION TREATY**
Gorbachev's point that Presidents Vladimir Putin and Donald Trump need to take the first steps toward banning nuclear warfare is well taken, but neither Trump nor Putin appear ready to move in this direction, given mutual imprecations between the United States and Russia, American accusations of Russian meddling and use of cyber-espionage in the US election process, and Russian counteraccusations of US meddling in Russian affairs, among the many other reasons discussed in this book. (See chapters and .) The dilemma is that US-Russian mutual imprecations only serve to enhance the power and influence of the US military-industrial-congressional complex and its Russian equivalent, Siloviki—Putin's prime supporters—as they both seek to build more and more powerful weapons in a very unbalanced game of terror.
Nevertheless, this fact should not prevent the world's populations and legislatures from pressing Washington, Moscow, Beijing, France, and the United Kingdom, among the other declared and non-declared nuclear powers, to work toward the reduction, if not the elimination, of nuclear weaponry—as is, in fact, demanded of nuclear-weapons states by Article VI of the Nuclear Non-Proliferation Treaty (NPT). And if the nuclear arms races continues unabated, then Congress should pass an act that would require a congressional declaration of war for the president to authorize a nuclear first strike or else declare that the United States will never use nuclear weapons first. If domestic pressures on governments can achieve global peace, then such steps could ultimately lead to the reconversion of the military-industrial complexes of all countries toward the development of alternative technologies for peaceful social and ecological purposes.
**TOWARD DUAL SOVEREIGNTY ARRANGEMENTS**
But the problem here is that even if state legislatures and populations do insist that their leaderships take the concrete steps to end the nuclear arms race and to settle international disputes, the devil is still in the details. A general political settlement with Russia, China, Iran, and other states will not be achieved by slogans calling for INSTANT PEACE, but only by a concerted US-European-Japanese strategy that possesses concrete proposals as to how negotiate arms reductions and eliminations—and that, in turn, could result in the withering away of the military-industrial complexes of the United States, Russia, China, and Europe. These proposals need to be coupled with complex negotiations in multilateral Contact Groups that deal with territorial and political-economic disputes as well. Given China's rejection of the Permanent Court of Arbitration ruling in favor of the Philippines in July 2016 (see chapter ) over island disputes in the South China Sea, Beijing's promise in November 2017 to abide by the nonbinding 2002 Code of Conduct that seeks to peacefully regulate island disputes and prevent conflict represents a tiny step forward toward the possibility of peace in the region.
One concrete proposal to augment the possibility of peace would be to implement an international legal agreement that would work to settle a number of territorial disputes. In some cases, this could be accomplished by either finding ways to establish zones that would be governed by an international administration or through a resurrection of the United Nations' trusteeship administration. Or states could find ways to implement systems of joint sovereignty and power-sharing or free-trade accords that provide certain regions greater autonomy, combined with agreements involving the mutual renunciation of the use of force. What is needed is an international legal agreement that would either establish internationally administered zones or else joint sovereignty agreements over specific territories.
Some of these proposed accords could use a potential Cypriot settlement brokered by the United Nations as a model, in addition to the power-sharing accords that were reached in the 1998 Belfast agreement of Northern Ireland, for example. The approach could seek out free-trade accords or even joint sovereignty and power-sharing agreements, combined with the agreements involving the mutual renunciation of the use of force, for Crimea between Ukraine and Russia; Kaliningrad between Germany and Russia; the Kurile Islands/Northern Territories between Japan and Russia; Senkaku/Diaoyu Islands between China, Taiwan, and Japan; the differing islands in the South China Sea; the islands Abu Musa, Greater Tunb, and Lesser Tunb between the United Arab Emirates and Iran; Gibraltar between Spain and the United Kingdom; and the Falklands Islands between Argentina and the United Kingdom. Such power-sharing accords could also apply to India and Pakistan over Kashmir. These accords could also apply to islands in the French empire or those in the American empire, including Guam, Diego Garcia, American Samoa, and other islands in the western Pacific, along with Puerto Rico and the Virgin Islands in the Caribbean, plus Guantánamo Bay.
In some cases, such accords could also serve to crack down on the island tax havens of the super-wealthy, as was revealed in the Paradise Papers in November 2017, at the same time that a global Tobin tax could be implemented on international financial transactions, among other possible items. In the United States, such a tax could possibly generate as much as $190 billion a year (or 1 percent of GDP). A Tobin tax could additionally help stabilize global financial transactions, but it needs to be accepted by a vast majority of countries, which will then need to close financial paradises.
The funds raised from a global tax base would be invested with strict international regulations and controls as to how and where such funds are to be distributed, with priorities given to assist the development of the poorest regions of the planet. On average—and taking into account population size—income inequality increased by 11 percent in developing countries between 1990 and 2010. A significant majority of households in developing countries—more than 75 percent of the population—are living today in societies where income is more unequally distributed than it was in the 1990s.
The Tobin tax, which had been proposed as a means to assist the underdeveloped world, had been rejected at the 2011 G20 summit. One of the reasons for its rejection was due to already-high taxation in a number of countries. So, to be made more politically acceptable, such a Tobin tax proposal would need to be accompanied by reductions in national taxation, so that international taxation would be given priority in the new era of globalization. The concept of an international Tobin tax should not entirely be an anathema to Trump. One pro-Trump proposal is a tax based on profits made by US overseas corporations as a means to support infrastructure development in the United States itself. Other options could include higher personal income taxes for CEOs—but Trump would dubiously support such an option. Nevertheless, if it can be shown that certain regions and urban areas in the United States are highly impoverished, there is no reason why a percentage of international tax revenues for regional development and infrastructure could not go to the United States or to other developed countries as well. The political question, of course, is precisely what kind of development and infrastructure assistance would prove most beneficial. (See discussion, this chapter.)
**CONTACT GROUP DIPLOMACY**
As argued in chapter 9, a grand compromise between the United States, Europeans, and Russia that would seek to draw Russia into a new relationship with NATO and the European Union—by means of establishing a regional system of peace and development for the entire Black Sea and Caucasus region—should be in the interests of all parties. But such a proposal, to be implemented under the auspices of the OSCE (Organization for Security and Co-operation in Europe) or the United Nations, will work only if it is given a real testing by truly engaged diplomacy in which US, EU, Russian, and Ukrainian vital interests are eventually redefined and reconciled.
This is an argument not for global government, but for interacting global, regional, national, and local governance. It is an argument for greater multilateral cooperation within and among states through multilateral accords and through differing international organizations. During his election campaign, Trump stopped short of threatening to leave the United Nations, but he nevertheless denounced the international regime as "just a club for people to get together, talk and have a good time." He asserted that it was an organization that possessed "enormous potential" but did not always live up to its potential, thus wasting time and money. In Trump's view, the United Nations did not always solve problems, but "caused them."
The Trump-Pence administration will probably not leave the United Nations altogether, but it has already begun to seek ways to cut US funding for a number UN programs, and it has dropped out of UNESCO, which oversees World Heritage Sites, in October 2017 due to mounting arrears and a purportedly "anti-Israeli bias." While international organizations such as the United Nations and the OSCE are far from perfect, they nevertheless provide a forum for dialogue between the United States, Europeans, Russia, China, other states, and non-state actors, when relations between states and peoples are strained. And it is clear that the United Nations is caught up in concepts of bureaucratic hierarchy and cannot implement a new UN Security Council that takes into account the rise of new powers such as India, Brazil, South Africa, and Japan since the end of World War II. The United Nations generally works only when the UN Security Council wants it to work. And UN operations and peacekeeping, for example, can be very complicated if officers on the ground are not given strong political support and assistance for what is needed to accomplish their task.
Nevertheless, UN or OSCE backing for more flexible multilateral Contact Groups can effectively deal with many key issues. One major example is the group of states that negotiated the Iran Joint Comprehensive Plan of Action (JCPOA) nuclear accord that was negotiated by the five UN Security Council members, plus Germany and later the EU. Another is the six-party talks dealing with North Korea's nuclear program, which are negotiated by the United States, China, Russia, Japan, and North and South Korea. Another is the Minsk accords, negotiated by Germany, France, Ukraine, and Russia; but that will eventually need US and Turkish participation, as argued in chapter 9.
The Contact Group formula has proved to be a very effective way to get around UN bureaucratic hierarchy and engage the key state actors involved in the pursuit of peace. The process, even if it may take years to reach a solid accord, helps create international legitimacy for the decisions reached, which should make those accords easier to implement—even if this is not proving the case with Trump's critique of the JCPOA Iran nuclear accord.
**TRUMP'S POLICIES TOWARD PEACEKEEPING AND CLIMATE CHANGE**
Of concern are Trump's negative attitudes toward UN Peacekeeping and climate change. The United Nations and the OSCE are absolutely essential in order to engage in peacekeeping operations. The deployment of peacekeepers for the protection of refugees and immigrants under UN or OSCE mandates or other international organizations may soon be needed in Syria, eastern Ukraine, the Caucasus, and between Israel and a new Palestine, as well as in other regions, such as the Sudan, the Congo, and possibly North Korea, among many others. This will prove a major endeavor, but it can be compared to the expansion of UN (and NATO) peacekeeping missions at the end of the Cold War.
The Trump-Pence administration plans to cut costs or limit the role of the UN peacekeeping operations represents a major error on the part of the new administration. These proposed cuts (in which the Trump administration was able to cut $500 million, but not $1 billion as it intended, for 2017) are not necessarily in US interests, as UN operations can serve US interests by providing a political buffer between US policy and the differing political and social interests that are in dispute. In short, these international regimes help provide legitimacy for multilateral actions to achieve peace and development. This helps the United States and other countries not to get dragged unilaterally into intractable conflicts. And the burden can be shared to a larger extent.
**T HE UNITED NATIONS AND PEACEKEEPING**
According to the United Nations, the approved budget for UN Peacekeeping operations for the fiscal year July 1, 2016–June 30, 2017, is $7.87 billion. This amount finances fourteen of the sixteen United Nations Peacekeeping missions. By way of comparison, the UN Peacekeeping budget is way below the nominal US defense budget of $639 billion (which, in reality, is probably double that amount—see chapter 2). The United States pays almost three times more than the amount China now pays for UN Peacekeeping operations. It is true that these relative dues could perhaps be better assessed given China's significant sovereign funds. Nevertheless, investing in UN and OSCE peacekeeping could save billions in defense expenditure—if accompanied by effective multinational diplomacy intended to bring lasting resolutions to conflict, or at least working to better manage those conflicts.
For example, UN Peacekeeping proved absolutely necessary for the Democratic Republic of Congo, which became for the focal point for war between major and regional powers in the period between 1994 and 2003. What was once called World War III in Africa, and which resulted in the loss of some five million lives, was sparked, in part, by the genocide that took place in Rwanda, when some two million Hutus fled into the Democratic Republic of Congo. The horrific conflict that ensued eventually drew the Congo, Rwanda, Zimbabwe, Namibia, and Angola into a proxy war.
The issue raised here is that the Trump-Pence administration suspended legislation that was intended to require transparency in the supply chain for "conflict minerals" in April 2017. The Securities and Exchange Commission (SEC) had implemented the Conflict Mineral Rule of 2010 so that companies and consumers knew that they were not contributing to illegal activities in purchasing "conflict minerals" either from the Democratic Republic of the Congo (DRC) or from those countries adjoining the DRC. The minerals found in this region (worth an estimated $24 trillion) include diamonds, gold, copper, cobalt, and zinc, plus tin, tungsten, and tantalum (the 3TGs). The regions also possess coltan, which is used in mobile phones and other electronic gadgets, as well as cassiterite, used in food packaging. The countries involved include Angola, Burundi, the Central African Republic, the Republic of the Congo, Rwanda, South Sudan, Tanzania, Uganda, and Zambia.
Yet under the Trump-Pence administration, the SEC has ruled that it will no longer enforce key parts of the Conflict Mineral Rule. US corporations will no longer be required to conduct a diligence review or an audit regarding sourcing of conflict minerals. In addition to the change in SEC rules—which could set the grounds for renewed conflict over conflict minerals—the Trump-Pence administration is also hoping to reduce the US contribution to UN Peacekeeping—even if renewed conflicts could ultimately hurt both US political and economic interests and actually make it harder for US businesses to invest. The UN Organization Stabilization Mission in DRC (MONUSCO) is the world's largest UN Peacekeeping operation, with about 18,750 uniformed personnel.
In effect, the Trump-Pence administration efforts in late March 2017 to cut the MONUSCO Peacekeeping operation down to only three thousand personnel could open the Congo to new conflicts and exploitation by rival mining industries and additionally exacerbate tensions in other countries in Africa. It was MONUSCO that had fought the M23 rebel group, the largest rebel force in the Congo. The Congolese government itself has spoken out against the potential reversal of this rule. UN Peacekeepers are still necessary, as the conflict is not altogether resolved, particularly in the eastern areas of Congo, where some forty to fifty conflicting rebel groups have been financed by extracting resources with the backing of external states and major corporations. It is true that MONUSCO and other UN Peacekeepers have been accused of sexual abuses, due to improper vetting of individuals involved in the peacekeeping operations. UN Peacekeeping operations have also been accused ineffectiveness; yet such ineffectiveness can be attributed to lack of resources and lack of clear political mandates, plus the need to more carefully vet the political purposes of the countries involved and the need to work more closely with the local population. UN Peacekeeping is run on a shoestring budget: UN Peacekeeping has had its successes and failures, but it is not necessarily worse than NATO or EU peacekeeping, which often interact with the United Nations and have had different kinds of weaknesses.
This rivalry in Africa to obtain access strategic minerals in the Congo region that are needed for the new global communications economy, and for other purposes, could well forewarn of an even more perverse global conflict ahead—one that could use nuclear weaponry if the major and regional powers cannot soon reach accords on the differences that divide them in their own quest for energy and other strategic resources across the planet.
**T RUMP AND GLOBAL WARMING**
In addition to his efforts to cut UN Peacekeeping costs, Trump has called global climate change a "hoax" created by and for the Chinese to undercut American business. One of Trump's first acts as president was to sign decrees to cut the funding of the following US governmental agencies: the Environmental and Natural Resources Division of the Department of Justice; the UN Intergovernmental Panel on Climate Change; the Office of Energy Efficiency and Renewable Energy; and the Office of Fossil Energy. In his first G7 meeting on the subject at the end of May 2017, Trump opposed the views of the six major industrial democracies: Germany, France, Italy, Canada, the United Kingdom, and Japan. Of positive note, Trump did agree to "promote multilateralism" in the G7 communique. On the negative side, Trump appeared more concerned with the German trade surplus than with global climate change. A few days later, Trump dropped out of the Paris COP 21 Accords on global environmental issues. Ironically, he claimed that he made his decision against the Paris COP 21 because he was elected by the people of Pittsburgh (which is in the heart of the coal mining and steel industry region of Pennsylvania) and "not by Paris"; yet the mayor of Pittsburgh then tweeted: "Pittsburgh stands with the world & will follow [the] Paris Agreement." Fortunately, American states and cities have declared that they will actively support the goals of the Paris Agreement; for example, in July 2017, the California legislature stated its plans to extend its COP 21 commitments to 2030.
The risk here is that a lack of US support for the COP21 will tend to undermine global cooperation in the struggle against global climate change and carbon pollution in general, which is choking urban areas throughout the world and poisoning the environment. Although not perfect, the COP21 represents an important environmental accord reached in Paris, in which the two major polluters of the planet, the United States and China, finally reached an agreement that brings together most of the countries in the world into cooperation. The Paris environmental accord needs complete interstate and interregional cooperation and implementation if it is to succeed.
Trump appears to have based his decision to drop out of the Paris COP 21 Accords, at least in part, on reports by the right-wing Heritage Foundation and NERA Consulting. The Heritage Foundation report presented a negative assessment of the impact of those accords on American society. According to the report, the Paris COP 21 accord will supposedly result in: (1) An overall average shortfall of nearly 400,000 jobs; (2) An average manufacturing shortfall of over 200,000 jobs; (3) A total income loss of more than $20,000 for a family of four; (4) An aggregate gross domestic product (GDP) loss of over $2.5 trillion; and (5) Increases in household electricity expenditures between 13 percent and 20 percent. The NERA Consulting report argued that "meeting the Obama administration's requirements in the Paris Accord would cost the U.S. economy nearly $3 trillion over the next several decades. So that by 2040, our economy would lose 6.5 million industrial sector jobs—including 3.1 million manufacturing sector jobs." Yet these studies make worst-case assumptions that tend to inflate the cost of meeting US targets under the Paris Accord while largely ignoring the economic benefits to US businesses from building and operating renewable energy projects.
By contrast with the Heritage Foundation and NERA Consulting reports, a number of major multinational corporations strongly support investment in alternative energy. More than 360 major American corporations—including DuPont, General Mills, Levi Strauss, Nike, and Starbucks—have supported the Paris accord. These major corporations signed a statement called "Business Supports a Low Carbon USA," which urged President Trump to honor the US commitments to the COP 21. These major companies declared that "failure to build a low-carbon economy puts American prosperity at risk," and that the "right action now will create jobs and boost US competitiveness." One pro–alternative energy study by the International Renewable Energy Agency argued that renewable energy could generate over 24 million jobs worldwide by 2030. And that if environmental and human health externalities are priced into the global energy mix over time, the renewable energy transition would result in net savings.
Once president, Trump moved immediately in an antediluvian pro–fossil fuels direction by issuing five executive orders that would start the process of developing the Keystone and Dakota Access Pipelines, which had been blocked by environmentalists and the Obama administration. Trump likewise signed executive actions intended to facilitate the process of obtaining permits for manufacturing projects and to speed environmental reviews for infrastructure projects, while removing a moratorium on coal-mining leases on federal lands. He also began to undermine a number of Obama-era efforts to prevent coal-mining debris from being dumped into nearby waters. It should not be surprising that the coal industry has been one of the major contributors to the Republican Party over many years. And in 2016, Trump and other politicians in the Republican Party received 97 percent out of a total of $13,461,828 of coal-mining contributions to the election campaign.
Yet Trump's own economic adviser, Gary Cohn, has downplayed coal and argued that the United States could export natural gas as "a cleaner fuel" and by investing in wind and solar energy. Cohn was quoted as saying that the country "can be a manufacturing powerhouse and still be environmentally friendly." And both Trump's daughter Ivanka Trump and her husband, Jared Kushner, were said to have pressed Trump to support the COP 21. Coal mining accounts for less than 75,000 US jobs. Fossil fuel generation employment accounts for only 22 percent of total electric power generation employment and supports 187,117 workers across coal, oil, and natural gas generation technologies.
As promised during his campaign, Trump hopes to create a mere 28,000 to 29,000 jobs in the process of producing up to $36 billion annually in increased energy production. Yet the sustainable energy sector already employs far more than the coal or oil or gas industries. Renewable energy—including wind, solar, and biofuels—accounts for more than 650,000 US jobs. Roughly 2.2 million Americans are employed, full- or part-time, in the area of energy efficiency products and services, while almost 1.4 million energy efficiency jobs are in the construction industry. Sustainable energies currently support hundreds of thousands of manufacturing and construction jobs around the country: Wind energy employs 101,738. Solar energy employed 300,192 American jobs in 2015 and grew to 373,807 in 2016, or 43 percent of the electric power generation workforce—more than the fossil fuel and nuclear energy industries. Employment in the US solar industry "grew 12 times as fast as overall job creation in the US economy, and surpassed those in oil and gas extraction (187,200) or coal mining (67,929)." The US Department of Energy has stated that three million Americans worked in the clean energy sector in 2016—a number that would be threatened by a Paris pullout.
There are weaknesses to the COP 21, but Trump is not concerned with them. One of the weaknesses of the COP 21 is not at all the costs of the accord that Trump points out. One problem is that the COP 21 emissions targets are not legally binding and thus there is no guarantee that all countries will be able to curb carbon dioxide emissions soon enough to prevent significant environmental damage. The next concern with the Paris COP 21 Accord is that the agreement does not directly oppose subsidies for fossil fuel industries—even if that is the heart of the problem of global climate change.
For his part, Trump has opposed international governmental subsidies for the COP 21 agreement, which would mount to $100 billion for the developed countries by 2020. Yet, conversely, Trump does not appear to be opposed to fossil fuel subsidies to major energy companies. These latter subsidies, which are intended to lower the price of oil, coal, and gas, cost world governments around US$500–$600 billion per year. Of this amount, the twenty major economic powers (the G20) were responsible for the majority of the subsidies—averaging $444 billion in 2013 and 2014. Russia, the United States, Australia, Brazil, China, and the United Kingdom all had significant national subsidies for fossil fuel production. The United Kingdom has been one of the few G20 countries that has been augmenting its fossil fuel subsidies while cutting back on support for the renewable energy investments. How UK environmental policy might develop after Brexit remains to be seen. South Korea and Germany, and then the United States, Russia, and France, possessed the next largest sources of funding for coal. On the positive side, Japan, South Korea, and Australia have been leading the effort to try to mandate limits on coal subsidies in international discussions.
The problem is that these G20 subsidies enforce the general trend toward high-carbon energy development and thus divert investments away from low-carbon energy alternatives, such as solar power, wind power, and hydropower, among others. President Barack Obama did demand that Congress try to eliminate fossil fuel industry subsidies, but to no avail. If the United States and other major countries do not lead, it is dubious the rest of world will act—except, perhaps, for France and China (the latter, because its major cities are literally choking in coal dust and pollution).
With Trump and pro–fossil fuel Cabinet members in power, ExxonMobil (and other energy companies) could stand to gain as much as $1 trillion overall if the Trump administration does continue to eliminate federal environmental restrictions while also undercutting the Obama's push to develop energy alternatives to fossil fuels. In turn, this could press ExxonMobil to invest in the United States rather than in Russia (See chapters and .) Trump is fighting real progress in the name of poisonous, outmoded sources of energy—when the future can create a full-employment, ecologically viable society.
More and more countries have been investing in renewables. It is estimated that global electricity production from renewable sources plus hydropower will increase by nearly 60 percent between 2011 and 2017. In part because of its extremely high levels of urban pollution, China has led investment in alternative non-carbon and renewable energy sources, with the United States, Japan, and Germany following. As costs continue to fall, alternative energies can provide even greater employment in housing, transport, and other needed infrastructure. Ironically, now that he has dropped the United States out of the Paris climate accord, in part to please his American First nationalist supporters and coal and shale energy producers, Trump will be handing sustainable energy innovation, including solar energy, wind power, and carbon-capture technology, over to the China and Europeans—instead of helping American firms take the lead in these very promising industries. In effect, Trump's anti-COP 21 decision will draw Europe and China closer together in political-economic, financial, and technological cooperation—which could effectively isolate the United States from the innovations of two largest economies in the world.
**THE SECURITY-ECONOMIC BENEFITS OF DECENTRALIZED ALTERNATIVE ENERGY**
Trump intends to reinstate heavily polluting and environmentally destructive coal and to invest in shale oil and gas. This is instead of investing in solar, wind, geothermal, and other less-polluting energies—possibilities that could also create healthier jobs—while concurrently decentralizing energy production in such a way so that the United States would be even less dependent on polluting carbon industries and overseas suppliers. Multiple decentralized energy units with different backup systems in case of supply shortage can better match local needs. This decentralized approach also results in greater energy savings and overall security than dependence on large, centralized nuclear power systems—which, when they break down, can cut power for millions of consumers all at once. In addition, in the new games of cyber-sabotage, large, centralized energy and power systems, such as nuclear power plants, could become the ultimate target that militant hackers could attempt to shut down. Or that terrorists and states may seek to destroy.
Global climate change and pollution is not a joke. Militaries throughout the world have been studying the potential security and political-economic implications of a warming planet. Global climate change has already begun create major humanitarian catastrophes due exceptionally severe weather conditions: climate-related droughts, floods, crop failures, hurricanes, and the like. These catastrophes have resulted in mass migrations and refugees—while directly or indirectly causing social suffering and conflicts. One wonders if the Trump-Pence administration might finally understand the issues involved if Manhattan (or other coastal cities such as Miami, New Orleans, London, Venice, Shanghai, Hong Kong, and Sydney) begin to flood, as predicted in some catastrophic scenarios. All of this is at a time when the Earth's temperature has hit a record high level for three years in a row—with the expected melting of the Arctic and Antarctic ice caps. By 2045, Trump's own "Winter White House" at Mar-a-Lago could be under at least a foot of water for 210 days a year because of increased tidal flooding. But this possibility seems to be too far in the future for Trump to think about.
The fundamental way to deal with this crucial, not-so-long existential threat is by fostering global social-ecological-aesthetic consciousness—in conjunction with the development of self-sustaining alternative-energy infrastructure. This is the true alternative to Trump's antediluvian path.
**TOWARD PEACE AND DEVELOPMENT COMMUNITIES**
Also as an alternative to Trump's essentially unilateralist and nationalist America First strategy, the policy approach argued in this book demands concerted multilateral efforts to formulate new international geopolitical, defense, political-economic, and environmental accords. The establishment of new regional systems dedicated to peace and development simply cannot be accomplished unilaterally or even bilaterally. Such an irenic peace and diplomacy-oriented strategy needs the support of all the major and regional actors, combined with concerted efforts to prevent "spoilers," such as anti-state "terrorist" groups, from trying to undermine negotiated peace accords.
In this peace-oriented strategy, Washington will need to fully engage in diplomacy with the major and regional actors of the world. Global peace and human development can only be achieved by redefining the US national interest in such a way as to reach compromises not just with US allies and friends but also with American rivals, including Russia, China, and Iran. Rather than asserting presumed national interests first, as the Trump administration appears to believe, each of the actors involved in a dispute or conflict needs to make trade-offs and possible sacrifices. Each state needs to redefine or reframe national interests in such a way that all involved can eventually benefit as much as possible, so that peace, or at least a _modus vivendi_ , can be established in a healthy and ecologically balanced global environment.
In an effort to prevent the global system from polarizing into two rival alliances, and in working with the major and regional actors, Washington needs to look toward ways to forge new systems of cooperative-collective security, involving international administrations or agreements of joint sovereignty in both the Euro-Atlantic region (with a focus on the Black Sea and Caucasus region) and in the Indo-Pacific region. In Europe, this means forging a new Euro-Atlantic security pact with Russia, based in large part on a new regional "peace and development community" for the Black Sea region under the auspices of the OSCE.
In the Indo-Pacific, this means supporting Japan's efforts to forge the new rapprochement with Russia that was initiated in 2016, while concurrently finding areas where the United States, Japan, Russia, and India can cooperate with China with respect to North Korea, island disputes, and Taiwan. In addition, despite their apparent reluctance to do so, India and Pakistan, with US, Russian, and Chinese diplomatic backing, need to negotiate their differences over Kashmir, among other issues. Washington likewise needs to back the efforts of the new South Korean leadership to forge a rapprochement with Russia, so that Seoul can pursue a new Sunshine Policy—in the effort to establish a long-term peace between North and South Korea. (See chapter 7.)
Establishing a system of peace in the Indo-Pacific could mean the creation of a new Organization for Security and Cooperation in the Indo-Pacific (OSCIP) modeled after the OSCE and interacting with the Association of Southeast Asian Nations (ASEAN). A proposed OSCIP could find international, legal ways to move toward differing forms of confederation and systems of joint sovereignty over specific regions of contention. All of this implies a US-led multilateral approach to conflict resolution, which means that Washington will also need to make significant compromises, as well as concessions in some cases—if peace is to be sustained in the long term. Here, one of Trump's more positive proposals on his trip to Asia in November 2017 was his offer to negotiate the island disputes in the South China Sea between China and its neighbors. Whether this offer will work out remains to be seen, but it represents a very positive sign that Trump may begin to be actually thinking of the greater good, instead of just America First.
In this perspective, Washington will concurrently need to work collectively through the United Nations, the OSCE and other multilateral Contact Groups with the major powers and regional partners, as well as with the states and certain anti-state actors involved, in order to quell the turmoil throughout the wider Middle East. Here the United States and its partners should engage in a multidimensional strategy designed to co-opt differing Islamist movements, while seeking to isolate and destroy IS and al-Qaeda offshoots where possible. This multidimensional strategy involves concerted multilateral diplomacy that is sometimes US-led and sometimes not. Such diplomacy would be intended to transform issues of dispute and conflicts in a more positive direction, if not resolve them altogether.
With respect to the wider Middle East, there are several increasingly interrelated conflicts that threaten to merge with major power rivalries and explode into war. The first is the ongoing regional conflict between Saudi Arabia and Iran, which manifests itself in Syria, Iraq, and Yemen, and which has begun to implicate Qatar and Lebanon and divide countries and political movements throughout the region. The second is the indirectly related conflict between Turkey, Syria, Iran, Iraq, and the Kurds. The third is the conflict between India and Pakistan over Kashmir, which needs to be resolved through power-sharing accords. And the fourth is the Israeli-Palestinian conflict, which helps fuel pan-Sunni and pan-Shi'a pan-Islamist movements in the background. All of these above regions need the concerted attention of Contact Groups that work with the United Nations and the major actors involved. (See chapter 8.)
Without making reasonable compromises—or even some significant concessions—with both rivals and allies, Trump's nationalistic America First project risks stumbling into a dangerous unilateralism that will result in a dangerous and destabilizing arms race and will further provoke not only US rivals but also those states that are presently US allies as well—and whose pro-American stance cannot be absolutely taken for granted in the future.
**TOWARD A REEVALUATION OF "AMERICA FIRST"**
Donald Trump may have initially hoped to change the geopolitical map and fulfill many of his campaign promises in his first one hundred days in office, but disputes over Crimea, Taiwan, Iran, and North Korea, among other crucial concerns, including the natural environment, are not issues that can be rapidly altered by threats to use force or hastily conceived real-estate-like "deals." This is because the complex and interwoven geo-strategic, military, political-economic, and even sociocultural and ideological issues that surround these areas represent the tip of the now-melting iceberg.
Each dispute will require very careful attention and patient irenic diplomacy—if they are not to continue to fester and ultimately result in a crisis, like an iceberg that does sink US warships. The problem is that traditional, realistic prudence and the need for careful long-term negotiations is not a trademark of the Trump casino dynasty. Trump definitely needs to sail with greater tact toward these issues and others. He also needs to sail more carefully if he wants to retain US allies, who could either float into neutrality or else look toward either Russia or China, as is the case with Bulgaria, Hungary, the Philippines, Qatar, Turkey, and others, if the United States opts to apply tougher protectionist and other nationalist measures.
It will prove nearly impossible for the United States to reach international agreements if all countries continue to assert their presumed national interests _above_ the interests of other states—as Trump put it in his inauguration speech, "It is the right of all nations to put their own interests first." On the one hand, Trump stated his preference for the United States to work positively with all countries; on the other, he has also promised to assert presumed US national self-interests first, with respect to all other states. This means that US interests may be seen as literally trumping the interests of second and third parties, against their will—rather than making fair and equitable compromises or even concessions.
These latter states will generally argue in response to Trump that the United States, as the predominant hegemonic power at least since the end of the Cold War, has always had an unfair advantage over a large number of areas—so his assertions appear completely false even if the United States may appear to be losing some of its economic advantages and ideological support for American-style democracy in recent years. Trump's assertion of an American First doctrine could consequently make international agreements and compromises either impossible to achieve or else very difficult to maintain in the long term.
In response to America First, some US allies could drift into neutrality or else seek out trade, political-economic, and financial deals from China, or seek out energy, resource, and arms deals from Russia—thus ignoring US interests as much as possible. A number of states, including China, could opt to nationalize or even expropriate US or European multinational firms, for example—and then militarize in the assertion of their own presumed vital interests—"first."
If the United States wants to retain its position of global diplomatic leadership, Trump's America First policies are not the right way to do it—that is, unless Trump ultimately proves willing to compromise, if not make concessions, on a number of issues with Russia and China, among other secondary powers, and only if his administration shows a willingness to engage in multilateral processes that involve both states and societies. What is needed is not a rush toward absolute military superiority but prudent steps toward an engaged and concerted US diplomacy that seeks to mitigate, if not resolve, through mutual compromise and concessions, the disputes and conflicts that divide various states and sociopolitical movements. Without engaged and concerted diplomacy that works to defuse significant disputes among both major and regional powers, and that incorporates the interests of both US allies and potential rivals, Russia and China, US military superiority alone will not prove capable of preserving the peace.
Moreover, US global hegemony cannot be sustained if the United States becomes entrenched in conflicts in its own hemisphere. The possibility of conflict in Mexico and the Caribbean region becomes increasingly plausible if the Trump administration tries to repel Mexican and Latin American immigrants (meaning that their remittances would no longer help float the economies of these foreign countries), while threatening to intervene militarily in Mexico's drug wars or in Venezuela's social strife—all without engaging in regional Contact Group diplomacy, that could involve Cuba, for example, in the case of Venezuela. (See chapter 3.)
With respect to immigration, there is another way to deal with that question, through the effective use of the Earnings Suspense File. One factor that needs consideration is that even undocumented workers have often paid into Social Security using fake or stolen Social Security numbers. This money goes into what is called the Earnings Suspense File (ESF). In 2010, unauthorized immigrants worked and contributed as much as $13 billion in payroll taxes to the Social Security Old Age, Survivors and Disability Insurance (OASDI) program; at that time, only about $1 billion in benefit payments during 2010 were attributable to unauthorized work. Moreover, the ESF reached a whopping $1.2 trillion in uncredited wages for the tax years 1937 to 2012—even if the number of W-2s posted to the ESF declined by 36 percent from 2007 to 2012; the latter is in part because of high unemployment, which contributed to the decrease in suspended wage items. And the decline in W-2s was also due to Obama's efforts to crack down on false Social Security numbers. What is to be done with this money needs an open public debate, for it could fund numerous social programs, including helping to pay for better integrating undocumented workers into American society, as well as other educational "infrastructure" projects.
Another issue is the need for a new approach to the War on Drugs that was initiated by President Richard Nixon, and which has spread from Columbia throughout Latin America and the world, much like the Global War on Terrorism has spread beyond the wider Middle East. It is clear that the War on Drugs has failed miserably, whether in Latin America, Afghanistan, or elsewhere. And the November 2016 peace accord between the Revolutionary Armed Forces of Colombia (FARC) and the Colombian government offers some hope for ending the drug wars throughout the region, even though new social and political tensions have come to the surface in the aftermath of that agreement. (See chapter 3.)
An approach that should help stabilize the general Latin American region economically, given the spread of drug production primarily for the American market, would be the legalization of less addictive drugs such as marijuana, hashish, and possibly cocaine, among other drugs—but with clear warnings about health effects, as is already the case for cigarettes and alcohol. There should be very strong laws against driving while under the influence, for example, but not against drug use and drug possession itself. Such an approach would be accompanied by a major police crackdown on more-dangerous drugs, including crack cocaine, PCP, scopolamine, crystal meth, among others. The legalization of some drugs, but not others, would represent an effort to push drug mafias into legal business as much as possible—and to obtain significant tax gains with significant cuts in law enforcement costs as well. At a minimum, drug use and possession should be decriminalized so the police can focus on terrorism and high-level crimes. Ireland, for example, is considering decriminalizing heroin possession, among other drugs. But decriminalization alone would not be sufficient to deal with the depth of the drug epidemic, which should be considered a health, and not a criminal, issue.
As was the case for legalizing alcohol after the Great Depression, which helped to eliminate at least some of the more pernicious influences and crimes of the Mafia, the drug issue is a major social and health issue and needs to be dealt with realistically. Drug legalization would additionally help to reduce gun violence, in that many urban shootings are a result of wars between drug gangs. With 1.5 million arrests for drug infractions, more persons are incarcerated for drug infractions than for all violent crimes combined, so that 50 percent of the US federal prison population comprises narcotics violators. This draws the police and the courts away from other, more vital, concerns. Despite US government efforts since the Nixon administration, drugs continue to flow into the country or else are produced in basement labs. Many dangerous drugs are formulated with the use of readily available chemicals and pharmaceutical products, helping to create the opioid epidemic in the United States, for example, which killed 64,000 people in 2016. A new approach is imperative.
**REVIVING US DEMOCRATIC PRINCIPLES AND PRACTICES**
Finally, as pointed out in the first chapters of this book, the United States will not be able to continue to engage in global leadership unless it truly begins to practice what it preaches. Trump himself recognized a number of the incongruences in the American system of governance. But instead of attempting to expand democratic practices, he has attempted to shift the country toward a more authoritarian form of governance under his leadership.
In his inaugural address in January 2017, Trump declared that "we do not seek to impose our way of life on anyone, but rather to let it shine as an example for everyone to follow." Yet the very nature of his own presidential victory—in which the billionaire Trump did not possess a clear mandate from the American people themselves—has raised questions as to whether many countries and peoples in the world will follow the American lead on various issues.
Ironically, in 2012, Trump himself had demanded some sort of reform of the democratic process—if not a "revolution"—when he mistakenly believed that Obama had won the electoral college vote but not the popular vote. He declared by tweet: "The phoney [ _sic_ ] electoral college made a laughing stock [ _sic_ ] out of our nation." Trump then called for a "revolution." By revolution, Trump was certainly referring an authoritarian, right-wing revolution that serves the interests of the military-industrial-congressional complex and the major fossil fuel industries. After Trump's victory (during which he lost the popular vote by a colossal 2.8 million votes), the electoral college system has indeed appeared to have made the United States a laughingstock—as Trump himself had put it.
The issue raised here is that the complexity of American governance fuels the propaganda machines of US rivals, terrorist groups, and other enemies—who seek to justify their own forms of illiberal, authoritarian, or theocratic systems of governance—by the failure of the American system of democracy to live up to its own principles and values. Yet it appears highly unlikely that these crucial of issues of democratic governance will be addressed by the Trump administration, which appears more interested in challenging the US judicial system than in tackling problems raised by liberal majoritarian democracy.
Even though it was only the fifth time in US history that a president won the election without winning the popular vote, some form of electoral reforms appear absolutely necessary, given huge imbalances in the population sizes across the fifty American states. One option—which would not require a constitutional amendment—is for every state to cast its votes for whomever won the popular vote within that state. A more radical option would be to reduce the number of states to thirty-eight in such a way as to better balance rural and urban areas. This option would provide fairer local and regional governance, while cutting state and federal governmental costs significantly, given the rising debt crisis. It would help simplify the process of American governance. If the French can reduce the number of regions from twenty-two to thirteen, as it did in 2014–2016, so can the Americans!
Another radical option intended to cut costs, strengthen local regional representation, improve efficiency, and attempt to ameliorate incapacity to act resulting from "vetocracy" (which seems to depict many issues except for defense appropriations), is to eliminate the aristocratic Senate and augment the power of the House of Representatives by making each member run for a single term of office, without chance of reelection, for four to six years, instead of two years. As discussed in chapter 2 in relation to defense appropriations, this would lessen the chances that crucial decisions would be made for demagogic purposes in order for congresspersons to be reelected, and it would also strengthen the role of the House. Because there would be sufficient checks and balances between regions and individual congresspersons inside such a unicameral system, a bicameral system is not necessary. Another proposal would be to limit the presidency to one five-year term of office, once again to minimize election-year demagoguery.
The goal of these proposed constitutional reforms is to make the US government less costly, more effective, and more responsive to the needs and interest of the American people. If the US government is to provide positive leadership for its own citizens, while likewise working in good faith with all other peoples and countries of the world community, it will need to rebuild its credibility and its democratic legitimacy that has been lost in the aftermath of its post–Cold War military interventions abroad, and further desecrated by the unstatesmanlike and untrustworthy nature of Donald Trump's presidency. (See chapters , , and .) The US Constitution has not been altered significantly for more than two hundred years, so it may be time to do so!
Even more crucially, and what should be given priority, is the fact that Trump's domestic policies do nothing to address the crucial issue of growing inequity and gross disproportion in incomes. One way to do this would be a global tax on financial transactions, as previously discussed, coupled with a reduction in national taxation. Another way to do this—which should become part of the national and international debate—is to better distribute the income of major corporations through employee stock-ownership plans combined with greater employee power-sharing in product and investment decisions through shared capitalism and "workplace democracy." There are, after all, times when employees possess more common sense and innovative ideas than their managers do—and they should be better rewarded for their contributions.
The above is, of course, just a sketch of what kind of steps need to be taken domestically and internationally if the United States, and the world with it, is to literally weather through this global financial, geopolitical, and ecological crisis. Given the significant amount of the US budget that is devoted toward defense purposes, the national debt keeps mounting without apparent end, in large part because of costs directly or indirectly related to military spending and military interventions. It is primarily by seeking ways to forge diplomatic and geopolitical compromises that lead to significant reductions in military expenditure and reconversion of the military-industrial complex to peaceful purposes that the United States will eventually be able to truly devote greater attention to the full development of its own citizens, and those of the world.
Trump could prove to be like Richard Nixon, who was forced to step down as president under the threat of impeachment. Or he could possibly prove to be more like Ronald Reagan, who reversed hardline American policies in his second term by seeking to improve US relations toward the Soviet Union and by working to put the conflicts of the Cold War to rest in the ash heap of history. In that era, it was the Soviet Union that made major compromises and concessions over nuclear arms and Germany, for example. In this era, it is the United States that must make major compromises and concessions over nuclear arms and policy toward Ukraine, for example, among other compromises with Russia, China, Iran, North Korea, and other countries discussed in this book.
If the Trump-Pence team or the next administration (whether or not Trump is impeached) does not take a radical about-face away from its present America First course, those nationalist policies will actually accelerate tendencies toward the polarization of Amerian society and the division of the world into rival camps—while concurrently setting the conditions for World War Trump.
In the midst of his presidential campaign, Trump denied that he was an isolationist, but he did affirm that he liked the expression, "America First."
It is accordingly not certain whether the Trump campaign's America First slogan was inspired by pre–World War II "isolationism" of the America First Committee, the largest anti-war organization in US history. The organization dissolved after the December 7, 1941, Japanese attack on Pearl Harbor. A number of prominent Americans were associated with the anti-war America First movement, which supported a strong national defense, but which opposed US intervention in the then-ongoing war in Europe. These prominent individuals included future US presidents Gerald Ford and John F. Kennedy, the controversial aviator Charles Lindbergh, who acted as its spokesperson, and many others. But even if the precise meaning of "America First" for Trump and his team is not entirely clear, the term has certainly lost any anti-war significance that it might have had prior to the arrival of the Trump-Pence administration to power.
Trump's conception of America First can nevertheless be ascertained by his choice of individuals whom he considers to be great leaders: General George S. Patton, General Douglas MacArthur, and President Teddy Roosevelt.
In choosing these individuals, who cannot really be considered anti-war, Trump indicates that he possesses a predilection not only for military power and unilateral actions but also for those maverick leaders who act outside of the chain of command and official hierarchy, or who, in some way, challenge established norms and preconceived ideas.
George S. Patton's speech to American soldiers appears to parallel Trump's views of America First and his own philosophy of supporting only "winners," however defined: "All real Americans love the sting and clash of battle.... Americans love a winner and will not tolerate a loser. Americans despise cowards. Americans play to win—all the time. I wouldn't give a hoot in hell for a man who lost and laughed. That's why Americans have never lost, not ever will lose a war, for the very thought of losing is hateful to an American."
Trump also venerates General MacArthur. In doing so, Trump advanced the dubious thesis that MacArthur was merely using the _threat_ to use nuclear weaponry as leverage to negotiate with the People's Republic of China during the 1950–1953 Korean War. In fact, however, MacArthur had drawn up the actual plans to use such weaponry, after already having used the destructive potential of Napalm and other weapons against North Korean and Chinese forces, which did not possess nuclear weapons at that time. In addition, and contrary to the general understanding, MacArthur was not removed from his position as supreme commander of the US-led UN force as a result of his public threat to use nuclear weaponry against North Korea and China; MacArthur was removed from duty for insubordination. The US Joint Chiefs of Staff were still considering the possible use of nuclear weaponry even after MacArthur's dismissal—an action that could have escalated hostilities with China and could have possibly brought the Soviet Union more directly into the war.
Contrary to Trump's argument, MacArthur's threat to use nuclear weapons did not dissuade Beijing from sending in one million troops across the Yalu River and pursuing the war in support of North Korea after MacArthur had ordered UN forces to attack north of the 38th parallel. And the brutal way in which the United States waged the war—US B-29 bombers had dropped 866,914 gallons of Napalm onto North Korea from June to late October 1950—did not deter either North Korea or China from fighting. These facts help to explain the militant pro-nuclear-weapons policy of the contemporary North Korean regime.
In terms of his favorite US president, Trump points to the Republican maverick Teddy Roosevelt. It was Roosevelt who had established the "Roosevelt corollary" to the Monroe Doctrine that permitted US military intervention in Latin American affairs while also strengthening the power of the presidency. It was Roosevelt who forced Congress to fund the American Great White Fleet so as to augment US naval capabilities across the globe. To his credit, Teddy Roosevelt also helped broker the 1904–1905 Russo-Japanese War. This fact could possibly inspire the Trump-Pence administration to mediate between Russia and Japan in contemporary circumstances—that is, if it will eventually prove possible to engage in a general rapprochement with Russia, given strong domestic American and international opposition to dealing with Russian President Vladimir Putin.
Roosevelt advocated "speaking softly and carrying a big stick." But this represents a maxim that Trump has not fully ingested. Given his loose and combative tongue, his emotional outbursts and reactions to critical media reports, and his apparent inability to control his "tweets" in the new social media that permits him to comment directly to his followers and denounce his critics outside of both governmental and corporate media outlets, Trump appears incapable of speaking softly, or even in a more controlled manner. Trump is well known for his tweeting; in fact, in a French political cartoon, one White House counselor laments to another: "You will see, he Trump] will end up tweeting the nuclear codes." Further underscoring his seeming inability to control his speech, Trump has also been accused of inadvertently leaking secret information to Moscow in a White House meeting. (See [chapter 3.)
The issue raised here is that Trump's tweets oversimplify complex policy issues that require complex behind-the-scenes discussion and debate before bringing them out into the public. If not appropriately articulated, the emotional impact of such simplified policy statements could provoke strong negative reactions among national populations concerned—particularly as Trump is the leader of the most powerful and influential country in the world. It may also prove difficult for Trump's own official spokespersons and leaders of other countries to play down some of Trump's more inflammatory remarks. Popular outbursts abroad against Trump could, in turn, force the leaderships of those countries to take strong stands—even if those leaders may actually prefer possible compromise approaches. Insulting leaders, if not whole populations, can create deep resentment. Yet Trump has done this with Mexico and China repeatedly; with Iran, which he considers the major cause of terrorism; and with North Korea, which he has threatened to "totally destroy." Calls for retribution could take a long time to die out.
Moreover, Trump's contemporary domestic policies appear to be totally at odds with the domestic policies of Roosevelt's era. Teddy Roosevelt had fought fiercely for environmental and consumer protection. Roosevelt doubled the number of US national parks to ten; created eighteen national monuments (including the Grand Canyon), through the 1906 Antiquities Act; and set aside fifty-one federal bird sanctuaries, four national game refuges, and more than 100 million acres of national forests. Yet it is Teddy Roosevelt's legacy and the Antiquities Act that Trump has begun to undermine by opening public lands (which were not entirely closed) to the mining industry. On March 28, 2017, Trump signed an executive order that could allow companies to mine and drill for oil at all national monuments designated after 1996. Companies are now demanding, for example, that the Grand Canyon be opened to uranium mining, in opposition to Obama's 2012 ban on mining. By November 2017, Trump was, in fact, reconsidering the ban on mining in the Grand Canyon.
Teddy Roosevelt had also fought for governmental regulation of industry and anti-monopoly reforms under the Sherman Antitrust act—an issue that Trump has largely ignored. And Roosevelt possessed a generally strong, even if mixed, record in support for the right of women to vote and for the rights of African Americans and minorities; this is in contrast to a number of Trump's outrageously crude remarks about women and his blatant sexism. Moreover, Trump does not even appear to be offering a Roosevelt-like Square Deal that will eventually compress the burgeoning gap between the very, very rich 1 percent (of which Trump and many members of his cabinet are prime examples) and the rest of the American public, including his own blue-collar supporters. (See chapters , , and .)
Trump's form of nationalism and populism—and his inauguration promises to the American people that "your voice, your hopes, and your dreams will define our American destiny"—can be traced back even further, to the frontier populism of Andrew Jackson. President Jackson, to a certain extent like President Trump, had been elected by the vote of the frontiersmen of the western states of that era once suffrage was extended. Jackson had hoped to assert federal government control over Native Americans while starting the brutal process of implementing American coast-to-coast claims of Manifest Destiny. The latter was a term, in effect, used by Jackson as early as 1835 in an editorial affirming that the United States was "manifestly called by the Almighty to a destiny" that would have been envied by Greece and Rome.
One of Andrew Jackson's many negative legacies was his signing of the democidal Indian Removal Act of 1830, which was responsible for the Cherokee Trail of Tears. This action involved the forced removal of and brutal wars with the Five Civilized Tribes (the Chickasaw, Choctaw, Muscogee or Creek, Seminole, and the original Cherokee Nations), in addition to other native peoples. One could argue that Trump, although he is an urban New Yorker, exhibits a frontiersmen mentality somewhat similar to that of Andrew Jackson, but in reference to conflicts overseas against non-European cultures that lie on the "frontier" of the American empire in the wider Middle East. In this respect, the American struggle against the Native American peoples on the US-conquered continent throughout the nineteenth century possesses some key parallels with the contemporary Global War on Terrorism against a number of Islamist movements, but now on a global scale.
In this analogy, the wars between the Texas Rangers (before Texas became a state) and with the US military against the Comanches (with their claims to Comancheria, which ranged from eastern New Mexico to northwestern Texas and parts of Mexico), the Apaches, and other native peoples who opposed American Manifest Destiny have now been replaced by US-led wars with al-Qaeda, the Islamic State, and other Islamist political movements, which generally possess the ultimate goal to create an Islamic caliphate, much like the Ottoman Empire.
In many ways, the ultimate goals of Trump's inauguration call—to "unite the civilized world against radical Islamic terrorism, which we will eradicate from the face of the Earth"—do not appear to have changed that significantly from the brutal democidal anti-Native American campaigns of Andrew Jackson and Texas President Mirabeau B. Lamar. In effect, much as the United States expanded its continental empire into the Wild West, Trump has hoped to expand US global hegemony into the wider Middle East. But this effort to assert control over that region is taking place in a geopolitical situation involving a complex mix of rivalry and collaboration with Russia, China, the Europeans, Israel, Iran, Turkey, and Saudi Arabia, among other powers.
Trump's support for America First nationalism (in large part by means of undermining systems of interstate governance, including the European Union, trade pacts such as NAFTA and the Trans-Pacific Partnership, or TPP), plus his decision to drop out of the Paris climate agreement, could actually work to undermine the very "civilized world" that he has hoped will work together for a number of causes and against a number of potential threats. And, given deeper civilizational divisions that Trump himself has emphasized, there is a real danger that despite his expressed hope that the "three Abrahamic Faiths...can join together in cooperation, then peace," Trump's form of Judeo-Christian messianism—as it clashes with Russian Orthodoxy, Chinese Communist revanchism, and differing apocalyptic pan-Islamist movements—risks provoking World War Trump in a self-fulfilling prophecy.
**TOWARD A WORLD EVEN BLEAKER THAN _1984_?**
Both _1984_ by George Orwell and _It Can't Happen Here_ by Sinclair Lewis surged in sales after Trump's victory. In many ways, Trump is more like Sinclair Lewis's demagogic and opportunist con man Buzz Windrip (purportedly based on the real-life US Senator Huey Long), who takes total control of the US government in the establishment of a plutocratic state, than he is like Orwell's omnipresent Big Brother in his novel _1984_. Moreover, Trump's version of political language is more that of "no think" than like Big Brother's more sophisticated and manipulative "doublethink." Trump's often vulgar language—and particularly his inability to articulate the reasons for his frequent policy flip-flops—makes his rule very destabilizing. Unlike Orwell's _1984_ , there is no logic or carefully constructed ideology to his often-unpredictable off-the-cuff remarks. He says or tweets spontaneously what he believes his public supporters will believe. At the same time, Trump appears to be expert at manipulating the news media in his own version of "doublespeak" by denouncing "fake news" even when that news is based on fact, while also proclaiming certain things to be "true" when they are, in fact, false. Trump claims to support America First, that is, to speak for presumed American values, but really he is speaking for the interests of an American empire of which he and his plutocratic associates possess a significant portfolio.
Yet let us assume that the militarists of the Trump-Pence administration do possess a more or less carefully conceived foreign and defense policy and that they are correct (or just lucky) that a global war will not result from a major arms race. But let us also assume that no significant steps are taken to defuse political-military and economic tensions throughout the world either. Let us assume that global interstate relations can remain in a precarious position of "no peace, no war," much like that depicted in Orwell's _1984_ , more so than Lewis's _It Can't Happen Here_. What will such a world look like? And how long will it last?
Trump had proclaimed in his January 2017 inauguration that he hoped to cultivate "friendship and goodwill with the nations of the world." This appeared to represent a utopian, Mazzini-like vision in which the republics and nations of the world would be able to agree to form a coalition to fight together in the Global War on Terrorism, while somehow also cooperating on other important geostrategic and political-economic issues, despite their differing national interests.
Yet, in accord with the new logic of Trump's version of doublespeak, precisely the opposite is happening. On the domestic side, Trump's inaugural address reflected his own version of doublespeak when he stated: "At the bedrock of our politics will be a total allegiance to the United States of America, and through our loyalty to our country, we will rediscover our loyalty to each other. When you open your heart to patriotism, there is no room for prejudice." Yet Trump's claim that allegiance, loyalty, and patriotism will somehow minimize prejudice is contradictory, as it is not at all grounded in actual historical experience. This is particularly true because appeals to patriotism can be very divisive—and even more so if the so-called patriotic cause is not considered by all to be just. Quite the contrary, it is in historical periods of extreme nationalism and patriotism that prejudice runs rampant.
The question remains: What will happen to those, such as FBI Director James Comey, who do not pledge "total allegiance" to the president or to the US government? And what happens if individuals do find themselves subject to discrimination and prejudice because they are not believed to be upholding "national" or "patriotic" values? What happens to those who do not pledge allegiance to the American flag? And if individuals are subject to discrimination and prejudice, will that fact then lead to dissent, repression, and social conflict at home?
Trump's public support during his presidential campaign for the use of torture and extrajudicial killing is very worrisome in this regard. (Both _1984_ and _It Can't Happen Here_ deal with themes of torture.) Trump has, for example, stated that he would consider tactics like waterboarding of suspected terrorists—and their families—in the struggle against terrorist groups. He has also threatened to engage in drone attacks on the families of terrorists and other civilians who are not directly involved in hostilities, thus expanding the military's mandate to use extrajudicial force. Trump has consequently promised to go beyond the efforts of the George W. Bush administration to legalize torture under the euphemism of "enhanced interrogation techniques;" by contrast, Senator John McCain, who was tortured as a captured US serviceman during the Vietnam War, vowed to oppose any Trump effort to revive the use of torture against detained terrorism suspects.
Trump's support for torture denigrates traditional American values and constitutional rights. The American military knows that it should not willingly do what is known to be illegal and immoral—and that they should resist illegal orders in order to preserve the civilian control over the military. The use of torture is counterproductive and does not necessarily obtain useful information for "actionable intelligence," despite what its proponents have claimed. A Trump-Pence administration decision to relegitimize the use of torture would furthermore feed into the propaganda machines of the Islamic State, al-Qaeda, and other anti-state sociopolitical movements and countries that oppose US foreign policy and US interests.
On the international side, Trump's calls for "patriotism" and "loyalty" have sounded like President George W. Bush, who declared to the countries and populations of the world, "either you are with us or you are with the terrorists"—in initiating the war against the Iraqi regime of Saddam Hussein in 2003. Yet despite Trump's disclaimers, as long as Washington continues to assert its own hegemonic interests first, the United States will not find itself cultivating the friendship of all countries. Instead, it will soon find itself as acting alone in an even more grotesque geopolitical context—in which conflicts, with unclear or complex causes, could break out unexpectedly.
In such a world, each major country, ruled by plutocratic elites, will be destabilized by the burgeoning gap between the very, very rich and the poor. Millions will be concentrated in overcrowded, highly polluted, and crime-ridden megacities—and those megacities along the coasts will increasingly be impacted by Arctic and Antarctic flooding while other regions will become increasingly arid. In addition to the massive pollution and overheating of the environment caused by carbon emissions, the tons of plastic waste dumped directly into the ocean is already causing unresolvable damage to aquatic ecosystems and to our bodies. As plastic breaks up, it, along with other poisonous pollutions, is ingested by fish and climbs up the food chain and into humans.
It will be a world in which the national security apparatus of states and their cyber wizards possess almost total informational control over their citizens, while seeking to root out more information, from friends and foes alike—in rivalry with "enemy" states and anti-state actors and mad computer hackers. In this global context, each government will seek to control media and information that might delegitimize or oppose the official line and policies.
In such a global situation, "friend-enemy" distinctions will not be as crystal-clear as they at least appeared to be during the Cold War. Enemies in one situation could become friends, and vice versa, much as George Orwell stated in his book _1984_ : "The enemy of the moment always represented absolute evil, and it followed that any past or future agreement with him was impossible." Orwell's comment appears to fit the present situation of US-Russian relations.
And, just as anarchists once claimed that sticks of dynamite could equal the score between themselves and the superior force and manpower of the police and military, lesser sociopolitical anti-state actors will continue to undermine reasonable diplomatic efforts to achieve "peace." This will prove particularly true in situations in which those anti-state actors interpret the grandiose claims of major powers to seek peace as actually intended to preserve their own hegemony. Major powers may appear to talk peace, but only in Orwellian doublespeak. Their leaderships may not at all be concerned with actually resolving complex political, economic, and social problems. In such cases, authoritarianism and repression becomes the status quo.
The geopolitical rivalries of such a world could be even more abominable than the fictional dystopia depicted in Orwell's _1984_. In that book, the planet is plagued by potentially nuclear conflicts between three major powers: Oceania, Eurasia, and Eastasia. In the contemporary geopolitical context, in which the world is under the constant threat of hybrid, if not nuclear, warfare, Oceania can be considered the United States, plus the Europeans, Japan, and its remaining allies; Eurasia is Russia and its allies; while an increasingly powerful Eastasia is now China—in a geopolitical rivalry in which Eurasia and Eastasia are presently threatening a full-fledged military alliance against Oceania for control over the world and over outer space as well. Concurrently, each of these powers are also be competing for the political-economic and military allegiance of a fourth power, amphibious India—the rising power of "Southasia," which was not foreseen in Orwell's otherwise-prescient geopolitical vision. In this view, India could play a potentially positive role of mediator, or a negative one of antagonist—that is, if India does not succumb to its own significant internal and regional conflicts.
Such a world is in obvious contrast to the liberal vision of a world of total interdependence. Nevertheless, such a horrific world could become closer to reality if the United States takes an extreme America First position. The Trump administration dumped the TPP accord, which will assist China's rise as a major authoritarian and anti-democratic actor. It also dumped the important COP 21 agreement, which unilaterally relinquishes American leadership in the area of alternative and sustainable energies and carbon emissions control. And it is not impossible for the European Union to break up into nationalist rivalries in the next five to ten years, in part with Trump's blessing. Moreover, Moscow and Beijing could both refuse to accept geo-economic compromises with the United States and Europeans, while the United States and North Korea could continue to threaten nuclear war against each other. And all sides could concurrently refuse to reduce, if not eliminate, their nuclear weaponry. An arms race and buildup of military forces would remain a permanent feature of the highly polluted geopolitical landscape.
In such a global context, any number of overseas conflicts could drag the United States in to protect a strategically significant ally against a presumed threat. But what if an opposing regional major power—or even a lesser state potentially backed by a major nuclear power—decides to call the American bluff? Or what if anti-state terrorist organizations, with differing social and political ideologies, not just Islamist, ostensibly operating alone, purposely seek to spark conflict between major and regional powers—which is even more plausible if alliances continue to polarize?
A self-isolated United States, as it attempts to bully its allies and rivals (and citizens) alike, could then find itself pressed to choose which international conflicts represent an existential priority and which do not. The country would be pushed to engage in a form of strategic triage that would seek to determine which areas and countries might be worth the risk of "hybrid" conflict against rival nuclear powers, and which areas and countries would not.
**INTRODUCTION: A SELF-FULFILLING PROPHECY**
. See Hall Gardner, _Dangerous Crossroads: Europe, Russia and the Future of NATO_ (Westport, CT: Praeger, 1997). The original adage has been attributed to Lord Ismay but probably came from one of his assistants.
. Stan Resor, "Opposition to NATO Expansion: Open Letter to President Clinton," Arms Control Association, June 26, 1997, <https://www.armscontrol.org/act/1997_06-07/natolet> (accessed May 22, 2017).
. On alternatives to NATO enlargement considered in the late 1990s, see Gardner, _Dangerous Crossroads_.
. See Mikhail Zygar, _All the Kremlin's Men_ (New York: Public Affairs, 2016). See also Hall Gardner, _NATO Expansion and the US Strategy in Asia: Surmounting the Global Crisis_ (New York: Palgrave Macmillan, 2013); Hall Gardner, "The Genesis of NATO Enlargement and of War 'Over' Kosovo," _Central and Southeastern Europe in Transition: Perspectives on Success and Failure Since 1989_ , ed. Hall Gardner (Westport, CT: Praeger, March 1999).
. Most of my books have addressed the question of Russian revanche since 1994: _Dangerous Crossroads_ ; _Surviving the Millennium: American Global Strategy, the Collapse of the Soviet Empire, and the Question of Peace_ (Westport, CT: Praeger, 1994); _American Global Strategy and the "War on Terrorism"_ (Ashgate, 2005; revised and updated, 2007); _Averting Global War: Regional Challenges, Overextension, and Options for American Strategy_ (New York: Palgrave Macmillan, 2007); _NATO Expansion and the US Strategy in Asia: Surmounting the Global Crisis_ (New York: Palgrave Macmillan, 2013); Hall Gardner, _Crimea, Global Rivalry, and the Vengeance of History_ (New York: Palgrave Macmillan, 2015).
. On the costs of post–September 11, 2001, wars, see the Watson Institute, "Costs of War" Project, <http://watson.brown.edu/costsofwar/> (accessed October 17, 2017).
. Ibid. These huge costs are, to a large extent, due to the new form of post–Cold War "short war illusion" in which the initial military interventions are rapid, but the peacekeeping and peacemaking have proven to be very long term. See Gardner, _American Global Strategy_.
. Donald J. Trump, "Remarks of President Donald J. Trump, as Prepared for Delivery, Inaugural Address," Washington, DC, January 20, 2017, <https://www.whitehouse.gov/inaugural-address> (accessed November 13, 2017).
**CHAPTER 1: THE PERILS OF THE NEW "AMERICA FIRST" NATIONALISM**
. Nick Gass, "Trump: Taking Back Crimea Would Trigger World War III," _Politico_ , August 1, 2016, <http://www.politico.com/story/2016/08/donald-trump-crimea-ukraine-war-226522> (accessed May 22, 2017).
. Jordan Fabian and Evelyn Rupert, "Trump Promises Chinese President He'll Honor 'One China' Policy," _Hill_ , February 10, 2017, <http://thehill.com/homenews/administration/318874-trump-to-honor-one-china-policy> (accessed May 22, 2017).
. Austin Ramzy, "Kim Jong-un Called Trump a 'Dotard.' What Does That Even Mean?" _New York Times_ , September 22, 2017, <https://www.nytimes.com/2017/09/22/world/asia/trump-north-korea-dotard.html> (accessed November 5, 2017).
. Robert Helbig and Guillaume Lasconjarias, "Winning Peace and Exporting Stability: Colombia as NATO's Next Global Partner?" _NATO Research Paper_ 138 (May 2017), <http://www.ndc.nato.int/news/news.php?icode=1056> (accessed October 17, 2017).
. Ruth Sherlock, "America's Allies Are 'Ripping Us Off' Says Donald Trump," _Telegraph_ , March 27, 2016, <http://www.telegraph.co.uk/news/2016/03/27/americas-allies-are-ripping-us-off-says-donald-trump/> (accessed November 5, 2017).
. Peter Navarro, who was appointed as head of the newly created National Trade Council, blames Beijing for the loss of 57,000 American factories and 25 million jobs. He has called China a "global pollution factory" and "disease incubator." Tom Phillips, "'Brutal, Amoral, Ruthless, Cheating': How Trump's New Trade Tsar Sees China," _Guardian_ , December 22, 2016, <https://www.theguardian.com/world/2016/dec/22/brutal-amoral-ruthless-cheating-trumps-trade-industrial-peter-navarro-views-on-china> (accessed October 17, 2017).
. On the failure of new technological innovation to produce jobs relative to previous epochs, see Robert J. Gordon, _The Rise and Fall of American Growth: The US Standard of Living since the Civil War_ (Princeton, NJ: Princeton University Press, 2016); Elena Holodny, "Trump Vows to 'Crack Down' on Anyone Who Violates Trade Agreements," _Business Insider_ , January 20, 2017, http://uk.businessinsider.com/trump-trade-deal-plans-on-whitehousegov-2017-1?r=US&IR=T (accessed October 23, 2017).
. Ken Moak, "A US-China Trade War Is the Last Thing the World Needs," _Asian Times_ , August 9, 2017, <http://www.atimes.com/us-china-trade-war-last-thing-world-needs/> (accessed October 23, 2017).
. Tom Murse, "How Much U.S. Debt Does China Really Own?" ThoughtCo, February 28, 2017, <https://www.thoughtco.com/how-much-debt-does-china-own-3321769> (accessed November 5, 2017).
. "There is surely something odd about the world's greatest power being the world's greatest debtor. In order to finance prevailing levels of consumption and investment, must the United States be as dependent as it is on the discretionary acts of what are inevitably political entities in other countries?" Lawrence H. Summers, "The United States and the Global Adjustment Process," speech at the Third Annual Stavros S. Niarchos Lecture, March 23, 2004 (Washington DC: Peterson Institute for International Economics, 2004), <https://piie.com/commentary/speeches-papers/united-states-and-global-adjustment-process> (accessed November 16, 2017).
. President Donald J. Trump, quoted by CNN on its Twitter account @CNN, February 24, 2017, <https://twitter.com/cnn/status/835157246212460546?lang=en> (accessed November 13, 2017).
. David Brunnstrom and Matt Spetalnick, "Tillerson Says China Should Be Barred from South China Sea islands," Reuters, January 11, 2017, <http://www.reuters.com/article/us-congress-tillerson-china-idUSKBN14V2KZ> (accessed October 23, 2017). Tillerson later softened his position; Jesse Johnson "Behind the Scenes, Tillerson Tones down Rhetoric on South China Sea," _Japan Times_ , February 7, 2017, <http://www.japantimes.co.jp/news/2017/02/07/asia-pacific/behind-scenes-tillerson-tones-rhetoric-south-china-sea/#.WR7k6PqGP8Q> (accessed October 23, 2017).
. Alec Luhn, "Russia Bans Siberia Independence March," _Guardian_ , August 5, 2014, <https://www.theguardian.com/world/2014/aug/05/russia-bans-siberia-independence-march-extremism-law> (accessed October 23, 2017).
. Ben Aris, "Moscow Blog: Is Russia Seeing the Start of a Colour Revolution?" _Intellinews_ , March 26, 2017, http://www.intellinews.com/moscow-blog-is-russia-seeing-the-start-of-a-colour-revolution-118327/?source=blogs&inf_contact_key=a4781fd4783dd3593aaa41c8da6700a29bf3206cfe5b7795e3d64b33000606fb (accessed October 23, 2017).
. Donald J. Trump Presidential Campaign, "Donald J. Trump Military Readiness Remarks," press release, September 7, 2016, <https://warsclerotic.com/2016/09/07/donald-j-trump-%e2%80%8bmilitary-readiness-remarks/> (October 23, 2017).
. In Trump's October 2016 debate with Hillary Clinton: "Our nuclear program has fallen way behind and they have gone wild with their nuclear program. Not good. Our government shouldn't have allowed that to happen. Russia is new in terms of nuclear and we are old and tired and exhausted in terms of nuclear. A very bad thing." Staff, "Full Transcript: Second 2016 Presidential Debate," _Politico_ , October 10, 2016, <http://www.politico.com/story/2016/10/2016-presidential-debate-transcript-229519> (accessed May 22, 2017).
. Ploughshares Fund, "Obama's Prague Speech: A World Without Nuclear Weapons," Ploughshares Fund, December 8, 2016, <http://www.ploughshares.org/issues-analysis/article/obamas-prague-speech-world-without-nuclear-weapons> (accessed May 22, 2017).
. Dov H. Levin, "When the Great Power Gets a Vote: The Effects of Great Power Electoral Interventions on Election Results," _International Studies Quarterly_ 60, no. 2 (2016): 189–202, <https://www.isanet.org/Publications/ISQ/Posts/ID/5027/When-the-Great-Power-Gets-a-Vote-The-Effects-of-Great-Power-Electoral-Interventions-on-Election-Results> (accessed October 23, 2017); Dov Levin, "Database Tracks History of US Meddling in Foreign Elections," interview by Ari Shapiro, NPR, _All Things Considered_ , December 22, 2016, <http://www.npr.org/2016/12/22/506625913/database-tracks-history-of-u-s-meddling-in-foreign-elections> <https://academic.oup.com/isq/article-abstract/60/2/189/1750842/When-the-Great-Power-Gets-a-Vote-The-Effects-of?redirectedFrom=fulltext> (accessed October 23, 2017).
. Hall Gardner, "Iranian and Russian Versions of 'Little Green Men' and Contemporary Conflict," NATO Defense College Research Paper 123, December 15, 2015, <http://www.ndc.nato.int/news/news.php?icode=885> (accessed November 5, 2017).
. Darya Korsunskaya, "Putin Says Russia Must Prevent 'Color Revolution,'" Reuters, November 20, 2014, <https://www.yahoo.com/news/putin-says-russia-must-guard-against-color-revolutions-135807378.html> (accessed October 23, 2017).
. Brendan I. Koener, "Inside the Cyberattack That Shocked the US Government," _Wired_ , October 23, 2016, <https://www.wired.com/2016/10/inside-cyberattack-shocked-us-government/> (accessed October 23, 2017).
. Andy Greenberg, "The WannaCry Ransomware Has a Link to Suspected North Korean Hackers," _Wired_ , May 15, 2017, <https://www.wired.com/2017/05/wannacry-ransomware-link-suspected-north-korean-hackers/> (October 23, 2017).
. "At this time, roughly 30 nations employ offensive cyber programs.... [The] future is burdened by an irony: Stuxnet started as nuclear counter-proliferation and ended up to open the door to proliferation that is much more difficult to control: The proliferation of cyber weapon technology." Ralph Langner, _To Kill a Centrifuge: A Technical Analysis of What Stuxnet's Creators Tried to Achieve_ (Arlington, Hamburg, Munich: Langner Group, November 2013), <http://www.langner.com/en/wp-content/uploads/2013/11/To-kill-a-centrifuge.pdf> (accessed October 31, 2017).
. Hall Gardner, _The Failure to Prevent World War I: The Unexpected Armageddon_ (Farnham, UK: Ashgate, 2013).
. See Hans Morgenthau and Kenneth Thompson, _Politics Among Nations_ , 6th ed. (New York: McGraw-Hill, 1985); Hall Gardner, _American Global Strategy and the "War on Terrorism"_ (Farnham, UK: Ashgate, 2007).
**CHAPTER 2: INAUGURATION TREMORS**
. In addition to expanding sales of George Orwell's 1984, Trump's unexpected presidential victory brought back the book _It Can't Happen Here_ , by Sinclair Lewis (Garden City, NY: Doubleday, Doran, 1935). In the book, the "Respectables" (respectable individuals) are unable to accept the fact that "justified discontent...against the smart politicians and the Plush Horses of Plutocracy" had permitted the largely unexpected rise to power in the United States of a right-wing authoritarian leader, Buzz Windrip, who bears some resemblance to Trump.
. Trump won the electoral college votes by 306 to Clinton's 232. Clinton won the popular vote by 48.2 percent to Trump's 46.1 percent, with roughly 58 percent of eligible voters voting. Trump was supported by roughly 58 percent of all white voters, as compared to only 8 percent of African American voters, and 29 percent each of Hispanics and Asian Americans, according to exit polls at the time of the vote. More males voted for Trump than did females (but he still obtained 42 percent of the women's vote); more people over forty and with incomes higher than $50,000 voted for Trump than Clinton; and Trump also obtained more rural and suburban votes than Clinton, who obtained more urban votes. Skye Gould, "7 Charts Show Who Propelled Trump to Victory," _Business Insider_ , November 11, 2016 http://nordic.businessinsider.com/exit-polls-who-voted-for-trump-clinton-2016-11?r=UK&IR=T#while-people-living-in-urban-areas-predictably-voted-democrat-and-those-in-rural-areas-voted-republican-its-interesting-to-see-that-trump-captured-more-votes-from-people-living-in-the-suburbs-than-clinton-did-6 (accessed May 23, 2017); Philip Bump, "Donald Trump Will Be President Thanks to 80,000 People in Three States," _Washington Post_ , December 1, 2016, <https://www.washingtonpost.com/news/the-fix/wp/2016/12/01/donald-trump-will-be-president-thanks-to-80000-people-in-three-states/?utm_term=.19386fdf51fa> (accessed May 26, 2017).
. Drew DeSilver, "Trump's Victory Another Example of How Electoral College Wins Are Bigger than Popular Vote Ones," Pew Research Center, Washington, DC, December 20, 2016, <http://www.pewresearch.org/fact-tank/2016/12/20/why-electoral-college-landslides-are-easier-to-win-than-popular-vote-ones/> (accessed May 26, 2017).
. Jonah Engel Bromwich, "Felony Charges for Journalists Arrested at Inauguration Protests Raise Fears for Press Freedom," _New York Times_ , January 25, 2017, <https://www.nytimes.com/2017/01/25/business/media/journalists-arrested-trump-inauguration.html> (accessed May 26, 2017).
. On January 19, 2017, Trump promised that he would cut funding for a large number of governmental programs. For the sixty-six programs he proposed to cut in May, see Niv Elis, "Here Are the 66 programs Eliminated in Trump's Budget," _Hill_ , May 23, 2017, <http://thehill.com/policy/finance/334768-here-are-the-66-programs-eliminated-in-trumps-budget> (accessed October 23, 2017).
. Andrews Wilson, Kenan Davis, Adam Pearce, and Nadia Popovich, "What Trump's Tax Proposal Will Cost," _New York Times_ , April 26, 2017, <https://www.nytimes.com/interactive/2017/04/26/us/politics/what-trumps-tax-proposal-will-cost.html?_r=0> (accessed October 23, 2017).
. "Economic Impact of Immigration: Why Is Labor Important to Farmers?" American Farm Bureau Federation, <http://www.fb.org/issues/immigration-reform/agriculture-labor-reform/economic-impact-of-immigration> (accessed October 23, 2017).
. Devin Henry and Timothy Cama, "Trump Using Executive Orders at Unprecedented Pace," _Hill_ , April 29, 2017, <http://thehill.com/policy/energy-environment/331134-trump-using-executive-orders-at-unprecedented-pace> (accessed May 26, 2017).
. "Global Inequality," Institute for Policy Studies, <http://inequality.org/global-inequality/> (accessed October 23, 2017).
. Lawrence Misahl and Alyssa Davis, "Top CEOs Make 300 Times More than Typical Workers," Economics Policy Institute, June 21, 2015, <http://www.epi.org/publication/top-ceos-make-300-times-more-than-workers-pay-growth-surpasses-market-gains-and-the-rest-of-the-0-1-percent/> (accessed October 23, 2017).
. Tim Mullaney, "Why Corporate CEO Pay Is So High, and Going Higher," CNBC, May 18, 2015, <http://www.cnbc.com/2015/05/18/why-corporate-ceo-pay-is-so-high-and-going-higher.html> (accessed October 23, 2017).
. Bill Allison, Mira Rojanasakul, Brittany Harris, and Cedric Sam, "Tracking the 2016 Presidential Money Race," Bloomberg, December 9, 2016, <https://www.bloomberg.com/politics/graphics/2016-presidential-campaign-fundraising/> (accessed May 26, 2017).
. Jared Kushner and Ivanka Trump have each received top secret security clearances for their work in the White House, but purportedly they omitted to mention their meetings with Russian officials in their security-clearance application forms. Nor did they mention their friendship with Russian billionaire Roman Abramovich and his wife, who have close ties to Putin. Eliza Relman, "Kushner Omits Contacts with Russian Officials in Application for Security Clearance," _Business Insider_ , April 7, 2017, <http://www.businessinsider.com/kushner-omits-contacts-with-russian-officials-in-application-for-security-clearance-2017-4> (accessed May 26, 2017); Marvin Zonis, "The Strange Case of Jared and Ivanka Trump and Their Friendship with Roman Abramovich and Dasha Zhukova and Their Top-Secret Security Clearances," Marvin Zonis, May 12, 2017, <http://www.marvinzonis.com/posts/the-strange-case-of-jared-and-ivanka-trump-and-their-friendship-with-roman-abramovich-and-dasha-zhukova-and-their-top-secret-security-clearances> (accessed May 26, 2017).
. Jay Solomon, "Military Brass Fill Donald Trump's National Security Council," _Wall Street Journal_ , January 26, 2017, <https://www.wsj.com/articles/military-brass-fill-national-security-council-1485478127> (accessed May 26, 2017).
. Morgan Chalfant, "Worries Mount about Vacancies in Trump's State Department," _Hill_ , May 21, 2017, <http://thehill.com/policy/international/334327-worries-mount-about-vacancies-in-trumps-state-department> (accessed October 23, 2017).
. Jim Tankersley and Ana Swanson, "Donald Trump Is Assembling the Richest Administration in Modern American History," _Washington Post_ , November 30, 2016, https://www.washingtonpost.com/news/wonk/wp/2016/11/30/donald-trump-is-assembling-the-richest-administration-in-modern-american-history/?tid=a_inl&utm_term=.6a6dc4b308fa (accessed May 26, 2017). George W. Bush's disastrous cabinet in 2001 possessed an inflation-adjusted net worth of about $250 million—which is roughly one-tenth the wealth of Donald Trump's nominee for commerce secretary alone or $2.5 billion.
. Christopher Hayes, _Twilight of Elites: America after Meritocracy_ (New York: Crown, 2012).
. Ismael Hossein-zadeh, _The Political Economy of US Militarism_ (New York: Palgrave Macmillan, 2006).
. Bouree Lam, "Trump's Promises to Corporate Leaders: Lower Taxes and Fewer Regulations," _Atlantic_ , January 23, 2017, <https://www.theatlantic.com/business/archive/2017/01/trump-corporate-tax-cut/514148/> (accessed October 23, 2017).
. As Trump claimed that he would not cut entitlements, he will most likely run yet another US government budgetary deficit, as the costs of social services payments mount along with rising interest payments on the national debt, which had crossed over 100 percent of US GDP in 2012. By the end of FY2017, the total government debt in the United States, including federal, state, and local, is expected to reach $23.2 trillion, if not more. David Lawder, "No Cuts to US Entitlement Programs in Trump Budget: Mnuchin," Reuters, February 26, 2017, <http://www.reuters.com/article/us-usa-trump-economy-idUSKBN1650LL> (accessed October 23, 2017).
. Mike Lillis, "Dems, Not Trusting Trump, Want Permanent ObamaCare Fix," _Hill_ , April 30, 2017, <http://thehill.com/homenews/house/331172-dems-not-trusting-trump-want-permanent-obamacare-fix> (accessed October 23, 2017); Kate Fritzsche, Sarah Masi, et al., _How Repealing Portions of the Affordable Care Act Would Affect Health Insurance Coverage and Premiums_ (Washington, DC: Congressional Budget Office, January 2017), <https://www.cbo.gov/sites/default/files/115th-congress-2017-2018/reports/52371-coverageandpremiums.pdf> (accessed October 23, 2017).
. A major issue in the new dynamic workplace means that moving from one job to another can not only result in a loss of insurance, but the issue of preexisting conditions will be reintroduced in the new job—even if one had already paid into insurance premiums at the previous job. Peter Sullivan and Rachel Roubein, "Republicans Go to Battle over Pre-Existing Conditions," _Hill_ , May 26, 2017, <http://thehill.com/policy/healthcare/335218-republicans-go-to-battle-over-pre-existing-conditions> (accessed October 23, 2017).
. _Congressional Budget Office Cost Estimate: American Health Care Act_ (Washington, DC: Congressional Budget Office, March 9, 2017), <https://www.cbo.gov/sites/default/files/115th-congress-2017-2018/costestimate/americanhealthcareact.pdf> (accessed October 23, 2017); Peter Sullivan, "GOP Hits the Gas on ObamaCare Repeal," _Hill_ , March 6, 2017, <http://thehill.com/policy/healthcare/322609-gop-releases-bill-to-repeal-and-replace-obamacare> (October 23, 2017); Niall Stanage, "THE MEMO: Trump Faces Long War on Healthcare," _Hill_ , May 6, 2017, <http://thehill.com/homenews/administration/332169-the-memo-trump-faces-long-war-on-healthcare> (accessed October 23, 2017).
. Drew DeSilver, "What the Unemployment Rate Does—and Doesn't—Say about the Economy," Pew Research Center, Washington, DC, March 7, 2017, <http://www.pewresearch.org/fact-tank/2017/03/07/employment-vs-unemployment-different-stories-from-the-jobs-numbers/> (accessed May 26, 2017).
. Graphics, Reuters, "Anxieties about Racism," Reuters, <http://fingfx.thomsonreuters.com/gfx/rngs/USA-TRUMP-POLL-RACE/010040W71X6/index.html> (accessed May 26, 2017).
. According to an eyewitness, it was the white supremacists and neo-fascists, who were shouting "blood and soil" and "Jews will not replace us," who initiated the violence against peaceful counterprotesters promoting antiracism, feminism, LGBTQ rights, and equity. See Jason Wilson, "I Was in Charlottesville. Trump Was Wrong about Violence on the Left," _Guardian_ , August 16, 2017, <https://www.theguardian.com/us-news/2017/aug/16/charlottesville-violence-right-left-trump> (accessed November 5, 2017).
. Sheryl Gay Stolberg and Brian M. Rosenthal, "Man Charged after White Nationalist Rally in Charlottesville Ends in Deadly Violence," _New York Times_ , August 12, 2017, <https://www.nytimes.com/2017/08/12/us/charlottesville-protest-white-nationalist.html?mcubz=1> (accessed October 26, 2017).
. Editorial Board, "Steve 'Turn on the Hate' Bannon, in the White House," _New York Times_ , November 15, 2016. Bannon was removed from his position as National Security Advisor to Trump in April 2017, but he remains a confidant of the president. Roberta Costa and Abby Philip, "Stephen Bannon Removed from National Security Council," _Washington Post_ , April 5, 2017.
. Donald Braman, "Stop-and-Frisk Didn't Make New York Safer," _Atlantic_ , March 26, 2014, <https://www.theatlantic.com/national/archive/2014/03/stop-and-frisk-didnt-make-new-york-safer/359666/> (accessed October 24, 2017); Christina Sterbenz, "Donald Trump Claims Stop-and-Frisk Had a 'Very, Very Big Impact' on New York City's Crime—Here's What the Data Really Says," _Business Insider_ , September 27, 2016, http://uk.businessinsider.com/donald-trump-stop-and-frisk-debate-2016-9?r=US&IR=T (accessed October 24, 2017); David F. Greenberg, "Studying New York City's Crime Decline: Methodological Issues," _Justice Quarterly_ 31, no. 1 (2014), <http://www.tandfonline.com/doi/abs/10.1080/07418825.2012.752026> (accessed May 29, 2017).
. Trump is on the record for banning assault weaponry: "I generally oppose gun control, but I support the ban on assault weapons and I support a slightly longer waiting period to purchase a gun. With today's Internet technology we should be able to tell within 72-hours if a potential gun owner has a record." "Donald Trump on Gun Control," On the Issues, last updated June 15, 2017, <http://www.ontheissues.org/Celeb/Donald_Trump_Gun_Control.htm> (accessed May 29, 2017).
. Trump: "I'm very much in favor of making all concealed-carry permits valid in every state." Ibid.
. Lois Beckett, "'Our Moment to Go on Offense': NRA Makes Big Plans for Trump Presidency," _Guardian_ , November 28, 2016, <https://www.theguardian.com/us-news/2016/nov/28/nra-gun-control-donald-trump-republicans> (accessed October 24, 2017);
. Jessica Schulberg, "Trump Sought Military Equipment for Inauguration, Granted 20-Plane Flyover," _Huffington Post_ , January 19, 2017, <http://www.huffingtonpost.com/entry/trump-military-equipment-inauguration_us_58811f4ae4b096b4a23091f7> (accessed October 24, 2017).
. Donald J. Trump, "Remarks of President Donald J. Trump, as Prepared for Delivery, Inaugural Address," Washington, DC, January 20, 2017, <https://www.whitehouse.gov/inaugural-address> (accessed November 13, 2017).
. South Front, "Trump's Vice President Mike Pence Wants War against Syria and Russia?" Global Research, November 13, 2016, <http://www.globalresearch.ca/trumps-vice-president-mike-pence-wants-war-against-syria-and-russia/5556752> (accessed May 26, 2017).
. _America First: A Budget Blueprint to Make America Great Again_ (Washington, DC: Office of Management and Budget, 2018), <https://www.whitehouse.gov/sites/whitehouse.gov/files/omb/budget/fy2018/2018_blueprint.pdf> (accessed October 24, 2017).
. Obama administration's Omnibus Spending bill sought over $1.1 trillion, split more or less evenly between defense and domestic spending. William J. Broad and David E. Sanger, "Race for Latest Class of Nuclear Arms Threatens to Revive Cold War," _New York Times_ , April 16, 2016, http://www.nytimes.com/2016/04/17/world/europe/atom-bomb-nuclear-weapons-hgv-arms-race-russia-china.html?emc=edit_th_20160417&nl=todaysheadlines&nlid=70196410&_r=0 (accessed October 24, 2017); in addition, Kingston Reif, "Fact Sheets & Briefs: US Nuclear Modernization Programs," Arms Control Association, August, 2017, <https://www.armscontrol.org/factsheets/USNuclearModernization> (accessed May 23, 2017); Fred Kaplan, "Obama's Whopping New Military Budget: Forget What the GOP Says. Obama Loves Big Military Budgets," _Slate_ , February 9, 2016, <http://www.slate.com/articles/news_and_politics/war_stories/2016/02/president_obama_s_military_budget_is_still_one_of_the_biggest_ever.html> (accessed May 23, 2017); Maj. Gen. James F. Martin, "Department of Defense Briefing by Maj. Gen. James Martin on the Fiscal Year 2016 Air Force Budget in the Pentagon Briefing Room," news transcript, United States Department of Defense, February 2, 2015, <http://archive.defense.gov/Transcripts/Transcript.aspx?TranscriptID=5586> (accessed May 23, 2017); James Drew, "Concept of a Nuclear-Armed F-35C Divides Opinion," FlightGlobal.com, August 4, 1970, <https://www.flightglobal.com/news/articles/concept-of-a-nuclear-armed-f-35c-divides-opinion-415353/> (accessed May 23, 2017); James Drew, "US Conducts First Flight Test of Guided B61-12 Nuclear Bomb," FlightGlobal.com, July 10, 1970, <https://www.flightglobal.com/news/articles/us-conducts-first-flight-test-of-guided-b61-12-nuclear-414484/> (accessed May 23, 2017).
. Amy F. Woolf, _The New START Treaty: Central Limits and Key Provisions_ (Washington, DC: Congressional Research Service, February 1, 2017), <https://fas.org/sgp/crs/nuke/R41219.pdf> (accessed October 24, 2017).
. Michael R. Gordon, "Russia Deploys Missile, Violating Treaty and Challenging Trump," _New York Times_ , February 14, 2017, <https://www.nytimes.com/2017/02/14/world/europe/russia-cruise-missile-arms-control-treaty.html> (accessed November 5, 2017); Franz-Stefan Gady, "Russia Tests Topol-M Intercontinental Ballistic Missile," _Diplomat_ , September 28, 2017, <https://thediplomat.com/2017/09/russia-tests-topol-m-intercontinental-ballistic-missile/> (accessed November 5, 2017).
. "Highlights from the 2017 Index," 2017 Index of Military Strength, <http://index.heritage.org/military/2017/assessments/> (accessed October 24, 2017).
. Here, US Secretary of Defense Jim Mattis has discussed the possible deployment of tactical nuclear weaponry with South Korean Defense Minister Song Young-moo, who told his parliament that he had requested that the United States consider the return of tactical nuclear weapons to the Korean Peninsula. No further information was provided. Tara Copp, "Mattis: Use of Tactical Nuclear Weapons Discussed with South Korea," _Defense News_ , September 18, 2017, <https://www.defensenews.com/news/your-military/2017/09/18/mattis-use-of-tactical-nuclear-weapons-discussed-with-south-korea/> (accessed November 5, 2017).
. For Trump proposals: Brendan McGarry, "Trump Unveils Plan to Boost Military with More Troops, Weapons," Military.com, September 7, 2016, <http://www.military.com/daily-news/2016/09/07/trump-unveils-plan-to-boost-military-with-more-troops-weapons.html> (accessed October 24, 2017).
. Secretary of Defense Mattis was cited as stating that the BCA and sequestration have "done more damage to our readiness than the enemies in the field." "DoD Releases Fiscal Year 2018 Budget Proposal," Release No: NR-192-17, May 23, 2017, <https://www.defense.gov/News/News-Releases/News-Release-View/Article/1190216/dod-releases-fiscal-year-2018-budget-proposal/> (accessed November 5, 2017).
. Adam Taylor and Laris Karklis, "This Remarkable Chart Shows How US Defense Spending Dwarfs the Rest of the World," _Washington Post_ , February 9, 2016, <https://www.washingtonpost.com/news/worldviews/wp/2016/02/09/this-remarkable-chart-shows-how-u-s-defense-spending-dwarfs-the-rest-of-the-world/?utm_term=.1df2b5f8eebd> (accessed October 24, 2017).
. Nan Tian, Aude Fleurant, Pieter D. Wezeman, and Siemon T. Wezeman, _Trends in World Military Expenditure, 2016: SIPRI Fact Sheet_ (Solna, Sweden: Stockholm International Peace Research Institute, April 2017), <https://www.sipri.org/sites/default/files/Trends-world-military-expenditure-2016.pdf> (accessed October 24, 2017).
. Hossein-zadeh, _Political Economy_.
. For details of budget, see Neta C. Crawford, _US Budgetary Costs of Wars through 2016: $4.79 Trillion and Counting Summary of Costs of the US Wars in Iraq, Syria, Afghanistan and Pakistan and Homeland Security_ (Boston: Watson Institute, International & Public Affairs, Brown University, September 2016), <http://watson.brown.edu/costsofwar/files/cow/imce/papers/2016/Costs%20of%20War%20through%202016%20FINAL%20final%20v2.pdf> (accessed October 24, 2017).
. After major tax cuts, Alan Greenspan had dropped US interest rates in 2001–2002 after the dot-com 2000–2002 crisis, instead of raising them, and those rates were kept low from 2002 to 2004, which assisted US borrowing for the military interventions after September 11, 2001, while also providing additional cheap money for China's boom. For the view that the crisis was Greenspan's fault for not raising interest rates in 2002: Susan Lee, "It Really Is All Greenspan's Fault," _Forbes_ , April 3, 2009, <http://www.forbes.com/2009/04/02/greenspan-john-taylor-fed-rates-china-opinions-columnists-housing-bubble.html> (accessed May 26, 2017). For the view it was China's fault: Heleen Mees, "How China's Boom Caused the Financial Crisis," _Foreign Policy_ , January 17, 2012, <http://foreignpolicy.com/2012/01/17/how-chinas-boom-caused-the-financial-crisis/> (accessed May 26, 2017).
. Yochi Dreazen, "Trump Says He's Boosting Defense Spending by $54 Billion. The Real number Is $18 Billion," _Vox_ , February 28, 2017, <http://www.vox.com/policy-and-politics/2017/2/28/14765492/trump-pentagon-budget-billion-state-department-misleading-54-billion> (accessed May 26, 2017); Kristina Wong, "Trump's Navy Build-Up Comes with Steep Price Tag," _Hill_ , January 16, 2017, <http://thehill.com/policy/defense/314311-trumps-navy-build-up-comes-with-steep-price-tag> (accessed May 26, 2017).
For options to cut budget deficit: "Chapter 3: Discretionary Spending Options," _Options for Reducing the Deficit: 2017 to 2026_ (Washington, DC: Congressional Budget Office, February 1, 2017), <https://www.cbo.gov/publication/52142;> <https://www.cbo.gov/sites/default/files/114th-congress-2015-2016/reports/52142-breakout-chapter32.pdf> (accessed May 26, 2017).
. Catherine A. Theohary, _Conventional Arms Transfers to Developing Nations, 2008-2015_ (Washington, DC: Congressional Research Service, December 19, 2016), <https://fas.org/sgp/crs/weapons/R44716.pdf> (accessed October 24, 2017).
. Melvin A. Goodman, _National Insecurity: The Cost of American Militarism_ (San Francisco: City Lights, 2013).
. Mike De Bonis, "The Pentagon Found $125 Billion in Waste. Now a GOP Chairman is Asking Other Agencies What They've Found," _Washington Post_ , February 10, 2017, <https://www.washingtonpost.com/news/powerpost/wp/2017/02/10/the-pentagon-found-125-billion-in-waste-now-a-gop-chairman-is-asking-other-agencies-what-theyve-found/?utm_term=.819a68f3904b> (accessed October 24, 2017).
. John M. Donnelly, "Atomic Arsenal Costs Ballooning by Billions of Dollars," _Roll Call_ , January 9, 2017, <http://www.rollcall.com/news/policy/atomic-arsenal-costs-ballooning-billions-dollars> (accessed October 24, 2017).
. Geoff Ziezulewicz, "B61-12 Life Extension Program Receives NNSA Approval," UPI, August 2, 2016, <http://www.upi.com/Business_News/Security-Industry/2016/08/02/B61-12-life-extension-program-receives-NNSA-approval/3261470147434/> (accessed October 24, 2017).
. Will Saetren, "3 Nuclear Weapons Programs Obama Should Kill," _National Interest_ , September 11, 2016, <http://nationalinterest.org/feature/3-nuclear-weapons-programs-president-obama-should-kill-17654> (accessed October 24, 2017).
. Amanda Macias, "The Legacy of the 2011 Debt Ceiling Fight Is the Biggest Issue the Next President Will Face on Day One," _Business Insider_ , August 14, 2016, http://www.businessinsider.com/budget-control-act-2016-8?r=UK&IR=T/#what-is-the-budget-control-act-1 (accessed October 24, 2017).
. See William Hartung, _Prophets of War: Lockheed Martin and the Making of the Military Industrial Complex_ (New York: Nation Books, 2012).
. Andrea Shalal-Esa, "Exclusive: US Sees Lifetime Cost of F-35 Fighter at $1.45 Trillion," Reuters, March 29, 2012, <http://www.reuters.com/article/us-lockheed-fighter-idUSBRE82S03L20120329> (accessed October 24, 2017).
. _F-35 Strike Fighter: DOD Needs to Complete Developmental Testing before Making Significant New Investments_ (Washington, DC: US Government Accountability Office, April 2017), <http://www.gao.gov/assets/690/684208.pdf> (accessed October 24, 2017); "DoD Releases Fiscal Year 2018 Budget Proposal."
. Dave Majumdar, "America's F-35 Stealth Fighter vs. Russia's Su-35: Who Wins?" _National Interest_ , September 15, 2015, <http://nationalinterest.org/blog/the-buzz/americas-f-35-stealth-fighter-vs-russias-su-35-who-wins-13855> (accessed October 24, 2017); Rakesh Krishnan Simha "Stealth Troubles: Why Leading Air Forces Want More Traditional Warplanes," _Russia Beyond_ , January 23, 2017, <https://in.rbth.com/blogs/stranger_than_fiction/2017/01/23/stealth-troubles-why-leading-air-forces-want-more-traditional-warplanes_686613> (accessed October 25, 2017).
. During the Reagan years, the gap between the expanding economy and the shrinking investment in public works/capital meant that the growing private economy augmented demands for public services that could not be supplied by a public sector that was decreasing in size. Hossein-zadeh, _Political Economy_. Here, however, Trump hopes to replace publicly supplied services and infrastructure with private financing.
. Alex Ward, "What America's New Arms Deal with Saudi Arabia Says about the Trump Administration," _Vox_ , May 20, 2017, <https://www.vox.com/2017/5/20/15626638/trump-saudi-arabia-arms-deal> (accessed October 25, 2017).
. Lingling Wei, "Push by China's Sovereign-Wealth Fund," _Wall Street Journal_ , May 17, 2017, <https://www.wsj.com/articles/trump-trade-reboot-spurs-u-s-push-by-chinas-sovereign-wealth-fund-1495022339> (accessed October 25, 2017); Viola Zhou, "China's Sovereign Wealth Fund Wants to Invest in the US Infrastructure Rebuild, Chairman Says," _South China Morning Post_ , January 16, 2017, <http://www.scmp.com/business/companies/article/2062593/chinas-sovereign-wealth-fund-wants-invest-us-infrastructure> (accessed October 25, 2017).
. Jaclyn Reiss, "Bernie Sanders Rails against Trump after Speech," _Boston Globe_ , March 1, 2017, <http://www.bostonglobe.com/news/politics/2017/03/01/bernie-sanders-rails-against-trump-after-speech/zFBd9Ff88rSdszRVrdlCJL/story.html?p1=Article_Trending_Most_Viewed> (accessed May 26, 2017). According to critics such as Bernie Sanders, Trump also wants to give tax breaks worth as much as $3 trillion to the very rich.
. This was the case of arms firms who pushed NATO enlargement as a means to expand arms sales in the late 1990s. Katherine Q. Seelye, "Arms Contractors Spend to Promote an Expanded NATO," _New York Times_ , March 30, 1998, http://www.nytimes.com/1998/03/30/world/arms-contractors-spend-to-promote-an-expanded-nato.html?pagewanted=all&_r=0US (accessed October 25, 2017).
. Mike Pence has called for reviving two wasteful weapons programs: the Army's Future Combat Systems and the Air Force's F-22 Raptor fighter jet, which was dumped by the Obama administration. Then Defense Secretary Robert Gates's decision to scrap the F-22 was in part the result of major cost overruns. Austin Wright, "Pence Could Undermine Key Trump War Argument," _Politico_ , July 14, 2016, <http://www.politico.com/story/2016/07/mike-pence-donald-trump-war-argument-225565> (accessed October 25, 2017); Jeremiah Gertler, _Air Force F-22 Fighter Program_ (Washington, DC: Congressional Research Service, July 11, 2013), <https://www.fas.org/sgp/crs/weapons/RL31673.pdf> (accessed October 25, 2017).
. Staff, "Trump's Remarks on Military Readiness," _Washington Examiner_ , September 7, 2016, <http://www.washingtonexaminer.com/text-trumps-remarks-on-military-readiness/article/2601173> (accessed October 25, 2017).
. McCain's plan would add $430 billion more than FY2017 projections, thus bringing the defense budget to $800 billion in fiscal 2022. McGarry, "Trump Unveils Plan."
. Daniel Bukszpan, "Why Bernie Sanders Is Backing a $1.5 Trillion Military Boondoggle," CNBC, July 12, 2016, <http://www.cnbc.com/2016/07/12/why-bernie-sanders-is-backing-a-15-trillion-military-boondoggle.html> (accessed May 26, 2017). Other boondoggles include the excessive expense of the USS _Enterprise_ and USS _Zumwald_.
**CHAPTER 3: THE NEW BOGEYMAN**
. Ruth Marcus, "Count on Trump to Be a Sore Loser," _Washington Post_ , August 5, 2016, <https://www.pressreader.com/usa/the-washington-post/20160805/282050506447036> (accessed May 29, 2017).
. These agencies included the CIA, the National Security Agency, the Justice Department, the Treasury Department's Financial Crimes Enforcement Network, and representatives of the director of national intelligence. Former CIA Director John Brennan testified that he had warned Alexander Bortnikov, head of the FSB Russian intelligence service, against tampering with the US elections in August 2016. Bortnikov, however, denied that Russia was meddling, but said he would raise the issue with President Vladimir Putin. Brennan could not describe Russian-Trump team interactions as "collusion." Tom LoBianco, "Ex-CIA Chief John Brennan: Russians Contacted Trump Campaign," CNN, May 23, 2017, <http://edition.cnn.com/2017/05/23/politics/john-brennan-house-intelligence-committee/> (accessed October 25, 2017).
. Evelyn Rupert, "Senate Intel Panel to Probe Trump Team's Ties to Russia," _Hill_ , January 13, 2017, <http://thehill.com/policy/cybersecurity/314298-senate-intel-committee-to-probe-russian-interference-in-election> (accessed October 25, 2017).
. Donald Trump (@realDonaldTrump): "Senators should focus their energies on ISIS, illegal immigration and border security instead of always looking to start World War III," Twitter, January 29, 2017, 1:49 p.m., <https://twitter.com/realdonaldtrump/status/825823217025691648?lang=en> (accessed October 25, 2017).
. Richard Gonzales, "Senate Panel Plans to Investigate Russian Activities during US Elections," NPR, January 13, 2017, <http://www.npr.org/sections/the-two-way/2017/01/13/509762116/senate-panel-plans-to-investigate-russian-activities-during-u-s-elections> (accessed October 25, 2017).
. Ellen Nakashima, "Russia's Apparent Meddling in US Election Is Not an Act of War, Cyber Expert Says," _Washington Post_ , February 7, 2017, https://www.washingtonpost.com/news/checkpoint/wp/2017/02/07/russias-apparent-meddling-in-u-s-election-is-not-an-act-of-war-cyber-expertsays/?postshare=1741486560691485&tid=ss_mail&utm_term=.ed8edfa98213 (accessed May 29, 2017).
. Zack Beauchamp and Andrew Prokop, "Robert Mueller's Russia Investigation, and Why Trump Is so Afraid of It," Vox News, August 3, 2017, <https://www.vox.com/world/2017/7/24/16008272/robert-mueller-fbi-trump-russia-explained> (accessed November 5, 2017).
. David Voreacos, Stephanie Baker, and Shannon Pettypiece, "Three Trump Associates Charged in Russia Collusion Probe," Bloomberg News, October 30, 2017, <https://www.bloomberg.com/news/articles/2017-10-30/trump-s-ex-campaign-chairman-manafort-told-to-surrender-to-u-s> (accessed November 5, 2017).
. "Statement by President Donald J. Trump on the Signing of H.R. 3364," news release, White House, Office of the Press Secretary, August 2, 2017, <https://www.whitehouse.gov/the-press-office/2017/08/02/statement-president-donald-j-trump-signing-hr-3364> (accessed October 25, 2017).
. Steve Mollman, "What Would Actually Happen If Donald Trump Shot That Russian Ship 'Right out of the Water,'" _Quartz_ , February 17, 2017, <https://qz.com/913443/what-would-actually-happen-if-trump-shot-that-russian-ship-right-out-of-the-water/> (accessed October 25, 2017).
. Darlene Superville, "Trump Defends Seeking Better Ties with Russia," _Intelligencer_ , February 6, 2017, <http://www.theintelligencer.net/news/top-headlines/2017/02/trump-u-s-isnt-so-innocent/> (accessed October 25, 2017).
. Ibid. This answer (which is not very presidential, but crudely honest) enraged a number of political elites, as if they had forgotten CIA/KGB assassinations during the Cold War and after. The grim reality is that the US foreign policy elites must deal with many leaders that are "killers," whether Washington likes it or not, and whether those countries like dealing with the United States.
. Numerous critics of Putin have died in mysterious circumstances, but it is not clear who paid for the killings. Mary Louise Kelly, "The Curious Deaths of Kremlin Critics," NPR, August 30, 2016, <http://www.npr.org/sections/parallels/2016/08/30/491898040/the-curious-deaths-of-kremlin-critics> (accessed November 5, 2017).
. "President Putin 'Probably' Approved Litvinenko Murder," BBC, January 21, 2016, <http://www.bbc.com/news/uk-35370819> (accessed November 5, 2017).
. Noam Chomsky, "On the NATO Bombing of Yugoslavia," interview by Danilo Mandic, RTS Online, April 25, 2006, <https://chomsky.info/20060425/> (accessed October 26, 2017).
. In November 2005, a British newspaper leaked a thus far unconfirmed secret report that purported that Tony Blair was able to convince George Bush not to bomb the Qatari TV station Al Jazeera in Doha in April 2004 during the US siege of Falluja in Iraq. Bush purportedly threatened to bomb Al Jazeera due to the fact that unembedded Al Jazeera journalists had been accusing the US military of atrocities. Jeremy Scahill, "Did Bush Really Want to Bomb Al Jazeera?" _Nation_ , November 23, 2005, <https://www.thenation.com/article/did-bush-really-want-bomb-al-jazeera/> (accessed October 26, 2017). Seymour M. Hersh, "Chain of Command," _New Yorker_ , May 17, 2004. In any case, the purpose of the leak seemed to be to make Blair look like a good guy who stood up to Bush, and not Bush's "poodle," as was often claimed at the time.
. Veteran Intelligence Professionals for Sanity (VIPS), "A Demand for Russian 'Hacking' Proof," _Consortium News_ , January 17, 2017, <https://consortiumnews.com/2017/01/17/a-demand-for-russian-hacking-proof/>. Here there should be a distinction between "cyber-leaks" and "cyber-hacking," as it is not clear which the Kremlin is allegedly responsible for.
. "Russia: The 'Cloud' over the White House," BBC News, September 29, 2017, <http://www.bbc.com/news/world-us-canada-38966846> (accessed October 26, 2017).
. Michelle Ye Hee Lee, "Julian Assange's Claim That There Was No Russian Involvement in WikiLeaks Emails," _Washington Post_ , January 5, 2017, <https://www.washingtonpost.com/news/fact-checker/wp/2017/01/05/julian-assanges-claim-that-there-was-no-russian-involvement-in-wikileaks-emails/?utm_term=.563247ffca74> (accessed May 29, 2017).
. Andrew Higgins, "Maybe Private Russian Hackers Meddled in Election, Putin Says," _New York Times_ , June 1, 2017, https://www.nytimes.com/2017/06/01/world/europe/vladimir-putin-donald-trump-hacking.html?emc=edit_na_20170601&nl=breaking-news&nlid=70196410&ref=cta&_r=0 (accessed October 26, 2017).
. Sean Gallagher, "WikiLeaks to US Government: Stop Leaking Secrets! WikiLeaks Joins Trump in Decrying NBC 'Exclusive' on Top Secret Intelligence Report," _Ars Technica_ , January 6, 2017, <https://arstechnica.com/tech-policy/2017/01/wikileaks-to-us-government-stop-leaking-secrets/> (accessed November 5, 2017). NBC, for example, illegally received a leaked document before Trump saw it, with no threat of punishment; yet Chelsea Manning was accused of "aiding the enemy" by handing over secret documents to Julian Assange.
. FBI Director James Comey had announced a new inquiry into Clinton's private email server only eleven days before the presidential election, but then he dropped the issue just two days before Americans voted. It is possible that the Kremlin was involved in hacking Democratic National Committee emails, but other groups could have been involved as well, possibly in an effort to make Moscow look like the culprit. Alex Thompson, "Why Nothing Was Done," Vice News, December 16, 2016, <https://news.vice.com/story/obama-explains-why-he-didnt-retaliate-against-russia-for-hacking-hillary-clintons-campaign> (accessed October 26, 2017).
. Pamela Engel, "Clinton Never Set Foot in Wisconsin—Then She Lost It, and It Helped Cost Her the Presidency," _Business Insider_ , November 9, 2016, http://uk.businessinsider.com/clinton-losing-wisconsin-results-2016-11?r=US&IR=T (accessed October 26, 2017).
. Jacob Heilbrunn, "It Was Inevitable That Trump Would Fire James Comey," _National Interest_ , May 9, 2017, <http://nationalinterest.org/feature/it-was-inevitable-trump-would-fire-james-comey-20589> (accessed October 26, 2017). Ali Vitali and Corky Siemaszko, "Trump Interview with Lester Holt: President Asked Comey If He Was Under Investigation."
For an analysis of Trump's interview with Lester Holt on NBC, see Amy Davidson Sorkin, "The Threat in President Trump's Interview with Lester Holt," _New Yorker_ , May 12, 2017, <https://www.newyorker.com/news/amy-davidson/the-threat-in-president-trumps-interview-with-lester-holt> (accessed November 5, 2017).
. Maggie Haberman, Mark Mazzetti, and Matt Apuzzomay, "Kushner Is Said to Have Discussed a Secret Channel to Talk to Russia," _New York Times_ , May 26, 2017, https://www.nytimes.com/2017/05/26/us/politics/kushner-talked-to-russian-envoy-about-creating-secret-channel-with-kremlin.html?emc=edit_th_20170527&nl=todaysheadlines&nlid=70196410&_r=0 (accessed October 26, 2017). Jo Becker, Matt Apuzzo, and Adam Goldman, "Trump's Son Met with Russian Lawyer after Being Promised Damaging Information on Clinton," _New York Times_ , July 9, 2017, <https://www.nytimes.com/2017/07/09/us/politics/trump-russia-kushner-manafort.html> (accessed November 5, 2017).
. See controversial report written by former British MI6 intelligence agent Christopher Steele, "Company Intelligence Report 2016/080: US Presidential Election: Republican Candidate Donald Trump's Activities in Russia and Compromising Relationship with the Kremlin," available at <https://www.documentcloud.org/documents/3259984-Trump-Intelligence-Allegations.html> (accessed October 26, 2017). This unconfirmed report was passed around Washington before it was placed on _BuzzFeed_ 's website: Ken Bensinger, Miriam Elder, and Mark Schoofs, "These Reports Allege Trump Has Deep Ties to Russia," _BuzzFeed_ , January 10, 2017, <https://www.buzzfeed.com/kenbensinger/these-reports-allege-trump-has-deep-ties-to-russia?utm_term=.mljnBG5rbp#.la3vMX57eJ> (accessed November 2, 2017). By November 2017, "Clinton Campaign, DNC Paid for Research That Led to Russia Dossier," _Washington Post_ , <https://www.washingtonpost.com/world/national-security/clinton-campaign-dnc-paid-for-research-that-led-to-russia-dossier/2017/10/24/226fabf0-b8e4-11e7-a908-a3470754bbb9_story.html?utm_term=.597105078b6b> (accessed November 5, 2017).
. For a critique of the Fusion GPS report, see Philip Bump, "What the Trump Dossier Says—and What It Doesn't," _Washington Post_ , October 25, 2017, https://www.washingtonpost.com/news/politics/wp/2017/10/25/what-the-trump-dossier-says-and-what-it-doesnt/?tid=a_inl&utm_term=.a63e2e8cf802 (accessed November 5, 2017).
. Mikhail Fishman, Daria Litvinova, "Why Putin Fired His Chief of Staff and Longtime Ally," _Moscow Times_ , August 25, 2016, <https://themoscowtimes.com/articles/why-putin-replaced-head-of-presidential-administration-54978> (accessed October 26, 2017).
. For a cautious view, see Fydor Lukyanov, "Like Obama, Trump Is Unlikely to Lead as the World's Policeman," _Russia in Global Affairs_ , November 18, 2016, <http://eng.globalaffairs.ru/redcol/Like-Obama-Trump-Is-Unlikely-to-Lead-as-the-Worlds-Policeman-18466> (accessed October 26, 2017).
. Bryan Logan, "Nikki Haley Just Delivered the Trump Administration's Most Hawkish Words Yet toward Russia," _Business Insider_ , February 3, 2017, http://uk.businessinsider.com/trump-nikki-haley-russia-ukraine-2017-2?r=US&IR=T (accessed October 26, 2017).
. David M. Herszenhorn and Ellen Barrydec, "Putin Contends Clinton Incited Unrest Over Vote," _New York Times_ , December 8, 2011, <http://www.nytimes.com/2011/12/09/world/europe/putin-accuses-clinton-of-instigating-russian-protests.html> (accessed October 26, 2017).
. Priyanka Boghani, "Putin's Legal Crackdown on Civil Society," _Frontline_ , January 13, 2015, <http://www.pbs.org/wgbh/frontline/article/putins-legal-crackdown-on-civil-society/> (accessed October 26, 2017). Former US Ambassador Michael McFaul stated he personally heard Putin complain about Clinton's comments. Michael Crowley and Julia Ioffe, "Why Putin Hates Hillary," _Politico_ , July 25, 2016, <http://www.politico.com/story/2016/07/clinton-putin-226153> (accessed October 26, 2017). In an effort to delegitimize Putin before the Russian people, Washington elites accused him of being a billionaire: Stephen Grey, Andrey Kuzmin, and Elizabeth Piper, "Putin's Daughter, a Young Billionaire and the President's Friends," Reuters, November 10, 2015, <http://www.reuters.com/investigates/special-report/russia-capitalism-daughters/> (accessed October 26, 2017).
. "Ukraine Crisis: Transcript of Leaked Nuland-Pyatt Call," BBC, February 7, 2014 <http://www.bbc.com/news/world-europe-26079957> (accessed October 26, 2017).
. Josh Meyer, "DNC Email Hack: Why Vladimir Putin Hates Hillary Clinton," NBC News, July 27, 2016, <http://www.nbcnews.com/news/us-news/why-putin-hates-hillary-clinton-n617236> (accessed October 26, 2017).
. Jon Swaine and Ed Pilkington, "The Wealthy Men in Trump's Inner Circle with Links to Tax Havens," _Guardian_ , November 5, 2017, <https://www.theguardian.com/news/2017/nov/05/wealthy-men-donald-trump-inner-circle-links-tax-havens> (accessed November 6, 2017).
. Charles Tiefer, "'Paradise Papers' Disclosures of Trump Administration-Russia Ties Warrant Congressional Hearings," _Forbes_ , November 5, 2017, <https://www.forbes.com/sites/charlestiefer/2017/11/05/new-paradise-papers-disclosures-of-trump-russia-ties-warrant-immediate-congressional-hearings/#720d71d14baa> (accessed November 13, 2017).
. ExxonMobil has lost at least $1 billion on its investments in Russia, as of early 2015. Ed Crooks, "Rex Tillerson, ExxonMobil, and the Separation of Oil and State," _Financial Times_ , January 22, 2017, <https://www.ft.com/content/2acabb7a-def9-11e6-9d7c-be108f1c1dce> (accessed October 16, 2017).
. Karoun Demirjian, "Tillerson Says US Should Have Used Other Options in Crimea," _Washington Post_ , January 11, 2017, <https://www.washingtonpost.com/politics/2017/live-updates/trump-white-house/confirmation-hearings-trump-speaks-and-vote-a-rama-analysis-and-updates/tillersons-says-u-s-should-have-used-other-options-in-crimea/?utm_term=.66a0f1f32078> (accessed October 26, 2017).
. In his testimony before Congress, Rex Tillerson affirmed that he would keep US sanctions in place and consider new penalties related to Russian meddling in the presidential election. Ibid.
. Crooks, "Rex Tillerson, ExxonMobil, and the Separation of Oil and State"; Samuel Rubenfeld, Lynn Cook, and Ian Talley, "Exxon Sues US over $2 Million Russian Sanctions Fine," _Market Watch_ , July 21, 2017, <http://www.marketwatch.com/story/exxon-sues-us-over-2-million-russian-sanctions-fine-2017-07-21> (accessed October 26, 2017); Peter Baker and Sophia Kishkovsky, "Trump Signs Russian Sanctions into Law, with Caveats," _New York Times_ , August 2, 2017, <https://www.nytimes.com/2017/08/02/world/europe/trump-russia-sanctions.html> (accessed October 26, 2017).
. Flynn was accused of acting illegally as a foreign agent, paid by Russian Television (RT) and by the Turkish government, without registering as a foreign agent during the Trump campaign. It was argued that his actions could lead him to be "compromised" by Moscow. Katie Bo Williams, "Yates Warned Flynn Was Vulnerable to Blackmail by Russia," _Hill_ , May 8, 2017, <http://thehill.com/policy/national-security/332416-yates-flynn-conduct-created-a-compromise-situation> (accessed October 26, 2017). Philip Giraldi put the issue this way: "...the destruction of Flynn, involving as it may have a number of leakers coming from all across the intelligence community, might be part of a coordinated effort to narrow the Trump White House's options for dealing with Russia. Many in Washington do not want a comfortable working relationship with Putin in spite of the fact that a reset with Moscow should be the No. 1 national-security objective." Philip Giraldi, "More about Russia and Less about Flynn?" _American Conservative_ , February 16, 2017, <http://www.theamericanconservative.com/articles/more-about-russia-and-less-about-flynn/> (accessed October 26, 2017).
. Megan Twohey and Scott Shane, "A Back-Channel Plan for Ukraine and Russia, Courtesy of Trump Associates," _New York Times_ , February 19, 2017, <https://www.nytimes.com/2017/02/19/us/politics/donald-trump-ukraine-russia.html?> (accessed May 29, 2017); Danielle Kurtzleben, "Michael Flynn Left the Trump White House This Week. Here's How That Happened," NPR, February 14, 2017, <http://www.npr.org/2017/02/14/515233669/michael-flynn-left-the-trump-white-house-this-week-heres-how-that-happened> (accessed May 29, 2017). Other allegations included Trump team efforts to discuss with Moscow a change in US policy toward Syria.
. "Transcript: Michael Flynn on ISIL," Al Jazeera, January 13, 2016, <http://www.aljazeera.com/programmes/headtohead/2016/01/transcript-michael-flynn-160104174144334.html> (accessed November 6, 2017).
. Elana Schor and Nolan D. McCaskill, "Republicans Warn Trump against Lifting Russia Sanctions," _Politico_ , January 27, 2017, <http://www.politico.com/story/2017/01/trump-russia-sanctions-john-mccain-response-234268> (accessed May 29, 2017).
. Russia essentially has had three sets of sanctions put on it by the US since Crimea became part of Russia. US sectoral sanctions on Russia (on energy) are due to expire in December 2017 and could be renewed—if the US Senate does not try to block such a move. "Ukraine/Russia Related Sanctions Program," US Department of the Treasury, last updated September 29, 2017, <https://www.treasury.gov/resource-center/sanctions/Programs/Pages/ukraine.aspx> (accessed October 26, 2017).
. Aaron Blake, "Donald Trump Claims None of Those 3 to 5 Million Illegal Votes Were Cast for Him. Zero," _Washington Post_ , January 26, 2017, <https://www.washingtonpost.com/news/the-fix/wp/2017/01/25/donald-trump-claims-none-of-those-3-to-5-million-illegal-votes-were-cast-for-him-zero/?utm_term=.c260ce102f66> (accessed November 6, 2017).
. Adam Liptak, "Supreme Court Rejects Challenge on 'One Person One Vote,'" _New York Times_ , April 4, 2016, <https://www.nytimes.com/2016/04/05/us/politics/supreme-court-one-person-one-vote.html?_r=0> (accessed October 26, 2017). The 2016 Supreme Court decision that states must count all residents in each voting district, whether they can vote or not, tends to enhance the power of urban areas, thus generally benefitting Democrats, while if only eligible voters would be counted in the census, then political power would shift from cities to rural areas, generally benefiting Republicans. In the near future, Trump supporters will be overseeing the 2020 census and could attempt to manipulate redistricting in each state according to Republican interests through gerrymandering.
. "Remittances to Mexico Jump by Most in 10 Years after Trump Win," Reuters, January 2, 2017, <http://www.reuters.com/article/us-mexico-economy-remittances/remittances-to-mexico-jump-by-most-in-10-years-after-trump-win-idUSKBN14M115> (accessed November 5, 2017).
. The income from remittances reduces poverty in families and discourages their members from engaging in criminal behavior. Remittance flows are an important source of income for many households, communities, and LAC countries. They also enable households to invest more in the education and safety of young people, which helps to prevent crime and provide better job opportunities in the future. Steve Brito, Ana Corbacho, and René Osorio, _Remittances and the Impact on Crime in Mexico_ (Washington, DC: Inter-American Development Bank, May 2014), working paper series no. 514, <https://publications.iadb.org/bitstream/handle/11319/6482/IFD%20WP%20Remittances%20and%20the%20Impact%20on%20Crime%20in%20Mexico.pdf?sequence=1> (accessed October 26, 2017).
. Associated Press, "Trump Says He'll Send Military to Mexico to 'Take Care of' 'Bad Hombres,' But Officials Say Comments Were Lighthearted," _Denver Post_ , February 2, 2017, <http://www.denverpost.com/2017/02/02/trump-mexico-bad-hombres/> (accessed October 26, 2017).
. See argument of Robert A. Levy, "Reflections on Gun Control by a Second Amendment Advocate," _National Law Journal_ , February 12, 2013, <https://www.law.com/nationallawjournal/almID/1202587764936/?slreturn=20171007060056> (accessed November 13, 2017).
. Ali Vitali, "President Trump Signs New Immigration Executive Order," NBC News, March 6, 2017, <http://www.nbcnews.com/politics/white-house/president-trump-signs-new-immigration-executive-order-n724276> (accessed October 26, 2017).
. The Justice Department insisted that ban on immigrants from certain countries was justified for national security reasons. Unless the law can be proved to be discriminatory, the ban may not violate the Establishment and Equal Protection Clause of the Constitution, but it could violate the Immigration and Nationality Act of 1965, which had banned all discrimination against immigrants on the basis of national origin—unless "countries of concern" can be considered exempt. Nolan Rappaport, "If Immigration Ban Goes to Supreme Court, Trump Is Shoo-In to Win," _Hill_ , February 13, 2017, <http://thehill.com/blogs/pundits-blog/immigration/319212-if-immigration-ban-goes-to-supreme-court-trump-is-shoo-in> (accessed October 26, 2017); Andrew Chung and Mica Rosenberg, "Courts Likely to Probe Trump's Intent in Issuing Travel Ban," Reuters, February 13, 2017, <http://www.reuters.com/article/us-usa-trump-immigration-legal-idUSKBN15S14H> (accessed October 26, 2017).
. Trump does, however, appear to believe gun control laws that could impact those "deemed to be a threat to themselves or others" is "something to look into—people with mental health problems are on the streets who shouldn't be." "Donald Trump on Gun Control," _On the Issues_ , last updated June 15, 2017, <http://www.ontheissues.org/Celeb/Donald_Trump_Gun_Control.htm> (accessed May 29, 2017).
. As of April 2016 (prior to the July 14, 2016, Nice attacks) research found that lone right-wing extremists represent a substantial aspect of the lone-actor threat and must not be overlooked. Clare Ellis, Raffaello Pantucci, Jeanine de Roy van Zuijdewijn, Edwin Bakker, Benoît Gomis, Simon Palombi, and Melanie Smith, _Lone-Actor Terrorism: Final Report_ , Countering Lone-Actor Terrorism Series, no. 11 (London: Royal United Services Institute for Defence and Security Studies, 2016), <https://rusi.org/sites/default/files/201604_clat_final_report.pdf> (accessed October 31, 2017); Tom Keatinge and Florence Keen, "Lone-Actor and Small Cell Terrorist Attacks: A New Front in Counter-Terrorist Finance?" RUSI Publications, January 24, 2017, <https://rusi.org/publication/occasional-papers/lone-actor-and-small-cell-terrorist-attacks-new-front-counter> (accessed October 26, 2017).
. For the ideological development of Breivik's anti-Marxist ideology, see his 2011 manifesto: Anders Breivik [Andrew Berwick, pseud.], "2083: A European Declaration of Independence" <https://fas.org/programs/tap/_docs/2083_-_A_European_Declaration_of_Independence.pdf> (accessed October 26, 2017).
. Asne Seierstad, "Is Norwegian Mass Murderer Anders Breivik Still a Threat to Europe?" _Newsweek_ , May 13, 2016, <http://europe.newsweek.com/anders-breivik-neo-nazi-suing-norway-asne-seierstad-447247?rm=eu> (accessed October 26, 2017).
. Sheryl Gay Stolberg and Brian M. Rosenthal, "Man Charged after White Nationalist Rally in Charlottesville Ends in Deadly Violence," _New York Times_ , August 12, 2017, <https://www.nytimes.com/2017/08/12/us/charlottesville-protest-white-nationalist.html?mcubz=1> (accessed October 26, 2017).
. Ibid.
. Joshua Berlinger, "Gavin Long: Who Is Baton Rouge Cop Killer?" CNN, August 4, 2016, <http://edition.cnn.com/2016/07/18/us/who-is-gavin-long/> (accessed October 26, 2017). Long stated in a video that he had no affiliation with the Nation of Islam or ISIS.
. Other issues that could lead to impeachment are based on two clauses in the US Constitution: The foreign-emoluments clause forbids any gifts or benefits from foreign governments and a second that forbids gifts or benefits from the US government or any US state. These concerns have been raised as a result from Donald Trump's refusal to separate himself from his major business interests as past presidents have done. David Swanson, "Why Impeach Donald Trump," _Counterpunch_ , January 24, 2017, <http://www.counterpunch.org/2017/01/24/why-impeach-donald-trump/> (accessed October 26, 2017).
. Ken Bensinger, Miriam Elder, and Mark Schoofs, "These Reports Allege Trump Has Deep Ties to Russia," _BuzzFeed_ , January 10, 2017, <https://www.buzzfeed.com/kenbensinger/these-reports-allege-trump-has-deep-ties-to-russia?utm_term=.qxL4y3W2O3#.ldwb6E173E> (accessed May 29, 2017).
. See controversial report written by former British MI6 intelligence agent Christopher Steele, Fusion GPS report. Interestingly enough, the report indicates that the Russian government did want to sell 19 percent of Rosneft shares to foreigners, which it did, but not all of the names of share owners have been revealed. The Qatar Investment Corporation and the Swiss firm Glencore are listed as purchasers of 19.5 shares of Rosneft, but the figures do not add up. An unknown entity in the Cayman Islands may hold as much as $2.2 billion. Katya Golubkova, Dmitry Zhdannikov, and Stephen Jewkes, "How Russia Sold Its Oil Jewel: Without Saying Who Bought It," Reuters, January 24, 2017, <http://www.reuters.com/article/us-russia-rosneft-privatisation-insight-idUSKBN1582OH> (accessed October 26, 2017).
. If Trump receives any special treatment in securing trademark rights, it might be considered a violation of the US Constitution, which bans public servants from accepting anything of value from foreign governments unless approved by Congress. Associated Press in Shanghai, "China Provisionally Grants Trump 38 Trademarks—Including for Escort Service," _Guardian_ , March 8, 2017, <https://www.theguardian.com/us-news/2017/mar/08/china-approves-trump-trademarks-businesses> (accessed October 26, 2017). One of the theories of the secret GPS Fusion report is that Trump's political dealings with Russia were intended to deflect attention from his even bigger business deals with China. If so, the Russia affair has exploded way beyond Trump's expectations!
. Greg Miller and Greg Jaffe, "Trump Revealed Highly Classified Information to Russian Foreign Minister and Ambassador," _Washington Post_ , May 15 2017, <https://www.washingtonpost.com/world/national-security/trump-revealed-highly-classified-information-to-russian-foreign-minister-and-ambassador/2017/05/15/530c172a-3960-11e7-9e48-c4f199710b69_story.html?utm_term=.731d6f4738e4> (accessed October 26, 2017).
. Aristotle, _Politics_.
**CHAPTER 4: RISKS OF THE NEW AMERICAN NATIONALISM FOR THE EUROPEAN UNION**
. Lewis Sanders, "World Leaders Tackle Uncertainty at the Heart of Global Order," _Deutsche Welle_ , February 17, 2017, <http://www.dw.com/en/world-leaders-tackle-uncertainty-at-the-heart-of-global-order/a-37608782> (accessed May 29, 2017).
. Lionel Barber, "Juncker Tells Trump to Stop 'Annoying' Praise for Brexit," _Financial Times_ , March 24, 2017, <https://www.ft.com/content/938452b6-1072-11e7-a88c-50ba212dce4d> (accessed October 26, 2017).
. Tom Batchelor, "Donald Trump Says He Is 'Totally in Favour' of 'Wonderful' EU," _Independent_ , February 23, 2017, <http://www.independent.co.uk/news/world/americas/us-politics/donald-trump-european-union-eu-us-president-totally-in-favour-wonderful-steve-bannon-mike-pence-a7596731.html> (accessed October 26, 2017).
. Gardiner Harris and James Kanter, "Mike Pence, in Europe, Says Trump Supports Partnership with EU," _New York Times_ , February 20, 2017, https://www.nytimes.com/2017/02/20/world/europe/pence-european-union-trump.html?rref=collection%2Fnewseventcollection%2FThe%20Trump%20White%20House&action=click&contentCollection=Politics&module=Collection®ion=Marginalia&src=me&version=newsevent&pgtype=article (accessed May 30, 2017); "Remarks by President Donald Tusk after His Meeting with Vice President of the United States Mike Pence," press release, European Council: Council of the European Union, February 20, 2017, <http://www.consilium.europa.eu/en/press/press-releases/2017/02/20-tusk-remarks-meeting-us-vice-president-pence/> (accessed May 30, 2017).
. Batchelor, "Donald Trump Says."
. Mathew Nussbaum, "Pence to Europe: We're Still with You," _Politico_ , February 18, 2017, <https://www.politico.com/story/2017/02/mike-pence-nato-european-leaders-235172> (accessed November 7, 2017).
. Michael Birnbaum, "Trump's Calls for Europe to Increase Defense Spending Could Force Other Upheaval," _Washington Post_ , February 15, 2017, https://www.washingtonpost.com/world/europe/trumps-calls-for-europe-to-increase-defense-spending-could-force-other-upheaval/2017/02/15/fe257b44-efc1-11e6-a100-fdaaf400369a_story.html?tid=a_inl&utm_term=.53f37519044d (accessed October 26, 2017).
. US Donald Trump, "President Trump's Remarks at 9/11 and Article 5 Memorial Unveiling," US Mission to NATO, May 25, 2017, <https://nato.usmission.gov/may-25-2017-president-trumps-remarks-911-article-5-memorial-unveiling/> (accessed October 26, 2017).
. Germany's foreign minister, Sigmar Gabriel, retorted, "I don't know where Germany can find billions of euros to boost defense spending if politicians also want to lower taxes." Ewan MacAskill, "Pence's Speech on NATO Leaves European Leaders Troubled over Alliance's Future," _Guardian_ , February 18, 2017, <https://www.theguardian.com/world/2017/feb/18/trump-pence-eu-nato-munich-conference-germany-britain> (accessed October 26, 2017); Steve Eder and Thomas Kaplan, "Donald Trump and Mike Pence: One Ticket, Two Worldviews," _New York Times_ , July 17, 2016, <https://www.nytimes.com/2016/07/18/us/politics/donald-trump-mike-pence.html?_r=0> (accessed October 26, 2017).
. The United States, the United Kingdom, and France are global hegemonic powers, but Germany is not. Its political-economic influence is largely European, although it does a large business in conventional arms. From 2014 to 2015, German arms sales almost doubled to €7.5 billion, with sales to the United Kingdom but also to Qatar and Saudi Arabia. "German Arms Exports Keep Rising in 2016," Deutsche Welle, July 5, 2016, <http://www.dw.com/en/german-arms-exports-keep-rising-in-2016/a-19377912> (accessed October 26, 2017).
. "Ombudsman: German Army Is 'Short of Almost Everything,'" Deutsche Welle, January 26, 2016, <http://www.dw.com/en/ombudsman-german-army-is-short-of-almost-everything/a-19005841> (accessed May 29, 2017).
. "Réduite de 50% sous Sarkozy et Hollande, l'armée française est au bord de l'explosion," _Le Salon Beige_ , October 27, 2015, <http://www.lesalonbeige.fr/reduite-de-50-sous-sarkozy-et-hollande-larmee-francaise-est-au-bord-de-lexplosion/> (accessed May 29, 2017).
. Ivana Kottasova, "How NATO Is Funded and Who Pays What," CNN, March 20, 2017, <http://money.cnn.com/2017/03/20/news/nato-funding-explained/index.html> (accessed November 7, 2017). On demands that the Europeans, not the Americans, pay more for defense, see Doug Bandow, "U.S. to Spend More on Europe's Defense: Let the Europeans Pay Instead," _Huffington Post_ , February 12, 2017, <https://www.huffingtonpost.com/doug-bandow/us-to-spend-more-on-europ_b_9219754.html> (accessed November 7, 2017).
. "Donald Trump: More Countries Will Leave the EU Following Brexit—Video," _Guardian_ , June 24, 2016, <https://www.theguardian.com/politics/video/2016/jun/24/donald-trump-more-countries-will-leave-eu-following-brexit-video> (accessed November 13, 2017).
. The Greek financial crisis is not only due to excessive covert expenditure, but also a result of the 2008 US financial crisis. On the relationship to the US mortgage crisis and European banking crisis, see "James K. Galbraith: The Final Death (and Next Life) of Maynard Keynes," transcript, _Shadowproof_ , August 5, 2011, <https://shadowproof.com/2011/08/01/james-k-galbraith-the-final-death-and-next-life-of-maynard-keynes/> (accessed May 30, 2017).
. Roberto Savio, "Merkel's Defeat Confirms Dismal Trend for Europe," _Other News_ , September 29, 2017, <http://www.other-news.info/2017/09/merkels-defeat-confirms-dismal-trend-for-europe/> (accessed November 6, 2017).
. John Mauldin, "Italy's Banking Crisis Is Nearly upon Us," _Forbes_ , December 8, 2016, <https://www.forbes.com/sites/johnmauldin/2016/12/08/italys-banking-crisis-is-nearly-upon-us/#6c235e6f6c23> (accessed October 26, 2017); Philip Molyneux, "Will Italy's Failing Banks Trigger Financial Collapse across Europe?" _Guardian_ , November 28, 2016, <https://www.theguardian.com/commentisfree/2016/nov/28/italy-failing-banks-new-japan> (accessed October 26, 2017).
. Nicole Koenig and Marie Walter-Franke, "France and Germany: Spearheading a European Security and Defence Union?" (policy paper; Berlin: Jacques Delors Institut, July 19, 2017), <http://www.institutdelors.eu/media/franceandgermanyspearheadingaeuropeansecurityanddefenceunion-koenigwalter-jdib-july2017.pdf?pdf=ok> (accessed October 26, 2017).
. _Joint Declaration Issued at the British-French Summit_ (Saint-Malo, France: EU Institute for Security Studies, February 2000), <https://www.cvce.eu/en/obj/franco_british_st_malo_declaration_4_december_1998-en-f3cd16fb-fc37-4d52-936f-c8e9bc80f24f.html> (accessed October 26, 2017).
. Daniel Keohane, "Three's Company? France, Germany, the UK, and European Defence Post-Brexit," Real Instituto Elcano, January 5, 2017, <http://www.realinstitutoelcano.org/wps/portal/rielcano_en/contenido?WCM_GLOBAL_CONTEXT=/elcano/elcano_in/zonas_in/ari1-2017-keohane-threes-company-france-germany-uk-european-defence-post-brexit>(accessed May 30, 2017).
. Tom Batchelor, "Scotland 'Would Not Get Automatic Membership of NATO If It Voted for Independence,'" _Independent_ , March 13, 2017, <http://www.independent.co.uk/news/uk/politics/scottish-referendum-nicola-sturgeon-nato-membership-independent-scotland-a7627481.html> (accessed May 30, 2017).
. Rainer Buergin and Toluse Olorunnipa, "Trump Slams NATO, Floats Russia Nuke Deal in European Interview," Bloomberg, January 15, 2017, <https://www.bloomberg.com/politics/articles/2017-01-15/trump-calls-nato-obsolete-and-dismisses-eu-in-german-interview> (accessed May 30, 2017).
. Kevin Liptak and Michelle Kosinski, "After Combative Meeting, Trump Tries Phone Flattery to Win over Merkel," CNN, March 28, 2017, <http://edition.cnn.com/2017/03/28/politics/donald-trump-angela-merkel-call/> (accessed October 26, 2017).
. "Donald Trump Slams Angela Merkel's Refugee Policy," Deutsche Welle, January 15, 2017, <http://www.dw.com/en/donald-trump-slams-angela-merkels-refugee-policy/a-37141791> (accessed May 30, 2017).
. "New US Russia Sanctions Bill Risks EU Anger," Deutsche Welle, July 28, 2017, <http://www.dw.com/en/new-us-russia-sanctions-bill-risks-eu-anger/a-39867060> (accessed October 26, 2017).
. Shannon Tiezzi, "China and Germany's 'Special Relationship,'" _Diplomat_ , July 08, 2014, <https://thediplomat.com/2014/07/china-and-germanys-special-relationship/> (accessed November 7, 2017).
. Robin Harding, "Japan Fears Brexit Blow to EU Arms Embargo on China," _Financial Times_ , <https://www.ft.com/content/219af680-41c6-11e6-b22f-79eb4891c97d> (accessed October 26, 2017).
. Daniel Twining, "Leveraging America's Asian Pivot to Reinforce the Transatlantic Alliance and Vice Versa," Asan Forum, June 30, 2016, <http://www.theasanforum.org/leveraging-americas-asian-pivot-to-reinforce-the-transatlantic-alliance-and-vice-versa/> (accessed May 30, 2017).
. For details, see "Migrant Crisis: Migration to Europe Explained in Seven Charts," BBC, March 4, 2016, <http://www.bbc.com/news/world-europe-34131911> (accessed November 7, 2017).
. "BP-Rosneft Deal," _Russia Beyond the Headlines_ , January 15, 2011, <http://rbth.com/bp-rosneft> (accessed May 30, 2017).
. Justin Sink, "Trump, Merkel Discussed NATO, Mideast in First Phone Call," Bloomberg, January 28, 2017, <https://www.bloomberg.com/politics/articles/2017-01-28/trump-merkel-discussed-nato-middle-east-in-first-phone-call> (accessed October 26, 2017).
. "New CAP Is Still Struggling to Find New Export Markets," _EurActiv: Main Challenges Facing the CAP_ , special report, July 25–29, 2016, <http://en.euractiv.eu/wp-content/uploads/sites/2/special-report/EurActiv-Multilingual-Special-Report-The-main-challenges-facing-the-CAP-1.pdf> (accessed November 7, 2017).
. Jake Rudnitsky and Ilya Arkhipov, "Putin's Reliance on American Commerce Has Never Been Greater," Bloomberg, June 15, 2016, <https://www.bloomberg.com/news/articles/2016-06-15/putin-s-reliance-on-american-commerce-has-never-been-greater> (accessed October 26, 2017).
. David Shamah, "Sanctions Propel Israel, Russia to Expand Agriculture Ties" _Times of Israel_ , September 14, 2014, <http://www.timesofisrael.com/sanctions-propel-israel-russia-to-expand-agriculture-ties/> (accessed October 26, 2017).
. Aleksandra Eriksson, _EU Observer_ , June 23, 2017, <https://euobserver.com/foreign/138334> (accessed November 6, 2017). Daniel P. Ahn and Rodney D. Ludema, "Measuring Smartness: Understanding the Economic Impact of Targeted Sanctions," working paper 2017-01, December 2016, <https://www.state.gov/documents/organization/267590.pdf> (accessed, November 6, 2017). Sarantis Michalopoulos, "Russia Extends Embargo on EU Food Products," _EurActiv_ , June 30, 2016, <https://www.euractiv.com/section/agriculture-food/news/russia-extends-embargo-on-eu-food-products/> (accessed May 30, 2017). See impact of sanctions on Russia and the EU. Anastasia Nevskaya, "Russia-EU Economic Relations: Assessing Two Years of Sanctions," _Russia Direct_ , June 16, 2016, <http://www.russia-direct.org/analysis/russia-eu-economic-relations-assessing-two-years-sanctions> (accessed October 26, 2017).
. Raphaël Proust, "Jean-Luc Mélenchon veut une « conférence sur la sécurité » en Europe," _l'Opinion_ , March 31, 2017, <http://www.lopinion.fr/edition/politique/jean-luc-melenchon-veut-conference-securite-en-europe-123375> (accessed October 26, 2017).
. _Wikipedia_ , s.v. "German Federal Election, 2017," last modified November 21, 2017, <https://en.wikipedia.org/wiki/German_federal_election,_2017> (accessed November 21, 2017).
. Trump's adviser and strategist Steve Bannon has been seen as organizing a new international order based on "illiberal democracy." His allies include Nigel Farage (United Kingdom Independence Party), Matteo Salvini (Italian Northern League), Beppe Grillo (Italian 5 Star Movement), Marine Le Pen (French National Front), and Geert Wilders (Netherlands Freedom Party)—who, at the beginning of the Trump administration, had Washington (and not so much Moscow) as their point of reference. Roberto Savio, "Trump Marks the End of a Cycle," Inter Press Service, February 21, 2017, <http://www.ipsnews.net/2017/02/trump-marks-the-end-of-a-cycle/> (accessed May 30, 2017). Given the shift back toward neoconservativism and neo-liberalism in the Trump administration, whether this alliance can hold together is another question.
. Dan Stewart, "Donald Trump's Meeting with Nigel Farage Leaves Britain's Leaders Red-Faced," _Time_ , Nov 14, 2016, <http://time.com/4569416/donald-trump-nigel-farage-meeting-theresa-may/> (accessed October 27, 2017).
. Ben Jacobs, "Donald Trump: Marine Le Pen Is 'Strongest Candidate' in French Election," _Guardian_ , April 21, 2017, <https://www.theguardian.com/us-news/2017/apr/21/donald-trump-marine-le-pen-french-presidential-election> (accessed October 27, 2017).
. Fredrik Wesslau, "Putin's Friends in Europe," European Council on Foreign Relations, October 19, 2016, <http://www.ecfr.eu/article/commentary_putins_friends_in_europe7153> (accessed May 30, 2017).
. Gašper Završnik, "Beata Szydło: Poland May Drop Rome Declaration," _Politico_ , March 23, 2017, <http://www.politico.eu/article/beata-szydlo-poland-may-drop-rome-declaration/> (accessed October 27, 2017).
. Péter Krekó and Lóránt Györi, "Don't Ignore the Left! Connections between Europe's Radical Left and Russia," _OpenDemocracy_ , June 13, 2016, <https://www.opendemocracy.net/od-russia/peter-kreko-lorant-gyori/don-t-ignore-left-connections-between-europe-s-radical-left-and-ru> (accessed May 30, 2017).
. Stephanie Kirchgaessner, "Italy's Five Star Movement Part of Growing Club of Putin Sympathisers in West," _Guardian_ , January 5, 2017, <https://www.theguardian.com/world/2017/jan/05/five-star-movement-beppe-grillo-putin-supporters-west> (accessed May 30, 2017).
. Two major events have worked to further destabilize the region, leading to a mass exodus: French and UK-led military intervention in Libya (backed by NATO) in March 2011 and then the Syrian civil war which began in 2011, but escalated even further with direct Russian military involvement in September 2015.
. Roberto Savio, "Merkel's Defeat Confirms Dismal Trend for Europe," _Other News_ , September 29, 2017, <http://www.other-news.info/2017/09/merkels-defeat-confirms-dismal-trend-for-europe/> (accessed November 6, 2017).
. IPSOS, Résultats du 2nd tour, 2017, <https://www.ipsos.com/fr-fr/presidentielle-2nd-tour-les-estimations-ipsos-sopra-steria-dune-grande-precision> (accessed November 6, 2017).
. Saim Saeed, "US Intelligence Chief: Russia Interfering in French, German Elections," _Politico_ , March 30, 2017, <http://www.politico.eu/article/us-intelligence-chief-russia-interfering-in-french-german-elections/> (accessed October 27, 2017).
. Helene Fouquet, Gregory Viscusi, and Henry Meyer, "Le Pen Struggling to Fund French Race as Russian Bank Fails," Bloomberg, December 22, 2016, <https://www.bloomberg.com/news/articles/2016-12-22/le-pen-struggling-to-fund-french-race-after-russian-backer-fails> (accessed November 13, 2017).
. Gideon Resnick, "Steve Bannon Knew about Marine Le Pen's Trump Tower Party, Organizer Claims," _Daily Beast_ , January 30, 2017, <http://www.thedailybeast.com/articles/2017/01/12/steve-bannon-knew-about-marine-le-pen-s-trump-tower-party-organizer-claims.html> (accessed May 30, 2017).
. Eric Auchard and Bate Felix, "French Candidate Macron Claims Massive Hack as Emails Leaked," Reuters, May 6, 2017, <http://www.reuters.com/article/us-france-election-macron-leaks-idUSKBN1812AZ> (accessed October 27, 2017).
. Andrew Higgins, "Maybe Private Russian Hackers Meddled in Election, Putin Says," _New York Times_ , June 1, 2017, <https://www.nytimes.com/2017/06/01/world/europe/vladimir-putin-donald-trump-hacking.html> (accessed November 7, 2017).
. Auchard and Felix, "French Candidate Macron."
. "Dutch Referendum a Difficult Result for EU and Ukraine," BBC News, April 7, 2016, <http://www.bbc.com/news/world-europe-35984821> (accessed May 30, 2017).
. Jennifer Rankin, "EU Lifts Most Sanctions against Belarus Despite Human Rights Concerns," _Guardian_ , February 15, 2016, <https://www.theguardian.com/world/2016/feb/15/eu-lifts-most-sanctions-against-belarus-despite-human-rights-concerns> (accessed May 30, 2017).
. See for example, Andrew Wilson, "Europe, Keep an Eye on Minsk: If the Belarus President Is to Survive, He Will Have to Walk a Narrow Path between Pressure from Demonstrators and the Kremlin," _Politico_ , March 17, 2017, <http://www.politico.eu/article/europe-keep-an-eye-on-minsk-belarus-alexander-lukashenko-vladimir-putin-russia/> (accessed May 30, 2017).
. Colin Brose, "Serbia Maneuvers between the EU and EEU," Jamestown Foundation, November 11, 2016, <https://jamestown.org/serbia-maneuvers-eu-eeu/> (accessed May 30, 2017).
. Rikard Jozwiak, "Brussels Notebook: Elections Aren't Only Things Keeping NATO, EU Folks up at Night," RadioFreeEurope/RadioLiberty, January 4, 2017, <http://www.rferl.org/a/brussels-notebook-elections-not-only-things-keeping-nato-eu-up-at-night/28214010.html?ltflags=mailer> (accessed October 27, 2017).
. Nick Gass, "Trump: Taking Back Crimea Would Trigger World War III," _Politico_ , August 1, 2016, <http://www.politico.com/story/2016/08/donald-trump-crimea-ukraine-war-226522> (accessed May 30, 2017).
. Christina Boyle, "European Council President Includes United States as Threat to Europe," _LA Times_ , January 31, 2017, <http://www.latimes.com/world/europe/la-fg-europe-trump-20170131-story.html> (accessed October 27, 2017); M. Emmanuel Macron, "United Nations General Assembly—Speech by M. Emmanuel Macron, President of the Republic," (speech; New York: United Nations, September 19, 2017), <http://www.diplomatie.gouv.fr/en/french-foreign-policy/united-nations/united-nations-general-assembly-sessions/unga-s-72nd-session/article/united-nations-general-assembly-speech-by-m-emmanuel-macron-president-of-the> (accessed October 27, 2017).
**CHAPTER 5: THE RISK OF WAR OVER CRIMEA, THE BLACK SEA, AND EASTERN EUROPE**
. Nick Gass, "Trump: Taking Back Crimea Would Trigger World War III," _Politico_ , August 1, 2016, <http://www.politico.com/story/2016/08/donald-trump-crimea-ukraine-war-226522> (accessed October 27, 2017).
. Bryan Logan, "Nikki Haley Just Delivered the Trump Administration's Most Hawkish Words Yet toward Russia," _Business Insider_ , February 3, 2017, http://uk.businessinsider.com/trump-nikki-haley-russia-ukraine-2017-2?r=US&IR=T (accessed May 29, 2017). Christopher Miller, "Anxious Ukraine Risks Escalation in 'Creeping Offensive,'" RadioFreeEurope/RadioLiberty, January 30, 2017, <http://www.rferl.org/a/ukraine-russia-creeping-offensive-escalation-fighting/28268104.html> (accessed October 27, 2017). See chapter 3.
. Nikita Vladimirov, "Trump Vows to Restore 'Peace' Along Russia, Ukraine Border," _Hill_ , February 4, 2017, <http://thehill.com/policy/international/317943-trump-vows-to-restore-peace-along-russia-ukraine-border> (accessed May 29, 2017).
. It is not absolutely clear whether Haley's speech implied a complete end to sectoral sanctions. Russia essentially has had three sets of sanctions put on it by the United States since Crimea became part of Russia. US sectoral sanctions on Russia are due to expire in December 2017. European sanctions are due to expire in January 2018. It is more likely that the Europeans will eventually end sanctions earlier than the Americans, given the impact of sanctions on European agriculture. Kenneth Rapoza, "What UN Ambassador Haley's Comment on Russia Really Means," _Forbes_ , February 3, 2017, <http://www.forbes.com/sites/kenrapoza/2017/02/03/what-u-n-ambassador-haleys-comment-on-russia-really-means/#63320ee74405> (accessed May 29, 2017).
. Andrew E. Kramer and Clifford Krauss, "Rex Tillerson's Company, Exxon, Has Billions at Stake over Sanctions on Russia," _New York Times_ , December 12, 2016, <https://www.nytimes.com/2016/12/12/world/europe/rex-tillersons-company-exxon-has-billions-at-stake-over-russia-sanctions.html?_r=1> (accessed October 27, 2017). For projected benefits of the Trump presidency for Exxon, see Jenny Rowland et al., "How Exxon Won the 2016 Election," Center for American Progress, January 10, 2017, <https://www.americanprogress.org/issues/green/news/2017/01/10/296277/how-exxon-won-the-2016-election/> (accessed October 27, 2017).
. If, for example, the ExxonMobil and Rosneft joint venture is forced to break up, then Exxon and Rosneft could swap assets in the United States and Canada and in Russia. But the gas deposits in the Arctic Circle are probably worth much more than the US/Canadian deposits. And the Bazhenov shale could prove ten times bigger than the Bakken shale of North Dakota. See Christopher Helman, "Why Forcing ExxonMobil out of Russia Isn't Going to Help Anything," _Forbes_ , September 14, 2014, <https://www.forbes.com/sites/christopherhelman/2014/09/14/why-forcing-exxonmobil-out-of-russia-isnt-going-to-help-anything/#2f7240121a20> (accessed October 27, 2017).
. Nick Cunningham, "Chevron Pulls out of $10 Billion Gas Deal with Ukraine," OilPrice.com, December 15, 2014, <http://oilprice.com/Energy/Natural-Gas/Chevron-Pulls-Out-Of-10-Billion-Gas-Deal-With-Ukraine.html> (accessed May 29, 2017).
. Christopher Hellie, "Should Russia's Internal Issues Scare Long-Term Investors?" _Global Risks Insights_ , January 31, 2017, <http://globalriskinsights.com/2017/01/russia-internal-issues/> (accessed October 27, 2017); Kenneth Rapoza, "Russia Investors Get No Love From Trump, But..." _Forbes_ , April 24, 2017, <https://www.forbes.com/sites/kenrapoza/2017/04/24/russia-investors-get-no-love-from-trump-but/#787ebf505470> (accessed October 27, 2017); Jason Corcoran, "Russia Set to Remain out in the Cold," _bne IntelliNews Daily_ , May 9, 2017, <http://online.flipbuilder.com/myab/ihpb/mobile/index.html?inf_contact_key=fe813ce84d777d1d7ae6e339ad06b82609b749d2ca1cd0f1867aca4c553ba280> (accessed October 27, 2017).
. Reuters, "The Art of the Deal: Why Putin Needs Trump More Than Trump Needs Putin," _Business Insider_ , February 1, 2017, http://uk.businessinsider.com/r-the-art-of-the-deal-why-putin-needs-one-more-than-trump-2017-2?r=US&IR=T (accessed May 30, 2017).
. There is a statistical dispute as to whether oil and gas account for around 34 percent or 50 percent of government revenues. Jon Hellevig, "Oil and Gas Revenue Accounts for 21% of Russia's Budget Not 50%," _Russia Insider_ , March 28, 2016, <http://russia-insider.com/en/business/oil-and-gas-revenue-accounts-just-21-russias-budget-not-over-50-routinely-misreported-west> (accessed May 29, 2017); Alexander Metelitsa, "Oil and Natural Gas Sales Accounted for 68% of Russia's Total Export Revenues in 2013," US Energy Information Administration (EIA), July 23, 2014, <http://www.eia.gov/todayinenergy/detail.php?id=17231> (accessed May 29, 2017).
. Donald Trump, "Statement by President Donald J. Trump on the Signing of H.R. 3364," press release, White House, Office of the Press Secretary, August 2, 2017, <https://www.whitehouse.gov/the-press-office/2017/08/02/statement-president-donald-j-trump-signing-hr-3364> (accessed October 27, 2017).
. Ibid.
. James Osborne, "EIA: US Shale Oil Production to Fall Sharply through 2017," _Fuel Fix_ , August 22, 2016, <http://fuelfix.com/blog/2016/08/22/eia-u-s-shale-oil-production-to-fall-sharply-through-2017/> (accessed May 29, 2017); Reuters, "US Shale Firms Go Back to Work after Donald Trump's Victory," _Fortune_ , November 14, 2016, <http://fortune.com/2016/11/14/donald-trump-victory-us-shale-oil/> (accessed October 27, 2017).
. ICMN Staff, "Rosebud Sioux Tribe Calls House Keystone XL Passage an 'Act of War,' Vows Legal Action," Indian Country Media Network, November 17, 2014, <https://indiancountrymedianetwork.com/news/politics/rosebud-sioux-tribe-calls-house-keystone-xl-passage-an-act-of-war-vows-legal-action/> (accessed May 29, 2017).
. Glenn Ellis and Katerina Barushka, "A Very Montenegrin Coup," Al Jazeera, March 2, 2017, <http://www.aljazeera.com/programmes/peopleandpower/2017/03/montenegrin-coup-170302060130440.html> (accessed October 27, 2017).
. It is possible that Finland and the Yeltsin administration may have talked about Russia selling Karelia back to Finland, but the idea was dumped. Putin has stated that territorial exchanges are not the best way to bring peace! Martti Valkonen, "President Ahtisaari Unlocked a Door towards Debate about Lost Karjala," _Prokarelia_ , November 28, 2002, http://prokarelia.net/en/?x=artikkeli&article_id=314&author=1 (accessed October 27, 2017).
. Rapoza, "What UN Ambassador Haley's Comment"; "Ukraine/Russia Related Sanctions Program," US Department of the Treasury, last updated September 29, 2017, <https://www.treasury.gov/resource-center/sanctions/Programs/Pages/ukraine.aspx> (accessed October 26, 2017).
. Kiev could have tried to sustain its nuclear weaponry; Kiev had the missiles, but needed to build the warheads as the control systems were in Moscow (at a cost of $65 billion). Yet the Clinton Administration threatened Ukraine with isolation and sanctions if Kiev sustained the nuclear weapons while Russia threatened preemptive strikes. See Hall Gardner, _Surviving the Millennium: American Global Strategy, the Collapse of the Soviet Empire, and the Question of Peace_ (Westport, CT: Praeger, 1994).
. NATO, "Founding Act on Mutual Relations, Cooperation and Security between NATO and the Russian Federation Signed in Paris, France," North Atlantic Treaty Organization, May 27, 1997, <http://www.nato.int/cps/en/natohq/official_texts_25468.htm> (accessed May 29, 2017).
. Ibid.
. Ibid.
. Ibid.
. Dave Majumdar, "Revealed: Russian Invasion Could Overrun NATO in 60 Hours," _National Interest_ , February 4, 2016, <http://nationalinterest.org/blog/the-buzz/revealed-russian-invasion-could-overrun-nato-60-hours-15112> (accessed May 29, 2017). It was estimated that a proposed deployment of seven brigades to deter potential Russian aggression in the Baltic region could cost about $2.7 billion. But such a force could only slow, not defeat, Moscow—if the latter were truly determined to overrun these countries. Lisabeth Gronlund, "How Much Does It Cost to Create a Single Nuclear Weapon?" Union of Concerned Scientists, November 2013, <http://www.ucsusa.org/publications/ask/2013/nuclear-weapon-cost.html#.V9VZ4z595wc> (accessed May 29, 2017); On Obama's military buildup: Dov S. Zakheim, "The Great Reversal: Obama's Military Buildup," _National Interest_ , February 9, 2016, <http://nationalinterest.org/feature/the-great-reversal-obamas-military-buildup-15151> (accessed May 29, 2017).
. Jeffrey Goldberg, "The Obama Doctrine," _Atlantic_ , March 17, 2016, <http://www.theatlantic.com/magazine/archive/2016/04/the-obama-doctrine/471525/#3> (accessed May 29, 2017).
. NATO, "Ukraine Commission," North Atlantic Treaty Organization, May 21, 2014, <http://www.nato.int/cps/iw/natohq/topics_50319.htm> (accessed May 29, 2017).
. Emily Tamkin, Dan De Luce, Robbie Gramer, "Ukraine Expects Trump to Approve Arms Deliveries," _Foreign Policy_ , October 26, 2017, <http://foreignpolicy.com/2017/10/26/ukraine-expects-trump-to-approve-arms-deliveries/> (accessed November 7, 2017).
. See Hall Gardner, _Crimea, Global Rivalry and the Vengeance of History_ (Basingstoke, UK: Palgrave MacMillan, 2015).
. "Minsk Agreement: Full Text in English," UNIAN, February 12, 2015, <http://www.unian.info/politics/1043394-minsk-agreement-full-text-in-english.html> (accessed May 30, 2017).
. Tim Judah, "Will Ukraine Ever Change?" _New York Review of Books_ , May 25, 2017, <http://www.nybooks.com/articles/2017/05/25/will-ukraine-ever-change/> (accessed November 7, 2017). See also Gordon M. Hahn, "Getting Ukraine Wrong," Gordon Hahn, September 8, 2017, <https://gordonhahn.com/2017/09/08/getting-ukraine-wrong/> (accessed October 27, 2017).
. Mikhail Minakov and Maryna Stavniichuk, "Ukraine's Constitution: Reform or Crisis?" _Open Democracy_ , February 16, 2016, <https://www.opendemocracy.net/od-russia/mikhail-minakov-maryna-stavniichuk/ukrainian-constitution-reform-or-crisis> (accessed November 7, 2017)
. "Poroshenko's proposal is not approved by the separatists, nor by the Kremlin. It does not really give any 'special status' to separatist areas, and any specific details on autonomous rule in Donbass may later be revised by a simple majority vote in Ukrainian parliament. Moreover, the so-called 'decentralisation' is accompanied by a strengthening of the presidential control over local self-government via centrally assigned 'prefects' with broad powers." Volodymyr Ishchenko, "Ukraine's Government Bears More Responsibility for Ongoing Conflict Than the Far-Right," _Guardian_ , September 4, 2015, <http://www.theguardian.com/world/2015/sep/04/ukraine-government-svoboda-clashes-conflict> (accessed October 27, 2017).
. "Corruption Perceptions Index 2016," _Transparency International_ , January 25, 2017, <http://www.transparency.org/news/feature/corruption_perceptions_index_2016> (accessed October 27, 2017).
. The self-proclaimed Union was established in May 2014. Yet the Novorossiya pan-nationalist movement was not officially recognized internationally, even by Russia, and was labeled as a "terrorist organization" by Kiev. The movement came to a sudden end in May 2015 after meetings between US and European officials with Moscow. Andrei Kolesnikov, "Why the Kremlin Is Shutting Down the Novorossiya Project," Carnegie.ru/commentary, May 29, 2015, <http://carnegieendowment.org/2015/05/29/why-kremlin-is-shutting-down-novorossiya-project/i96u> (accessed November 14, 2017).
. Pavel Podvig, "What the Crimea Crisis Will Do to US-Russia Relations," _Bulletin of the Atomic Scientists_ , March 27, 2014, <http://thebulletin.org/what-crimea-crisis-will-do-us-russia-relations7009> (accessed May 30, 2017).
. NATO, "Wales Summit Declaration," press release, issued by the heads of state and government participating in the meeting of the North Atlantic Council in Wales, September 5, 2014, <http://www.nato.int/cps/ic/natohq/official_texts_112964.htm> (accessed October 27, 2017).
. Joshua Yaffa, "The Unaccountable Death of Boris Nemtsov," _New Yorker_ , February 26, 2016, <http://www.newyorker.com/news/news-desk/the-unaccountable-death-of-boris-nemtsov> (accessed May 30, 2017).
. Miller, "Anxious Ukraine Risks Escalation."
. Roman Olearchyk, "Ukraine Imposes Cargo Blockade on Breakaway East," _Financial Times_ , March 15, 2017, <https://www.ft.com/content/276f3fd8-098c-11e7-ac5a-903b21361b43> (accessed October 27, 2017); Nicolai Petro, "The Bizarre Reason Ukraine Could Be Facing a Legitimacy Crisis," _National Interest_ , March 15, 2017, <http://nationalinterest.org/feature/the-bizarre-reason-ukraine-could-be-facing-legitimacy-crisis-19787> (accessed October 27, 2017); Miller, "Anxious Ukraine Risks Escalation."
. Andrew Roth, "Russia Accuses Ukraine of Igniting Border Clash in Crimea," _Washington Post_ , August 10, 2016, <https://www.washingtonpost.com/world/russia-accuses-ukraine-of-igniting-border-clashes-in-crimea/2016/08/10/f8e1641a-5f00-11e6-84c1-6d27287896b5_story.html?tid=a_inl> (accessed May 30, 2017); Nicolai N. Petro and David C. Speedie, "Update from Ukraine," Carnegie Council for Ethics in International Affairs, August 18, 2016, <http://www.carnegiecouncil.org/studio/multimedia/20160818b/index.html> (accessed May 30, 2017).
. Nina Sorokopud, _Ukraine UNHCR Operational Update_ (Geneva, Switzerland: UN Refugee Agency, January 2017), <http://unhcr.org.ua/attachments/updates/2017%2001%20Update%20FINAL%20EN.pdf> (accessed October 27, 2017).
. Halya Coynash, "Russia's Crimea Bridge Could Collapse Anytime," The Atlantic Council (blog), January 10, 2017, <http://www.atlanticcouncil.org/blogs/ukrainealert/russia-s-crimea-bridge-could-collapse-anytime> (accessed October 27, 2017).
. Lily Hyde, "Crimea's Water Troubles," _Euromaidan Press_ , February 8, 2017, <http://euromaidanpress.com/2017/02/10/crimeas-water-troubles/#arvlbdata> (accessed October 27, 2017); "Dam Leaves Crimea Population in Chronic Water Shortage," Al Jazeera, January 4, 2017, <http://www.aljazeera.com/indepth/features/2016/12/dam-leaves-crimea-population-chronic-water-shortage-161229092648659.html> (accessed October 27, 2017).
. Writer, Staff. "F-35s to Participate in NATO Exercises Near Russian Border," _Popular Military_ , April 26, 2017, <http://popularmilitary.com/f-35s-participate-nato-exercises-near-russian-border/> (accessed May 29, 2017).
**CHAPTER 6: THE GLOBAL IMPACT OF THE CHINA-RUSSIA EURASIAN ALLIANCE**
. Laura Zhou, "China and Russia Criticise THAAD Missile Defence System as Destabilising Region," _South China Morning Post_ , July 8, 2016, <http://www.scmp.com/news/china/diplomacy-defence/article/1987103/china-and-russia-criticise-thaad-missile-defence-system> (accessed May 30, 2017).
. Bob Savic, "Behind China and Russia's 'Special Relationship'" _Diplomat_ , December 7, 2016, <http://thediplomat.com/2016/12/behind-china-and-russias-special-relationship/> (accessed May 30, 2017). The revised, official translated name of the initiative is "The Belt and Road Initiative"—with "BRI" as its acronym instead of the "OBOR."
. China's per capita disposable personal income was $3,469 in 2016. Xinhua, "China's personal Income Rises 6.3% in 2016," _China Daily_ , January 1, 2017, <http://www.chinadaily.com.cn/business/2017-01/20/content_28010029.htm> (accessed November 7, 2017).
. Franz-Stefan Gady, "China and Russia Conclude Naval Drill in Mediterranean," _Diplomat_ , May 22, 2015, <http://thediplomat.com/2015/05/china-and-russia-conclude-naval-drill-in-mediterranean/> (accessed May 30, 2017).
. Joshua Kucera, "Armenia Nixes Pakistan's Ties with CSTO," EurasiaNet.org, November 29, 2016, <http://www.eurasianet.org/node/81476> (accessed May 30, 2017). Pakistan not only supports Azerbaijan in its conflict with Armenia over Nagorno Karabakh, but goes so far as refusing to recognize Armenia's existence until it gives Karabakh back to Azerbaijan.
. Vladimir Radyuhin, "The Dragon Gets a Bear Hug," _The Hindu_ , March 7, 2013, <http://www.thehindu.com/opinion/op-ed/the-dragon-gets-a-bear-hug/article4485335.ece> (accessed May 30, 2017).
. Charles Clover, "Russia Resumes Advanced Weapons Sales to China," _Financial Times_ , November 16, 2016, <https://www.ft.com/content/90b1ada2-a18e-11e6-86d5-4e36b35c3550> (accessed November 11, 2017).
. According to Konstantin Makienko, "the Chinese mainly need the Su-35 to obtain access to the aircraft's new 117S engine, and Russia's latest and extremely powerful aircraft-based IRBIS radar system." Beyond that, Russia has a very limited catalogue of military hardware that it can sell to Beijing at this point. Russia has already sold most everything else. Matthew Bodner, "In Arms Trade, China Is Taking Advantage of Russia's Desperation," _Moscow Times_ , November 1, 2016, <https://themoscowtimes.com/articles/in-arms-trade-china-is-taking-advantage-of-russian-desperation-55965> (accessed May 30, 2017).
. Charles Clover, "Russia Resumes Advanced Weapons Sales to China," _Financial Times_ , November 3, 2016, <https://www.ft.com/content/90b1ada2-a18e-11e6-86d5-4e36b35c3550> (accessed May 30, 2017). In November 2016, China unveiled its own advanced stealth fighter, deployments to come later.
. China's main focus is indigenizing foreign technologies and systems in order to create a self-sufficient defense industry that can both produce weaponry platforms for the PLA and compete successfully in the global arms market. Stephen Blank, "Moscow Talks Business, Beijing Answers with Geo-strategy," _China Brief_ 13, no. 22 (Washington, DC: Jamestown Foundation, November 07, 2013), http://www.jamestown.org/single/?tx_ttnews[tt_news]=41596&no_cache=1#.U3t4EyhauZQ (accessed May 30, 2017).
. Bodner, "In Arms Trade, China Is Taking Advantage."
. Stephen Blank and Younkyoo Kim, "Russian Arms Sales and Its Future as an Asian Power," _Asian Politics & Policy_ 6, no. 2 (2013): 267–84.
. Jakobson et al., "China's Energy and Security Relations with Russia: Hopes, Frustrations and Uncertainties," (policy paper; Solna: Stockholm International Peace Research Institute, 2011), pp. 27 and 35.
. Neal Buckley, "Sino-Russian Gas Deal: Smoke without Fire," _Financial Times_ , May 11, 2016, <https://www.ft.com/content/eea4f2ec-16c0-11e6-b197-a4af20d5575e> (accessed October 27, 2017); Henry Foy and Neil Hume, "CEFC China Energy Buys $9bn Stake in Rosneft," _Financial Times_ , September 8, 2017, <https://www.ft.com/content/25b18d2e-94a4-11e7-a9e6-11d2f0ebb7f0?mhq5j=e6> (accessed October 27, 2017). The deal reduces the shares of the Swiss Glencore mining and the Qatar Investment Authority.
. Blank and Kim, "Russian Arms Sales."
. "China, Russia Sign Joint Statement on Strengthening Global Strategic Stability," Xinhuanet, June 25, 2016, <http://news.xinhuanet.com/english/2016-06/26/c_135466187.htm> (accessed May 30, 2017). Savic, "Behind China and Russia's 'Special Relationship.'"
. Robert D. Kaplan is correct to call the ongoing Sino-Soviet disputes in Eurasia a "quiet rivalry," but this rivalry is going beyond a marriage of convenience, as he argues, that might break up in the near future and toward a proto-alliance due to the US unwillingness to forge an entente with Moscow over Ukraine while concurrently seeking to _channel_ China's power potential through a US-China rapprochement, as argued in this book. For Kaplan's views, see "The Quiet Rivalry Between China and Russia," _New York Times_ , November 3, 2017, <https://www.nytimes.com/2017/11/03/opinion/china-russia-rivalry.html> (accessed November 8, 2011).
. Le Hong Hiep, UNSW ADFA, and VNU, "Defence Cooperation Underpins Vietnam–Russia Push for Renewed Economic Cooperation," East Asia Forum, November 13, 2013, <http://www.eastasiaforum.org/2013/11/13/defence-cooperation-underpins-vietnam-russia-push-for-renewed-economic-cooperation/> (accessed May 30, 2017). "China, Russia Sign Joint Statement."
. Philip S. Golub, _East Asia's Reemergence_ (Cambridge, UK: Polity Press, 2016).
. "SFWI, "Fund Rankings," <https://www.swfinstitute.org/fund-rankings/> (accessed November 25, 2017). US and Japanese pension funds rank first and second in terms of total amount of assets.
. On a number of energy partnerships, see Savic, "Behind China and Russia's 'Special Relationship.'" The economic relationship between China and Russia has been driven by a variety of bilateral intergovernmental commissions, including 26 sub-commissions.
. Rem Korteweg, "Unfreezing TTIP: Why a Transatlantic Trade Pact Still Makes Strategic Sense," Center for European Reform, May 2017, <http://www.cer.eu/sites/default/files/pb_ttip_rk_10.5.17.pdf> (accessed November 14, 2017).
. "Goldman Sachs, China's CIC to Launch up to $5 Billion Fund: Sources," Reuters, November 6, 2017, <http://www.reuters.com/article/us-goldman-sachs-cic/goldman-sachs-chinas-cic-to-launch-up-to-5-billion-fund-sources-idUSKBN1D61H7> (accessed November 8, 2017).
. Prior to Trump's decision to dump the TPP, Australian Prime Minister Malcolm Turnball had stated that the "effectiveness of the TPP will be considerably enhanced by the inclusion of China whose constructive participation in regional elements is a central element in its peaceful rise." Malcolm Turnball, "Assessing the Future of the Asia-Pacific–US/Australia Dialogue," MalcolmTurnball.com, January 31, 2015, <http://www.malcolmturnbull.com.au/media/future-of-the-asia-pacific> (accessed October 27, 2017).
. Xi Jinping, "President Xi's Speech to Davos in Full," World Economic Forum, January 17, 2017, <https://www.weforum.org/agenda/2017/01/full-text-of-xi-jinping-keynote-at-the-world-economic-forum> (accessed May 30, 2017).
. Barack Obama, "Statement by the President on the Signing of the Trans-Pacific Partnership," press release, White House, Office of the Press Secretary, February 3, 2016, National Archives and Records Administration, <https://obamawhitehouse.archives.gov/the-press-office/2016/02/03/statement-president-signing-trans-pacific-partnership> (accessed May 30, 2017).
. Emanuele Scimia, "Taiwan in the TPP: How to Break the Cross-Strait Status Quo," _Asian Times_ , July 3, 2017, http://www.atimes.com/taiwan-tpp-break-cross-strait-status-quo/?utm_source=The+Daily+Brief&utm_campaign=68d50bf894-EMAIL_CAMPAIGN_2017_07_03&utm_medium=email&utm_term=0_1f8bca137f-68d50bf894-31523549 (accessed October 27, 2017).
. According to the Organization of Economic Cooperation and Development (OECD), Chile is the first most inequitable society; Mexico is second; and the United States is third, as compared to other European, North American, and South American states.
. Steve Benen, "China Is Eager to Capitalize on Trump's Early Missteps," MSNBC, February 7, 2017, (accessed May 22, 2017).
. Paul Coyer, "Undermining America While Washington Sleeps: China in Latin America," _Forbes_ , February 5, 2016, <http://www.forbes.com/sites/paulcoyer/2016/01/31/undermining-america-while-washington-sleeps-china-in-latin-america/#505fca7d6694> (accessed May 30, 2017).
. "Landbridge Group Connects Three Ports with Acquisition of Margarita Island Port," _ChinaGoAbroad_ , May 25, 2016, <http://www.chinagoabroad.com/en/recent_transaction/20530> (accessed May 30, 2017); Andreea Brînză, "How a Greek Port Became a Chinese 'Dragon Head,'" _Diplomat_ , April 25, 2016, <http://thediplomat.com/2016/04/how-a-greek-port-became-a-chinese-dragon-> (accessed May 30, 2017). Peter Jennings, "Darwin: Storm in a Port," _Strategist_ , November 6, 2015, <https://www.aspistrategist.org.au/darwin-storm-in-a-port/> (accessed May 30, 2017); Geoff Wade, "Landbridge, Darwin, and the PRC," _Strategist_ , November 9, 2015, <https://www.aspistrategist.org.au/landbridge-darwin-and-the-prc/> (accessed May 30, 2017).
. Suzanne Daley, "Lost in Nicaragua, a Chinese Tycoon's Canal Project," _New York Times_ , April 3, 2016, <https://www.nytimes.com/2016/04/04/world/americas/nicaragua-canal-chinese-tycoon.html?_r=0> (accessed May 30, 2017). François Lafargue, "China's Presence in Latin America. Strategies, Aims and Limits," _China Perspectives_ 68, June 1, 2007, <https://chinaperspectives.revues.org/3053> (accessed May 30, 2017); David Z. Morris, "Why China and Nicaragua's Canal Project Is Floundering?" _Fortune_ , February 29, 2016, <http://fortune.com/2016/02/29/china-nicaragua-canal/> (accessed May 30, 2017); Jonathan Watts, "Nicaragua Canal: In a Sleepy Pacific Port, Something Stirs," _Guardian_ , November 24, 2016, <https://www.theguardian.com/world/2016/nov/24/nicaragua-canal-interoceanic-preparations> (accessed May 30, 2017).
. Peter Jennings, "Darwin: Storm in a Port," _Strategist_ , November 6, 2015, <https://www.aspistrategist.org.au/darwin-storm-in-a-port/> (accessed May 30, 2017); Wade, "Landbridge, Darwin, and the PRC."
. _Russia and the Caribbean_ (London: Caribbean Council, March 24, 2015), <http://www.caribbean-council.org/wp-content/uploads/2015/08/Russia-and-the-Caribbean.pdf> (accessed October 27, 2017).
. Karen de Young, "White House Implements New Cuba Policy Restricting Travel and Trade," _Washington Post_ , November 8, 2017.
. Negative economic effects of Chinese investment in Latin America and elsewhere include the hollowing out of manufacturing in the region and the enlarging of regional dependence on raw material exports, which increases the regional economies' sensitivity to fluctuations in commodity prices. The countries can also become dependent on fluctuations of Chinese demand. Paul Coyer, "Undermining America While Washington Sleeps: China in Latin America," _Forbes_ , January 31, 2016, <http://www.forbes.com/sites/paulcoyer/2016/01/31/undermining-america-while-washington-sleeps-china-in-latin-america/#505fca7d6694> (accessed October 27, 2017). Further, because Chinese firms often bring their own labor, Chinese investments do not always increase job opportunities.
. Andrew Rosati and Jose Orozco, "Trump Calls Venezuela 'Horrible' as US Expands Sanctions," Bloomberg, May 18, 2017, <https://www.bloomberg.com/politics/articles/2017-05-18/trump-calls-venezuela-horrible-as-treasury-expands-sanctions> (accessed October 27, 2017).
. Robert Helbig and Guillaume Lasconjarias, "Winning Peace and Exporting Stability: Colombia as NATO's Next Global Partner?" North Atlantic Treaty Organization Research Division Research Paper 138, May 23, 2017, <http://www.ndc.nato.int/news/news.php?icode=1056> (accessed October 27, 2017).
. "More Than 100 Terrorists in TT," _Newsday_ , February 24, 2016, <http://archives.newsday.co.tt/news/print,0,224465.html> (accessed October 27, 2017). The Financial Intelligence Unit of Trinidad and Tobago observed that suspected terrorist financial transactions had tripled in 2015.
. Steve Holland and Anthony Boadle, "Trump Says Democracy Must Be Restored in Venezuela Soon," Reuters, September 19, 2017, <https://www.reuters.com/article/us-usa-trump-latin-america/trump-says-democracy-must-be-restored-in-venezuela-soon-idUSKCN1BT2R7> (accessed November 14, 2017).
. Jorge G. Castaneda, "Trump and Castro Can Save Venezuela," _New York Times_ , September 21, 2017, <https://www.nytimes.com/2017/09/21/opinion/trump-castro-venezuela.html> (accessed November 7, 2017).
. AFP, "India, Pakistan Edge Closer to Joining SCO Security Bloc," _Express Tribune_ , June 24, 2016, <http://tribune.com.pk/story/1129533/india-pakistan-edge-closer-joining-sco-security-bloc/> (accessed May 30, 2017).
. _Joint Communiqué of the 14th Meeting of the Foreign Ministers of the Russian Federation, the Republic of India and the People's Republic of China_ (New Delhi: Ministry of External Affairs, Government of India, April 18, 2016), <http://mea.gov.in/bilateral-documents.htm?dtl/26628/Joint_Communiqu_of_the_14th_Meeting_of_the_Foreign_Ministers_of_the_Russian_Federation_the_Republic_of_India_and_the_Peoples_Republic_of_China> (accessed October 29, 2017).
. Alexander Korablinov, "China Snubs Russian Request for Trilateral Defense Meeting with India," _Russia Beyond_ , April 12, 2017, <https://www.rbth.com/international/2017/04/12/china-snubs-russian-request-for-trilateral-defense-meeting-with-india_740367> (accessed October 29, 2017).
. Mohan Malik, "Balancing Act: The China-India-US Triangle," _World Affairs_ , Spring 2016, <http://www.worldaffairsjournal.org/article/balancing-act-china-india-us-triangle> (accessed October 29, 2017).
. Wade Shapard, "China's Jewel in the Heart of the Indian Ocean: The Colombo Port City Project Was Conceived by Sri Lanka, But Is Now an Almost 100 Percent Chinese Undertaking," _Diplomat_ , May 9, 2016, <http://thediplomat.com/2016/05/chinas-jewel-in-the-heart-of-the-indian-ocean/> (accessed October 29, 2017).
. Brigadier Vinod Anand, "India's Defence Cooperation with South East Asian Countries," _SP's Land Forces_ , no. 3 (2013), <http://www.spslandforces.com/story.asp?id=258> (accessed May 30, 2017).
. Nirmala Ganapathy, "India, Thailand to Boost Defence Ties," _Indian Express_ , June 18, 2016, <http://indianexpress.com/article/india/india-news-india/india-thailand-to-boost-defence-ties-2859804/> (accessed May 30, 2017).
. "India's Partnership with Southeast Asia Nears Its Limits," _Stratfor Worldview_ , September 20, 2016, <https://www.stratfor.com/analysis/indias-partnership-southeast-asia-nears-its-limits> (accessed May 30, 2017).
. Russia and India have held annual naval and land exercises since 2003. In 2016 exercises were hosted by Russia which interestingly chose to host them in Ussuriysk, near Vladivostok and very close to the Chinese border.
. Loro Hort, "From Russia without Love: Russia Resumes Weapons Sales to China," _PacNet_ , no. 89 (Pacific Forum CSIS Honolulu, Hawaii, December 12, 2013), <https://csis-prod.s3.amazonaws.com/s3fs-public/legacy_files/files/publication/Pac1389_0.pdf> (accessed October 29, 2017).
. Malik, "Balancing Act."
. Blank and Kim, "Russian Arms Sales."
. Different experts dispute whether the Indian Rafales to be acquired from France can match the Russian SU-35s. If the SU-35 is better, India might look back to Russia, unless the latter starts backing Pakistan, which is acquiring the JF-17 fighter jet from China. Franz-Stefan Gady, "Pakistan to Order 50 More Fighter Jets in 2017," _Diplomat_ , February 8, 2017, <https://thediplomat.com/2017/02/pakistan-to-order-50-more-fighter-jets-in-2017/> (accessed November 26, 2017).
. Ipsita Chakravarty, "Why Has India Been Silent about the Chemical Attacks in Syria?" Scroll.in, April 8, 2017, <https://scroll.in/article/834039/why-has-india-been-silent-about-the-chemical-attacks-in-syria> (accessed October 29, 2017).
. Devirupa Mitra, "Pakistan Critical to Defeating ISIS: Russian Special Rep to Afghanistan," _Wire_ , December 5, 2016, <https://thewire.in/84672/pakistan-isis-afghanistan-russia/> (accessed May 29, 2017).
. David Brewster, "India Plays the Balochistan Card with China," _Interpreter_ , August 22, 2016, <https://www.lowyinstitute.org/the-interpreter/india-plays-balochistan-card-china> (accessed May 29, 2017).
. M. Ilyas Khan, "India's 'Surgical Strikes' in Kashmir: Truth or Illusion?" _BBC News_ , October 23, 2016, <http://www.bbc.com/news/world-asia-india-37702790> (accessed May 29, 2017). About ten million people live in Indian-administrated Jammu and Kashmir and 4.5 million in Pakistani-run Azad Kashmir. There are 1.8 million people in the Gilgit-Baltistan autonomous territory, which Pakistan created from northern Kashmir and the two small princely states of Hunza and Nagar in 1970.
. "Is India Planning to Cut Off Pakistan's Water Supply?" _American Interest_ , September 27, 2016, <https://www.the-american-interest.com/2016/09/27/is-india-planning-to-cut-off-pakistans-water-supply/> (accessed May 29, 2017).
. David Brewster, "India Plays the Balochistan Card with China."
. "Highlights of PM Modi's Independence Day speech," _Times of India_ , August 15, 2017, <https://timesofindia.indiatimes.com/india/highlights-of-pm-modis-independence-day-speech/articleshow/60067783.cms> (assessed November 7, 2016).
. "Pakistan Prime Minister Slams India's 'Expansionist Designs' in Independence Day Address," _First Post_ , August 16, 2017, <http://www.firstpost.com/world/pakistan-prime-minister-slams-indias-expansionist-designs-in-independence-day-address-full-text-of-his-speech-3932719.html> (assessed November 7, 2016).
. Emanuele Scimia, "India Is Buying 36 Rafale Fighters from France (and Pakistan Should Worry)," _National Interest_ , October 3, 2016, <http://nationalinterest.org/blog/the-buzz/indias-buying-36-rafale-fighters-france-pakistan-should-17911> (accessed May 29, 2017). Gady, "Pakistan to Order 50 More Fighter Jets."
. Arif Rafiq, "India's Modi Is Playing the Wrong Game against China and Pakistan," _National Interest_ , August 21, 2016, <http://nationalinterest.org/feature/indias-modi-playing-the-wrong-game-against-china-pakistan-17411?page=3> (accessed May 29, 2017).
**CHAPTER 7: CHINA, NORTH KOREA, AND THE RISK OF WAR IN THE INDO-PACIFIC**
. Will Ripley, "North Korean Official: Take Hydrogen Bomb Threat 'Literally,'" CNN, October 26, 2017, <http://edition.cnn.com/2017/10/25/politics/north-korea-us-hydrogen-bomb-threat/index.html> (accessed November 14, 2017).
. Nahal Toosi and David Cohen, "Trump Undercuts Tillerson's Efforts on North Korea," _Politico_ , October 1, 2017, <https://www.politico.com/story/2017/10/01/trump-tillerson-korea-twitter-243339> (accessed November 14, 2017).
. Some diplomats argue that Tillerson's advisers have isolated him from State Department expertise and see this restructuring as creating bottlenecks. Eliana Johnson and Michael Crowley, "The Bottleneck in Rex Tillerson's State Department," _Politico_ , June 4, 2017, <https://www.politico.com/story/2017/06/04/rex-tillerson-state-department-bottleneck-239107> (accessed November 14, 2017). Others see the restructuring as a politically motivated Republican purge against the predominantly Democratic State Department bureaucracy, while still others argue that Tillerson is not downsizing as much as Trump desires. Nahal Toosi, "Diplomats Fear Tillerson Transparency Push Is Linked to Clinton Emails," _Politico_ , November 7, 2017, <https://www.politico.com/story/2017/11/07/tillerson-state-clinton-emails-244626> (accessed November 14, 2017).
. Caren Bohan and David Brunnstrom, "Trump Says US Not Necessarily Bound by 'One China' Policy," Reuters, December 12, 2016, <http://www.reuters.com/article/us-usa-trump-china-idUSKBN1400TY> (accessed May 30, 2017).
. _Politico_ magazine revealed that Bob Dole, the former Senate Majority Leader and 1996 Republican presidential nominee, had lobbied the Trump team for months on behalf of the Taiwanese government.
. Tyler Durden, "Chinese Carrier Sails by Taiwan, Enters Contested South China Sea," _ZeroHedge_ , December 26, 2016, <http://www.zerohedge.com/news/2016-12-26/chinese-carrier-sails-taiwan-enters-contested-south-china-sea> (accessed May 30, 2017).
. Matthew M. Burke and Chiyomi Sumida, "China Reportedly Responds to Trump's Taiwan Call by Flying Nuclear-Capable Bomber," _Stars and Stripes_ , December 12, 2016, <http://www.stripes.com/news/china-reportedly-responds-to-trump-s-taiwan-call-by-flying-nuclear-capable-bomber-1.443999> (accessed May 30, 2017).
. David Brunnstrom and Matt Spetalnick, "Tillerson Says China Should Be Barred from South China Sea Islands," Reuters, January 11, 2017, <http://www.reuters.com/article/us-congress-tillerson-china-idUSKBN14V2KZ> (accessed October 29, 2017).
. Jordan Fabian and Evelyn Rupert, "Trump Promises Chinese President He'll Honor 'One China' Policy," _Hill_ , February 10, 2017, <http://thehill.com/homenews/administration/318874-trump-to-honor-one-china-policy> (accessed May 30, 2017). Tillerson later softened his statement about stopping Chinese access to the islands, saying that in the event of an unspecified "contingency," the United States and its allies "must be capable of limiting China's access to and use of" those islands to pose a threat. Idrees Ali, "Exclusive: China Finishing South China Sea Buildings That Could House Missiles—US Officials," Reuters, February 21, 2017, <http://www.reuters.com/article/us-china-usa-southchinasea-exclusive-idUSKBN161029> (accessed May 30, 2017).
. "Projected GDP Ranking (2016–2020)," _Statistics Times_ , December 16, 2016, <http://statisticstimes.com/economy/projected-world-gdp-ranking.php> (accessed May 30, 2017); Liyan Chen, "2015 Global 2000: The World's Largest Banks," _Forbes_ , May 6, 2015, <http://www.forbes.com/sites/liyanchen/2015/05/06/2015-global-2000-the-worlds-largest-banks/#63c5166724f1> (accessed May 30, 2017). In 2015, the top ten productive countries in the world in nominal terms were: the United States, China, Japan, Germany, United Kingdom, France, India, Brazil, Italy, and Canada. In this measure, the US is first and China second; Russia did not even appear in the top ten! But in terms of purchasing power parity (PPP), China came out on top over the US, with Russia in 6th place.
. Mike Patton, "China's Economy Will Overtake the U.S. in 2018," _Forbes_ , <https://www.forbes.com/sites/mikepatton/2016/04/29/global-economic-news-china-will-surpass-the-u-s-in-2018/#32bf7839224a> (accessed November 8, 2017).
. Robert D. Kaplan, "Why the South China Sea Is So Crucial," _Business InsiderAustralia_, February 20, 2015, <https://www.businessinsider.com.au/why-the-south-china-sea-is-so-crucial-2015-2/> (accessed November 8, 2017).
. Harold Raveche, "How Trump, Tillerson Could Bring Peace to South China Sea," _Hill_ , March 23, 2017, <http://thehill.com/blogs/pundits-blog/international-affairs/325620-how-trump-tillerson-could-bring-peace-to-south-china> (accessed October 29, 2017). Superoptimists claim that it could possess oil reserves as high as 130 billion barrels, which would make it the second largest reserve, under Saudi Arabia. See Kaplan, "Why the South China Sea Is So Crucial."
. Jesse Johnson, "Behind the Scenes, Tillerson Tones down Rhetoric on South China Sea," _Japan Times_ , February 7, 2017, <http://www.japantimes.co.jp/news/2017/02/07/asia-pacific/behind-scenes-tillerson-tones-rhetoric-south-china-sea/#.WR7rcvqGP8Q> (accessed October 29, 2017).
. Minnie Chan, "South China Sea Air Strips' Main Role Is 'to Defend Hainan Nuclear Submarine Base,'" _South China Morning Post_ , July 24, 2016, <http://www.scmp.com/news/china/diplomacy-defence/article/1993754/south-china-sea-air-strips-main-role-defend-hainan> (accessed October 29, 2017).
. Permanent Court of Arbitration, "The South China Sea Arbitration (The Republic of Philippines v. The People's Republic of China)," Case View: Case no. 2013-19, <http://www.pcacases.com/web/view/7> (accessed May 30, 2017); see also "PCA Case No. 2013-19: The South China Sea Arbitration Award of 12 July 2016," <http://www.pcacases.com/web/sendAttach/2086> (accessed November 2, 2017).
. Andrew S. Erickson, "China's Blueprint for Sea Power," _China Brief_ 16, no. 11 (July 6, 2016), Jamestown, <https://jamestown.org/program/chinas-blueprint-for-sea-power/#sthash.iBD4wLgr.dpuf> (accessed May 30, 2017). Beijing has been developing "killer" anti-aircraft carrier missiles and other asymmetrical forms of weaponry so as to check US, Japanese, and Taiwanese naval and air forces from moving close to the Chinese mainland in case of conflict.
. Nidhi Prasad, "Yes, Japan Could Build Nuclear Weapons (But at What Cost?)," _National Interest_ , October 12, 2016, <http://nationalinterest.org/blog/the-buzz/yes-japan-could-build-nuclear-weapons-what-cost-18019?page=2> (accessed May 30, 2017).
. Motoko Rich, "Japanese Government Urges Another Increase in Military Spending," _New York Times_ , August 30, 2016, <https://www.nytimes.com/2016/08/31/world/asia/japan-defense-military-budget-shinzo-abe.html> (accessed May 30, 2017).
. Ibid.
. Justin McCurry, "Japan Increases Defence Budget Amid Tensions with China," _Guardian_ , December 17, 2013, <https://www.theguardian.com/world/2013/dec/17/japan-increases-defence-budget-tensions-china> (accessed May 30, 2017).
. Other recent incidents include the overflight of a Xian H-6 aircraft over the South China sea in March 2016 and two Xian H-6 bombers and two escort planes additionally flying around Taiwan in late November. These flights are largely in response to US efforts to sustain its hegemony over the South China Sea's international waters using its own nuclear-capable bombers. The Pentagon has repeatedly deployed its own nuclear capable B-52 and B-2 bombers to Guam, which is easily in range of the South China Sea, and which are seen as potential threats to both North Korea and China.
. Reinhard Drifte, "The Japan-China Confrontation over the Senkaku/Diaoyu Islands—Between 'Shelving' and 'Dispute Escalation,'" _Asia Pacific Journal_ 12, no. 30/3 (2014); available online at: Global Research, July 28, 2014, <http://www.globalresearch.ca/the-japan-china-confrontation-over-the-senkakudiaoyu-islands-between-shelving-and-dispute-escalation/5393760> (accessed May 30, 2017).
. Burke and Sumida, "China Reportedly Responds."
. Brunnstrom and Spetalnick, "Tillerson Says China Should Be Barred."
. These deployments took place just weeks before scheduled talks between Putin and Japanese Prime Minister Shinzo Abe in Yamaguchi Prefecture on December 15. The Bal complex, armed with the X-35 anti-ship missile, can hit targets at a range of 120 kilometers (seventy-five miles). The Bastion complex, however, is equipped with supersonic Onyx missiles and can strike not only battleships but also destroy land-based targets within a range of 600 kilometers. This makes it not only a defensive but also an offensive weapon. Vladimir Mikheev, "Military Build-Up in the Kuril Islands: Bad Timing or a Signal from Moscow?" _Russia Beyond_ , November 24, 2016, <http://rbth.com/international/2016/11/24/military-build-up-in-the-kuril-islands-bad-timing-or-a-signal-from-moscow_650715> (accessed October 29, 2017). This buildup includes the Borei-class nuclear-powered ballistic-missile submarines (SSBN), test-launching the Bulava intercontinental ballistic missile (ICBM) in the Kamchatka Peninsula.
. White House, "Fact Sheet: Advancing the Rebalance to Asia and the Pacific," Office of the Press Secretary, November 16, 2017, <https://obamawhitehouse.archives.gov/the-press-office/2015/11/16/fact-sheet-advancing-rebalance-asia-and-pacific> (November 14, 2017).
. Douglas H. Paal, "How Trump Should Deal with China," Carnegie Endowment for International Peace, December 12, 2016, <http://carnegieendowment.org/2016/12/12/how-trump-should-deal-with-china-pub-66418> (accessed May 30, 2017). Andrew S. Erickson, "China's Blueprint for Sea Power," Jamestown Foundation, July 6, 2016, <https://jamestown.org/program/beijing-talks-tough-new-cold-war-asia/> (accessed May 30, 2017).
. Not all extrajudicial killing is related to drugs. See "License to Kill," Human Rights Watch, March 2, 2017, <https://www.hrw.org/report/2017/03/02/license-kill/philippine-police-killings-dutertes-war-drugs> (November 14, 2017).
. _Phone Call between Trump and Duterte_ (Pasas City, Philippines: Office of American Affairs, April 29, 2017), <https://www.documentcloud.org/documents/3729123-POTUS-RD-Doc.html#document/p1> (accessed October 30, 2017).
. Stephen Blank, "Russia's Growing Ties with Vietnam," _Diplomat_ , September 19, 2013, <http://thediplomat.com/2013/09/russias-growing-ties-with-vietnam/> (accessed May 30, 2017).
. Phil Stewart and Nobuhiro Kubo, "Trump's Defense Chief Heads to Asia, Eying China, North Korea Threat," Reuters, February 1, 2017, <http://www.reuters.com/article/us-usa-trump-mattis-asia-idUSKBN15G3FG> (accessed May 30, 2017).
. Robert S. Litwak, "An Iran-Style Nuclear Deal with North Korea Is the Best America Can Hope For," _Atlantic_ , May 4, 2017, <https://www.theatlantic.com/international/archive/2017/05/iran-deal-north-korea-jcpoa/525372/?utm_source=nl-atlantic-daily-050417> (accessed May 30, 2017); Wit and Ahn, _North Korea's Nuclear Futures_.
. Austin Ramzysept, "Kim Jong-un Called Trump a 'Dotard.' What Does That Even Mean?" _New York Times_ , September 22, 2017, <https://www.nytimes.com/2017/09/22/world/asia/trump-north-korea-dotard.html> (accessed November 14, 2017).
. Zachary Cohen, "North Korea Accuses Trump of Declaring War," CNN, September 26, 2017, <http://edition.cnn.com/2017/09/25/politics/north-korea-fm-us-bombers/index.html> (accessed November 14, 2017).
. Steve Holland, "Trump Wants to Make Sure US Nuclear Arsenal at 'Top of the Pack,'" Reuters, February 24, 2017, <http://www.reuters.com/article/us-usa-trump-idUSKBN1622IF?il=0> (accessed May 30, 2017).
. Lily Hay Newman, "All about the US Missile Defense That'll Protect South Korea—and Tick Off China," _Wired_ , April 23, 2017, <https://www.wired.com/2017/04/missile-defense-will-protect-south-korea-make-china-nervous/> (accessed October 30, 2017).
. Lily Hay Newman, "South Korea's New Missile Defense Tech Isn't a Cure-All for North Korea," _Wired_ , May 5, 2017, <https://www.wired.com/2017/05/south-koreas-new-missile-defense-tech-isnt-cure-north-korea/> (accessed October 30, 2017). In its current placement, the THAAD can defend a number of US military bases, like Camp Walker in Daegu and Kunsan Air Base in Gunsan, along with ports in Busan and the southern tip of South Korea; but it cannot defend Seoul. So Washington will pay for the one that defends US interests, but not all of South Korea. Seoul would need to purchase its own system.
. Kingston Reif, "Moon Reverses THAAD Decision," _Arms Control Today_ , September 2017, <https://www.armscontrol.org/act/2017-09/news/moon-reverses-thaad-decision> (accessed November 14, 2017).
. See Hall Gardner, _Averting Global War_ (New York: Palgrave MacMillan, 2007).
. Kristin Huang, "The 10 Minutes with Xi That Changed Trump's Mind on North Korea," _South China Morning Post_ , April 14, 2017, <http://www.scmp.com/news/china/diplomacy-defence/article/2087518/10-minutes-xi-jinping-changed-donald-trumps-mind-north> (accessed May 30, 2017).
. Gerry Mullany, Chris Buckley, and David E. Sanger, "China Warns of 'Storm Clouds Gathering' in US-North Korea Standoff," _New York Times_ , April 14, 2017, https://www.nytimes.com/2017/04/14/world/asia/north-korea-china-nuclear.html?emc=edit_th_20170415&nl=todaysheadlines&nlid=70196410&_r=0 (accessed May 30, 2017).
. Veronica Rocha, "US Air Force to Launch Test Missile off Central California Coast," _Los Angeles Times_ , April 25, 2017, <http://www.latimes.com/local/lanow/la-me-ln-missile-test-launch-vandenberg-20170425-story.html> (accessed May 30, 2017).
. Kristin Huang, "China's Nuclear Get-Out Clause over Defence of North Korea," _South China Morning Post_ , April 14, 2017, <http://www.scmp.com/news/china/diplomacy-defence/article/2087320/china-not-obliged-defend-n-korea-if-its-attacked-say> (accessed May 30, 2017).
. Ryan Pickrell, "What Would Happen If North Korea Fired Off a Nuclear Weapon?" _National Interest_ , April 14, 2017, <http://nationalinterest.org/blog/the-buzz/what-would-happen-if-north-korea-fired-nuclear-weapon-20205?page=2> (accessed May 30, 2017).
. "All North Korean Missile Tests," Nuclear Threat Initiative, April 24, 2017, <http://www.nti.org/newsroom/news/new-database-documents-all-north-korean-missile-tests/> (accessed May 30, 2017).
. Litwak, "An Iran-Style Nuclear Deal."
. "North Korea Demands Recognition as Legitimate Nuclear State," _Guardian_ , September 11, 2016, <https://www.theguardian.com/world/2016/sep/11/north-korea-demands-recognition-as-legitimate-nuclear-state-pyongyang> (accessed November 14, 2017).
. Merrit Kennedy, "Pence Tells North Korea: 'The Era Of Strategic Patience Is Over'" (April 17, 2017), <http://www.npr.org/sections/thetwo-way/2017/04/17/524316419/pence-tells-north-korea-the-era-of-strategic-patience-is-over> (accessed November 16, 2017).
. Franz-Stefan Gady, "Time to Go 'Huge'? What Will Trump's Defense Policy in Asia Be?" _Diplomat_ , November 10, 2016, <http://thediplomat.com/2016/11/time-to-go-huge-what-will-trumps-defense-policy-in-asia-be/> (accessed May 30, 2017).
. Harry J. Kazianis, "Why China Could Declare a South China Sea ADIZ Right about Now," _National Interest_ , February 1, 2017, <http://nationalinterest.org/blog/the-buzz/why-china-could-declare-south-china-sea-adiz-right-about-now-19273> (accessed May 30, 2017).
. See realistic pre-war scenario: Michael Auslin, "How China Could Respond to Trump Call," CNN, December 5, 2016, <http://edition.cnn.com/2016/12/05/opinions/china-relations-after-trumps-taiwan-call-auslin/index.html> (accessed May 30, 2017).
**CHAPTER 8: SYRIA AND WIDENING WARS IN THE "WIDER MIDDLE EAST"**
. Syrian Observatory for Human Rights, <http://www.syriahr.com/en/> (accessed November 11, 2017). For a skeptical view of death toll estimates, see Alex Ray, "The Death Toll in Syria: What Do the Numbers Really Say?" _Counterpunch_ , May 26, 2016, <https://www.counterpunch.org/2016/05/26/the-death-toll-in-syria-what-do-the-numbers-really-say/> (accessed November 11, 2017).
. UN OCHA, "About the Crisis," <http://www.unocha.org/syrian-arab-republic/syria-country-profile/about-crisis> (accessed October 30, 2017).
. See the Global Coalition official website: <http://theglobalcoalition.org/en/home/>.
. Hall Gardner, "The Geopolitical Convolutions of Fighting the Global War on Terror (GWOT)," in _A New Global Agenda: Priorities, Practices, and Pathways of the International Community_ , ed. by Diana Ayton-Shenker (Lanham, MD: Rowman & Littlefield, 2018); Hall Gardner, "The Russian Annexation of Crimea: Regional and Global Ramifications," _European Politics and Society_ 17, no. 4, _Ukraine in Crisis_ , ed. by Nicolai Petro (March 15, 2016): 490–505, DOI: 10.1080/23745118.2016.1154190.
. Jordan Fabian, "Trump Hits Obama after Syrian Gas Attack," _Hill_ , April 4, 2017, <http://thehill.com/homenews/administration/327259-trump-hits-obama-after-syrian-gas-attack> (accessed May 30, 2017).
. The Tomahawk strike took place after national-populist ideologue Steve Bannon was removed from National Security Council staff and after heavy consultation between Trump and his National Security Advisor, H. R. McMaster. Jordan Fabian, "McMaster Shows Clout in Trump's First Crisis," _Hill_ , April 8, 2017, <http://thehill.com/homenews/administration/327882-mcmaster-shows-clout-in-trumps-first-crisis> (accessed May 30, 2017).
. Gabriel Sherman, "Trump's Syria Strike Is Latest Sign of Steve Bannon's Waning Influence," _New York Magazine_ , April 7, 2017, <http://nymag.com/daily/intelligencer/2017/04/trumps-syria-strike-is-sign-of-bannons-waning-influence.html?mid=twitter-share-di> (accessed October 30, 2017).
. Claudia Assis, "Here's How Much It Costs to Replace the 59 Tomahawk Missiles Trump Fired on Syria," _MarketWatch_ , April 15, 2017, <http://www.marketwatch.com/story/this-is-how-much-it-will-cost-to-replace-the-tomahawks-used-in-syria-2017-04-07> (accessed May 22, 2017).
. White House, "Press Briefing by Secretary of State Rex Tillerson and National Security Advisor General H. R. McMaster," Office of the Press Secretary, June 4, 2017, <https://www.whitehouse.gov/the-press-office/2017/04/06/press-briefing-secretary-state-rex-tillerson-and-national-security> (accessed November 11, 2017).
. Brian Barrett, "The US Strike on Syria Underscores Trump's Media-Fueled Worldview," _Wired_ , April 7, 2017, https://www.wired.com/2017/04/us-strike-syria-underscore-trumps-media-fueled-worldview/?mbid=nl_4717_p3&CNDID=49332341 (accessed May 30, 2017).
. Mary Atkinson, "Latest US Arms Sales to Saudi Arabia Raise Eyebrows about IS War," _Middle East Eye_ , October 2, 2014, last updated May 11, 2015, <http://www.middleeasteye.net/news/saudi-arabia-1295679323> (accessed November 14, 2017).
. "Daesh Launches New Palmyra Push after US Strike," _PressTV_ , April 7, 2017, <http://www.presstv.ir/Detail/2017/04/07/517062/Syria-Daesh-Palmyra-US-Russia-airbase> (accessed May 30, 2017).
. Jonathan Steele, "US-Russia Talks: No Love-In, but No Break-Up Either," _Middle East Eye_ , April 13, 2017, <http://www.middleeasteye.net/columns/us-russia-tillerson-putin-lavrov-trump-moscow-959468392> (accessed October 30, 2017).
. Euan McKirdy, Jason Hanna, and Barbara Starr, "Syria Strikes: Site of Chemical Attack Hit Again," CNN, April 8, 2017, <http://edition.cnn.com/2017/04/08/middleeast/syria-strikes-russia-donald-trump/> (accessed May 30, 2017). "ISIL Takes Advantage of US Attack on Government to Storm Western Palmyra," MEMPSI, April 7, 2017, <http://www.mempsi.net/2017/04/07/isil-takes-advantage-of-us-attack-on-government-to-storm-western-palmyra/> (accessed October 30, 2017).
. Emma Graham-Harrison, "Syria Nerve Agent Attack: Why It Made Sense to Assad," _Guardian_ , April 7, 2017, <https://www.theguardian.com/world/2017/apr/07/syria-nerve-agent-attack-why-it-made-sense-to-assad> (accessed May 30, 2017).
. "Seventh Report of the Organization for the Prohibition of Chemical Weapons–United Nations Joint Investigative Mechanism," October 25, 2017, <https://drive.google.com/file/d/0ByLPNZ-eSjJdcGZUb0hqalFOa0hhdEZ3WlBvZmRnajFRV3pr/view> (accessed November 14, 2017).
. Robert Parry, "Did Al-Qaeda Dupe Donald Trump on Alleged Syrian Sarin Gas Attack on April 4, 2017," _Consortium News_ , November 9, 2017, <https://www.newcoldwar.org/al-qaeda-dupe-trump-alleged-syrian-sarin-gas-attack-april-4-2017/> (accessed November 14, 2017); Daniel Lazare, "Luring Trump into Mideast Wars," _Consortium News_ , April 8, 2017, <https://consortiumnews.com/2017/04/08/luring-trump-into-mideast-wars/> (accessed May 30, 2017); Robert Parry, "UN Team Heard Claims of 'Staged' Chemical Attacks," _Consortium News_ , September 8, 2016, <https://consortiumnews.com/2016/09/08/un-team-heard-claims-of-staged-chemical-attacks/> (accessed May 30, 2017).
. Human Rights Watch, "Syria: Barrage of Barrel Bombs: Attacks on Civilians Defy UN Resolution," July 30, 2014, <https://www.hrw.org/news/2014/07/30/syria-barrage-barrel-bombs> (accessed November 14, 2017); _Wikipedia_ , s.v. "List of Syrian Civil War Barrel Bomb Attacks," last edited October 9, 2017, <https://en.wikipedia.org/wiki/List_of_Syrian_Civil_War_barrel_bomb_attacks> (accessed November 14, 2017).
. Hamidreza Azizi, "Will US Missile Strike Shift Iran-Russia Partnership in Syria?" _Al-Monitor_ , April 12, 2017, <http://www.al-monitor.com/pulse/originals/2017/04/iran-russia-syria-partnership-us-missile-strike-impact.html#ixzz4eKCz40hj> (accessed May 30, 2017).
. Iran and Russia have had a legal battle over payments involving the S-300 antimissile system. But Moscow proposed the S-400 as an option in 2015. "Russian S-400 Missiles for Iran," Investmentwatchblog, March 20, 2015, <http://investmentwatchblog.com/russian-s-400-missiles-for-iran/> (accessed November 11, 2017).
. Alec Luhn, "Russia Sends Missile Cruiser to Mediterranean as Syria Tension Mounts," _Guardian_ , September 12, 2013, <https://www.theguardian.com/world/2013/sep/12/russia-sends-ships-mediterranean-syria> (accessed May 30, 2017).
. Senator Joni Ernst, who is on the Senate Armed Services Committee, stated that "this was a one-time attack on the assets that were used in a chemical weapons attack against the people of Syria," and "not an on-going operation." Todd Beamon, "Joni Ernst on Syrian Strike: 'This Was a One-Time Attack,'" _Newsmax_ , April 7, 2017, <http://www.newsmax.com/Politics/joni-ernst-syrian-strike-one-time/2017/04/07/id/783224/> (accessed November 14, 2017).
. Rebecca Savransky, "Top Trump Officials Turn Up Heat on Russia," _Hill_ , April 9, 2017, <http://thehill.com/homenews/administration/328019-top-trump-officials-turn-up-heat-on-russia> (accessed May 30, 2017).
. Ibid.
. AFP, Reuters, "Trump Strikes on Syria: How World Leaders Reacted," _Khaleej Times_ , April 7, 2017, <http://www.khaleejtimes.com/region/mena/trump-strikes-on-syria-how-world-leaders-reacted> (accessed May 30, 2017).
. White House, "Press Briefing by Secretary of State Rex Tillerson and National Security Advisor General H. R. McMaster," June 4, 2017. Peter Baker, Neil MacFarquhar, and Michael R. Gordon, "Syria Strike Puts US Relationship with Russia at Risk," _New York Times_ , April 7, 2017, https://www.nytimes.com/2017/04/07/world/middleeast/missile-strike-syria-russia.html?emc=edit_th_20170408&nl=todaysheadlines&nlid=70196410&_r=0 (accessed May 30, 2017).
. Parvez Jabri, "China Warns of Deterioration in Syria with Xi in US," _Business Recorder_ , April 7, 2017, <http://www.brecorder.com/2017/04/07/342347/china-warns-of-deterioration-in-syria-with-xi-in-us/> (accessed November 11, 2017).
. Liu Jieyi, China's permanent representative to the United Nations, said military actions will only worsen the suffering of the Syrian people and called for all countries to support the efforts of UN Special Envoy for Syria Staffan de Mistura to implement a "political solution." "Political Solution Only Way Out for Syrian Issue: Chinese Envoy," Xinhua, April 8, 2017, <http://news.xinhuanet.com/english/2017-04/08/c_136191497.htm> (accessed November 14, 2017).
. "Secretary-General Urges 'Restraint' after United States Air Strikes against Syria, Stressing Risk of Escalation, Need for Renewed Commitment to Political Solution," UN Press Release, April 7, 2017, <https://www.un.org/press/en/2017/sgsm18487.doc.htm> (accessed November 14, 2017).
. "World Leaders React to the US Attack on Syrian Military Bases," _Gulf News Syria_ , April 7, 2017, <http://gulfnews.com/news/mena/syria/world-leaders-react-to-the-us-attack-on-syrian-military-bases-1.2007394> (accessed May 30, 2017).
. Ibid.
. Tom Miles, "Yemen's Cholera Epidemic Hits 600,000," Reuters, September 5, 2017, <https://www.reuters.com/article/us-yemen-cholera/death-toll-in-yemen-cholera-outbreak-hits-nearly-700-who-idUSKBN18X1LG> (accessed November 14, 2017).
. "Death Toll in Yemen Conflict Passes 10,000," Al Jazeera, January 17, 2017, <http://www.aljazeera.com/news/2017/01/death-toll-yemen-conflict-passes-10000-170117040849576.html> (accessed November 14, 2017).
. Patrick W. Ryan, "The Yemen Crisis and the Bab el-Mandeb Maritime Chokepoint," Saudi-US-Relations Information Service (SUSRIS), April 14, 2015, <http://susris.com/2015/04/14/the-bab-el-mandeb-maritime-chokepoint/> (accessed November 14, 2017).
. "Djibouti Naval Base," sinodefence.com, August 20, 2017, <http://sinodefence.com/djibouti-naval-base/> (accessed November 14, 2017).
. Ramin Mostaghim and Shashank Bengali, "Syrian Ally Iran Blasts U.S. Missile Strikes as 'Dangerous, Destructive and a Violation of International Law,'" _LA Times_ , April 5, 2017, <http://www.latimes.com/world/la-fg-iran-syria-20170407-story.html> (accessed November 14, 2017).
. Jonathan Stempel, "Saudi Arabia Faces $6 Billion US Lawsuit by Sept. 11 Insurers," Reuters, March 24, 2017, <http://www.reuters.com/article/us-usa-saudi-sept-idUSKBN16V1ZP> (accessed October 30, 2017).
. Thomas Gibbons-Neff, "Trump's First Arms Sales, Holdovers from the Obama Era, Are Business as Usual," _Washington Post_ , January 24, 2017, <https://www.washingtonpost.com/news/checkpoint/wp/2017/01/24/trumps-first-arms-sales-holdovers-from-the-obama-era-are-business-as-usual/?utm_term=.de21160e81f6> (accessed May 30, 2017).
. Aaron Mehta, "Revealed: Trump's $110 Billion Weapons List for the Saudis," _Defense News_ , June 8, 2017, <https://www.defensenews.com/breaking-news/2017/06/08/revealed-trump-s-110-billion-weapons-list-for-the-saudis/> (accessed November 14, 2017).
. Baker, MacFarquhar, and Gordon, "Syria Strike Puts US."
. Michael Flynn, "Transcript: Michael Flynn on ISIL," Al Jazeera, January 13, 2016, <http://www.aljazeera.com/programmes/headtohead/2016/01/transcript-michael-flynn-160104174144334.html> (accessed November 14, 2017).
. "US Missile Strike on Syria: Timeline of Reactions," _PressTV_ , April 7, 2017, <http://www.presstv.com/Detail/2017/04/07/517085/US-military-attack-Syria-timeline> (accessed October 30, 2017).
. Ibid.
. Patrick Kingsley, "Erdogan Claims Vast Powers in Turkey After Narrow Victory in Referendum," _New York Times_ , April 16, 2017, https://www.nytimes.com/2017/04/16/world/europe/turkey-referendum-polls-erdogan.html?mtrref=fr.search.yahoo.com&gwh=D4531B929DC49A1CFB7B39524DD062DF&gwt=pay (accessed November 14, 2017).
. Zeina Karam and Sarah El Deeb, "Trump Launches US Missile Strike against Syria," _CTVNews_ , April 7, 2017, <http://www.ctvnews.ca/world/trump-launches-u-s-missile-strike-against-syria-1.3358473> (accessed May 30, 2017).
. For an outline of Kurdish parties in the region: Rodi Hevian, "The Main Kurdish Political Parties in Iran, Iraq, Syria, and Turkey: A Research Guide," Rubin Center, August 19, 2013, <http://www.rubincenter.org/2013/08/the-main-kurdish-political-parties-in-iran-iraq-syria-and-turkey-a-research-guide/> (accessed May 29, 2017).
. Ryan Browne and Elise Labott, "US 'Deeply Concerned' after Turkey Bombs Allies in Iraq and Syria," CNN, April 25, 2017, <http://edition.cnn.com/2017/04/25/politics/turkey-bombs-kurds-iraq-us-concerned/index.html> (accessed November 14, 2017).
. Michael R. Gordon and Eric Schmitt, "Trump to Arm Syrian Kurds, Even as Turkey Strongly Objects," _New York Times_ , May 9, 2017, https://www.nytimes.com/2017/05/09/us/politics/trump-kurds-syria-army.html?emc=edit_na_20170509&nl=breaking-news&nlid=70196410&ref=cta&_r=0 (accessed May 29, 2017).
. Staff, "Iraqi Kurdish Leader Calls for Non-Binding Independence Referendum," Reuters, February 2, 2016, <http://www.reuters.com/article/us-iraq-kurds-idUSKCN0VB2EY> (accessed May 30, 2017).
. Nikolai Pakhomov, "Russia and Turkey: The Arms Deal That Signals the Age of Pragmatism," _LobeLog_ , September 9, 2017, <http://lobelog.com/russia-and-turkey-the-arms-deal-that-signals-the-age-of-pragmatism/> (accessed October 30, 2017).
. "UN Envoy Urges Russia, Iran, Turkey to Convene Further Syrian Talks," RadioFreeEurope/RadioLiberty, March 25, 2017, <https://www.rferl.org/a/syria-un-envoy-urges-talks-russia-iran-turkey/28390318.html> (accessed October 30, 2017).
. Margaret Talev and Jennifer Jacobs, "Trump Praises Erdogan for 'High Marks' amid Crackdown Concerns," Bloomberg Politics, September 21, 2017, <https://www.bloomberg.com/news/articles/2017-09-21/trump-praises-erdogan-for-high-marks-amid-crackdown-concerns> (accessed November 14, 2017).
. Ipsita Chakravarty, "Why Has India Been Silent about the Chemical Attacks in Syria?" Scroll.in, April 8, 2017, <https://scroll.in/article/834039/why-has-india-been-silent-about-the-chemical-attacks-in-syria> (accessed October 30, 2017).
. Richard Nephew, "How the Iran Deal Prevents a Covert Nuclear Weapons Program," _Arms Control Today_ , September 2, 2015; Martin Zonas, "Iran Nuclear Deal: There Is No Alternative," _Economonitor_ , April 7, 2015, <http://www.economonitor.com/blog/2015/04/iran-nuclear-deal-there-is-no-alternative/> (accessed October 30, 2017).
. Sarah Begley, "Read Donald Trump's Full Speech to AIPAC," _Time_ , March 21, 2016, <http://time.com/4267058/donald-trump-aipac-speech-transcript/> (accessed May 30, 2017). Benjamin Netanyahu immediately denounced the Iranian nuclear accord and continued to threaten a potential military strike against Iranian nuclear infrastructure. See, for example, analysis by Ben Caspit, "Netanyahu Threatens to 'Kill Himself' in Order to Stop Iran Deal," _Al-Monitor_ , July 15, 2015, <http://www.al-monitor.com/pulse/originals/2015/07/benjamin-netanyahu-iran-nuclear-deal-inspection-clauses.html#> (accessed May 30, 2017).
. Bozorgmehr Sharafedin, "Iran Confirms Missile Test, Drawing Tough Response from Trump Aide," Reuters, February 1, 2017, <http://www.reuters.com/article/us-usa-iran-missiles-idUSKBN15G3ZO> (accessed October 30, 2017).
. Peter Kenyon, "Did Iran's Ballistic Missile Test Violate a UN Resolution?" NPR, February 3, 2017, <http://www.npr.org/sections/parallels/2017/02/03/513229839/did-irans-ballistic-missile-test-violate-a-u-n-resolution> (accessed May 30, 2017).
. As of October 15, 2017, the US Congress is to decide in sixty days whether Iran is actually cheating on the Iran nuclear accord, the JCPOA. On the one hand, President Trump had been urged to decertify the treaty by Senators Tom Cotton, Ted Cruz, David A. Perdue, and Marco Rubio, and by former Ambassador to the United Nations John Bolton, for example. Israeli Prime Minister Benjamin Netanyahu also demanded renegotiation of JCPOA. On the other hand, US Secretary of State Rex Tillerson and US Secretary of Defense James Mattis, and the Chairman of the Joint Chiefs of Staff Gen. Joseph Dunford, have all argued that the maintenance of the JCPOA is in the US national security interest, and have been reluctant to decertify it. Former US Secretaries of State John Kerry and Madeleine Albright have both strongly supported the JCPOA. Uzi Arad, the former National Security Advisor of Israeli Prime Minister Netanyahu, has urged the White House and Congress not to abandon the JCPOA. In addition, a group of over 180 Democrats led by Representatives Ted Deutch (FL) and David Price (NC) sent a letter to President Trump urging him to recertify the Iran nuclear accord to Congress even before the October 15 deadline. It is possible that Congress could decide to maintain the JCPOA. But this possibility appears unlikely due to strong opposition to JCPOA among both Republicans and Democrats. See interview with Hall Gardner and Majid Golpour (in French), "Trois questions sur l'accord nucléaire iranien," _Contrepoints_ , October 15, 2017, <https://www.contrepoints.org/2017/10/12/300772-trois-questions-laccord-nucleaire-iranien> (accessed November 14, 2017).
. Thomas Erdbrink, "As Iran and US Leaders Trade Barbs, Big Deals Proceed," _New York Times_ , May 28, 2017, <https://www.nytimes.com/2017/05/28/world/middleeast/iran-nuclear-deal-hassan-rouhani-donald-trump.html> (accessed October 30, 2017).
. Ken Bredemeier, "Iran Warns US of Possible Missile Attack If It Imposes New Sanctions," VOA, October 08, 2017, <https://www.voanews.com/a/iran-guard-chief-warns-us-against-imposing-new-sanctions/4061320.html> (accessed November 15, 2017).
. Christopher Woody, "Trump: The US Is Ready to Leave One of its Most Important Military Bases If the Gulf Crisis Worsens," _Business Insider_ , July 19, 2017, <http://www.businessinsider.fr/us/trump-us-is-ready-to-leave-al-udeid-military-base-amid-gulf-crisis-2017-7/> (accessed November 15, 2017).
. "How the Battle for Mosul Unfolded," BBC News, July 10, 2017, <http://www.bbc.com/news/world-middle-east-37702442> (accessed May 29, 2017).
. Angela Dewan and Tim Lister, "Mosul Completely Freed from ISIS: What's Next for the City Left in Ruins?" CNN, July 10, 2017 <http://edition.cnn.com/2017/07/10/middleeast/mosul-what-next/index.html> (accessed November 15, 2017).
. _SIGAR's High-Risk List_ (Arlington, VA: Office of the Special Inspector General for Afghanistan Reconstruction), <https://www.sigar.mil/interactive-reports/high-risk-list/index.html> (accessed May 29, 2017).
. Missy Ryan Greg Jaffe, "Donald Trump to Declare War on Taliban in Afghanistan Despite Experts' Concerns," _Independent_ , May 9, 2017, <http://www.independent.co.uk/news/world/americas/donald-trump-afghanstan-taliban-us-troop-surge-islamist-militants-talks-battle-fight-general-a7725601.html> (accessed May 29, 2017).
. David Corn, "Here's More Evidence That Trump Did Not Oppose the Iraq War Before It Began," _Mother Jones_ , September. 2, 2016, <http://www.motherjones.com/politics/2016/09/heres-more-evidence-trump-did-not-oppose-iraq-war/> (accessed November 15, 2017).
. Hallie Jackson and Erik Ortiz, "Trump Weighs Sending as Many as 5,000 More Troops to Afghanistan," NBC News, May 9, 2017, <http://www.nbcnews.com/news/world/trump-weighs-sending-many-5-000-more-troops-afghanistan-n756751> (accessed May 29, 2017).
. Devirupa Mitra, "Pakistan Critical to Defeating ISIS: Russian Special Rep to Afghanistan," _Wire_ , December 5, 2016, <https://thewire.in/84672/pakistan-isis-afghanistan-russia/> (accessed May 29, 2017).
. Masood Saifullah, "Trump's Afghanistan Policy Will Face Big Limitations," Deutsche Welle, May 2, 2017, <http://www.dw.com/en/trumps-afghanistan-policy-will-face-big-limitations/a-38661290> (accessed May 29, 2017).
. A high-ranking Russian official, Zamir Kubalov, was quoted as saying that "our (Russian) interests are the same as Taliban in fighting Daesh." Indrani Bagchi, "Russia's Stand on Taliban Is Trouble for India," _India Times_ , December 15, 2016, <http://economictimes.indiatimes.com/articleshow/56001501.cms> (accessed October 30, 2017). Kubalov is seen as the brains behind the new Russian rapprochement with Pakistan.
. Tillerson: "The ongoing commitment of NATO Allies and partners to peace in Afghanistan, including to an eventual settlement between the Afghan government and the Taliban, protects this Alliance's interests, and, when successful, ensures that Afghanistan never again becomes a safe haven for terrorists." Rex W. Tillerson, "NATO Foreign Ministerial Intervention Remarks" (speech; Brussels, Belgium: US Department of State, March 31, 2017), <https://www.state.gov/secretary/remarks/2017/03/269339.htm> (accessed May 29, 2017).
. Saifullah, "Trump's Afghanistan Policy."
. Peter Baker and Michael D. Shearmay, "Trump Softens Tone on Islam but Calls for Purge of 'Foot Soldiers of Evil,'" _New York Times_ , May 21, 2017, <https://www.nytimes.com/2017/05/21/world/middleeast/trump-saudi-arabia-islam-speech.html?_r=0> (accessed October 30, 2017).
. Zainab Fattah, "Guide to $400 Billion in Saudi-U.S. Deals: Black Hawks to Oil," Bloomberg, May 31, 2017, <https://www.bloombergquint.com/markets/2017/05/22/guide-to-400-billion-in-saudi-u-s-deals-black-hawks-to-oil> (accessed November 15, 2017).
. For a Qatari perspective, "Saudi Crown Prince Mohammed bin Salman Widens Purge," Al Jazeera, November 6, 2017, <http://www.aljazeera.com/news/2017/11/saudi-crown-prince-mohammed-bin-salman-widens-purge-171106104312835.html> (accessed November 15, 2017). For a Yemeni perspective, "Saudi Prince Mohammad bin Salman Consolidates Power & Purges Rivals under 'Anti-Corruption' Pretense Story, Interview with Toby Jones and Afrah Nasser," _Democracy Now_ , November 9, 2017, <https://www.democracynow.org/2017/11/9/saudi_prince_mohammad_bin_salman_consolidates> (accessed November 15, 2017).
**CHAPTER 9: PEACE THROUGH STRENGTH? OR WORLD WAR TRUMP?**
. Donald Trump, _The Art of the Deal_ (New York: Random House, 1987).
. The lifting of sanctions on Moscow—without reinforcing NATO's eastern flank—will be interpreted by Moscow "as consent" to "further expansion," and "the result of a policy not to 'aggravate' Russia and instead to seek 'constructive dialogue' will be war." Przemyslaw Zurawski vel Grajekski, in Kinga Redlowska, ed., _NATO: Rethink, Realign, React_ (Warsaw: Institute for Eastern Studies, 2016), p. 9.
. Gilbert Doctorow, "Trump Quiets Some Russian Doubts," _Consortium News_ , January 30, 2017, <https://consortiumnews.com/2017/01/30/trump-quiets-some-russian-doubts/> (accessed May 29, 2017).
. I. William Zartman, "The Timing of Peace Initiatives: Hurting Stalemates and Ripe Moments," _Global Review of Ethnopolitics_ 1, no. 1 (September 2001): 8–18.
. For Minsk II accords: "Full Text of the Minsk Agreement," _Financial Times_ , February 15, 2015, <https://www.ft.com/content/21b8f98e-b2a5-11e4-b234-00144feab7de> (accessed May 29, 2017).
. Gwendolyn Sasse, "Constitution Making in Ukraine: Refocusing the Debate," Carnegie Europe, April 12, 2016, <http://carnegieeurope.eu/2016/04/12/constitution-making-in-ukraine-refocusing-debate-pub-63304> (accessed May 29, 2017).
. Vladimir Frolov, "Russia Looks On as Ukraine Hangs in the Balance," _Moscow Times_ , April 13, 2016, <http://www.themoscowtimes.com/opinion/opinion/article/russia-looks-on-as-ukraine-hangs-in-the-balance-op-ed/565762.html> (accessed May 29, 2017). See also Brian Milakovsky, "Understanding the 'Under Control' Donbas," Woodrow Wilson Center, _Kennan Cable_ , no. 16 (April 2016), <https://www.wilsoncenter.org/publication/kennan-cable-no16-understanding-the-under-control-donbas> (accessed October 30, 2017); Nicolai Petro, "Bringing Ukraine Back into Focus: How to End the New Cold War and Provide Effective Political Assistance to Ukraine," Carnegie Council, August 19, 2015, <https://www.carnegiecouncil.org/publications/articles_papers_reports/742> (accessed October 30, 2017).
. "Savchenko Meets Russia-Backed Separatist Leaders, Stirring Outrage," RadioFreeEurope/RadioLiberty, December 12, 2016, <http://www.rferl.org/a/ukraine-savchenko-meets-separatists-minsk-plotnitsky-zakharchenko/28172257.html> (accessed May 30, 2017).
. Philip Karber and Phillip Petersen, in Redlowska, ed., _NATO Rethink, Realign, React_ , p. 36.
. One proposal is that Russia and the United States and Europeans "agree to disagree" for the indefinite future over the diplomatic status of Crimea; that the status quo in Donbas as part of Ukraine continues, with an effectively enforced cease-fire; and that a multibillion-dollar aid package be assembled from international sources directed to economic recovery in Ukraine. And, finally, a full and frank exchange on Russian involvement in US domestic elections must be addressed. See Jeffrey Burt, James Hitch, Peter Pettibone, and Thomas Shillinglaw, "Trump, Eisenhower and Russia: A Chance for Peace," _National Interest_ , November 5, 2017, <http://nationalinterest.org/feature/trump-eisenhower-russia-chance-peace-23051?page=2> (accessed November 15, 2017). In the view of this author, an agreement in which Russia and the West "agree to disagree" for the indefinite future on the diplomatic status of Crimea, just as the Soviet Union and United States did during the Cold War with respect to the Baltic states, will not prove sufficient to guarantee peace as long as Kiev retains its irredentist claims to Crimea and does not adopt a formally neutral stance with respect to NATO and the Russian-led CSTO.
. The North Atlantic Treaty Organization (Washington DC: NATO, April 4, 1949), <http://www.nato.int/cps/en/natolive/official_texts_17120.htm> (accessed October 30, 2017). Hall Gardner, _NATO Expansion and US Strategy in Asia_ (Basingstoke: Palgrave MacMillan, 2013); Hall Gardner, _Crimea, Global Rivalry and the Vengeance of History_ (New York: Palgrave Macmillan, 2015).
. Verkhovna Rada Staff, "Declaration of State Sovereignty of Ukraine," Verkhovna Rada of the Ukrainian Soviet Socialist Republic, <http://static.rada.gov.ua/site/postanova_eng/Declaration_of_State_Sovereignty_of_Ukraine_rev1.htm> (accessed May 29, 2017).
. "Gorbachev: Ukraine Should Sign Neutrality into Constitution," _Moscow Times_ , August 18, 2016, <https://themoscowtimes.com/news/ukraine-must-sign-neutrality-into-its-constitution-gorbachev-55027> (accessed May 29, 2017).
. The plan may have been originally proposed by Opposition Bloc, a parliamentary faction that formed in 2014 from remnants of the old party of the ousted pro-Kremlin President Viktor Yanukovych. Nick Paton Walsh, Salma Abdelaziz, and Victoria Butenko, "Lawmaker: Trump Lawyer Discussed Ukraine Deal," CNN, February 24, 2017, <http://edition.cnn.com/2017/02/23/politics/trump-lawyer-ukraine-peace-deal/> (accessed May 30, 2017).
The Ukrainian ambassador, Mr. Chaly, rejected a lease of that kind. "It is a gross violation of the Constitution.... Such ideas can be pitched or pushed through only by those openly or covertly representing Russian interests." Megan Twohey and Scott Shane, "A Back-Channel Plan for Ukraine and Russia, Courtesy of Trump Associates," _New York Times_ , February 19, 2017, https://www.nytimes.com/2017/02/19/us/politics/donald-trump-ukraine-russia.html?hp&action=click&pgtype=Homepage&clickSource=story-heading&module=a-lede-package-region®ion=top-news&WT.nav=top-news&_r=1 (accessed May 30, 2017).
. This is what the British did after seizing Hong Kong in 1847—and then leasing it for 150 years before returning Hong Kong to China in 1997, a fact which has not pleased all Hong Kong residents. The US had leased the Panama Canal Zone after supporting and recognizing Panama's independence from Colombia. In the 1921 Thomson–Urrutia Treaty, the US then paid off Colombia and granted it special privileges in the Canal Zone for recognizing Panama's independence.
. Twohey and Shane, "A Back-Channel Plan"; Julia Ioffe, "The Mystery of the Ukraine Peace Plan," _Atlantic_ , February 20, 2017, <https://www.theatlantic.com/international/archive/2017/02/ukraine-peace-plan/517275/> (accessed May 30, 2017).
. Przemyslaw Zurawski vel Grajekski, in Redlowska, ed., _NATO Rethink, Realign, React_ , p. 15.
. Andrew E. Kramer, "Ethnic Conflict between Armenia and Azerbaijan Flares Anew," _New York Times_ , April 4, 2016, <https://www.nytimes.com/2016/04/05/world/europe/ethnic-conflict-between-armenia-and-azerbaijan-flares-anew.html?_r=0> (accessed October 30, 2017).
. US Ambassador Vershbow has rightly argued for such deployments, but NATO has done little to develop the PfP. Alexander Vershbow, "Trump to Lavrov: Get Out of Ukraine or Face Stiffer Sanctions," _Newsweek_ , May 9, 2015, <http://www.newsweek.com/trump-lavrov-get-out-ukraine-or-face-stiffer-sanctions-606246> (accessed May 30, 2017).
. Eleni Fotiou, "Caucasus Stability and Cooperation Platform: What Is at Stake for Regional Cooperation?" _ICBSS Policy Brief_ , no. 16 (June 2009), <https://www.files.ethz.ch/isn/104737/PB_16.pdf> (accessed October 30, 2017).
. Henry Kissinger, "How the Ukraine Crisis Ends," _Washington Post_ , March 5, 2014. See also Des Browne, Wolfgang Ischinger, Igor S. Ivanov, Sam Nunn, and Adam Daniel Rotfeld, "Ukraine Must Not Become a New Berlin Wall," Nuclear Threat Institute, March 13, 2014.
. Gabriela Baczynska and Robin Emmott, "Germany, France Seek Stronger EU Defense after Brexit: Document," Reuters, September 12, 2016, <http://www.reuters.com/article/us-europe-defense-idUSKCN11I1XU> (accessed October 30, 2017).
. See Sven Biscop, "How the EU Can Save NATO," _Security Policy Brief_ , no. 83 (Brussels, Belgium: Egmont Institute, March 2017), www.egmontinstitute.be/wp-content/uploads/2017/03/SBP83.pdf (accessed October 30, 2017).
. NATO, "Relations with the European Union" North Atlantic Treaty Organization, March 30, 2017, <http://www.nato.int/cps/en/natohq/topics_49217.htm> (accessed October 30, 2017).
. _White Paper on the Future of Europe_ (Brussels, Belgium: European Commission, March 1, 2017), <https://ec.europa.eu/commission/sites/beta-political/files/white_paper_on_the_future_of_europe_en.pdf> (accessed October 30, 2017).
. Y. Varoufakis, S. Holland, and J. K. Galbraith, "A Modest Proposal for Resolving the Eurozone Crisis," _Genius_ , <https://genius.com/Y-varoufakis-s-holland-and-jk-galbraith-a-modest-proposal-for-resolving-the-eurozone-crisis-annotated> (accessed May 30, 2017).
And even this 2008 crisis stemmed, at least in part, from the US Federal Reserve's decision to keep Federal interest rates artificially low (in the view of many economists) in the period from 2002–2004 during the George W. Bush administration.
. Jorge Rodríguez, "EU's New Thinking on Decentralisation and Territorial Development," European Centre for Development Policy Management, June 2015, <http://ecdpm.org/great-insights/territorial-development-2/eus-new-thinking-on-decentralisation-and-territorial-development/> (accessed May 30, 2017).
. Hall Gardner, _NATO Expansion and US Strategy in Asia_ (New York: Palgrave MacMillan, 2013), chap. 7.
. Ayhan Simsek, "Germany Opposes Call to End Turkey's EU Accession Talks," AA, November 25, 2016, <http://aa.com.tr/en/europe/germany-opposes-call-to-end-turkey-s-eu-accession-talks/693243> (accessed November 15, 2017).
. Christopher De Bellaigue, "Welcome to Demokrasi: How Erdogan Got More Popular than Ever," _Guardian_ , August 30, 2016, <https://www.theguardian.com/world/2016/aug/30/welcome-to-demokrasi-how-erdogan-got-more-popular-than-ever> (accessed May 29, 2017).
. Paul Tugwell and Selcan Hacaoglu, "Why the World's Watching Cyprus Unification Talks," Bloomberg, January 9, 2017, <https://www.bloomberg.com/news/articles/2017-01-08/why-the-world-s-watching-cyprus-unification-talks-quicktake-q-a> (accessed October 30, 2017).
. Patrick Wintour, "Cyprus Peace Talks:—All You Need to Know," _Guardian_ , January 9, 2017, <https://www.theguardian.com/news/2017/jan/09/cyprus-peace-talks-all-you-need-to-know> (accessed May 31, 2017); "Publications," European Union Institute for Security Studies, <http://www.iss.europa.eu/publications/detail/article/toward-a-new-euro-atlantic-security-framework/> (accessed October 30, 2017).
. Laura Rozen, "Syria Talks Pulled Back from Brink," _Al Monitor_ , April 18, 2016, <http://www.al-monitor.com/pulse/originals/2016/04/syria-talks-geneva-opposition-assad-mistura.html> (accessed October 30, 2017). Ian Black, "All Eyes on US and Russia as Syria Deadline Passes," _Guardian_ , February 19, 2016, <http://www.theguardian.com/world/2016/feb/19/all-eyes-on-us-and-russia-as-syria-deadline-passes> (accessed October 30, 2017).
. "The PKK's Fateful Choice in Northern Syria," _International Crisis Group_ , Report 176, May 4, 2017, <https://www.crisisgroup.org/middle-east-north-africa/eastern-mediterranean/syria/176-pkk-s-fateful-choice-northern-syria> (accessed October 30, 2017).
. Chase Winter, "Iraq Sweeps Up More Territories as Kurds Quarrel amongst Themselves," DW, October 17, 2017, <http://www.dw.com/en/iraq-sweeps-up-more-territories-as-kurds-quarrel-amongst-themselves/a-40996992> (accessed November 15, 2017).
. US policy has traditionally supported limited Israeli expansions into the West Bank, but in exchange for trade off of other lands with the Palestinians. John Podhoretz, "Media Gets Trump's Settlements Policy Wrong," _Commentary Magazine_ , February 3, 2017, <https://www.commentarymagazine.com/foreign-policy/middle-east/israel/israel-settlements-media/> (accessed May 31, 2017).
. Jack Khoury and Amir Tibon, "Jordan's King Warns Trump against Moving US Embassy in Israel to Jerusalem, State Media Reports," _Haaretz_ , February 2, 2017, <http://www.haaretz.com/us-news/1.769266> (accessed May 31, 2017).
. See for example, Stephen Walt, "Making the Middle East Worse, Trump-Style," _Foreign Policy_ , June 9, 2017, <http://foreignpolicy.com/2017/06/09/making-the-middle-east-worse-trump-style-saudi-arabia-qatar-iran-israel/> (accessed November 15, 2017). See also Ibrahim Fraihat, "Why Saudi-Israeli Normalisation Could Be Dangerous," Al Jazeera, November 19, 2017, <http://www.aljazeera.com/indepth/opinion/saudi-israeli-normalisation-dangerous-171119083143078.html> (accessed November 22, 2017).
. Lubna Masarwa and Arwa Ibrahim, "EXCLUSIVE: Abbas to Offer Large Land Swap with Israel in Trump Talks," _Middle East Eye_ , May 21, 2017, <http://www.middleeasteye.net/news/abbas-propose-unprecedented-land-exchange-israel-during-trump-visit-703500509> (accessed October 30, 2017).
. See Hall Gardner, _Averting Global War_ (New York: Palgrave Macmillan, 2008).
. Masarwa and Ibrahim, "Abbas to Offer Large Land Swap."
. For a skeptical view, see Harsh V. Pant, "The SCO Illusion Takes India: As India Joins the SCO, It Must Keep in Mind Certain Geopolitical Realities," _Diplomat_ , June 09, 2017, <https://thediplomat.com/2017/06/the-sco-illusion-takes-india/> (accessed November 15, 2017).
. India's trade with the members of the Eurasian Economic Union stands at about $10 billion. "India to Speed Up FTA with Eurasian Economic Union," _Hindu Business Line_ , February 28, 2017, <http://www.thehindubusinessline.com/economy/india-to-speed-up-fta-with-eurasian-economic-union/article9564225.ece> (accessed November 15, 2017).
. For a clear outline of seven different proposals to resolve the Kashmir question, thus indicating its complexity, see "The Future of Kashmir?" BBC News, <http://news.bbc.co.uk/2/shared/spl/hi/south_asia/03/kashmir_future/html/7.stm> (accessed October 30, 2017).
. Srinagar, "India's Kashmir Problem Is Getting Worse," _Economist_ , May 25, 2017, <http://www.economist.com/news/asia/21722666-shunning-separatists-will-not-make-it-better-indias-kashmir-problem-getting-worse> (accessed October 30, 2017).
. On May 6, 2016, Prime Minister Abe had traveled to Russia to meet with President Vladimir Putin in Sochi. As this was seen as breaking the G7's policy of isolating Russia in response to the 2014 annexation of Crimea, President Barack Obama purportedly phoned Abe in an effort to dissuade him from making the visit. James D. J. Brown, "Japan's 'New Approach' to Russia," _Diplomat_ , June 18, 2016, <http://thediplomat.com/2016/06/japans-new-approach-to-russia/> (accessed October 30, 2017).
. "Putin, Abe Agree on Joint Russia-Japan Activities on Kuril Islands," RT, December 15 2016, <https://www.rt.com/news/370452-putin-visits-japan-talks/> (accessed October 30, 2017). See also Hall Gardner, _NATO Expansion and US Strategy in Asia_ (New York: Palgrave MacMillan, 2013).
. Robin Harding in Tokyo and Kathrin Hille, "Russia and Japan Agree Economic Deal on Disputed Islands," _Financial Times_ , December 16, 2016, <https://www.ft.com/content/1905fc24-c360-11e6-9bca-2b93a6856354> (accessed October 30, 2017).
. Jennifer Jacobs and Andreo Calonzo, "Trump Offers to Play South China Sea Peacemaker as Trip Wraps Up," Bloomberg, November 12, 2017, <https://www.bloomberg.com/news/articles/2017-11-12/trump-offers-to-broker-deal-to-resolve-south-china-sea-dispute> (accessed November 15, 2017).
. Office of US Trade Representative, "U.S. -Korea Free Trade Agreement," Office of the United States Trade Representative, <https://ustr.gov/trade-agreements/free-trade-agreements/korus-fta> (accessed November 15, 2017).
. For timeline of diplomacy dealing with North Korea, see _Wikipedia_ , s.v. "Timeline of the North Korean Nuclear Program," last edited October 17, 2017, <https://en.wikipedia.org/wiki/Timeline_of_the_North_Korean_nuclear_program> (accessed October 30, 2017).
. Ben Kamisar, "Trump Praises China for Abstaining from UN Vote," _Hill_ , April 12, 2017, <http://thehill.com/homenews/administration/328573-trump-praises-china-for-abstaining-from-un-vote> (accessed May 31, 2017).
. Sam Nunn, "Former Senator Sam Nunn on CNN re: North Korea," interview by Wolf Blitzer, _Nuclear Threat Initiative_ , April 25, 2017, <http://www.nti.org/newsroom/news/former-senator-sam-nunn-cnn-re-north-korea-interview-wolf-blitzer/> (accessed May 31, 2017).
. "China Warns Conflict Could Erupt 'Any Moment' over North Korea," _South China Morning Post_ , April 14, 2017, <http://www.scmp.com/news/china/diplomacy-defence/article/2087726/conflict-could-break-out-any-moment-over-north-korea> (accessed May 31, 2017).
. Anna Fifield and Simon Denyer, "North Korea Shows off New Missiles in Huge Military Parade, but Doesn't Test Nuke," _Washington Post_ , April 15, 2017, <https://www.washingtonpost.com/world/north-korea-blames-trump-and-hisaggressive-tweets-for-tensions/2017/04/14/6932c9aa-20e1-11e7-bcd6-6d1286bc177d_story.html?utm_term=.bc5ea434be8f> (accessed May 31, 2017).
. "The way you are going to make a good trade deal is to help us with North Korea. Otherwise, we are just going to go it alone, but going it alone means going at it with a lot of other nations. President Xi, I think he means well and I think he wants to help. We'll see whether or not he does." Alex Lockie, "Trump Says He Put Economic Pressure on China's President to Help with North Korea," _Business Insider_ , April 12, 2017, <http://www.businessinsider.fr/us/trump-trade-pressure-north-korea-china-2017-4/> (accessed October 30, 2017).
. Secretary of Defense James Mattis was said to be opposed to deploying tactical nuclear weapons in South Korea. Robert Burns, "James Mattis: North Korea Has 'Accelerated' Threat of Nuclear Attack," _Time_ , October 28, 2017, <http://time.com/5001305/james-mattis-north-korea-acclerated-nuclear-threat/> (accessed November 15, 2017).
. "Donald Trump's South Korea Speech: The Key Points," _Guardian_ , November 8, 2017, <https://www.theguardian.com/world/2017/nov/08/donald-trumps-south-korea-speech-key-points-kim-jong-un> (accessed November 15, 2017).
. Staff, "China Cautions against Use of Force on North Korea," Reuters, April 28, 2017, <http://uk.reuters.com/article/uk-northkorea-usa-un-china-idUKKBN17U29C?il=0> (accessed October 30, 2017).
. AFP, "North Korea Fires Another Missile, Trump Tweets His Anger," _TheJournal_ , April 29, 2017, <http://www.thejournal.ie/north-korea-trump-2-3365580-Apr2017/> (accessed May 31, 2017).
. Barbara Plett Usher, "North Korea: US Not Seeking Regime Change, Says Rex Tillerson," BBC, August 2, 2017, <http://www.bbc.com/news/world-us-canada-40797613> (accessed November 15, 2017).
. Nick Wadhams, "Tillerson Says He Envisions US-North Korea Talks," Bloomberg, November 10, 2017, <https://www.bloomberg.com/news/articles/2017-11-10/tillerson-sees-u-s-north-korea-agreeing-to-start-conversation> (accessed November 15, 2017).
. Nunn, "Former Senator Sam Nunn," interview by Wolf Blitzer.
. Steve Holland, Stephen J. Adler, and Jeff Mason, "Exclusive: Trump Says 'Major, Major' Conflict with North Korea Possible, but Seeks Diplomacy," Reuters, April 27, 2017, <http://www.reuters.com/article/us-usa-trump-exclusive-idUSKBN17U04E> (accessed May 31, 2017).
. Matthew Little, "China 'Unequivocal' on North Korea Not Getting Nuclear Weapons: Tillerson, Trump and Xi Affirm Position on North Korea during State Visit," _Epoch Times_ , November 9, 2017, <https://www.theepochtimes.com/china-unequivicol-on-north-korea-not-getting-nuclear-weapons-tillerson_2353648.html> (accessed November 15, 2017).
. See Gardner, _Averting Global War_. See also proposals of Rajan Menon, "What Would War Mean in Korea?" _TomDispatch_ , June 4, 2017, <http://www.tomdispatch.com/post/176291/tomgram%3A_rajan_menon%2C_what_would_war_mean_in_korea/> (accessed October 30, 2017).
. Eleanor Albert, "North Korea's Military Capabilities," Council of Foreign Relations, September 5, 2017, <https://www.cfr.org/backgrounder/north-koreas-military-capabilities> (accessed November 15, 2017).
. Holland, Adler, and Mason, "Trump Says 'Major, Major' Conflict."
. Bruce Cumings, "Korea: Forgotten Nuclear Threats," _Le Monde Diplomatique_ , December 2004, <http://mondediplo.com/2004/12/08korea> (accessed October 30, 2017). While the US has blamed Soviet- and Chinese-backed North Korea for initiating the Korean War, the security forces of the US-backed government of President Syngman Rhee (1948–1960) were responsible for killing more than 100,000 people, which included between 30,000 to 60,000 in the infamous 1948 Cheju massacre alone, in an effort to eradicate left-wing opposition in the country. Menon, "What Would War Mean in Korea?" The US needs to dialogue with the North Korean regime whether Washington likes it or not!
**CHAPTER 10: DEFUSING THE GLOBAL CRISIS**
. Mikhail Gorbachev, "It All Looks as If the World Is Preparing for War," _Time Magazine_ , January 26, 2017, <http://time.com/4645442/gorbachev-putin-trump/> (accessed October 30, 2017).
. Julian Borger, "America's New, More 'Usable,' Nuclear Bomb in Europe," _Guardian_ , November 10, 2015, <https://www.theguardian.com/world/julian-borger-global-security-blog/2015/nov/10/americas-new-more-usable-nuclear-bomb-in-europe> (accessed November 15, 2017).
. Gregory Kulacki, _China's Military Calls for Putting Its Nuclear Forces on Alert_ (Cambridge, MA: Union of Concerned Scientists, 2016), <http://www.ucsusa.org/nuclear-weapons/us-china-relations/china-hair-trigger#.WNuCRWSGNwc> (accessed October 30, 2017).
. In 2010, the NATO foreign ministers agreed that "no nuclear weapons would be removed from Europe unless all 28-member states of NATO agreed." Amy F. Woolf, _Nonstrategic Nuclear Weapons_ (Washington, DC: Congressional Research Service, February 21, 2017), <https://fas.org/sgp/crs/nuke/RL32572.pdf> (accessed October 30, 2017). See also Ernie Regehr, _Canadian Defence Policy and NATO's Nuclear Weapons_ (Vancouver: Simons Foundation, August 23, 2016), <http://www.thesimonsfoundation.ca/sites/default/files/Canadian%20Defence%20Policy%20and%20NATO%E2%80%99s%20Nuclear%20Weapons%2C%20Defence%20Policy%20Review%20briefing%20paper%20-%20Aug%2023%2C%202016.pdf> (accessed October 30, 2017).
. Hall Gardner, _NATO Expansion and US Strategy in Asia_ (New York: Palgrave MacMillan, 2013).
. Treaty on the Non-Proliferation of Nuclear Weapons (Department for Disarmament Affairs, United Nations, 2000), <http://www.un.org/en/conf/npt/2005/npttreaty.html> (accessed October 30, 2017). Article VI: Each of the Parties to the Treaty undertakes to pursue negotiations in good faith on effective measures relating to cessation of the nuclear arms race at an early date and to nuclear disarmament, and on a treaty on general and complete disarmament under strict and effective international control.
. Katrina vanden Heuvel, "The Escalating Nuclear Threat Finally Has the Public's Attention," _Washington Post_ , October 24, 2017, https://www.washingtonpost.com/opinions/the-escalating-nuclear-threat-finally-has-the-publics-attention-now-what/2017/10/24/504fd5c4-b80b-11e7-9e58-e6288544af98_story.html?tid=ss_mail&utm_term=.bf695991b73c (accessed November 15, 2017).
. James Pomfret, Neil Jerome Morales, "South China Sea Code of Conduct Talks to Be 'Stabilizer' for Region: China Premier," Reuters, November 14, 2017, <http://www.reuters.com/article/us-asean-summit-southchinasea/south-china-sea-code-of-conduct-talks-to-be-stabilizer-for-region-china-premier-idUSKBN1DE05K?il=0> (accessed November 16, 2017). But Beijing's promise does not touch the larger questions of North Korea and rivalries between China, Taiwan, and Japan.
. Martin Kettle, "The World's Powers Have to Resolve Their Remnants of Empire," _Guardian_ , December 23, 2017, <https://www.theguardian.com/commentisfree/2016/dec/23/post-imperial-territories> (accessed October 30, 2017).
. See Will Fitzgibbon and Scilla Alecci, "Paradise Papers Firm Managed Millions for a Carousel of Millionaires and Fraudsters," International Consortium of Investigative Journalism, November 15, 2017, <https://www.icij.org/investigations/paradise-papers/> (accessed November 15, 2017).
. Dean Baker, "A Job-Killing Robot for Rich People," _Jacobin_ , June 27, 2017, https://jacobinmag.com/2017/06/financial-transactions-tax-finance-inequality-bernie-sanders?cn=cmV0d2VldA== UN (accessed October 30, 2017). "Goal 10: Reduce Inequality within and among Countries," United Nations, <http://www.un.org/sustainabledevelopment/inequality/> (accessed October 30, 2017).
. Lesley Wroughton, "G20 Fails to Endorse Financial Transaction Tax," Reuters, November 4, 2011, <http://www.reuters.com/article/g20-tax-idUSN1E7A302520111104> (accessed October 30, 2017).
. Jennifer Epstein and Mark Niquette, "Trump Wants $200 Billion for Infrastructure, Mulvaney Says," Bloomberg, April 20, 2017, <https://www.bloomberg.com/politics/articles/2017-04-20/trump-wants-200-billion-for-infrastructure-mulvaney-says> (accessed May 26, 2017); Adie Tomer, Joseph Kane, and Robert Puentes, "How Historic Would a $1 Trillion Infrastructure Program Be?" Brookings Institution, May 12, 2017, https://www.brookings.edu/blog/the-avenue/2017/05/12/how-historic-would-a-1-trillion-infrastructure-program-be/?utm_campaign=Brookings%20Brief&utm_source=hs_email&utm_medium=email&utm_content=51917420 (accessed May 26, 2017).
. John Wagner, "Trump Re-Ups Criticism of United Nations, Saying It's Causing Problems, Not Solving Them," _Washington Post_ , December 28, 2016, <https://www.washingtonpost.com/news/post-politics/wp/2016/12/28/trump-re-ups-criticism-of-united-nations-saying-its-causing-problems-not-solving-them/?utm_term=.85d14cebfef3> (accessed May 31, 2017).
. Robert Schroeder, "U.S. Drops Out of UNESCO over Arrears, 'Anti-Israel Bias,'" _MarketWatch_ , Oct 12, 2017, <https://www.marketwatch.com/story/us-drops-out-of-unesco-over-arrears-anti-israel-bias-2017-10-12> (accessed November 15, 2017).
. "Financing Peacekeeping," United Nations, <http://www.un.org/en/peacekeeping/operations/financing.shtml> (accessed October 30, 2017). Percentages of costs: The US and its allies: United States (28.57 %); Japan (9.68 %); Germany (6.39 %); France (6.31 %); United Kingdom (5.80 %); Italy (3.75 %); Canada (2.92 %); Spain (2.44 %); Russia and China: Russian Federation (4.01 %); China (10.29 %).
. _Democratic Republic of Congo: Background and US Relations_ (Washington, DC: Congressional Research Service, July 24, 2015–February 27, 2017), <https://www.everycrsreport.com/reports/R43166.html> (accessed October 30, 2017).
. Sarah N. Lynch, "SEC Halts Some Enforcement of Conflict Minerals Rule amid Review," Reuters, April 7, 2017, <https://www.reuters.com/article/us-usa-sec-conflictminerals/sec-halts-some-enforcement-of-conflict-minerals-rule-amid-review-idUSKBN1792WX> (accessed November 15, 2017). Michael S. Piwowar, "Reconsideration of Conflict Minerals Rule Implementation," (public statement; Washington, DC: US Securities and Exchange Commission, January 31, 2017), <https://www.sec.gov/news/statement/reconsideration-of-conflict-minerals-rule-implementation.html> (accessed October 30, 2017); for opposing views see: "Implementation of US Dodd-Frank Act Rule on Conflict Minerals: Commentaries, Guidance, Company Actions," Business and Human Rights Resource Center, <https://business-humanrights.org/en/conflict-peace/conflict-minerals/implementation-of-us-dodd-frank-act-rule-on-conflict-minerals-commentaries-guidance-company-actions> (accessed October 30, 2017).
. In 2006, the US Senate passed the S. 2125, the Democratic Republic of Congo Relief, Security, and Democracy Promotion Act. See Pub. L. 109-456 (December 22, 2006). This legislation had concluded that disease, war, and desperate poverty in Africa threatens both the United States' core value of preserving human dignity and the United States' strategic priority of combating global terror. The legislation accordingly committed the United States to work toward peace, prosperity, and good governance in the Congo. Securities and Exchange Commission, "Release No. 34-67716; File No. S7-40-10: Conflict Minerals," <https://www.sec.gov/rules/final/2012/34-67716.pdf> (accessed November 2, 2017).
. Lynch, "SEC Halts Some Enforcement of Conflict Minerals Rule amid Review."
. On the UN Organization Stabilization Mission in the DRC (MONUSCO), see <https://monusco.unmissions.org/en> (accessed November 15, 2017).
. John Calvelli, "Only Transparency Can Prevent Conflict Minerals from Harming People and Wildlife," _Hill_ , April 18, 2017, <http://thehill.com/blogs/pundits-blog/energy-environment/329347-only-transparency-can-prevent-conflict-minerals-from> (accessed October 30, 2017).
. _Conflict Minerals and the Democratic Republic of Congo: Responsible Action in Supply Chains, Government Engagement and Capacity Building_ (Washington, DC: Business for Social Responsibility, May 2010), <https://www.bsr.org/reports/BSR_Conflict_Minerals_and_the_DRC.pdf> (accessed October 30, 2017); Tomi Oladipo, "The UN's Peacekeeping Nightmare in Africa," BBC News, January 5, 2017, <http://www.bbc.com/news/world-africa-38372614> (accessed October 30, 2017).
. In 2007, the United States was spending some $4.5 billion per month to support its military operations in Iraq. This was about the same as the United Nations spent to run all eighteen of its current peacekeeping missions for a year. See James Dobbins, "A Comparative Evaluation of United Nations Peacekeeping," testimony presented before the House Committee on Foreign Affairs, Subcommittee on International Organizations, Human Rights, and Oversight, June 13, 2007, <https://www.rand.org/content/dam/rand/pubs/testimonies/2007/RAND_CT284.pdf> (accessed November 15, 2017).
. According to Trump, the concept of global warming was created by and for the Chinese in order to make US manufacturing non-competitive. Jeremy Diamond, "Trump Nominees Say Climate Change Is No Hoax, but Still Invite Skepticism," CNN, January 19, 2017, <http://edition.cnn.com/2017/01/17/politics/donald-trump-cabinet-picks-climate-change/> (accessed May 31, 2017).
. "Donald Trump at Loggerheads with Rest of G7 over Climate Change," _Financial Times_ , May 27, 2017, <https://www.ft.com/content/d6ad0050-42cd-11e7-ab92-4c27fbc26eed> (login required for access).
. Mayor Bill Peduto issued an executive order a day after pledging Pittsburgh would continue to follow the guidelines of the Paris Climate Agreement. "Pittsburgh Mayor Issues Executive Order in Response to Trump's Paris Climate Decision," CBS Pittsburgh, June 2, 2017, <http://pittsburgh.cbslocal.com/2017/06/02/pittsburgh-paris-climate-executive-order/> (accessed November 15, 2017).
. Jacqueline Thomsen, "Pittsburgh Mayor Fires Back at Trump: My City Will Follow Paris Agreement," _Hill_ , June 1, 2017, <http://thehill.com/blogs/blog-briefing-room/news/335994-pittsburgh-mayor-fires-back-at-trump-my-city-will-follow-paris> (accessed October 30, 2017).
. On NERA Consulting, see "Factchecking President Trump's Claims about the Paris Agreement," _Fortune_ , June 2, 2017, <http://fortune.com/2017/06/02/paris-agreement-factchecking-trump/> (accessed November 15, 2017). See also the Heritage Foundation Report, Kevin Dayaratna, Nicolas Loris, and David Kreutzer, _Consequences of Paris Protocol: Devastating Economic Costs, Essentially Zero Environmental Benefits_ (Washington, DC: Heritage Foundation, April 13, 2016), <http://www.heritage.org/environment/report/consequences-paris-protocol-devastating-economic-costs-essentially-zero> (accessed October 30, 2017).
. "Factchecking President Trump's Claims."
. Alanna Petroff, "The Heat Is On: President Trump Says He Will Decide This Week Whether to Stick with the Landmark Paris Climate Accord," CNN, May 29, 2017, <http://money.cnn.com/2017/05/29/news/trump-paris-climate-change-business/> (accessed October 30, 2017). Tomás Carbonell, "What Do the 2016 Elections Mean for the Clean Power Plan?" Pardon Our Interruption, December 6, 2016, <http://www.renewableenergyworld.com/articles/2016/12/what-do-the-2016-elections-mean-for-the-clean-power-plan.html> (accessed May 31, 2017).
. International Renewable Energy Agency (IRENA), "Rethinking Energy: Renewable Energy and Climate Change," 2015, http://www.irena.org/-/media/Files/IRENA/Agency/Publication/2014/IRENA-_REthinking_Energy_2nd_report_2015.pdf?la=en&hash=35AF7434755915D342D41966EF595175CB0AE738 (accessed November 15, 2017).
. Timothy Cama and Devin Henry, "Trump Takes Action to Move Forward with Keystone, Dakota Access Pipelines," _Hill_ , January 24, 2017, <http://thehill.com/policy/energy-environment/315852-trump-orders-keystone-dakota-access-pipeline-applications-to-move> (accessed May 31, 2017).
. "Coal Mining: Long-Term Contribution Trends," OpenSecrets.org: Center for Responsive Politics, <https://www.opensecrets.org/industries/summary.php?ind=E1210> (accessed October 30, 2017). In 2016, general mining interests gave Republicans roughly ten times more in direct campaign contributions than Democrats, electrical utilities gave roughly two times more to Republicans; natural gas industries four times more; oil and gas industries gave nine times more, and Republicans even obtained slightly more than Democrats from alternative energy firms.
. Associated Press, "Trump Has Promised to Revive the Coal Industry, But His Economic Advisor Says That Doesn't Make Much Sense," CNBC, May 26, 2017, <http://www.cnbc.com/2017/05/26/donald-trump-and-economic-advisor-gary-cohn-differ-on-coal.html> (accessed October 30, 2017).
. Ibid.
. Alexander C. Kaufman, "Trump Signs Executive Orders on Keystone XL, Dakota Access Pipelines," _Huffington Post_ , January 24, 2017, <http://www.huffingtonpost.com/entry/trump-keystone-dakota-access_us_58877e02e4b070d8cad57814> (accessed October 30, 2017).
. Ibid.
. _Renewable Energy and Jobs Annual Review (2016)_ (Abu Dhabi, United Arab Emirates: International Renewable Energy Agency, 2016), <http://www.se4all.org/sites/default/files/IRENA_RE_Jobs_Annual_Review_2016.pdf> (accessed October 30, 2017); Linda Pentz Gunter, "Trump Is Foolish to Ignore the Flourishing Renewable Energy Sector," _Truthout_ , February 5, 2017, <http://www.truth-out.org/opinion/item/39306-trump-is-foolish-to-ignore-the-flourishing-renewable-energy-sector> (accessed May 31, 2017).
. _US Energy and Employment Report_ (Washington, DC: Department of Energy, January 2017), <https://energy.gov/sites/prod/files/2017/01/f34/2017%20US%20Energy%20and%20Jobs%20Report_0.pdf> (accessed October 30, 2017).
. Suzanne Goldenberg, "Rich Countries' $100bn Promise to Fight Climate Change 'Not Delivered,'" _Guardian_ , June 29, 2015, <https://www.theguardian.com/environment/2015/jun/29/rich-countries-100bn-promise-fight-climate-change-not-delivered> (accessed October 30, 2017).
. Elizabeth Bast, Alex Doukas, Sam Pickard, Laurie Van Der Burg, and Shelagh Whitley, _Empty Promises: G20 Subsidies to Oil, Gas and Coal Production_ (Washington, DC: Oil Change International, November 2015), <https://www.odi.org/sites/odi.org.uk/files/odi-assets/publications-opinion-files/9958.pdf> (accessed October 30, 2017); Elizabeth Bast, Sebastien Godinot, Stephen Kretzmann, and Jake Schmidt, _Under the Rug: How Governments and International Institutions Are Hiding Billions in Support to the Coal Industry_ (Washington, DC: Oil Change International, June 2015), <http://priceofoil.org/content/uploads/2015/05/Under_The_Rug_NRDC_OCI_WWF_Jun_2015.pdf> (accessed October 30, 2017).
. Bast, Doukas, Pickard, Van Der Burg, and Whitley, _Empty Promises._
. Ibid.
. Ibid.
. Jenny Rowland, Myriam Alexander-Kearns, Erin Auel, Matt Lee-Ashley, and Howard Marano, "How Exxon Won the 2016 Election," Center for American Progress, January 10, 2017, <https://www.americanprogress.org/issues/green/news/2017/01/10/296277/how-exxon-won-the-2016-election/> (accessed May 31, 2017). Natasha Bertrand, "Rex Tillerson's Confirmation Hearing Is Today: Here's How His Company, ExxonMobil, Could Benefit from a Trump Presidency," _Business Insider_ , January 11, 2017, http://uk.businessinsider.com/how-exxon-mobil-trump-presidency-benefits-2017-1?r=US&IR=T (accessed May 31, 2017).
. For a study of renewable energies, see Cédric Philibert, "Renewable Energy for Industry: From Green Energy to Green Materials and Fuels," International Energy Agency: Insights Series 2017, <http://www.iea.org/publications/insights/insightpublications/Renewable_Energy_for_Industry.pdf> (accessed November 15, 2017).
. Nigel Purvis and Joshua Busby, _The Security Implications of Climate Change for the UN System_ (Washington, DC: Wilson Center, United Nations and Environmental Security, 2004); Justin Gillis, "Climate Model Predicts West Antarctic Ice Sheet Could Melt Rapidly," _New York Times_ , March 30, 2016, <https://www.nytimes.com/2016/03/31/science/global-warming-antarctica-ice-sheet-sea-level-rise.html?_r=1> (accessed October 31, 2017); Ian Urbina, "Perils of Climate Change Could Swamp Coastal Real Estate," _New York Times_ , November 24, 2016, https://www.nytimes.com/2016/11/24/science/global-warming-coastal-real-estate.html?module=Promotron®ion=Body&action=click&pgtype=article (accessed May 31, 2017); Justin Gillis, "Earth Sets a Temperature Record for the Third Straight Year," _New York Times_ , January 18, 2017, https://www.nytimes.com/2017/01/18/science/earth-highest-temperature-record.html?emc=edit_na_20170118&nlid=70196410&ref=cta (accessed May 31, 2017).
. Adam Gabbatt, "How Hurricanes and Sea-Level Rise Threaten Trump's Florida Resorts," _Guardian_ , September 9, 2017, <https://www.theguardian.com/us-news/2017/sep/09/trump-florida-mar-a-lago-hurricane-irma> (accessed October 31, 2017).
. Jennifer Jacobs and Andreo Calonzo, "Trump Offers to Play South China Sea Peacemaker as Trip Wraps Up," Bloomberg, November 12, 2017, <https://www.bloomberg.com/news/articles/2017-11-12/trump-offers-to-broker-deal-to-resolve-south-china-sea-dispute> (accessed November 15, 2017).
. Stephen Goss et al., _Effects of Unauthorized Immigration on the Actuarial Status of the Social Security Trust Funds_ , Actuarial Note No. 151 (Baltimore, MD: Social Security Administration Office of the Actuary, April 2013), <https://www.ssa.gov/oact/NOTES/pdf_notes/note151.pdf> (accessed October 31, 2017).
. Inspector General, _Status of the Social Security Administration's Earnings Suspense File_ (Baltimore, MA: Social Security Administration, September 2015), <https://oig.ssa.gov/sites/default/files/audit/full/pdf/A-03-15-50058.pdf> (accessed October 31, 2017).
. "War on Drugs an Epic Fail, BMJ Editors Say," Global Commission on Drug Policy, November 17, 2016, <https://www.globalcommissionondrugs.org/reports/war-on-drugs/> (accessed October 31, 2017); _War on Drugs_ (Geneva, Switzerland, Global Commission on Drug Policy, June 2011), <https://www.globalcommissionondrugs.org/wp-content/uploads/2012/03/GCDP_WaronDrugs_EN.pdf> (accessed October 31, 2017).
. Jeffrey A. Miron, _The Budgetary Implications of Drug Prohibition_ (Cambridge, MA: Harvard University, February 2010), <https://scholar.harvard.edu/files/miron/files/budget_2010_final_0.pdf> (accessed October 31, 2017).
Miron's report estimates that legalizing drugs would save roughly $48.7 billion per year in government expenditure on enforcement of prohibition. $33.1 billion of this savings would accrue to state and local governments, while $15.6 billion would accrue to the federal government. Approximately $13.7 billion of the savings would result from legalization of marijuana, $22.3 billion from legalization of cocaine and heroin, and $12.8 from legalization of other drugs. The report also estimates that drug legalization would yield tax revenue of $34.3 billion annually, assuming legal drugs are taxed at rates comparable to those on alcohol and tobacco. Approximately $6.4 billion of this revenue would result from legalization of marijuana, $23.9 billion from legalization of cocaine and heroin, and $4.0 billion from legalization of other drugs.
. Kitty Holland, "Decriminalization of All Drugs for Personal Use Considered," _Irish Times_ , April 18, 2017.
. Robert A. Levy, "Reflections on Gun Control by a Second Amendment Advocate," _National Law Journal_ , February 11, 2013, available at CATO Institute, <https://www.cato.org/publications/commentary/reflections-gun-control-second-amendment-advocate> (accessed November 15, 2017).
. National Institute on Drug Abuse, "Opioid Crisis," last updated June 2017, <http://www.drugabuse.gov/drugs-abuse/opioids/opioid-crisis> (accessed November 15, 2017).
. Kim Stephens "Trump's Tweets Return to Haunt Him. Again," News.com.au, January 23, 2017, <http://www.news.com.au/technology/online/social/donald-trump-mocks-protesters-four-years-after-unsuccessfully-calling-for-protests-against-barack-obama/news-story/e114739c33fabb80fd34fef91ce238fa> (accessed May 26, 2017). Many of Trump's tweets were later deleted.
. Drew DeSilver, "Trump's Victory Another Example of How Electoral College Wins Are Bigger than Popular Vote Ones," Pew Research Center, December 20, 2016, <http://www.pewresearch.org/fact-tank/2016/12/20/why-electoral-college-landslides-are-easier-to-win-than-popular-vote-ones/> (accessed May 26, 2017).
. Richard Dawkins, "Can the Electoral College System Be Reformed?" Richard Dawkins Foundation, February 9, 2017, <https://richarddawkins.net/2017/02/can-the-electoral-college-system-be-reformed/> (accessed May 26, 2017).
. Carol Orsag, "A 38-State Nation," The Thirty-Eight States, <http://www.tjc.com/38states/> (accessed May 26, 2017), originally published in David Wallenchinsky and Irving Wallace, _The People's Almanac_ (New York: Doubleday, 1975). This was proposed in 1972 by C. Etzel Pearcy. Other proposals reduce the number to ten to twelve states, but that might give too much power to certain regions.
. While one might think politicians would not want to give up power, the highly bureaucratic French were able to reduce the number of regions in mainland France from twenty-two to thirteen in the period 2014 to 2016. "La carte à 13 régions définitivement adoptée," _Le Monde_ , December 17, 2014.
. Francis Fukuyama, _The End of History and the Last Man_ (New York: Avon, 1993). See my critique of Fukuyama's work in Hall Gardner, _Crimea, Global Rivalry and the Vengeance of History_ (New York: Palgrave Macmillan, 2015).
. For a brief history of the concept of workplace democracy (what the French call "autogestion") which can take different forms from employee management and control to employee stock ownership without control, see Markus Pausch, "Workplace Democracy from a Democratic Ideal to a Managerial Tool and Back," _The Innovation Journal: The Public Sector Innovation Journal_ 19, no. 1 (2013), article 3, <http://www.innovation.cc/scholarly-style/19_1_3_pausch_workplace-democracy.pdf> (accessed October 31, 2017). For a brief positive description of a generally not well-known form of shared-capitalist management that can function effectively, see Jerry L. Ripperger, "How Employee Ownership Benefits Executives, Companies, and Employees," American Management Association, <http://www.amanet.org/training/articles/how-employee-ownership-benefits-executives-companies-and-employees.aspx> (accessed October 31, 2017). See more detailed analysis, Douglas L. Kruse, Richard B. Freeman, and Joseph R. Blasi, ed., _Shared Capitalism at Work: Employee Ownership, Profit and Gain Sharing, and Broad-Based Stock Options_ (Chicago: University of Chicago Press, 2010), available online at: <http://www.nber.org/chapters/c8085.pdf> (accessed October 31, 2017). I was national office manager of the Association for Workplace Democracy (AWD) in Washington, DC for two years in the early 1980s, which produced the journal _Workplace Democracy_. Unfortunately, AWD soon died out in the Reagan period, but many of its practical proposals are still relevant in today's socio-political and financial crisis.
**POSTSCRIPT: IT CAN HAPPEN HERE**
. Trump: "I'm not isolationist, but I am 'America First.' So I like the expression. I'm 'America First.'" David E. Sanger and Maggie Haberman, "In Donald Trump's Worldview, America Comes First, and Everybody Else Pays," _New York Times_ , March 26, 2016, <https://www.nytimes.com/2016/03/27/us/politics/donald-trump-foreign-policy.html?_r=0> (accessed October 31, 2017).
. Ibid.
. "George S. Patton's Speech to the Third US Army," Patton Museum of Calvary and Armor, March 24, 1944, <https://web.archive.org/web/20060616031308/http://www.knox.army.mil/museum/pattonsp.htm> (accessed October 31, 2017).
. Sanger and Haberman, "In Donald Trump's Worldview."
. After dropping "30 or so atomic bombs...strung across the neck of Manchuria," MacArthur planned to introduce half a million Chinese Nationalist troops at the Yalu and then "spread behind us—from the Sea of Japan to the Yellow Sea—a belt of radioactive cobalt." MacArthur was certain that the Russians would have done nothing about this extreme strategy: "My plan was a cinch." MacArthur was not removed from duty because he advocated the use of nuclear weapons, but because he could not be fully trusted to carry out orders that might involve their use. See Bruce Cumings, "Why Did Truman Really Fire MacArthur?...The Obscure History of Nuclear Weapons and the Korean War Provides the Answer," _History News Network_ , January 10, 2005, <http://historynewsnetwork.org/article/9245> (accessed October 31, 2017).
. Sanger and Haberman, "In Donald Trump's Worldview."
. Trump's November 2017 offer to negotiate disputes between China and its neighboring countries over the South China Sea could prove positive, as would a settlement between the United States, Japan, and Russia, if Trump is truly willing to lead the negotiations. See chapters and .
. Guillaume Bouzard, "Inquiétude à la Maison Blanche," _Le Canard Enchainé_ , January 25, 2017, p. 4.
. David Horsey, "Trump Leaks State Secrets and Self-Incriminating Boasts," _Los Angeles Times_ , May 17, 2017, <http://www.latimes.com/opinion/topoftheticket/la-na-tt-trump-leaks-20170516-story.html> (accessed November 16, 2017).
. Tatiana Schlossberg, "What Is the Antiquities Act and Why Does President Trump Want to Change It?" _New York Times_ , April 26, 2017, <https://www.nytimes.com/2017/04/26/climate/antiquities-act-federal-lands-donald-trump.html> (accessed October 31, 2017).
. Presidential Documents, "Promoting Energy Independence and Economic Growth," Exec. Order No. 13, 783, 82 C.F.R. 16093 (March 28, 2017), _Federal Register_ 82, no. 61, March 31, 2017, <https://www.gpo.gov/fdsys/pkg/FR-2017-03-31/pdf/2017-06576.pdf> (accessed November 2, 2017).
. Timothy Cama, "Trump to Reconsider Grand Canyon Uranium Mining Ban," _Hill_ , November 1, 2017, <http://thehill.com/policy/energy-environment/358311-trump-admin-to-reconsider-grand-canyon-uranium-mining-ban> (accessed November 16, 2017); Joanna Walters, "Grand Canyon at Risk as Arizona Officials Ask Trump to End Uranium Mining Ban," _Guardian_ , June 5, 2017, <https://www.theguardian.com/environment/2017/jun/05/public-lands-uranium-mining-arizona-grand-canyon> (accessed October 31, 2017).
. Nina Bahadur, "18 Real Things Donald Trump Has Actually Said about Women," _Huffington Post_ , October 10, 2016, <https://www.huffingtonpost.com/entry/18-real-things-donald-trump-has-said-about-women_us_55d356a8e4b07addcb442023> (accessed November 16, 2017).
. John William Ward, _Andrew Jackson: A Symbol for an Age_ (New York: Oxford University, Press, 1962).
. In contemporary circumstances, many Native American peoples will be impacted by mining interests the more that federal lands originally set aside by the Antiquities Act are opened, as has also been the case for the Keystone pipeline. See chapter 10.
. Donald Trump, "Transcript of Trump's Speech in Saudi Arabia," CNN, May 21, 2017, <http://edition.cnn.com/2017/05/21/politics/trump-saudi-speech-transcript/index.html> (accessed October 31, 2017).
. Trump's alleged statement that he expects "loyalty" from the FBI director James Comey, but who promised "honesty" in return, has not only been interpreted by critics of Trump as obstructing justice, but also as undermining the FBI's relative independence. Katie Bo Williams, "Comey's Dramatic Account on Trump Rocks Washington," _Hill_ , June 7, 2017, <http://thehill.com/policy/national-security/336848-comeys-dramatic-account-rocks-washington> (accessed October 31, 2017). Trump's lawyer, has, however, denied that Trump ever told "Mr. Comey, 'I need loyalty, I expect loyalty' in form or substance." "President Trump's Lawyer's Statement on Comey Hearing," CNN, June 8, 2017. <http://edition.cnn.com/2017/06/08/politics/marc-kasowitz-statement-trump-comey/index.html> (accessed October 31, 2017). If Comey's position can be somehow verified, the incident raises the threat of "loyalty oaths" as during the early Cold War McCarthy period.
. NFL player Colin Kaepernick's protest against police brutality and racism during the singing of the national anthem in 2016 is a case in point. President Trump challenged the constitutional right of Kaepernick and others to symbolic protest, urging in a tweet that such individuals be "FIRED" for disrespecting the "Great American Flag (or country)." Trump's threats could become reality if waves of dishonest and hypocritical patriotism sweep the country against honest protest of telling truth to power. Bryan Flaherty, "From Kaepernick Sitting to Trump's Fiery Comments: NFL's Anthem Protests Have Spurred Discussion," _Washington Post_ , September 24, 2017, <https://www.washingtonpost.com/graphics/2017/sports/colin-kaepernick-national-anthem-protests-and-NFL-activism-in-quotes/?utm_term=.88e10ee1ac46> (accessed October 31, 2017).
. Stephen Feller, "Trump Supports Waterboarding, Says Intel Officials Told Him It 'Works,'" UPI, January 26, 2017, <https://www.upi.com/Top_News/US/2017/01/26/Trump-supports-waterboarding-says-intel-officials-told-him-it-works/4251485406127/> (accessed November 16, 2017). See critique by Vanessa Schipani, "Trump on Torture," FactCheck.org, July 28, 2016, last updated August 1, 2016, <http://www.factcheck.org/2016/07/trump-torture/> (accessed November 16, 2017).
. Igor Bobic, "Donald Trump Says He Would 'Take Out' Families of Terrorists," _Huffington Post_ , December 12, 2015, <https://www.huffingtonpost.com/entry/donald-trump-families-terrorists_us_565ef81ae4b072e9d1c41f99> (accessed November 16, 2017).
. Tom LoBianco, "Trump Softens Tone on McCain, Stands by Waterboarding Support," CNN, February 9, 2016, <http://edition.cnn.com/2016/02/09/politics/donald-trump-john-mccain-waterboarding/index.html> (accessed November 16, 2017). Having supported waterboarding during the Bush administration, CIA Director Mike Pompeo made ambiguous statements in his testimony to Congress as to whether he was open to the use of waterboarding techniques, if legalized. Pompeo was also ambiguous about the collection of phone metadata and other information about Americans, including about their "lifestyle"—by the use of broad powers that were granted through the expansion of the Foreign Intelligence Surveillance Act in 2008. Amy Davidson Sorkin, "Mike Pompeo and the Question of Torture," _New Yorker_ , January 12, 2017, <http://www.newyorker.com/news/amy-davidson/mike-pompeo-and-the-question-of-torture> (accessed October 31, 2017). By contrast, in opposition to Trump's campaign rhetoric, Secretary of State Rex Tillerson, homeland security director John F. Kelly, and Attorney General Jeff Sessions, all rejected reviving the use of torture.
. Kenneth Allard et al., "Defending the Honor of the US Military from Donald Trump," _Foreign Policy_ , March 16, 2016, <http://foreignpolicy.com/2016/03/04/defending-the-honor-of-the-u-s-military-from-donald-trump/> (accessed October 31, 2017). "Refusing to carry out such orders will protect the rule of law and the constitutional order, of which civilian control of the military is fundamental."
. George W. Bush, address to a joint session of Congress, September 20, 2001; a transcript of his address is available at "Transcript of President Bush's Address," CNN, September 21, 2001, <http://edition.cnn.com/2001/US/09/20/gen.bush.transcript/> (accessed November 10, 2017).
. UN, _World Cities Report_ (Nairobi, Kenya: UN Habitat, 2016), <http://wcr.unhabitat.org/main-report/> (accessed October 31, 2017).
. While China has been attempting to reduce its use of coal and investing more in renewable energies, Trump has sought to boost the use of coal and shale oil. (See chapter 10.) China is nevertheless responsible for almost one-third of all oceanic plastic waste. It is time for all countries to reduce use of plastics as much as possible. Chen Ronggang, "China Is the World's Largest Consumer of Fast Food. And It's Causing Major Damage," World Economic Forum, October 25, 2017, <https://www.weforum.org/agenda/2017/10/china-is-the-worlds-largest-consumer-of-fast-food-and-its-causing-major-damage> (accessed November 16, 2017).
. George Orwell, _1984_ (Planet E-Book), chapter 3, p. 43.
. Against this Orwellian vision, Russian analyst Andrei Kortunov was more or less correct to argue that "in today's conditions of total interdependence...neither the UK's withdrawal from the EU, nor China's non-participation in the Trans-Pacific Partnership (TPP) will have the disastrous economic consequences predicted by those who love geopolitical horror stories." Andrey Kortunov, "The Inevitable, Weird World," Russia in Global Affairs, September 25, 2016, <http://eng.globalaffairs.ru/number/The-Inevitable-Weird-World-18385> (accessed October 31, 2017). Yet, this optimistic perspective was articulated prior to the Trump administration's decision to drop out of the TPP and out of the COP 21 Global Climate Agreement. The latter actions could represent just the beginning of the unravelling of other multilateral and international accords.
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forward edge defense,
Nine Dash Map, ,
Regional Comprehensive Economic Partnership (RCEP), , , –, , , , , ,
relations with Germany ( _see under_ Germany)
relations with India, , , , –, , –, , , –, , , , –,
relations with Japan ( _see under_ Japan)
relations with Russia ( _see under_ Russia)
relations with Taiwan ( _see under_ Taiwan)
relations with the United States ( _see_ Trump administration)
"string of pearls,"
Clinton, Bill, ,
Clinton, Hillary, , –, , , –, –, , ,
Cold War, , , , , , , , , , , , , , , , , , , , ,
Collins, Susan,
Colombia, , –,
Comey, James, , –, ,
Community of Latin American and Caribbean States (CELAC),
confederation/confederal, , , , , ,
Congo, Democratic Republic of, –
Congressional Budget Office (CBO), ,
contact groups, , , , , –
COP 21. _See_ United Nations, United Nations Climate Change Conference
cosmopolitanism,
Crimea, , –, , –, , , , –, , –, , , –, , –, , –, , –, –, , , –, , , , , – , –, –, , , , . _See also_ Ukraine
Croatia, , ,
Cruz, Ted,
Cuba, , –,
cyber-warfare (cyber-sabotage), –, –, –, , –, –, , , , ,
Stuxnet malware,
Cyprus, –, , –
Czech Republic, –, , , , , ,
Dakota Access pipeline, ,
Defense Intelligence Agency, ,
democracy engineering, , , , ,
democracy movements, , –, ,
Democratic Party, ,
democratic reforms (US), –
Djibouti, ,
Donbas, , , , –, , –, –
Donetsk,
dual sovereignty, –
Duke, David, ,
Duterte, Rodrigo, –
Egypt, , , , , , , –, , ,
Eisenhower, Dwight D., , , , ,
farewell address (1961), ,
energy companies, , , ,
Chevron,
Chornomornaftogaz,
ExxonMobil, –, , , ,
Petrochina,
Repsol,
Rosneft, , , , , ,
Royal Dutch Shell,
Erdoğan, Tayyip, , , , , , –
Eritrea,
Estonia, , , , ,
Ethiopia, ,
Euromaidan movement, , , , ,
European Union (EU), , –, , , , –, , , , , –, –, –, , –, , , , , , –, –, , , ,
arms embargo on China (since 1989),
Common Foreign and defense policy,
Eastern Partnership and neighborhood program,
EU enlargement, –, –, –, –, –
European Coal and Steel Community,
relations with China, , ,
relations with Russia/Ukraine, , , , , , –, –, , –, , , , –, –,
relations with the United States/Trump, , –, , , –, , –, , , –, –, , ,
Facebook, ,
fake news, , , , ,
Falklands/Malvinas, ,
Farage, Nigel,
Fatah,
Ferrand, Richard,
Fillon, François,
financial crisis (2008), , , , , , , –, , , , –
cause of, –
Finland, , , –,
Flynn, Michael, , , ,
Ford, Gerald,
fossil fuels, , , , , –,
France, , , , , , , , , –, –, –, –, –, –, , , , , , –, , , , ,
relations with China, , , ,
relations with Germany ( _see under_ Germany)
relations with India, , , ,
relations with Russia, , , –, , , –, –, , , , , , –, ,
relations with the United Kingdom,
relations with the United States/NATO, , , , , , –, –, , , –
F-35 fighter, –, ,
Fusion GPS Report, ,
Gaddafi, Muammar, ,
Gaza, –
Georgia, , , , , , , , , –, , , , ,
Germany, , , , , , , , , , –, , –, –, , –, , , , , , –, , , , , , –,
relations with China, , –, ,
relations with France, , , , –, , –, , , –,
relations with NATO/the United States, , , , , –, , –
relations with Trump ( _see_ Trump administration)
Gibraltar, , ,
global warming, , , , ,
Global War on Terrorism (GWOT), –, , , , –, , –, , , , , , , , , , , , ,
Gorbachev, Mikhail, , , , , ,
Gore, Al,
Graham, Lindsey, ,
Greece, –, , , , –, , , , –,
financial crisis, , ,
Port of Piraeus,
relations with China,
relations with Russia, ,
relations with Turkey, , , –
Group of 7 (G7), ,
Group of 8 (G8),
Group of 20 (G–20), ,
Guantanamo Bay, ,
Guinea,
Gülen, Fethullah, –, ,
Gulf Cooperation Council (GCC),
Hadi, Abd-Rabbu Mansour,
Haley, Nikki, , –,
Hamas, , –
hegemony, , , , –, , , , , , , , –, , , –
Hezbollah, , , , , ,
Hitler, Adolf, ,
Hofer, Norbert,
Hollande, Francois, , ,
Hong Kong, ,
Umbrella Revolution,
Hungary, –, , , –, –, ,
Hussein, Saddam, , ,
hybrid warfare, , –, , –, , ,
hypersonic warheads,
illiberal democracy, ,
immigration, , , , , , , –, , –, , , , ,
European Union and immigration, –
remittances (to Mexico), –,
India, , , , , , , , , , , , –, , –, –, , , , , , –, , , –, , , –,
Kargil crisis (India/Pakistan),
Kashmir (India/Pakistan), , , –, –, –, , –
Indonesia, , , ,
intercontinental ballistic missiles (ICBMs), , , , ,
Internet/World Wide Web,
Iran, , , –, –, –, –, , , –, , , , , , , , , , , , –, –, –, –, , , –, , –, , , –, , , –, , ,
Joint Comprehensive Plan of Action (JCPOA), –, –, –, , ,
Natanz processing plant,
nuclear program, , , , –, , , ,
relations with Israel, , , , , , , , –, , –, –, –, , ,
relations with Saudi Arabia, , , , , , , , –, –, –, –, , , –,
relations with Syria, , , , –, –, –, –, ,
Iraq, –, , , , , , –, , , , –, , –, , –, –, –, –, , –, ,
war with Islamic State in Mosul, ,
Ischinger, Wolfgang,
Islamic State (IS or Daesh), , , , , , , , , , , –, , , –, , –, –, , , , –, ,
Islamism,
Israel, , , , , –, , , –, , , , –, , –, , –, –, , –, –, ,
Israeli-Palestinian relations, , , , ,
relations with Iran ( _see under_ Iran)
two-state solution, –,
Jabhat al-Nusra, –,
Japan, , –, , –, , , –, , , –, , , , –, , –, –, , –, , –, –, –, –, , , –, –, –, –, –, , , – , , , ,
relations with China, –, , , –, –, –, –, –, –, –, –, , –,
relations with Germany, –,
relations with India, , , , , , –, , –, , , , –, ,
relations with Russia, , –, –, , –, , , –, , , , –, , –, , , –, , , ,
relations with the United States, –, –, –, –, –, –, –, –, –, –, , ,
self-defense force, ,
Jaysh al-Islam,
Johnson, Micah,
joint sovereignty, , , , , –
Jordan, ,
Juncker, Jean-Claude,
Kaliningrad, , , , , , , ,
Kazakhstan, , ,
Kelly, John,
Kennedy, John F.,
Kenny, Enda,
Keystone XL pipeline, , –,
Khan Shaykhun, –
Kim, Dae-jung,
Kim, Il-Sung,
Kim, Jung Un, , –, , , , –,
Korea, North, , –, –, –, –, –, –, –, , , , , , , –, , , , –, , –, , , , , , –, , –, , , , ,
Kaesong complex,
Korea, South, , , , , –, , , –, –, , , –, –, –, –, , ,
Free Trade Agreement (Korean-US),
relations with China, , , , , –, , , –, –, ,
relations with Trump/United States, , –, , , –, –, –, , ,
Sunshine policy, ,
Korean War, , , ,
Kosovo, , , ,
Ku Klux Klan (KKK), ,
Kurdistan, ,
Kurdistan Regional Government (KRG),
Kurdistan Worker's Party (PKK), –
Kurds, , –, , –,
Kurile/Northern Territories, , ,
Kushner, Jared, , ,
Lamar, Mirabeau B.,
Lanza, Adam,
Lashkar-e-Taiba,
Latin America, , , , –, , –, –, –,
Lavrov, Sergei, , ,
Le Pen, Marine, –, –
Lewis, Sinclair, ,
LGBTQ communities,
Libya, , , , , , , , ,
Lindbergh, Charles,
Lithuania, ,
Litvinenko, Alexander, ,
Long, Gavin,
Long-Range Standoff Cruise Missile (LRSO), ,
Luhansk,
Lukashenka, Alyaksandr, , ,
MacArthur, Douglas, –
Macedonia,
Macron, Emmanuel, , –, –, ,
Malaysia, , , , ,
Manafort, Paul J.,
Manning, Chelsea,
Mao, Zedong,
Mariupol,
Mattis, James, ,
May, Theresa, –,
McCain, John, , , , , ,
McMaster, H. R., , , ,
Medvedev, Dmitry, , , , ,
Mélenchon, Jean-Luc,
Merkel, Angela, , , , –,
Mexico, , , , –, , , –, –, , ,
Middle East, , , , , , , , , , , , , , , –, , , –, , , , , , , –, –, , –
military Keynesianism, , ,
missile defense (MD), , , –, , , , , , , , –
missile program, , , –, ,
Modi, Narendra, , , –
Moldova, –, , , , ,
Monroe Doctrine, , , ,
Montenegro, , ,
Moon, Jae-In, –
Mueller, Robert, –
Murkowski, Lisa,
Muslim, , , , –, –, , –, , , , ,
Muslim Brotherhood, ,
mutual assured destruction (MAD), ,
National Coalition of Syrian Revolution and Opposition Forces,
nationalism, –, , , , –, , , , , , , , , , , , , , , , , , , , , –, , , , –, , , , –,
National Rifle Association (NRA),
NATO Article V, , , ,
NATO enlargement, –, –, –, , –
NATO membership, , , , –, , –,
NATO-Russia Founding Act, –, –
natural gas, –, , –
"near abroad," , ,
Nemtsov, Boris, ,
Netanyahu, Benjamin, , –
Netherlands, ,
New Zealand, , ,
Nitze, Paul,
Nixon, Richard, , , , , –,
North American Free Trade Agreement (NAFTA), ,
North Atlantic Treaty Organization (NATO), –, , , –, , , , , , , –, , , –, –, –, –, –, –, –, –, , –, –, , , –, , –, –, –, –, , , –, , , . _See also_ NATO Article V; NATO enlargement; NATO membership; NATO-Russia Founding Act
Northern Ireland, –, ,
Good Friday Agreement,
Norway, ,
nuclear power, , , , , ,
Nuclear Supplier Group (NSG),
nuclear weapons, , , , , , , –, –, –, , , , , , –, , , , , –, , –, –, , , –, –, , –, –, , , , –, –, , , ,
Nuland, Victoria,
Obama, Barack, , , , –, , , –, –, , , , , , , –, , –, –, , –, , , –, –, , , –, , , , , –, , , ,
administration of, , –, –, , , , , , , , , , , , , –
Okinawa, ,
Old Age Survivor and Disability Insurance (OASDI),
One-China policy, , , , –,
one-state solution,
open NATO enlargement, –
Orbán, Viktor, ,
Organization for Security and Cooperation in Europe (OSCE), , –, , , , –, , –, –, –, –
Organization for Security and Cooperation in Indo-Pacific (OSCIP),
Organization of Oil Exporting Countries (OPEC),
Orwell, George, , , –
Overseas Contingency Operations (OCO), –
Paddock, Stephen C., –
Palestine, , –,
Palestinian authority, –,
pan-Islamist movements, , , , , , , , , , , ,
Paradise Papers, ,
Park, Geun-hye,
Partnership for Peace (PfP),
Patton, George S., –
Paul, Rand,
peace and development communities, , ,
Pence, Mike, –, –, –, –, , , , , –, , , , –, , , , , –, , , , , , , , , –, , –, , ,
Pentagon, , , , –, , , , –, , , , , , , –, –, , ,
petroleum, ,
Philippines, , , , , , , , –, –, , ,
PIIGS (Portugal, Italy, Ireland, Greece, Spain),
pipelines, , , ,
piracy, , ,
Poland, , , , , , , , –, , , –,
polarization, –, –, , , ,
Politkovskaya, Anna,
Poroshenko, Petro, , , , ,
protectionism, , , , –
Putin, Vladimir, , , , –, , –, –, , , –, –, –, , –, , , , –, –, , , –, , , , ,
Qatar, , , , , , , , –, , , –, –
relations with Saudi Arabia, –
Quadrilateral Security Dialogue (QUAD), ,
Reagan, Ronald, , –,
Regional Comprehensive Economic Partnership (RCEP), , , –, –, , , , ,
Romania, , , ,
Roosevelt, Teddy, , –
national parks,
Rosebud Sioux Tribe,
Rouhani, Hassan,
Russia, –, –, –, , –, , , –, , –, , , –, –, –, –, –, , , –, –, , –, –, –, –, , –, –, , –, –, –, –, , –, –, , , ,
capital flight,
Collective Security Treaty Organization (CSTO), , –, , , , –, , , ,
foreign agent law,
NATO-Russia Founding Act, –, –
Novorossiya,
sanctions on, , , , , , –, –, , , –, , , , –, –, , , –, –, , –, , , , , , , , , –, , –, , –, –, –, –
relations with China (Eurasian alliance), , , –, –, –, ,
relations with France ( _see under_ France)
relations with Iran, , , , , –, , , , –, –, , , –,
relations with Japan ( _see under_ Japan)
relations with Syria, , , , , , , , , –, , –,
Sakhalin Island, ,
Russian Federation, –, , , , , , ,
Rwanda,
Saipov, Sayfullo,
Salafist, ,
Salman (king),
Salman, Mohammed bin,
Sanders, Bernie,
Sarkozy, Nicolas,
satellite, ,
Saudi Arabia, , , , , , , , , , , , , , –, –, –, , –, –, , , –, , ,
relations with Iran ( _see under_ Iran)
relations with Israel, , , , , –, –, –, –,
_See also_ Trump administration
Schultz, Martin,
Schumer, Chuck,
Scotland, –
Scottish Independence, –
sea lines of communication, , ,
Senkaku/Diaoyu Islands, –, ,
September 11, 2001, attacks, , , , , , ,
Serbia, , , , , , –, ,
Sevastopol, , ,
Seventh Report of the Organization for the Prohibition of Chemical Weapons,
shale energy, –,
Shanghai Cooperation Organization (SCO), , , , , ,
Sharif, Nawaz, –
Shi'a, , , , , , –, ,
Sindhudesh Liberation Army,
Singapore, , ,
Slovakia, , , ,
Somalia, , ,
South Africa, , , , ,
South China Sea, , , , , –, –, , –, , , , ,
South Ossetia, , , , –
South Stream,
South Sudan,
Soviet Union, , , , , , ,
Spain, –, , ,
Spencer, Richard B.,
Spratly Islands, ,
Sputnik, ,
Sri Lanka, –
Stalin, Joseph, ,
Steele, Christopher,
Straits of Malacca,
Sturgeon, Nicola,
Sudan, , , –
Sultan Murad group,
Sunni, , , , –, ,
SU-35 fighter, , ,
Sweden, , , , –,
Syria, , , , –, , , , –, , , , , , , , , , –, –, –, –, , , –, , , ,
Syrian Kurdish People's Protection Units (YPG), –
Syrian Union Party (PYD),
Szydło, Beata,
tactical nuclear weapons, , , , , –
Taiwan, –, , , , , –, –, –, –, , , , , ,
relations with China, –, –, –,
Taiwan independence, , , , ,
_See also_ Trump administration
Taliban, , , , –, , ,
Tanzania, ,
Tartus, port of,
Tatars, ,
Terminal High Altitude Area Defense (THAAD), , , , , , ,
terrorism, –, , , , , , –, –, , –, , –, , , , , , , , –, , , , , , , , –, –, –
Three Seas Initiative, ,
Tiananmen Square,
Tibet, , ,
Tillerson, Rex, , , –, , , , –, , –
Tobin tax, –
Tomahawk cruise missile, , , , –, , , ,
Topol-MR missile,
torture, , –
trade pacts, , , ,
Transatlantic Trade and Investment Partnership (TTIP),
Trans-Pacific Partnership (TPP), , –, ,
Trump, Donald, –, –, –, –, –, –, –, –, –, –, –, –, –, –, –, –, , , , –, –, , –, , , –, –, –, –, –, , –, –, , , –, –, –, , , –, –,
_Art of the Deal, The_ ,
impeachment, , , –, ,
_See also_ Trump administration
Trump, Donald Jr., ,
Trump administration, –, , , , , , –, , –, , , , , –, , , , , –, , , , , , , , , , , , , , , , ,
alleged collusion with Moscow, –
America First policy, –, –, –, –, –, –, –
energy policies, –, , ,
environmental policies, –, –
policy on China, –, , , , , , , , , ,
policy on Germany, ,
policy on Iran, , , , , , , , ,
policy on North Korea, , , , , –, –,
policy on Saudi Arabia, , , –
policy on Syria, , , ,
policy on Taiwan, , , , , , –,
policy on Turkey,
policy on Ukraine, , , , , , , –, –, , –, –, , , ,
sanctions on Russia, , , , , –, –, , , , –, , , –, , , , , , , , –, , ,
UN peacekeeping policies, –
Turkey, –, –, , , , , , , , , –, , , –, , , , , , –, –, , –,
Tusk, Donald, ,
Twitter, ,
Uganda,
Ukraine, , , , –, , , , –, –, –, –, –, –, –, –, , –, –, , –, –, , –, , , , –,
relations with NATO, , , , , –, , –, –, –,
relations with Russia, , , –, , , , –, –, , –, –, –, , ,
relations with Trump/United States ( _see under_ Trump administration)
United Arab Emirates (UAE), , , , , ,
United Kingdom (UK), , , , , , –, –, –, , –, , –, , , –, ,
Brexit, , , –, –, , , , –, –,
relations with Argentina,
relations with China,
relations with Cyprus,
relations with France,
relations with Ireland, –
relations with NATO, , ,
relations with Northern Ireland,
relations with Scotland, –
relations with the United States, , –, –, –, –, , –, –, , –
United Nations (UN), , –, , , , , , , –, , , , , –, , , , –, , ,
UN Convention on the Law of the Sea (UNCLOS), ,
UN Educational, Scientific and Cultural Organization (UNESCO), ,
United Nations Climate Change Conference (COP 21), , , , , –,
United Nations General Assembly (UNGA), , ,
UN Security Council (UNSC), , , , , , , , , , , , ,
United States (US), , –, –, , , , –, , –, , –, –, , –, , , , , , –, –, –, –, –, –, , , –, –, , –, –, , , –, –, –, –, , –, –, –, –, –, –, –, –, –, –, –, , , –, –, –, –, –, –, , –, –, , , –, –, –
Affordable Health Care Act,
causes of financial crisis, –
Congress, , , , , , , , , , , , , , , , , , , , , , , , ,
constitutional reforms, , , , ,
Countering America's Adversaries through Sanctions Act, , , , –,
defense spending, –, , , , , –, –, –, , ,
Department of Defense (DOD), , ,
Earnings Suspense File,
electoral college (reform of), , , , , ,
Environmental Protection Agency (EPA),
European Reassurance Initiative (2016),
Federal Bureau of Investigation (FBI), –, –, ,
Federal Reserve,
Federal Reserve Board,
Government Accountability Office (GAO),
gun control, , , , –, –
House of Representatives, ,
Justice Against Sponsors of Terrorism Act (JASTA), –
military-industrial-congressional complex, , –, , –, , –, , –, ,
Monroe Doctrine, , , ,
mother of all bombs (MOAB), , , ,
national debt, , , , ,
National Security Council (NSC), , ,
nuclear weapons program, , , , ,
One-China policy (1972 Shanghai Communique), , , , ,
Overseas Contingency Operations (OCO), –
"reset" relations with Russia, , , ,
sanctions on Russia ( _see under_ Russia)
Securities and Exchange Commission (SEC), ,
Senate, , , , , , ,
Social Security, , , ,
State Department, , , , , , , , , , , ,
Treasury Department, , , , , ,
US-China Shanghai Communiqué (1972),
US-Japan Defense Cooperation Guidelines in April 2015,
US "pivot" or "rebalancing" to Asia, , , ,
_See also_ Trump administration
Unite the Right demonstration, , , ,
Ushakov, Yuri,
Van der Bellen, Alexander,
Venezuela, , , , , , –, , –
Vietnam, , , , , , , –, –, , , ,
Vladivostok, ,
Wang, Yi, –
war on drugs, , , , ,
Warsaw Pact, , ,
weapons of mass destruction (WMD), ,
wider Middle East, , , , , , , , , , , , –, , , , , –, , , , –
Wilders, Geert,
World Heritage Sites, ,
World War III, , , , ,
Xi, Jinping, , , , , , ,
Xinjiang province, –, ,
Yanukovych, Viktor, , , –,
Yeltsin, Boris, ,
Yemen, , , , , , , –, , , –, ,
Zambia,
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 3,131
|
\section{Introduction}
Fix a prime $p>3$ and a positive integer $N$ prime to $p$. Let $\mathfrak{O}_L$ be the ring of integers of a finite extension $L/\mathbf{Q}_p$, and let
\[
{\boldsymbol{f}}=\sum_{n=1}^\infty\boldsymbol{a}_nq^n\in\mathbb{I}\pwseries{q}
\]
be a Hida family of tame level $N$, where $\mathbb{I}$ is a finite flat extension
of the one-variable Iwasawa algebra $\mathfrak{O}_L\pwseries{T}$ with fraction field $F_\mathbb{I}$. Throughout this paper, we shall assume that $\mathbb{I}$ is regular.
Let $\mathcal{K}$ be an imaginary quadratic field of discriminant prime to $Np$, and
let $\Gamma_\mathcal{K}:={\rm Gal}(\mathcal{K}_\infty/\mathcal{K})$ be the Galois group of the $\mathbf{Z}_p^2$-extension of $\mathcal{K}$.
From work of Hida \cite{hidaII},
there is a $3$-variable $p$-adic $L$-function
\[
L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}\pwseries{\Gamma_\mathcal{K}}
\]
interpolating critical values $L({\boldsymbol{f}}_\phi/\mathcal{K},\chi,j)$
for the Rankin--Selberg $L$-function attached to the classical specializations
of ${\boldsymbol{f}}$ twisted by
finite order characters $\chi:\Gamma_\mathcal{K}\rightarrow\mu_{p^\infty}$. Let
\[
\rho_{{\boldsymbol{f}}}:G_{\mathbf{Q}}:={\rm Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow{\rm Aut}_{F_{\mathbb{I}}}(V_{{\boldsymbol{f}}})\simeq{\rm GL}_2(F_\mathbb{I})
\]
be
the Galois representation associated to ${\boldsymbol{f}}$ (which we take to be the contragredient of the Galois representation first constructed in \cite{hida86b}),
and let $\bar{\rho}_{{\boldsymbol{f}}}:G_{\mathbf{Q}}\rightarrow{\rm GL}_2(\kappa_\mathbb{I})$, where $\kappa_\mathbb{I}=\mathbb{I}/\mathfrak{m}_\mathbb{I}$ is the residue field of $\mathbb{I}$, be the associated semi-simple residual representation.
By work of Mazur and Wiles \cite{MW-families,wiles88}, upon restriction to a decomposition group $D_p\subset G_{\mathbf{Q}}$ at $p$ we have
\[
\bar{\rho}_{{\boldsymbol{f}}}\vert_{D_p}\sim
\left(\begin{smallmatrix}\bar{\varepsilon}& *\\& \bar{\delta}\end{smallmatrix}\right)
\]
where the character $\bar{\delta}$ is unramified. Under the assumption that
\begin{equation}\label{ass:irred}
\textrm{$\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible and $\bar\varepsilon\neq\bar{\delta}$},\tag{MT}
\end{equation}
one knows that there exists a $G_\mathbf{Q}$-stable lattice $T_{{\boldsymbol{f}}}\subset V_{{\boldsymbol{f}}}$ which is free of rank two over $\mathbb{I}$ with the inertia coinvariants $\mathscr{F}^-T_{\boldsymbol{f}}$ being $\mathbb{I}$-free of rank one. Set
\[
A_{\boldsymbol{f}}:=T_{\boldsymbol{f}}\otimes_\mathbb{I}\cR^\vee,\quad
\mathscr{F}^-A_{\boldsymbol{f}}:=(\mathscr{F}^-T_{\boldsymbol{f}})\otimes_\mathbb{I}\cR^\vee,
\]
where $\mathbb{I}^\vee:={\rm Hom}_{\rm cts}(\mathbb{I},\mathbf{Q}_p/\mathbf{Z}_p)$ is the Pontryagin dual of $\mathbb{I}$, and define the Greenberg Selmer group ${\rm Sel}_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$ by
\begin{equation}\label{eq:2Gr-intro}
{\rm Sel}_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}):=\ker\biggl\{{\rm H}^1(\mathcal{K}_\infty,A_{\boldsymbol{f}})\rightarrow
\prod_{w\nmid p}{\rm H}^1(I_w,A_{\boldsymbol{f}})\times\prod_{w\vert p}{\rm H}^1(\mathcal{K}_{\infty,w},\mathscr{F}^-A_{\boldsymbol{f}})\biggr\},
\end{equation}
where $w$ runs over the places of $\mathcal{K}_\infty$. The Pontryagin dual
\[
X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}):={\rm Hom}_{\rm cts}({\rm Sel}_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}),\mathbf{Q}_p/\mathbf{Z}_p)
\]
is easily seen to be a finitely generated $\mathbb{I}\pwseries{\Gamma_\mathcal{K}}$-module.
In this paper we study the following instance of the Iwasawa--Greenberg main conjectures (see \cite{Greenberg55}).
\begin{intro-conj1}[{\bf Iwasawa--Greenberg main conjecture}]
The module $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\tt Gr}(\mathcal{K}_\infty,A_{{\boldsymbol{f}}}))=(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))
\]
as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}]]$.
\end{intro-conj1}
Many cases of this conjecture are known by the work of Skinner--Urban \cite{SU} and \cite{Kato295}. As we shall explain below, in this paper we place ourselves in a setting complementary to that in \cite{SU}, obtaining the following new result towards the Iwasawa--Greenberg main conjecture. The imaginary quadratic field $\mathcal{K}$ determines a factorization
\[
N=N^+N^-
\]
with $N^-$ being the largest factor of $N$ divisible only by primes inert in $\mathcal{K}$.
\begin{ThmA
In addition to {\rm (\ref{ass:irred})}, assume that:
\begin{itemize}
\item{} $N$ is squarefree,
\item{} some specialization ${\boldsymbol{f}}_\phi$ is the $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$,
\item{} $N^-$ is the product of a positive even number of primes,
\item{} $\bar\rho_{{\boldsymbol{f}}}$ is ramified at every prime $q\vert N^-$,
\item{} $p$ splits in $\mathcal{K}$.
\end{itemize}
Then $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}))=(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))
\]
as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}]]\otimes_{\mathbf{Z}_p}\mathbf{Q}_p$.
\end{ThmA}
As in \cite{SU}, the fact that $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion
follows easily from Kato's work, and the proof of Theorem~A is reduced to establishing the divisibility ``$\subset$'' as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}]]$ predicted by the main conjecture. For the proof of this divisibility,
in \cite{SU} the authors study congruences between $p$-adic families of cuspidal automorphic forms and Eisenstein series on
${\rm GU}(2,2)$,
and their method (in particular, their application of Vatsal's result \cite{vatsal-special}) relies crucially on their hypothesis that $N^-$ is the squarefree product of an \emph{odd} number of primes.
In contrast, for the proof of Theorem~A we first link the above main conjecture with another instance of the Iwasawa--Greenberg main conjectures, and exploit our assumption on $N^-$ to prove the latter using Heegner points.
As a consequence of our approach, we also obtain
an application to Greenberg's conjecture (see \cite[\S{0}]{Nekovar-Plater}, following \cite{GreenbergCRM}) on the generic order of vanishing at the center of the $p$-adic $L$-functions attached to cusp form in Hida families. To state this, assume for simplicity that $\mathbb{I}$ is just $\mathfrak{O}_L[[T]]$, and for each $k\in\mathbf{Z}_{\geqslant 2}$ let ${\boldsymbol{f}}_k$ be the $p$-stabilized newform on $\Gamma_0(Np)$ obtained by setting $T=(1+p)^{k-2}-1$ in ${\boldsymbol{f}}$. One can show that the $p$-adic $L$-functions $L_p^{\tt MTT}({\boldsymbol{f}}_k,s)$ of \cite{mtt} satisfy a functional equation
\[
L_p^{\tt MTT}({\boldsymbol{f}}_k,s)=-wL_p^{\tt MTT}({\boldsymbol{f}}_\kappa,k-s)
\]
with a sign $w=\pm{1}$ independent of $k\in\mathbf{Z}_{\geqslant 2}$ with $k\equiv 2\pmod{2(p-1)}$.
\begin{intro-conj4}[{\bf Greenberg's nonvanishing conjecture}]
Let $e\in\{0,1\}$ be such that $-w=(-1)^{e}$. Then
\[
\frac{L_p^{\tt MTT}({\boldsymbol{f}}_k,s)}{(s-k/2)^{e}}\biggr\vert_{s=k/2}\neq 0,
\]
for all but finitely many $k\in\mathbf{Z}_{\geqslant 2}$ with $k\equiv 2\pmod{2(p-1)}$.
\end{intro-conj4}
In other words, for all but finitely many $k$ as above, the order of vanishing of $L_p^{\tt MTT}({\boldsymbol{f}}_k,s)$ at the center should be the least allowed by the sign in the functional equation.
To state our result in the direction of this conjecture, let
\[
T_{\boldsymbol{f}}^\dagger:=T_{\boldsymbol{f}}\otimes\Theta^{-1}
\]
be the self-dual twist of $T_{\boldsymbol{f}}$. By work of Plater \cite{Plater} (and more generally Nekov{\'a}{\v{r}} \cite{nekovar310})
there is a cyclotomic $\mathbb{I}$-adic height pairing
\begin{equation}\label{eq:ht-intro}
\langle,\rangle_{\mathcal{K},\mathbb{I}}^{\rm cyc}:{\rm Sel}_{\tt Gr}(\mathcal{K},T_{\boldsymbol{f}}^\dagger)\times
{\rm Sel}_{\tt Gr}(\mathcal{K},T_{\boldsymbol{f}}^\dagger)\rightarrow F_\mathbb{I}
\end{equation}
interpolating the $p$-adic height pairings for the classical specialization of ${\boldsymbol{f}}$
as constructed by Perrin-Riou \cite{PR-109}.
It is expected that $\langle,\rangle_{\mathcal{K},\mathbb{I}}^{\rm cyc}$ is non-degenerate, in the sense that its kernel on either side should reduce to $\mathbb{I}$-torsion submodule of ${\rm Sel}_{\tt Gr}(\mathcal{K},T_{\boldsymbol{f}}^\dagger)$.
\begin{ThmB
In addition to {\rm (\ref{ass:irred})}, assume that:
\begin{itemize}
\item $N$ is squarefree,
\item
${\boldsymbol{f}}_2$ is old at $p$,
\item there are at least two primes $\ell\Vert N$ at which $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified.
\end{itemize}
If ${\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)$ has $\mathbb{I}$-rank one and
$\langle,\rangle_{\mathcal{K},\mathbb{I}}^{\rm cyc}$ is non-degenerate, then
\[
\frac{d}{ds}L_p^{\tt MTT}({\boldsymbol{f}}_k,s)\biggr\vert_{s=k/2}\neq 0,
\]
for all but finitely many $k\in\mathbf{Z}_{\geqslant 2}$ with $k\equiv 2\pmod{2(p-1)}$.
\end{ThmB}
\begin{rem}
The counterpart to Theorem~B in rank zero, i.e., the implication
\begin{equation}\label{eq:rank0}
{\rm rank}_{\mathbb{I}}\;{\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)=0\quad\Longrightarrow \quad L_p^{\tt MTT}({\boldsymbol{f}}_k,k/2)\neq 0,
\end{equation}
for all but finitely many $k$ as above,
follows easily from
\cite{SU} (see Theorem~\ref{thm:Gr+1}).
\end{rem}
\begin{rem}\label{rem:0or1}
By the control theorem for ${\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)$ (see e.g. \cite[Prop.~12.7.13.4(i)]{nekovar310})
and the $p$-parity conjecture for classical Selmer groups (see e.g. \cite[Thm.~6.4]{cas-hsieh1}), the hypothesis that ${\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)$ has $\mathbb{I}$-rank one (resp. zero) implies that $w=1$ (resp. $w=-1$).
Conversely, it is expected that the $\mathbb{I}$-rank of ${\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)$ is \emph{always} $0$ or $1$; more precisely,
\[
{\rm rank}_{\mathbb{I}}\;{\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)\overset{?}=\left\{
\begin{array}{ll}
1&\textrm{if $w=1$,}\\[0.1cm]
0&\textrm{if $w=-1$.}
\end{array}
\right.
\]
For example, by \cite[Cor.~3.4.3 and Eq.~(21)]{howard-invmath} this prediction is a consequence of Howard's ``horizontal nonvanishing conjecture''.
\end{rem}
\begin{rem}\label{rem:CM-case}
For Hida families ${\boldsymbol{f}}$ with CM (a case that is excluded by our hypotheses), the analogue of Theorem~B is due to Agboola--Howard and Rubin \cite[Thm.~B]{AHsplit}.
In this case, the rank one and non-degeneracy assumptions
follow from Greenberg's nonvanishing results \cite{greenberg-BSD} (see \cite[Prop.~2.4.4]{AHsplit})
and a transcendence result of Bertrand \cite{bertrand-AH} (see \cite[Thm.~A.1]{AHsplit}). In rank zero, the CM case of $(\ref{eq:rank0})$ follows from \cite{greenberg-BSD} and Rubin's proof of the Iwasawa main conjecture for imaginary quadratic fields \cite{rubin-IMC}.
\end{rem}
We conclude this Introduction with some of the ingredients that go into the proofs of the above results.
The proof of Theorem~A builds on the link that we establish in $\S\ref{sec:Iw}$ between different instances of the Iwasawa--Greenberg main conjectures involving Selmer groups differing from $(\ref{eq:2Gr-intro})$ in their local conditions at the places above $p$. In particular, letting $\mathfrak{p}$ be the prime of $\mathcal{K}$ above $p$ determined by a fixed embedding $\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$, and denoting by $\hat\mathbf{Z}_p^{\rm ur}$ the completion of the ring of integers of the maximal unramified extension of $\mathbf{Q}_p$, a central role is played by the Selmer group
defined by
\[
{\rm Sel}_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}):=\ker\biggl\{{\rm H}^1(\mathcal{K}_\infty,A_{\boldsymbol{f}})\rightarrow\prod_{w\nmid p}{\rm H}^1(I_w,A_{\boldsymbol{f}})\times
\prod_{w\mid\overline{\mathfrak{p}}}{\rm H}^1(\mathcal{K}_{\infty,w},A_{\boldsymbol{f}})\biggr\}
\]
whose Pontryagin dual is conjecturally generated by a $p$-adic $L$-function
\[
\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]],\quad\quad\textrm{where}\;\;\mathbb{I}^{\rm ur}:=\mathbb{I}\hat{\otimes}_{\mathbf{Z}_p}\hat{\mathbf{Z}}_p^{\rm ur},
\]
interpolating critical values $L({\boldsymbol{f}}_\phi/\mathcal{K},\chi,j)$
with $\chi$ running over characters of $\Gamma_\mathcal{K}$ with associated theta series of weight higher than the weight of ${\boldsymbol{f}}_\phi$.
This second instance of the Iwasawa--Greenberg main conjecture can be related on the one hand to the main conjecture for $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ by building on the explicit reciprocity laws for the Rankin--Eisenstein classes of Kings--Loeffler--Zerbes \cite{KLZ2}, and on the other hand (after restriction to the anticyclotomic line) to the big Heegner point main conjecture of Howard \cite[Conj.~3.3.1]{howard-invmath}
using the explicit reciprocity law for Heegner points of \cite{cas-hsieh1}\footnote{Itself a generalization of the celebrated $p$-adic Waldspurger formula of Bertolini--Darmon--Prasanna \cite{bdp1}.}, thereby allowing us to take the results of \cite{wanIMC} and \cite{Fouquet} towards the proof of those conjecture to bring to bear on the main conjecture for $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$.
On the other hand, a key ingredient in the proof of Theorem~B is the Birch and Swinnerton-Dyer type formula for
$L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ along $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$ that we obtain in Theorem~\ref{thm:3.1.5} by building on the ealier results of the paper, leading to a $p$-adic Gross--Zagier formula for Howard's system of big Heegner points $\mathfrak{Z}_\infty$ that we then apply for a suitably chosen imaginary quadratic field $\mathcal{K}$.
\vspace{0.1in}
\noindent{\bf Acknowledgements.}
It is a pleasure to thank Chris Skinner for several helpful conversations.
Subtantial progress on this paper occurred during visits of the first author to Fudan University in January 2019, the Morningside Center of Mathematics in June 2019, and Academia Sinica in December 2019, and he would like to thank these institutions
for their hospitality.
\section{$p$-adic $L$-functions}\label{sec:padicL}
\subsection{Hida families}\label{subsec:hida}
Let $\mathbb{I}$ be a local reduced normal extension of $\mathfrak{O}_L[[T]]$, where $\mathfrak{O}_L$ is the ring of integers of a finite extension $L$ of $\mathbf{Q}_p$, and denote by $\mathcal{X}_a(\mathbb{I})\subset{\rm Hom}_{\rm cts}(\mathbb{I},\overline{\mathbf{Q}}_p)$ the set of continuous $\mathfrak{O}_L$-algebra homomorphisms
$\phi:\mathbb{I}\rightarrow\overline{\mathbf{Q}}_p$ satisfying
\[
\phi(1+T)=\zeta(1+p)^{k-2}
\]
for some $p$-power root of unity $\zeta=\zeta_\phi$ and some integer $k=k_\phi\in\mathbf{Z}_{\geqslant 2}$ called the \emph{weight} of $\phi$. We shall refer to the elements of $\mathcal{X}_a(\mathbb{I})$ as \emph{arithmetic} primes of $\mathbb{I}$, and let $\mathcal{X}_a^o(\mathbb{I})$ denote the set consisting of arithmetic primes $\phi$ with $\zeta_\phi=1$ and weight $k_\phi\equiv 2\pmod{p-1}$.
Let $N$ be a positive integer prime to $p$, let $\chi$ be an even Dirichlet character modulo $Np$ taking values in $L$, and let ${\boldsymbol{f}}=\sum_{n=1}^\infty\boldsymbol{a}_nq^n\in\mathbb{I}[[q]]$ be an ordinary $\mathbb{I}$-adic cusp eigenform of tame level $N$ and character $\chi$, as defined in \cite[\S{3.3.9}]{SU}. In particular, for every $\phi\in\mathcal{X}_a(\mathbb{I})$ we have
\[
{\boldsymbol{f}}_\phi:=\sum_{n=1}^\infty\phi(\boldsymbol{a}_n)q^n\in S_{k}(\Gamma_0(p^{t}N),\chi\omega^{2-k_\phi}\psi_{\zeta}),
\]
where
\begin{itemize}
\item $t=t_\phi\geqslant 1$ is such that $\zeta$ is a primitive $p^{t-1}$-st root of unity,
\item $\omega$ is the Teichm\"uller character, and
\item
$\psi_{\zeta}:(\mathbf{Z}/p^{t}N\mathbf{Z})^\times\twoheadrightarrow(\mathbf{Z}/p^{t}\mathbf{Z})^\times\rightarrow\overline{\mathbf{Q}}_p^\times$ is determined by $\psi_{\zeta}(1+p)=\zeta=\zeta_\phi$.
\end{itemize}
Denote by $S^{\rm ord}(N,\chi;\mathbb{I})$ the space of such $\mathbb{I}$-adic eigenforms ${\boldsymbol{f}}$. If in addition ${\boldsymbol{f}}_\phi$ is $N$-new for all $\phi\in\mathcal{X}_a(\mathbb{I})$, we say that ${\boldsymbol{f}}$ is a \emph{Hida family} of tame level $N$ and character $\chi$.
We refer to ${\boldsymbol{f}}_\phi$ as the specialization of ${\boldsymbol{f}}$ at $\phi$. More generally, if $\phi\in{\rm Hom}_{\rm cts}(\mathbb{I},\overline{\mathbf{Q}}_p)$ is such that ${\boldsymbol{f}}_\phi$ is a classical eigenform, we say that ${\boldsymbol{f}}_\phi$ is a classical specialization of ${\boldsymbol{f}}$; this includes the specializations of ${\boldsymbol{f}}\in S^{\rm ord}(N,\chi;\mathbb{I})$ at $\phi\in\mathcal{X}_a(\mathbb{I})$, but possibly also specializations in weight $1$, for example.
\subsection{Congruence modules}\label{subsec:congr}
We recall the notion of congruence modules following the treatment of \cite[$\S{12.2}$]{SU} and \cite[\S{3.3}]{hsieh-triple}. Let ${\boldsymbol{f}}$ be a Hida family of tame level $N$ and character $\chi$ defined over $\mathbb{I}$, and let $\rho_{\boldsymbol{f}}:G_\mathbf{Q}\rightarrow{\rm GL}_2(F_\mathbb{I})$ be the Galois representation associated to ${\boldsymbol{f}}$, where $F_\mathbb{I}$ is the fraction field of $\mathbb{I}$. Let $\mathbb{T}(N,\chi,\mathbb{I})$ be the Hecke algebra acting $S^{\rm ord}(N,\chi;\mathbb{I})$, and let $\lambda_{{\boldsymbol{f}}}:\mathbb{T}(N,\chi,\mathbb{I})\rightarrow\mathbb{I}$ be the algebra homomorphism defined by $\mathbb{I}$, which factors through the local component $\mathbb{T}_{\mathfrak{m}_{{\boldsymbol{f}}}}$.
Since ${\boldsymbol{f}}$ is $N$-new, there is an algebra direct sum decomposition
\[
\lambda:\mathbb{T}_{\mathfrak{m}_{{\boldsymbol{f}}}}\otimes_\mathbb{I} F_\mathbb{I}\simeq\mathbb{T}'\times F_\mathbb{I}
\]
with the projection onto the second factor given by $\lambda_{{\boldsymbol{f}}}$. The \emph{congruence module} $C({\boldsymbol{f}})\subset\mathbb{I}$ is defined by
\[
C({\boldsymbol{f}}):=\lambda_{{\boldsymbol{f}}}\left(\mathbb{T}_{\mathfrak{m}_{{\boldsymbol{f}}}}\cap\lambda^{-1}(\{0\}\times F_\mathbb{I})\right).
\]
Following the convention in \cite[\S{7.7}]{KLZ2}, we shall also consider the \emph{congruence ideal} $I_{\boldsymbol{f}}$, defined as the fractional ideal $I_{\boldsymbol{f}}:=C({\boldsymbol{f}})^{-1}\subset F_\mathbb{I}$. As noted in \emph{loc.cit.}, if follows from \cite[Thm.~4.2]{hidaII} that elements of $I_{\boldsymbol{f}}$ define meromorphic functions on ${\rm Spec}(\mathbb{I})$ which are regular at all arithmetic points.
\subsection{Rankin--Selberg $p$-adic $L$-functions}\label{sec:2varL}
The next result on the construction of $3$-variable $p$-adic Rankin $L$-series is due to Hida.
Let $\Gamma$
be Galois group of the cyclotomic $\mathbf{Z}_p^\times$-extension of $\mathbf{Q}$, and set
\[
\Lambda_\Gamma=\mathbf{Z}_p[[\Gamma]].
\]
If $j\in\mathbf{Z}$ and $\chi$ is a Dirichlet character of $p$-power conductor, there is a unique $\phi\in{\rm Hom}_{\rm cts}(\Lambda_\Gamma,\overline{\mathbf{Q}}_p^\times)$ extending the character $z\mapsto z^j\chi(z)$ on $\mathbf{Z}_p^\times$.
\begin{thm}\label{thm:hida}
Let ${\boldsymbol{f}}_1, {\boldsymbol{f}}_2$ be Hida families of tame levels $N_1, N_2$, respectively, and let $N={\rm lcm}(N_1,N_2)$. Then there is an element
\[
L_p({\boldsymbol{f}}_1,{\boldsymbol{f}}_2)\in\left(I_{{\boldsymbol{f}}_1}\hat{\otimes}_{\mathbf{Z}_p}\mathbb{I}_{{\boldsymbol{f}}_2}\hat{\otimes}_{\mathbf{Z}_p}\Lambda_{\Gamma}\right)\otimes_{\mathbf{Z}}\mathbf{Z}[\mu_N]
\]
uniquely characterized by the following interpolation property. Let $f_1$, $f_2$ be classical specializations of ${\boldsymbol{f}}_1$, ${\boldsymbol{f}}_2$ of weights $k_1$, $k_2$, respectively, with $k_1>k_2\geqslant 1$, let $j$ be an integer in the range $k_2\leqslant j\leqslant k_1-1$, and let $\chi$ be a Dirichlet character of $p$-power conductor.
Suppose the automorphic representation $\pi_{f_1}$ is a principal series representation $\pi(\eta_1,\eta_1')$ with $\eta_1$ unramified and $\eta_1(p)$ a $p$-adic unit.
Then the value of $L_p({\boldsymbol{f}}_1,{\boldsymbol{f}}_2)$ at the corresponding specialization $\phi\in{\rm Spec}(\mathbb{I}_{{\boldsymbol{f}}_1}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}_{{\boldsymbol{f}}_2}\hat\otimes_{\mathbf{Z}_p}\Lambda_{\Gamma})$ is given by
\begin{align*}
\phi(L_p({\boldsymbol{f}}_1,{\boldsymbol{f}}_2))&=\frac{\mathcal{E}(f_1,f_2,\chi,j)}{\mathcal{E}(f_1)\mathcal{E}^*(f_1)}\cdot\frac{\Gamma(j)\Gamma(j-k_2+1)}{\pi^{2j+1-k_2}(-i)^{k_1-k_2}2^{2j+k_1-k_2}\left\langle f_1,f_1^c\vert_{k_1}\bigl(\begin{smallmatrix}&-1\\p^{t_1}N_1\end{smallmatrix}\bigr)\right\rangle_{N_1}}\\
&\quad\times L(
f_1,f_2,\chi^{-1},j),
\end{align*}
where if $\alpha_{i}$ and $\beta_{i}$ are the roots of the Hecke polynomial of $f_i$ at $p$, with $\alpha_{i}$ being the $p$-adic unit root, and $p^t$ is the conductor of $\chi$, the Euler factors are given by
\[
\mathcal{E}(f_1,f_2,\chi,j)=\left\{
\begin{array}{ll}
\left(1-\frac{p^{j-1}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j-1}}{\alpha_{1}\beta_{2}}\right)
\left(1-\frac{\beta_{1}\alpha_{2}}{p^j})(1-\frac{\beta_{1}\beta_{2}}{p^j}\right)&\textrm{if $t=0$,}\\[0.2cm]
G(\chi)^2\cdot\left(\frac{p^{2j-2}}{\alpha^2_1\alpha_2\beta_2}\right)^t &\textrm{if $t\geqslant 1$,}
\end{array}
\right.
\]
where $G(\chi)$ is the Gauss sum of $\chi$, and if $p^{t_1}$ is the $p$-part of the conductor of $\eta_1'$, then
\[
\mathcal{E}(f_1)\mathcal{E}^*(f_1)=\left\{
\begin{array}{ll}
\left(1-\frac{\beta_{1}}{p\alpha_{1}}\right)\left(1-\frac{\beta_{1}}
{\alpha_{1}}\right)&\textrm{if $t_1=0$,}\\[0.2cm]
G(\chi_1)\cdot\eta_1'\eta_1^{-1}(p^{t_1})p^{-t_1}&\textrm{if $t_1\geqslant 1$,}\\
\end{array}
\right.
\]
where $\chi_1$ is the nebentypus of $f_1$.
\end{thm}
\begin{proof}
This follows from \cite[Thm.~5.1]{hidaII}, which we have stated adopting the formulation in \cite[Thm.~7.7.2]{KLZ2} (slightly extended to include some more general specializations of the dominant Hida family ${\boldsymbol{f}}_1$).
\end{proof}
In this paper, we shall consider the $p$-adic $L$-functions $L_p({\boldsymbol{f}}_1,{\boldsymbol{f}}_2)$ of Theorem~\ref{thm:hida} in the cases where either ${\boldsymbol{f}}_1$ or ${\boldsymbol{f}}_2$ has CM.
Thus let ${\boldsymbol{f}}$ be a fixed Hida family of tame level $N$ defined over $\mathbb{I}$, and let $\mathcal{K}$ be an imaginary quadratic field of discriminant $-D_{\mathcal{K}}<0$ prime to $pN$ such that
\begin{equation}\label{eq:spl}
\textrm{$p=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $\mathcal{K}$,}\nonumber
\end{equation}
with $\mathfrak{p}$ denoting the prime of $\mathcal{K}$ above $p$ induced by our fixed embedding
$\imath_p:\overline\mathbf{Q}\hookrightarrow\mathbf{C}_p$.
Let $\mathcal{K}_\infty$ be the $\mathbf{Z}_p^2$-extension of $\mathcal{K}$, and denote by $\Gamma_\mathfrak{p}\simeq\mathbf{Z}_p$ the Galois group over $\mathcal{K}$ of the maximal subfield of $\mathcal{K}_\infty$ unramified outside $\overline{\mathfrak{p}}$. Let
\begin{equation}\label{eq:g-CM}
{\boldsymbol{g}}=\sum_{n=1}^\infty\boldsymbol{b}_nq^n
\in\mathbb{I}_{{\boldsymbol{g}}}[[q]]
\end{equation}
be the canonical Hida family of CM forms constructed in \cite[\S{5.2}]{JSW}, where $\mathbb{I}_{{\boldsymbol{g}}}=\mathbf{Z}_p[[\Gamma_\mathfrak{p}]]$. Specifically, denoting by $\theta_\mathfrak{p}:\mathbb{A}_{\mathcal{K}}^\times\rightarrow\Gamma_\mathfrak{p}$ the composition of the global reciprocity map ${\rm rec}_\mathcal{K}:\mathbb{A}_\mathcal{K}^\times\rightarrow G_\mathcal{K}^{\rm ab}$ with the natural projection $G_\mathcal{K}^{\rm ab}\twoheadrightarrow\Gamma_\mathfrak{p}$, we have
\[
\boldsymbol{b}_n=\sum_{\substack{N(\mathfrak{a})=n, (\mathfrak{a},\overline{\mathfrak{p}})=1}}\theta_\mathfrak{p}(x_{\mathfrak{a}}),
\]
summing over integral ideals $\mathfrak{a}\subset\mathcal{O}_\mathcal{K}$, and $x_\mathfrak{a}\in\mathbb{A}_{\mathcal{K}}^{\infty,\times}$ is any finite id\`ele of $\mathcal{K}$ with ${\rm ord}_w(x_{\mathfrak{a},w})={\rm ord}_{w}(\mathfrak{a})$ for all finite places $w$ of $\mathcal{K}$.
\subsection{Non-dominant CM: ${\boldsymbol{f}}_2={\boldsymbol{g}}$}\label{subsec:L-2}\label{sec:dom-CM}
Assume that the residual representation $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible and $p$-distinguished. Then by \cite[Cor.~2, p,~482]{Fermat-Wiles} the local ring $\mathbb{T}_{\mathfrak{m}_{{\boldsymbol{f}}}}$ introduced in $\S\ref{subsec:congr}$ is Gorenstein, and by Hida's results (see e.g.
\cite{hida-AJM88}) the congruence module $C({\boldsymbol{f}})$ is principal. Let $c_{\boldsymbol{f}}\in C({\boldsymbol{f}})$ be a generator, and set
\[
L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}):=c_{{\boldsymbol{f}}}\cdot L_p({\boldsymbol{f}},{\boldsymbol{g}}),
\]
viewed as an element in $\mathbb{I}[[\Gamma_\mathcal{K}]]$ (well-defined up to a unit in $\mathbb{I}^\times$), where $\Gamma_\mathcal{K}={\rm Gal}(\mathcal{K}_\infty/\mathcal{K})$.
The action of complex conjugation yields a decomposition
\[
\Gamma_\mathcal{K}\simeq
\Gamma_\mathcal{K}^{\rm ac}\times\Gamma_\mathcal{K}^{\rm cyc},
\]
where
$\Gamma_\mathcal{K}^{\rm ac}$ (resp. $\Gamma_\mathcal{K}^{\rm cyc}$) denotes the Galois group of the anticyclotomic (resp. cyclotomic) $\mathbf{Z}_p$-extension of $\mathcal{K}$. We next study the projections of $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ to $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$ and $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm cyc}]]$.
\subsubsection{Anticyclotomic restriction of $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$}\label{sec:anti-L}
Assume that ${\boldsymbol{f}}$ has trivial tame character, and following \cite[Def.~2.1.3]{howard-invmath} define the \emph{critical character} $\Theta:G_\mathbf{Q}\rightarrow\mathbb{I}^\times$ by
\begin{equation}\label{def:crit-char}
\Theta:=[\langle\varepsilon_{\rm cyc}\rangle^{1/2}],
\end{equation}
where $\varepsilon_{\rm cyc}:G_\mathbf{Q}\rightarrow\mathbf{Z}_p^\times$ is the cyclotomic character, $\langle\cdot\rangle:\mathbf{Z}_p^\times\rightarrow 1+p\mathbf{Z}_p$ is the natural projection, and
\[
[\cdot]:1+p\mathbf{Z}_p\hookrightarrow\mathbf{Z}_p[[1+p\mathbf{Z}_p]]^\times\simeq\mathbf{Z}_p[[T]]^\times\rightarrow\mathbb{I}^\times
\]
is the composition of the obvious maps. This induces the twist map
\begin{equation}\label{eq:tw-theta}
{\rm tw}_{\Theta^{-1}}:\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}]]
\end{equation}
defined by $\gamma\mapsto\Theta^{-1}(\gamma)\gamma$ for $\gamma\in\Gamma_\mathcal{K}$.
Write $N$ as the product
\[
N=N^+ N^-
\]
with $N^+$ (resp. $N^-$) divisible only by primes which are split (resp. inert) in $\mathcal{K}$, and consider the following generalized \emph{Heegner hypothesis}:
\begin{equation}\label{eq:gen-Heeg-f}
\textrm{$N^-$ is the squarefree product of an even number of primes.}\tag{gen-H}
\end{equation}
Whenever we assume that $\mathcal{K}$ satisfies the hypothesis (\ref{ass:gen-H}), we fix an integral ideal $\mathfrak{N}^+\subset\mathcal{O}_\mathcal{K}$ with $\mathcal{O}_\mathcal{K}/\mathfrak{N}^+\simeq\mathbf{Z}/N^+\mathbf{Z}$.
\begin{prop}\label{thm:hida-1}
Let $L_p^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ be the image of ${\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$ under the natural projection $\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}^{{\rm ac}}]]$. If $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{eq:gen-Heeg-f})},
then $L_p^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ is identically zero.
\end{prop}
\begin{proof}
Let $\phi\in{\rm Spec}(\mathbb{I}_{{\boldsymbol{f}}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}_{{\boldsymbol{g}}}\hat\otimes_{\mathbf{Z}_p}\Lambda_\Gamma)={\rm Spec}(\mathbb{I}[[\Gamma_\mathcal{K}]])$ be a specialization in the range specified in Theorem~\ref{thm:hida}, with $f_1={\boldsymbol{f}}_\phi$ the $p$-stabilization of a newform $f\in S_k(\Gamma_0(N))$ of weight $k\geqslant 2$ and $f_2={\boldsymbol{g}}_\phi$ a classical weight $1$ specialization.
By the interpolation property, the value $\phi(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$ is a multiple of
\[
L(f_1,f_2,\chi^{-1},j)=L(f/\mathcal{K},\psi,j),
\]
with $\psi$ a finite order character of $\Gamma_\mathcal{K}$ and $1\leqslant j\leqslant k-1$, and so $\phi({\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})))$ is also a multiple $L(f/\mathcal{K},\psi',k/2)$ for a finite order character $\psi'$ of $\Gamma_\mathcal{K}$. If $\psi'$ factors through the projection $\Gamma_\mathcal{K}\twoheadrightarrow\Gamma_{\mathcal{K}}^{{\rm ac}}$, then the $L$-function $L(f/\mathcal{K},\psi',s)$ is self-dual, with
a functional equation relating its values at $s$ and $k-s$, and if $\mathcal{K}$ satisfies the hypothesis (\ref{eq:gen-Heeg-f}), then the sign in this
functional equation is $-1$ (see e.g. \cite[\S{1}]{CV-dur}). Thus $L(f/\mathcal{K},\psi',k/2)=0$, and letting $\phi$ vary, the result follows.
\end{proof}
\subsubsection{Cyclotomic restriction of $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$}\label{sec:anti-L}
As above, we denote by $\Gamma_\mathcal{K}^{\rm cyc}$ the Galois group of the cyclotomic $\mathbf{Z}_p$-extension of $\mathcal{K}$, which we shall often identify with the maximal torsion-free quotient of $\Gamma$.
For any ordinary $p$-stabilized newform $f$ of tame level $N$ defined over $L$ (a finite extension of $\mathbf{Q}_p$), let $L_p^{\tt MTT}(f)\in\mathfrak{O}_L[[\Gamma]]$ be the cyclotomic $p$-adic $L$-function attached to $f$ in \cite{mtt}, where $\mathfrak{O}_L$ is the ring of integers of $L$ (see \cite[\S{3.4.4}]{SU} and the references therein). We refer the reader to \emph{loc.cit.} for the precise interpolation property satisfied by $L_p^{\tt MTT}(f)$, only noting here that the complex periods used for the construction are Shimura's periods $\Omega_f^\pm\in\mathbf{C}^\times/\mathfrak{O}_L^\times$ (as reviewed in \cite[\S{3.3.3}]{SU}).
\begin{thm}\label{thm:cyc-res}
Let $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})_{\rm cyc}$ be the image of $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ under the natural projection $\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}^{\rm cyc}]]$. Then for every $\phi\in\mathcal{X}_a^o(\mathbb{I})$
we have
\[
\phi(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})_{\rm cyc})=L_p^{\tt MTT}({\boldsymbol{f}}_\phi)\cdot L_p^{\tt MTT}({\boldsymbol{f}}_\phi\otimes\epsilon_\mathcal{K})
\]
up to a unit in $\phi(\mathbb{I})[[\Gamma]]^\times$, where $\epsilon_\mathcal{K}$ is the quadratic character associated to $\mathcal{K}$.
\end{thm}
\begin{proof}
Since we assume that $\bar{\rho}_{{\boldsymbol{f}}}$ satisfies the hypotheses (\ref{ass:irred}), by \cite[Thm.~0.1]{hida-AJM88} (see also \cite[Lem.~12.1]{SU}) for every $\phi\in\mathcal{X}_a^o(\mathbb{I})$ we have the relation
\[
\langle{\boldsymbol{f}}_\phi,{\boldsymbol{f}}_\phi\rangle_{N}\cdot\phi(c_{{\boldsymbol{f}}})^{-1}=u\cdot\Omega_{{\boldsymbol{f}}_\phi}^\pm\cdot\Omega_{{\boldsymbol{f}}_\phi\otimes\epsilon_\mathcal{K}}^\pm
\]
between the periods appearing in the interpolation property of the respective sides of the claimed equality, for some unit $u\in\phi(\mathbb{I})^\times$. Since by construction, $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ specializes at $\phi$ to the $p$-adic $L$-function $L_p({\boldsymbol{f}}_\phi/\mathcal{K})$ considered
in \cite{BL-ord},
the result thus follows from \cite[Cor.~2.2]{BL-ord} and \cite[Thm.~12.8]{SU}.
\end{proof}
\subsection{Dominant CM: ${\boldsymbol{f}}_1={\boldsymbol{g}}$}\label{subsec:L-1}\label{sec:nondom-CM}
As in $\S\ref{sec:dom-CM}$, let ${\boldsymbol{f}}\in\mathbb{I}[[q]]$ be a fixed Hida family of tame level $N$, and let ${\boldsymbol{g}}$ be the CM Hida family in (\ref{eq:g-CM}).
Let ${\hat{\bZ}_p^{\rm ur}}$ be the completion of the ring of integers of the maximal unramified extension of $\mathbf{Q}_p$, and set $\mathbb{I}^{\rm ur}:=\mathbb{I}\hat\otimes_{\mathbf{Z}_p}{\hat{\bZ}_p^{\rm ur}}$.
By \cite[\S{5.3.0}]{Katz49} (see also \cite[Thm.~II.4.14]{de-shalit}) there exists a $p$-adic $L$-function
$\mathscr{L}_{\mathfrak{p}}^{}(\mathcal{K})\in{\hat{\bZ}_p^{\rm ur}}[[\Gamma_\mathcal{K}]]$
such that if $\psi$ is a character of $\Gamma_\mathcal{K}$ corresponding to an algebraic Hecke character of $\mathcal{K}$ with trivial conductor
and infinity type $(a,b)$ with $0\leqslant-b<a$, then
\begin{equation}\label{eq:Katz}
\mathscr{L}^{}_{\mathfrak{p}}(\mathcal{K})(\psi)=\biggl(\frac{\sqrt{D_\mathcal{K}}}{2\pi}\biggr)^{a}
\cdot\Gamma(b)\cdot(1-\psi(\mathfrak{p}))\cdot(1-p^{-1}\psi^{-1}(\overline\mathfrak{p}))
\cdot\frac{\Omega_p^{b-a}}{\Omega_K^{b-a}}\cdot L_{}(\psi,0),
\end{equation}
where $\Omega_K\in\mathbf{C}^\times$ and $\Omega_p\in\mathbf{C}_p^\times$ are certain CM periods (as defined in e.g. \cite[\S{2.5}]{cas-hsieh1}).
Let $h_\mathcal{K}$ be the class number of $\mathcal{K}$, $w_\mathcal{K}:=\vert\mathcal{O}_\mathcal{K}^\times\vert$, and set
\begin{equation}\label{eq:factor-hida}
\mathscr{L}_{\mathfrak{p}}({\boldsymbol{f}}/\mathcal{K}):=\biggl(\frac{h_\mathcal{K}}{w_\mathcal{K}}\mathscr{L}_\mathfrak{p}(\mathcal{K})_{{\rm ac}}\biggr)\cdot L_{p}({\boldsymbol{g}},{\boldsymbol{f}}),
\end{equation}
where $\mathscr{L}_\mathfrak{p}(\mathcal{K})_{{\rm ac}}$ is the anticyclotomic projection of $\mathscr{L}_\mathfrak{p}(\mathcal{K})$. A priori, $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$ is an element in $I_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]$ but comparing its interpolating property with that of a different $3$-variable $p$-adic $L$-function, we can show its integrality.
\begin{prop}
The $p$-adic $L$-function in {\rm (\ref{eq:factor-hida})} is integral, i.e., $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]$
\end{prop}
\begin{proof}
For any finite set $\Sigma$ of places $\mathcal{K}$ outside $p$ and containing all the places dividing $N D_\mathcal{K}$, the results of \cite[\S{7.5}]{wanIMC} yield the construction of the ``$\Sigma$-imprimitive'' element
\[
\mathfrak{L}_\mathfrak{p}^\Sigma({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]
\]
characterized by the following interpolation property.
For a Zariski dense set of arithmetic points $\phi\in{\rm Spec}(\mathbb{I}[[\Gamma_\mathcal{K}]])$ with ${\boldsymbol{f}}_\phi$ of weight $2$ and conductor $p^tN$ generating a unitary $\pi_{{\boldsymbol{f}}_\phi}\simeq\pi(\chi_{1,p},\chi_{2,p})$ with $v_p(\chi_{1,p}(p))=-\frac{1}{2}$ and $v_p(\chi_{2,p}(p))=\frac{1}{2}$, and with $\boldsymbol{\psi}_\phi$ a Hecke character of $\mathcal{K}$ of infinity type $(-n,0)$ for some $n\geqslant 3$ and conductor $p^t$, we have:
\begin{align*}
\phi(\mathfrak{L}_{\mathfrak{p}}({\boldsymbol{f}}/\mathcal{K}))&=p^{(n-3)t}\boldsymbol{\psi}_{\phi,\mathfrak{p}}^2\chi_{1,p}^{-1}\chi_{2,p}^{-1}(p^{-t})G(\boldsymbol{\psi}_{\phi,\mathfrak{p}}\chi_{1,p}^{-1})G(\boldsymbol{\psi}_{\phi,\mathfrak{p}}\chi_{2,p}^{-1})\Gamma(n)\Gamma(n-1)\Omega_p^{2n}\\
&\quad\times \frac{L^\Sigma({\boldsymbol{f}}_\phi,\chi^{-1}_{\phi}\boldsymbol{\psi}_\phi,0)}{(2\pi i)^{2n-1}\Omega_\mathcal{K}^{2n}},
\end{align*}
where $\chi_{\phi}$ is the nebentypus of ${\boldsymbol{f}}_\phi$, $\Omega_p\in\mathbf{C}_p^\times$ and $\Omega_\mathcal{K}\in\mathbf{C}^\times$ are CM periods, and $L^\Sigma({\boldsymbol{f}}_\phi,\chi^{-1}_{\phi}\boldsymbol{\psi}_\phi,0)$ is the $\Sigma$-imprimitive Rankin--Selberg $L$-values. Setting
\begin{equation}\label{eq:prim}
\mathfrak{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K}):=\mathfrak{L}_\mathfrak{p}^\Sigma({\boldsymbol{f}}/\mathcal{K})\times\prod_{w\in\Sigma}P_w(\Psi_\mathcal{K}({\rm Frob}_w))^{-1},
\end{equation}
where $P_w$ is the Euler factor at $w$ and $\Psi_\mathcal{K}:G_\mathcal{K}\twoheadrightarrow\Gamma_\mathcal{K}$ is the natural projection, we thus obtain an element interpolating the Rankin--Selberg $L$-values themselves, but which \emph{a priori} is just an element in the fraction field of $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]$. To see the inclusion $\mathfrak{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]$ we shall compare $(\ref{eq:prim})$ with the product in the right-hand side of $(\ref{eq:factor-hida})$; the required integrality of $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$ will follow from this comparison.
Any arithmetic point $\phi\in\mathrm{Spec}(\mathbb{I}[[\Gamma_\mathcal{K}]])$ as above can be written as the product $\boldsymbol{\psi}_\phi'\cdot\boldsymbol{\psi}_\phi''$, with $\boldsymbol{\psi}_\phi'$ cyclotomic (i.e., factoring through $\Gamma_\mathcal{K}\twoheadrightarrow\Gamma_\mathcal{K}^{\rm cyc}$), and $\boldsymbol{\psi}_\phi''$ corresponding to a Hecke character unramified at $\overline{\mathfrak{p}}$ and of infinity type $(-n,0)$. Then $\chi^{-1}_{\phi}\boldsymbol{\psi}'_\phi$ (resp. the theta series of $\boldsymbol{\psi}''_\phi$) corresponds to $\chi\vert\cdot\vert^j$ (resp. $f_1={\boldsymbol{g}}_\phi$) in Theorem~\ref{thm:hida}, so that
\[
L({\boldsymbol{f}}_\phi,\chi_{\phi}^{-1}\boldsymbol{\psi}_\phi,0)=L(f_1,f_2,\chi^{-1},j)
\]
with $f_2={\boldsymbol{f}}_\phi$. Letting $p^tD_\mathcal{K}$
be the conductor of ${\boldsymbol{g}}_\phi$, according to \cite[Thm.~7.1]{HT-ENS} a direct calculation shows that the product $\mathcal{E}({\boldsymbol{g}}_\phi)\mathcal{E}^*({\boldsymbol{g}}_\phi)\cdot\langle{\boldsymbol{g}}_\phi,{\boldsymbol{g}}_\phi\rangle_{p^tD_\mathcal{K}}$ in Theorem~\ref{thm:hida} agrees with
\begin{equation}\label{eq:factor-RS}
\frac{\Gamma(n)G(\boldsymbol{\psi}''^{-1}_{\phi,\bar{\mathfrak{p}}})L(\boldsymbol{\psi}''_\phi(\boldsymbol{\psi}''_\phi)^{-c}, 1)}{(-2\pi i)^n}\cdot\frac{L(\epsilon_\mathcal{K},1)}{-2\pi i}
\end{equation}
up to a $p$-adic unit independent of $\phi$, where $\epsilon_\mathcal{K}$ is the quadratic character attached to $\mathcal{K}$.
By the class number formula, the second factor in $(\ref{eq:factor-RS})$ is given by $h_\mathcal{K}$ up to a $p$-adic unit, while by the interpolation property of the Katz $p$-adic $L$-function $\mathscr{L}_\mathfrak{p}(\mathcal{K})$ (see \cite[\S{5.3.0}]{Katz49}), the left factor multiplied by $(\Omega_p/\Omega_\mathcal{K})^{2n}$ is interpolated, for varying $\phi$, by the anti-cyclotomic projection of $\mathscr{L}_\mathfrak{p}(\mathcal{K})$ viewed as an element in ${\hat{\bZ}_p^{\rm ur}}[[\Gamma_{\mathfrak{p}}]]$. This shows the factorization that $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$ in $(\ref{eq:factor-hida})$ and $\mathfrak{L}_\mathfrak{p}({\boldsymbol{f}}/K)$ differ by a unit.
Finally, by the proof of \cite[Prop.~8.3]{wan-combined}, the only possible denominators of $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$ are powers of the augmentation ideal of $\mathbf{Z}_p[[\Gamma_{\mathfrak{p}}]]$, while by $(\ref{eq:prim})$ the possible denominators can only be either powers of $p$ or factors coming from Euler factors at primes $w\in\Sigma$. Since these two sets are disjoint, integrality of $\mathfrak{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$, and hence of $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$, follows.
\end{proof}
We conclude this section by discussing the anticyclotomic restriction of $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$, which contrary to $L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K})$ will be nonzero under the generalized Heegner hypothesis.
\begin{thm}\label{thm:bdp}
Assume
that $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{eq:gen-Heeg-f})}, and if $N^->1$ assume in addition that $N$ is squarefree. Then there exists an element $\mathscr{L}_{\mathfrak{p}}^{\tt BDP}({\boldsymbol{f}}/\mathcal{K})\in\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{{\rm ac}}]]$ such that for every $\phi\in\mathcal{X}_a(\mathbb{I})$ of weight $k$ and trivial nebentypus,
and every crystalline character $\psi$ of $\Gamma_\mathcal{K}^{\rm ac}$ corresponding to a Hecke character of $\mathcal{K}$ of infinity type $(n,-n)$ with $n\geqslant 0$, we have
\begin{align*}
\phi(\mathscr{L}_{\mathfrak{p}}^{\tt BDP}({\boldsymbol{f}}/\mathcal{K})^2)(\psi)
&=\mathcal{E}_\mathfrak{p}({\boldsymbol{f}}_\phi,\psi)^2\cdot
\psi(\mathfrak{N}^+)^{-1}\cdot 2^3\cdot\varepsilon({\boldsymbol{f}}_\phi)\cdot w_\mathcal{K}^2\sqrt{D_\mathcal{K}}\cdot\Gamma(k+n)\Gamma(n+1)\Omega_p^{2k+4n}\\
&\quad\times\frac{L({\boldsymbol{f}}_\phi/\mathcal{K},\psi,k/2)\cdot\alpha({\boldsymbol{f}}_\phi,{\boldsymbol{f}}_\phi^B)^{-1}}{(2\pi)^{k+2n+1}\cdot({\rm Im}\;\boldsymbol{\theta})^{k+2n}\cdot\Omega_\mathcal{K}^{2k+4n}},
\end{align*}
where $\mathcal{E}_\mathfrak{p}({\boldsymbol{f}}_\phi,\psi)=(1-\phi(\boldsymbol{a}_p)\psi_{\overline\mathfrak{p}}(p)p^{-k/2})
(1-\phi(\boldsymbol{a}_p)^{-1}\psi_{\overline\mathfrak{p}}(p)p^{k/2-1})$,
$\varepsilon({\boldsymbol{f}}_\phi)$ is the global root number of ${\boldsymbol{f}}_\phi$, $w_\mathcal{K}:=\vert\mathcal{O}_\mathcal{K}^\times\vert$, $\Omega_p\in\mathbf{C}_p^\times$ and $\Omega_\mathcal{K}\in\mathbf{C}^\times$ are CM periods attached to $\mathcal{K}$ as \cite[\S{2.5}]{cas-hsieh1}, $\boldsymbol{\theta}\in\mathcal{K}$ is as in {\rm (\ref{eq:vartheta})} below, and
\[
\alpha({\boldsymbol{f}}_\phi,{\boldsymbol{f}}_\phi^B)=\frac{\langle{\boldsymbol{f}}_\phi,{\boldsymbol{f}}_\phi\rangle}{\langle{\boldsymbol{f}}_\phi^B,{\boldsymbol{f}}_\phi^B\rangle}
\]
is a ratio of Petersson norms of ${\boldsymbol{f}}_\phi$ and its transfer ${\boldsymbol{f}}_\phi^B$ to a quaternion algebra, normalized as in \cite[\S{2.2}]{prasanna}.
\end{thm}
\begin{proof}
When $N^-=1$, this is \cite[Thm.~2.11]{cas-2var} (in which case $\alpha({\boldsymbol{f}}_\phi,{\boldsymbol{f}}_\phi^B)=1$). In the following we sketch how to extend that result to include the more general hypothesis (\ref{ass:gen-H}). Some of the notations used here will be introduced later in $\S\ref{sec:HP}$.
Let $\mathcal{O}_B$ be a maximal order of $B$, and let ${\rm Ig}_{N^+,N^-}$ be the Igusa scheme over $\mathbf{Z}_{(p)}$ classifying abelian surfaces with $\mathcal{O}_B$-multiplication and $U_\infty$-level structure (here $U_\infty$ is the open compact $U_r\subset\hat{R}_r^\times$ in $\S\ref{subsec:Sh}$ with $r=\infty$). For any valuation ring $W$ finite flat over $\mathbf{Z}_p$, denote by $V_p(W)$ the module of formal functions on ${\rm Ig}_{N^+,N^-}$ (i.e., $p$-adic modular forms) defined over $W$, and set
\[
V_p(\mathbb{I}):=V_p(W_0)\hat{\otimes}_{W_0}\mathbb{I},
\]
where $W_0=W(\kappa_\mathbb{I})$ is the ring of Witt vectors of the residue field of $\mathbb{I}$. For every $\mathcal{O}_\mathcal{K}$-ideal $\mathfrak{a}$ prime to $\mathfrak{N}^+\mathfrak{p}$, the construction of $\varsigma^{(s)}$ (for arbitrary $s\geqslant 0$) in $\S\ref{subsec:construct}$ determines CM points $x(\mathfrak{a})\in{\rm Ig}_{N^+,N^-}$, and the argument in \cite[Thm.~3.2.16]{hida-GME} with the use of $q$-expansions and the $q$-expansion principle replaced by Serre--Tate-expansions and the resulting $t$-expansion principle around any such $x(\mathfrak{a})$ (see e.g \cite[p.~107]{hida-mu})
shows that every element ${\boldsymbol{f}}^B\in V_p(\mathbb{I})$ defines a $p$-adic family (in fact, finite collections of such, since $\mathbb{I}$ is finite over $W_0[[T]]$) of $p$-adic modular forms ${\boldsymbol{f}}^B_z={\boldsymbol{f}}^B(u^z-1)\in V_p(W_0)$, where $u=1+p$, indexed by $z\in\mathbf{Z}_p$.
The Hida family ${\boldsymbol{f}}$ corresponds to minimal prime in the localized universal $p$-ordinary Hecke algebra $\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}$, and by the integral Jacquet--Langlands correspondence (see e.g. the discussion in \cite[\S{5.3}]{LV}), there exists a $p$-adic family ${\boldsymbol{f}}_B$ as above corresponding to ${\boldsymbol{f}}$, which we normalize by requiring that some Serre--Tate expansion ${\boldsymbol{f}}_z^B(t)$ does not vanish modulo $p$.
There are $U$- and $V$-operators acting on ${\boldsymbol{f}}_B$ defined as in \cite[\S{3.6}]{brooks}, and we set
\[
{\boldsymbol{f}}_B^\flat:={\boldsymbol{f}}_B\vert(VU-UV).
\]
With these, we may define $\mathbb{I}^{\rm ur}$-valued measures $\mu_{{\boldsymbol{f}}_B,x(\mathfrak{a})}$ and $\mu_{{\boldsymbol{f}}_{B,\mathfrak{a}}^\flat}$ on $\mathbf{Z}_p$ (with the latter supported on $\mathbf{Z}_p^\times$ by \cite[Prop.~4.17]{brooks}) as in \cite[\S{2.7}]{cas-2var}, and define $\mathscr{L}_{\mathfrak{p},\boldsymbol{\xi}}({\boldsymbol{f}})$ to be the $\mathbb{I}^{\rm ur}$-valued measure on ${\rm Gal}(H_{p^\infty}/\mathcal{K})$ given by
\[
\mathscr{L}_{\mathfrak{p},\boldsymbol{\xi}}({\boldsymbol{f}})(\phi)=
\sum_{[\mathfrak{a}]\in{\rm Pic}(\mathcal{O}_\mathcal{K})}
\boldsymbol{\xi}\boldsymbol{\chi}^{-1}(\mathfrak{a})\mathbf{N}(\mathfrak{a})^{-1}
\int_{\mathbf{Z}_p^\times}(\phi\vert[\mathfrak{a}])(z){\rm d}\mu_{{\boldsymbol{f}}^\flat_{B,\mathfrak{a}}}(z)
\]
for all $\phi:{\rm Gal}(H_{p^\infty}/\mathcal{K})\rightarrow\mathcal{O}_{\mathbf{C}_p}^\times$, where, if $\sigma_{\mathfrak{a}}$ corresponds to $\mathfrak{a}$ under the Artin reciprocity map, $\phi\vert[\mathfrak{a}]$ is the character on $z\in\mathbf{Z}_p^\times$ given by $\phi(\sigma_\mathfrak{a}{\rm rec}_{{\mathfrak{p}}}(z))$ for the local reciprocity map ${\rm rec}_\mathfrak{p}:\mathcal{K}_\mathfrak{p}^\times\rightarrow G_\mathcal{K}^{\rm ab}\rightarrow\Gamma_\mathcal{K}^{\rm ac}$, $\boldsymbol{\chi}:\mathcal{K}^\times\backslash\mathbb{A}_\mathcal{K}^\times\rightarrow\mathbb{I}^\times$ is the character given by $x\mapsto\Theta({\rm rec}_\mathbf{Q}({\rm N}_{\mathcal{K}/\mathbf{Q}}(x)))$ for the reciprocity map ${\rm rec}_\mathbf{Q}:\mathbf{Q}^\times\backslash\mathbb{A}^\times\rightarrow G_\mathbf{Q}^{\rm ab}$, and $\boldsymbol{\xi}$ is the auxiliary anticyclotomic $\mathbb{I}$-adic character constructed in \cite[Def.~2.8]{cas-2var}.
Still denoting by $\mathscr{L}_{\mathfrak{p},\boldsymbol{\xi}}({\boldsymbol{f}})$ its image under the natural projection $\mathbb{I}^{\rm ur}[[{\rm Gal}(H_{p^\infty}/\mathcal{K})]]\rightarrow\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]]$, and setting
\[
\mathscr{L}_\mathfrak{p}({\boldsymbol{f}})={\rm tw}_{\boldsymbol{\xi}^{-1}}(\mathscr{L}_{\mathfrak{p},\boldsymbol{\xi}}({\boldsymbol{f}})),
\]
one then readily checks as in the proof of \cite[Thm.~2.11]{cas-2var} that for every $\phi\in\mathcal{X}_a^o(\mathbb{I})$, the measure $\phi(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}))$ agrees with the measure constructed in \cite[\S{8.4}]{brooks} (in a formulation germane to that in \cite[\S{5.2}]{burungale-II}) attached to ${\boldsymbol{f}}_\phi$, from where the stated interpolation property follows from \cite[Prop.~8.9]{brooks}.
\end{proof}
\begin{cor}\label{cor:wan-bdp}
Let the hypotheses be as in Theorem~\ref{thm:bdp}, and denote by $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ the image of ${\rm tw}_{\Theta^{-1}}(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K}))$
under the natural projection $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{{\rm ac}}]]$. Then
\begin{equation}\label{eq:wan-bdp}
\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}=\mathscr{L}^{\tt BDP}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})^2
\end{equation}
up to a unit in $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]][1/p]^\times$. In particular, $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ is nonzero.
\end{cor}
\begin{proof}
In light of $(\ref{eq:factor-hida})$, the claimed equality up to a unit follows from a direct comparison of the interpolation properties in Theorem~\ref{thm:hida}, in $(\ref{eq:Katz})$, and in Theorem~\ref{thm:bdp} (\emph{cf.} \cite[\S{3.3}]{JSW}). On the other hand, by construction
for every $\phi\in\mathcal{X}_a^o(\mathbb{I})$ the $p$-adic $L$-function $\mathscr{L}_\mathfrak{p}^{\tt BDP}({\boldsymbol{f}}/\mathcal{K})$ specializes at $\phi$ to the $p$-adic $L$-functions
constructed in \cite[\S{3.3}]{cas-hsieh1} (for $N^-=1$), and in \cite[\S{5.2}]{burungale-II} and \cite[\S{8}]{brooks} (for $N^->1$);
since the latter are nonzero by \cite[Thm.~3.9]{cas-hsieh1} and \cite[Thm.~5.7]{burungale-II}, the last claim in the theorem follows.
\end{proof}
\section{Iwasawa theory}\label{sec:Iw}
Fix a prime $p>3$ and a positive integer $N$ prime to $p$, let
\[
{\boldsymbol{f}}=\sum_{n=1}^\infty\boldsymbol{a}_nq^n\in\mathbb{I}[[q]]
\]
be a Hida family
of tame level $N$ and trivial tame character, and let $\mathcal{K}$ be an imaginary quadratic field of discriminant prime of $Np$ in which $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits.
Let $\Sigma$ be a finite set of places of $\mathbf{Q}$ containing $\infty$ and the primes dividing $Np$, and for any number field $F$, let $\mathfrak{G}_{F,\Sigma}$ denote
the Galois group of the maximal extension of $F$ unramified outside
the places above $\Sigma$.
\subsection{Selmer groups}\label{sec:selmer}
Let $T_{\boldsymbol{f}}$ be the big Galois representation associated to ${\boldsymbol{f}}$,
for which we shall take the geometric realization denoted $M({\boldsymbol{f}})^*$ in \cite[Def.~7.2.5]{KLZ2}. In particular, $T_{{\boldsymbol{f}}}$ is a locally free $\mathbb{I}$-module of rank $2$, and letting $D_p\subset G_{\mathbf{Q}}$ be the decomposition group at $p$ determined by our fixed embedding $\iota_p:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$, it fits in an exact sequence of $\mathbb{I}[[D_p]]$-modules
\begin{equation}\label{eq:Gr-f}
0\rightarrow\mathscr{F}^+T_{{\boldsymbol{f}}}\rightarrow T_{{\boldsymbol{f}}}\rightarrow\mathscr{F}^-T_{{\boldsymbol{f}}}\rightarrow 0
\end{equation}
with $\mathscr{F}^\pm T_{{\boldsymbol{f}}}$ locally free of rank $1$ over $\mathbb{I}$, and with the $D_p$-action on the quotient $\mathscr{F}^-T_{{\boldsymbol{f}}}$ given by the unramified character sending an arithmetic Frobenius to $\boldsymbol{a}_p\in\mathbb{I}^\times$.
Let $k_\mathbb{I}:=\mathbb{I}/\mathfrak{m}_\mathbb{I}$ be the residue field of $\mathbb{I}$, and denote by $\bar\rho_{{\boldsymbol{f}}}:G_\mathbf{Q}\rightarrow{\rm GL}_2(\kappa_\mathbb{I})$ the semi-simple residual representation associated with $T_{{\boldsymbol{g}}}$, which by (\ref{eq:Gr-f}) is conjugate to an upper-triangular representation upon restriction to $D_p$:
\[
\bar\rho_{{\boldsymbol{f}}}\vert_{D_p}\sim\left(\begin{smallmatrix}\bar{\varepsilon}&*\\ &\bar\delta\end{smallmatrix}\right).
\]
If $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible and $\bar\varepsilon\neq\bar\delta$, as we shall assume from now on, then by work of Wiles \cite{wiles88} (see also \cite[Thm.~7.2.8]{KLZ2}), $T_{{\boldsymbol{f}}}$ is free of rank $2$ over $\mathbb{I}$, and each $\mathscr{F}^\pm T_{{\boldsymbol{f}}}$ is free of rank $1$.
Recall that we let
$\Gamma_\mathcal{K}={\rm Gal}(\mathcal{K}_\infty/\mathcal{K})$ denote the Galois group of the $\mathbf{Z}_p^2$-extension of $\mathcal{K}$, and
consider the $\mathbb{I}[[\Gamma_\mathcal{K}]]$-module
\[
\mathbf{T}_{}:=T_{{\boldsymbol{f}}}\otimes_{\mathbb{I}}\mathbb{I}[[\Gamma_\mathcal{K}]]\quad
\]
equipped with the $G_\mathcal{K}$-action via $\rho_{{\boldsymbol{f}}}\otimes\Psi_\mathcal{K}$,
where $\rho_{{\boldsymbol{f}}}$
is the $G_\mathbf{Q}$-representation afforded by $T_{\boldsymbol{f}}$, and $\Psi_\mathcal{K}$ is the tautological character $G_\mathcal{K}\twoheadrightarrow\Gamma_\mathcal{K}\hookrightarrow\mathbb{I}[[\Gamma_\mathcal{K}]]^\times$. Replacing $\Gamma_\mathcal{K}$ by $\Gamma_\mathcal{K}^{\rm ac}$ (resp. $\Gamma_\mathcal{K}^{\rm cyc}$), we define the $G_\mathcal{K}$-module $\mathbf{T}^{\rm ac}$ (resp. $\mathbf{T}^{\rm cyc}$) similarly.
As in \cite{howard-invmath}, we also define the critical twist
\begin{equation}\label{eq:dagger}
T_{{\boldsymbol{f}}}^\dagger:=T_{{\boldsymbol{f}}}\otimes\Theta^{-1},
\end{equation}
where $\Theta:G_\mathbf{Q}\rightarrow\mathbb{I}^\times$ is the character (\ref{def:crit-char}), and define its deformations $\mathbf{T}^\dagger, \mathbf{T}^{\dagger,{\rm ac}}$, and $\mathbf{T}^{\dagger,{\rm cyc}}$ similarly as before.
In the definitions that follow, we let $M$ denote either of the Galois modules just introduced,
for which we naturally define $\mathscr{F}^\pm{M}$ using $(\ref{eq:Gr-f})$.
Consider the \emph{$p$-relaxed} Selmer group defined by
\[
{\rm Sel}^{\{p\}}(F,M)={\rm ker}\Biggl\{{\rm H}^1(\mathfrak{G}_{F,\Sigma},M
\rightarrow\prod_{v\in\Sigma,\;v\nmid p}\frac{{\rm H}^1(F_v,M)}{{\rm H}^1_{\rm ur}(F_v,M)}\Biggr\},\nonumber
\]
where ${\rm H}^1_{\rm ur}(F_v,M)={\rm ker}\{{\rm H}^1(F_v,M)\rightarrow {\rm H}^1(F_v^{\rm ur},M)\}$ is the unramified local condition.
\begin{defn}
For $v\vert p$ and $\mathscr{L}_v\in\{\emptyset,{\tt Gr},0\}$, set
\[
{\rm H}^1_{\mathscr{L}_v}(F_v,M):=
\left\{
\begin{array}{ll}
{\rm H}^1(F_v,M)&\textrm{if $\mathscr{L}_v=\emptyset$,}\\[0.1cm]
{\rm ker}\{{\rm H}^1(F_v,M)\rightarrow{\rm H}^1(F_v^{\rm ur},\mathscr{F}^-M)\}&\textrm{if $\mathscr{L}_v={\tt Gr}$,}\\[0.1cm]
\{0\}&\textrm{if $\mathscr{L}_v=0$,}
\end{array}
\right.
\]
and for $\mathscr{L}=\{\mathscr{L}_v\}_{v\vert p}$, define
\begin{equation}\label{def:auxsel}
{\rm Sel}_{\mathscr{L}}(F,M):={\rm ker}\Biggr\{{\rm Sel}^{\{p\}}(F,M)
\rightarrow\prod_{v\vert p}
\frac{{\rm H}^1(F_{v},M)}{{\rm H}_{\mathscr{L}_v}^1(F_{v},M)}\Biggr\}.\nonumber
\end{equation}
\end{defn}
Thus, for example ${\rm Sel}_{0,\emptyset}(\mathcal{K},M)$ is the subspace of ${\rm Sel}^{\{p\}}(\mathcal{K},M)$ consisting of classes which satisfy no condition (resp. are locally trivial) at $\overline{\mathfrak{p}}$ (resp. $\mathfrak{p}$). For the ease of notation, we let ${\rm Sel}_{\tt Gr}(F,M)$ denote the Selmer group ${\rm Sel}_{\mathscr{L}}(F,M)$ given by $\mathscr{L}_v={\tt Gr}$ for all $v\vert p$.
We shall also need to consider Selmer groups for the discrete module
\[
A_{{\boldsymbol{f}}}:={\rm Hom}_{\rm cts}(T_{{\boldsymbol{f}}},\mu_{p^\infty}).
\]
To define these, recall that by Shapiro's lemma there is a canonical isomorphism
\begin{equation}\label{eq:Shapiro}
{\rm H}^1(\mathcal{K},\mathbf{T})\simeq\varprojlim_
{\mathcal{K}\subset_{\rm f}F\subset\mathcal{K}_\infty}{\rm H}^1(F,T_{\boldsymbol{f}}),
\end{equation}
where $F$ runs over the finite extensions of $\mathcal{K}$ contained in $\mathcal{K}_\infty$ and the limits is with respect to the corestriction maps. By the compatibility of $(\ref{eq:Shapiro})$ with the local restriction maps (see e.g. \cite[\S{3.1.2}]{SU}), the Selmer groups ${\rm Sel}_\mathscr{L}(\mathcal{K},\mathbf{T})$ are defined by local conditions ${\rm H}^1_{\mathscr{L}_v}(F_v,T_{{\boldsymbol{f}}})\subset{\rm H}^1(F_v,T_{{\boldsymbol{f}}})$ for all primes $v$. Thus we may let
\[
{\rm Sel}_{\mathscr{L}}(\mathcal{K}_\infty,A_{\boldsymbol{f}})\subset\varinjlim_{\mathcal{K}\subset_{\rm f} F\subset\mathcal{K}_\infty}{\rm H}^1(F,A_{\boldsymbol{f}})
\]
be the submodule cut out by
the orthogonal complements of ${\rm H}^1_{\mathscr{L}_v}(F_v,T_{{\boldsymbol{f}}})$ under the perfect Tate duality
\[
{\rm H}^1(F_v,T_{\boldsymbol{f}})\times{\rm H}^1(F_v,A_{\boldsymbol{f}})\rightarrow
\mathbf{Q}_p/\mathbf{Z}_p.
\]
This also defines the Selmer groups ${\rm Sel}_\mathscr{L}(F,A_{\boldsymbol{f}})\subset{\rm H}^1(F,A_{\boldsymbol{f}})$ for any number field $F$, and we shall also consider their variants for the twisted module
\[
A_{{\boldsymbol{f}}}^\dagger:={\rm Hom}_{\mathbf{Z}_p}(T_{{\boldsymbol{f}}}^\dagger,\mu_{p^\infty}),
\]
or their specializations.
Finally, if $W$ denotes any of the preceding discrete modules, we set
\[
X_{\mathscr{L}}(F,W):={\rm Hom}_{\mathbf{Z}_p}({\rm Sel}_{\mathscr{L}}(F,W),\mathbf{Q}_p/\mathbf{Z}_p),
\]
which we simply denote by $X_{\tt Gr}(F,W)$ when $\mathscr{L}_v={\tt Gr}$ for all $v\vert p$.
We now record a number of lemmas for our later use.
\begin{lem}\label{lem:eq-ranks}
Assume that $\overline{\rho}_{{\boldsymbol{f}}}\vert_{G_F}$ is absolutely irreducible. Then ${\rm Sel}_{\tt Gr}(F,T_{{\boldsymbol{f}}}^\dagger)$ and $X_{\tt Gr}(F,A_{{\boldsymbol{f}}}^\dagger)$ have the same $\mathbb{I}$-rank.
\end{lem}
\begin{proof}
For any height one prime $\mathfrak{P}\subset\mathbb{I}$, let $\mathbb{I}_{\mathfrak{P}}$ be the localization of $\mathbb{I}$ at $\mathfrak{P}$, and let $F_\mathfrak{P}=\mathbb{I}_{\mathfrak{P}}/\mathfrak{P}$ be the residue field. It suffices to show that for all but finitely many $\mathfrak{P}\in\mathcal{X}_a(\mathbb{I})$, the spaces ${\rm Sel}_{\tt Gr}(F,T_{{\boldsymbol{f}}}^\dagger)_{\mathfrak{P}}/\mathfrak{P}$ and $X_{\tt Gr}(F,A_{{\boldsymbol{f}}}^\dagger)_{\mathfrak{P}}/\mathfrak{P}$ have the same $F_{\mathfrak{P}}$-dimension.
As noted in \cite[\S{12.7.5}]{nekovar310} (see also \cite[Lem.~2.1.6]{howard-invmath}), Hida's results imply that
the localization $\mathbb{I}_\mathfrak{P}$ of $\mathbb{I}$ at any $\mathfrak{P}\in\mathcal{X}_a(\mathbb{I})$ is a discrete valuation ring. Let $\pi\in\mathbb{I}_{\mathfrak{P}}$ be a uniformizer. From Nekov{\'a}{\v{r}}'s theory (see \cite[Prop.~12.7.13.4(i)]{nekovar310}) and the identification \cite[(21)]{howard-invmath},
multiplication by $\pi$
induces natural maps
\begin{align*}
{\rm Sel}_{\tt Gr}(F,T_{{\boldsymbol{f}}}^\dagger)_{\mathfrak{P}}/\pi&\hookrightarrow{\rm Sel}_{\tt Gr}(F,T_{{\boldsymbol{f}},\mathfrak{P}}^\dagger/\pi),\\
{\rm Sel}_{\tt Gr}(F,A_{{\boldsymbol{f}},\mathfrak{P}}^\dagger[\pi])&\twoheadrightarrow{\rm Sel}_{\tt Gr}(F,A_{{\boldsymbol{f}}}^\dagger)_{\mathfrak{P}}[\pi]
\end{align*}
which are isomorphisms for all but finitely many $\mathfrak{P}\in\mathcal{X}_a(\mathbb{I})$.
Since by \cite[Lem.~1.3.3]{howard-PhD-I} the spaces ${\rm Sel}_{\tt Gr}(F,T_{{\boldsymbol{f}},\mathfrak{P}}^\dagger/\pi)$ and ${\rm Sel}_{\tt Gr}(F,A_{{\boldsymbol{f}},\mathfrak{P}}^\dagger[\pi])$ have the same $F_\mathfrak{P}$-dimension, the result follows.
\end{proof}
\begin{lem}\label{lem:no-tors}
If $\bar{\rho}_{{\boldsymbol{f}}}\vert_{G_\mathcal{K}}$ is irreducible, then the modules
${\rm H}^1(\mathfrak{G}_{\mathcal{K},\Sigma},\mathbf T^\dagger)$ and ${\rm H}^1(\mathfrak{G}_{\mathcal{K},\Sigma},{\mathbf{T}^{\dagger,{\rm ac}}})$ are torsion-free
over $\mathbb{I}_{}[[\Gamma_\mathcal{K}]]$ and $\mathbb{I}_{}[[\Gamma^{\rm ac}_\mathcal{K}]]$, respectively.
\end{lem}
\begin{proof}
This follows immediately from \cite[\S{1.3.3}]{PR:Lp}, since ${\rm H}^0(\mathcal{K}_\infty,\bar{\rho}_{\boldsymbol{f}})={\rm H}^0(\mathcal{K}_\infty^{\rm ac},\bar{\rho}_{\boldsymbol{f}})=\{0\}$ by the irreducibility of $\bar{\rho}_{\boldsymbol{f}}\vert_{G_\mathcal{K}}$.
\end{proof}
\begin{lem}\label{lem:str-rel}
We have ${\rm rank}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},\emptyset}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger))=1+{\rm rank}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger}))$. Moreover, if $\mathbb{I}$ is regular then
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},\emptyset}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger})_{\rm tors})={\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{0,{\tt Gr}}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger})_{\rm tors}),
\]
where the subscript {\rm tors} denotes the $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion submodule.
\end{lem}
\begin{proof}
The first claim follows from an argument similar to that in Lemma~\ref{lem:eq-ranks} using part (2) of \cite[Lem.~2.3]{cas-BF}. For the second, note that the regularity of $\mathbb{I}$ implies that of $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$. Thus by \cite[Lem.~6.18]{Fouquet} the second claim follows from part (3) of \cite[Lem.~2.3]{cas-BF} (see also \cite[Prop.~3.16]{BL-ord}).
\end{proof}
We conclude this section with the following useful commutative algebra lemma from \cite{SU}, which will be used repeatedly in the proof of our main results in $\S\ref{sec:main}$.
\begin{lem}\label{lem:3.2}
Let $R$ be a local ring and $\mathfrak{a}\subset R$ a proper ideal such that $R/\mathfrak{a}$ is a domain. Let $I\subset R$ be an ideal and $\mathcal{L}$ an element of $R$ with $I\subset (\mathcal{L})$. Denote by a `bar' the image under the reduction map $R\rightarrow R/\mathfrak{a}$. If $\overline{\mathcal{L}}\in R/\mathfrak{a}$ is nonzero and $\overline{\mathcal{L}}\in\overline{I}$, then $I=(\mathcal{L})$.
\end{lem}
\begin{proof}
This is a special case of \cite[Lem.~3.2]{SU}.
\end{proof}
\subsection{Explicit reciprocity laws}
Let $G_\mathbf{Q}$ act on $\Lambda_\Gamma$ via the tautological character $G_\mathbf{Q}\twoheadrightarrow\Gamma\hookrightarrow\Lambda_\Gamma^\times$. In \cite{KLZ2}, Kings--Loeffler--Zerbes constructed special elements
\[
_c\mathcal{BF}_m^{{\boldsymbol{f}},{\boldsymbol{g}}}\in{\rm H}^1(\mathbf{Q}(\mu_m),T_{{\boldsymbol{f}}}\hat\otimes_{\mathbf{Z}_p} T_{{\boldsymbol{g}}}\hat\otimes_{\mathbf{Z}_p}\Lambda_\Gamma)
\]
attached to pairs of Hida families ${\boldsymbol{f}},{\boldsymbol{g}}$, and related the image of $_c\mathcal{BF}_1^{{\boldsymbol{f}},{\boldsymbol{g}}}$ under a Perrin-Riou big logarithm map to the $p$-adic $L$-functions $L_p({\boldsymbol{f}},{\boldsymbol{g}})$ and $L_p({\boldsymbol{g}},{\boldsymbol{f}})$ of Theorem~\ref{thm:hida}. In this section we describe the variant of their results that we shall need.
Since $\mathbf T^\dagger=\mathbf T\otimes\Theta^{-1}$ by definition, the twist map ${\rm tw}_{\Theta^{-1}}:\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}]]$ of $(\ref{eq:tw-theta})$ induces a $\mathbb{I}$-linear isomorphism
\[
\widetilde{\rm tw}_{\Theta^{-1}}:{\rm H}^1(\mathcal{K},\mathbf{T})\rightarrow{\rm H}^1(\mathcal{K},\mathbf{T}^\dagger)
\]
satisfying $\widetilde{\rm tw}_{\Theta^{-1}}(\lambda x)={\rm tw}_{\Theta^{-1}}(\lambda)\widetilde{\rm tw}_{\Theta^{-1}}(x)$ for all $\lambda\in\mathbb{I}[[\Gamma_\mathcal{K}]]$.
\begin{thm}[Kings--Loeffler--Zerbes]
\label{thm:Col}
There exists a class $\mathcal{BF}_{}^\dagger\in{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf{T}^\dagger)$ and $\mathbb{I}[[\Gamma_\mathcal{K}]]$-linear injections with pseudo-null cokernel
\begin{align*}
{\rm Col}^{(1),\dagger}:{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^{-}\mathbf{T}^\dagger)
\;&\rightarrow\;I_{\boldsymbol{f}}\otimes_{\mathbb{I}}\mathbb{I}[[\Gamma_\mathcal{K}]],\\
{\rm Col}^{(2),\dagger}:{\rm H}^1(\mathcal{K}_{\mathfrak{p}},\mathscr{F}^{+}\mathbf{T}^\dagger)
\;&\rightarrow\;I_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}[[\Gamma_\mathcal{K}]],
\end{align*}
where ${\boldsymbol{g}}$ is the CM Hida family in $(\ref{eq:g-CM})$, such that
\begin{align*}
{\rm Col}^{(1),\dagger}({\rm loc}_{\overline{\mathfrak{p}}}(\mathcal{BF}^\dagger))&={\rm tw}_{\Theta^{-1}}(L_p^{}({\boldsymbol{f}},{\boldsymbol{g}}))\\
{\rm Col}^{(2),\dagger}({\rm loc}_\mathfrak{p}(\mathcal{BF}^\dagger))&={\rm tw}_{\Theta^{-1}}(L_p^{}({\boldsymbol{g}},{\boldsymbol{f}})).
\end{align*}
In particular, for every prime $v$ of $\mathcal{K}$ above $p$, the class ${\rm loc}_v(\mathcal{BF}^\dagger)\in{\rm H}^1(\mathcal{K}_v,\mathbf{T}^\dagger)$ is non-torsion over $\mathbb{I}[[\Gamma_\mathcal{K}]]$.
\end{thm}
\begin{proof}
This follows from the results of \cite{KLZ2}, as explained in \cite[Thm.~2.4]{cas-BF}, and to which one needs to add some of the analysis in \cite{BST}.
Indeed, taking $m=1$ in \cite[Def.~8.1.1]{KLZ2} (and using \cite[Lem.~6.8.9]{LLZ} to dispense with an auxiliary $c>1$ needed for the construction), one obtains a cohomology class
\begin{equation}\label{eq:KLZ-class}
\mathcal{BF}^{{\boldsymbol{f}},{\boldsymbol{g}}}\in{\rm H}^1(\mathbf{Q},T_{\boldsymbol{f}}\hat\otimes_{\mathbf{Z}_p}T_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\Lambda_\Gamma) \nonumber
\end{equation}
attached to our fixed Hida family ${\boldsymbol{f}}$ and a second Hida family ${\boldsymbol{g}}$.
Taking for ${\boldsymbol{g}}$ the canonical CM family in (\ref{eq:g-CM}), by \cite{BST} we have an isomorphism
\[
T_{\boldsymbol{g}}\simeq{\rm Ind}_\mathcal{K}^\mathbf{Q}\mathbf{Z}_p[[\Gamma_\mathfrak{p}]]
\]
as $G_\mathbf{Q}$-modules,
where the $G_\mathcal{K}$-action on $\mathbf{Z}_p[[\Gamma_\mathfrak{p}]]$ is given by the tautological character $G_\mathcal{K}\twoheadrightarrow\Gamma_\mathfrak{p}\hookrightarrow\mathbf{Z}_p[[\Gamma_\mathfrak{p}]]^\times$. By Shapiro's lemma, the class $\mathcal{BF}^{{\boldsymbol{f}},{\boldsymbol{g}}}$ therefore defines a class $\mathcal{BF}\in{\rm H}^1(\mathcal{K},\mathbf{T})$ whose image under $\widetilde{\rm tw}_{\Theta^{-1}}$ will be our required $\mathcal{BF}^\dagger$.
Indeed, the inclusion $\mathcal{BF}^\dagger\in {\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)$ follows from \cite[Prop.~8.1.7]{KLZ2},
and by the explicit reciprocity law of \cite[Thm.~10.2.2]{KLZ2}, the maps
\[
{\rm Col}^{(1)}:=\langle\mathcal{L}(-),\eta_{{\boldsymbol{f}}}\otimes\omega_{{\boldsymbol{g}}}\rangle,\quad
{\rm Col}^{(2)}:=\left\langle\mathcal{L}(-),\eta_{{\boldsymbol{g}}}\otimes\omega_{{\boldsymbol{f}}}\right\rangle
\]
described in the proof of \cite[Thm.~2.4]{cas-BF} send
the restriction at $\overline{\mathfrak{p}}$ and $\mathfrak{p}$ of
$\mathcal{BF}$ to the $p$-adic $L$-functions $L_p({\boldsymbol{f}},{\boldsymbol{g}})$ and $L_p^{}({\boldsymbol{g}},{\boldsymbol{f}})$, respectively. Thus letting ${\rm Col}^{(1),\dagger}$ and ${\rm Col}^{(2),\dagger}$ be the $\mathbb{I}[[\Gamma_\mathcal{K}]]$-linear maps defined by the commutative diagrams
\[
\xymatrix{
{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^{-}\mathbf{T})\ar[r]^-{{\rm Col}^{(1)}}\ar[d]^-{\widetilde{\rm tw}_{\Theta^{-1}}}&I_{\boldsymbol{f}}\otimes_{\mathbb{I}}\mathbb{I}[[\Gamma_\mathcal{K}]]\ar[d]^-{{\rm tw}_{\Theta^{-1}}}&
{\rm H}^1(\mathcal{K}_{\mathfrak{p}},\mathscr{F}^{+}\mathbf{T})\ar[r]^-{{\rm Col}^{(2)}}\ar[d]^-{\widetilde{\rm tw}_{\Theta^{-1}}}&I_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}[[\Gamma_\mathcal{K}]]\ar[d]^-{{\rm tw}_{\Theta^{-1}}}\\
{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^{-}\mathbf{T}^\dagger)\ar[r]^-{{\rm Col}^{(1),\dagger}}&I_{\boldsymbol{f}}\otimes_{\mathbb{I}}\mathbb{I}[[\Gamma_\mathcal{K}]]&
{\rm H}^1(\mathcal{K}_{\mathfrak{p}},\mathscr{F}^{+}\mathbf{T}^\dagger)\ar[r]^-{{\rm Col}^{(2),\dagger}}&I_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}[[\Gamma_\mathcal{K}]],
}
\]
the result follows, with the last claim being an immediate consequence of the nonvanishing of the $p$-adic $L$-functions $L_p({\boldsymbol{f}},{\boldsymbol{g}})$ and $L_p({\boldsymbol{g}},{\boldsymbol{f}})$ (see e.g. \cite[Rem.~1.3]{cas-BF}).
\end{proof}
We shall also need to consider anticyclotomic variants of the maps ${\rm Col}^{(i),\dagger}$ in Theorem~\ref{thm:Col}. Letting $\mathcal{I}_{\rm cyc}$ be the kernel of the natural projection $\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_{\mathcal{K}}^{\rm ac}]]$, the map
\[
{\rm Col}_{\rm ac}^{(1),\dagger}:{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^{\dagger,{\rm ac}})\rightarrow I_{\boldsymbol{f}}\otimes_\mathbb{I}\cR[[\Gamma_\mathcal{K}^{\rm ac}]]
\]
is defined by reducing ${\rm Col}^{(1),\dagger}$ modulo the ideal $\mathcal{I}_{\rm cyc}$, using the fact that by the vanishing of ${\rm H}^0(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^{\dagger,{\rm ac}})$ the restriction map induces a natural isomorphism
\[
{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^{\dagger})/\mathcal{I}_{\rm cyc}\simeq{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^{\dagger,{\rm ac}}).
\]
The map ${\rm Col}_{\rm ac}^{(2),\dagger}:{\rm H}^1(\mathcal{K}_{\mathfrak{p}},\mathscr{F}^+\mathbf T^{\dagger,{\rm ac}})\rightarrow I_{\boldsymbol{g}}\hat\otimes_{\mathbf{Z}_p}\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$ is defined in the same manner.
Note that since the maps ${\rm Col}^{(i),\dagger}$ are injective with pseudo-null cokernel, the same is true for the maps ${\rm Col}_{\rm ac}^{(i),\dagger}$.
\begin{cor}\label{cor:str-Gr}
Let $\mathcal{BF}^{\dagger,{\rm ac}}$ be the image of the class $\mathcal{BF}^\dagger$ under the natural map ${\rm H}^1(\mathcal{K},\mathbf{T}^\dagger)\rightarrow{\rm H}^1(\mathcal{K},\mathbf{T}^{\dagger,{\rm ac}})$.
Assume that $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{eq:gen-Heeg-f})}. Then we have the inclusion
\[
{\rm loc}_{\overline{\mathfrak{p}}}(\mathcal{BF}_{}^{\dagger,{\rm ac}})\in{\rm ker}\{{\rm H}^1(\mathcal{K}_{\overline\mathfrak{p}},{\mathbf{T}^{\dagger,{\rm ac}}})\rightarrow{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-{\mathbf{T}^{\dagger,{\rm ac}}})\};
\]
in particular, $\mathcal{BF}_{}^{\dagger,{\rm ac}}\in{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})$. Moreover, if we assume in addition that $N$ is squarefree when $N^->1$, then ${\rm loc}_{\mathfrak{p}}(\mathcal{BF}_{}^{\dagger,{\rm ac}})$ is non-torsion over $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$.
\end{cor}
\begin{proof}
The combination of Theorem~\ref{thm:Col} and Proposition~\ref{thm:hida-1} yields the vanishing of the image of ${\rm loc}_{\overline{\mathfrak{p}}}(\mathcal{BF}^{\dagger,{\rm ac}})$ under the map ${\rm Col}_{\rm ac}^{(1),\dagger}$, so the first claim follows from its injectivity. The second claim follows from Theorem~\ref{thm:Col} together with the novanishing result of Corollary~\ref{cor:wan-bdp}.
\end{proof}
\subsection{Iwasawa main conjectures}\label{sec:ES}
We now use the reciprocity laws of Theorem~\ref{thm:Col} to relate different variants of the Iwasawa main conjecture.
\begin{thm}\label{thm:2-varIMC}
Assume that $\overline{\rho}_{{\boldsymbol{f}}}\vert_{G_\mathcal{K}}$ is irreducible.
Then the following are equivalent:
\begin{enumerate}
\item[(i)]{} $X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger})$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, ${\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf{T}^{\dagger})$ has $\mathbb{I}[[\Gamma_\mathcal{K}]]$-rank one,
and
\begin{equation}\label{eq:BF-IMC}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger}))=
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}\bigg(\frac{{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf{T}^{\dagger})}{\mathbb{I}[[\Gamma_\mathcal{K}]]\cdot\mathcal{BF}^{\dagger}}\biggr)
\nonumber
\end{equation}
up to powers of $p$.
\item[(ii)]{} Both $X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger})$ and ${\rm Sel}_{0,\emptyset}(\mathcal{K},\mathbf{T}^{\dagger})$ are $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger}))\cdot\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]=({\rm tw}_{\Theta^{-1}}(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})))
\]
up to powers of $p$.
\item[(iii)]{} Both $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger})$ and ${\rm Sel}_{\tt Gr}(\mathcal{K},\mathbf{T}^{\dagger})$ are $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^{\dagger}))=({\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))).
\]
up to powers of $p$.
\end{enumerate}
Moreover, if in addition $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{eq:gen-Heeg-f})}, with $N$ being squarefree when $N^->1$, then the following are equivalent:
\begin{enumerate}
\item[(i)']{} $X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger})$ is $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion, ${\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf{T}^{\dagger,{\rm ac}})$ has $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one,
and
\begin{equation}\label{eq:BF-IMC}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger}))=
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\bigg(\frac{{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf{T}^{\dagger,{\rm ac}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathcal{BF}^{\dagger,{\rm ac}}}\biggr)
\nonumber
\end{equation}
up to powers of $p$.
\item[(ii)']{} Both $X_{\emptyset,0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger})$ and ${\rm Sel}_{0,\emptyset}(\mathcal{K},\mathbf{T}^{\dagger,{\rm ac}})$ are $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{\emptyset,0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger}))\cdot\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]]=(\mathscr{L}^{\tt BDP}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})^2)
\]
up to powers of $p$.
\end{enumerate}
\end{thm}
\begin{proof}
Consider the exact sequence coming form Poitou--Tate duality
\begin{align*}\label{eq:ES-1b}
0\rightarrow{\rm Sel}_{0,\emptyset}(\mathcal{K},\mathbf T^\dagger)\rightarrow{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)\xrightarrow{{\rm loc}_\mathfrak{p}}
&{\rm H}^1_{\tt Gr}(\mathcal{K}_{\mathfrak{p}},\mathbf T^\dagger)\\
&\rightarrow
X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow 0.\nonumber
\end{align*}
By Theorem~\ref{thm:Col}, the cokernel of the map ${\rm loc}_\mathfrak{p}$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and so the equivalence between the claimed ranks in (i) and (ii) follows. By Lemma~\ref{lem:no-tors}, if ${\rm Sel}_{0,\emptyset}(\mathcal{K},\mathbf T^\dagger)$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion then it is trivial, and so the above yields
\begin{equation}\label{eq:div-1}
0\rightarrow\frac{{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)}{\mathbb{I}[[\Gamma_\mathcal{K}]]\cdot\mathcal{BF}^\dagger}\xrightarrow{{\rm loc}_\mathfrak{p}}
\frac{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\mathfrak{p}},\mathbf T^\dagger)}{\mathbb{I}[[\Gamma_\mathcal{K}]]\cdot{\rm loc}_\mathfrak{p}(\mathcal{BF}^\dagger)}\\
\rightarrow
X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow 0.
\end{equation}
By \cite[Cor.~4.13]{betina-dimitrov-CM}, the congruence ideal of the CM Hida family ${\boldsymbol{g}}$ in $(\ref{eq:g-CM})$ is generated by $\mathscr{L}_\mathfrak{p}(\mathcal{K})_{\rm ac}$ after inverting $p$, and therefore by Theorem~\ref{thm:Col} and (\ref{eq:factor-hida}) the map ${\rm Col}_{}^{(2),\dagger}$ multiplied by this generator yields an injection
\begin{equation}\label{eq:katz-col}
\frac{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\mathfrak{p}},\mathbf T^\dagger)\cdot\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]][1/p]}{\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]][1/p]\cdot{\rm loc}_\mathfrak{p}(\mathcal{BF}^\dagger))}\hookrightarrow\frac{\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]][1/p]}{({\rm tw}_{\Theta^{-1}}(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})))}\nonumber
\end{equation}
with pseudo-null cokernel, which combined with $(\ref{eq:div-1})$ completes the proof of the equivalence of (i) and (ii). The equivalence between (i)' and (ii)' when $\mathcal{K}$ satisfies the hypothesis (\ref{ass:gen-H}) is shown in the same way, using the nonvanishing of ${\rm loc}_\mathfrak{p}(\mathcal{BF}^{\dagger,{\rm ac}})$ from Corollary~\ref{cor:str-Gr}.
Now consider the exact sequence
\begin{align*}\label{eq:ES-2b}
0\rightarrow{\rm Sel}_{\tt Gr}(\mathcal{K},\mathbf T^\dagger)\rightarrow{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)\xrightarrow{{\rm loc}_{\overline\mathfrak{p}}}
&\frac{{\rm H}^1(\mathcal{K}_{\overline\mathfrak{p}},\mathbf T^\dagger)}{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\overline\mathfrak{p}},\mathbf T^\dagger)}\simeq{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^\dagger)\\
&\rightarrow
X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow 0,\nonumber
\end{align*}
which similarly as before implies the equivalence between the claimed $\mathbb{I}[[\Gamma_\mathcal{K}]]$-ranks in (ii) and (iii), and by Theorem~\ref{thm:Col} and Lemma~\ref{lem:no-tors} yields the exact sequence
\[
0\rightarrow\frac{
{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)}{\mathbb{I}[[\Gamma_\mathcal{K}]]\cdot\mathcal{BF}^\dagger}\xrightarrow{{\rm loc}_{\overline\mathfrak{p}}}
\frac{{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^\dagger)}{\mathbb{I}[[\Gamma_\mathcal{K}]]\cdot{\rm loc}_{\mathfrak{p}}(\mathcal{BF}^\dagger)}\rightarrow
X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow X_{{\tt Gr},0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)\rightarrow 0.\nonumber
\]
Since by Theorem~\ref{thm:Col} the map ${\rm Col}_{}^{(1),\dagger}$ multiplied by a generator of the congruence ideal of ${\boldsymbol{f}}$ yields an injection ${\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},\mathscr{F}^-\mathbf T^\dagger)\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}]]$ with pseudo-null cokernel sending ${\rm loc}_{\mathfrak{p}}(\mathcal{BF}^\dagger)$ into ${\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$ up to a unit in $\mathbb{I}^\times$,
the equivalence between (ii) and (iii) follows.
\end{proof}
\subsection{Rubin's height formula}\label{sec:main}
Fix a topological generator $\gamma_{\rm cyc}\in\Gamma_\mathcal{K}^{\rm cyc}$, and using the identification $\mathbb{I}[[\Gamma_\mathcal{K}]]\simeq(\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]])[[\Gamma_\mathcal{K}^{\rm cyc}]]$, expand
\begin{equation}\label{eq:2-exp}
{\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))=L_{p,0}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}+L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}\cdot(\gamma_{\rm cyc}-1)+\cdots
\end{equation}
as a power series in $\gamma_{\rm cyc}-1$. The constant term $L_{p,0}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ thus corresponds to the image of ${\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$ under the natural projection $\mathbb{I}[[\Gamma_\mathcal{K}]]\rightarrow\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$.
By Shapiro's lemma,
we may consider the class $\mathcal{BF}^\dagger\in{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},\mathbf T^\dagger)$ of Theorem~\ref{thm:Col} as a system of classes
$\mathcal{BF}_F^\dagger\in{\rm Sel}_{{\tt Gr},\emptyset}(F,T_{{\boldsymbol{f}}}^\dagger)$,
indexed by the finite extensions $F$ of $\mathcal{K}$ contained in $\mathcal{K}_\infty$, compatible under the corestriction maps. For any intermediate extension $\mathcal{K}\subset L\subset \mathcal{K}_\infty$, we then set
\[
\mathcal{BF}^\dagger_{}(L):=\{\mathcal{BF}^\dagger_{F}\}_{\mathcal{K}\subset_{\rm f} F\subset L}
\]
with $F$ running over the finite extensions of $\mathcal{K}$ contained in $L$,
so in particular $\mathcal{BF}^\dagger_{}(\mathcal{K}_\infty)$ is nothing but $\mathcal{BF}^\dagger_{}$.
Let $\mathcal{K}_n^{\rm ac}$ be the subextension of $\mathcal{K}_\infty^{\rm ac}$ with $[\mathcal{K}_n^{\rm ac}:\mathcal{K}]=p^n$, define $\mathcal{K}_k^{\rm cyc}$ similarly, and set $L_{n,k}=\mathcal{K}_n^{\rm ac}\mathcal{K}_k^{\rm cyc}$ for all $k\leqslant\infty$.
\begin{lem}\label{lem:3.1.1}
Assume that $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{eq:gen-Heeg-f})} and that $\bar{\rho}_{\boldsymbol{f}}\vert_{G_\mathcal{K}}$ is irreducible. Then there is a unique element
\[
\beta^\dagger_{n}\in {\rm H}^1_{}(\mathcal{K}^{\rm ac}_{n,\overline\mathfrak{p}},\mathscr{F}^-\mathbf{T}^{\dagger,{{\rm cyc}}})
\]
such that ${\rm loc}_{\overline{\mathfrak{p}}}(\mathcal{BF}^\dagger_{}(L_{n,\infty}))=(\gamma_{{\rm cyc}}-1)\beta_n^\dagger$.
Furthermore, the natural images of the classes $\beta_n^\dagger$ in ${\rm H}^1(\mathcal{K}_{n,\overline\mathfrak{p}}^{\rm ac},\mathscr{F}^-T_{{\boldsymbol{f}}}^\dagger)$ are norm-compatible, defining a class
\[
\{\beta_n^\dagger(\mathds{1})\}_n\in\varprojlim_n{\rm H}^1(\mathcal{K}_{n,\overline\mathfrak{p}}^{{\rm ac}},\mathscr{F}^-T_{{\boldsymbol{f}}}^\dagger)\simeq{\rm H}^1(\mathcal{K}_{\overline\mathfrak{p}},\mathscr{F}^-{\mathbf{T}^{\dagger,{\rm ac}}})
\]
that is sent to the linear term $L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ under the
map ${\rm Col}_{{\rm ac}}^{(1),\dagger}$.
\end{lem}
\begin{proof}
By the explicit reciprocity law of Theorem~\ref{thm:Col}, the first claim follows from the vanishing of $L_{p,0}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ (see Proposition~\ref{thm:hida-1}) and the injectivity of ${\rm Col}^{(1),\dagger}$, with the uniqueness claim being an immediate consequence of Lemma~\ref{lem:no-tors}; the last claim is a direct consequence of the definitions of $\beta^\dagger_n$ and $L^{\tt Hi}_{p,1}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$.
\end{proof}
Let $\mathcal{I}^{\rm cyc}=(\gamma_{\rm cyc}-1)\subset\mathbb{I}[[\Gamma_\mathcal{K}^{\rm cyc}]]$ be the augmentation ideal,
and set $\mathcal{J}^{\rm cyc}=\mathcal{I}^{\rm cyc}/(\mathcal{I}^{\rm cyc})^2$. By work of Plater \cite{Plater}, and more generally Nekov{\'a}{\v{r}} \cite[\S{11}]{nekovar310}, for every $n$ there is a canonical (up to sign) $\mathbb{I}$-adic height pairing
\begin{equation}\label{eq:I-ht}
\langle,\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{{\rm cyc}}:{\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)\times{\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)
\rightarrow\mathcal{J}^{\rm cyc}\otimes_\mathbb{I} F_\mathbb{I}.
\end{equation}
(Note that the local indecomposability hypothesis (H1) in \cite[p.~107]{Plater} is only used to ensure the existence of well-defined sub and quotients at the places above $p$, which for $T_{\boldsymbol{f}}^\dagger$ is automatic, while hypotheses (H2) and (H3) in \emph{loc.cit.} follow from \cite[Lem.~2.4.4]{howard-invmath} for $T_{\boldsymbol{f}}^\dagger$.)
Denoting by ${\rm H}^1_{\tt Gr}(L_{n,k,v},T_{\boldsymbol{f}}^\dagger)\subset{\rm H}^1(L_{n,k,v},T_{\boldsymbol{f}}^\dagger)$ the local condition defining ${\rm Sel}_{\tt Gr}(L_{n,k},T_{\boldsymbol{f}}^\dagger)$ at a place $v$,
Plater's definition of $\langle,\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{{\rm cyc}}$ (which we shall briefly recall in the proof of Proposition~\ref{thm:rubin-ht} below)
shows that $\langle,\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{{\rm cyc}}$ takes integral values in the submodule of ${\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)$ consisting of classes which are local cyclotomic universal norms at all places $v$ above $p$, i.e., classes in
\[
{\rm H}^1_{\tt Gr}(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger)^{\rm univ}:=\bigcap_{k}{\rm cor}_{L_{n,k,v}/\mathcal{K}_{n,v}^{\rm ac}}({\rm H}^1_{\tt Gr}(L_{n,k,v},T_{\boldsymbol{f}}^\dagger)),
\]
and so by
\cite[Lem.~2.3.1]{PR-109} the denominators of (\ref{eq:I-ht}) are abounded independently of $n$.
The next result generalizes the height formula of \cite[Thm.~3.2(ii)]{rubin-ht} to our context.
\begin{prop}\label{thm:rubin-ht}
Assume that $\mathcal{K}$ satisfies the hypothesis {\rm (\ref{ass:gen-H})} and that $\bar{\rho}_{\boldsymbol{f}}\vert_{G_\mathcal{K}}$ is irreducible. Then the classes $\mathcal{BF}^\dagger_{\mathcal{K}_n^{\rm ac}}$ land in ${\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)$, and for every $x\in{\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)$ we have
\begin{equation}\label{eq:rubin-ht}
\langle\mathcal{BF}^\dagger_{\mathcal{K}_n^{\rm ac}},x\rangle^{\rm cyc}_{\mathcal{K}_n^{\rm ac},\mathbb{I}}=(\beta_n^\dagger(\mathds{1}),{\rm loc}_{\overline\mathfrak{p}}(x))_{\mathcal{K}_{n,\overline{\mathfrak{p}}}^{\rm ac}}\otimes(\gamma_{\rm cyc}-1),
\end{equation}
where $(,)_{\mathcal{K}_{n,\overline{\mathfrak{p}}}^{\rm ac}}$ is the local Tate pairing
\[
\frac{{\rm H}^1(\mathcal{K}_{n,\overline\mathfrak{p}}^{\rm ac},T_{\boldsymbol{f}}^\dagger)}{{\rm H}^1_{\tt Gr}(\mathcal{K}_{n,\overline\mathfrak{p}}^{\rm ac},T_{\boldsymbol{f}}^\dagger)}
\times {\rm H}^1_{\tt Gr}(\mathcal{K}_{n,\overline\mathfrak{p}}^{\rm ac},T_{\boldsymbol{f}}^\dagger)\rightarrow\mathbb{I}.
\]
\end{prop}
\begin{proof}
The first claim follows from the explicit reciprocity law of Theorem~\ref{thm:Col}, the vanishing of $L_{p,0}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$, and the injectivity of ${\rm Col}^{(1),\dagger}$. On the other hand, the proof of formula $(\ref{eq:rubin-ht})$ could be deduced from the general result \cite[(11.3.14)]{nekovar310}, but shall give a proof following the more direct generalization of Rubin's formula contained in \cite[\S{3}]{arnold-ht}.
We begin by recalling Plater's definition of the $\mathbb{I}$-adic height pairing (itself a generalization of Perrin-Riou's \cite[\S{1.2}]{PR-109} in the $p$-adic setting). Let $\lambda$ be the isomorphism $\Gamma_\mathcal{K}^{\rm cyc}\simeq\mathcal{J}^{\rm cyc}$ sending $\gamma_{\rm cyc}$ to the class of $\gamma_{\rm cyc}-1$. Composing with the natural isomorphism ${\rm Gal}(L_{n,\infty}/\mathcal{K}_n^{\rm ac})\simeq\Gamma^{\rm cyc}$ the map $\lambda$ defines a class in ${\rm H}^1(\mathcal{K}_n^{\rm ac},\mathcal{J}^{\rm cyc})$, where we equip $\mathcal{J}^{\rm cyc}$ with the trivial Galois action, and so taking cup product we get
\[
\rho_v:{\rm H}^1(K_{n,v}^{\rm ac},\mathbb{I}(1))\xrightarrow{\cup{\rm loc}_v(\lambda)}{\rm H}^2(K_{n,v}^{\rm ac},\mathcal{J}^{\rm cyc}(1))\simeq\mathcal{J}^{\rm cyc}
\]
for every place $v$.
Denote by ${\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)^{\rm univ}$ the submodule of ${\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)$ (with $\mathbb{I}$-torsion quotient, as noted earlier) consisting of classes lying in ${\rm H}^1_{\tt Gr}(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger)^{\rm univ}$ for all $v\mid p$, and let $x, y\in{\rm Sel}_{\tt Gr}(\mathcal{K}_n^{\rm ac},T_{\boldsymbol{f}}^\dagger)^{\rm univ}$. Then $x$ corresponds to an extension of Galois modules
\[
0\rightarrow T_{\boldsymbol{f}}^\dagger\rightarrow X\rightarrow\mathbb{I}\rightarrow 0.
\]
The Kummer dual of this sequence induces maps on cohomology
\[
{\rm H}^1(\mathcal{K}_{n}^{\rm ac},X^*(1))\rightarrow{\rm H}^1(\mathcal{K}_{n}^{\rm ac},T_{\boldsymbol{f}}^\dagger)\xrightarrow{\delta}{\rm H}^2(\mathcal{K}_{n}^{\rm ac},\mathbb{I}(1))
\]
such that $\delta(y)=0$ (since ${\rm H}^2(\mathcal{K}_{n}^{\rm ac},\mathbb{I}(1))$ injects into $\bigoplus_v{\rm H}^2(\mathcal{K}_{n,v}^{\rm ac},\mathbb{I}(1))$ and the $v$-th component of $\delta(y)$ is given by ${\rm loc}_v(y)\cup{\rm loc}_v(x)=0$ by the self-duality of Greenberg's local conditions). Thus $y$ is the image of some $y^{\rm glob}\in{\rm H}^1(\mathcal{K}_n^{\rm ac},X^*(1))$.
On the other hand, if $v$ is any place of $\mathcal{K}_{n}^{\rm ac}$, for every $k$ we can write ${\rm loc}_v(y)={\rm cor}_{L_{n,k,v}/\mathcal{K}_{n,v}^{\rm ac}}(y_{k,v})$ for some $y_{k,v}\in{\rm H}^1_{\tt Gr}(L_{n,k,v},T_{\boldsymbol{f}}^\dagger)$, and by a similar argument as above there exists a class $\widetilde{y}_{k,v}\in{\rm H}^1(L_{n,k,v},X^*(1))$ lifting $y_{k,v}$ under the natural map $\pi_v$ in the exact sequence
\begin{equation}\label{eq:local-v}
{\rm H}^1(L_{n,k,v},X^*(1))\xrightarrow{\pi_v}{\rm H}^1(L_{n,k,v},T_{\boldsymbol{f}}^\dagger)\xrightarrow{\delta_v}{\rm H}^2(L_{n,k,v},\mathbb{I}(1)).
\end{equation}
The difference ${\rm loc}_v(y^{\rm glob})-{\rm cor}_{L_{n,k,v}/\mathcal{K}_{n,v}^{\rm ac}}(\widetilde{y}_{k,v})$ is then the image of some class $w_{k,v}\in{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},\mathbb{I}(1))$, and we define
\[
\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}:=\lim_{k\to\infty}\sum_v\rho_v(w_{k,v}),
\]
a limit which is easily checked to exist and be independent of all choices. If in addition $y=y_0$ is the base class of a compatible system of classes
\[
y_\infty=\{y_k\}_k\in{\rm H}^1(\mathcal{K}_n^{\rm ac},\mathbf{T}^{\dagger,{\rm cyc}})=\varprojlim_k{\rm H}^1(L_{n,k},T_{\boldsymbol{f}}^\dagger),
\]
then one easily checks (see e.g. \cite[Lem.~3.2.2]{AHsplit}) that there are classes $y_k^{\rm glob}\in{\rm H}^1(L_{n,k},X^*(1))$ lifting $y_k$. Similarly as above, for every place $v$ of $L_{n,k}$ the corestriction of ${\rm loc}_v(y_k^{\rm glob})-\widetilde{y}_{k,v}$ to ${\rm H}^1(K_{n,v}^{\rm ac},X^*(1))$ is the image of a class $w'_{k,v}\in{\rm H}^1(K_{n,v}^{\rm ac},\mathbb{I}(1))$, and with these choices we see that the above expression for $\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}$ reduces to
\begin{equation}\label{eq:ht-p}
\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}=\lim_{k\to\infty}\sum_{v\mid p}\rho_v(w'_{k,v}).
\end{equation}
As in \cite[\S{3.8}]{arnold-ht}, division by $\gamma_{\rm cyc}-1$ defines a natural \emph{derivative map}
\[
\mathfrak{Der}_{}:{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger\otimes_\mathbb{I}\mathcal{I}^{\rm cyc})\rightarrow{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger)
\]
whose composition with the natural projection ${\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger)\rightarrow{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},\mathscr{F}^-T_{\boldsymbol{f}}^\dagger)$ factors as
\begin{equation}\label{eq:der-v}
\begin{aligned}
\xymatrix{
{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger\otimes_\mathbb{I}\mathcal{I}^{\rm cyc})\ar@{->>}[r]\ar[d]_-{\mathfrak{Der}_{}}&
{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},\mathscr{F}^-T_{\boldsymbol{f}}^\dagger\otimes_\mathbb{I}\mathcal{I}^{\rm cyc})\ar[d]^-{\mathfrak{Der}_{-}}\\
{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger)\ar@{->>}[r]&{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},\mathscr{F}^-T_{\boldsymbol{f}}^\dagger).
}
\end{aligned}
\end{equation}
Letting ${\rm pr}_{\mathds{1}}$ be the natural projection ${\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},X^*(1)\otimes_\mathbb{I}\cR[[\Gamma^{\rm cyc}]])\rightarrow{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},X^*(1))$, the expression $(\ref{eq:ht-p})$ for $\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}$ can be rewritten as
\[
\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}=\sum_{v\vert p}{\rm pr}_{\mathds{1}}({\rm loc}_v(y_\infty^{\rm glob})-\widetilde{y}_{\infty,v}),
\]
where ${\rm loc}_v(y_\infty^{\rm glob})-\widetilde{y}_{\infty,v}\in{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},X^*(1)\otimes_\mathbb{I}\cR[[\Gamma^{\rm cyc}]])$ is a lift of ${\rm loc}_v(y_\infty)-y_{\infty,v}\in{\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},T_{\boldsymbol{f}}^\dagger\otimes\mathcal{I}^{\rm cyc})$, and hence by \cite[Prop.~3.10]{arnold-ht} we obtain
\begin{equation}\label{eq:der}
\begin{aligned}
\langle y,x\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}&=\sum_{v\mid p}\delta_v\left(\mathfrak{Der}({\rm loc}_v(y_\infty)-y_{\infty,v})\right)\otimes(\gamma^{\rm cyc}-1)\\
&=\sum_{v\mid p}\left(\mathfrak{Der}({\rm loc}_v(y_\infty)-y_{\infty,v}),{\rm loc}_v(x)\right)_{\mathcal{K}_{n,v}^{\rm ac}}\otimes(\gamma^{\rm cyc}-1)\\
&=\sum_{v\mid p}\left(\mathfrak{Der}_{-}({\rm loc}_v(y_\infty)),{\rm loc}_v(x)\right)_{\mathcal{K}_{n,v}^{\rm ac}}\otimes(\gamma^{\rm cyc}-1),
\end{aligned}
\end{equation}
where the last equality follows from the commutativity of $(\ref{eq:der-v})$ and the fact that $y_{\infty,v}=\{y_{k,v}\}_k$ has trivial image in ${\rm H}^1(\mathcal{K}_{n,v}^{\rm ac},\mathscr{F}^-\mathbf{T}^{\dagger,{\rm cyc}})$.
Now taking $y_\infty=\mathcal{BF}^\dagger(L_{n,\infty})$ in $(\ref{eq:der})$ we see that the contribution to $\langle\mathcal{BF}^\dagger_{\mathcal{K}_n^{\rm ac}},x\rangle^{\rm cyc}_{\mathcal{K}_n^{\rm ac},\mathbb{I}}$ from $\mathfrak{p}$ is zero, since $\mathcal{BF}^\dagger(L_{n,\infty})\in{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K}_n^{\rm ac},\mathbf{T}^{\dagger,{\rm cyc}})$ is finite at the places above $\mathfrak{p}$, while at $\overline{\mathfrak{p}}$ chasing through the definitions we see that
\[
\mathfrak{Der}_{-}({\rm loc}_{\overline{\mathfrak{p}}}(\mathcal{BF}^\dagger(L_{n,\infty}))=\beta_n^\dagger(\mathds{1}),
\]
thus concluding the proof of the height formula (\ref{eq:rubin-ht}).
\end{proof}
\section{Big Heegner points}\label{sec:HP} \label{sec:bigHP}
In this section, we recall the construction of big Heegner points and classes \cite{howard-invmath,LV} with some complements.
Fix a prime $p>3$ and a positive integer $N$ prime to $p$. Let $\mathcal{K}$ be an imaginary quadratic field with ring of integers $\mathcal{O}_\mathcal{K}$ and
discriminant $-D_\mathcal{K}<0$ prime to $Np$, and write
\[
N=N^+N^-
\]
with $N^+$ (resp. $N^-$) divisible only by primes which are split (resp. inert) in $\mathcal{K}$. Throughout, we assume the following \emph{generalized Heegner hypothesis}:
\begin{equation}\label{ass:gen-H}
\textrm{$N^-$ is the squarefree product of an even number of primes,}\tag{gen-H}
\end{equation}
and fix an integral ideal $\mathfrak{N}^+$ of $\mathcal{K}$ with $\mathcal{O}_\mathcal{K}/\mathfrak{N}^+\simeq\mathbf{Z}/N^+\mathbf{Z}$.
\subsection{Towers of Shimura curves}\label{subsec:Sh}
Let $B/\mathbf{Q}$ be an
indefinite quaternion algebra of discriminant $N^-$. We fix a $\mathbf Q$-algebra embedding
$\iota_\mathcal{K}:\mathcal{K}\hookrightarrow B$, which we shall use to identify $\mathcal{K}$ with a subalgebra of $B$.
Let $z\mapsto\overline{z}$ be the non-trivial automorphism of $\mathcal{K}$, and choose a basis $\{1,j\}$
of $B$ over $\mathcal{K}$ such that:
\begin{itemize}
\item $j^2=\beta\in\mathbf Q^\times$ with $\beta<0$ and $jt=\bar tj$ for all $t\in \mathcal{K}$,
\item $\beta\in (\mathbf Z_q^\times)^2$ for $q\mid pN^+$, and $\beta\in\mathbf Z_q^\times$ for $q\mid D_\mathcal{K}$.
\end{itemize}
Fix a square-root $\delta=\sqrt{-D_\mathcal{K}}$, and define $\boldsymbol{\theta}\in \mathcal{K}$ by
\begin{equation}\label{eq:vartheta}
\boldsymbol{\theta}:=\frac{D_\mathcal{K}'+\delta}{2},\quad\textrm{where}\;\;
D_\mathcal{K}':=\left\{
\begin{array}{ll}
D_\mathcal{K} &\textrm{if $2\nmid D_\mathcal{K}$,}\\[0.1cm]
D_\mathcal{K}/2 &\textrm{if $2\mid D_\mathcal{K}$,}
\end{array}
\right.
\end{equation}
so that $\mathcal O_\mathcal{K}=\mathbf Z+\boldsymbol{\theta}\mathbf{Z}$. For every prime $q\mid pN^+$,
define the isomorphism $i_q:B_q:=B\otimes_\mathbf Q\Q_q \simeq \M_2(\mathbf Q_q)$ by
\[
i_q(\boldsymbol{\theta})=\mat{\mathrm{Tr}(\boldsymbol{\theta})}{-\mathrm{Nm}(\boldsymbol{\theta})}10,
\quad\quad
i_q(j)=\sqrt\beta\mat{-1}{\mathrm{Tr}(\boldsymbol{\theta})}01,
\]
where $\mathrm{Tr}$ and $\mathrm{Nm}$ are the reduced trace and norm maps on $B$.
For primes $q\nmid Np$, we fix any isomorphism $i_q:B_q\simeq \M_2(\mathbf Q_q)$
with $i_q(\mathcal O_\mathcal{K}\otimes_\mathbf Z\Z_q)\subset\M_2(\mathbf Z_q)$.
Let $\hat{\mathbf{Z}}$ denote the profinite completion of $\mathbf{Z}$, and for any abelian group $M$ set $\hat{M}=M\otimes_{\mathbf{Z}}\hat{\mathbf{Z}}$.
For each $r\geqslant 0$, let $R_{r}$ be the Eichler order of $B$ of level $N^+p^r$ with respect to
the isomorphisms $\{i_q:B_q\simeq{\rm M}_2(\mathbf Q_q)\}_{q\nmid N^-}$, and let $U_{r}\subset\hat{R}_{r}^\times$ be the compact open
subgroup
\[
U_{r}:=\left\{(x_q)_q\in\hat{R}_{r}^\times\;\colon\;i_p(x_p)\equiv\mat 1*0*\pmod{p^r}\right\}.
\]
Consider the double coset spaces
\begin{equation}\label{def:gross-curve}
X_{r}=B^\times\backslash\bigl(\Hom_\mathbf Q(\mathcal{K},B)\times\hat{B}^\times/U_{r}\bigr),
\end{equation}
where $b\in B^\times$ acts on $(\Psi,g)\in\Hom_\mathbf Q(\mathcal{K},B)\times\hat B^\times$ by
\[
b\cdot(\Psi,g)=(b\Psi b^{-1},bg),
\]
and $U_{r}$ acts on $\hat{B}^\times$ by right multiplication. As is well-known (see e.g. \cite[\S\S{2.1-2}]{LV}), $ X_{r}$ can be identified
with a set of algebraic points on the Shimura curve with complex uniformization
\[
X_{r}(\mathbf{C})=B^\times\backslash\bigl(\Hom_\mathbf Q(\mathbf{C},B)\times\hat{B}^\times/U_{r}\bigr).
\]
Let ${\rm rec}_\mathcal{K}:\mathcal{K}^\times\backslash\hat{\mathcal{K}}^\times\rightarrow{\rm Gal}(\mathcal{K}^{\rm ab}/\mathcal{K})$ be the reciprocity map of class field theory. By Shimura's reciprocity law, if $P\in X_{r}$
is the class of a pair $(\Psi,g)$, then $\sigma\in{\rm Gal}(\mathcal{K}^{\rm ab}/\mathcal{K})$ acts on $P$ by
\[
P^{\sigma}:=[(\Psi,\hat{\Psi}(a)g)],
\]
where $a\in \mathcal{K}^\times\backslash\hat{\mathcal{K}}^\times$ is such that ${\rm rec}_\mathcal{K}(a)=\sigma$, and $\hat\Psi:\hat \mathcal{K}\rightarrow\hat B$ is the adelization of $\Psi$. We extend this to an action of
$G_\mathcal{K}:={\rm Gal}(\overline{\mathbf Q}/\mathcal{K})$ in the obvious manner.
The curves $ X_{r} $ are also equipped
with natural actions of Hecke operators $T_\ell$ for $\ell\nmid Np$, $U_\ell$ for $\ell\vert Np$, and diamond
operators $\langle d \rangle$ for $d\in(\mathbf Z/p^r\mathbf Z)^\times$, as described in e.g. \cite[\S{2.4}]{LV} and \cite[\S{2.1}]{ChHs2}.
\subsection{Compatible systems of Heegner points}\label{subsec:construct}
For each $c\geqslant 1$, let $\mathcal{O}_c=\mathbf Z+c\mathcal{O}_\mathcal{K}$ be the order of $\mathcal{K}$ of conductor $c$ and denote
by $H_c$ the ring class field of $\mathcal{K}$ of conductor $c$, so that ${\rm Pic}(\mathcal{O}_c)\simeq{\rm Gal}(H_c/\mathcal{K})$ by class field theory. In particular, $H_1$ is the Hilbert class field of $\mathcal{K}$.
\begin{defn}\label{def:HP}
A point $P\in X_{r}$ is a \emph{Heegner point of conductor $c$}
if it is the class of a pair $(\Psi,g)$ with
\[
\Psi(\mathcal{O}_c)=\Psi(\mathcal{K})\cap(B\cap g\hat{R}_{r}g^{-1})
\]
and
\[
\Psi_p\left((\mathcal{O}_c\otimes\mathbf Z_p)^\times\cap(1+p^r\mathcal{O}_c\otimes\mathbf Z_p)^\times\right)
=\Psi_p\left((\mathcal{O}_c\otimes\mathbf Z_p)^\times\right)\cap g_pU_{r,p}g_p^{-1},
\]
where $\Psi_p$ and $U_{r,p}$ denote the $p$-components of $\Psi$ and $U_{r}$, respectively.
\end{defn}
For each prime $q\neq p$ define
\begin{itemize}
\item{} $\varsigma_q=1$, if $q\nmid N^+$,
\item{} $\varsigma_q=\delta^{-1}\begin{pmatrix}\boldsymbol{\theta} & \overline{\boldsymbol{\theta}} \\ 1 & 1 \end{pmatrix}
\in{\rm GL}_2(\mathcal{K}_{\mathfrak{q}})={\rm GL}_2(\mathbf Q_q)$, if
$q=\mathfrak{q}\overline{\mathfrak{q}}$ splits with $\mathfrak{q}\mid\mathfrak{N}^+$,
\end{itemize}
and for each $s\geqslant 0$, let
\begin{itemize}
\item{} $\varsigma_p^{(s)}=\begin{pmatrix}\boldsymbol{\theta}&-1\\1&0\end{pmatrix}\begin{pmatrix}p^s&0\\0&1\end{pmatrix}
\in{\rm GL}_2(\mathcal{K}_{\mathfrak{p}})={\rm GL}_2(\mathbf Q_p)$,
if $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $\mathcal{K}$,
\item{}
$\varsigma_p^{(s)}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}p^s&0\\0&1\end{pmatrix}$, if $p$ is inert in $\mathcal{K}$.
\end{itemize}
\begin{rem} We shall ultimately assume that $p$ splits in $\mathcal{K}$, but it is worth-noting that, just as in \cite{howard-invmath, LV}, the constructions in this section also allow the case $p$ inert in $\mathcal{K}$.
\end{rem}
Set $\varsigma^{(s)}:=\varsigma_p^{(s)}\prod_{q\neq p}\varsigma_q$, which we view as an element in $\hat{B}^\times$ via the isomorphisms
$\{i_q:B_q\simeq{\rm M}_2(\mathbf Q_q)\}_{q\nmid N^-}$ introduced in $\S\ref{subsec:Sh}$. With the $\mathbf{Q}$-algebra embedding $\iota_\mathcal{K}:\mathcal{K}\hookrightarrow B$ fixed there,
one easily checks that for all $s\geqslant r$ the points
\[
{P}_{s,r}^{}:=[(\iota_\mathcal{K},\varsigma^{(s)})]\in X_{r}
\]
are Heegner points of conductor $p^{s}$ in the sense of Definition~\ref{def:HP} with the following properties:
\begin{itemize}
\item{} \emph{Field of definition}: ${P}_{s,r}\in H^0(H_{p^s}({\mu}_{p^r}),{X}_{r})$.
\item{} \emph{Galois equivariance}: For all $\sigma\in{\rm Gal}(H_{p^s}({\mu}_{p^r})/H_{p^s})$,
\[
{P}_{s,r}^\sigma=\langle\vartheta(\sigma)\rangle\cdot {P}_{s,r},
\]
where $\vartheta:{\rm Gal}(H_{p^s}({\mu}_{p^r})/H_{p^{s}})\rightarrow\mathbf Z_p^\times/\{\pm{1}\}$ is such that
$\vartheta^2=\varepsilon_{\rm cyc}$.
\item{} \emph{Horizontal compatibility}: If $s\geqslant r> 1$, then
\[
\sum_{\sigma\in{\rm Gal}(H_{p^s}({\mu}_{p^r})/H_{p^{s-1}}({\mu}_{p^r}))}
{\alpha}_r({P}_{s,r}^{{\sigma}})
=U_p\cdot{P}_{s,r-1},
\]
where ${\alpha}_r: X_{r}\rightarrow {X}_{r-1}$ is the map
induced by the inclusion $U_{r}\subset U_{r-1}$.
\item{} \emph{Vertical compatibility}: If $s\geqslant r\geqslant 1$, then
\[
\sum_{\sigma\in{\rm Gal}(H_{p^s}({\mu}_{p^r})/H_{p^{s-1}}({\mu}_{p^r}))}
{P}_{s,r}^{{\sigma}}
=U_p\cdot{P}_{s-1,r}.
\]
\end{itemize}
(See \cite[Thm.~1.2]{cas-longo} and the references therein.)
\subsection{Big Heegner points}\label{subsec:bigHP}
Let $\mathbb{B}_r$ the $\mathbf{Z}_p$-algebra generated by the Hecke operators
$T_\ell$, $U_\ell$, and $\langle a\rangle$ acting on the Shimura curve ${X}_{r}$ from $\S$\ref{subsec:Sh},
let $\mathfrak{h}_{r}$ be the $\mathbf{Z}_p$-algebra generated by the usual Hecke operators $T_\ell$,
$U_\ell$, and $\langle a\rangle$ acting on the space $S_2(\Gamma_{0,1}(N,p^r))$
of classical modular form of level $\Gamma_{0,1}(N,p^r):=\Gamma_0(N)\cap\Gamma_1(p^r)$,
and let $\mathbb{T}^{N^-}_{N,r}$ be the quotient of $\mathfrak{h}_r$ acting faithfully on the
subspace of $S_2(\Gamma_{0,1}(N,p^r))$ consisting of $N^-$-new forms.
The Jacquet--Langlands correspondence
yields $\mathbf{Z}_p$-algebra isomorphisms
\begin{equation}\label{eq:JL}
\mathbb{B}_r\simeq\mathbb{T}^{N^-}_{N,r}
\end{equation}
(see e.g. \cite[\S{2.4}]{HMI}). In particular, letting $e_{\rm ord}=\lim_{n\to\infty}U_p^{n!}$ be Hida's ordinary projector, the $\mathbf{Z}_p$-module
\[
\mathfrak{D}_{r}^{\rm ord}:=e_{\rm ord}({\rm Div}({X}_{r})\otimes_{\mathbf Z}\mathbf Z_p)
\]
is naturally endowed with an action of $\mathbb{T}_{r}^{\rm ord}:=e_{\rm ord}\mathbb{T}^{N^-}_{N,r}$.
Denote by $\mathbb T_{r}^\dagger$ be the free $\mathbb T_{r}^{\rm ord}$-module of rank one
equipped with the Galois action via the inverse of the critical character $\Theta$, and set $\mathfrak{D}_{r}^\dagger:=\mathfrak{D}_{r}^{\rm ord}\otimes_{\mathbb{T}_{r}^{\rm ord}}\mathbb T_{r}^\dagger$.
Let ${P}_{s,r}\in{X}_{r}$ be the Heegner point of conductor $p^s$ ($s\geqslant r$) constructed in $\S$\ref{subsec:construct}, and denote by $\mathcal{P}_{s,r}$ the image of
$e_{\rm ord}{P}_{s,r}^{}$ in $\mathfrak{D}_{r}^{\rm ord}$.
It follows from the Galois-equivariance property of ${P}_{s,r}$ that
\[
\mathcal{P}_{s,r}^\sigma=\Theta(\sigma)\cdot\mathcal{P}_{s,r}
\]
for all $\sigma\in{\rm Gal}(H_{p^s}({\mu}_{p^r})/H_{p^{s}})$ (see \cite[\S{7.1}]{LV}),
and hence $\mathcal{P}_{s,r}$ defines an element
\begin{equation}\label{eq:pt-dag}
\mathcal{P}_{s,r}\otimes\zeta_r\in{\rm H}^0(H_{p^{s}},\mathfrak{D}_{r}^\dagger).
\end{equation}
Let ${\rm Pic}({X}_{r})$ be the Picard variety of ${X}_{r}$, and set
\[
\mathfrak{J}_{r}^{\rm ord}:=e_{\rm ord}({\rm Pic}({X}_{r})\otimes_{\mathbf Z}\mathbf Z_p),\quad\quad\mathfrak{J}_{r}^\dagger:=\mathfrak{J}_{r}^{\rm ord}\otimes_{\mathbb{T}_r^{\rm ord}}\mathbb{T}_r^\dagger.
\]
Since the $U_p$-operator has degree $p$, taking ordinary parts yields an isomorphism $\mathfrak{D}_r^{\rm ord}\simeq\mathfrak{J}_r^{\rm ord}$, and so we may also view (\ref{eq:pt-dag}) as $\mathcal{P}_{s,r}\otimes\zeta_r\in{\rm H}^0(H_{p^{s}},\mathfrak{J}_{r}^\dagger)$.
Let $t\geqslant 0$, and denote by $\mathfrak{G}_{H_{p^t}}$ the Galois group of the maximal
extension of $H_{p^t}$ unramified outside the primes above $pN$. Consider the twisted Kummer map
\[
{\rm Kum}_r:{\rm H}^0(H_{p^t},\mathfrak{J}_{r}^\dagger)
\rightarrow{\rm H}^1(\mathfrak{G}_{H_{p^t}},{\rm Ta}_p(\mathfrak{J}_{r}^\dagger))
\]
explicitly defined in \cite[p.~101]{howard-invmath}. This map is equivariant for the Galois and $U_p$ actions, and hence by horizontal compatibility the classes
\begin{equation}\label{eq:HP-r}
\mathfrak{X}_{p^t,r}:={\rm Kum}_r({\rm Cor}_{H_{p^{r+t}/H_{p^t}}}(\mathcal{P}_{r+t,r}\otimes\zeta_r))
\end{equation}
satisfy ${\alpha}_{r,*}(\mathfrak{X}_{p^t,r})=U_p\cdot\mathfrak{X}_{p^t,r-1}$ for all $r>1$, where
\[
{\alpha}_{r,*}:{\rm H}^1(\mathfrak{G}_{H_{p^t}},{\rm Ta}_p(\mathfrak{J}_{r}^\dagger))\rightarrow{\rm H}^1(\mathfrak{G}_{H_{p^t}},{\rm Ta}_p(\mathfrak{J}_{r-1}^\dagger))
\]
is the map induced by the covering ${X}_r\rightarrow{X}_{r-1}$ by Albanese functoriality.
Now let ${\boldsymbol{f}}\in\mathbb{I}[[q]]$ be a Hida family of tame level $N$. To define big Heegner points attached to ${\boldsymbol{f}}$ from the system of Heegner classes (\ref{eq:HP-r}) for varying $r$, we need to recall the following result realizing the big Galois representation $T_{\boldsymbol{f}}$ attached to ${\boldsymbol{f}}$ in the \'etale cohomology of the $p$-tower of Shimura curves
\[
\cdots\rightarrow{X}_r\rightarrow{X}_{r-1}\rightarrow\cdots
\]
(rather than classical modular curves, as implicitly taken in $\S\ref{sec:selmer}$).
Let $\kappa_\mathbb{I}:=\mathbb{I}/\mathfrak{m}_\mathbb{I}$ be the residue field of $\mathbb{I}$, and denote by $\bar{\rho}_{{\boldsymbol{f}}}:G_\mathbf{Q}\rightarrow{\rm GL}_2(\kappa_\mathbb{I})$ the associated (semi-simple) residual representation. Set
\[
\mathbb{T}_\infty^{\rm ord}:=\varprojlim_r\mathbb{T}_r^{\rm ord}.
\]
By (\ref{eq:JL}) (see also the discussion in \cite[\S{5.3}]{LV}), there is a maximal ideal $\mathfrak{m}\subset\mathbb{T}^{\rm ord}_\infty$ associated with $\bar{\rho}_{{\boldsymbol{f}}}$, and ${\boldsymbol{f}}$ corresponds to a minimal prime in the localization $\mathbb{T}^{\rm ord}_{\infty,\mathfrak{m}}$.
\begin{thm}\label{thm:helm}
Assume that
\begin{itemize}
\item[(i)] $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible and $p$-distinguished,
\item[(ii)] $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified at every prime $\ell\vert N^-$ with $\ell\equiv\pm{1}\pmod{p}$,
\end{itemize}
and let $\mathfrak{m}\subset\mathbb{T}_\infty^{\rm ord}$ be the maximal ideal associated with $\bar{\rho}_{{\boldsymbol{f}}}$. Then the module
\[
\mathbf{Ta}_{\mathfrak{m}}^{\rm ord}:=\biggl(\varprojlim_r{\rm Ta}_p(\mathfrak{J}_r^{\rm ord})\biggr)\otimes_{\mathbb{T}_\infty^{\rm ord}}\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}
\]
is free of rank $2$ over $\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}$, and if ${\boldsymbol{f}}$ corresponds to the minimal prime $\mathfrak{a}\subset\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}$, then there is an isomorphism
\[
T_{\boldsymbol{f}}\simeq\mathbf{Ta}_{\mathfrak{m}}^{\rm ord}\otimes_{\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}}\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}/\mathfrak{a}
\]
as $(\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}/\mathfrak{a})[G_\mathbf{Q}]$-modules.
\end{thm}
\begin{proof}
This is shown in \cite[Thm.~3.1]{Fouquet} assuming the ``mod $p$ multiplicity one'' hypothesis in [\emph{loc.cit.}, Prop.~3.7]. Since by \cite[Cor.~8.11]{helm} that hypothesis is ensured by our ramification condition on $\bar\rho_{{\boldsymbol{f}}}$, the result follows.
\end{proof}
Let $\mathfrak{m}\subset\mathbb{T}_\infty^{\rm ord}$ be a maximal ideal satisfying the hypotheses of Theorem~\ref{thm:helm}, and suppose that the Hida family ${\boldsymbol{f}}$ corresponds to a minimal prime of $\mathbb{T}_{\infty,\mathfrak{m}}^{\rm ord}$, so by Theorem~\ref{thm:helm} there is a quotient map $\mathbf{Ta}_{\mathfrak{m}}^{\rm ord}\rightarrow T_{\boldsymbol{f}}$. Note also that immediately from the definitions there are natural maps
${\rm Ta}_p(\mathfrak{J}_r^\dagger)\rightarrow\mathbf{Ta}_{\mathfrak{m}}^{\rm ord}\otimes\Theta^{-1}\rightarrow T_{\boldsymbol{f}}^\dagger$.
\begin{defn}
The \emph{big Heegner point} of conductor $p^t$ is the class
\[
\mathfrak{X}_{p^t}\in{\rm H}^1(H_{p^t},T_{{\boldsymbol{f}}}^\dagger)
\]
given by the image of $\varprojlim_rU_p^{-r}\cdot\mathfrak{X}_{p^t,r}$ under the composite map
\[
\varprojlim_r{\rm H}^1(\mathfrak{G}_{H_{p^t}},{\rm Ta}_p(\mathfrak{J}_{r}^\dagger))
\rightarrow{\rm H}^1(\mathfrak{G}_{H_{p^t}},\mathbf{Ta}_{\mathfrak{m}}^{\rm ord}\otimes\Theta^{-1})
\rightarrow{\rm H}^1(H_{p^t},T_{{\boldsymbol{f}}}^\dagger).
\]
\end{defn}
We conclude this section with the following result essentially due to Howard, showing that the big Heegner points are Selmer classes under mild hypotheses.
\begin{prop}\label{prop:HPinSel}
Assume that $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified at every prime $\ell\vert N^-$.
Then the classes $\mathfrak{X}_{p^t}$ lie in ${\rm Sel}_{\tt Gr}(H_{p^t},T_{{\boldsymbol{f}}}^\dagger)$.
\end{prop}
\begin{proof}
The argument in \cite[Prop.~2.4.5]{howard-invmath} (see also \cite[Prop.~10.1]{LV}) shows that for every prime $w$ of $H_{p^t}$ the localization
${\rm loc}_w(\mathfrak{X}_{p^t})$ lies in the subspace ${\rm H}_{\tt Gr}^1(H_{p^t,w},T_{{\boldsymbol{f}}}^\dagger)\subset{\rm H}^1(H_{p^t,w},T_{{\boldsymbol{f}}}^\dagger)$ defining ${\rm Sel}_{\tt Gr}(H_{p^t},T_{{\boldsymbol{f}}}^\dagger)$, except when $w\vert\ell\vert N^-$, in which case it is shown that
\[
{\rm loc}_{w}(\mathfrak{X}_{p^t})\in{\rm ker}\biggl\{{\rm H}^1(H_{p^t,w},T_{{\boldsymbol{f}}}^\dagger)\rightarrow\frac{{\rm H}^1(H_{p^t,w}^{\rm ur},T_{{\boldsymbol{f}}}^\dagger)}{{\rm H}^1(H_{p^t,w}^{\rm ur},T_{{\boldsymbol{f}}}^\dagger)_{\rm tors}}\biggr\},
\]
where ${\rm H}^1(H_{p^t,w}^{\rm ur},T_{{\boldsymbol{f}}}^\dagger)_{\rm tors}$ denotes the $\mathbb{I}$-torsion submodule of ${\rm H}^1(H_{p^t,w}^{\rm ur},T_{{\boldsymbol{f}}}^\dagger)$. However,
such primes $\ell$ are inert in $\mathcal{K}$, so
$H_{p^t,w}=\mathcal{K}_\ell$, and since our hypothesis on $\bar\rho_{\boldsymbol{f}}$ implies that ${\rm H}^1(\mathcal{K}_\ell^{\rm ur},T_{{\boldsymbol{f}}}^\dagger)$ is $\mathbb{I}$-torsion free (see e.g. \cite[Lem.~3.12]{buy-bigHP}), the result follows.
\end{proof}
Recall that $\mathcal{K}_\infty^{\rm ac}=\cup_n \mathcal{K}_n^{\rm ac}$ is the anticyclotomic $\mathbf{Z}_p$-extension of $\mathcal{K}$, with $\mathcal{K}_n^{\rm ac}$ denoting the subextension of $\mathcal{K}_\infty^{\rm ac}$ with $[\mathcal{K}_n^{\rm ac}:\mathcal{K}]=p^n$. Similarly as in \cite[\S{3.3}]{howard-invmath} and \cite[\S{10.3}]{LV},
we set
\[
\mathfrak{Z}_n:={\rm Cor}_{H_{p^t}/\mathcal{K}^{\rm ac}_n}(U_p^{-t}\cdot\mathfrak{X}_{p^t})\in{\rm H}^1(\mathcal{K}_n^{\rm ac},T_{{\boldsymbol{f}}}^\dagger),
\]
where $t\gg 0$ is chosen so that $\mathcal{K}_n^{\rm ac}\subset H_{p^t}$. By horizontal compatibility, the definition of $\mathfrak{Z}_n$ is independent of the choice of $t$, and for varying $n$ they define a system
\[
\mathfrak{Z}_\infty:=\{\mathfrak{Z}_n\}_n\in\varprojlim_n{\rm H}^1(\mathcal{K}_n^{\rm ac},T_{{\boldsymbol{f}}}^\dagger)\simeq{\rm H}_{}^1(\mathcal{K},\mathbf{T}^{\dagger,{\rm ac}})
\]
which is not $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion
by the work of Cornut--Vatsal \cite{CV-dur} (see also \cite[Cor.~3.1.2]{howard-invmath}).
\section{Main results} \label{sec:main}
In this section we conclude the proof of the main results of this paper.
Fix a prime $p>3$ and let
\[
{\boldsymbol{f}}=\sum_{n=1}^\infty\boldsymbol{a}_nq^n\in\mathbb{I}[[q]]
\]
be a primitive
Hida family of tame level $N$, and let $\mathcal{K}$ be an imaginary quadratic field of discriminant prime to $Np$ satisfying the generalized Heegner hypothesis (\ref{ass:gen-H}) relative to $N$.
Our results will require some of the technical hypotheses below, which we record here for our later reference.
\begin{itemize}
\item[(h0)] $\mathbb{I}$ is regular,
\item[(h1)]{} some specialization ${\boldsymbol{f}}_\phi$ is the $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$,
\item[(h2)] $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible,
\item[(h3)] $N$ is squarefree,
\item[(h4)] $N^-\neq 1$,
\item[(h5)] $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified at every prime $\ell\vert N^-$,
\item[(h6)] $p$ splits in $\mathcal{K}$.
\end{itemize}
As usual, here $N^-$ denotes the largest factor of $N$ divisible only by primes which are inert in $\mathcal{K}$.
\subsection{Converse to a theorem of Howard}
As shown by Howard \cite[\S\S{2.3-4}]{howard-invmath},
for varying $c$ prime to $N$ the big Heegner points $\mathfrak{X}_c\in{\rm H}^1(H_c,T_{{\boldsymbol{f}}}^\dagger)$ form an anticyclotomic Euler system for $T_{{\boldsymbol{f}}}^\dagger$. Setting
\[
\mathfrak{Z}_0:={\rm Cor}_{H_1/\mathcal{K}}(\mathfrak{X}_1)\in{\rm H}^1(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger),
\]
Kolyvagin's methods thus yield a proof of the implication
\[
\mathfrak{Z}_0\not\in{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)_{\rm tors}
\quad\Longrightarrow\quad
{\rm rank}_\mathbb{I}\;{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)=1,
\]
where the subscript ${\rm tors}$ denotes the $\mathbb{I}$-torsion submodule
(see \cite[Cor.~3.4.3]{howard-invmath}).
In the spirit of Skinner's celebrated converse to the theorem of Gross--Zagier and Kolyvagin \cite{skinner}, in this section we prove a result in the converse direction (see Theorem~\ref{thm:converse-How} below).
Following the approach of \cite{wan}, this will be deduced from progress on the Iwasawa main conjecture for big Heegner points (\cite[Conj.~3.3.1]{howard-invmath})
exploiting the non-triviality of $\mathfrak{Z}_\infty$.
\begin{thm}\label{thm:HP-MC}
Assume hypotheses {\rm (h0)--(h6)}. Then both $X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)$ and ${\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})$ have $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one, and
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)_{\rm tors})=
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathfrak{Z}_\infty}\biggr)^2,
\]
where the subscript {\rm tors} denotes the $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion submodule.
\end{thm}
\begin{proof}
Since $\mathfrak{Z}_\infty$ is not $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion,
part (iii) of \cite[Thm.~B]{Fouquet} implies that $X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)$ and ${\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})$ have $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one, and that the divisibility
\begin{equation}\label{eq:fouquet}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)_{\rm tors})\supset
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathfrak{Z}_\infty}\biggr)^2
\end{equation}
holds in $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$. Concerning the additional hypotheses in Fouquet's result, we note that:
\begin{itemize}
\item Assumption~3.4, that $\bar\rho_{\boldsymbol{f}}$ is irreducible, is our (h2),
\item Assumption~3.5, that $\bar\rho_{\boldsymbol{f}}$ is $p$-distinguished, follows from (h1) (see \cite[Rem.~7.2.7]{KLZ2}),
\item Assumption~3.10, that the tame character of ${\boldsymbol{f}}$ admits a square-root, is satisfied by (h1),
\item Assumption~5.10, that all primes $\ell\vert N$ for which $\bar\rho_{\boldsymbol{f}}$ is not ramified have infinite decomposition group in $\mathcal{K}_\infty^{\rm ac}/\mathcal{K}$, is a reformulation of (h5),
\item Assumption~5.13, that $\bar{\rho}_{{\boldsymbol{f}}}\vert_{G_\mathcal{K}}$ is irreducible, follows from (h2), (h4) and (h5) (see \cite[Lem.~2.8.1]{skinner}).
\end{itemize}
Let $\phi\in\mathcal{X}_a(\mathbb{I})$ be such that ${\boldsymbol{f}}_\phi$ is the ordinary $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$ as in hypothesis (h1).
Letting $X\supset Y$ stand for the divisibility $(\ref{eq:fouquet})$, by \cite[Thm.~3.4]{cas-BF} (see also \cite[Thm.~1.2]{wan}) we have the equality
\[
X=Y\pmod{{\rm ker}(\phi)\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]},
\]
(note that this requires the additional hypotheses (h3) and (h6)), from where the result follows by an application of Lemma~\ref{lem:3.2}.
\end{proof}
\begin{thm}\label{thm:converse-How}
Assume hypotheses {\rm (h0)--(h6)}. Then the following implication holds:
\[
{\rm rank}_\mathbb{I}\;{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)=1\quad\Longrightarrow\quad
\mathfrak{Z}_0\not\in{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)_{\rm tors}
\]
where the subscript ${\rm tors}$ denotes the $\mathbb{I}$-torsion submodule.
\end{thm}
\begin{proof}
Let $\gamma_{\rm ac}\in\Gamma_\mathcal{K}^{{\rm ac}}$ be a topological generator. The restriction map for the extension $\mathcal{K}_\infty^{\rm ac}/\mathcal{K}$ induces a surjective homomorphism
\[
X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)/(\gamma_{\rm ac}-1)X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)\twoheadrightarrow X_{\tt Gr}(\mathcal{K},A_{{\boldsymbol{f}}}^\dagger
\]
with $\mathbb{I}$-torsion kernel. Since $X_{\tt Gr}(\mathcal{K},A_{{\boldsymbol{f}}}^\dagger)$ and ${\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)$ have the same $\mathbb{I}$-rank by Lemma~\ref{lem:eq-ranks}, we thus see from Theorem~\ref{thm:HP-MC} that our assumption implies that
\[
(\gamma_{\rm ac}-1)\nmid{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathfrak{Z}_\infty}\biggr).
\]
By \cite[Cor.~3.8]{SU} (with $F=\mathcal{K}_\infty^{\rm ac})$, it follows that
the image of $\mathfrak{Z}_\infty$ in ${\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})/(\gamma_{{\rm ac}}-1){\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})$ is not $\mathbb{I}$-torsion; since this image is sent to $\mathfrak{Z}_0$ under the natural injection
\[
{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})/(\gamma_{{\rm ac}}-1){\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})\hookrightarrow{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger),
\]
the result follows.
\end{proof}
\begin{rem}
Replacing the application of \cite[Thm.~3.4]{cas-BF} or \cite[Thm.~1.2]{wan} in the proof of Theorem~\ref{thm:HP-MC} by an application of \cite[Thm.~1.2]{BCK} or \cite[Thm.~1.1.5]{zanarella} the above argument gives a proof of Theorems~\ref{thm:HP-MC}
and \ref{thm:converse-How}
with hypotheses (h3)--(h6) replaced by ``Hypothesis~$\heartsuit$'' from \cite{zhang-kolyvagin}, i.e.,
letting ${\rm Ram}(\bar{\rho}_{\boldsymbol{f}})$ be the set of primes $\ell\Vert N$ such that
$\bar{\rho}_{\boldsymbol{f}}$ is ramified at $\ell$:
\begin{itemize}
\item ${\rm Ram}(\bar{\rho}_{\boldsymbol{f}})$ contains all primes $\ell\Vert N^+$, and all primes $\ell\vert N^-$ such that $\ell\equiv\pm 1 \pmod{p}$,
\item If $N$ is not squarefree, then ${\rm Ram}(\bar{\rho}_{\boldsymbol{f}})\neq\emptyset$, and either ${\rm Ram}(\bar{\rho}_{\boldsymbol{f}})$ contains a prime $\ell\vert N^-$ or there are at least two primes $\ell\Vert N^+$,
\item If $\ell^2\vert N^+$, then ${\rm H}^0(\mathbf{Q}_\ell,\bar{\rho}_{\boldsymbol{f}})=\{0\}$,
\end{itemize}
and the assumption that $\bar{\rho}_{{\boldsymbol{f}}}$ is surjective and $\boldsymbol{a}_p\not\equiv\pm{1}\pmod{p}$.
\end{rem}
\subsection{Iwasawa--Greenberg main conjectures}
Now we can upgrade the main result
in \cite{wanIMC} towards the Iwasawa--Greenberg main conjecture for $\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K})$
to a proof of the full equality.
\begin{thm}\label{thm:3-IMC-BDP}
Assume hypotheses {\rm (h0)--(h6)}.
Then $X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$
is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion and
\begin{equation}\label{eq:3-IMC-bdp}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}))\cdot\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]=(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K}))\nonumber
\end{equation}
as ideals in $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}]]$.
\end{thm}
\begin{proof}
Clearly (see \cite[Lem.~1.2]{Rubin-ES}), it suffices to show that the twisted module $X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, with characteristic ideal generated by ${\rm tw}_{\Theta^{-1}}(\mathscr{L}_\mathfrak{p}({\boldsymbol{f}}/\mathcal{K}))$ after extension of scalars to $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]]$. This in turn can be shown by a similar argument to that in the proof of Theorem~\ref{thm:HP-MC}, so we shall be rather brief.
Taking $\phi\in\mathcal{X}_a(\mathbb{I})$ such that ${\boldsymbol{f}}_\phi$ is the ordinary $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$, we deduce that $X_{\emptyset,0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger,{\rm ac}})$ is $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion and that the equality
as ideals in $\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]]$
\begin{equation}\label{eq:BDP-IMC}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{\emptyset,0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^{\dagger,{\rm ac}}))\cdot\mathbb{I}^{\rm ur}[[\Gamma_\mathcal{K}^{\rm ac}]]=(\mathscr{L}_\mathfrak{p}^{\tt BDP}({\boldsymbol{f}}/\mathcal{K})^2)
\end{equation}
holds by applying Lemma~\ref{lem:3.2} to the combination of the divisibility in \cite[Thm.~1.1]{wanIMC} (projected under $\Gamma_\mathcal{K}\twoheadrightarrow\Gamma_\mathcal{K}^{\rm ac}$) with the equality in \cite[Thm.~3.4]{cas-BF} for $f={\boldsymbol{f}}_\phi$.
The $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsionness of $X_{\emptyset,0}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)$ then follows from that of $X_{\emptyset,0}(\mathcal{K}_\infty^{\rm ac},A_{\boldsymbol{f}}^\dagger)$ over $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$, and using the comparison of $p$-adic $L$-functions in Corollary~\ref{cor:wan-bdp}, the three-variable divisibility
in \cite[Thm.~1.1]{wanIMC} combined with the equality (\ref{eq:BDP-IMC})
yields the desired three-variable equality by another application of
Lemma~\ref{lem:3.2}.
\end{proof}
We can now deduce from Theorem~\ref{thm:3-IMC-BDP} the proof of Theorem~A in the Introduction.
\begin{cor}\label{thm:3-IMC}
Assume hypotheses {\rm (h0)--(h6)}.
Then $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}})$
is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, and
\begin{equation}\label{eq:3-IMC}
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}]]}(X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}))=(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))\nonumber
\end{equation}
as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}]]\otimes_{\mathbf{Z}_p}\mathbf{Q}_p$.
\end{cor}
\begin{proof}
As in the proof of Theorem~\ref{thm:3-IMC-BDP}, it suffices to show that the twisted module $X_{\tt Gr}(\mathcal{K}_\infty,A_{\boldsymbol{f}}^\dagger)$ is $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion with characteristic ideal generated by ${\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$, which by the equivalence between main conjectures in Theorem~\ref{thm:2-varIMC}, follows from Theorem~\ref{thm:3-IMC-BDP}.
\end{proof}
\subsection{Greenberg's nonvanishing conjecture for derivatives}
\label{sec:appl-greenberg}
As in the Introduction, let $-w\in\{\pm{1}\}$ be the generic sign in the functional equation of the $p$-adic $L$-functions $L_p^{\tt MTT}({\boldsymbol{f}}_\phi,s)$ for varying $\phi\in\mathcal{X}_a^o(\mathbb{I})$.
By \cite[Cor.~3.4.3 and Eq.~(21)]{howard-invmath}, Howard's horizontal nonvanishing conjecture implies that
\[
{\rm rank}_{\mathbb{I}}\;{\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)=\left\{
\begin{array}{ll}
1&\textrm{if $w=1$,}\\
0&\textrm{if $w=-1$.}
\end{array}
\right.
\]
In the case $w=-1$, a result in the converse direction follows from \cite{SU}:
\begin{thm}[Skinner--Urban]\label{thm:Gr+1}
Assume that:
\begin{itemize}
\item{} $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible and $p$-distinguished,
\item{} ${\boldsymbol{f}}$ has trivial tame character,
\item{} there is a prime $\ell\Vert N$ such that $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified at $\ell$.
\end{itemize}
If ${\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger)$ is $\mathbb{I}$-torsion, then $L({\boldsymbol{f}}_\phi,k_\phi/2)\neq 0$ for all but
finitely many $\phi\in\mathcal{X}_a^o(\mathbb{I})$.
\end{thm}
\begin{proof}
Since the $\mathbb{I}$-modules ${\rm Sel}_{\tt Gr}(\mathbf{Q},T_{{\boldsymbol{f}}}^\dagger)$ and $X_{\tt Gr}(\mathbf{Q},A_{{\boldsymbol{f}}}^\dagger)$ have the same rank by Lemma~\ref{lem:eq-ranks}, our hypothesis implies that $X_{\tt Gr}(\mathbf{Q},A_{{\boldsymbol{f}}}^\dagger)$ is $\mathbb{I}$-torsion. Thus in particular
${\rm Sel}_{\tt Gr}(\mathbf{Q},A_{{\boldsymbol{f}}_\phi}(1-k_\phi/2))$ is finite for all but finitely many $\phi$ as in the statement, and so the result follows from \cite[Thm.~3.6.13]{SU}.
\end{proof}
Our application to Greenberg's nonvanishing conjecture (in the case $w=1$) will build on an $\mathbb{I}$-adic Gross--Zagier formula for the big Heegner point $\mathfrak{Z}_0$. In fact, we shall prove a formula of this type for the $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-adic family $\mathfrak{Z}_\infty$, and deduce the result for $\mathfrak{Z}_0$ by specialization at the trivial character.
Define the cyclotomic $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-adic height pairing
\begin{equation}\label{eq:lambda-ht}
\langle,\rangle_{\mathcal{K}^{\rm ac}_\infty,\mathbb{I}}^{{\rm cyc}}:{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})
\otimes_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})^\iota
\rightarrow\mathcal{J}^{\rm cyc}\otimes_{\mathbb{I}}\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\otimes_\mathbb{I} F_\mathbb{I}
\end{equation}
by
\[
\langle a_\infty,b_\infty\rangle^{\rm cyc}_{\mathcal{K}_\infty^{\rm ac},\mathbb{I}}=
\varprojlim_n\sum_{\sigma\in{\rm Gal}(\mathcal{K}_n^{\rm ac}/\mathcal{K})}\langle a_n,b_n^\sigma\rangle^{\rm cyc}_{\mathcal{K}_n^{\rm ac},\mathbb{I}}\cdot\sigma
\]
(using the fact that the $\mathbb{I}$-adic height pairing $\langle,\rangle_{\mathcal{K}_n^{\rm ac},\mathbb{I}}^{\rm cyc}$ have denominators that are bounded independently of $n$), and let the cyclotomic regulator $\mathcal{R}_{\rm cyc}\subset\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\otimes_\mathbb{I} F_\mathbb{I}$
be the characteristic ideal of the cokernel of $(\ref{eq:lambda-ht})$ (after dividing by the image of $(\gamma_{\rm cyc}-1)$ in $\mathcal{J}^{\rm cyc}$).
Recall that since we assume that $\mathcal{K}$ satisfies the generalized Heegner hypothesis (\ref{ass:gen-H}), the constant term $L_{p,0}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ in the expansion $(\ref{eq:2-exp})$ vanishes by Proposition~\ref{thm:hida-1}. The next result provides a first interpretation of the linear term $L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$.
\begin{thm}\label{thm:3.1.5}
Assume hypotheses {\rm (h0)--(h6)}, and denote by $\mathcal{X}^{}_{\rm tors}$ the characteristic ideal of $X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)_{\rm tors}$. Then
\[
\mathcal{R}_{\rm cyc}^{}\cdot\mathcal{X}_{\rm tors}
=(L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}})
\]
as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\otimes_{\mathbb{I}}F_\mathbb{I}$.
\end{thm}
\begin{proof}
The height formula of Theorem~\ref{thm:rubin-ht} and Lemma~\ref{lem:3.1.1} immediately
yield the equality
\begin{equation}\label{eq:cor-ht}
\mathcal{R}_{\rm cyc}\cdot
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathcal{BF}^{\dagger,{\rm ac}}}\biggr)
=(L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}})\cdot\eta^\iota,
\end{equation}
where $\eta\subset\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$ is the characteristic ideal of
${\rm H}^1_{\tt Gr}(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})/{\rm loc}_{\overline{\mathfrak{p}}}({\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}}))$.
We shall argue below that $\eta\neq 0$. Global duality yields the exact sequence
\begin{equation}\label{eq:PT2}
0\rightarrow\frac{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\mathfrak{p}},{\mathbf{T}^{\dagger,{\rm ac}}})}
{{\rm loc}_{\mathfrak{p}}({\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}}))}
\rightarrow X_{\emptyset,{\tt Gr}}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)
\rightarrow X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)\rightarrow 0.
\end{equation}
Note that the left-most term in $(\ref{eq:PT2})$ is $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion, since by Corollary~\ref{cor:str-Gr} the image of the map ${\rm loc}_{\mathfrak{p}}:{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})\rightarrow {\rm H}^1_{\tt Gr}(\mathcal{K}_\mathfrak{p},{\mathbf{T}^{\dagger,{\rm ac}}})$ is nonzero and the target has $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one. By Theorem~\ref{thm:HP-MC}, it follows that the middle term has $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one, and by the action of complex conjugation the same is true for $X_{{\tt Gr},\emptyset}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)$. Thus the nonvanishing of $\eta$ follows from the analogue of $(\ref{eq:PT2})$ for the prime $\overline{\mathfrak{p}}$ (see (\ref{eq:PT2bar}) below).
By Lemma~\ref{lem:str-rel} the above also shows that $X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)$ is $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion, and counting ranks in the exact sequence
\begin{align*}
0\rightarrow {\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})\rightarrow{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})\rightarrow&\frac{{\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})}{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})}\\
&\rightarrow
X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)
\rightarrow X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)\rightarrow 0,
\end{align*}
we conclude that the first two terms in this sequence have $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one. Since the quotient ${\rm H}^1(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})/{\rm H}^1_{\tt Gr}(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})$ has no $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion, it follows that
\begin{equation}\label{eq:Gr-rel}
{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})=
{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}}).
\end{equation}
Taking $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-torsion in the analogue of $(\ref{eq:PT2})$ for $\overline{\mathfrak{p}}$, that is
\begin{equation}\label{eq:PT2bar}
0\rightarrow\frac{{\rm H}^1_{\tt Gr}(\mathcal{K}_{\overline{\mathfrak{p}}},{\mathbf{T}^{\dagger,{\rm ac}}})}
{{\rm loc}_{\overline{\mathfrak{p}}}({\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}}))}
\rightarrow X_{{\tt Gr},\emptyset}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)
\rightarrow X_{\tt Gr}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)\rightarrow 0,
\end{equation}
and applying Lemma~\ref{lem:str-rel} and the ``functional equation'' $\mathcal{X}_{\rm tors}^\iota=\mathcal{X}_{\rm tors}$ of \cite[p.~1464]{howard-PhD-I} we obtain
\begin{equation}\label{eq:takechar}
\begin{split}
\eta^\iota\cdot\mathcal{X}_{\rm tors}
&={\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger)).
\end{split}
\end{equation}
On the other hand, by the equivalence (i)'$\Longleftrightarrow$(ii)' in Theorem~\ref{thm:2-varIMC}, the equality (\ref{eq:BDP-IMC}) implies that
\[
{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}(X_{{\tt Gr},0}(\mathcal{K}_\infty^{\rm ac},A_{{\boldsymbol{f}}}^\dagger))={\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{{\tt Gr},\emptyset}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathcal{BF}^{\dagger,{\rm ac}}}\biggr)
\]
as ideals in $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\otimes_{\mathbf{Z}_p}\mathbf{Q}_p$, and so the result follows from the combination of (\ref{eq:cor-ht}), (\ref{eq:Gr-rel}), and (\ref{eq:takechar}).
\end{proof}
The aforementioned $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-adic Gross--Zagier formula for $\mathfrak{Z}_\infty$ is the following.
\begin{cor}\label{cor:I-GZ}
Assume hypotheses {\rm (h0)--(h6)}. Then we have the equality
\[
(L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}})=(\langle\mathfrak{Z}_\infty,\mathfrak{Z}_\infty\rangle_{\mathcal{K}_\infty^{{\rm ac}},\mathbb{I}}^{\rm cyc})
\]
as ideals of $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\otimes_{\mathbb{I}}\mathbb{I}$.
\end{cor}
\begin{proof}
Since ${\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})$ has $\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]$-rank one by Theorem~\ref{thm:HP-MC} and $\mathfrak{Z}_\infty$ is not $\mathbb{I}[[\Gamma_\mathcal{K}]]$-torsion, the regulator $\mathcal{R}_{\rm cyc}$ of $(\ref{eq:lambda-ht})$ satisfies
\[
(\langle\mathfrak{Z}_\infty,\mathfrak{Z}_\infty\rangle_{\mathcal{K}_\infty^{{\rm ac}},\mathbb{I}}^{\rm cyc})=\mathcal{R}_{\rm cyc}\cdot {\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathfrak{Z}_\infty}\biggr)\cdot{\rm Char}_{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]}\biggl(\frac{{\rm Sel}_{\tt Gr}(\mathcal{K},{\mathbf{T}^{\dagger,{\rm ac}}})}{\mathbb{I}[[\Gamma_\mathcal{K}^{\rm ac}]]\cdot\mathfrak{Z}_\infty}\biggr)^\iota.
\]
By the ``functional equation'' of \cite[p.~1464]{howard-PhD-I}, the result thus follows from the combination of Theorem~\ref{thm:3.1.5} and the equality of characteristic ideals in Theorem~\ref{thm:HP-MC}.
\end{proof}
Now we can conclude the proof of our application to Greenberg's nonvanishing conjecture.
\begin{thm}\label{thm:Gr-1}
Assume that:
\begin{itemize}
\item[(i)] $\mathbb{I}$ is regular,
\item[(ii)] $\bar{\rho}_{{\boldsymbol{f}}}$ is irreducible,
\item[(iii)]{} some specialization ${\boldsymbol{f}}_\phi$ is the $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$,
\item[(iv)] $N$ is squarefree,
\item[(v)] there are at least two primes $\ell\vert N$ at which $\bar{\rho}_{{\boldsymbol{f}}}$ is ramified.
\end{itemize}
If ${\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger)$ has $\mathbb{I}$-rank one
and the $\mathbb I$-adic height pairing
$\langle,\rangle_{\mathbf Q,\mathbb I}^{\rm cyc}$ is non-degenerate, then
\begin{equation}\label{eq:gen}
\frac{d}{ds}L_p^{\tt MTT}({\boldsymbol{f}}_\phi,s)\biggr\vert_{s=k_\phi/2}\neq 0,
\nonumber
\end{equation}
for all but finitely many $\phi\in\mathcal{X}_a^o(\mathbb{I})$.
\end{thm}
\begin{proof}
Let $\phi\in\mathcal{X}_a^o(\mathbb{I})$ be such that ${\boldsymbol{f}}_\phi$ is the ordinary $p$-stabilization of a newform $f\in S_2(\Gamma_0(N))$. Let $\ell_1$ and $\ell_2$ be two distinct primes as in hypothesis (v), and choose an imaginary quadratic field $\mathcal{K}$ such that the following hold:
\begin{itemize}
\item $\ell_1$ and $\ell_2$ are inert in $\mathcal{K}$,
\item every prime dividing $N^+:=N/\ell_1\ell_2$ splits in $\mathcal{K}$,
\item $p$ splits in $\mathcal{K}$,
\item $L(f\otimes\epsilon_\mathcal{K},1)\neq 0$, where $\epsilon_\mathcal{K}$ is the quadratic character corresponding to $\mathcal{K}$.
\end{itemize}
Note that the existence of $\mathcal{K}$ is ensured by \cite{FH}, and that, so chosen, $\mathcal{K}$ satisfies (\ref{ass:gen-H}) with $N^-=\ell_1\ell_2$. Now, the action of a complex conjugation $\tau$ combined with the restriction map induces an isomorphism
\begin{equation}\label{eq:dec}
{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)\simeq{\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger)\oplus{\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger\otimes\epsilon_\mathcal{K}),
\end{equation}
where the first and second summands are identified with the $+$ and $-$ eigenspaces for the action of $\tau$, respectively (see \cite[Lem.~3.1.5]{SU}). By Kato's work \cite{Kato295}, the nonvanishing of $L(f\otimes\epsilon_\mathcal{K},1)$ implies that ${\rm Sel}(\mathbf{Q},T_{f}\otimes\epsilon_\mathcal{K})$ is finite, and so by the control theorem for ${\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger\otimes\epsilon_\mathcal{K})$ (see the exact sequence in \cite[Cor.~3.4.3]{howard-invmath}) we conclude that ${\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger\otimes\epsilon_\mathcal{K})$ is $\mathbb{I}$-torsion, and so
\[
{\rm rank}_\mathbb{I}\;{\rm Sel}_{\tt Gr}(\mathcal{K},T_{{\boldsymbol{f}}}^\dagger)={\rm rank}_\mathbb{I}\;{\rm Sel}_{\tt Gr}(\mathbf Q,T_{{\boldsymbol{f}}}^\dagger)=1
\]
by (\ref{eq:dec}) and our assumption. In particular, since by (i)--(iv) we are assuming (h0)--(h3), and (h4)--(h6) hold by our choice of $\mathcal{K}$, Theorem~\ref{thm:converse-How} yields the non-triviality of the class $\mathfrak{Z}_0$, and so the element $\langle\mathfrak{Z}_0,\mathfrak{Z}_0\rangle_{\mathcal{K},\mathbb{I}}^{\rm cyc}\in\mathbb{I}$ is non-zero by our hypothesis of non-degeneracy.
Let $L_p^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{\rm cyc}$ be the image of ${\rm tw}_{\Theta^{-1}}(L_p^{\tt Hi}({\boldsymbol{f}}/\mathcal{K}))$ under the natural projection $\mathbb{I}[[\Gamma_\mathcal{K}]]\twoheadrightarrow\mathbb{I}[[\Gamma_\mathcal{K}^{\rm cyc}]]$. By Theorem~\ref{thm:cyc-res}, for every $\phi\in\mathcal{X}_a^o(\mathbb{I})$ we have the factorization
\begin{equation}\label{eq:factor-cyc}
\phi(L_p^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{\rm cyc})={\rm tw}_{\Theta_\phi^{-1}}(L_p^{\tt MTT}({\boldsymbol{f}}_\phi))\cdot {\rm tw}_{\Theta_\phi^{-1}}(L_p^{\tt MTT}({\boldsymbol{f}}_\phi\otimes\epsilon_\mathcal{K}))
\end{equation}
up to a unit in $\phi(\mathbb{I})[[\Gamma^{\rm cyc}]]^\times$. Expand
\begin{align*}
\phi(L_p^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{\rm cyc})&=L_{p,0}^{\tt Hi}({\boldsymbol{f}}_\phi^\dagger/\mathcal{K})+L_{p,1}^{\tt Hi}({\boldsymbol{f}}_\phi^\dagger/\mathcal{K})\cdot(\gamma_{\rm cyc}-1)+\cdots,\\
{\rm tw}_{\Theta_\phi^{-1}}(L_p^{\tt MTT}({\boldsymbol{f}}_\phi))&=L_{p,0}^{\tt MTT}({\boldsymbol{f}}_\phi^\dagger)+L_{p,1}^{\tt MTT}({\boldsymbol{f}}_\phi^\dagger)\cdot(\gamma_{\rm cyc}-1)+\cdots,\\
{\rm tw}_{\Theta_\phi^{-1}}(L_p^{\tt MTT}({\boldsymbol{f}}_\phi\otimes\epsilon_\mathcal{K}))&=L_{p,0}^{\tt MTT}({\boldsymbol{f}}_\phi^\dagger\otimes\epsilon_\mathcal{K})+L_{p,1}^{\tt MTT}({\boldsymbol{f}}_\phi^\dagger\otimes\epsilon_\mathcal{K})\cdot(\gamma_{\rm cyc}-1)+\cdots,
\end{align*}
as power series in $\gamma_{\rm cyc}-1$, and note that by the $p$-adic Mellin transform we have
\[
\frac{d}{ds}L_p^{\tt MTT}({\boldsymbol{f}}_\phi,s)\biggr\vert_{s=k_\phi/2}\neq 0\quad\Longleftrightarrow\quad L_{p,1}^{\tt MTT}({\boldsymbol{f}}^\dagger_\phi)\neq 0
\]
(see \cite[(24)]{venerucci-p-conv}). The constant term $L_{p,0}^{\tt Hi}({\boldsymbol{f}}_\phi^\dagger/\mathcal{K})\in\mathbb{I}$ vanishes by Proposition~\ref{thm:hida-1}, and so the factorization $(\ref{eq:factor-cyc})$ yields the following equality up to unit in $\mathcal{O}_\phi^\times$:
\begin{equation}\label{eq:desc}
L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger_\phi/\mathcal{K})=L_{p,1}^{\tt MTT}({\boldsymbol{f}}_\phi^\dagger)\cdot L_{p,0}^{\tt MTT}({\boldsymbol{f}}^\dagger_\phi\otimes\epsilon_\mathcal{K}).
\end{equation}
Finally, since by definition $L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})\in\mathbb{I}$ agrees with the image of the linear term $L_{p,1}^{\tt Hi}({\boldsymbol{f}}^\dagger/\mathcal{K})_{{\rm ac}}$ in $(\ref{eq:2-exp})$ under the augmentation map $\mathbb{I}[[\Gamma_\mathcal{K}^{{\rm ac}}]]\rightarrow\mathbb{I}$, from Corollary~\ref{cor:I-GZ} specialized at the trivial character of $\Gamma_\mathcal{K}^{\rm ac}$ and $(\ref{eq:desc})$ we see that
\begin{align*}
\langle\mathfrak{Z}_0,\mathfrak{Z}_0\rangle^{\rm cyc}_{\mathcal{K},\mathbb I}\neq 0\quad&\Longrightarrow\quad
L_{p,1}^{\tt Hi}({\boldsymbol{f}}_\phi^\dagger/\mathcal{K})\neq 0,\quad\textrm{for almost all $\phi\in\mathcal{X}_a^o(\mathbb{I})$}\\\
&\Longrightarrow\quad L_{p,1}^{\tt MTT}({\boldsymbol{f}}^\dagger_\phi)\neq 0,\quad\quad\textrm{for almost all $\phi\in\mathcal{X}_a^o(\mathbb{I})$},
\end{align*}
concluding the proof of Theorem~\ref{thm:Gr-1}.
\end{proof}
\bibliographystyle{amsalpha}
|
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Wind Turbine Operation in Electric Power Systems
Wind Turbine Operation in Electric Power Systems pp 73-186 | Cite as
Mathematical Modeling of WTGS Components
Zbigniew Lubosny
Part of the Power Systems book series (POWSYS)
Models of reality (objects, processes, phenomena, etc.) are usually an element of the process presented in Fig. 5.1.
For a properly designed WTGS, within the wind turbine operating speed range, the curves showing eigenfrequencies as a function of the rotor speed should not cross. This prevents resonance self-excitation.Google Scholar
Therefore, these two values cannot be subtracted directly. When computing rotor-shaft twist, it is necessary to divide or multiply the relevant torsion angle.Google Scholar
The mean value of wind velocity at partial load is usually assumed as equal to the value which on the WTGS power characteristic p=f(v) gives the highest slope, e.g. v0=9 m/s.Google Scholar
In general, in WTGS modeling, the drive-train model described in sect.5.2 should be used.Google Scholar
Other types of transformation are also utilized.Google Scholar
The subscript on defines the on-state of the swich, e.g. thyristor, while the subscript off defines the off-state of the switch.Google Scholar
The network consists of conductance-type branches only (except for current and voltage sources).Google Scholar
In some types of control systems the WTGS characteristic between points 2 and 3 in Fig. 5.44 is defined by the function P = K•w with high slope (high value of coefficient K).Google Scholar
When a non-realistic step change of the wind velocity is assumed.Google Scholar
In fact, energy storage is not utilized for this purpose today.Google Scholar
Various types of WTGS utilize supervisory algorithms that can differ from that presented here. Those algorithms should be considered as examples only.Google Scholar
Taking into account the number of conditions being checked.Google Scholar
The controller reaction to a given input signal depends, of course, on the controller structure. Here, a controller with integration block is considered.Google Scholar
The control system can determine the turbine power-speed characteristics.Google Scholar
Whether the real power is proportional to the d-axis rotor current and the reactive power is proportional to the q-axis rotor current, or whether the opposite state takes place depends on the dq-reference frame definition.Google Scholar
Insuch a case, the maximum point power tracking (MPPT) scheme, based on the dp/dw=0 rule, is not utilized.Google Scholar
The generator stator winding can be (and is) switched between delta and star connection. The area of operation of the generator with the given connection is marked in Fig.5.61. The generator operation with the star connection reduces losses when the wind speed is lower, A t higher winds, when the rated power is achieved, the generator operates with delta-connected stator windings.Google Scholar
Because the power flows here in one direction only, converters with non-controlled rectifiers can be (and usually are) utilized.Google Scholar
The rated voltage of the network depends on the power system.Google Scholar
Usually, in such a type of WTGS, the power factor cos is controlled, which means that present wind turbines are not utilized for voltage control.Google Scholar
The power network is modeled by a set of algebraic equations, while the generating units (and sometime loads), FACTS and AC/DC systems are modeled by a set of differential and algebraic equations.Google Scholar
Its multi-modality is the positive feature of the multi-machine produce many more complex and difficult operating conditions (characteristic of the real system) for turbine and generator controllers than those obtainable in the single-machine system.Google Scholar
The presented controlled rectifier model described by (5.237) is a simplified model of the one defined by (5.243).Google Scholar
This assumed that s=3v I, which causes the inductive power to be positive (Q 0). When equation s=3VI is used, the inductive power becomes negative (Q 0).Coefficient 3 results from utilizing the phase-to-neutral rms voltageV.Google Scholar
That is 10kv,15kv, 20kv or 60kv, depending on the country grid type (voltage level and the WTGS location).Google Scholar
It is usually possible to make these networks closed.Google Scholar
When the feeding bus voltage value is imposed, the WTGES voltage and current can be computed iteratively by using the WTGS f(P,Q,V) characteristics.Google Scholar
Terminal voltage (and mechanical torque) is the generator model input and the current is the output. Inverse models of generator.Google Scholar
A power system model in which the dynamic elements (e.g. generators) are modeled with current as outpur and voltage as input is considered.Google Scholar
All quantities are per unit quantities.Google Scholar
For load flow computation purposes (especially for computing bulk power systems), the methods using sparse matrix techniques are widely utilized.Google Scholar
The problem can be solved also in rectangular coordinates a,b (V=Va+jvb)Google Scholar
There are set as initial in the computing procedure.Google Scholar
When the loads are modeled as constant admittances then the power P.Google Scholar
1.Dept. Electrical Power SystemsGdansk University of TechnologyGdanskPoland
Lubosny Z. (2003) Mathematical Modeling of WTGS Components. In: Wind Turbine Operation in Electric Power Systems. Power Systems. Springer, Berlin, Heidelberg
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{"url":"https:\/\/studysoup.com\/tsg\/1033365\/statics-and-mechanics-of-materials-5-edition-chapter-14-8-problem-14-87","text":"\u00d7\nGet Full Access to Statics And Mechanics Of Materials - 5 Edition - Chapter 14.8 - Problem 14-87\nGet Full Access to Statics And Mechanics Of Materials - 5 Edition - Chapter 14.8 - Problem 14-87\n\n\u00d7\n\n# Solve Prob. 14-86 for an element oriented 8 = 30\" clockwise\n\nISBN: 9780134382593 479\n\n## Solution for problem 14-87 Chapter 14.8\n\nStatics and Mechanics of Materials | 5th Edition\n\n\u2022 Textbook Solutions\n\u2022 2901 Step-by-step solutions solved by professors and subject experts\n\u2022 Get 24\/7 help from StudySoup virtual teaching assistants\n\nStatics and Mechanics of Materials | 5th Edition\n\n4 5 1 278 Reviews\n29\n5\nProblem 14-87\n\nSolve Prob. 14-86 for an element oriented 8 = 30\" clockwise.\n\nStep-by-Step Solution:\nStep 1 of 3\n\nENGR 121 B Lecture Notes for 10\/17\/2016 Spencer Kociba \u25cf Quiz on vector operations \u25cf Truth tables \u25cb Both variables must be true (logical output) X Y xandy 0 0 0 0 1 0 1 0 0 1 1 1 \u25cb At least one must be true X Y xory 0 0 0 0 1 1 1 0 1 1 1 1 \u25cb Only 1 is true X Y x+y 0 0 0 0 1 1 1 0 1 1 1 0 \u25cf If\/else statements \u25cb Nested ifelse command \u25a0 If condition 1 \u25cf Action 2 \u25a0 Elseif condition 2 \u25a0 Elseif condition 3 \u25cf Action 3 \u25a0 (etc) \u25a0 Else \u25cf Action final \u25a0 End\n\nStep 2 of 3\n\nStep 3 of 3\n\n## Discover and learn what students are asking\n\nCalculus: Early Transcendental Functions : Preparation for Calculus\n?In Exercises 1\u20134, find any intercepts. $$y=\\frac{x-3}{x-4}$$\n\nCalculus: Early Transcendental Functions : Conservative Vector Fields and Independence of Path\n?In Exercises 5 - 10, determine whether the vector field is conservative. $$\\mathbf{F}(x, y)=e^{x}(\\sin y \\mathbf{i}+\\cos y \\mathbf{j})$$\n\nStatistics: Informed Decisions Using Data : Estimating a Population Mean\n?Explain what is meant by degrees of freedom.\n\n#### Related chapters\n\nUnlock Textbook Solution","date":"2022-06-26 04:20:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.21543875336647034, \"perplexity\": 4291.800212209787}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103037089.4\/warc\/CC-MAIN-20220626040948-20220626070948-00751.warc.gz\"}"}
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<title>Range Checking - Debugging with GDB</title>
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Previous: <a rel="previous" accesskey="p" href="Type-Checking.html#Type-Checking">Type Checking</a>,
Up: <a rel="up" accesskey="u" href="Checks.html#Checks">Checks</a>
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<h4 class="subsection">15.3.2 An Overview of Range Checking</h4>
<p>In some languages (such as Modula-2), it is an error to exceed the
bounds of a type; this is enforced with run-time checks. Such range
checking is meant to ensure program correctness by making sure
computations do not overflow, or indices on an array element access do
not exceed the bounds of the array.
<p>For expressions you use in <span class="sc">gdb</span> commands, you can tell
<span class="sc">gdb</span> to treat range errors in one of three ways: ignore them,
always treat them as errors and abandon the expression, or issue
warnings but evaluate the expression anyway.
<p>A range error can result from numerical overflow, from exceeding an
array index bound, or when you type a constant that is not a member
of any type. Some languages, however, do not treat overflows as an
error. In many implementations of C, mathematical overflow causes the
result to “wrap around” to lower values—for example, if <var>m</var> is
the largest integer value, and <var>s</var> is the smallest, then
<pre class="smallexample"> <var>m</var> + 1 ⇒ <var>s</var>
</pre>
<p>This, too, is specific to individual languages, and in some cases
specific to individual compilers or machines. See <a href="Supported-Languages.html#Supported-Languages">Supported Languages</a>, for further details on specific languages.
<p><span class="sc">gdb</span> provides some additional commands for controlling the range checker:
<p><a name="index-set-check-range-850"></a><a name="index-show-check-range-851"></a>
<dl>
<dt><code>set check range auto</code><dd>Set range checking on or off based on the current working language.
See <a href="Supported-Languages.html#Supported-Languages">Supported Languages</a>, for the default settings for
each language.
<br><dt><code>set check range on</code><dt><code>set check range off</code><dd>Set range checking on or off, overriding the default setting for the
current working language. A warning is issued if the setting does not
match the language default. If a range error occurs and range checking is on,
then a message is printed and evaluation of the expression is aborted.
<br><dt><code>set check range warn</code><dd>Output messages when the <span class="sc">gdb</span> range checker detects a range error,
but attempt to evaluate the expression anyway. Evaluating the
expression may still be impossible for other reasons, such as accessing
memory that the process does not own (a typical example from many Unix
systems).
<br><dt><code>show range</code><dd>Show the current setting of the range checker, and whether or not it is
being set automatically by <span class="sc">gdb</span>.
</dl>
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Talent management | July/August 2020
Investing in employee development is a must for companies today
By Journal of Property Management staff
The working world is getting remade, from globalization and new technologies to shifting employee demographics and work models. Such change was prevalent before the coronavirus reshaped the world and altered how companies and employees view remote work forever. Now, in the face of even more rapid shifts, organizations need skilled, capable workers to meet nascent challenges.
Training is the way forward. Of all HR expenditures in 2019, a PayScale report found that professional development was the single biggest area of investment.
That's for good reason. The global economy will need to reskill more than 1 billion people by 2030, according to the World Economic Forum. In the next two years alone, 42% of core skills required to perform existing jobs are expected to change. Failing to add these new skills could cost the global economy $11.5 trillion in potential GDP growth over the next decade, according to an Accenture report.
"The most successful companies realize they have to invest in their future."
—Shannon Alter, CPM, Leaders Exceed
The good news is that investing in employees can be a cure for that ailing bottom line. It boosts retention, increases profitability and helps create autonomy. "The most successful companies realize they have to invest in their future," says Shannon Alter, CPM, the owner of Leaders Exceed in Santa Ana, California. "And people are the future."
Retention deficit
COVID-19 has changed the labor market, and we're still getting clarity on the pandemic's effects. In property management, attracting and retaining employees are ongoing priorities for every company, no matter the economy, and some positions are a revolving door for talent. Prior to the pandemic, voluntary turnover was skyrocketing, and in pandemic conditions, you still have those employees to keep.
Companies struggling to keep employees might be neglecting the important truth that more than ever, today's employees want to learn and grow. Many workers value development as much, if not more, than salary. According to a 2019 Work Institute report of people who quit their jobs, 22% of employees left to improve their career development—the top reason for leaving. Only 9% left because of compensation and benefits. The real kicker? More than 75% of employees who quit were willing to stay at the company if only the company made efforts to keep them.
"If individuals are not seeing companies invest in them, they will seek out other organizations that will," says Leo Turley, CEO of H Two National, a real estate executive search firm in Charlotte, North Carolina.
The financial hit from turnover can't be understated. For lower-level employees, a single instance of turnover can cost roughly $3,300, according to a 2019 Seedco report. And those costs can climb substantially with more senior employees. In the U.S. alone, turnover cost companies an estimated $600 billion in 2018, according to a Work Institute report.
Companies need to let employees know that they care and have a professional development plan for them, Alter says. That should begin the second they walk in the door on their first day, when their buy-in and enthusiasm are the highest.
"I think that 99.9% of people want to do things the right way, and employees want to know how you do it—what your best practices and guidelines are," Alter says. "Training lets people know you're invested and willing to put the financial support behind them, which is crucial to retention."
Value adds
"We really feel that the CPM designation elevates the mindset of a regional manager to that of an asset manager."
—Topher Olsen, MFA, MEd, Roscoe Property Management, AMO
Of course, training isn't only about avoiding loss. It's also about generating value, and development programs can create engaged employees who are better at their jobs. That's especially true if organizations tie training directly to business goals.
"If an associate doesn't understand what the core of the organization is about, then they're not going to feel connected to it," says Topher Olsen, MFA, MEd, the vice president of learning and development at Roscoe Property Management, AMO, in Austin. He was previously the senior director of learning and culture at Alliance Residential Company, AMO.
One such program that Olsen oversaw was focused on developing the regional manager role, with the goal of having 90% of regional managers earn an IREM CPM designation. "We really feel that the CPM designation elevates the mindset of a regional manager to that of an asset manager," Olsen says.
In part, the IREM CPM courses help regional managers talk to clients at a much higher level, Olsen says. "Our clients are giving us more business because they feel more adequately supported due to the level of knowledge our CPMs are bringing to the table."
As a result of this ability to help the company's bottom line, many of its internal promotions, such as to regional vice president or vice president, have gone to regional managers who have earned the CPM designation, Olsen says.
Benefits abound
For Turley, having his employees generate greater sales numbers is a profit booster, which is, of course, a plus. But he also sees a personal benefit. If employees are capable, confident and engaged, he doesn't need to provide as much oversight, freeing Turley to spend more time on his own work. It's a win-win.
"If individuals are not seeing companies invest in them, they will seek out other organizations that will."
—Leo Turley, H Two National
"It is important to train employees so that one day the supervisor will be able to let them fly on their own," Turley says. "Supervisors are able to focus on other things—improving operations, new business development, etc."
Getting to this level of autonomy doesn't require grand gestures. Small training investments can reap big rewards. For Turley, the goal is to teach people how to think like leadership does. That can be accomplished by having employees sit it in on meetings with higher-ups, letting people see how senior leaders do their job. "We're a smaller organization and I tell my people, 'We could sit down today, and I could train 24 hours a day for a long, long time, and I may not touch on some of the experiences or tasks that you might encounter.'"
Instead, by teaching people how he thinks, all they have to do is put themselves in Turley's shoes, and they're probably going to make the right decision, he says.
Like any financial investment, training people is a risk. Employees might not master the skills. Or, worse yet, a company could spend all this money developing an employee only to see them bail for another job. Canceling training because of that, Olsen says, is leading with fear. "We want to lead with confidence."
"Why would I hold back on this amount of information that would prevent somebody from becoming the best version of themselves?" Topher asks. "If we hold back, we are doing ourselves a disservice."
Companies have to understand that, yes, people leave, Olsen says. "But we want to do everything possible for them to stay."
Issue: July/August 2020
Written by Journal of Property Management staff
In 2020, talent and property management took an unexpected turn
Training in progress
Property management degrees offer well-rounded preparation for the job
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Q: htaccess - direct jpeg requests to specific php page and pass through parameters I need to direct all jpeg requests from a particular folder resize on my website to a specific php script that resizes and compresses the image with the GD library, and at the same time passes parameters to the script. I have no issue with the GD resize/compressing part but it is directing all requests to jpegs to the php script with the necessary parameters.
Is it correct that the best way of doing so is by using htaccess's QUERY_STRING?
In my php script i have the following parameters needed:
path
size
ratio
So on my page i would call something like:
<img src="http://www.website.com/[folder]/[size]/[ratio]/[path]"/>
i.e
<img src="http://www.website.com/resize/200/square/the-image.jpg"/>
I imagine my htaccess file would be something like so but i don't know what rule i would use:
RewriteEngine On
RewriteCond %{QUERY_STRING} (.*)
RewriteRule XXXXXXXXXXXXXXXXXXXX
A: You can use this code in your DOCUMENT_ROOT/.htaccess file:
RewriteEngine On
RewriteBase /
RewriteRule ^(resize)/(\d+)/([^/]+)/(.+)$ $1/index.php?size=$2&ratio=$3&path=$4 [L,QSA]
This is assuming /resize/index.php is the php file handling all the images. This will let you have image tag as;
<img src="http://www.website.com/resize/200/square/the-image.jpg"/>
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{"url":"http:\/\/colinreimerdawson.com\/stat209\/labs\/lab2.html","text":"Since we want to be working with data, let\u2019s load a dataset into our environment.\n\nThere are three main ways to load a dataset for use in R:\n\n\u2022 Accessing data that is bundled with an R package\n\u2022 Reading in data that is posted on the web\n\n### Accessing data from a package\n\nSome R packages come with example datasets. There is one such package, Lock5Data developed as a companion to our textbook for this course. I will also use examples from the Stat2Data package.\n\n1. Create a new code chunk, in which you load the Lock5Data package using the library() command. Run the line. Note: R is case-sensitive, so commands do not work (or worse, may run but do something different) if you use different capitalization. Pay attention to detail!\n\nOnce we have loaded the package that contains the dataset we want, we then (usually) have to load the dataset. We can do this using the data() command.\n\nFor example, the \"Pollster08\" dataset provided by the Stat2Data package contains data about some political polls taken during the 2008 U.S. Presidential campaign.\n\nWe can load the package and then the dataset as follows:\n\nlibrary(Stat2Data) #loads the package\ndata(Pollster08) #loads the dataset\n\nNotice that Pollster08 now appears in our Environment.\n\nIn the future, any time you want to use a dataset from an R package, you will first need to load the package, then the dataset, as above.\n\n### Accessing Documentation\n\nFor packaged datasets like this, as well as for R functions, we can get some information about the data using R\u2019s help interface. At the console, type\n\nhelp(Pollster08)\n\nThis will pop up a documentation window in the Help tab in the lower right. Here you can read about the source of this data, what each case is, what each variable means, how it is measured, etc.\n\nThis documentation of a dataset is what\u2019s called the code book that accompanies the data. When collecting your own data, you should also create an accompanying code book (which might just be a text file) to give context to anyone using your dataset.\n\n### Reading data directly from the web\n\nReading data bundled from an R package is nice for a course, but in \u201creal life\u201d the data we want is usually not so conveniently packaged. More often it is stored in a file somewhere; either on the web or on our computer itself.\n\nWe can load a data set from a file on the web if we know the URL.\n\nThe read.file() command makes this simple for well formatted data files.\n\nThis command instructs R to fetch the data from the given URL, and put it in an object in our environment called Depression, which will now appear in the Environment pane.\n\nDepression <- read.file(\"http:\/\/colindawson.net\/data\/depression.csv\")\n\nNote that the read.file() produces a result; namely a data frame object, which we store in a variable which we give a name that tells us something about what that data frame contains data about.\n\nThis is a bit different from the way the data() command worked for loading data from an R package. There, we didn\u2019t have to create a variable in R; there was already one existing; we just had to tell R that we were going to be using it, and that it should be added to our environment.\n\nYou can also point to a file on your computer by supplying the file\u2019s location on the computer as a path, in place of the URL.\n\n#### Absolute paths\n\nIn your \u201cFiles\u201d tab you should see a folder called stat209. Double click it to see its contents in the Files tab. It should have several subdirectories, one of which is called data. If you double click on data you should see the file depression.csv inside.\n\nIf instead of a URL you put the path to this file, which should be \"~\/stat209\/data\/depression.csv\" inside the parentheses in read.file(), it would read in this file. The ~ symbol represents your \u201cHome\u201d folder.\n\nA location that starts with a forward slash \/ or a tilde ~ is an absolute path. That is, it doesn\u2019t matter what directory (folder) you\u2019re currently working in; it will look in that specific location.\n\n#### Relative paths\n\nSometimes it is convenient to specify a file using a relative path instead. Particularly when you want your code to run on another computer, where your files will not be in the exact same location they\u2019re in on the computer you were working on.\n\nInstead, we can tell R to look in a location which is relative to the location you\u2019re currently working in (which might be the location where your script is).\n\nFor example, if I am currently working in my home directory (abbreviated in paths as ~), I could replace the absolute path above with \"stat209\/data\/depression.csv\". This tells R to look for a folder called stat209 inside the current working directory; a folder called data inside that one; and a file called depression.csv inside that.\n\nIf my file were directly inside my home folder, I could just type \"depression.csv\" inside read.file().\n\nIf you have been following along, the code below will probably give you an error:\n\n## Probably generates an error\nDepression <- read.file(\"stat209\/data\/depression.csv\")\n\nThat\u2019s because R is not looking inside your home directory, but inside your current working directory, which should be where your project is located.\n\nR interprets relative paths relative to a starting point called your working directory. It is not, unfortunately, necessarily the same as the directory you are currently viewing in the Files tab.\n\nYou can see what directory is currently set as your working directory (relative to which all relative paths are interpreted) by typing getwd() in the console.\n\nYou can change your working directory (for example, to ~\/stat209) by typing setwd(\"~\/stat209\") (with the quotes).\n\n1. Make a code chunk in which you change your working directory to stat209, and then write a command using a relative path to read in the depression.csv dataset using the read.file() command. Don\u2019t forget to store the resulting data frame in a named variable.\n\n## Interacting with data sets\n\nThe Pollster08 data that we read in earlier consists of several variables about various opinion polls taken during the 2008 U.S. Presidential election.\n\nYou should see that in the Environment tab that Pollster08 consists of 102 observations (cases), each with 11 variables recorded.\n\nWe could have R print out the entire data table by simply typing the name of the data object:\n\nPollster08\n\nHowever printing the whole dataset is not that useful, especially if the data contains a lot of cases.\n\nOne advantage of RStudio is that it comes with a built-in data viewer.\n\n1. Click on the name Pollster08 in the Environment pane. This will bring up a \u201cspreadsheet\u201d-style display of the data set in the Data Viewer (upper left window). What you should see are 11 columns, each row representing a poll, and each column representing a variable (PollTaker, PollDates, etc.) The first entry in each row is simply the row number. Use the scrollbar on the right side of the console window to examine the complete data set.\n\nYou can close the data viewer by clicking on the \u201cX\u201d in the upper lefthand corner.\n\nR has stored this data in a data frame. This is the sort of table we have sketched in class: each row is a case, each column is a variable.\n\nMost of what we will do in R will consist, in one way or another, of taking actions on data frames.\n\n1. Run the command below, and then examine its structure. Identify all of the function names, argument names, and argument values. (Note that sometimes argument values, particularly for the first argument of a function, are supplied without an argument name: R can assign values to names according to the order they\u2019re given in. However, the reverse is never true: if you give an argument name you need to supply a value.) What does this command appear to do?\npull(Pollster08, var = PollTaker)\n\n### Getting Help\n\nThe same help interface works for functions just as for data frames. We can get more information about the pull() or head() functions by typing\n\nhelp(pull)\nhelp(head)\n\nat the console. We could also have done\n\n?pull\n?head\n\nto achieve the same result.\n\n1. Look up the nrow and ncol functions using the Help interface. Add a sentence to this document briefly describing what they do.\n\n1. Load the HoneybeeCircuits dataset from the Lock5Data package (see \u201cReading in data from a package\u201d). Use the Help interface to look up what the dataset is about and add a brief summary in the text of this document below this question. Then, write an R command that returns the number of cases in the data (hint: use what you found in Exercise 9).\n\n2. Make a code chunk below and add code to read in the women_in_labor_force.csv dataset from the data directory on your server account using the read.file() with an absolute path, storing the result as WomenInLabor. Make a second chunk to read the same dataset with a relative path from your project directory, storing the result as WomenInLabor2. What are the cases and variables in this dataset?\n\n3. Knit your modified .Rmd file. Save a copy of your edited version of this lab (both the .Rmd and Knitted .html files) in the hw2 turnin folder.\n\n## Environment and Session Information\n\n\u2022 File creation date: 2021-06-03\n\u2022 R version 3.6.0 (2019-04-26)\n\u2022 R version (short form): 3.6.0\n\u2022 mosaic package version: 1.5.0\n\u2022 tidyverse package version: 1.3.1\n## R version 3.6.0 (2019-04-26)\n## Platform: x86_64-pc-linux-gnu (64-bit)\n## Running under: Ubuntu 18.04.5 LTS\n##\n## Matrix products: default\n## BLAS: \/usr\/lib\/x86_64-linux-gnu\/blas\/libblas.so.3.7.1\n## LAPACK: \/usr\/lib\/x86_64-linux-gnu\/lapack\/liblapack.so.3.7.1\n##\n## locale:\n## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C\n## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8\n## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8\n## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C\n## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C\n##\n## attached base packages:\n## [1] stats graphics grDevices utils datasets methods base\n##\n## other attached packages:\n## [1] Stat2Data_2.0.0 mosaic_1.5.0 Matrix_1.2-17\n## [4] mosaicData_0.17.0 ggformula_0.9.1 ggstance_0.3.1\n## [7] lattice_0.20-38 forcats_0.5.1 stringr_1.4.0\n## [13] tidyr_1.1.3 tibble_3.1.1 ggplot2_3.3.3\n## [16] tidyverse_1.3.1\n##\n## loaded via a namespace (and not attached):\n## [1] ggdendro_0.1-20 httr_1.4.2 jsonlite_1.7.2 splines_3.6.0\n## [5] modelr_0.1.8 shiny_1.3.2 assertthat_0.2.1 cellranger_1.1.0\n## [9] yaml_2.2.0 ggrepel_0.8.1 pillar_1.6.0 backports_1.1.4\n## [13] glue_1.4.2 digest_0.6.21 promises_1.0.1 rvest_1.0.0\n## [17] colorspace_1.4-1 htmltools_0.3.6 httpuv_1.5.1 pkgconfig_2.0.3\n## [21] broom_0.7.6 haven_2.4.1 xtable_1.8-4 scales_1.0.0\n## [25] later_0.8.0 generics_0.0.2 ellipsis_0.3.0 withr_2.4.2\n## [29] lazyeval_0.2.2 cli_2.5.0 magrittr_2.0.1 crayon_1.4.1\n## [33] readxl_1.3.1 mime_0.7 evaluate_0.14 fs_1.3.1\n## [37] fansi_0.4.0 MASS_7.3-51.4 xml2_1.3.2 tools_3.6.0\n## [41] hms_1.0.0 lifecycle_1.0.0 munsell_0.5.0 reprex_2.0.0\n## [45] compiler_3.6.0 rlang_0.4.11 grid_3.6.0 rstudioapi_0.13\n## [49] htmlwidgets_1.3 crosstalk_1.0.0 mosaicCore_0.6.0 rmarkdown_2.5\n## [53] gtable_0.3.0 DBI_1.0.0 R6_2.4.0 gridExtra_2.3\n## [57] lubridate_1.7.10 knitr_1.25 utf8_1.1.4 stringi_1.4.3\n## [61] Rcpp_1.0.2 vctrs_0.3.8 leaflet_2.0.2 dbplyr_2.1.1\n## [65] tidyselect_1.1.1 xfun_0.19","date":"2021-10-25 10:32:32","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\": 0.4004817605018616, \"perplexity\": 3231.5628358570534}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587659.72\/warc\/CC-MAIN-20211025092203-20211025122203-00351.warc.gz\"}"}
| null | null |
/* eslint-env browser */
// 'use strict';
//import * as wavesAudio from 'waves-audio';
import * as wavesLoaders from 'waves-loaders';
import WaveSurfer from 'wavesurfer.js';
import GranularBackend from './backend_granular_player';
import SegmentBackend from './backend_segment_player';
//let wavesAudio = require('../waves-audio.umd.js');
import * as wavesAudio from '../../waves-audio/src/index';
const globalTransporter = new wavesAudio.Transport();
const globalPlayControl = new wavesAudio.PlayControl(globalTransporter);
const Modulo = (num, mod) => ((num % mod) + mod) % mod;
let globalTransportedEngines = [];
// Init & load audio file
document.addEventListener('DOMContentLoaded', () => {
const button1 = document.querySelector('[data-action="play1"]');
const button2 = document.querySelector('[data-action="stop1"]');
const loader = new wavesLoaders.SuperLoader(); // instantiate loader
const assets = [
'./assets/piano_v2.json',
'./assets/footstomps_v2.json',
//'./assets/3_4_guitar-loop_v2.json',
'./assets/strings_v2.json',
];
/* Resampling Slider */
let r = document.createElement("INPUT");
r.setAttribute("type", "range");
r.min = -40;
r.max = 40;
r.step = 10;
r.value = 0;
document.getElementById('waveform').appendChild(r);
// load audio and marker files
loader.load(assets).then((jsonfiles) => {
console.log(jsonfiles);
var sequenceData = {
"time": [0, 6, 12, 18, 24, 30, 36, 42, 48, 54],
"identifier": ["Am", "G", "C", "F", "Dm", "Am", "G", "C", "F", "Dm"],
"duration": [3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0],
"offset": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
};
/* Setup Sequencer */
let sequencer = new wavesAudio.SequencerEngine({
// buffer not used, but requires more changes in source to be able to remove.
// perhaps I will use it as a recording sink...
buffer: wavesAudio.audioContext.createBuffer(2, 22050, 44100),
positionArray: sequenceData.time,
durationArray: sequenceData.duration,
offsetArray: sequenceData.offset,
cyclic: false,
callback: function(seqIndex) {
// Ok, so right now just outputting random chords,
// but this will be controlled by the sequencer in the future...
console.log("Sequencer says: " + sequenceData.identifier[seqIndex]); // print sequencer chords, (not what's playing, though)
console.log(globalTransportedEngines);
let pos = globalTransporter.currentPosition;
var rand = Math.floor(Math.random() * 6);
let transportedObject1 = globalTransportedEngines[2];
transportedObject1.setBoundaries(pos, 6, rand * -12, 1);
let transportedObject2 = globalTransportedEngines[1];
transportedObject2.setBoundaries(pos, 6, 0, 1);
let transportedObject3 = globalTransportedEngines[0];
transportedObject3.setBoundaries(pos, 6, rand * -12, 1);
}
});
globalTransporter.add(sequencer);
/* Add wavesurfer object for each json-file */
//let index = 0;
for (const jsonitem of jsonfiles) {
const waveform = document.createElement('div');
const demo = document.getElementById('waveform');
demo.appendChild(waveform);
let wavesurfer = WaveSurfer.create({
container: waveform,
plugins: [
//GranularBackend.create({
SegmentBackend.create({
transport: globalTransporter,
playctrl: globalPlayControl,
json: jsonitem
})
]
});
console.log(jsonitem);
const fname = jsonitem.file_metadata.identifiers.url;
console.log('FileName: ' + fname);
//
wavesurfer.on('ready', () => {
console.log('ready');
//let engine = wavesurfer.backend.getEngine();
//let transportedObject = globalTransporter.add(engine, 99999);
let engine = wavesurfer.backend.getTransported();
let transportedObject = globalTransporter.add(engine, 99999);
console.log("adding engine at transport pos 99999");
transportedObject.advancePosition = function(time, position, speed) {
position = this.__offsetPosition + this.__engine.advancePosition(time, position - this.__offsetPosition, speed);
if (speed > 0 && position < this.__endPosition || speed < 0 && position >= this.__startPosition)
return position;
// Returning Infinite causes a halt, again trying a big number instead.
return 99999; //Infinity * speed;
}.bind(transportedObject);
globalTransportedEngines.push(transportedObject);
});
wavesurfer.load(fname);
/* Gain Slider */
let x = document.createElement("INPUT");
x.setAttribute("type", "range");
x.min = 0;
x.max = 1;
x.step = 0.01;
x.value = 1;
demo.insertBefore(x, waveform);
x.oninput = () => {
wavesurfer.setVolume(x.value);
//wavesurfer.backend.transportedSegmentEngine.polyCutoff = x.value;
};
r.addEventListener("mouseup", function() {
wavesurfer.backend.transportedEngine.resampling = r.value;
console.log(r.value);
}, false);
}
const slider = document.querySelector('#slider');
slider.oninput = () => {
const playBackRate = Number(slider.value) / 100.0;
globalPlayControl.speed = playBackRate * 2;
//globalPlayControl.seek(slider.value/10);
};
globalPlayControl.speed = 1.5 * 2;
slider.value = globalPlayControl.speed * 100 / 2;
button1.addEventListener('click', () => {
globalPlayControl.seek(0);
globalPlayControl.loop = true;
globalPlayControl.setLoopBoundaries(0, 60);
globalPlayControl.start();
});
button2.addEventListener('click', () => {
globalPlayControl.stop();
});
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,168
|
SEO, or Search Engine Optimization, is an important and often misunderstood aspect of web design. Search engine findability is obviously an important factor in the success of any site. You want a person who is conducting a search for the terms that match the products or services that your company offers to find your website, right?
That makes perfect sense, but the application of SEO practices is unfortunately open to abuse and outright scams, either by outdated practitioners who are not up-to-date on the latest trends and industry best practices, or actual scam artists who are out to take your money in exchange for services that could actually harm, rather than help, your website.
Let's take a look at what keywords in web design are, including how they can help your site and what practices you should avoid.
In the most general of terms, keywords in HTML are words you are targeting on a web page. They are typically short phrases that represent what the page is about. They are also the words that someone might type into a search engine to find your page.
In general, HTML keywords are found whether you intend them to be there or not. Keywords are just text like any other text, and when a search engine views your page, it looks at the text and attempts to make a decision regarding what the page is about based on the text it sees. It reads the content of your page and sees what important words are contained in that text.
The best way to use keywords is by making sure that they are naturally included on your page. You do not want to overdo this, however. Remember, your content should be written for humans, not search engines. The text should read and feel natural and not be peppered with every possible keyword. Not only does overusing keyword, called keyword stuffing, make your site hard to read, but it can also get your site penalized by search engines so your site is actually pushed deeper into search engine results.
Search engines today don't use the keywords meta tag in their ranking algorithms because it can be manipulated so easily by the web page writer. In other words, many page writers used to put random keywords into the keywords tag, in the hopes that the page would be optimized for those (probably more popular) phrases. If you are speaking to someone about SEO and they talk about meta keywords being important, they are probably out of touch with current practices!
If you are going to include metadata on your web pages, ignore the keywords tag and instead use the meta description tag. This is metadata that nearly all search engine use to describe your web page in their indexes. It does not impact rankings, but it does impact what a person sees when your listing appears. That extra information could mean the difference from a customer clicking on your site for information or on someone else's.
Instead of relying on the keywords meta tag, think about keywords in the actual content of your web page. These are the terms that the search engines will use to evaluate what the page is about, and thus where it should appear in their search results. First, write content that is useful, and then focus on search engine optimization to optimize that content for the keywords you are focusing on for that page.
When you are choosing the keyword phrase for a web page, you should first focus on just one phrase or main idea per web page. It is not a good idea to try to optimize one web page for many different things, as this could confuse not only the search engines but more importantly your readers.
One strategy that may seem counterintuitive but works well for many sites is to choose "long-tail" keywords. These are the keywords that don't receive huge amounts of search traffic. Because they aren't as popular with searchers, they aren't as competitive, and it's possible to rank higher in a search for them. This gets your site noticed and you gain credibility. As your site gains credibility, it will start ranking higher for the popular terms.
One thing to be aware of is that Google and other search engines are really good at understanding synonyms. This means that you do not need to include every variation of a keyword on your site. Google will often know that certain phrases mean the same thing.
For example, you may optimize a page for the phrase "mold cleanup," but Google knows that "mold removal" and "mold abatement" mean the same thing, so your site will likely rank for all 3 terms even if only 1 is actually included in the site's content.
Another way to determine the keywords in your HTML is to use a keyword generator. Many online tools will analyze your web page content and tell you how many times various phrases are used on your page. These are typically called keyword density analyzers. Check out keyword density tools recommended by others online.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,937
|
\section{Introduction}
Dynamical systems are called $\mathcal{PT}$-symmetric if they are invariant with
respect to the combined parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) transformations.
A non-Hermitian $\mathcal{PT}$-symmetric linear operator may have a real spectrum
and may define a unitary time evolution of the linear $\mathcal{PT}$-symmetric system \cite{bender}. A
non-Hamiltonian $\mathcal{PT}$-symmetric nonlinear system may have a continuous family
of stationary states parameterized by their energy \cite{ptreview, suchkov}.
Originated in quantum mechanics~\cite{bender2005,mostafazadeh}, the topic of
$\mathcal{PT}$-symmetry was later boosted by applications in optics~\cite{makris, musslimani}
and other areas of physics~\cite{benderExp,rubinsteinExp,Schindler}.
Recent applications include single-mode $\cal{PT}$ lasers~\cite{feng14,hodaei}
and unidirectional reflectionless $\cal{PT}$-symmetric metamaterials at
optical frequencies~\cite{feng13}.
The non-Hermitian $\mathcal{PT}$-symmetric linear operator may lose real eigenvalues at
the so-called $\cal{PT}$-phase transition point, where two real eigenvalues coalesce and
bifurcate off to the complex plane, creating instability. A stationary state
of the non-Hamiltonian $\mathcal{PT}$-symmetric nonlinear system may
exist beyond the $\mathcal{PT}$-phase transition point but may become spectrally unstable
due to coalescence of purely imaginary eigenvalues and their bifurcation off to the complex plane.
Examples of such instabilities have been identified for many $\mathcal{PT}$-symmetric
linear and nonlinear systems \cite{bender,ptreview, suchkov}.
In Hamiltonian systems, instabilities arising due to coalescence of purely imaginary eigenvalues
can be predicted by computing the {\em Krein signature} for each eigenvalue, which is defined as
the sign of the quadratic part of Hamiltonian restricted to the associated eigenspace of the linearized problem.
When two purely imaginary eigenvalues coalesce,
they bifurcate off to the complex plane only if they
have opposite Krein signatures prior to collision~\cite{kapitulabook}.
The concept of Krein signature was introduced by MacKay~\cite{mackay} in the case of
finite-dimensional Hamiltonian systems, although the idea dates back to the works of Weierstrass~\cite{weierstrass}.
There have been several attempts to extend the concept of Krein signature
to the non-Hamiltonian $\mathcal{PT}$-symmetric systems.
Nixon and Yang~\cite{yang} considered the linear Schr\"{o}dinger equation
with a complex-valued $\mathcal{PT}$-symmetric potential and introduced
the indefinite $\mathcal{PT}$-inner product with the induced $\mathcal{PT}$-Krein signature,
in the exact correspondence with the Hamiltonian-Krein signature.
In our previous works \cite{CP1,CP2}, we considered a Hamiltonian
version of the $\mathcal{PT}$-symmetric system of coupled oscillators
and introduced Krein signature of eigenvalues by using the corresponding Hamiltonian.
In the recent works \cite{AB,PTL,SS}, a coupled non-Hamiltonian $\mathcal{PT}$-symmetric system
was considered and the linearized system was shown to be block-diagonalizable
to the form where Krein signature of eigenvalues can be introduced.
All these cases were too special, the corresponding Krein signatures
cannot be extended to a general $\mathcal{PT}$-symmetric system.
In this work, we address the following nonlinear Schr\"{o}dinger's equation (NLSE) with
a general complex potential:
\begin{equation}
i\partial_t \psi + \partial^2_x \psi - ( V(x) + i\gamma W(x) )\psi + g|\psi|^2 \psi = 0,
\label{NLS}
\end{equation}
where $\gamma\in\mathbb{R}$ is a gain-loss parameter, $g=+1$ ($g=-1$) defines
focusing (defocusing) nonlinearity, and the real potentials $V$ and $W$
satisfy the even and odd symmetry, respectively:
\begin{equation}
\label{potentials}
V(x) = V(-x), \quad W(-x) = -W(x), \quad x \in \mathbb{R}.
\end{equation}
In quantum physics, the complex potential $V + i \gamma W$ is used to
describe effects observed when quantum particles are loaded in an open
system~\cite{wunner,dast}. The intervals with
positive and negative imaginary part correspond to the gain and loss of quantum particles,
respectively. When gain exactly matches loss, which happens under the symmetry
condition (\ref{potentials}), the potential $V + i \gamma W$ is
$\cal{PT}$-symmetric with respect to the parity operator $\mathcal{P}$ and
the time reversal operator $\cal{T}$ acting on a function $\psi(x,t)$ as follows:
\begin{equation}
\label{operators}
\mathcal{P} \psi(x,t) = \psi(-x,t), \quad
\mathcal{T} \psi(x,t) = \overline{\psi(x,-t)}.
\end{equation}
The NLSE (\ref{NLS}) is $\mathcal{PT}$-symmetric under the condition (\ref{potentials})
in the sense that if $\psi(x,t)$ is a solution to (\ref{NLS}), then
\begin{equation*}
\widetilde{\psi}(x,t) = \mathcal{PT} \psi(x,t) = \overline{\psi(-x,-t)}
\end{equation*}
is also a solution to (\ref{NLS}).
The NLSE (\ref{NLS}) with a $\mathcal{PT}$-symmetric potential is also used in the paraxial nonlinear optics.
In that context, time and space have a meaning of longitudinal and transverse coordinates,
and complex potential models the complex refractive index~\cite{delgado}.
Another possible application of the NLSE (\ref{NLS})
is Bose-Einstein condensate, where it models the dynamics of the self-gravitating boson gas trapped in a
confining potential $V$. Intervals, where $W$ is positive and negative,
allow one to compensate atom injection and particle leakage, correspondingly~\cite{wunner}.
Here we deal with the stationary states in the NLSE (\ref{NLS}) and introduce
Krein signature of isolated eigenvalues in the spectrum of their linearization.
We prove that the necessary condition for the onset
of instability of the stationary states from a defective eigenvalue of algebraic multiplicity two
is the {\em opposite} Krein signature of the two simple isolated eigenvalues prior to
their coalescence. Compared to the Hamiltonian system in \cite{CP1}
or the linear Schr\"{o}dinger equation in \cite{yang}, the Krein signature
of eigenvalues cannot be computed from the eigenvectors in the linearized
problem, as the adjoint eigenvectors need to be computed
separately and the sign of the adjoint eigenvector needs to be chosen by a continuity argument.
We show how to compute Krein signature numerically for several examples of
the $\cal{PT}$-symmetric potentials. In the focusing case $g = 1$, we consider the Scarf II
potential studied in~\cite{ahmed,wadati,Kev,yang} with
\begin{equation}
\label{pot-Wadati}
V(x) = -V_0 \sech^2(x), \quad
W(x) = \sech(x)\tanh(x),
\end{equation}
where $V_0 > 0$ is a parameter. This potential is a complexification of the real hyperbolic
Scarf potential~\cite{Bagchi}. The nonhyperbolic version of the latter first appeared in~\cite{Scarf},
where the linear Schr\"{o}dinger equation with Scarf potential was solved.
In the defocusing case $g = -1$,
we consider the confining potential studied in
\cite{kevrekidis} with
\begin{equation}
\label{pot-BEC}
V(x) = \Omega^2 x^2, \quad W(x) = xe^{-\frac{x^2}{2}},
\end{equation}
where $\Omega > 0$ is a parameter. In agreement with the theory,
we show that the coalescence of two isolated eigenvalues
in the linearized problem associated with the stationary states in the NLSE (\ref{NLS})
leads to instability only if the Krein signatures of the
two eigenvalues are opposite to each other.
The paper is organized as follows. Section \ref{sec-theory} introduces the stationary states,
eigenvalues of the linearization, and the Krein signature of eigenvalues for the NLSE (\ref{NLS})
under some mild assumptions. Section \ref{sec-proof} gives the proof of the necessary condition
for the instability bifurcation from a defective eigenvalue of algebraic multiplicity two.
Section \ref{sec-numerics} explains details of the numerical technique.
Section \ref{sec-examples} presents outcomes of numerical approximations
for the two potentials (\ref{pot-Wadati}) and (\ref{pot-BEC}). Section \ref{sec-conclusion}
concludes the paper with open questions.
\vspace{0.25cm}
{\bf Acknowledgements.} We thank P.G. Kevrekidis for suggesting the problem back in 2012 and for useful discussions.
A. Chernyavsky is supported by the McMaster graduate scholarship. D.E. Pelinovsky is supported from
the state task of Russian Federation in the sphere of scientific activity (Task No. 5.5176.2017/8.9).
\section{Stationary states, eigenvalues, and Krein signature}
\label{sec-theory}
Let us define the stationary state of the NLSE (\ref{NLS}) by $\psi(x,t) = \Phi(x) e^{-i \mu t}$, where $\mu \in \mathbb{R}$
is a parameter. In the context of BECs, $\mu$ has the meaning of the chemical potential~\cite{dast}.
The function $\Phi(x) : \mathbb{R} \to \mathbb{C}$ is
a suitable solution of the stationary NLSE in the form
\begin{equation}
-\Phi''(x) + ( V(x) + i\gamma W(x) ) \Phi(x) - g |\Phi(x)|^2 \Phi(x) = \mu \Phi(x), \quad x \in \mathbb{R}.
\label{NLSstat}
\end{equation}
We say that $\Phi$ is a $\mathcal{PT}$-symmetric stationary state if $\Phi$ satisfies the $\mathcal{PT}$ symmetry:
\begin{equation}
\label{PT-sym}
\Phi(x) = \mathcal{PT} \Phi(x) = \overline{\Phi(-x)}, \quad x \in \mathbb{R}.
\end{equation}
In addition to the symmetry constraints on the potentials $V$ and $W$ in (\ref{potentials}),
our basic assumptions are given below. Here and in what follows, we denote the Sobolev space of square integrable functions
with square integrable second derivatives by $H^2(\mathbb{R})$ and the weighted
$L^2$ space with a finite second moment by $L^{2,2}(\mathbb{R})$.
\begin{assumption}
\label{assumption-1}
We assume that the linear Schr\"{o}dinger operator $L_0 := -\partial_x^2 + V(x)$ in $L^2(\mathbb{R})$
admits a self-adjoint extension with a dense domain $D(L_0)$ in $L^2(\mathbb{R})$.
\end{assumption}
\begin{remark}
If $V \in L^2(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ as in (\ref{pot-Wadati}),
then Assumption~\ref{assumption-1} is satisfied with $D(L_0) = H^2(\mathbb{R})$
(see \cite{sigal}, Ch. 14, p.143).
If $V$ is harmonic as in (\ref{pot-BEC}),
then Assumption~\ref{assumption-1} is satisfied with
$D(L_0) = H^2(\mathbb{R}) \cap L^{2,2}(\mathbb{R})$ (see ~\cite{heffler}, Ch. 4, p.37).
\end{remark}
\begin{assumption}
\label{assumption-2}
We assume that $W$ is a bounded and exponentially decaying potential satisfying
\begin{equation*}
|W(x)| \leq C e^{-\kappa |x|}, \quad x \in \mathbb{R},
\end{equation*}
for some $C > 0$ and $\kappa > 0$.
\end{assumption}
\begin{remark}
Both examples in (\ref{pot-Wadati}) and (\ref{pot-BEC}) satisfy Assumption~\ref{assumption-2}.
By Assumption~\ref{assumption-2}, the potential $i \gamma W$ is
a relatively compact perturbation to $L_0$ (see \cite{reed4}, Ch. XIII, p.113).
This implies that the continuous spectrum of $L_0 + i\gamma W$ is the same as $L_0$.
If $V \in L^2(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$, then the continuous spectrum of $L_0$
is located on the positive real line. If $V$ is harmonic, then the continuous spectrum of $L_0$
is empty (see~\cite{reed4}, Ch. XIII, Theorem 16 on p.120).
\end{remark}
\begin{assumption}
\label{assumption-3}
We assume that for a given $\mu \in \mathbb{R}$, there exist $\gamma_* > 0$ and
a bounded, decaying, and $\mathcal{PT}$-symmetric solution $\Phi \in D(L_0) \subset L^2(\mathbb{R})$
to the stationary NLSE (\ref{NLSstat}) with $\gamma \in (-\gamma_*,\gamma_*)$ satisfying (\ref{PT-sym}) and
\begin{equation*}
|\Phi(x)| \leq C e^{-\kappa |x|}, \quad x \in \mathbb{R},
\end{equation*}
for some $C > 0$ and $\kappa > 0$. Moreover, the map $(-\gamma_*,\gamma_*) \ni \gamma \mapsto \Phi \in D(L_0)$
is real-analytic.
\end{assumption}
\begin{remark}
Since the nonlinear equation~\eqref{NLSstat} is real-analytic in $\gamma$, the Implicit Function Theorem
(see~\cite{Zeidler}, Ch. 4, Theorem 4.E on p.250) provides real analyticity of the map
$(-\gamma_*,\gamma_*) \ni \gamma \mapsto \Phi \in D(L_0)$ as long as the Jacobian operator
\begin{equation}
\label{jacobian}
\mathcal{L} := \left[\begin{array}{cc}
-\partial_x^2 + V + i \gamma W - \mu - 2g |\Phi|^2 & -g\Phi^2 \\
-g\overline\Phi^2 & -\partial_x^2 + V - i \gamma W - \mu - 2g |\Phi|^2
\end{array}\right]
\end{equation}
is invertible in the space of $\mathcal{PT}$-symmetric functions in $L^2(\mathbb{R})$.
\end{remark}
\begin{remark}
Under Assumption~\ref{assumption-3}, we think about $\mu$ as a fixed parameter and $\gamma$ as a varying parameter
in the interval $(-\gamma_*,\gamma_*)$. The interval includes the Hamiltonian case $\gamma = 0$.
In the context of the example of $V$ in (\ref{pot-Wadati}), it will be more natural to fix the value of $\gamma$
and to consider the parameter continuation of $\Phi \in D(L_0)$ with respect to $\mu$.
The results are analogous to what we present here under Assumption \ref{assumption-3}.
\end{remark}
We perform the standard linearization of the NLSE (\ref{NLS})
near the stationary state $\Phi$ by substituting
\begin{equation*}
\psi(x,t) = e^{-i\mu t} \left[ \Phi(x) + u(t,x) \right]
\end{equation*}
into the NLSE ~\eqref{NLS} and truncating at the linear terms in $u$:
\begin{equation*}
\begin{cases}
i u_t = (-\partial^2_x + V + i\gamma W - \mu - 2g|\Phi|^2)u - g\Phi^2 \overline{u}, \\
-i \overline{u}_t = (-\partial^2_x + V - i\gamma W - \mu - 2g|\Phi|^2)\overline{u} - g\overline\Phi^2 u.
\end{cases}
\end{equation*}
Using $u = Ye^{-\lambda t}$ and $\overline{u} = Ze^{-\lambda t}$ with the spectral parameter $\lambda$
yields the spectral stability problem in the form
\small
\begin{equation}
\arraycolsep=0pt\def1{1}
\mathcal{L} \left[\begin{array}{c} Y \\ Z \end{array}\right] = -i\lambda\sigma_3
\left[\begin{array}{c} Y \\ Z \end{array}\right],
\label{original}
\end{equation}
\normalsize
where $\sigma_3 = {\rm diag}(1,-1)$ is the third Pauli's matrix and $\mathcal{L}$
is given by (\ref{jacobian}). Note that if $\lambda\not\in\mathbb{R}$, then $Z \ne \overline{Y}$.
\begin{lemma}
\label{lemma-continuous-spectrum}
The continuous spectrum of the operator $i \sigma_3 \mathcal{L} : D(L_0) \times D(L_0) \to L^2(\mathbb{R}) \times L^2(\mathbb{R})$,
if it exists, is a subset of $i \mathbb{R}$.
\end{lemma}
\begin{proof}
Thanks to the Assumptions~\ref{assumption-1},~\ref{assumption-2} and~\ref{assumption-3},
$W$ and $\Phi^2$ terms are relatively compact perturbations to the diagonal unbounded
operator $\mathcal{L}_0 := {\rm diag}(L_0-\mu I,L_0- \mu I)$, where
$L_0 = -\partial^2_x + V$ is introduced in (A1) and $I$ is an identity
$2\times 2$ matrix. Therefore,
\begin{equation*}
\sigma_c(i \sigma_3 \mathcal{L}) = \sigma_c(i \sigma_3 \mathcal{L}_0) \subset i \mathbb{R},
\end{equation*}
where $\sigma_c(A)$ denotes the absolutely continuous part of the spectrum of the operator
$A : D(A) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R})$.
\end{proof}
\begin{remark}
If $V \in L^2(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$, then $\mu < 0$ and
\begin{equation*}
\sigma_c(i \sigma_3 \mathcal{L}) = i (-\infty,-|\mu|] \cup i[|\mu|,\infty).
\end{equation*}
If $V$ is harmonic, then $\sigma_c(i \sigma_3 \mathcal{L})$ is empty.
\end{remark}
\begin{definition}
We say that the stationary state $\Phi$ is spectrally stable if every nonzero solution
$(Y,Z) \in D(L_0) \times D(L_0)$ to the spectral problem (\ref{original})
corresponds to $\lambda \in i \mathbb{R}$.
\end{definition}
We note the quadruple symmetry of eigenvalues in the spectral problem (\ref{original}).
\begin{lemma}
\label{lem-symmetry}
If $\lambda_0$ is an eigenvalue of the spectral problem ~\eqref{original},
so are $-\lambda_0$, $\bar\lambda_0$, and $-\bar\lambda_0$.
\end{lemma}
\begin{proof}
We note the symmetry of $\mathcal{L}$ and $\sigma_3$:
\begin{equation}
\label{symm1}
\mathcal{L} = \sigma_1 \overline{\mathcal{L}} \sigma_1, \quad
\sigma_3 = -\sigma_1 \sigma_3 \sigma_1,
\end{equation}
where $\sigma_1 = \antidiag(1,1)$ is the first Pauli's matrix.
If $\lambda_0$ is an eigenvalue of the spectral problem~\eqref{original} with the eigenvector
$v_0 := (Y,Z)$, then so is $\overline{\lambda}_0$ with the eigenvector
$\sigma_1 \overline{v}_0 = (\overline{Z},\overline{Y})$.
We note the second symmetry of $\mathcal{L}$ and $\sigma_3$:
\begin{equation}
\label{symm2}
\mathcal{L} = \mathcal{P} \overline{\mathcal{L}} \mathcal{P}, \quad
\sigma_3 = \mathcal{P} \sigma_3 \mathcal{P},
\end{equation}
where $\mathcal{P}$ is the parity transformation given by (\ref{operators}).
If $\lambda_0$ is an eigenvalue of the spectral problem~\eqref{original} with the eigenvector
$v_0 := (Y,Z)$, then so is $-\overline{\lambda}_0$ with the eigenvector
$\mathcal{P} \mathcal{T} v_0(x) = (\overline{Y(-x)},\overline{Z(-x)})$.
As a consequence of the two symmetries (\ref{symm1}) and (\ref{symm2}),
$-\lambda_0$ is also an eigenvalue with the eigenvector
$\mathcal{P} \sigma_1 v_0(x) = (Z(-x),Y(-x))$.
\end{proof}
Besides the spectral problem (\ref{original}), we also introduce the adjoint
spectral problem with the adjoint eigenvector denoted by $(Y^\#,Z^\#)$:
\small
\begin{equation}
\arraycolsep=-1.7pt\def1{1}
\mathcal{L}^* \left[\begin{array}{c} Y^\# \\ Z^\# \end{array}\right] = -i\lambda \sigma_3
\left[\begin{array}{c} Y^\# \\ Z^\# \end{array}\right],
\label{Adj}
\end{equation}
where
\begin{equation*}
\mathcal{L}^* := \left[\begin{array}{cc}
-\partial_x^2 + V - i \gamma W - \mu - 2 g |\Phi|^2 & - g \Phi^2 \\
- g \overline\Phi^2 & -\partial_x^2 + V + i \gamma W - \mu - 2 g |\Phi|^2
\end{array}\right].
\end{equation*}
\normalsize
\begin{remark}
Unless $\gamma = 0$ or $\Phi = 0$, the adjoint eigenvector $(Y^\#,Z^\#)$ cannot be
related to the eigenvector $(Y,Z)$ for the same eigenvalue $\lambda$.
\end{remark}
Our next assumption is on the existence of a nonzero isolated eigenvalue of the spectral problem (\ref{original}).
\begin{assumption}
\label{assumption-4}
We assume that there exists a simple isolated eigenvalue $\lambda_0 \in \mathbb{C} \backslash \{0\}$
of the spectral problems (\ref{original}) and (\ref{Adj})
with the eigenvector $v_0 := (Y,Z) \in D(L_0) \times D(L_0)$
and the adjoint eigenvector $v_0^\# := (Y^\#,Z^\#) \in D(L_0) \times D(L_0)$, respectively.
\end{assumption}
\begin{lemma}
\label{lem-PT-symmetry}
Under Assumption \ref{assumption-4}, if $\lambda_0\in i\mathbb{R}$, then
the corresponding eigenvectors $v_0 := (Y,Z)$ and $v_0^\# := (Y^\#,Z^\#)$
can be normalized to satisfy
\begin{equation}
\label{PT-sym-eigenvector}
Y(x) = \overline{Y(-x)}, \quad Z(x) = \overline{Z(-x)}, \quad x \in \mathbb{R}
\end{equation}
and
\begin{equation}
\label{PT-sym-eigenvector-adjoint}
Y^\#(x) = \overline{Y^\#(-x)}, \quad Z^\#(x) = \overline{Z^\#(-x)}, \quad x \in \mathbb{R}.
\end{equation}
\end{lemma}
\begin{proof}
By Lemma \ref{lem-symmetry}, if $\lambda_0 \in i\mathbb{R}$ is a nonzero
eigenvalue with the eigenvector $v_0 := (Y,Z)$,
so is $-\overline{\lambda}_0 = \lambda_0$ with the eigenvector $\mathcal{P T} v_0$.
Since $\lambda_0$ is a simple eigenvalue, there is a constant $C \in \mathbb{C}$ such that
$v_0 = C \mathcal{P T} v_0$. Taking norms on both sides, we have $|C|=1$.
Therefore $C = e^{i\alpha}$ for some $\alpha\in [0,2\pi]$, and $\alpha$
can be chosen so that $v_0$ satisfy $v_0 = \mathcal{P T} v_0$ as in (\ref{PT-sym-eigenvector}).
The same argument applies to the adjoint eigenvector $v^\#_0 := (Y^\#,Z^\#)$.
\end{proof}
We shall now introduce the main object of our study, the Krein signature of the simple nonzero
isolated eigenvalue $\lambda_0$ in Assumption~\ref{assumption-4}.
\begin{definition}
\label{def-Krein}
The Krein signature of the eigenvalue $\lambda_0$ in Assumption \ref{assumption-4}
is the sign of the Krein quantity $K(\lambda_0)$ defined by
\begin{equation}
K(\lambda_0) = \langle v_0, \sigma_3 v_0^\# \rangle
= \int_{\mathbb{R}} \left[ Y(x) \overline{Y^\#(x)} - Z(x)\overline{Z^\#(x)} \right] dx.
\label{pt-krein}
\end{equation}
\end{definition}
The following lemma states the main properties of the Krein quantity $K(\lambda_0)$.
\begin{lemma}
Assume (A4) and define $K(\lambda_0)$ by (\ref{pt-krein}). Then,
\begin{enumerate}
\item $K(\lambda_0)$ is real if $\lambda_0 \in i\mathbb{R} \backslash \{0\}$.
\item $K(\lambda_0) \ne 0$ if $\lambda_0 \in i\mathbb{R} \backslash \{0\}$.
\item $K(\lambda_0) = 0$ if $\lambda_0 \in \mathbb{C}\backslash\{i\mathbb{R}\}$.
\end{enumerate}
\label{lem-krein}
\end{lemma}
\begin{proof}
First, we prove that if $f$ and $g$ are $\mathcal{PT}$-symmetric functions,
then their inner product $\langle f, g\rangle$ is real-valued. Indeed, this follows from
\begin{align*}
\langle f,g\rangle &= \int_{\mathbb{R}} f(x)\overline{g(x)} dx =
\int_0^{+\infty} \bigl( f(x)\overline{g(x)} + f(-x)\overline{g(-x)}\bigr)dx
\\ &= \int_0^{+\infty}
\bigl( f(x)\overline{g(x)} + \overline{f(x)}g(x) \bigr) dx.
\end{align*}
Since $\lambda_0 \in i \mathbb{R} \backslash \{0\}$ is simple by Assumption~\ref{assumption-4},
then the eigenvectors $v_0 := (Y,Z)$ and $v_0^\# := (Y^\#,Z^\#)$ satisfy the $\mathcal{PT}$-symmetry
(\ref{PT-sym-eigenvector}) and (\ref{PT-sym-eigenvector-adjoint}) by
Lemma \ref{lem-PT-symmetry}. Hence, the inner products in the definition of $K(\lambda_0)$ in (\ref{pt-krein})
are real.
Next, we prove that $K(\lambda_0) \ne 0$ if $\lambda_0 \in i\mathbb{R} \backslash \{0\}$ is simple.
Consider a generalized eigenvector problem for the spectral problem ~\eqref{original}:
\begin{equation}
(\mathcal{L} + i \lambda_0 \sigma_3 ) \left[\begin{array}{c} Y_g \\ Z_g \end{array}\right]
= \sigma_3 \left[\begin{array}{c} Y \\ Z \end{array}\right].
\label{eqn-gen}
\end{equation}
Since $\lambda_0 \notin \sigma_c(i\sigma_3\mathcal{L})$ is isolated and simple by Assumption \ref{assumption-4},
there exists a solution $v_g := (Y_g,Z_g) \in D(L_0) \times D(L_0)$ to the nonhomogeneous
equation~\eqref{eqn-gen} if and only if
$\sigma_3 v_0$ is orthogonal to $v_0^\#$, which is
the kernel of adjoint operator $\mathcal{L}^* + i\lambda_0\sigma_3$.
The orthogonality condition coincides with $K(\lambda_0) = 0$.
However, no $v_g$ exists since $\lambda_0 \in i \mathbb{R} \backslash \{0\}$
is simple by Assumption~\ref{assumption-4}. Hence $K(\lambda_0) \neq 0$.
Finally, we show that $K(\lambda_0) = 0$ if $\lambda_0 \in \mathbb{C}\backslash\{i\mathbb{R}\}$.
Taking inner products for the spectral problems \eqref{original} and \eqref{Adj} with the corresponding
eigenvectors yields
\begin{equation*}
\begin{cases}
\langle \mathcal{L} v_0, v_0^\# \rangle \!\!\!\! &= -i\lambda_0
\langle\sigma_3 v_0, v_0^\#\rangle, \\
\langle v_0, \mathcal{L}^* v_0^\# \rangle
\!\!\!\! &= i\overline{\lambda}_0
\langle v_0, \sigma_3 v_0^\# \rangle,
\end{cases}
\end{equation*}
hence
\begin{equation*}
i (\lambda_0 + \overline{\lambda}_0) K(\lambda_0) = 0.
\end{equation*}
If $\lambda_0 \in \mathbb{C}\backslash\{i\mathbb{R}\}$, then $\lambda_0 + \overline{\lambda}_0 \neq 0$ and $K(\lambda_0) = 0$.
\end{proof}
We shall now compare the Krein quantity $K(\lambda_0)$ in (\ref{pt-krein})
for simple eigenvalues of the $\mathcal{PT}$-symmetric spectral problem
(\ref{original}) with the corresponding definition of the Krein quantity
in the Hamiltonian case $\gamma = 0$ and in the linear $\mathcal{PT}$-symmetric case $\Phi = 0$.
In the Hamiltonian case ($\gamma = 0$), the operator $\mathcal{L}$ in the spectral problem (\ref{original})
is self-adjoint in $L^2(\mathbb{R})$, that is, $\mathcal{L} = \mathcal{L}^*$.
The standard definition of Krein quantity \cite{kapitulabook,mackay} is given by
\begin{equation}
\label{Krein-Ham}
\gamma = 0: \quad K(\lambda_0) =
\left\langle \mathcal{L} v_0, v_0\right\rangle =
-i\lambda_0 \int_\mathbb{R} \left[ |Y(x)|^2 - |Z(x)|^2 \right] dx.
\end{equation}
If $\gamma=0$ and $\lambda_0 \in i \mathbb{R}$, then the adjoint eigenvector $(Y^\#,Z^\#)$ satisfies
the same equation as $(Y,Z)$. Therefore, it is natural to choose the adjoint eigenvector in the form:
\begin{equation}
\label{Ham-case}
\gamma = 0: \quad Y^\#(x) = Y(x), \quad Z^\#(x) = Z(x), \quad x \in \mathbb{R},
\end{equation}
in which case the definition~\eqref{pt-krein} yields the integral
in the right-hand side of (\ref{Krein-Ham}). Note that the signs
of $K(\lambda_0)$ in (\ref{pt-krein}) and (\ref{Krein-Ham})
are the same if $\lambda_0 \in i \mathbb{R}_+$.
\begin{remark}
Since the potential $V$ is even in (\ref{potentials}),
the eigenvector $v_0 := (Y,Z)$ of the spectral problem (\ref{original})
for a simple eigenvalue $\lambda_0 \in i \mathbb{R} \backslash\{0\}$ is either even or odd
in the Hamiltonian case $\gamma=0$
by the parity symmetry. It follows from the $\mathcal{PT}$-symmetry (\ref{PT-sym-eigenvector})
that the $\mathcal{PT}$-normalized eigenvector $v_0$ is real if it is even and is purely imaginary if it is odd.
\label{remark-real-odd}
\end{remark}
\begin{remark}
Since the adjoint eigenvector $v_0^\# := (Y^\#,Z^\#)$ satisfying the
$\mathcal{PT}$-symmetry condition (\ref{PT-sym-eigenvector-adjoint})
is defined up to an arbitrary sign, the Krein quantity $K(\lambda_0)$ in (\ref{pt-krein}) is defined up to
the sign change. In the continuation of the NLSE (\ref{NLS})
with respect to the parameter $\gamma$ from the Hamiltonian case $\gamma = 0$,
the sign of the Krein quantity $K(\lambda_0)$ in (\ref{pt-krein}) can be chosen so that
it matches the sign of $K(\lambda_0)$ in (\ref{Krein-Ham}) for $\lambda_0 \in i \mathbb{R}_+$ and $\gamma = 0$.
In other words, the choice (\ref{Ham-case}) is always made for $\gamma = 0$ and the Krein
quantity $K(\lambda_0)$ is extended continuously with respect to the parameter $\gamma$.
\label{remark-cont}
\end{remark}
In the linear $\mathcal{PT}$-symmetric case ($\Phi = 0$), the spectral problem (\ref{original})
becomes diagonal. If $Z = 0$, then $Y$ satisfies the scalar Schr\"{o}dinger equation
\begin{equation}
\label{scalar-spectral}
\left[-\partial^2_x + V(x) + i\gamma W(x) - \mu \right] Y(x) = -i\lambda Y(x).
\end{equation}
The $\mathcal{PT}$-Krein signature for the simple eigenvalue $\lambda_0 \in i \mathbb{R}$
of the scalar Schr\"{o}dinger equation (\ref{scalar-spectral}) is defined in
\cite{yang} as follows:
\begin{equation}
\label{Krein-Jang}
\Phi = 0, \quad Z = 0: \qquad K(\lambda_0) = \int_\mathbb{R} Y(x)\overline{Y(-x)}dx.
\end{equation}
If $\lambda_0 \in i \mathbb{R}$, then the adjoint eigenfunction $Y^\#$ satisfies
a complex-conjugate equation to the spectral problem (\ref{scalar-spectral}),
which becomes identical to (\ref{scalar-spectral}) after the parity transformation.
Therefore, it is natural to choose the adjoint eigenfunction $Y^\#$ in the form:
\begin{equation*}
\Phi = 0, \quad Z = 0: \qquad Y^\#(x) = Y(-x), \quad x \in \mathbb{R},
\end{equation*}
after which the definition~\eqref{pt-krein} with $Z = 0$ corresponds to the
definition (\ref{Krein-Jang}). If $Y=0$, then $Z$ satisfies the scalar Schr\"{o}dinger equation
\begin{equation}
\label{scalar-spectral-Z}
\left[-\partial^2_x + V(x) - i\gamma W(x) - \mu \right] Z(x) = i\lambda Z(x).
\end{equation}
The $\mathcal{PT}$-Krein signature for the simple eigenvalue $\lambda_0 \in i \mathbb{R}$
of the scalar Schr\"{o}dinger equation (\ref{scalar-spectral-Z}) is defined by
\begin{equation}
\label{Krein-Z-Jang}
\Phi = 0, \quad Y = 0: \qquad K(\lambda_0) = \int_\mathbb{R} Z(x)\overline{Z(-x)}dx,
\end{equation}
which coincides with the definition~\eqref{pt-krein} for $Y = 0$ if the
adjoint eigenfunction $Z^\#$ is chosen in the form:
\begin{equation}
\label{Jang-Z}
\Phi = 0,\quad Y = 0: \qquad Z^\#(x) = -Z(-x), \quad x \in \mathbb{R}.
\end{equation}
Note that if the choice $Z^\#(x) = Z(-x)$ is made instead of (\ref{Jang-Z}),
then the definition (\ref{pt-krein}) with $Y = 0$
is negative with respect to the definition (\ref{Krein-Z-Jang}).
\section{Necessary conditions of the instability bifurcation}
\label{sec-proof}
Recall that the eigenvalue is called {\em semi-simple} if algebraic and geometric multiplicities coincide
and {\em defective} if algebraic multiplicity exceeds geometric multiplicity.
Here we consider the case when the nonzero eigenvalue $\lambda_0 \in i \mathbb{R}$ of the spectral problem
(\ref{original}) is defective with geometric multiplicity {\em one} and algebraic multiplicity {\em two}.
This situation occurs in the parameter continuations of the NLSE (\ref{NLS}) when two simple
isolated eigenvalues $\lambda_1, \lambda_2 \in i \mathbb{R} \backslash \{0\}$ coalesce at the point $\lambda_0 \neq 0$
and split into the complex plane resulting in the {\em instability bifurcation}.
We will use the parameter $\gamma$ to control the coalescence of two simple eigenvalues $\lambda_1, \lambda_2 \in i \mathbb{R}$.
Our main result states that the instability bifurcation occurs from the defective eigenvalue
$\lambda_0 \in i \mathbb{R}$ of algebraic multiplicity two
only if the Krein signatures of $K(\lambda_1)$ and $K(\lambda_2)$ for the two simple isolated eigenvalues $\lambda_1, \lambda_2 \in i \mathbb{R}$
before coalescence are opposite to each other. Therefore, we obtain the necessary condition
for the instability bifurcation in the $\mathcal{PT}$-symmetric spectral problem (\ref{original}),
which has been proven for the Hamiltonian spectral problems~\cite{kapitulabook,mackay}.
\begin{remark}
The necessary condition for instability bifurcation allows us to predict
the transition from stability to instability when a pair of imaginary eigenvalues collide.
Pairs with the same Krein signature do not bifurcate off the imaginary axis if they collide.
In the contrast, pairs with the opposite Krein signature may bifurcate off the imaginary axis
under a technical non-degeneracy condition (\ref{non-degeneracy}) below.
\end{remark}
First, we state why the perturbation theory can be applied to the spectral problem (\ref{original}).
\begin{lemma}
\label{lem-operator-L}
Under Assumptions \ref{assumption-1}, \ref{assumption-2}, and \ref{assumption-3},
the operator
\begin{equation*}
\mathcal{L} : D(L_0) \times D(L_0) \to L^2(\mathbb{R}) \times L^2(\mathbb{R})
\end{equation*}
in the spectral problem (\ref{original}) is real-analytic with respect to
$\gamma \in (-\gamma_*,\gamma_*)$.
Consequently, if $\mathcal{L}(\gamma_0)$ with $\gamma_0 \in (-\gamma_*,\gamma_*)$
has a spectrum consisting of two separated parts, then the subspaces of $L^2(\mathbb{R}) \times L^2(\mathbb{R})$
corresponding to the separated parts are also real-analytic in $\gamma$.
\end{lemma}
\begin{proof}
Operator $\mathcal{L}$ depends on $\gamma$ via the potential $i \gamma W$ and the bound state $\Phi$,
the latter is real-analytic for $\gamma \in (-\gamma_*,\gamma_*)$ by Assumption~\ref{assumption-3}.
The assertion of the lemma follows from Theorem 1.7 in Chapter VII on p. 368 in \cite{Kato}.
\end{proof}
By Lemma \ref{lem-operator-L}, simple isolated eigenvalues
$\lambda_1, \lambda_2 \in i \mathbb{R}$ of the spectral problem (\ref{original})
and their eigenvectors $v_1 := (Y_1,Z_1)$ and $v_2 := (Y_2,Z_2)$ are continued analytically
in $\gamma$ before the coalescence point. Similarly,
the adjoint eigenvectors $v_1^\# := (Y_1^\#,Z_1^\#)$ and $v_2^\# := (Y_2^\#,Z_2^\#)$ of the adjoint spectral
problem (\ref{Adj}) for $\lambda_1, \lambda_2 \in i \mathbb{R}$ are continued analytically in $\gamma$. Therefore, the
Krein quantities $K(\lambda_1)$ and $K(\lambda_2)$ are continued analytically in $\gamma$.
Let $\gamma_0$ denote the bifurcation point when the two eigenvalues coalesce:
$\lambda_1 = \lambda_2 = \lambda_0 \in i \mathbb{R} \backslash \{0\}$.
For this $\gamma_0 \in \mathbb{R}$, we can define a small parameter $\varepsilon\in\mathbb{R}$ such that
$\gamma = \gamma_0 + \varepsilon$. If $\mathcal{L}$ is denoted by $\mathcal{L}(\gamma)$,
then $\mathcal{L}(\gamma)$ can be represented by the Taylor expansion:
\begin{equation}
\mathcal{L}(\gamma) = \mathcal{L}(\gamma_0)
+ \varepsilon \mathcal{L'}(\gamma_0) + \varepsilon^2 \hat{\mathcal{L}}(\varepsilon),
\label{operator:series}
\end{equation}
where $\hat{\mathcal{L}}(\varepsilon)$ denotes the remainder terms,
\begin{equation}
\mathcal{L}'(\gamma_0) =
\left[\begin{array}{cc}
iW - 2 g \partial_\gamma |\Phi(\gamma_0)|^2 & - g \partial_\gamma \Phi^2(\gamma_0) \\[2pt]
-g \partial_\gamma \overline{\Phi^2(\gamma_0)} & -iW - 2 g \partial_\gamma |\Phi(\gamma_0)|^2
\end{array}\right],
\label{operator:lprime}
\end{equation}
and $\partial_\gamma$ denotes a partial derivative with respect to the parameter $\gamma$.
Since the remainder terms in $\hat{\mathcal{L}}(\varepsilon)$ come from the second derivative of $\Phi$ in $\gamma$ near $\gamma_0$,
then $\hat{\mathcal{L}}(\varepsilon) \in L^2(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ thanks to Assumption \ref{assumption-3}.
Instead of Assumption~\ref{assumption-4}, we shall now use the following assumption.
\setcounter{assumption}{3}
\renewcommand{\theassumption}{(A\arabic{assumption}$^\prime$)}
\begin{assumption}
\label{assumption-5}
For $\gamma = \gamma_0$, we assume that there exists a
defective isolated eigenvalue $\lambda_0 \in i \mathbb{R} \backslash \{0\}$
of the spectral problems (\ref{original}) and (\ref{Adj})
with the eigenvector $v_0 := (Y_0,Z_0) \in D(L_0) \times D(L_0)$,
the generalized eigenvector $v_g := (Y_g,Z_g) \in D(L_0) \times D(L_0)$
and the adjoint eigenvector $v_0^\# := (Y^\#_0,Z^\#_0) \in D(L_0) \times D(L_0)$,
the adjoint generalized eigenvector $v_g^\# := (Y^\#_g,Z^\#_g) \in D(L_0) \times D(L_0)$,
respectively.
\end{assumption}
\renewcommand{\theassumption}{(A\arabic{assumption})}
By setting $\lambda_0 = i \Omega_0$, we can write the linear equations
for the eigenvectors and generalized eigenvectors in Assumption~\ref{assumption-5}:
\begin{gather}
\mathcal{L}(\gamma_0)v_0 = \Omega_0 \sigma_3 v_0, \qquad
\mathcal{L}(\gamma_0)v_g = \Omega_0 \sigma_3 v_g + \sigma_3 v_0,
\label{eig:omega:orig} \\
\mathcal{L}^*(\gamma_0)v_0^\# =
\Omega_0 \sigma_3 v_0^\#, \qquad
\mathcal{L}^*(\gamma_0)v_g^\# = \Omega_0 \sigma_3 v_g^\# + \sigma_3 v_0^\#.
\label{eig:omega:adj}
\end{gather}
The solvability conditions for the inhomogeneous equations in (\ref{eig:omega:orig}) and (\ref{eig:omega:adj})
yield the following elementary facts.
\begin{lemma}
\label{lem-facts}
Under Assumption~\ref{assumption-5}, we have
\begin{equation}
\label{fact-1}
K(\lambda_0) = \langle v_0,\sigma_3 v_0^\#\rangle = 0.
\end{equation}
and
\begin{equation}
\label{fact-2}
\langle v_g,\sigma_3 v_0^\#\rangle = \langle v_0,\sigma_3 v_g^\#\rangle \neq 0.
\end{equation}
\end{lemma}
\begin{proof}
Since $v_g$ exists by Assumption~\ref{assumption-5}, the
solvability condition for (\ref{eig:omega:orig}) implies (\ref{fact-1}),
see similar computations in Lemma \ref{lem-krein}.
Since the eigenvalue $\lambda_0$ is double, no
second generalized eigenvector $\tilde{v}_g$ exists such that
\begin{equation*}
\mathcal{L}(\gamma_0) \tilde{v}_g = \Omega_0 \sigma_3 \tilde{v}_g + \sigma_3 v_g.
\end{equation*}
The nonsolvability condition for this equation implies $\langle v_g, \sigma_3 v_0^\# \rangle \ne 0$.
Finally, equations ~\eqref{eig:omega:orig} and \eqref{eig:omega:adj} yield
\begin{align*}
\langle v_g, \sigma_3 v_0^\#\rangle &=
\langle v_g, (\mathcal{L}^*-\Omega_0 \sigma_3)v_g^\#\rangle =
\langle (\mathcal{L}-\Omega_0 \sigma_3)v_g, v_g^\#\rangle \\ &=
\langle \sigma_3 v_0, v_g^\#\rangle =
\langle v_0, \sigma_3 v_g^\#\rangle,
\end{align*}
which proves the symmetry in (\ref{fact-2}).
\end{proof}
\begin{remark}
Since the generalized eigenvectors are given by solutions of the inhomogeneous linear equations
(\ref{eig:omega:orig}) and (\ref{eig:omega:adj}) and the eigenvectors satisfy
the $\mathcal{PT}$-symmetry (\ref{PT-sym-eigenvector}) and
(\ref{PT-sym-eigenvector-adjoint}), the generalized eigenvectors also satisfy the same $\mathcal{PT}$-symmetry
(\ref{PT-sym-eigenvector}) and (\ref{PT-sym-eigenvector-adjoint}).
\end{remark}
The following result gives the necessary condition that the defective eigenvalue $\lambda_0$ in Assumption~\ref{assumption-5}
splits into the complex plane in a one-sided neighborhood of the bifurcation point $\gamma_0$.
\begin{theorem}
\label{theorem-main}
Assume \ref{assumption-1}, \ref{assumption-2}, \ref{assumption-3}, \ref{assumption-5}, and
the non-degeneracy condition
\begin{equation}
\label{non-degeneracy}
\langle \mathcal{L'}(\gamma_0) v_0, v_0^\#\rangle \ne 0.
\end{equation}
There exists $\varepsilon_0 > 0$ such that two simple eigenvalues $\lambda_1,\lambda_2$
of the spectral problem (\ref{original}) exist near $\lambda_0$ for every $\varepsilon \in (-\varepsilon_0,\varepsilon_0) \backslash \{0\}$
with $\lambda_{1,2} \to \lambda_0$ as $\varepsilon \to 0$.
On one side of $\varepsilon = 0$, the eigenvalues are $\lambda_1, \lambda_2 \in i \mathbb{R}$ and
\begin{equation}
\label{opposite-Krein}
{\rm sign} K(\lambda_1) = -{\rm sign} K(\lambda_2).
\end{equation}
On the other side of $\varepsilon = 0$, the eigenvalues are $\lambda_1,\lambda_2 \notin i \mathbb{R}$.
\end{theorem}
\begin{proof}
We are looking for an eigenvalue $\Omega(\varepsilon)$ of the perturbed spectral problem
\begin{equation}
\left[ \mathcal{L}_0
+ \varepsilon \widetilde{\mathcal{L}}(\varepsilon) \right] v(\varepsilon) =
\Omega(\varepsilon) \sigma_3 v(\varepsilon),
\label{original:perturbed}
\end{equation}
such that $\Omega(\varepsilon) \to \Omega_0$ as $\varepsilon \to 0$.
Here we denote operators from the decomposition \eqref{operator:series} as
$\mathcal{L}_0 = \mathcal{L}(\gamma_0)$ and $\widetilde{\mathcal{L}}(\varepsilon) =
\mathcal{L}'(\gamma_0) + \varepsilon \hat{\mathcal{L}}(\varepsilon)$.
Since $\Omega_0$ is a defective eigenvalue of geometric multiplicity {\em one}
and algebraic multiplicity {\em two}, we apply Puiseux expansions~\cite{Knopp}:
\begin{equation}
\left\{ \begin{array}{l}
\Omega(\varepsilon) = \Omega_0 + \varepsilon^{1/2} \widetilde\Omega(\varepsilon), \\
v(\varepsilon) = v_0 + \varepsilon^{1/2} a(\varepsilon) v_g
+ \varepsilon \widetilde{v_1}(\varepsilon),
\end{array} \right.
\label{Puiseux}
\end{equation}
where $v_0$ and $v_g$ are the eigenvector and the generalized
eigenvector of the spectral problem (\ref{eig:omega:orig}),
$a(\varepsilon)$ is the projection coefficient to be defined, and
$\widetilde\Omega(\varepsilon)$ and $\widetilde{v_1}(\varepsilon)$ are the remainder terms.
To define $\widetilde{v_1}(\varepsilon)$ uniquely, we add the orthogonality condition
\begin{equation}
\langle \widetilde{v_1}(\varepsilon), \sigma_3 v_0^\# \rangle
= \langle \widetilde{v_1}(\varepsilon), \sigma_3 v_g^\# \rangle = 0.
\label{ve-orthogonal}
\end{equation}
Plugging~\eqref{Puiseux} into~\eqref{original:perturbed} and dropping the dependence on $\varepsilon$
for $\widetilde{\mathcal{L}}$, $\widetilde{v}_1$, $a$ and $\widetilde{\Omega}$ gives us
the nonhomogeneous equation
\begin{gather}
\left( \mathcal{L}_0 - \Omega_0 \sigma_3
+ \varepsilon \widetilde{\mathcal{L}} - \varepsilon^{1/2} \widetilde{\Omega} \sigma_3 \right) \widetilde{v}_1 =
\varepsilon^{-1/2} (\widetilde{\Omega} - a) \sigma_3 v_0
- \widetilde{\mathcal{L}} v_0 + a
\bigl( \widetilde{\Omega} \sigma_3
- \varepsilon^{1/2} \widetilde{\mathcal{L}} \bigr) v_g.
\label{ve-equation}
\end{gather}
By Assumption~\ref{assumption-5}, the limiting operator $\sigma_3 (\mathcal{L}_0 - \Omega_0 \sigma_3)$ has the two-dimensional
generalized null space $X_0 = {\rm span}\{v_0,v_g\} \subset L^2(\mathbb{R}) \times L^2(\mathbb{R})$.
Since $\Omega_0 \notin \sigma_c(\sigma_3 \mathcal{L}_0)$ is isolated from the rest of the spectrum of $\sigma_3 \mathcal{L}_0$,
the range of $\sigma_3 (\mathcal{L}_0 - \Omega_0 \sigma_3)$ is orthogonal with respect to generalized null space
$Y_0 = {\rm span}\{ \sigma_3 v_0^\#, \sigma_3 v_g^\#\} \subset L^2(\mathbb{R}) \times L^2(\mathbb{R})$
of the adjoint operator $(\mathcal{L}_0^* - \Omega_0 \sigma_3) \sigma_3$. As a result, $\sigma_3 (\mathcal{L}_0 - \Omega_0 \sigma_3)$ is
invertible on an element of $Y_0^\perp$ and the inverse operator
is uniquely defined and bounded in $Y_0^{\perp}$. In other words,
there exist positive constants $\varepsilon_0$, $\Omega_0$, and $C_0$ such that for
all $|\varepsilon| \leq \varepsilon_0$, $|\widetilde{\Omega}| \leq \Omega_0$, and all
$\sigma_3 f\in Y_0^\perp$, there exists a unique
$(\mathcal{L}_0 - \Omega_0 \sigma_3)^{-1} f \in D(L_0) \times D(L_0)$
satisfying the orthogonality conditions (\ref{ve-orthogonal}) and the bound
\begin{equation}
\label{resolvent-estimate}
\| (\mathcal{L}_0 - \Omega_0 \sigma_3)^{-1} f\|_{L^2} \le C_0 \|f\|_{L^2}.
\end{equation}
In order to provide existence of a unique $(\mathcal{L}_0 - \Omega_0 \sigma_3)^{-1} f$,
we add the orthogonality constraints $\langle f,v_0^\#\rangle = \langle f,v_g^\#\rangle = 0$.
By using~\eqref{fact-2} and ~\eqref{ve-orthogonal}, we obtain two equations:
\begin{gather}
\varepsilon \langle \widetilde{\mathcal{L}} \widetilde{v}_1, v_0^\#\rangle +
\langle \widetilde{\mathcal{L}} v_0, v_0^\# \rangle =
\widetilde\Omega a \langle v_g, \sigma_3 v_0^\#\rangle
- \varepsilon^{1/2} a \langle \widetilde{\mathcal{L}} v_g, v_0^\# \rangle,
\label{proj-v0s}
\end{gather}
and
\begin{align}
\varepsilon \langle \widetilde{\mathcal{L}} \widetilde{v_1}, v_g^\#\rangle +
\langle \widetilde{\mathcal{L}} v_0,v_g^\#\rangle =
& \, \widetilde\Omega a \langle v_g,\sigma_3 v_g^\#\rangle \notag \\
&+ \varepsilon^{-1/2} (\widetilde\Omega - a) \langle v_0,\sigma_3 v_g^\#\rangle
- \varepsilon^{1/2} a \langle \widetilde{\mathcal{L}} v_g,v_g^\#\rangle.
\label{proj-v1s}
\end{align}
Since $\widetilde{\mathcal{L}}$ and $\widetilde\Omega \sigma_3$
are relatively compact perturbations to $(\mathcal{L}_0 - \Omega_0 \sigma_3)$,
under the constraints~\eqref{proj-v0s} and \eqref{proj-v1s}, there exists a unique
solution of the nonhomogeneous equation~\eqref{ve-equation}
satisfying the orthogonality conditions (\ref{ve-orthogonal})
and the resolvent estimate~\eqref{resolvent-estimate}. In particular, there exist positive constants
$\varepsilon_0$, $\Omega_0$, $A_0$, and $C_0$ such that for
all $|\varepsilon| \leq \varepsilon_0$, $|\widetilde{\Omega}| \leq \Omega_0$, and $|a| \leq A_0$,
the solution $\widetilde{v}_1 \in D(L_0) \times D(L_0)$ of equation (\ref{ve-equation}) satisfies the estimate
\begin{equation}
\| \widetilde{v_1} \|_{L^2} \le C_0 \left( \varepsilon^{-1/2}|a - \widetilde{\Omega}| + 1 + |\widetilde{\Omega}|^2\right).
\label{ve-estimate}
\end{equation}
Equation~\eqref{proj-v1s} yields
\begin{align*}
\varepsilon^{-1/2} (a - \widetilde{\Omega}) =
\frac{1}{\langle v_0, \sigma_3 v_g^\#\rangle}
&\left( \widetilde\Omega a \langle v_g, \sigma_3 v_g^\#\rangle
- \varepsilon^{1/2} a \langle \widetilde{\mathcal{L}} v_g,v_g^\#\rangle\right. \notag \\
&\quad - \left.\langle \widetilde{\mathcal{L}} v_0, v_g^\#\rangle
- \varepsilon \langle \widetilde{\mathcal{L}} \widetilde{v}_1, v_g^\#\rangle \right),
\end{align*}
where~$\langle v_0, \sigma_3 v_g^\#\rangle\ne 0$ due to Lemma~\ref{lem-facts}.
Combining with the estimate (\ref{ve-estimate}), we obtain for some $C_1>0$
\begin{equation}
\label{ve-estimate-a}
|a - \widetilde\Omega | \le C_1 \varepsilon^{1/2} (1 + |\widetilde{\Omega}|^2) \quad \text{and} \quad
\|\tilde{v}_1\|_{L^2} \le C_1 (1 + |\widetilde{\Omega}|^2).
\end{equation}
Equation~\eqref{proj-v0s} yields
\begin{gather*}
\widetilde\Omega a = \frac{1}{\langle v_g, \sigma_3 v_0^\#\rangle}
\left( \langle \widetilde{\mathcal{L}} v_0, v_0^\#\rangle
+ \varepsilon^{1/2} a \langle \widetilde{\mathcal{L}} v_g,v_0^\#\rangle
+ \varepsilon \langle \widetilde{\mathcal{L}} \widetilde{v_1}, v_0^\# \rangle \right),
\end{gather*}
where $\langle v_g, \sigma_3 v_0^\#\rangle\ne 0$ due to Lemma~\ref{lem-facts}.
Thanks to (\ref{ve-estimate-a}), we obtain
\begin{equation*}
| \widetilde\Omega - \Omega_g | \le C_2 \varepsilon^{1/2},
\end{equation*}
where $C_2>0$ is a constant, and $\Omega_g$ is a root of the quadratic equation
\begin{equation}
\Omega_g^2 = \frac{\langle \mathcal{L}'(\gamma_0) v_0,v_0^\#\rangle }
{\langle v_g,\sigma_3 v_0^\#\rangle},
\label{eq:omegag}
\end{equation}
with $\mathcal{L}'(\gamma_0)$ given by~\eqref{operator:lprime}.
Since $\mathcal{L}'(\gamma_0) v_0$, $v_g$, and $v_0^\#$ satisfy the
$\mathcal{PT}$-conditions (\ref{PT-sym}), (\ref{PT-sym-eigenvector}), and (\ref{PT-sym-eigenvector-adjoint}),
both the nominator and the denominator of (\ref{eq:omegag}) are real-valued by the same computations as in
the proof of Lemma \ref{lem-krein}. By the assumption (\ref{non-degeneracy}),
$\Omega_g^2$ is nonzero, either positive or negative.
Let us assume that $\Omega_g^2 > 0$ without loss of generality and pick $\Omega_g > 0$.
Then $\varepsilon^{1/2} \Omega_g \in \mathbb{R}$ if $\varepsilon > 0$ and we obtain the expansions
for the two simple eigenvalues:
\begin{equation*}
\begin{cases}
\Omega_1(\varepsilon) = \Omega_0 + \varepsilon^{1/2} \Omega_g
+ \mathcal{O}(\varepsilon), \\
\Omega_2(\varepsilon) = \Omega_0 - \varepsilon^{1/2} \Omega_g
+ \mathcal{O}(\varepsilon)
\end{cases}
\end{equation*}
and their corresponding eigenvectors:
\begin{equation*}
\begin{cases}
v_1(\varepsilon) = v_0 + \varepsilon^{1/2} \Omega_g v_g
+ \mathcal{O}(\varepsilon), \\
v_2(\varepsilon) = v_0 - \varepsilon^{1/2} \Omega_g v_g
+ \mathcal{O}(\varepsilon).
\end{cases}
\end{equation*}
The same expansions hold for eigenvectors of the adjoint spectral problems
corresponding to the same eigenvalues $\Omega_1,\Omega_2$:
\begin{equation*}
\begin{cases}
v_1^\#(\varepsilon) = v_0^\# + \varepsilon^{1/2} \Omega_g v_g^\#
+ \mathcal{O}(\varepsilon), \\
v_2^\#(\varepsilon) = v_0^\# - \varepsilon^{1/2} \Omega_g v_g^\#
+ \mathcal{O}(\varepsilon).
\end{cases}
\end{equation*}
The leading order of Krein quantitites for eigenvalues $\lambda_1 = i\Omega_1$ and $\lambda_2 = i\Omega_2$
is given by
\begin{equation*}
\begin{cases}
K(\lambda_1) = \langle v_1,\sigma_3 v_1^\#\rangle
= \varepsilon^{1/2} \Omega_g
\langle v_g, \sigma_3 v_0^\#\rangle + \overline{\varepsilon^{1/2} \Omega_g}
\langle v_0, \sigma_3 v_g^\#\rangle + \mathcal{O}(\varepsilon),
\\
K(\lambda_2) = \langle v_2,\sigma_3 v_2^\#\rangle =
-\varepsilon^{1/2} \Omega_g \langle v_g,\sigma_3 v_0^\#\rangle
- \overline{\varepsilon^{1/2} \Omega_g} \langle v_0,\sigma_3 v_g^\#\rangle
+ \mathcal{O}(\varepsilon),
\end{cases}
\end{equation*}
which is simplified with the help of (\ref{fact-2}) to
\begin{equation*}
\begin{cases}
K(\lambda_1) = 2\varepsilon^{1/2} \Omega_g \langle v_g, \sigma_3 v_0^\#\rangle
+ \mathcal{O}(\varepsilon), \\
K(\lambda_2) = -2\varepsilon^{1/2} \Omega_g \langle v_g,\sigma_3 v_0^\#\rangle
+ \mathcal{O}(\varepsilon).
\end{cases}
\end{equation*}
Since $\epsilon^{1/2} \Omega_g \in \mathbb{R}$ and $\langle v_g, \sigma_3 v_0^\# \rangle \neq 0$,
we obtain (\ref{opposite-Krein}).
If $\varepsilon < 0$, then $\epsilon^{1/2} \Omega_g \in i \mathbb{R}$,
so that $\lambda_1, \lambda_2 \notin i \mathbb{R}$.
\end{proof}
\begin{remark}
If the non-degeneracy assumption (\ref{non-degeneracy}) is not satisfied, then
$\Omega_g = 0$ and the perturbation theory must be extended to the next order. In this case,
the defective eigenvalue $\lambda_0 = i \Omega_0$ may split along $i \mathbb{R}$
both for $\varepsilon > 0$ and $\varepsilon < 0$.
\end{remark}
\section{Numerical Approximations}
\label{sec-numerics}
We approximate nonlinear modes $\Phi$ of the stationary NLSE~\eqref{NLSstat} and eigenvectors
$(Y,Z)$ of the spectral problem~\eqref{original} with the Chebyshev interpolation method~\cite{trefethen}.
This method was recently applied to massive Dirac equations in~\cite{yusuke}.
Chebyshev polynomials are defined on the interval $[-1,1]$. The stationary NLSE~\eqref{NLSstat} is defined
on the real line, therefore we make a coordinate transformation for the Chebyshev grid points
$\{ z_j = \cos(\frac{j\pi}{N})\}_{j=0}^{j=N}$:
\begin{equation}
x_j = L \arctanh(z_j), \quad j = 1,2,\ldots,N-1,
\label{eq:transform}
\end{equation}
where $x_0 = +\infty$ and $x_N = -\infty$.
The scaling parameter $L$ is chosen so that the grid points $\{ x_j \}_{j =1}^{j = N-1}$ are
concentrated in the region where the nonlinear mode $\Phi$ changes fast.
We apply the chain rule for the second derivative:
\begin{gather*}
\frac{d^2 u}{dx^2} = \frac{d}{dx} \left(\frac{du}{dx}\right) =
\frac{d}{dz} \left(\frac{du}{dz} \frac{dz}{dx}\right)=
\frac{d^2 u}{dz^2} \left(\frac{dz}{dx}\right)^2 + \frac{du}{dz} \frac{d^2 z}{dx^2},
\end{gather*}
where
\begin{eqnarray*}
\frac{dz}{dx} = \frac{1}{L} \sech^2 \left(\frac{x}{L}\right) = \frac{1}{L} (1-z^2)
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{d^2 z}{dx^2} = -\frac{2}{L^2} \sech^2\left(\frac{x}{L}\right)
\tanh\left(\frac{x}{L}\right) = -\frac{2}{L^2} z(1-z^2).
\end{eqnarray*}
The first and second derivatives for $\partial_z$ and $\partial^2_z$ are approximated by
the Chebyshev differentiation matrices $D_N$ and $D_N^2$, respectively (see~\cite{trefethen}, p.53).
The stationary NLSE~\eqref{NLSstat} is written in the form:
\begin{equation}
\label{root-finding}
F(\Phi) := (- \partial_x^2 + V + i\gamma W - \mu - g|\Phi|^2) \Phi = 0.
\end{equation}
We fix $\mu$, $\gamma$, $g$, $V(x)$, $W(x)$ and use Newton's method to look for a solution
$\Phi$ satisfying Assumption \ref{assumption-3}:
\begin{equation}
\left[\begin{array}{c} \Phi_{n+1} \\ \bar\Phi_{n+1} \end{array}\right] =
\left[\begin{array}{c} \Phi_n \\ \bar\Phi_n \end{array}\right]
- \mathcal{L}^{-1}_n \left[\begin{array}{c} F(\Phi_n) \\ \bar F(\Phi_n) \end{array}\right],
\label{newton}
\end{equation}
where $\mathcal{L}_n$ is the Jacobian operator to the nonlinear problem (\ref{root-finding}),
which coincides with~\eqref{jacobian} computed at $\Phi_n$.
Since $\Phi(x_0) = \Phi(x_N) = 0$,
the Jacobian operator $\mathcal{L}_n$ is represented by the $2(N-1) \times 2(N-1)$ matrix.
It follows by the gauge transformation that
\begin{equation*}
\mathcal{L} \left[ \begin{array}{c} i\Phi \\ -i\bar\Phi \end{array} \right]
= \left[ \begin{array}{c} 0 \\ 0 \end{array} \right],
\end{equation*}
where $\mathcal{L}$ is given by (\ref{jacobian}).
Therefore, $\mathcal{L}$ is a singular operator for every parameter choice of equation~\eqref{root-finding}.
However, if the eigenvector satisfies the symmetry $\bar{Z} = Y$, then the eigenvector does not
satisfy the $\mathcal{PT}$-symmetry:
\begin{equation*}
\mathcal{PT}\left[ \begin{array}{c} i\Phi \\ -i\bar\Phi \end{array} \right]
= \left[ \begin{array}{c} -i\overline{\Phi(-x)} \\ i\Phi(-x) \end{array} \right]
= - \left[ \begin{array}{c} i\Phi \\ -i\bar\Phi \end{array} \right].
\end{equation*}
Hence, $\mathcal{L}$ is invertible on the space of $\cal{PT}$-symmetric functions satisfying (\ref{PT-sym}).
In terms of the coefficients of Chebyshev polynomials, the restriction means that
the even-numbered coefficients are purely real, whereas the odd-numbered
coefficients are purely imaginary.
Choosing a first guess for the iterative procedure~\eqref{newton} depends on the choice of
the potentials $V$ and $W$. For the Scarf II potential~\eqref{pot-Wadati}, one can use a scalar multiple of
the $\sech(x)$ function for the first branch of solutions and a scalar multiple of
the $\sech(x)\tanh(x)$ function for the second branch of solutions~\cite{ahmed}. For the confining potential~\eqref{pot-BEC},
one can use the corresponding Gauss-Hermite functions of the linear system for each branch~\cite{zezyulin}.
The spectral problem~\eqref{original} uses the same operator $\mathcal{L}$ and can
be discretized similarly. One looks for eigenvalues and eigenvectors of the discretized
matrix by using the standard numerical methods for non-Hermitian matrices.
For example, MATLAB$^\text{\textregistered}$ performs these computations by using the QZ algorithm.
Throughout the numerical results, we pick the value of a scaling parameter $L$ to be $L=10$.
This choice ensures that $\Phi$ remains nonzero up to $16$ decimals on
the interior grid points $\{ x_j \}_{j=1}^{j=N-1}$. The algorithm was tested on the exact
solution derived in~\cite{wadati} for the Scarf II potential~\eqref{pot-Wadati} with $V_0=1$
and $\mu=\gamma=-1$:
\begin{equation}
\label{exact-sol}
\Phi_{exact}(x) = \sin \alpha \sech(x) \exp\left( \frac{i}{2} \cos\alpha \arctan(\sinh(x)) \right),
\end{equation}
where $\alpha = \arccos(2/3)$. Table~\ref{err-table} shows a good agreement between exact and numerical
results.
\begin{table}
\center
\begin{tabular}{cc}
\hline
& $\|\Phi_{exact} - \Phi_{numerical} \|^2$ \\
\hline
N = 50 & $1.5 \times 10^{-6}$ \\
\hline
N = 100 & $2.4 \times 10^{-13}$ \\
\hline
N = 500 & $2.2\times 10^{-13}$ \\
\hline
\end{tabular}
\caption{The numerical error for the exact solution (\ref{exact-sol})
versus $N$.}
\label{err-table}
\end{table}
Once we computed eigenvalues and eigenvectors for the spectral problem~\eqref{original},
we proceed to computations of the Krein quantity defined by ~\eqref{pt-krein}.
Several obstacles arise in the definition of the Krein quantity:
\begin{enumerate}
\item Eigenvectors of the Chebyshev discretization matrices are normalized with respect to $z$.
\item Eigenvectors are not necessarily $\cal{PT}$-symmetric.
\item The sign of the adjoint eigenvectors relative to the eigenvectors is undefined.
\end{enumerate}
Here we explain how to deal with these technical difficulties.
\begin{enumerate}
\item The eigenvectors are normalized in the $L^2([-1,1])$ norm with respect to the variable $z$.
In order to normalize them in the $L^2(\mathbb{R})$ norm with respect to the variable $x$,
we perform the change of coordinates~\eqref{eq:transform}. In particular, we use integration with
the composite trapezoid method on the grid points $\{ x_j \}_{j=1}^{j=N-1}$ and neglect integrals
for $(-\infty,x_{N-1})$ and $(x_1,+\infty)$.
\item In order to restore the $\mathcal{PT}$-symmetry condition~\eqref{PT-sym-eigenvector},
we multiply the component $Y$ returned from the eigenvector by $e^{i \theta}$ with $\theta \in [0,2\pi]$
and require
\begin{gather*}
e^{i\theta} Y(x) = e^{-i\theta} \overline{Y(-x)} \quad \Rightarrow \quad
2i\theta = \log\frac{\overline{Y(-x)}}{Y(x)},
\end{gather*}
where the point $x$ is chosen so that $Y(x)$ and $Y(-x)$ are nonzero.
For example, we compute $\theta$ for all interior grid points $\{ x_j \}_{j=1}^{j=N-1}$
for which $Y(x_j) \neq 0$ and take the average. Both $Y$ and $Z$ in the same eigenvector
are rotated with the same angle $\theta$. Similarly,
this step is performed for $Y^\#$ and $Z^\#$ according to the
$\mathcal{PT}$-symmetry condition~\eqref{PT-sym-eigenvector-adjoint}.
\item We fix the sign of the adjoint eigenvectors at the Hamiltonian case $\gamma = 0$
by using (\ref{Ham-case}). Then we continue the eigenvectors and the adjoint
eigenvectors for simple eigenvalues before coalescence points.
Numerically, we take two steps in $\gamma$: $\gamma_1 < \gamma_2$,
with $|\gamma_2 - \gamma_1| \ll 1$. Suppose that the sign of eigenvector for $\gamma_1$ has been
chosen already. We take eigenvectors for $\gamma_1$ and $\gamma_2$ and compare them.
If eigenvectors have been made $\mathcal{PT}$-symmetric and properly normalized,
then the norm of their difference is either small
(the eigenvectors are almost the same) or close to $2$ (the eigenvectors are negatives of each other). We choose
the sign of the eigenvector so that the norm of their difference is small.
\end{enumerate}
With the refinements described above, we can now compute the Krein quantity $K(\lambda)$ defined by
(\ref{pt-krein}) using the same numerical method as the one used for computing the norms of eigenvectors.
In numerical computations, we have often encountered situations when eigenvalues nearly coalesce,
but the standard MATLAB$^\text{\textregistered}$ numerical routines do not approximate well
the coalescence of eigenvalues. In order to check if the eigenvectors
are linearly dependent near the possible coalescence point, we compute the norm
of the difference between the two eigenvectors (or opposites of each other)
for the two simple eigenvalues and plot it with respect to the parameter $\gamma$.
If the difference between the two eigenvectors vanishes as $\gamma$ is increased towards the
coalescence point, we say that the defective eigenvalue arises at the bifurcation
point. If the difference remains finite,
either we are dealing with the semi-simple eigenvalue at the coalescence point or
the two simple eigenvalues pass each other without coalescence.
\section{Numerical Examples}
\label{sec-examples}
In the numerical examples, we set $N=500$. This gives enough
accuracy for computing eigenvalues, as it was shown in~\cite{yusuke}.
We will demonstrate numerical results on Figures~\ref{fig:yang},\ref{fig:wadati},\ref{fig:kev3}
and~\ref{fig:kev4}. Each figure displays branches of the nonlinear modes $\Phi$
versus a parameter used in the numerical continuations
(either $\mu$ or $\gamma$), where the blue solid line corresponds to stable modes and
the red dashed line denotes unstable ones. The top and middle panels show
the power curves of $\| \Phi \|^2$, a sample profile
of the nonlinear mode $\Phi$, and the spectrum of linearization before and after the instability
bifurcation. The bottom panels show the imaginary part of eigenvalues
$\lambda$ and the Krein quantity of isolated eigenvalues. Green color corresponds to
eigenvalues $\lambda \in i \mathbb{R}$ with the positive Krein
signature, red -- to those with the negative Krein signature, and
black color is used for complex eigenvalues $\lambda \notin i \mathbb{R}$ and
for the continuous spectrum.
\begin{figure}[htbp]
\centerline{
\includegraphics[max size={1.3\textwidth}{0.95\textheight}]{w-2-paper}}
\caption{Scarf II potential~\eqref{pot-Wadati} with $V_0 = 2$, $\gamma = -2.21$.
(a) Power curves versus $\mu$.
(b) Amplitude profile for point $A$.
(c) Spectrum of linearization for point $A$.
(d) Same for point $B$.
(e) ${\rm Im}(\lambda)$ for the spectrum of linearization versus $\mu$.
(f) Krein quantities for isolated eigenvalues versus $\mu$. }
\label{fig:yang}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[max size={1.3\textwidth}{0.95\textheight}]{w-1-paper}}
\caption{Scarf II potential~\eqref{pot-Wadati} with $V_0 = 3$, $\gamma = -3.7$.
(a) Power curves versus $\mu$.
(b) Amplitude profile for point $A$.
(c) Spectrum of linearization for point $A$.
(d) Same for point $B$.
(e) ${\rm Im}(\lambda)$ for the spectrum of linearization versus $\mu$.
(f) Krein quantities for isolated eigenvalues versus $\mu$. }
\label{fig:wadati}
\end{figure}
Figure~\ref{fig:yang} (a)-(f) shows the instability bifurcation for the Scarf II potential (\ref{pot-Wadati})
studied in~\cite{yang} in the focusing case with $g=1$. Here $V_0 = 2$, $\gamma = -2.21$, and
the first branch of the nonlinear modes $\Phi$ is considered.
As two eigenvalues with different Krein signatures coalesce, they bifurcate into
a complex quadruplet, in agreement with Theorem \ref{theorem-main}.
Note that there's a small region of stability for the nonlinear modes
$\Phi$ of small amplitudes, as it was shown in~\cite{yang}.
Figure~\ref{fig:wadati} (a)-(f) shows the instability bifurcation for the Scarf II
potential~\eqref{pot-Wadati} studied in~\cite{wadati} in the focusing case with $g=1$.
Here $V_0=3$, $\gamma=-3.7$, and the second branch of the nonlinear modes $\Phi$
is considered. The second branch is unstable with at least one complex quadruplet
for all values of parameter $\mu$ used. The imaginary part of this complex quadruplet
is not visible on Figure~\ref{fig:wadati} (e) as it coincides with the location
of the continuous spectrum. In the presence of this complex quadruplet,
we observe a coalescence of two simple eigenvalues $\lambda_1,\lambda_2 \in i \mathbb{R}$
and the instability bifurcation into another complex quadruplet. Numerical evidence confirms
that the eigenvalues have the opposite Krein signatures prior to collision,
allowing us to predict the instability bifurcation, in agreement with Theorem \ref{theorem-main}.
\begin{figure}[htbp]
\centerline{
\includegraphics[max size={1.3\textwidth}{0.95\textheight}]{k-1-paper}}
\caption{Confining potential~\eqref{pot-BEC}, scaled as in~\eqref{pot-BEC-scaled}.
(a) Power curves versus $\gamma$.
(b) Amplitude profile for point $A$.
(c) Spectrum of linearization for point $A$.
(d) Same for point $B$.
(e) ${\rm Im}(\lambda)$ for the spectrum of linearization versus $\gamma$.
(f) Krein quantities for isolated eigenvalues versus $\gamma$. }
\label{fig:kev3}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[max size={1.3\textwidth}{0.95\textheight}]{k-2-paper}}
\caption{Confining potential~\eqref{pot-BEC}, scaled as in~\eqref{pot-BEC-scaled}.
(a) Power curves versus $\gamma$.
(b) Amplitude profile for point $A$.
(c) Spectrum of linearization for point $A$.
(d) Same for point $B$.
(e) ${\rm Im}(\lambda)$ for the spectrum of linearization versus $\gamma$.
(f) Krein quantities for isolated eigenvalues versus $\gamma$. }
\label{fig:kev4}
\end{figure}
Figures~\ref{fig:kev3},\ref{fig:kev4} (a)-(f) show the confining potential~\eqref{pot-BEC} studied in~\cite{kevrekidis},
in the defocusing case with $g=-2$. Compared to~\eqref{pot-BEC}, we use a scaled version of this
potential to match the one in~\cite{kevrekidis}:
\begin{equation}
\label{pot-BEC-scaled}
V(x) = x^2, \quad W(x) = 2\Omega^{-3/2}xe^{-\frac{x^2}{2\Omega}},
\end{equation}
where $\Omega=10^{-1}$ is a scaling parameter.
There are four branches of the nonlinear modes $\Phi$ shown,
out of which we highlight only the third and fourth branches.
The first branch is stable, whereas the second branch becomes unstable because of a coalescence
of a pair of eigenvalues $\pm \lambda \in i \mathbb{R}$ with the negative Krein signature
at the origin \cite{kevrekidis}. The third and fourth branches are studied in Figures~\ref{fig:kev3} and~\ref{fig:kev4}.
In Figure~\ref{fig:kev3} we can see that there are three bifurcations occurring at $\gamma_1\approx 0.07$,
$\gamma_2 \approx 0.1031$ and $\gamma_3\approx 0.1069$. For each bifurcation two eigenvalues with different Krein
signatures collide and bifurcate off to the complex plane in accordance with Theorem \ref{theorem-main}.
In addition, two simple eigenvalues with
different Krein signatures nearly coalesce near $\gamma_4 \approx 0.1$.
Figure~\ref{fig:comp} (a) shows the norm
of the difference between the two eigenvectors and two adjoint eigenvectors for the two simple eigenvalues
while $\gamma$ is increased towards $\gamma_4$. As the difference does not vanish,
we rule out this point as the bifurcation point for the defective eigenvalue. Consequently,
the eigenvalues are continued past this point with preservation of their Krein signatures.
In Figure~\ref{fig:kev4} we can see three bifurcations occurring at $\gamma_1\approx 0.1303$,
$\gamma_2\approx 0.1427$, and $\gamma_3\approx 0.2078$. At $\gamma_1$,
an eigenvalue pair with negative Krein signature coalesce at zero and become a pair of
real (unstable) eigenvalues. As $\gamma$ is increased towards $\gamma_2$,
two eigenvalues with opposite Krein signature move towards each other.
Figure~\ref{fig:comp} (b) illustrates that the norm of the difference between the two eigenvectors
and the two adjoint eigenvectors vanishes at the coalescence point.
Therefore, we conclude that at $\gamma_2$ we have a defective eigenvalue
which does not split into a complex quadruplet. According to Theorem \ref{theorem-main},
the defective eigenvalue does not split into complex unstable eigenvalues only if
the non-degeneracy condition~\eqref{non-degeneracy} is not satisfied. Similar safe passing
of eigenvalues of opposite Krein signature through each other is observed in~\cite{yang}.
The behavior near $\gamma_2$ shows that having opposite Krein signatures
prior to coalescence of two simple eigenvalues into a defective eigenvalue
is a \emph{necessary but not sufficient} condition for the instability bifurcation.
At $\gamma_3$, two eigenvalues with opposite Krein signatures coalesce and
bifurcate into a complex quadruplet according to Theorem \ref{theorem-main}.
\begin{figure}[t]
\centerline{
\includegraphics[width=0.7\textwidth]{kev34-v-compare}}
\caption{The norm of the difference between the two eigenvectors and the two adjoint
eigenvectors prior to a possible coalescence point: (a) for Figure~\ref{fig:kev3}
(b) for Figure~\ref{fig:kev4}.}
\label{fig:comp}
\end{figure}
\section{Discussion}
\label{sec-conclusion}
In this work, we introduced the Krein quantity for simple isolated eigenvalues
in the linearization of the nonlinear modes in the $\mathcal{PT}$-symmetric NLS equation.
We proved that the Krein quantity is zero for complex eigenvalues and nonzero for simple
purely imaginary eigenvalues. When two simple eigenvalues coalesce
on the imaginary axis in a defective eigenvalue,
the Krein quantity vanishes and we proved
under the non-degeneracy assumption
that this bifurcation point produces
complex unstable eigenvalues on one side of the bifurcation point.
This result shows that the main feature of the instability bifurcation
in Hamiltonian systems is extended to the $\mathcal{PT}$-symmetric NLS equation.
There are nevertheless limitations of this theory in the $\mathcal{PT}$-symmetric systems.
First, the adjoint eigenvectors are no longer related to the eigenvectors of the spectral
problem, which opens up a problem of normalizing the adjoint eigenvector relative to the
eigenvector. We fixed the sign of the adjoint eigenvector in the Hamiltonian limit
and continue the sign off the Hamiltonian limit by using continuity of eigenvectors
along the parameters of the model.
Second, if the bifurcation point corresponds to a semi-simple eigenvalue, then the bifurcation theory
does not lead to the same conclusion as in the Hamiltonian case.
The first-order perturbation theory results in the non-Hermitian matrices, hence
it is not clear how to conclude on the splitting
of the semi-simple eigenvalues on each side of the bifurcation point.
Finally, coalescence of the simple purely imaginary eigenvalues at the origin and the related instability
bifurcations are observed frequently in the $\mathcal{PT}$-symmetric systems and they are not
predicted from the Krein quantity. Therefore, we conclude that the stability theory
of Hamiltonian systems cannot be fully extended to the $\mathcal{PT}$-symmetric NLS equation,
only the necessary condition for the instability bifurcation can be, as is shown in this work.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,771
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Żurawica Rozrządowa – przystanek osobowy PKP Polskich Linii kolejowych znajdujący się w Żurawicy, w województwie podkarpackim, w Polsce.
W skład przystanku wschodzą dwa perony: 200-metrowy, jednokrawędziowy przy torze w kierunku Przemyśla oraz 300-metrowy, jednokrawędziowy przy torze w stronę Rzeszowa. Dojście na dłuższy peron (peron 2.) jest możliwy dzięki kładce.
W roku 2017 stacja obsługiwała 50–99 pasażerów na dobę.
Połączenia
Z przystanku można dojechać elektrycznymi pociągami regionalnymi do Przemyśla, Jarosławia, Przeworska, Rzeszowa, Dębicy oraz Tarnowa.
Zobacz też
Katastrofa kolejowa w Żurawicy
Przypisy
Linki zewnętrzne
Stacje i przystanki kolejowe w powiecie przemyskim
Żurawica
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,414
|
I don't think that in the history of television there has been or will be such a significant show as the series Modern Marvels - Season 22, which, in my opinion, literally cuts space and time. From the opening credits to the final scenes, everything about Modern Marvels - Season 22's story in the Documentary genre is overwhelming: the brutality, the romance, the duality of all life on Earth. The Modern Marvels - Season 22 series, which debuted in 2021, deserves a detailed study by all fans of the History genre, as the story is coherent, exciting, worthy of re-watching. Bruce Nash tries himself in different genres, but Documentary is a direction where he manages to realize his potential especially effectively, which once again underlines the series Modern Marvels - Season 22. Choosing a series for the next viewing, I often give preference to the products of a particular country, as a rule, this is United States, and Modern Marvels - Season 22 has strengthened my conviction in the correctness of such a selection.... Show more
Being the Queen
My Truth: The Rape of Two Coreys
AC/DC: No Bull
Turtle Power: The Definitive History of the Teenage Mutant Ninja Turtles
JFK Revisited: Through the Looking Glass
Tech Billionaires: Mark Zuckerberg
Mega Shippers - Season 1
Space Junk 3D
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 410
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Rocker Benji Madden's Australian radio show boast he once fooled a police officer into thinking he was his twin brother Joel has landed both Good Charlotte stars in trouble back in Los Angeles.
During a recent appearance on "The Kyle and Jackie O Show," Madden confessed to switching licenses after he was pulled over for speeding, revealing he panicked when he realized he was close to losing his driving privileges following a string of similar traffic offences.
Madden told the radio show hosts he pretended to be Joel and it appears he got away with the scam – his brother received the traffic ticket instead of him.
The representative says, "We will look into this," adding that any future traffic stops will be scrutinized and the brothers could face fraud charges and jail time if they ever pull a similar stunt.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}\label{Sec:1}
A ternary $[n,k]$ code $C$ is a $k$-dimensional vector subspace
of $\FF_3^n$,
where $\FF_3$ denotes the finite field of order $3$.
The weight $\wt(x)$ of a vector $x$ is
the number of non-zero components of $x$.
The minimum non-zero weight of all codewords in $C$ is called
the minimum weight of $C$.
A ternary $[n,k,d]$ code is a ternary $[n,k]$ code with minimum weight $d$.
Throughout this note,
we denote the minimum weight of a code $C$ by $d(C)$.
Shimada and Zhang~\cite{Shimada} studied
the existence of
polarizations on
the supersingular $K3$ surfaces in characteristic $3$
with Artin invariant $1$
(see~\cite[Theorem~1.5]{Shimada} for the details).
This was done by
reducing the problem of the existence of
the polarizations
to a problem of the existence of ternary $[12,5]$ codes $C$
satisfying the following
conditions:
\begin{align}
\label{C1}
& \wt((x_1,x_2,\ldots,x_{10}))\equiv y_1y_2 \pmod 3,
\\
\label{C2}
&
\text{ if } c
\text{ is not the zero vector, } \text{ then }
\wt((x_1,x_2,\ldots,x_{10})) \ge 3,
\\
\label{C3}
&
\text{ if }\wt((x_1,x_2,\ldots,x_{10}))=3,
\text{ then } (y_1,y_2) \ne (0,0),
\end{align}
for any codeword
$c=(x_1,x_2,\ldots,x_{10},y_1,y_2) \in C$ (see~\cite[Claim~5.2]{Shimada}).
Seven ternary $[12,5]$ codes satisfying the conditions
(\ref{C1})--(\ref{C3}) were found by
Shimada and Zhang~\cite{Shimada}.
This motivates us to classify all such ternary $[12,5]$ codes.
For ternary $[12,5]$ codes satisfying the conditions
(\ref{C1})--(\ref{C3}),
the following equivalence is considered in~\cite{Shimada}.
We say that two ternary $[12,5]$ codes satisfying the conditions
(\ref{C1})--(\ref{C3})
are {\em SZ-equivalent} if one can be obtained from the
other by using the following:
\begin{equation}\label{eq:eq}
\begin{split}
&(x_1,\ldots,x_{10},y_1,y_2)
\\
&\mapsto
((-1)^{\alpha_1} x_{\sigma(1)},\ldots,
(-1)^{\alpha_{10}} x_{\sigma(10)},
(-1)^\beta y_{\tau(1)},(-1)^\beta y_{\tau(2)}),
\end{split}
\end{equation}
where $\alpha_1,\ldots,\alpha_{10},\beta \in \{0,1\}$
and $\sigma \in S_{10}, \tau \in S_2$
(see~\cite[Remark~5.3]{Shimada}).
Here, $S_n$ denotes the symmetric group of degree $n$.
The main aim of this note is to give the following
classification, which is based on a computer calculation.
\begin{thm}\label{thm}
Any ternary $[12,5]$ code satisfying the conditions
{\rm (\ref{C1})--(\ref{C3})} is SZ-equivalent to
one of the seven codes given in {\rm \cite[Remark~5.3]{Shimada}}.
\end{thm}
To complete the above classification,
ternary $[10,5,d]$ codes are classified for
the cases $d=3$ and $4$.
\section{Characterization of ternary $[12,5]$ codes
satisfying (\ref{C1})--(\ref{C3})}\label{Sec:pre}
Let $C$ be a ternary $[n,k]$ code.
The code obtained from $C$ by deleting some coordinates $I$
in each codeword is called the {\em punctured code} of $C$ on $I$.
Throughout this note, we denote
the punctured code of a ternary $[12,5]$ code
$C$ on $\{11,12\}$ by $Pun(C)$.
Let $d_{max}(n,k)$ denote the largest minimum weight among
ternary $[n,k]$ codes.
It is known that $d_{max}(10,5)=5$ and
$d_{max}(12,5)=6$ (see~\cite{Brouwer-Handbook}, \cite{Grassl}).
\begin{lem}\label{lem:pun}
If $C$ is a ternary $[12,5]$ code satisfying the condition
{\rm (\ref{C2})},
then $Pun(C)$ is a
ternary $[10,5]$ code and $d(Pun(C)) \in \{3,4,5\}$.
\end{lem}
\begin{proof}
Suppose that $Pun(C)$ has dimension at most $4$.
Then we may assume without loss of generality that
$C$ has generator matrix whose first row is
$(0,0,\ldots,0,y_1,y_2)$, where
$(y_1,y_2) \ne (0,0)$.
This contradicts with the condition (\ref{C2}).
Hence, $Pun(C)$ is a ternary $[10,5]$ code.
Again, by the condition (\ref{C2}),
$Pun(C)$ has minimum weight at least $3$.
Since $d_{max}(10,5)=5$, the result follows.
\end{proof}
\begin{lem}\label{lem:d}
Let $C$ be a ternary $[12,5]$ code satisfying the conditions
{\rm (\ref{C1})--(\ref{C3})}.
\begin{itemize}
\item[\rm (i)] $d(Pun(C))\in \{4,5\}$ if and only if $d(C)=6$.
\item[\rm (ii)] $d(Pun(C))=3$ if and only if $d(C) =4$.
\end{itemize}
\end{lem}
\begin{proof}
By Lemma~\ref{lem:pun},
$Pun(C)$ is a
ternary $[10,5]$ code and $d(Pun(C)) \in \{3,4,5\}$.
It is trivial that $d(C)-d(Pun(C)) \in \{0,1,2\}$.
Suppose that $d(Pun(C))\in \{4,5\}$.
Let $x=(x_1,\ldots,x_{10})$ be a codeword of $Pun(C)$.
If $\wt(x)=4$ (resp.\ $5$), then any corresponding codeword
$(x_1,\ldots,x_{10},y_1,y_2)$ of $C$
has weight $6$ (resp.\ $7$), by the condition (\ref{C1}).
Since $d_{max}(12,5)=6$, we have that $d(C)=6$.
Conversely, if $d(C)=6$, then it follows from
$d_{max}(10,5)=5$ that $d(Pun(C))\in \{4,5\}$.
Suppose that $d(Pun(C))=3$.
Let $x=(x_1,\ldots,x_{10})$ be a codeword of $Pun(C)$.
If $\wt(x)=3$, then any corresponding codeword
$(x_1,\ldots,x_{10},y_1,y_2)$ of $C$
has weight $4$, by the conditions (\ref{C1}) and (\ref{C3}).
Hence, we have that $d(C) =4$.
Conversely, suppose that $d(C)=4$.
Then $d(Pun(C)) \in \{2,3,4\}$. By the condition (\ref{C2}),
$d(Pun(C)) \in \{3,4\}$. From the statement (i),
$d(Pun(C)) = 3$.
\end{proof}
Recall that
two ternary codes are equivalent if one can be obtained from the
other by permuting the coordinates and (if necessary) changing
the signs of certain coordinates.
For ternary $[10,5]$ codes, we consider this
usual equivalence.
\begin{lem}\label{lem:puneq}
Let $C$ and $C'$ be ternary $[12,5]$
codes satisfying the conditions
{\rm (\ref{C1})--(\ref{C3})}.
Suppose that
$C$ and $C'$ are SZ-equivalent.
Then $Pun(C)$ and $Pun(C')$ are equivalent.
\end{lem}
\begin{proof}
Suppose that $C$ is obtained from $C'$
by (\ref{eq:eq}).
Then $Pun(C)$ can be obtained from $Pun(C')$ by
$
(x_1,\ldots,x_{10}) \mapsto
((-1)^{\alpha_1} x_{\sigma(1)},\ldots,
(-1)^{\alpha_{10}} x_{\sigma(10)}).
$
\end{proof}
By considering the inverse operation of puncturing,
one can construct ternary $[12,5]$
codes satisfying the conditions
(\ref{C1})--(\ref{C3}) as follows.
Throughout this note, we denote the ternary code
having generator matrix $G$ by $C(G)$.
Suppose that $C(G)$ is a
ternary $[10,5]$ code and $d(C(G)) \in \{3,4,5\}$.
Let $g_i$ denote the $i$th row of $G$.
Consider the following generator matrix:
\begin{equation}\label{eq:gmat}
\left(\begin{array}{ccccc|cc}
& & & & & a_1 & b_1\\
& & G& & & \vdots & \vdots\\
& & & & & a_5 & b_5\\
\end{array}\right),
\end{equation}
where
\[
(a_i,b_i)=
\begin{cases}
(0,0),(0,1),(0,2),(1,0),(2,0) &\text{ if } \wt(g_i)\equiv 0 \pmod 3,\\
(1,1),(2,2) &\text{ if } \wt(g_i)\equiv 1 \pmod 3,\\
(1,2),(2,1) &\text{ if } \wt(g_i)\equiv 2 \pmod 3.
\end{cases}
\]
We denote this generator matrix
by $G(a,b)$, where $a=(a_1,\ldots,a_5)$ and $b=(b_1,\ldots,b_5)$.
The set of the codes $C(G(a,b))$ contains all
ternary $[12,5]$ codes $C$ satisfying the conditions
(\ref{C1}) and $Pun(C(G(a,b)))=C(G)$.
Hence, in this way, every
ternary $[12,5]$ code satisfying the conditions
(\ref{C1})--(\ref{C3}) can be obtained from some ternary
$[10,5]$ code.
Here, by Lemma~\ref{lem:d}, its minimum weight is $3,4$ or $5$.
In addition, if $C(G)$ and $C(G')$ are equivalent $[10,5]$
codes, then
the sets of all codes $C(G(a,b))$
satisfying the conditions (\ref{C1})--(\ref{C3})
is obtained from the set of all codes $C(G'(a,b))$
satisfying the same conditions
by considering (\ref{eq:eq}) with $\beta=0$ and $\tau$ is the identity
permutation.
Hence, it is sufficient to consider only inequivalent
ternary $[10,5,d]$ codes with $d \in \{3,4,5\}$
for the classification of
ternary $[12,5]$ codes satisfying the conditions
(\ref{C1})--(\ref{C3}).
This is a reason why we consider
the classification of
ternary $[10,5,d]$ codes with $d \in \{3,4,5\}$
in the next section.
\section{Ternary $[10,5,d]$ codes with $d \in \{3,4,5\}$}
\label{sec:10}
There is a unique ternary $[10,5,5]$ code, up to equivalence~\cite{GS}.
In this section, we give a classification of
ternary $[10,5,d]$ codes with
$d \in \{3,4\}$, which is based on a computer calculation.
We describe how ternary $[10,5,3]$ codes and $[10,5,4]$ codes
were classified.
Let $C$ be a ternary $[10,5,3]$ code
(resp.\ $[10,5,4]$ code).
We may assume without loss of generality that
$C$ has generator matrix
of the following form:
\[
G=
\left(\begin{array}{cccccc}
&I_{5}& & A &\\
\end{array}\right),
\]
where $A$ is a $5 \times 5$ matrix over $\FF_3$
and $I_5$ denotes the identity matrix of order $5$.
Thus, we only need consider the set of $A$,
rather than the set of generator matrices.
The set of matrices $A$ was constructed, row by row,
as follows, by a computer calculation.
Let $r_i$ be the $i$th row of $A$.
Then, we may assume without loss of generality that
$r_1=(0,0,0,1,1)$ (resp.\ $r_1=(0,0,1,1,1)$),
by permuting and (if necessary) changing
the signs of the columns of $A$.
Let $e_1,\ldots,e_5$ denote the vectors $(1,0,0,0,0),\ldots,
(0,0,0,0,1)$, respectively.
We denote the ternary code
generated by vectors $y_1,y_2,\ldots,y_s$
by $\langle y_1,y_2,\ldots,y_s \rangle$.
For $x=(x_1,\ldots,x_5) \in \FF_3^5$, consider
the following conditions:
\begin{itemize}
\item
the first nonzero element of $x$ is $1$,
\item
$\wt(x)\ge 2$ (resp.\ $\wt(x)\ge 3$),
\item
the ternary code $\langle (e_1,r_1),(e_2,x) \rangle$
has minimum weight $3$ (resp.\ $4$),
\item
$x_{1} \le x_{2} \le x_{3} \le 1$ and $x_{4} \le x_{5}$
(resp.\
$x_{1} \le x_{2} \le 1$ and $x_{3} \le x_{4} \le x_{5}$),
where we consider a natural order on
the elements of $\FF_3=\{0,1,2\}$ by $0<1<2$.
\end{itemize}
The determination of the minimum weights
was done by a computer calculation for all codes in this note.
Let $X_1$ be the set of vectors $x \in \FF_3^5$
satisfying the first three conditions.
Let $X_2$ be the set of vectors $x \in X_1$
satisfying the fourth condition.
Our computer calculation shows that
$(\#X_1,\#X_2) = (115,18)$
(resp.\ $(88,14)$).
Define a lexicographical order on
$X_1$ induced by the above order of $\FF_3$,
that is, $(a_1,\dots,a_5) < (b_1,\dots,b_5)$
if $a_1 < b_1$, or
$a_1=b_1,\dots,a_{k}=b_{k}$ and $a_{k+1}<b_{k+1}$
for some $k \in \{1,2,3,4\}$.
The matrices $A$ were constructed, row by row,
satisfying
the following conditions:
\begin{itemize}
\item
the ternary code
$\langle (e_s,r_s) \mid s=1,2,3 \rangle$
has minimum weight $3$ (resp.\ $4$), where
$r_2 \in X_2,\ r_3 \in X_1$,
\item
the ternary code
$\langle (e_s,r_s) \mid s=1,2,3,4 \rangle$
has minimum weight $3$ (resp.\ $4$), where
$r_2 \in X_2,\ r_3,r_4 \in X_1\ (r_3 < r_4)$,
\item
the ternary code
$\langle (e_s,r_s) \mid s=1,2,3,4,5 \rangle$
has minimum weight $3$ (resp.\ $4$), where
$r_2 \in X_2,\
r_3, r_4, r_5 \in X_1\ (r_3<r_4<r_5)$.
\end{itemize}
It is obvious that the set of the matrices $A$
which must be checked to achieve
a complete classification, can be obtained in this way.
Then, by a computer calculation,
we found $4328352$ (resp.\ $650051$) matrices $A$.
Our computer calculation shows the $4328352$ ternary $[10,5,3]$ codes
(resp.\ $650051$ ternary $[10,5,4]$ codes)
are divided into $527$ (resp.\ $64$)
classes by comparing their Hamming weight enumerators.
For each Hamming weight enumerator,
to test equivalence of codes, we use the
algorithm given in~\cite[Section 7.3.3]{KO} as follows.
For a ternary $[n,k]$ code $C$, define
the digraph $\Gamma(C)$ with vertex set
\[
(C-\{\0\}) \cup (\{1,2,\dots,n\}\times (\FF_3-\{0\}))
\]
and arc set
\begin{align*}
&
\{(c,(j,c_j)) \mid c=(c_{1},\ldots,c_{n}) \in C-\{\0\}, c_j \ne 0,
1 \le j \le n\}
\\ &
\cup \{((j,1),(j,2)),((j,2),(j,1)) \mid 1 \le j \le n\}.
\end{align*}
Then, two ternary $[n,k]$ codes $C$ and $C'$ are equivalent
if and only if $\Gamma(C)$ and $\Gamma(C')$ are isomorphic.
We use the package {\tt GRAPE}~\cite{GRAPE}
of {\tt GAP}~\cite{GAP4} for digraph isomorphism testing.
After checking whether codes are equivalent or not
by a computer calculation for each
Hamming weight enumerator, we have the following:
\begin{prop}
There are $135$ ternary $[10,5,4]$ codes, up to equivalence.
There are $1303$ ternary $[10,5,3]$ codes, up to equivalence.
\end{prop}
We denote the $135$ ternary $[10,5,4]$ codes by
$C_{10,4,i}$ $(i=1,2,\ldots,135)$, and
we denote the $1303$ ternary $[10,5,3]$ codes by
$C_{10,3,i}$ $(i=1,2,\ldots,1303)$.
Generator matrices of all codes
can be obtained electronically from~\cite{Araya}.
The unique ternary $[10,5,5]$ code $C_{10,5}$ is
formally self-dual,
that is, the Hamming weight enumerators of the code and its
dual code are identical.
In addition, the supports of the codewords of minimum weight in
$C_{10,5}$ form a $3$-design~\cite{DGH}.
We verified by a computer calculation that
$38$ ternary $[10,5,4]$ codes and
$242$ ternary $[10,5,3]$ codes are formally self-dual.
In addition, we verified by a computer calculation that
the supports of the codewords of minimum weight in
only the code $C_{10,4,132}$ form a $2$-design and
the supports of the codewords of minimum weight in
$C_{10,4,i}$ form a $1$-design
for only $i=6,86,87,89,132$.
\section{Ternary $[12,5]$ codes satisfying (\ref{C1})--(\ref{C3})}
In this section,
we give a classification of
ternary $[12,5]$ codes satisfying the conditions
{\rm (\ref{C1})--(\ref{C3})}, which is based on a computer calculation.
This is obtained from the classification of
ternary $[10,5,d]$ codes with $d \in \{3,4,5\}$,
by using the method given in Section~\ref{Sec:pre}.
\subsection{From the $[10,5,5]$ code and the $[10,5,4]$ codes}
As described in the previous section,
there is a unique ternary $[10,5,5]$ code, up to equivalence~\cite{GS}.
It follows from~\cite{DGH} that
this code $C_{10,5}$ has generator matrix
$G_{10,5}=
\left(\begin{array}{cccccc}
I_{5} & A \\
\end{array}\right)$,
where $A$ is the following circulant matrix:
\[
A=
\left(\begin{array}{cccc}
12210\\
01221\\
10122\\
21012\\
22101\\
\end{array}\right).
\]
In order to construct all ternary $[12,5]$ codes $C$
satisfying the conditions (\ref{C1}) and $Pun(C)=C_{10,5}$,
we consider generator matrices $G_{10,5}(a,b)$ of the form (\ref{eq:gmat}).
Since the weight of each row of $G_{10,5}$ is $5$,
$(a_i,b_i)=(1,2)$ or $(2,1)$ for $i=1,2,3,4,5$.
By (\ref{eq:eq}), we may assume that $(a_1,b_1)=(1,2)$.
Since the weight of the sum of the first row and the second row
of $G_{10,5}$ is $5$, $(a_2,b_2)$ must be $(1,2)$.
Similarly, we have that
$(a_i,b_i)=(1,2)$ for $i=3,4,5$,
since $A$ is circulant.
In addition, we verified by a computer calculation that this code
satisfies the condition {\rm (\ref{C1})}.
Note that the code automatically satisfies
the conditions (\ref{C2}) and (\ref{C3}).
We denote the code by $\cC_{12,1}$.
Now, consider the ternary $[10,5,4]$ codes $C_{10,4,i}$ $(i=1,2,\ldots,135)$.
By considering generator matrices of the form (\ref{eq:gmat}),
we found all ternary $[12,5]$ codes $C$
satisfying the conditions (\ref{C1}) and $Pun(C)=C_{10,4,i}$.
This was done by a computer calculation.
We denote by $G_{10,4,i}$ the generator matrix
$\left(\begin{array}{cccccc}
I_{5}& A \\
\end{array}\right)$
of $C_{10,4,i}$ for each $i$.
Since the weight of the first row of $A$ is $3$ (see Section~\ref{sec:10}),
by (\ref{eq:eq}), we may assume that $(a_1,b_1)=(1,1)$ in (\ref{eq:gmat}).
Under this situation,
we verified by a computer calculation that only the codes
$C_{10,4,60}$ and $C_{10,4,132}$ give
ternary $[12,5]$ codes satisfying the condition (\ref{C1}).
Note that these codes automatically satisfy
the conditions (\ref{C2}) and (\ref{C3}).
In Table~\ref{Tab:d4mat}, we list the matrices $A$ and
$(a^T,b^T)$ in $G_{10,4,i}(a,b)$ for $i=60,132$,
where $a^T$ denotes the transposed of a vector $a$.
It can be seen by hand that
the two codes $C(G_{10,4,60}(a,b))$ are SZ-equivalent.
By Lemma~\ref{lem:puneq},
there are two ternary $[12,5]$ codes $C$
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}
and the condition that $Pun(C)$ is a ternary $[10,5,4]$ code.
We denote the two codes by $\cC_{12,2}$ and $\cC_{12,3}$,
respectively (note that take the first $(a^T,b^T)$ for $i=60$).
\begin{table}[thb]
\caption{Generator matrices $G_{10,4,i}(a,b)$ $(i=60,132)$}
\label{Tab:d4mat}
\begin{center}
{\footnotesize
\begin{tabular}{c|c|cc}
\noalign{\hrule height0.8pt}
$i$& $A$ & \multicolumn{2}{c}{$(a^T,b^T)$} \\
\hline
&&&\\
60 &
$\left(\begin{array}{c}
0 0 1 1 1\\
0 1 0 1 1\\
1 0 1 0 1\\
1 1 0 0 1\\
1 2 2 1 0
\end{array}\right)$&$\left(\begin{array}{c}
1 1\\
2 2\\
2 2\\
1 1\\
1 2\\
\end{array}\right),$&$\left(\begin{array}{c}
1 1\\
2 2\\
2 2\\
1 1\\
2 1\\
\end{array}\right)$ \\
&&&\\
\hline
&&&\\
132 &
$\left(\begin{array}{c}
0 0 1 1 1\\
0 1 0 1 1\\
1 0 1 0 1\\
1 1 0 0 1\\
1 1 1 1 1
\end{array}\right)$&$\left(\begin{array}{c}
1 1\\
2 2\\
2 2\\
1 1\\
0 0\\
\end{array}\right)$ \\
&&&\\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
Lemma~\ref{lem:d} shows that there are no other
ternary $[12,5,6]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}.
Hence, we have the following:
\begin{lem}\label{lem:d6}
Up to SZ-equivalence,
there are three ternary $[12,5,6]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}.
\end{lem}
\subsection{From the $[10,5,3]$ codes}
By considering generator matrices of the form (\ref{eq:gmat}),
we found all ternary $[12,5]$ codes $C$
satisfying the conditions (\ref{C1}) and
$Pun(C)=C_{10,3,i}$ $(i=1,2,\ldots,1303)$.
This was done by a computer calculation.
We denote by $G_{10,3,i}$ the generator matrix
$\left(\begin{array}{cccccc}
I_{5}& A \\
\end{array}\right)$
of $C_{10,3,i}$ for each $i$.
Since the weight of the first row of $A$ is $2$ (see Section~\ref{sec:10}),
by (\ref{eq:eq}), we may assume that $(a_1,b_1)=(0,1)$ in (\ref{eq:gmat}).
Under this situation,
we verified by a computer calculation that only the codes
$C_{10,3,i}$ give
ternary $[12,5]$ codes satisfying the condition (\ref{C1})
for
\begin{align*}
i=&302, 639, 662, 666, 667, 756, 878, 957, 958, 987, \\
&1210, 1215, 1241, 1245, 1263, 1285, 1297, 1298, 1299.
\end{align*}
In this case, there are codes
satisfying the condition (\ref{C1}), but not (\ref{C3}).
We verified by a computer calculation that only the codes
$C_{10,3,i}$ give
ternary $[12,5]$ codes satisfying the conditions
(\ref{C1}) and (\ref{C3}) for $i=302,666,987,1245$.
Note that these four codes automatically satisfy the condition (\ref{C2}).
In Table~\ref{Tab:d3mat}, we list the matrices $A$ and
$(a^T,b^T)$ in $G_{10,4,i}(a,b)$ for $i=302,666,987,1245$.
By Lemma~\ref{lem:puneq},
there are four ternary $[12,5]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}
and the condition that $Pun(C)$ is a ternary $[10,5,3]$ code.
We denote the four codes by $\cC_{12,i}$ $(i=4,5,6,7)$,
respectively.
\begin{table}[thb]
\caption{Generator matrices $G_{10,3,i}(a,b)$ $(i=302,666,987,1245)$}
\label{Tab:d3mat}
\begin{center}
{\footnotesize
\begin{tabular}{c|c|c||c|c|c}
\noalign{\hrule height0.8pt}
$i$& $A$ & $(a^T,b^T)$ &
$i$& $A$ & $(a^T,b^T)$ \\
\hline
&&&&&\\
302 &
$\left(\begin{array}{c}
0 0 0 1 1\\
0 1 1 0 0\\
1 0 1 0 1\\
1 1 0 1 0\\
1 1 2 2 1
\end{array}\right)$&$\left(\begin{array}{c}
0 1\\0 1\\2 2\\2 2\\0 1\\
\end{array}\right)$ &
987&
$\left(\begin{array}{c}
0 0 0 1 1\\
0 0 1 0 1\\
0 1 0 1 0\\
0 1 1 0 0\\
1 1 2 2 1
\end{array}\right)$&$\left(\begin{array}{c}
0 1\\2 0\\2 0\\0 1\\0 0\\
\end{array}\right)$ \\
&&&&&\\
\hline
&&&&&\\
666 &
$\left(\begin{array}{c}
0 0 0 1 1\\
0 1 1 0 0\\
1 0 1 0 1\\
1 1 0 1 0\\
1 2 2 2 2
\end{array}\right)$&$\left(\begin{array}{c}
0 1\\0 1\\2 2\\2 2\\2 0\\
\end{array}\right)$ &
1245&
$\left(\begin{array}{c}
0 0 0 1 1\\
0 0 1 0 1\\
0 1 0 1 0\\
0 1 1 0 0\\
0 1 1 1 1
\end{array}\right)$&$\left(\begin{array}{c}
0 1\\2 0\\2 0\\0 1\\1 2\\
\end{array}\right)$ \\
&&&&&\\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
Lemma~\ref{lem:d} shows that there are no other
ternary $[12,5,4]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}.
Hence, we have the following:
\begin{lem}\label{lem:d5}
Up to SZ-equivalence,
there are four ternary $[12,5,4]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}.
\end{lem}
Up to SZ-equivalence,
seven ternary $[12,5]$ codes
satisfying the conditions (\ref{C1})--(\ref{C3}) are known
(see~\cite[Remark~5.3]{Shimada}).
Lemmas~\ref{lem:d6} and~\ref{lem:d5} show that there are no other
ternary $[12,5]$ codes
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})}.
Therefore, we have Theorem~\ref{thm}.
\subsection{Some properties}
For the ternary $[12,5]$ codes $C$
satisfying the conditions {\rm (\ref{C1})--(\ref{C3})},
instead of the Hamming weight enumerators,
we consider the weight enumerators
$\displaystyle{
\sum_{(x_1,\ldots,x_{10},y_1,y_2) \in C} x^{\wt((x_1,\ldots,x_{10}))}
y^{n_1}z^{n_2}
}$,
where $n_1$ and $n_2$ are the numbers of $1$'s and $2$'s in
$(y_1,y_2)$, respectively.
We verified by a computer calculation
that the codes $\cC_{12,i}$ $(i=1,2,\ldots,7)$ have
the following weight enumerators $W_i$:
\begin{align*}
W_1=& 1+ 72 x^5 y z+ 60 x^6 + 90 x^8 y z+ 20 x^9,
\\
W_2=&
1+ 9x^4 z^2 + 9x^ 4 y^2 + 18x^ 5 y z + 24x^ 6 + 36x^ 6 z + 36x^ 6 y + 18x^ 7 z^2
+ 18x^ 7 y^2
\\ &
+ 36x^ 8 y z + 2x^ 9 + 18x^ 9 z + 18x^ 9 y,
\\ &
1+ 15x^ 4 z^2 + 15x^ 4 y^2 + 60x^ 6 + 60x^ 7 z^2 + 60x^ 7 y^2 + 20x^ 9 + 6x^{10} z^2
+ 6x^{10} y^2,
\\ &1 + 2x^3z + 2x^3y + 4x^4z^2 + 4x^4y^2 + 24x^5yz + 18x^6 + 38x^6z+ 38x^6y
\\ &
+ 22x^7z^2 + 22x^7y^2 + 30x^8yz + 8x^9 + 14x^9z + 14x^9y
+ x^{10}z^2 + x^{10}y^2,
\\ &
1+ 3x^3z+ 3x^3y+ 3x^4z^2+ 3x^4y^2+ 18x^5yz+ 24x^6+ 39x^6z+ 39x^6y
\\ &
+ 21x^7z^2+ 21x^7y^2+ 36x^8yz+ 2x^9+ 12x^9z+ 12x^9y+ 3x^{10}z^2+ 3x^{10}y^2,
\\ &
1+ 4x^3z+ 4x^3y+ 5x^4z^2+ 5x^4y^2+ 24x^5yz+ 18x^6+ 34x^6z+ 34x^6y
\\ &
+ 20x^7z^2+ 20x^7y^2+ 30x^8yz+ 8x^9+ 16x^9z+ 16x^9y+ 2x^{10}z^2+2x^{10}y^2,
\\ &
1+ 6x^3z+ 6x^3y+ 9x^4z^2+ 9x^4y^2+ 36x^5yz+ 24x^6+ 42x^6z+ 42x^6y
\\ &
+ 18x^7z^2+ 18x^7y^2+ 18x^8yz+ 2x^9+ 6x^9z+ 6x^9y,
\end{align*}
respectively.
These weight enumerators guarantee that the codes
$\cC_{12,i}$ $(i=1,2,\ldots,7)$
satisfy the conditions {\rm (\ref{C1})--(\ref{C3})}.
By putting $y=z=1$,
the above weight enumerators
determine the Hamming weight enumerators of $Pun(\cC_{12,i})$ $(i=1,2,\ldots,7)$.
This implies that $\cC_{12,1}$
is SZ-equivalent to
${\mathcal C}_7$ in~\cite[Table~5.1]{Shimada}.
In addition,
by comparing generator matrices, it is easy to see that
$\cC_{12,i}$ $(i=2,3,\ldots,7)$ are equal to
${\mathcal C}_6$,
${\mathcal C}_5$,
${\mathcal C}_3$,
${\mathcal C}_4$,
${\mathcal C}_2$ and
${\mathcal C}_1$ in~\cite[Table~5.1]{Shimada}, respectively.
\begin{rem}
Shimada and Zhang~\cite{Shimada} also considered
the existence of ternary $[12,4,6]$ codes
satisfying the condition that
all codewords have weight divisible by three,
in the proof of Theorem~1.4
(see~\cite[Claim~6.2]{Shimada}).
We point out that a code satisfying the condition
is self-orthogonal.
There is a unique
self-orthogonal ternary $[12,4,6]$ code,
up to equivalence~\cite[Table~1]{MPS}.
\end{rem}
\bigskip
\noindent {\bf Acknowledgments.}
The authors would like to thank the anonymous referee for useful comments.
This work is supported by JSPS KAKENHI Grant Number 23340021.
In this work, the supercomputer of ACCMS, Kyoto University was partially used.
|
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{"url":"http:\/\/www.cjss.ac.cn\/EN\/volumn\/volumn_1154.shtml","text":"CNKI\u671f\u520a\u5168\u6587\u6570\u636e\u5e93\n\n\u300a\u4e2d\u56fd\u5b66\u672f\u671f\u520a\u6587\u6458\u300b\n\u300a\u4e2d\u56fd\u7269\u7406\u6587\u6458\u300b\n\u300a\u4e2d\u56fd\u5929\u6587\u5b66\u6587\u6458\u300b\n\n#### Table of Content\n\n15 May 2009, Volume 29 Issue 3\n Feature of the Martian Magnetic Field Structure ZHANG Yiteng;LI Lei 2009, 29 (3):\u00a0 257-261.\u00a0 doi: 10.11728\/cjss2009.03.257 Abstract ( 3371 )\u00a0\u00a0 PDF (603KB) ( 1676 )\u00a0\u00a0 Based on a single-fluid MHD model, this paper has studied the global magnetic field structure in the near-Mars space, and investigated the influence of the Martian crustal magnetic fields on the magnetic field structure. Mars bow shock, Magnetic Pileup Region are produced when the solar wind flows over Mars. The interplanetary magnetic field lines are curved and draping' around the planet. The majority of magnetic field lines bypass over two poles, leaving V shaped' structure in the wake behind Mars. Near the surface of Mars, the local crustal magnetic fields also have noticeable influence on the magnetic field structure. The crustal magnetic fields at different positions, with different intensities form the dissimilar magnetic filed structure and mini-magnetospheres when interacting with the solar wind. The towed mini-magnetosphere and the mini-magnetosphere with open magnetic lines are illustrated in the paper. The local crustal magnetic fields change the morphology of the magnetic field, and accordingly may play an important role in determining the plasma distribution.\n\u8bba\u6587\n Study on Langmuir Wave Activities Within the Magnetic Cloud Boundary Layers Zuo Pingbing;Zhu Chunming;Wei Fengsi;Feng Xueshang;Li Huijun 2009, 29 (3):\u00a0 262-267.\u00a0 doi: 10.11728\/cjss2009.03.262 Abstract ( 3441 )\u00a0\u00a0 PDF (805KB) ( 2625 )\u00a0\u00a0 Two particular types of Langmuir wave activities are found within the Magnetic Cloud Boundary Layers (MCBL): Langmuir wave enhancements in entire region of MCBL compared with the adjacent magnetic cloud body and sheath region for majority MCBL and the rapid Langmuir wave burst phenomena associated with broad-band Doppler shifted ion-coustic wave activities for a few MCBL. On 3 Oct., 2000, WIND detected a typical MCBL inside which rapid Langmuir waves burst was observed. The Langmuir waves burst was right corresponded to the magnetic field minimum within the MCBL. The analysis based on high resolution electron distribution function data indicates that the bump-on-tail instability, resulting from the electron beam with beam velocity vb about7\u00d710^3 km\/s, is responsible for the rapid Langmuir waves burst.\n Ionospheric Responses to IMF Southward Turnings in Mid- and Low-latitudes Sun Shuji;Chen Chun;Ding Zonghua;Ban Panpan;Xi Dilong 2009, 29 (3):\u00a0 268-274.\u00a0 doi: 10.11728\/cjss2009.03.268 Abstract ( 2470 )\u00a0\u00a0 PDF (677KB) ( 1153 )\u00a0\u00a0 Using the historical data from ionosonde stations and satellite, the response of ionospheric f0F2 to major Interplanetary magnetic Field (IMF) southward turnings in mid- and low-latitudes is studied. It shows that the IMF southward turnings can cause disturbing response in the ionosphere, which depends on latitude, season and local time when the turning occurs. In mid-latitude, the ionospheric response to southward turnings in summer and equinox and by night is stronger, which fades out with latitude decreases. During the recovery phase the irregular fluctuation in the ionosphere comes forth. In low-latitude, the disturbance caused by the turnings is strong in summer and equinox and weak in winter, although it tends to be immersing into the background disturbances. It is also found that the maximal negative responses of the ionosphere is well related with the maximal negative Bz after the southward turnings linearly.\n\u8bba\u6587","date":"2021-11-27 04:09:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.4233902096748352, \"perplexity\": 5509.416942324056}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358078.2\/warc\/CC-MAIN-20211127013935-20211127043935-00622.warc.gz\"}"}
| null | null |
// BenStack.h: interface for the CBenStack class.
//
//////////////////////////////////////////////////////////////////////
/*********************************************************************
////////////////////////////
CBenStack Class Description/
////////////////////////////
When this class is instantiated, it will create a BenStack object.
A BenStack is different than a normal STL Stack object in that
it allows the user to peek n-levels down into the stack.
The standard stack functions, such as Pop, Push, and Top, Sizeof,
are also provided.
/// Nice idea - very clean
This class was created using the STL vector library.
ADDED ResetStack in order to let user reset the table at the beginning of
the program.
*********************************************************************/
#ifndef CBenstacker
#define CBenstacker
#include <afxtempl.h> // for MFC classes CMap and CArray
#include <vector.h> // for STL vector object
#include <iterator.h> // for vector iterator
#include "Attr.h"
typedef CMap <CString, LPCSTR, CAttr*, CAttr*> CMAP; /// Nice!
typedef vector <CMAP*> STACK;
class CBenStack
{
public:
//Class constructor and Deconstructor
CBenStack();
virtual ~CBenStack();
//Public Member functions
//***************************************************************************
//PRE: Must have class object created.
//POST: Returns true if the vector is empty.
//***************************************************************************
bool isEmpty();
//***************************************************************************
//PRE: Must have class object created.
//POST: Returns the number of objects currently in the stack
//***************************************************************************
int SizeofStack();
//***************************************************************************
//PRE: Must have class object created.
//POST: Takes a pointer to a CMap object. Will insert the CMap at the front
// of the vector.
//***************************************************************************
void Push(CMAP* CurrScope);
//***************************************************************************
//PRE: Must have at least one CMap on the Stack
//POST: Will iterate to the first position in the vector, and erase its
// contents
//***************************************************************************
bool Pop();
//***************************************************************************
//PRE: Must have at least one CMap on the stack
//POST: Will return a pointer to the first CMap object on the stack
//***************************************************************************
CMAP* Top();
//***************************************************************************
//PRE: Must have at least one CMap on the stack
//POST: Returns a pointer to a CMap object of the object a ScopeDepth into
// the stack.
//***************************************************************************
CMAP* Peek(int ScopeDepth);
//***************************************************************************
//PRE: Invoked by user
//POST: Erases the contents of the Stack.
//***************************************************************************
void ResetStack();
//Public Member Data
STACK m_vcmScopeStack; //Scope Stack
private:
//Private Member Data
CMAP* m_pcmCurrScope;
};
#endif /*CBenstacker*/
|
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| 5,130
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{"url":"https:\/\/physics.stackexchange.com\/questions\/477305\/is-the-relative-number-of-electric-field-lines-between-two-charges-proportional","text":"# Is the relative number of electric field lines between two charges proportional to the difference between those two charges?\n\nLet us consider a system of two unlike charges and suppose that the magnitude of the positive charge is greater that of the negative charge and call them a and b respectively.\n\nIf that(asked in the title) is true then if we keep them close together then more field lines will be generated from the a but less field lines will end in charge b. What happens to those field lines that don't end ? We are only considering those field lines that are not away from the system (or those that can actually end\n\n\u2022 Hint: What happens to the field lines when you just have one point charge? \u2013\u00a0The Photon May 2 '19 at 5:42\n\nThe number of field lines will be the same.\n\nThe electric field lines are pictorial depictions of the electric vector field. The best way to imagine it is to leave a test charge in the field and track its movement. If you perform this mental exercise, you will find that the field lines get denser near the charge with the higher magnitude and are rarer near the on with lower magnitude. This happens since the charge with higher magnitude attracts\/repels the test charge with a greater force than the other charge.\n\nWhat happens to those field lines that don't end ?\n\nThe field lines don't end if the charge is isolated. They simply radiate outward or inward. If the charge is not isolated, some of the lines will end at the other opposite charge. Others will not if they are not sufficiently influenced by the other charge.\n\nTo illustrate this, the figure below shows the field lines produced by positive and negative charge where the magnitude of the positive charge (call it charge $$a$$) is greater than the negative charge (call it charge $$b$$) as in the example you cite. The arrows point to the path that a positive test charge placed in the field will take. The density of the field lines at any location indicates the relative strength of the field (Newtons per coulomb or volts per meter) at that location.\n\nNotice that some of the field lines of the positive charge end at the negative charge. But most do not because the lower magnitude negative charge has less influence on the field of the positive charge. On the other hand most of the field lines of the negative charge end at the positive charge, because of the greater influence of the positive charge on the field of the negative charge. Even those that don't end at the positive charge are \"bent\" more due to the influence of the positive charge.\n\nIf the charges are far enough apart all the field lines of each would radiate out (for the positive charge) or in (for the negative charge) without ending.\n\nHope this helps.","date":"2020-11-29 08:16:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7831557989120483, \"perplexity\": 236.23931662959072}, \"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-2020-50\/segments\/1606141197278.54\/warc\/CC-MAIN-20201129063812-20201129093812-00437.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{sec:intro}
Instance segmentation is a task that jointly estimates class labels and segmentation masks of individual objects.
As in other visual recognition tasks, supervised learning of Convolutional Neural Networks (CNNs) has driven recent advances in instance segmentation~\cite{Sds,DAICVPR15,Dai16_SDS,mask_rcnn,instancecut,Semiconv,Dai2016,Liu2016}.
Due to the data-hungry nature of deep CNNs, this approach demands an enormous number of training images with groundtruth labels, which are given by hand in general.
However, manual annotation of instance-wise segmentation masks is prohibitively time-consuming, which results in existing datasets limited in terms of both class diversity and the amount of annotated data.
It is thus not straightforward to learn instance segmentation models that can handle diverse object classes in the real world.
One way to alleviate this issue is weakly supervised learning that adopts weaker and less expensive labels than instance-wise segmentation masks as supervision.
Thanks to low annotation costs of weak labels, approaches in this category can utilize more training images of diverse objects, although they have to compensate for missing information in weak labels.
For instance segmentation, bounding boxes have been widely used as weak labels since they provide every property of objects except shape~\cite{SDI,Cutnpaste}.
However, it is still costly to obtain box labels for a variety of classes in a large number of images as they are manually annotated.
To further reduce the annotation cost, one may utilize image-level class labels for learning instance segmentation since such labels are readily available in large-scale image classification datasets, \emph{e.g.}, ImageNet~\cite{Russakovsky2015}.
Furthermore, although image-level class labels indicate only the existence of object classes, they can be used to derive strong cues for instance segmentation, called \emph{Class Attention Maps} (CAMs)~\cite{Cam,Oquab15,Wei_2017_CVPR,Selvaraju_2017_ICCV}.
A CAM roughly estimates areas of each class by investigating the contribution of local image regions to the classification score of the class.
However, CAMs cannot be directly utilized as supervision for instance segmentation since they have limited resolution, often highlight only partial areas of objects, and most importantly, cannot distinguish different instances of the same class.
To resolve this issue, a recent approach~\cite{PRM} incorporates CAMs with an off-the-shelf segmentation proposal technique~\cite{MCG}, which however has to be trained separately on an external dataset with additional supervision.
In this paper, we present a novel approach for learning instance segmentation using image-level class labels, which outperforms the previous state-of-the-art trained with the same level of supervision~\cite{PRM} and even some of approaches relying on stronger supervision~\cite{SDI,Sds}.
Moreover, it requires neither additional supervision nor any segmentation proposals unlike the previous approaches~\cite{Sds,PRM}.
Our method generates pseudo instance segmentation labels of training images given their image-level labels and trains a known CNN model with the pseudo labels.
For generating the pseudo labels, it utilizes CAMs, but as mentioned earlier, they can neither distinguish different instances nor find entire instance areas with accurate boundaries.
To overcome these limitations of CAMs, we introduce \emph{Inter-pixel Relation Network} (IRNet) that is used to estimate two types of additional information complementary to CAMs: a class-agnostic instance map and pairwise semantic affinities.
A class-agnostic instance map is a rough instance segmentation mask without class labels nor accurate boundaries.
On the other hand, the semantic affinity between a pair of pixels is a confidence score for class equivalence between them.
By incorporating instance-agnostic CAMs with a class-agnostic instance map, we obtain instance-wise CAMs, which are in turn enhanced by propagating their attention scores to relevant areas based on the semantic affinities between neighboring pixels.
After the enhancement, a pseudo instance segmentation label is generated by selecting the instance label with the highest attention score in the instance-wise CAMS at each pixel.
The entire procedure for label synthesis is illustrated in \Fig{label_synthesis}.
IRNet has two branches estimating an instance map and semantic affinities, respectively.
The first branch predicts a displacement vector field where a 2D vector at each pixel indicates the centroid of the instance the pixel belongs to.
The displacement field is converted to an instance map by assigning the same instance label to pixels whose displacement vectors point at the same location.
The second branch detects boundaries between different object classes.
Pairwise semantic affinities are then computed from the detected boundaries in such a way that two pixels separated by a strong boundary are considered as a pair with a low semantic affinity.
Furthermore, we found that IRNet can be trained effectively with inter-pixel relations derived from CAMs.
Specifically, we collect pixels with high attention scores and train IRNet with the displacements and class equivalence between the collected pixels.
Thus, no supervision in addition to image-level class labels is required.
The contribution of this paper is three-fold:
\vspace{-2mm}
\begin{itemize}[leftmargin=5mm]
\itemsep=-1mm
\item We propose a new approach to identify and localize instances with image-level supervision through class-agnostic instance maps.
This enables instance segmentation without off-the-shelf segmentation proposals.
\item We propose a new way to learn and predict semantic affinities between pixels with image-level supervision through class boundary detection, which is more effective and efficient than previous work~\cite{affinitynet}.
\item On the PASCAL VOC 2012 dataset~\cite{Pascalvoc}, our model substantially outperforms the previous state-of-the-art trained with the same level of supervision~\cite{PRM}.
Also, it even surpasses previous models based on stronger supervision like SDI~\cite{SDI} that uses bounding box labels and SDS~\cite{Sds}, an early model that uses full supervision.
\end{itemize}
\iffalse
The rest of the paper is organized as follows.
\Sec{relatedwork} summarizes related work and \Sec{cams} reviews CAMs as an initial step of our framework.
We then describe details of IRNet in \Sec{irnet}, and our label synthesis method using IRNet in \Sec{labelsynth}.
\Sec{experiments} presents experimental results. Finally, \Sec{conclusion} concludes this paper.
\fi
\iffalse
To overcome these limitations of CAMs, we introduce \emph{Inter-pixel Relation Network} (IRNet) that estimates two types of additional information that are complementary to CAMs: a class-agnostic instance map and pairwise semantic affinities.
A class-agnostic instance map is a rough instance segmentation mask without class labels nor accurate boundaries.
On the other hand, semantic affinities provides information about potential instance boundaries.
By combining instance-agnostic CAMs, class-agnostic instance maps, and semantic affinities, we can generate an accurate pseudo instance segmentation label.
IRNet does not require any additional supervision, but it is trained with local inter-pixel relations derived from CAMs.
Specifically, to estimate an instance map, IRNet learns a displacement vector field of 2D vectors indicating the centroids of instances they belong to.
For pairwise semantic affinities, IRNet learns to predict semantic object class boundaries.
The contribution of this paper is three-fold:
\begin{itemize}[leftmargin=5mm]
\itemsep=0mm
\item
We propose a new approach based on the displacement field estimation to identify and localize instances in weakly supervised setting. This allows our model to perform instance segmentation without off-the-shelf segmentation proposals.
\item We propose a new way to learn semantic affinities between pixels with image-level supervision only, and it is more effective and memory-efficient compared to the previous approach~\cite{affinitynet}.
\item On the PASCAL VOC 2012 dataset~\cite{Pascalvoc}, our model substantially outperforms the previous state of the art trained with the same level of supervision~\cite{PRM}.
Also, it even surpasses previous models based on stronger supervision like SDI~\cite{SDI} using bounding box labels and SDS~\cite{Sds}, one of the earliest fully supervised model.
\end{itemize}
\fi
\iffalse
To overcome these limitations of CAMs, our method estimates two types of additional information that are complementary to CAMs: a class-agnostic instance map and pairwise semantic affinities.
A class-agnostic instance map is a rough instance segmentation mask without class labels nor accurate boundaries.
On the other hand, semantic affinities provides information about potential instance boundaries.
By combining instance-agnostic CAMs with class-agnostic instance maps, our method generates instance-wise CAMs.
Then, the method propagates attention scores of each instance-wise CAM to reveal the entire area of the corresponding instance using semantic propagation based on the pairwise semantic affinities.
After semantic propagation, a pseudo instance segmentation label is generated by selecting the instance-class combination with the highest attention score for each pixel.
The entire procedure for label synthesis is illustrated in \Fig{label_synthesis}.
The remaining issue is how to estimate class-agnostic instance maps and semantic affinities between pixels.
To this end, we design \emph{Inter-pixel Relation Network} (IRNet), which has two output branches for the two tasks respectively.
The first branch of IRNet predicts a displacement vector field where a 2D vector at each pixel indicates the centroid of the instance the pixel belongs to.
The displacement field is converted to a class-agnostic instance map by assigning the same instance label to pixels whose displacement vectors point out the same location.
The second branch predicts maps of potential boundaries of object classes.
Pairwise semantic affinities are then computed from the detected boundary maps in such a way that, given detected boundaries, two pixels separated by a strong boundary are considered as a pair with a low semantic affinity.
We train IRNet with local inter-pixel relations derived from CAMs without any additional supervision.
Specifically, we first collect pixels with high attention scores, and train IRNet with the displacements and class equivalence between the collected pixels.
Note that learning IRNet in principle demands instance-wise segmentation supervision as it predicts instance-aware information, but we found that it can be trained successfully with CAMs
since their class label prediction is often correct within local confident areas,
hence able to provide reliable evidences for local inter-pixel relations, as also shown in~\cite{affinitynet}.
\fi
\iffalse
The issue is then how to predict instance label maps and semantic affinities between pixels.
To this end, we design a CNN called \emph{Inter-pixel Relation Embedding Network} (IREN), which has two output branches conducting the two prediction tasks respectively.
The first branch predicts offset vectors from pixels to centroids of instances they belong to so that pixels converging to the same centroid are assigned the same instance label.
The second one detects boundaries between different classes so that two pixels separated by such boundaries are considered as a pair with a low semantic affinity.
Importantly, IREN is trained with inter-pixel relations derived from CAMs, thus demands image-level supervision only.
Specifically, we define two kinds of relations between neighboring pixels, class equivalent and location displacement, where the class labels of pixels are given by CAMs.
\fi
\section{Related Work}
\label{sec:relatedwork}
\iffalse
\subsection{Instance Segmentation}
One of the earlier work SDS~\cite{Sds} pioneer into instance segmentation with a simple solution in which a bounding box is generated by region proposal network, then the shape inside each region is identified with another module for mask prediction.
The basic principle given SDS still persists in most state-of-the-art methods ~\cite{DAICVPR15,Dai16_SDS,mask_rcnn,instancecut,Semiconv,Dai2016,Liu2016}.
Like ours, a few methods incorporate position-sensitive embedding, and instances are divided in a proposed box region box~\cite{kendall2017multi, Semiconv}, or a instance-agnostic segment~\cite{Liang_sds}.
\fi
This section reviews semantic and instance segmentation models closely related to our method.
We first introduce weakly supervised approaches for the two tasks, and discuss models that are based on ideas similar with the displacement field and pairwise semantic affinity of our framework.
\noindent \textbf{Weakly Supervised Semantic Segmentation:}
For weak supervision of semantic segmentation,
various types of weak labels such as bounding boxes~\cite{Boxsup, Wssl}, scribbles~\cite{scribblesup, Vernaza2017}, and points~\cite{Bearman16} have been utilized.
In particular, image-level class labels have been widely used as weak labels since they require minimal or no effort for annotation~\cite{oh17cvpr,Huang_2018_CVPR,Tokmakov16,Hong2017_webly,Wsl,wildcat,PRM,Wei_2017_CVPR,affinitynet}.
Most approaches using the image-level supervision are based on CAMs~\cite{Cam,Oquab15,Selvaraju_2017_ICCV} that roughly localize object areas by drawing attentions on discriminative parts of object classes.
However, CAMs often fail to reveal the entire object areas with accurate boundaries.
To address this issue, extra data or supervision have been exploited to obtain additional evidences like saliency~\cite{oh17cvpr,Huang_2018_CVPR}, motion in videos~\cite{Tokmakov16, Hong2017_webly} and class-agnostic object proposals~\cite{Wsl}.
Recent approaches tackle the issue without external information
by mining complementary attentions iteratively~\cite{Wei_2017_CVPR,Huang_2018_CVPR} or propagating CAMs based on semantic affinities between pixels~\cite{affinitynet}.
\noindent \textbf{Weakly Supervised Instance Segmentation:}
For instance segmentation, bounding boxes have been widely used as weak labels.
Since a bounding box informs the exact location and scale of an object, weakly supervised models using box labels focus mainly on estimating object shapes.
For example, in~\cite{SDI}, GraphCut is incorporated with generic boundary detection~\cite{HED} to better estimate object shapes by considering boundaries.
Also, in~\cite{Cutnpaste}, an object shape estimator is trained by adversarial learning~\cite{GAN} so that a pseudo image generated by cutting and pasting the estimated object area to a random background looks realistic.
Meanwhile, weakly supervised instance segmentation with image-level class labels has been rarely studied since this is a significantly ill-posed problem where supervision does not provide any instance-specific information.
To tackle this challenging problem, a recent approach~\cite{PRM} detects peaks of class attentions to identify individual instances and combines them with high-quality segmentation proposals~\cite{MCG} to reveal entire instance areas.
However, the performance of the method heavily depends on that of the segmentation proposals, which have to be trained with extra data with high-level supervision.
In contrast, our approach requires neither off-the-shelf proposals nor additional supervision and it surpasses the previous work~\cite{PRM} by a substantial margin.
\noindent \textbf{Pixel-wise Prediction of Instance Location:}
Pixel-wise prediction of instance location has been proven to be effective for instance segmentation in literature.
In~\cite{Liang_sds} the coordinates of the instance bounding box each pixel belongs to are predicted in a pixel-wise manner so that pixels with similar box coordinates are clustered as a single instance mask.
This idea is further explored in~\cite{kendall2017multi, Semiconv}, which predict instance centroids instead of box coordinates.
Our approach based on the displacement field share the same idea with~\cite{kendall2017multi, Semiconv}, but it requires only image-level supervision while the previous approaches are trained with instance-wise segmentation labels.
\noindent \textbf{Semantic Affinities Between Pixels:}
Pairwise semantic affinities between pixels have been used to enhance the quality of semantic segmentation.
In~\cite{randomwalk_net, ChengCVPR2017}, CNNs for semantic segmentation are incorporated with a differentiable module computing a semantic affinity matrix of pixels, and trained in an end-to-end manner with full supervision.
In \cite{randomwalk_net}, a predicted affinity matrix is used as a transition probability matrix for random walk, while in \cite{ChengCVPR2017}, it is embedded into a convolutional decoder~\cite{deconvnet} to encourage local pixels to have the same labels during inference.
Recently, a weakly supervised model has been proposed to learn pairwise semantic affinities with image-level class labels~\cite{affinitynet}.
This model predicts a high-dimensional embedding vector for each pixel, and the affinity between a pair of pixels is defined as the similarity between their embedding vectors.
Our approach shares the same motivation with~\cite{affinitynet}, but our IRNet can learn and predict affinities more effectively and efficiently by detecting class boundaries.
\section{Class Attention Maps}
\label{sec:cams}
CAMs play two essential roles in our framework.
First, they are used to define seed areas of instances, which are propagated later to recover the entire instance areas as in~\cite{sec,affinitynet}.
Second, they are a source of supervision for learning IRNet;
by exploiting CAMs carefully, we extract reliable inter-pixel relations, from which IRNet is trained.
To generate CAMs for training images, we adopt the method of~\cite{Cam} using an image classification CNN with global average pooling followed by a classification layer.
Given an image, the CAM of a groundtruth class $c$ is computed by
\begin{equation}
M_c (\mathbf{x}) = \frac{\phi_c^\top f (\mathbf{x})}{\max_{\mathbf{x}} \phi_c^\top f (\mathbf{x})},
\end{equation}
where $f$ is a feature map from the last convolution layer of the CNN, $\mathbf{x}$ is a 2D coordinate on $f$, and $\phi_c$ is the classification weights of the class $c$.
Also, CAMs for irrelevant classes are fixed to a zero matrix.
We adopt ResNet50~\cite{resnet} as the classification network,
and reduce the stride of its last downsampling layer from 2 to 1 to prevent CAMs from further resolution drop.
As a result, the width and height of CAMs are $1/16$ of those of the input image.
\section{Inter-pixel Relation Network}
\label{sec:irnet}
\begin{figure} [!t]
\begin{center}
\includegraphics[width=1.0 \linewidth] {figures/figure_net13.pdf}
\end{center}
\vspace{-0.3cm}
\caption{
Overall architecture of IRNet.
}
\label{fig:net}
\vspace{-0.1cm}
\end{figure}
IRNet aims to provide two types of information: a displacement vector field and a class boundary map, both of which are in turn used to estimate pseudo instance masks from CAMs.
This section describes the IRNet architecture
and the strategy for learning the model using CAMs as supervision.
How to use IRNet for pseudo label generation will be illustrated in \Sec{labelsynth}.
\subsection{IRNet Architecture}
IRNet has two output branches that predict a displacement vector field and a class boundary map, respectively.
Its architecture is illustrated in \Fig{net}.
The two branches share the same ResNet50 backbone, which is identical to that of the classification network in \Sec{cams}.
As inputs, both branches take feature maps from all the five levels\footnote{A level means a group of residual units sharing the same output size in~\cite{resnet}. However, in our backbone, the output sizes of \textsf{\scriptsize level4} and \textsf{\scriptsize level5} are identical since the stride of the last downsampling layer is reduced to 1.} of the backbone.
All the convolution layers of both branches are followed by group normalization~\cite{GroupNorm} and ReLU except the last layer.
Details of both branches are described below.
\noindent \textbf{Displacement Field Prediction Branch:}
A 1$\times$1 convolution layer is first applied to each input feature map, and the number of channels is reduced to 256 if it is larger than that.
On top of them, we append a top-down path way~\cite{lin2017_fpn} to merge all the feature maps iteratively in such a way that low resolution feature maps are upsampled twice, concatenated with those of the same resolution, and processed by a 1$\times$1 convolution layer.
Finally, from the last concatenated feature map, a displacement field is decoded through three 1$\times$1 convolution layers, whose output has two channels.
\noindent \textbf{Boundary Detection Branch:}
We first apply 1$\times$1 convolution to each input feature map for dimensionality reduction.
Then the results are resized, concatenated, and fed into the last 1$\times$1 convolution layer, which produces a class boundary map from the concatenated features.
\subsection{Inter-pixel Relation Mining from CAMs}
\label{sec:interpixelrel}
Inter-pixel relations are the only supervision for training IRNet, thus it is important to collect them reliably.
We define two kinds of relations between a pair of pixels: the displacement between their coordinates and their class equivalence.
The displacement can be easily computed by a simple subtraction, but the class equivalence is not since pixel-wise class labels are not given in our weakly supervised setting.
\begin{figure} [!t]
\centering
\includegraphics[width = 0.9 \linewidth]{figures/figure_pair4.pdf}
\caption{
Visualization of our inter-pixel relation mining process. (a) CAMs. (b) Confident areas of object classes. (c) Pseudo class label map within a local neighborhood. (d) Class equivalence relations between the center and the others.
}
\vspace{-0.1cm}
\label{fig:pair}
\end{figure}
Thus, we carefully exploit CAMs to predict pixel-wise pseudo class labels and obtain reliable class equivalence relations from them.
The overall procedure of our method is illustrated in \Fig{pair}.
Since CAMs are blurry and often inaccurate, we first identify areas with confident foreground/background attention scores.
Specifically, we collect pixels with attention scores larger than $0.3$ as foreground pixels, and smaller than $0.05$ as background pixels.
Note that we do not care pixels outside of confident areas during the process.
Each confident area is then refined by dense CRF~\cite{Fullycrf} to better estimate object shapes.
After that, we construct a pseudo class map $\hat{M}$ by choosing the class with the best score for each pixel.
Finally, we sample pairs of neighboring pixels from the refined confident areas, and categorize them into two sets $\mathcal{P}^+$ and $\mathcal{P}^-$ according to their class equivalence by
\begin{align}
\mathcal{P} &= \big\{(i, j) \mid \lVert \mathbf{x}_i - \mathbf{x}_j \rVert_2 < \gamma, \forall i\neq j \big\},
\label{eq:pair_set}
\\
\mathcal{P}^+ &= \big\{(i, j) \mid \hat{M}(\mathbf{x}_i) = \hat{M}(\mathbf{x}_j), (i, j) \in \mathcal{P} \big\},
\label{eq:pair_set_plus}
\\
\mathcal{P}^- &= \big\{(i, j) \mid \hat{M}(\mathbf{x}_i) \neq \hat{M}(\mathbf{x}_j), (i, j) \in \mathcal{P} \big\},
\label{eq:pair_set_minus}
\end{align}
where $\gamma$ is a radius limiting the maximum distance of a pair.
We further divide $\mathcal{P}^+$ into $\mathcal{P}^+_\text{fg}$ and $\mathcal{P}^+_\text{bg}$, a set of foreground pairs and that of background pairs, respectively.
\subsection{Loss for Displacement Field Prediction}
The first branch of IRNet predicts a displacement vector field $\mathcal{D}\in\mathbb{R}^{w\times h\times 2}$, where each 2D vector points at the centroid of the associated instance.
Although ground truth centroids are not given in our setting,
we argue that $\mathcal{D}$ can be learned implicitly with displacements between pixels of the same class.
There are two conditions for $\mathcal{D}$ to be a displacement field.
First, for a pair of pixel locations $\mathbf{x}_i$ and $\mathbf{x}_j$ belonging to the same instance, their estimated centroids must be identical, \emph{i.e.},
$\mathbf{x}_i + \mathcal{D}(\mathbf{x}_i) = \mathbf{x}_j + \mathcal{D}(\mathbf{x}_j)$.
Second, by the definition of centroid, $\sum_\mathbf{x} \mathcal{D(\mathbf{x})} = 0$ for each instance.
To satisfy the first condition, we first assume that a pair of nearby pixels $(i, j)\in \mathcal{P}^+$ is likely to be of the same instance since they are sampled within a small radius $\gamma$.
Then, given such a pair $(i, j)$, our goal is to approximate their image coordinate displacement $\hat{\delta}(i, j) = \mathbf{x}_j - \mathbf{x}_i$ with their difference in $\mathcal{D}$ denoted by $\delta(i, j) = \mathcal{D}(\mathbf{x}_i) - \mathcal{D}(\mathbf{x}_j)$.
In the ideal case where $\delta = \hat\delta$, it will hold that $\mathbf{x}_i + \mathcal{D}(\mathbf{x}_i) = \mathbf{x}_j + \mathcal{D}(\mathbf{x}_j)$ for all $(i, j)$ of the same instance.
This implies that $\mathcal{D}(\mathbf{x})$ is the displacement vector indicating the corresponding centroid.
For learning $\mathcal{D}$ with the inter-pixel relations obtained in \Sec{interpixelrel},
we minimize $L_1$ loss between $\delta(i, j)$ and $\hat\delta(i, j)$:
\begin{equation}
\mathcal{L}^\mathcal{D}_\text{fg} = \frac{1}{| \mathcal{P}^+_\text{fg} |} \sum_{(i, j)\in \mathcal{P}^+_\text{fg}} \Big| \delta(i, j) - \hat\delta(i, j) \Big|.
\label{eq:displacement_loss}
\end{equation}
The second condition, on the other hand, is not explicitly encouraged by \Eq{displacement_loss}.
However, we argue that IRNet can still learn to predict displacement vectors pointing to rough centroids of instances due to the randomness of initial network parameters. Intuitively speaking, initial random displacement vectors are already likely to satisfy the second condition, and the training of IRNet converges to a local minimum that still satisfies the condition. A similar phenomenon is observed in~\cite{Semiconv}.
Displacement vectors are then further refined by subtracting the mean of $\mathcal{D}$ from $\mathcal{D}$.
Also, we eliminate trivial centroid estimation from background pixels since the centroid of background is indefinite and may interfere with the above process.
For the purpose, we minimize the following loss for background pixels:
\begin{equation}
\mathcal{L}^\mathcal{D}_\text{bg} = \frac{1}{| \mathcal{P}^+_\text{bg} |} \sum_{(i, j)\in \mathcal{P}^+_\text{bg}} | \delta(i, j) |.
\end{equation}
\subsection{Loss for Class Boundary Detection}
Given an image, the second branch of IRNet detects boundaries between different classes, and the output is denoted by $\mathcal{B}\in [0, 1]^{w\times h}$.
Although no ground truth labels for class boundaries are given in our setting, we can train the second branch with class equivalence relations between pixels through a Multiple Instance Learning (MIL) objective.
The key assumption is that a class boundary exists somewhere between a pair of pixels with different pseudo class labels.
To implement this idea, we express the semantic affinity between two pixels in terms of the existence of a class boundary.
For a pair of pixels $\mathbf x_i$ and $\mathbf x_j$,
we define their semantic affinity $a_{i j}$ as:
\begin{equation}
a_{i j} = 1 - \max_{k \in \Pi_{i j}}{\mathcal{B} (\mathbf{x}_k)}
\label{eq:affinity_def}
\end{equation}
where $\Pi_{i j}$ is a set of pixels on the line between $\mathbf x_i$ and $\mathbf x_j$.
We utilize class equivalence relations between pixels as supervision for learning $a_{i j}$.
Specifically, the class equivalence between two pixels is represented as a binary label whose value is 1 if their pseudo class labels are the same and 0 otherwise.
The affinity is then learned by minimizing cross-entropy between the one-hot vector of the binary affinity label and the predicted affinity in \Eq{affinity_def}:
\begin{align}
\mathcal{L}^\mathcal{B} = & - \sum_{(i, j)\in \mathcal{P}^+_\text{fg}} \frac{\log a_{i j}}{2| \mathcal{P}^+_\text{fg} |} \ \ - \sum_{(i, j)\in \mathcal{P}^+_\text{bg}} \frac{\log a_{i j}}{2 | \mathcal{P}^+_\text{bg} |} \nonumber \\
& - \sum_{(i, j)\in \mathcal{P}^-} \frac{\log( 1 - a_{i j} ) } {| \mathcal{P}^- |} \label{eq:boundary_objective}
\end{align}
where three separate losses are aggregated after normalization since populations of $\mathcal{P}^+_\text{fg}$, $\mathcal{P}^+_\text{bg}$, and $\mathcal{P}^-$ are significantly imbalanced in general.
Through the loss in \Eq{boundary_objective}, we can learn $\mathcal{B}$ implicitly with inter-pixel class equivalence relations.
In this aspect, \Eq{boundary_objective} can be regarded as a MIL objective where $\Pi_{i j}$ is a bag of potential boundary pixels.
\subsection{Joint Learning of the Two Branches}
The two branches of IRNet are jointly trained by minimizing all the losses we defined previously at the same time:
\begin{equation}
\mathcal{L} = \mathcal{L}^\mathcal{D}_\text{fg} + \mathcal{L}^\mathcal{D}_\text{bg} + \mathcal{L}^\mathcal{B}. \label{eq:loss_final}
\end{equation}
Note that the above loss is class-agnostic since $\mathcal{P}^+$ and $\mathcal{P}^-$ only consider class equivalence between pixels rather than their individual class labels.
This allows our approach to utilize more inter-pixel relations per class and helps to improve the generalization ability of IRNet.
\section{Label Synthesis Using IRNet}
\label{sec:labelsynth}
To synthesize pseudo instance labels, the two outputs $\mathcal{D}$ and $\mathcal{B}$ of IRNet are converted to a class-agnostic instance map and pairwise affinities, respectively.
Among them, semantic affinities can be directly derived from $\mathcal{B}$ by \Eq{affinity_def} as illustrated in \Fig{aff}, while the conversion of $\mathcal{D}$ is not straightforward due to its inaccurate estimation.
This section first describes how $\mathcal{D}$ is converted to an instance map, then how to generate pseudo instance segmentation labels with the instance map and semantic affinities.
\begin{figure} [!t]
\centering
\includegraphics[width = 0.97 \linewidth, trim={0 1.3cm 0 0}, clip]{figures/figure_aff2.pdf}
\caption{Deriving pairwise semantic affinities from a class boundary map. (left) Input Image. (center) A class boundary map. (right) Label propagation from the center after random walks.
}
\label{fig:aff}
\end{figure}
\subsection{Generating Class-agnostic Instance Map}
\label{sec:class_agnostic_instance_map}
\begin{figure} [!t]
\centering
\includegraphics[width = 0.97 \linewidth, trim={0 1.4cm 0 0},clip]{figures/figure_pos3.pdf}
\caption{Detecting instance centroids. (left) Input image. (center) An initial displacement field. (right) A refined displacement field and detected centroids.
}
\label{fig:centroid}
\end{figure}
A class-agnostic instance map $I$ is a $w\times h$ 2D map, each element of which is the instance label associated with the element.
If $\mathcal{D}$ is estimated with perfect accuracy, $I$ can be obtained simply by grouping pixels whose displacement vectors point at the same centroid.
However, $\mathcal{D}$ often fails to predict the exact offsets to centroids since IRNet is trained with incomplete supervision derived from CAMs.
To address this issue, $\mathcal{D}$ is refined iteratively by
\begin{eqnarray}
\mathcal{D}_{u+1}(\mathbf{x}) = \mathcal{D}_u(\mathbf{x}) + \mathcal{D}\left(\mathbf{x} + \mathcal{D}_u(\mathbf{x})\right) \ \ \forall \mathbf{x},\label{eq:displacement_update}
\end{eqnarray}
where $u$ is an iteration index and $\mathcal{D}_0$ is the initial displacement field given by IRNet.
Each displacement vector is refined iteratively by adding the displacement vector at the currently estimated centroid location.
As displacement vectors near centroids tend to be almost zero in magnitude, the refinement converges within a finite number of iterations.
The effect of the refinement is demonstrated in \Fig{centroid}.
Since centroids estimated via the refined $\mathcal{D}$ are still scattered in general,
we consider a small group of neighboring pixels, instead of a single coordinate, as a centroid.
To this end, we first identify pixels whose displacement vectors in $\mathcal{D}$ have small magnitudes, and regard them as candidate centroids since pixels around a true centroid will have near zero displacement vectors.
Then each connected component of the candidates is considered as a centroid.
Note that the candidates tend to be well grouped into a few connected components since displacement vectors change smoothly within a local neighborhood as can be seen in \Fig{centroid}.
\iffalse
\jiwo{this section overlaps with Section 4.4}
\subsection{Estimating Pairwise Semantic Affinities}
Given the semantic boundary map $B$, affinity matrix is derived by Equation~\ref{eq:affinity_def}.
If there is no boundary on the line, then $\mathbf{x}$ and $\mathbf{y}$ are from the semantically same area. Thus we set their affinity to be high.
Mathematically, we define $W(\mathbf{x},\mathbf{y})$ as:
\begin{equation}
W(\mathbf{x},\mathbf{y}) = 1 - \max_{\mathbf{z} \in \Pi_{\mathbf{x}\mathbf{y}}}{\mathcal{B} (\mathbf{z})},
\label{eq:affinity_def}
\end{equation}
where $\Pi_{\mathbf{x}\mathbf{y}}$ is the set of pixels on the line connecting $\mathbf{x}$ and $\mathbf{y}$.
\fi
\subsection{Synthesizing Instance Segmentation Labels}
\begin{figure} [!t]
\centering
\includegraphics[width = 1 \linewidth]{figures/figure_synth_res5.pdf}
\caption{Examples of pseudo instance segmentation labels on the PASCAL VOC 2012 \emph{train} set. (a) Input image. (b) CAMs. (c) Displacement field. (d) Class boundary map. (e) Pseudo labels.
}
\label{fig:qualitative_labels}
\end{figure}
For generating pseudo instance masks, we first combine CAMs with a class-agnostic instance map as follows:
\begin{equation}
\bar{M}_{c k}( \mathbf{x}) =
\begin{cases}
M_c (\mathbf x) & \text{if } I(\mathbf x) = k, \\
0 & \text{otherwise,}
\end{cases}
\end{equation}
where $\bar{M}_{c k}$ is the instance-wise CAMs of class $c$ and instance $k$.
Each instance-wise CAM is refined individually by propagating its attention scores to relevant areas.
Specifically, the propagation is done by random walk, whose transition probability matrix is derived from the semantic affinity matrix $A=[a_{ij}]\in \mathbb{R}^{wh \times wh}$ as follows:
\begin{eqnarray}
T = S^{-1} A^{\circ \beta}, \ \ \textrm{where} \ \ S_{i i} = \sum_j a_{i j}^\beta \label{eq:trans_mat}
\end{eqnarray}
and $A^{\circ \beta}$ is $A$ to the Hadamard power of $\beta$ and $S$ is a diagonal matrix for row-normalization of $A^{\circ \beta}$.
Also, $\beta>1$ is a hyper-parameter for smoothing out affinity values in $A$.
The random walk propagation with $T$ is then conducted by
\begin{equation}
\text{vec}(\bar{M}^*_{c k}) = T^t \cdot \text{vec}(\bar{M}_{c k} \odot (1 - \mathcal{B})),
\label{eq:rw_prop}
\end{equation}
where $t$ denotes the number of iterations, $\odot$ is the Hadamard product, and vec$(\cdot)$ means vectorization.
We penalize scores of boundary pixels by multiplying $(1 - \mathcal{B})$ since those isolated pixels do not propagate their scores to neighbors and have overly high scores compared to the others in consequence.
Then an instance segmentation label is generated by choosing the combination of $c$ and $k$ that maximizes $\bar{M}_{ck}^*(\mathbf{x})$
for each pixel $\mathbf{x}$.
If the maximum score is less than bottom 25\%, the pixel is regarded as background.
\section{Experiments}
\label{sec:experiments}
The effectiveness of our framework is demonstrated on the PASCAL VOC 2012 dataset~\cite{Everingham}, where our framework generates pseudo labels for training images and trains a fully supervised model with the images and their pseudo labels.
We evaluate the quality of our pseudo labels as well as the performance of the model trained with them.
The evaluation is done for both instance segmentation and semantic segmentation since our pseudo labels can be used to train semantic segmentation models as well.
\iffalse
\subsection{Network Architectures}
\subsubsection{Backbone Network}
The two networks of our framework, IRNet and the CNN computing CAMs, are all built upon the same ResNet-50~\cite{resnet} backbone network, where the stride of the last convolution layer is reduced from 2 to 1 for increasing the resolution of the last feature map.
\subsubsection{IRNet}
Fig.~\ref{fig:net} illustrates the overall architecture of IRNet.
Both branches of the network take feature maps from all the 5 levels\footnote{A level is a group of residual units sharing the same output resolution.} of the backbone network through skip connections as inputs.
Convolution layers appended on top of the input feature maps are all followed by group normalization~\cite{GroupNorm} and ReLU except the last layer.
Details of each branch are described below.
\noindent \textbf{Displacement Field Branch:}
First, a $1 \times 1$ convolution layer is applied to each input feature map, where the number of channels is reduced to 256 if it is larger than that.
Then, we add a top-down path way in ~\cite{lin2017_fpn} where low-resolution features are upsampled by a factor of 2, and merged with the corresponding features of the same resolution. Specifically, the feature maps with the lowest resolution (level4, level5) are upsampled by bilinear interpolation and concatenetaed with level3. After applying a 1x1 convolution, this output is again upsampled and merged with level1 and level2. Finally, displacement field is decoded through three 1$\times$1 convolution layers. Note that in our backbone network, level5 share the same feature map resoltuion as lavel4 in our network, and the resolution of level1 is halved by a max pooling operation.
\noindent \textbf{Boundary Detection Branch:}
We add 1x1 convolution layer with 32 channels to each layer of the levels for dimensionality reduction. These features are concatenated, and then additional 1x1 convolution layer is attached to produce a boundary map. We do not employ deep-supervision presented in ~\cite{HED} for the sake of brevity.
\subsubsection{Network Computing CAMs}
We simply replaced the last fully connected layer of the backbone with another one for PASCAL VOC 2012 classification.
\fi
\subsection{Experimental Setting}
\noindent \textbf{Dataset:}
We train and evaluate our framework on the PASCAL VOC 2012~\cite{Pascalvoc} dataset.
Although the dataset contains labels for semantic segmentation and instance segmentation, we only exploit image-level class labels.
Following the common practice, the training set is expanded by adding image set proposed in~\cite{Hariharan}.
In total, 10,582 images are used for training, and 1,449 images are kept for validation.
\noindent \textbf{Hyperparameter Settings:}
The radius that limits the search space of pairs $\gamma$ in \Eq{pair_set} is set to 10 when training, and reduced to 5 at inference for conservative propagation.
The number of random walk iterations $t$ in \Eq{rw_prop} is fixed to 256.
The hyperparameter $\beta$ in \Eq{trans_mat} is set to 10.
The iterative update of $\mathcal{D}$ in \Eq{displacement_update} is done 100 times.
\noindent \textbf{Network Parameter Optimization:}
We adopt the stochastic gradient descent for network optimization.
Learning rate is initially set to 0.1, and decreases at every iteration with polynomial decay~\cite{Liu2015ParseNetLW}.
The backbone of IRNet is frozen during training, and gradients that displacement field branch receives are amplified by a factor of 10.
\noindent \textbf{Comparison to AffinityNet:}
For a fair comparison, we modified AffinityNet~\cite{affinitynet} by replacing its backbone with ResNet50 as in our IRNet.
Then we compare IRNet with the modified AffinityNet in terms of the accuracy of pseudo segmentation labels (Table~\ref{tab:synth_segm_anno_acc}) and performance of DeepLab~\cite{deeplab_v2} trained with these pseudo labels (Table~\ref{tab:comparison_segm}).
\begin{table}[!t] \small
\centering
\begin{tabular}{l|c}
\hline
Method & mIoU \\
\hline
CAM & 8.6 \\
CAM + Class Boundary & 34.1 \\
CAM + Displacement Field + Class Boundary (Ours) & 37.7 \\
\hline
\end{tabular}
\vspace{0.1mm}
\caption{Quality of our pseudo instance segmentation labels in $\text{AP}^r_\text{50}$, evaluated on the PASCAL VOC 2012 \emph{train} set.}
\label{tab:synth_inst_anno_acc}
\end{table}
\iffalse
\begin{table}[!t] \small
\centering
\begin{tabular}{l|c}
\hline
Method & $\text{AP}^r_\text{50}$ \\
\hline
CAM & 48.3 \\
CAM + Prop. with AffinityNet~\cite{affinitynet} & 59.3 \\
CAM + Prop. with IRNet (Ours) & 66.5 \\
\hline
\end{tabular}
\vspace{0.1mm}
\caption{Quality of pseudo semantic segmentation labels in mIoU, evaluated on the PASCAL VOC 2012 \emph{train} set.
``Prop'' means the semantic propagation using predicted affinities.
}
\label{tab:synth_segm_anno_acc}
\end{table}
\fi
\begin{table}[!t] \small
\centering
\begin{tabular}{c||c|c}
\hline
CAM & Prop. w/ AffinityNet~\cite{affinitynet} & Prop. w/ IRNet (Ours) \\
\hline
48.3 & 59.3 & 66.5 \\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Quality of pseudo semantic segmentation labels in mIoU, evaluated on the PASCAL VOC 2012 \emph{train} set.
``Prop'' means the semantic propagation using predicted affinities.
}
\vspace{-3mm}
\label{tab:synth_segm_anno_acc}
\end{table}
\iffalse
\begin{table}[!t] \small
\centering
\begin{tabular}{c|c|c||c|c}
\hline
\multicolumn{3}{c||}{AffinityNet-ResNet50~\cite{affinitynet}} & \multicolumn{2}{c}{Ours}\\
\hline
CAM & Prop & CAM & Prop\\
\hline
48.0 & 59.3& 48.3 & 66.5 \\
\hline
\end{tabular}
\caption{Quality of pseudo semantic segmentation labels in mIoU, evaluated on the PASCAL VOC 2012 \emph{train} set.
``Prop'' means the semantic propagation using predicted affinities.
}
\label{tab:synth_segm_anno_acc}
\end{table}
\fi
\subsection{Analysis of Pseudo Labels}
\noindent \textbf{Instance Segmentation labels:}
A few qualitative examples of pseudo instance segmentation labels are presented in \Fig{qualitative_labels}, and the contribution of each branch of IRNet to the quality of the labels is analyzed in \Tbl{synth_inst_anno_acc}.
In the case of ``CAM'' in \Tbl{synth_inst_anno_acc}, we directly utilize raw CAMs to generate pseudo labels by thresholding their scores and applying connected component analysis while assuming that there are no instances of the same class attached to each other.
In the case of ``CAM + Class Boundary'' in \Tbl{synth_inst_anno_acc}, pseudo labels are obtained in the same manner, but we enhance CAMs by the semantic propagation based on the class boundary map before generating pseudo labels.
We evaluated the performance of each method in terms of average precision (AP).
For evaluating APs, the score of each detected instance is given as the maximum class score within its mask.
As shown in the table, exploiting a class boundary map effectively improves the quality of pseudo labels by more than 25\% as it helps to recover the entire areas of objects missing in CAMs.
Exploiting a displacement field further improves the performance by 3.6\% as it helps to distinguish different instances of the same class.
\noindent \textbf{Semantic Segmentation Labels:}
A reduced version of our framework, which skips the instance-wise CAM generation step, produces pseudo labels for semantic segmentation.
In this aspect, we compare our framework with the previous state-of-the-art in semantic segmentation label synthesis, AffinityNet~\cite{affinitynet}, in terms of mean Intersection-over-Union (mIoU).
Similar to ours, AffinityNet also conducts the semantic propagation to enhance CAMs using predicted pairwise semantic affinities.
\Tbl{synth_segm_anno_acc} compares the quality of our pseudo segmentation labels to that of AffinityNet~\cite{affinitynet}.
The accuracy of our pseudo labels is substantially higher than that of AffinityNet thanks to the superior quality of pairwise semantic affinities predicted by IRNet.
\iffalse
A reduced version of our framework, which skips the instance-wise CAM generation step, produces pseudo labels for semantic segmentation.
In this aspect, we compare our framework with previous state of the art in semantic segmentation label synthesis, called AffinityNet~\cite{affinitynet}, in terms of mean Intersection-over-Union (mIoU).
\Tbl{synth_segm_anno_acc} compares the quality of our pseudo labels to that of AffinityNet~\cite{affinitynet}.
Similar to ours, AffinityNet also conducts the semantic propagation to enhance CAMs using predicted pairwise semantic affinities.
Once a pseudo label is obtained, it is further refined by dense CRF~\cite{Fullycrf}.
Note that since our framework adopts a different network architecture from AffinityNet, the qualities of CAMs are slightly different.
Nevertheless, the accuracy of our labels is significantly higher than that of AffinityNet thanks to the superior quality of pairwise semantic affinities predicted by IRNet.
\fi
\begin{table}[!t] \footnotesize
\centering
\begin{tabular}{lcc|cc}
\hline
Method & Sup. & Extra data / Information & $\textrm{AP}^r_\textrm{50}$ & $\textrm{AP}^r_\textrm{70}$ \\
\hline
PRM~\cite{PRM} \raggedright & $\mathcal{I}$ & MCG~\cite{MCG} & 26.8 & - \\
SDI~\cite{SDI} \raggedright & $\mathcal{B}$ & BSDS~\cite{BSDS} & 44.8 & - \\
\hline
SDS~\cite{Sds} \raggedright & $\mathcal{F}$ & MCG~\cite{MCG} & 43.8 & 21.3 \\
MRCNN~\cite{mask_rcnn} \raggedright & $\mathcal{F}$ & MS-COCO~\cite{Mscoco} & 69.0 & - \\
\hline
\bf{Ours-ResNet50} \raggedright & $\mathcal{I}$ & - & 46.7 & 23.5 \\
\hline
\end{tabular}
\vspace{0.1mm}
\caption{Instance segmentation performance on the PASCAL VOC 2012 \emph{val} set. The supervision types (Sup.) indicate: $\mathcal{I}$--image-level label, $\mathcal{B}$--bounding box, and $\mathcal{F}$--segmentation label.}
\label{tab:comparison_inst}
\end{table}
\begin{table}[!t] \footnotesize
\centering
\begin{tabular}{lcc|cc}
\hline
Method & Sup. & Extra Data / Information & \emph{val} & \emph{test} \\
\hline
SEC~\cite{sec} \raggedright & $\mathcal{I}$ & - & 50.7 & 51.7 \\
AffinityNet~\cite{affinitynet} \raggedright & $\mathcal{I}$ & - & 58.7 & - \\
\hline
PRM~\cite{PRM} \raggedright & $\mathcal{I}$ & MCG~\cite{MCG} & 53.4 & - \\
CrawlSeg~\cite{Hong2017_webly} \raggedright & $\mathcal{I}$ & YouTube Videos & 58.1 & 58.7 \\
MDC~\cite{Wei_2018_CVPR} \raggedright & $\mathcal{I}$ & Ground-truth Backgrounds & 60.4 & 60.8 \\
DSRG~\cite{Huang_2018_CVPR} \raggedright & $\mathcal{I}$ & MSRA-B~\cite{LiuCVPR07} & 61.4 & 63.2 \\
\hline
ScribbleSup~\cite{scribblesup} \raggedright & $\mathcal{S}$ & - & 63.1 & - \\
BoxSup~\cite{Boxsup} \raggedright & $\mathcal{B}$ & - & 62.0 & 64.6 \\
SDI~\cite{SDI} \raggedright & $\mathcal{B}$ & BSDS~\cite{BSDS} & 65.7 & 67.5 \\
\hline
Upperbound \raggedright & $\mathcal{F}$ & - & 72.3 & 72.5 \\
\bf{Ours-ResNet50} \raggedright & $\mathcal{I}$ & - & 63.5 & 64.8 \\
\hline
\end{tabular}
\vspace{0.1mm}
\caption{Semantic segmentation performance on the PASCAL VOC 2012 \emph{val} and \emph{test} sets. The supervision type (Sup.) indicates: $\mathcal{I}$--image-level label, $\mathcal{B}$--bounding box, $\mathcal{S}$--scribble, and $\mathcal{F}$--segmentation label.}
\label{tab:comparison_segm}
\vspace{-1mm}
\end{table}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=1.00 \linewidth] {figures/figure_qualinst5.pdf}
\end{center}
\vspace{-0.4cm}
\caption{Qualitative results of our instance segmentation model on the PASCAL VOC 2012 \emph{val} set.
}
\label{fig:qualitative_inst}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=1.00 \linewidth] {figures/figure_qualsegm4.pdf}
\end{center}
\vspace{-0.4cm}
\caption{Qualitative results of smenatic segmentation on the PASCAL VOC 2012 \emph{val} set. (\emph{top}) Input images. (\emph{middle}) Groundtruth semantic segmentaton. (\emph{bottom}) Results of Ours-ResNet50.
}
\label{fig:qualitative_segm}
\end{figure*}
\subsection{Mask R-CNN for Instance Segmentation}
We evaluate the performance of an instance segmentation network trained with pseudo labels generated by our framework.
For evaluation, we adopt Mask R-CNN~\cite{mask_rcnn}, which is one of the state-of-the-art instance segmentation networks, with ResNet-50-FPN~\cite{lin2017_fpn} as its backbone.
\Fig{qualitative_inst} shows qualitative results of the Mask-RCNN trained with our pseudo labels, and \Tbl{comparison_inst} compares its performance to those of previous approaches in $\text{AP}^r$\footnote{$\text{AP}^r$ means average precision of masks at different IoU thresholds.}~\cite{Sds}.
As shown in \Tbl{comparison_inst}, ours largely outperforms PRM~\cite{PRM}, which is the state-of-the-art that also uses image-level supervision.
Our approach even outperforms SDI~\cite{SDI}, which uses bounding box supervision, by 1.9\%, and SDS~\cite{Sds}, which uses full supervision, by 2.9\% in $\text{AP}^r_{\text{50}}$.
\subsection{DeepLab for Semantic Segmentation}
We further explore the effectiveness of our framework by training DeepLab v2-ResNet50~\cite{deeplab_v2} with our pseudo semantic segmentation labels.
\Fig{qualitative_segm} visualizes semantic segmentation results obtained by our approach and \Tbl{comparison_segm} compares ours with other weakly supervised approaches.
Our approach outperforms previous arts relying on the same level of supervision,
and is even competitive with BoxSup~\cite{Boxsup}, which utilizes stronger bounding box supervision.
Also it recovers 88\% of its fully supervised counterpart, the upper bound that it can achieve.
\section{Conclusion}
\label{sec:conclusion}
Weakly supervised instance segmentation with image-level supervision is a significantly ill-posed problem due to the lack of instance-specific information.
To tackle this challenging problem, we propose IRNet, a novel CNN architecture that identifies individual instances and estimates their rough boundaries.
Thanks to the evidences provided by IRNet, simple class attentions can be significantly improved and used to train fully supervised instance segmentation models.
On the Pascal VOC 2012 dataset, models trained with our pseudo labels achieve the state-of-the-art performance in both instance and semantic segmentation.
\vspace{0.3cm}
\noindent \textbf{Acknowledgement:} This work was supported by Korea Creative Content Agency (KOCCA), Ministry of Culture, Sports, and Tourism (MCST) of Korea, Basic Science Research Program, and Next-Generation Information Computing Development Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT (NRF-2018R1C1B6001223, NRF-2018R1A5A1060031, NRF-2017M3C4A7066316). It was also supported by the DGIST Start-up Fund Program (2018010071).
}
\section{Appendix}
\label{sec:appendix}
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\def\emph{e.g.}{\emph{e.g.}}
\def\emph{et al.}{\emph{et al.}}
\def\emph{w.r.t.}{\emph{w.r.t.}}
\definecolor{brown}{rgb}{0.65, 0.16, 0.16}
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This appendix provides contents omitted in the regular sections for the sake of brevity.
\Sec{centroid} describes the centroid detection algorithm of \Sec{class_agnostic_instance_map} in more detail,
and \Sec{segnet} introduces the instance and semantic segmentation models trained with our synthetic labels for the final evaluation.
Additional qualitative results are then presented in \Sec{results}.
\subsection{Details of the Centroid Detection Algorithm}
\label{sec:centroid}
As discussed in \Sec{class_agnostic_instance_map} of the main paper, a small group of neighboring pixels, instead of a single coordinate, are considered as a centroid in practice.
To this end, we first identify pixels whose displacement vectors in $\mathcal{D}$ have magnitudes smaller than a certain threshold, and consider them as candidate centroids.
Specifically, the set of candidate centroids are defined as:
\begin{equation}
\mathcal{C} = \big\{\mathbf{x} \mid \lVert \mathcal{D} (\mathbf x) \rVert_2 < 2.5 \big\} = \hat{\mathcal{C}}_1 \cup \hat{\mathcal{C}}_2 \cup \cdots \cup \hat{\mathcal{C}}_K,
\end{equation}
where $\hat{\mathcal{C}}_i$ is a connected component of pixels in $\mathcal{C}$ and $K$ is the number of connected components.
Then a class-agnostic instance map $I$ is obtained by assigning each pixel a connected component index in the following manner:
\begin{equation}
I(\mathbf{x}) = k, \ \ \textrm{if} \ \big(\mathbf{x} + \mathcal{D}(\mathbf{x}) \big) \in \hat{\mathcal{C}}_k, \ \ \forall \mathbf{x}.
\end{equation}
\subsection{Details of Our Segmentation Networks}
\label{sec:segnet}
As our framework aims to generate synthetic labels for instance and semantic segmentation, we evaluated the efficacy of our framework by learning fully supervised models for the two tasks with our synthetic labels.
Specifically, we adopt Mask R-CNN~\cite{mask_rcnn} for instance segmentation and DeepLab v2~\cite{deeplab_v2} for semantic segmentation.
Both of them are first pretrained on ImageNet~\cite{Imagenet} then finetuned with the synthetic labels instead of groundtruth segmentation masks.
The rest of this section describes details of the two models.
\subsubsection{Mask R-CNN for instance Segmentation}
We use Detectron~\cite{Detectron2018}, which is the official implementation of~\cite{mask_rcnn}, to implement Mask R-CNN~\cite{mask_rcnn} with ResNet-50-FPN~\cite{lin2017_fpn} as its backbone.
We directly adopt the default training setting given in the provided source code, except the number of training steps that is adjusted for better adaptation to the PASCAL VOC 2012 dataset~\cite{Pascalvoc}.
\subsubsection{DeepLab v2 for Semantic Segmentation}
We manually implement DeepLab v2~\cite{deeplab_v2} in PyTorch~\cite{pytorch}.
Its architecture consists of ResNet-50~\cite{resnet} followed by an atrous spatial pyramid pooling module~\cite{deeplab_v2}.
The training setting of ours is identical to that of the original model.
We also employ the ensemble of multi-scale prediction during evaluation.
Specifically, a single input image is converted to a set of 8 images through resizing with 4 different scales \{0.5, 1.0, 1.5, 2.0\} and horizontal flip, and fed into the segmentation network so that the 8 outputs are aggregated by pixel-wise average pooling.
We also reproduce the performance of the fully supervised DeepLab v2, which is the \emph{upperbound} our segmentation model can achieve.
Note that, as summarized in Table 4 of the main paper, \emph{upperbound} we measured is lower than the performance reported in the original paper~\cite{deeplab_v2} as we did not tune the parameters of dense CRF~\cite{Fullycrf} carefully.
Thanks to the accurate segmentation labels synthesized in our framework, the DeepLab trained with our synthetic labels achieves 89.4\% of its fully supervised one on the PASCAL VOC 2012 \emph{test} set.
\subsection{More Qualitative Results of Our Approach}
\label{sec:results}
In this section, we provide additional qualitative results of our framework on the PASCAL VOC dataset. Although IRNet is trained with image-level supervision only, it successfully finds accurate class boundary and displacement field to instance centroids which are not directly available in CAMs, and synthesizes accurate instance segmentation masks from CAMs incorporating those two additional information as illustrated in \Fig{qualitative_synth}.
\Fig{qualitative_inst} and \Fig{qualitative_segm} show additional instance segmentation and semantic segmentation results of our models, respectively. Thanks to synthetic labels that are able to differentiate attached instances, our models not only find fine object shape, but also detect independent instances that are adjacent and of the same class.
\begin{figure*}[!ht]
\hspace{0.7cm}Input Image\hspace{1.5cm} CAM \hspace{0.9cm} Displacement Field \hspace{0.4cm}Class Boundary\hspace{0.6cm} Instance Labels \hspace{0.6cm} Class Labels
\begin{center}
\includegraphics[width=0.98 \linewidth] {figures/figure_qual_synth.pdf}
\end{center}
\vspace{-0.2cm}
\caption{Qualitative results of our instance segmentation model on the PASCAL VOC 2012 \emph{train} set.
}
\label{fig:qualitative_synth}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=0.97 \linewidth] {figures/figure_qual_inst.pdf}
\end{center}
\vspace{-0.2cm}
\caption{Qualitative results of our instance segmentation model on the PASCAL VOC 2012 \emph{val} set.
}
\label{fig:qualitative_inst}
\end{figure*}
\begin{figure*}[!ht]
\hspace{0.5cm}Input Image\hspace{1.0cm} Ground-truth \hspace{1.5cm} Ours \hspace{1.7cm}Input Image\hspace{1.0cm} Ground-truth \hspace{1.5cm} Ours
\begin{center}
\includegraphics[width=1.00 \linewidth] {figures/figure_qual_segm.pdf}
\end{center}
\vspace{-0.2cm}
\caption{Qualitative results of our semantic segmentation model on the PASCAL VOC 2012 \emph{val} set.
}
\label{fig:qualitative_segm}
\end{figure*}
$ $
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 5,304
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|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
\label{sec:intro}
As a result of the rapidly increasing popularity of omnidirectional videos on video-sharing platforms such as YouTube and Veer,
professional content producers are striving to produce better viewing experiences with higher resolution, new interaction mechanisms, as well as more degrees of viewing freedom,
using playback platforms such as the Google \textit{Welcome to Light Field}, where the viewer has
the freedom to move along three axes with motion parallax. A body of research has been dedicated to composing 6-DoF from footage captured by VR cameras,
extending toolsets for the professional content generators to produce more immersive contents. However, most of the proposed systems involve complex procedures such as
depth estimation and in-painting, that are both time- and resource- consuming and require tedious hand-optimizations.
Recently, Attal~\textit{et al.} proposed MatryODShka~\cite{attal2020matryodshka}, which used a convolutional neural network to predict multi-sphere images (MSIs) that
can be viewed in 6-DoF. This approach significantly simplified the overall pipeline complexity while producing promising visual results.
However, the system requires omnidirectional stereo (ODS) inputs that cannot be acquired directly from widely available VR cameras, as ODS must be produced
from raw panoramic VR footage using stitching with annoying visual artifacts.
In addition, as in the case for any system design, 6-DoF content production also requires a large volume of high-quality content that can serve as the
ground-truth, for both training purposes and performance evaluation. However, because 6-DoF contents are difficult to produce, there is no widely available high-quality
6-DoF content that can be used as a ground-truth dataset for the community.
The contribution of this paper is therefore two-fold: First, we propose a system that can produce large volumes of high-quality artifact-free 6-DoF data for training and performance evaluation
that serves as the basis for evaluation of the system proposed in this paper, as well as for future development by the community.
Second, we propose an algorithm for predicting MSI from VR camera footage directly, using a modified weighted sphere sweep volume fusing scheme
with 3D ConvNet without explicit depth map or segmentation mask, thereby improving quality while lowering complexity.
\begin{figure}[tb]
\centering
\includegraphics[width=0.75\hsize]{pic/text_shortpipeline.png}
\caption{Our proposed 6-DoF omnidirectional video composition framework.}
\label{fig:framework}
\end{figure}
\vspace{-9pt}
\begin{figure}[tb]
\centering
\includegraphics[width=0.75\hsize]{pic/text_common_pipeline.png}
\caption{An example of conventional 6-DoF content generation method}
\label{fig:conventional}
\end{figure}
\section{Related Works}
\label{sec:related_works}
Substantial research has been carried out on the reconstruction and representation of omnidirectional contents~\cite{peri1997generation}\cite{szeliski2006image}\cite{anderson2016jump}\cite{tang2018universal}\cite{bertel2020omniphotos}\cite{li2016novel}. In this section, we focus on
recent research aimed at 6-DoF reconstruction and view synthesis.
Six degrees of freedom content generation requires detailed scene depth information. Dynamic 3D reconstruction and content playback have
been extensively studied in the context of free-viewpoint video, with many approaches achieving real-time performance~\cite{collet2015high}\cite{dou2016fusion4d}.
Using the conventional multi-view stereo method, Google \textit{Welcome to Light field} ~\cite{overbeck2018welcome} and Facebook manifold system~\cite{pozo2019integrated}
both achieved realistic high-quality 6-DoF content composition, with hardware systems that are substantially more complex than VR cameras widely available on the market.
Lately, studies that used convolutional neural networking (CNN) show promising results for depth estimation and view synthesis~\cite{godard2017unsupervised}\cite{kalantari2016learning}\cite{srinivasan2017learning}.
CNN achieves excellent results for predicting multi-plane images (MPI) and representing the non-Lambertian
reflectance~\cite{zhou2018stereo}\cite{flynn2019deepview}\cite{mildenhall2019local}\cite{srinivasan2019pushing}\cite{choi2019extreme}. Many recent approaches adopt and
extend these studies to generate omnidirectional contents~\cite{attal2020matryodshka}\cite{lin2020panorama}\cite{broxton2020immersive}.
Broxton~\textit{et al.}~\cite{broxton2020immersive} designed a half-sphere camera rig with GoPro cameras and used the method in \cite{flynn2019deepview} to generate
pieces of MPIs and later converted it into 360 layered mesh representation for viewing. Lin~\textit{et al.}~\cite{lin2020panorama} generated multiple MPIs and formed them
into a multi-depth panorama using the MPI predicting techniques in \cite{mildenhall2019local}. Lai~\textit{et al.}~\cite{lai2019real} proposed to generate panorama depth
map for ODS images 6-DoF synthesization. More recently, MartyODShka~\cite{attal2020matryodshka} archived real time performance when
using the method in~\cite{zhou2018stereo} to synthesize MSI from ODS content. In spite of such successes, ODS images relaid by these approaches still require
stitching of original VR camera footage, which by itself is still a problem that has not been sufficiently solved. Many existing neural network based approaches were also designed
with a fixed number of input images that is not configurable.
In this paper, we propose an approach without stitching pre-processing or depth map estimation. In our approach, a weighted sphere sweep volume
is fused directly from camera footage using spherical projection, eliminating artifacts introduced by stitching. By utilizing 3D ConvNet, our framework is applicable to
different VR camera designs with different numbers of MSI layers. We also propose a high-quality 6-DoF dataset generation approach using UnrealCV~\cite{qiu2016unrealcv}
and Facebook Replica~\cite{straub2019replica} engine, so that our, and future systems for 6-DoF content composition can be designed and evaluated qualitatively
and quantitatively using our data generation scheme.
\vspace{-9pt}
\section{Omnidirectional 6-DoF Data Generation}
\label{sec:data}
Commercially available VR camera models provide a range of selection for omnidirectional content acquisition. A great amount of footage from these VR
cameras is captured by professional photographers and enthusiasts. However, it is hard to render a ground truth from these footage, as content
captured by such cameras need to be stitched, which by itself is still a problem that has not been completely solved. As a result, it has been extremely difficult to
find artifact-free high-quality 360 video datasets in this literature, which is required for the continued development and evaluation of VR and/or 6-DoF content. The few
datasets produced by companies like Google or Facebook are limited in size, content scenarios, resolutions etc., as they were constrained by the various factors
imposed by the time and equipment used for producing such datasets.
It is therefore highly desirable to be able to generate any number of test clips of any use cases and different resolutions and to continue this test data generation process
as new cameras with higher resolutions become available. In this study, we use two CG rendering frameworks UnrealCV~\cite{qiu2016unrealcv} and Replica~\cite{straub2019replica}
to generate high-quality VR datasets for both training and evaluation. The UnrealCV engine can render realistic images with lighting changes and reflections on handcrafted models.
On the other hand, Replica engine uses indoor reconstruction models, and produces rendered images with a similar distribution to real-world textures and dynamic ranges.
We use the combination of two datasets to mimic real-world challenges.
To compose an omnidirectional image using the above mentioned CG engines, we first generate six ${120}^{\circ}$ pin-hole images with different poses towards
the engine's x-y-z axes and their inverse directions. The optical centers of these virtual cameras are located at the same point, to avoid any parallax between cameras.
We then project six pin-hole images to equirectangular projection (ERP) and blend the overlapping area to avoid aliasing. To simulate camera footage from a n-sensor VR
camera, we render ERP footage on the poses of each sensor and mask off the external details according to the lens field of view.
In our experiments, we selected 2000 locations to generate 6-DoF contents for network training and evaluation. The generated datasets were
split into training subsets of 1600 locations and evaluation subsets with 400 locations. The split was implemented based on virtual camera locations with
no overlap across the training and evaluation subsets. Contents generated from UnrealCV and Replica were mixed together during training procedure
but evaluated separately. We rendered two resolutions (640$\times$320 and 400$\times$200) on each location for training with different numbers
of MSI layers due to GPU memory constraints. Upon that, we rendered two fields of view (${190}^{\circ}$ and ${220}^{\circ}$) to simulate fisheye images from different VR camera modules.
\vspace{-10pt}
\section{A Simplified System for Producing 6-DoF Content from Panoramic VR Footage}
\label{sec:methodology}
Our process of turning panoramic VR camera footage into 6-DoF playable multi-sphere images involves three steps
as shown in Fig.~\ref{fig:framework}. We first project the fisheye images onto concentric spheres using the weighted sphere sweep method and
generate weighted sphere sweep volume (WSSV) as described in Sec.~\ref{sec:wssv}. This 4D volume is input into the 3D ConvNet detailed in Sec.~\ref{sec:network}
to predict the $\alpha$ channel. We combine the predicted $\alpha$ channel with WSSV to form the MSI representation. Later inside the render engine,
the MSI is calculated following Equ.~\ref{equ:6dof} to infer per-eye views in VR.
\vspace{-9pt}
\subsection{Multi-Sphere Images}
\label{sec:msi}
\begin{figure}[tb]
\centering
\includegraphics[width=0.75\linewidth]{pic/msi.png}
\caption{Illustration of an MSI representation. Blue lines show the opacity at different area of the concentric spheres.}
\label{fig:msi}
\end{figure}
Inspired by multi-plane image (MPI) and its performance in view synthesis applications, multi-sphere images (MSIs) is generated for omnidirectional content representation.
Following the design of MPI, the MSI representation used in our work consists of $N$ images to represent $N$ concentric spheres. These concentric spheres are warped into planes
using equirectangular projection (ERP) for efficient processing. Each image inside an MSI contains 3 channels of color and an additional $\alpha$ channel of transparency information.
This character inherited from MPI empowers the MSI to represent scene occlusion and non-Lambertian reflectance. The form of $RGB\alpha$ data representation
also allows MSIs to be compressed using standard image compression algorithms.
The differentiable rendering scheme for the MPI also applies on the MSI. As described in Equ.~\ref{equ:alpha rendering}, in MPI differentiable rendering,
the position $\{p_i\}_{i=1}^{N}$ where a ray intersects with each layer $\{L_i\}_{i=1}^{N}$ is calculated and RGB value $\{c_{p_i}\}_{i=1}^{N}$ and $\alpha$ value $\{\alpha_{p_i}\}_{i=1}^{N}$
are interpolated and then used to calculate the output color $c$
\begin{equation}
c = \sum_{i=1}^{N}c_{p_i}\cdot{\alpha}_{p_i}\cdot \prod_{j=1}^{i-1} (1 - {\alpha}_{p_i}).
\label{equ:alpha rendering}
\end{equation}
The rendering procedure for viewing MSI in 6-DoF is similar, as described in Equ.~\ref{equ:6dof}. To render a ray inside of the inmost sphere of MSI with
viewing angle $(\theta, \phi)$ from camera location $(x,y,z)$, the intersection point $s_{i}$ with $i$-th layers of MSI is determined by ray-plane intersection function $q(\cdot)$ in graphic engines, where ${d}_{i}$ is the radius of $i$-th sphere.
The color of the rendered ray $c_{x,y,z,\theta,\phi}$ is computed by color values $\{c_{s_{i}}\}_{i=1}^{N}$ and transparency values $\{\alpha_{s_{i}}\}_{i=1}^{N}$ in each layer following:
\begin{equation}
\label{equ:6dof}
\begin{split}
&s_{i} = q(x,y,z,\theta, \phi, d_i)\\
&{c}_{x,y,z,\theta,\phi} = \sum_{i=1}^{N}c_{s_{i}}\cdot{\alpha}_{s_{i}}\cdot \prod_{j=1}^{i-1} (1 - {\alpha}_{s_{i}}).
\end{split}
\end{equation}
This procedure can be carried out efficiently on a common CG engine such as Unity or OpenGL.
\begin{figure}[tb]
\centering
\includegraphics[width=0.3\linewidth]{pic/equi2fisheye.png}~
\includegraphics[width=0.6\linewidth]{pic/equi_220fov.png}
\caption{Left: A fisheye image with ${220}^{\circ}$ FOV. Right: A fisheye image reprojected to ERP format}
\label{fig:fisheye}
\end{figure}
\vspace{-9pt}
\subsection{Weighted Sphere Sweep Volume Construction}
\label{sec:wssv}
A weighted sphere sweep volume is constructed with $N$ layers of concentric spheres. The radius of these spheres is chosen uniformly
in the reciprocal space between the closest object distance and the farthest distance. As images from VR cameras are often captured using fisheye lenses that compresses
the field of view and causes chroma aberration on the edges of images, we propose a weighted sphere sweep method to reduce such optical defects.
To this end, we first warp these input fisheye images into the ERP form as shown in Fig.~\ref{fig:fisheye}. Then $M$ input ERP images are projected onto $N$ layers of
sphere in ERP form using the intrinsic and extrinsic parameters of the camera.
In each layer, we fuse $M$ input images together on the overlapping area following :
\begin{equation}
{c}_{{p}} = \sum_{i=1}^{Q}\frac{{e}^{{\gamma}_{p,i} } \cdot {c}_{p,i}}{\sum_{j=1}^{Q}{e}^{{\gamma}_{p,j}}},
\label{equ:weighted}
\end{equation}
where $Q$ is the number of overlapping images
at position $p$, and ${c}_{p,i}$ is the color of the $i$-th image on position $p$. The parameter $\gamma$ is the optical distortion value.
Here we use ${\gamma}_{p} = 1 - r_p$, where $r_p$ is the distance of pixel $p$ to the optical center normalized to $[0,1]$. We can also use the lens MTF data to replace $\gamma$ values
Then, $N$ projected ERP images are stacked to form a 4D volume so that the first three dimensions are ERP image height (H), width (H) and number of layers (N), while
the 4-th dimension is the color channel. As each ERP image contains 3 channels of RGB color, the constructed WSSV has a shape of [H, W, N, 3].
The WSSV construction can be represented by Equ.~\ref{equ:wssv}, where the weighted and warp function ${\cal{W}}(\cdot)$ takes in a camera
pose ${\{\omega_j\}}_{j=1}^{M}$ and the camera center pose $\omega_c$, then projects an input image from ${\{I_j\}}_{j=1}^{M}$ to a set of concentric spheres
with predefined radius ${\{d_i\}}_{i=1}^{N}$. The ${\cal{S}}(\cdot)$ function stacks the warped results along 4-th dimension to form the weighted sphere sweep volume:
\begin{equation}
WSSV = {\cal{S}}_{j=1}^{N} ({\cal{W}}_{i=1}^{M} (I_i,\omega_i, \omega_c, {d}_{j})).
\label{equ:wssv}
\end{equation}
As each layer of sphere sweep volume is formed by multiple input images projected onto the same sphere, the combination of these images is similar to the result of light field refocusing.
Because multiple images of the same object are projected to a layer whose radius approximately equals to the distance of that object, the average color of such a combination of the
projected images will appear in-focus.
The 3D ConvNet in our framework is trained to distinguish such properties and predict the transparency values, which are further combined with WSSV to generate MSI.
\input{network_arch}
\vspace{-9pt}
\subsection{Network Architecture}
\label{sec:network}
Our 3D ConvNet architecture is a slight variation of Middle~\textit{et al.} 's work~\cite{mildenhall2019local}.
In our design, we modify the network to suit the WSSV input described in Sec.~\ref{sec:wssv}.
The output vector of the neural network has a shape of [H, W, D, 1], which is converted into the $\alpha$ channel of an MSI by a ReLu function. This $\alpha$ channel is then stacked with WSSV RGB
volume to form the MSI. As color information of an MSI comes directly from WSSV, which is the combination of projected input footage, during training, the network is
not learning to simulate the correct color but to select the correct layer by inferring the corresponding weight of the $\alpha$ channel to imply an object's distance.
The training objective is :
\begin{equation}
\begin{split}
&{MSI}_{\theta} = {f}_{\theta}(WSSV) \\
&{\cal{L}} = {\cal{L}}_{L1} + {\lambda}_{VGG}{\cal{L}}_{VGG}\\
&{\argmin}_{\theta}~\sum_{i=1}^{M}{\cal{L}}(Render({MSI}_{\theta},\omega_c, \omega_i), {\cal{W}}_{ERP}(I_i)),
\end{split}
\label{equ:train}
\end{equation}
where the goal is to minimize the difference between the
rendered MSI result and ground truth. In Equ.~\ref{equ:train} the 3D ConvNet $f_\theta(\cdot)$ takes in $WSSV$ and predicts ${MSI}_{\theta}$.
The $Render(\cdot)$ function follows the differentiable
rendering scheme in Equ.~\ref{equ:alpha rendering}, where the ${\cal{W}}_{ERP}(\cdot)$ is the fisheye to ERP warp function. The training loss $ \cal{L}(\cdot)$ is a weighted combination of
L1 loss ${\cal{L}}_{L1}(\cdot)$ and VGG loss $ {\cal{L}}_{VGG}(\cdot)$ proposed in~\cite{chen2017photographic}. We refer readers to Tab.~\ref{tab:network} for detailed network architecture.
\input{data}
\vspace{-9pt}
\section{Experiments}
\label{sec:experiments}
We trained and evaluated our network performance by generating MSI representations and rendering them in different input positions and computing
peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) quality scores against ground truth generated using our proposed approach in Sec~\ref{sec:data}.
In this section, we describe network implementation details, and experiment results.
\subsection{Implementation Details}
\label{sec:implementation}
We implement our proposed 3D ConvNet using TensorFlow API following the description in Tab.~\ref{tab:network}. During training,
Automatic Mixed Precision feature was applied to utilize GPU memory by using half precision (FP16) in computation. The network was optimized with an SGD optimizer
with the learning rate set to $2\times{10}^{-4}$. We employed the VGG loss~\cite{chen2017photographic} as the perceptual loss with weight $\lambda_{VGG}=200$.
Using the data generation method described in Sec.~\ref{sec:data}, we synthesized a set of 6-sensors VR camera footage on 2000 locations
and masked the field of view to ${190}^{\circ}$ and ${220}^{\circ}$ randomly. The entire dataset was split into 1600 locations for training subsets and the reset for evaluation.
The network was trained on 640$ \times$320 resolution with 32 layers of MSI and 400$\times$200 resolution with 64 layers of MSI simultaneously for 400k iterations on an Nvidia RTX 2080Ti GPU.
\input{fig_results}
\subsection{Evaluation}
\label{sec:evaluation}
We examined the performance of our approach with the evaluation subsets that contains 400 locations with two different resolutions and corresponding
color ground truth. We generated MSIs on evaluation subsets using our proposed method and rendered each MSI to novel viewpoints using input poses, then
computed PSNR and SSIM of these novel views with the ground truth color.
These quantitative results are reported in Tab.~\ref{tab:results} and some selected qualitative results are shown in Fig~\ref{fig:results}.
As demonstrated in Tab.~\ref{tab:results}, our system can generate high-quality 6-DoF contents from VR camera footage. Comparing with the ground truth color,
results rendered on input views achieve an average PSNR over 31 dB. After carefully examining the quantitative results in Tab.~\ref{tab:results},
we notice a minor performance difference between two datasets. A small variance between two render engines could be the cause, as UnrealCV engine renders reflectance
that varies from different angles while Replica uses static reconstructed object texture acquired from real-world scenes. Overall, results rendered
from MSIs show plausible visual details on complex textures like leaves
, books, floor textures as shown in Fig.~\ref{fig:results}.
We also notice there is a performance gain by increasing the number of layers in MSI. As each MSI is a set of concentric spheres, a denser set of spheres can represent
more detailed depth variances among scene objects.
In our experiments, we also confirmed that our network can be trained and utilized among variable image resolution and number of layers of an MSI.
We used 640x320 with 32-layer and 400x200 with 64-layer configuration on training and evaluated the network on the combination of both resolution and numbers of layers. Experiment results show that the network is not overfitting on one particular configuration.
\vspace{-9pt}
\section{Conclusions}
\label{sec:conclusion}
In this paper, we present an end-to-end deep-learning framework to compose 6-DoF omnidirectional with multi-sphere images. We use weighted sphere sweep volume to
unify inputs from various panoramic camera setups into one constant volume size, solving the compatibility issue in previous work. Combined with our proposed 3D ConvNet architecture,
we can process camera footage directly and reduce the systematic artifacts introduced in the ODS stitching process. We propose a high-quality 6-DoF dataset generation
method using UnrealCV and Facebook Replica engines for training and quantitive performance evaluations. A series of experiments were conducted to verify our system.
Experiment results show our system can operate on variable image resolution and MSI layers, as well as producing high-quality novel views that contain correct occlusion and detailed textures.
\bibliographystyle{IEEEbib}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{
"redpajama_set_name": "RedPajamaC4"
}
| 1,618
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The game is developed by Frozenbyte, the studio behind Trine.
We had the chance to talk with a couple developers of Frozenbyte that were present at Gamescom 2017 showcasing the game once again. Specifically, Perttu Hotakainen and Petri Ruuskanen talked to us about the game and its current state of development.
With the studio having reached its highest peak ever, at 111 people working for Frozenbyte (around 50 on the project of Nine Parchments) the company is ever growing and more than ready to move to the release phase of the game. Since the release is soon, the company is trying to build the hype for the game as much as possible by attending as many big game expos as possible, such as the Tokyo Game Show and PAX West which are both coming up after Gamescom 2017.
The core of the game has reportedly not changed since the last time we played it (which was at last year's Gamescom), but this time we see a few RPG elements that have been added to it. The co-op part of it is also one of its main premises since the start of it so it is still there and going more strong than ever.
You can learn more about Nine Parchments through the game's Steam page, website and Twitter.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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GameModeWidget::GameModeWidget(QWidget * parent) :QWidget(parent)
{
//This should be an object
QVBoxLayout *modeLayout = new QVBoxLayout;
QHBoxLayout *topLayout = new QHBoxLayout;
QHBoxLayout *bottomLayout = new QHBoxLayout;
QVBoxLayout *modeNameLayout = new QVBoxLayout;
QHBoxLayout *finalLayout = new QHBoxLayout;
m_playerControlAngle = new PlayerControlModeWidget(1, true);
m_playerControlPower = new PlayerControlModeWidget(2, false);
m_playerControlFire = new PlayerControlModeWidget(3, false);
topLayout->addWidget(m_playerControlAngle);
topLayout->addWidget(m_playerControlPower);
topLayout->addWidget(m_playerControlFire);
m_player1Indicator = new PlayerActivatedWidget;
m_player2Indicator = new PlayerActivatedWidget(2, false);
bottomLayout->addWidget(m_player1Indicator);
bottomLayout->addWidget(m_player2Indicator);
QLabel * mode = new QLabel("Mode :");
modeNameLayout->addWidget(mode);
QFont font = mode->font();
font.setPointSize(20);
font.setBold(true);
mode->setFont(font);
QLabel * joueur = new QLabel("Tour du joueur :");
modeNameLayout->addWidget(joueur);
QFont font1 = joueur->font();
font1.setPointSize(20);
font1.setBold(true);
joueur->setFont(font1);
modeLayout->addLayout(topLayout);
modeLayout->addLayout(bottomLayout);
finalLayout->addLayout(modeNameLayout);
finalLayout->addLayout(modeLayout);
setLayout(finalLayout);
}
GameModeWidget::~GameModeWidget()
{
}
void GameModeWidget::setCurrentPlayer(Player player)
{
if (player == Player::Player1)
{
m_player1Indicator->setActivated(true);
m_player2Indicator->setActivated(false);
}
else if(player == Player::Player2)
{
m_player1Indicator->setActivated(false);
m_player2Indicator->setActivated(true);
}
else
{
m_player1Indicator->setActivated(false);
m_player2Indicator->setActivated(false);
}
}
Player GameModeWidget::currentPlayer()const
{
if (m_player1Indicator->activated())
return Player::Player1;
else if (m_player2Indicator->activated())
return Player::Player2;
return Player::NoPlayer;
}
InputState GameModeWidget::currentMode()const
{
if (m_playerControlAngle->activated())
return InputState::Angle;
else if (m_playerControlFire->activated())
return InputState::Fire;
else if (m_playerControlPower->activated())
return InputState::Power;
return InputState::NoState;
}
void GameModeWidget::setCurrentMode(InputState gameMode)
{
if (gameMode == InputState::Angle)
{
m_playerControlAngle->setActivated(true);
m_playerControlFire->setActivated(false);
m_playerControlPower->setActivated(false);
}
else if (gameMode == InputState::Fire)
{
m_playerControlAngle->setActivated(false);
m_playerControlFire->setActivated(true);
m_playerControlPower->setActivated(false);
}
else if (gameMode == InputState::Power)
{
m_playerControlAngle->setActivated(false);
m_playerControlFire->setActivated(false);
m_playerControlPower->setActivated(true);
}
else
{
m_playerControlAngle->setActivated(false);
m_playerControlFire->setActivated(false);
m_playerControlPower->setActivated(false);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,045
|
NaClTerm.nmf = 'lua.nmf'
NaClTerm.env = ['NACL_DATA_MOUNT_FLAGS=manifest=/manifest.txt']
function log(message) {
document.getElementById('log').textContent = message;
}
function fsErrorHandler(error) {
log("Filesystem error: "+ error);
}
function uploadFile(file) {
fs.root.getFile(file.name, {create: true, exclusive: true},
function(fileEntry) {
fileEntry.createWriter(function(fileWriter) {
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"redpajama_set_name": "RedPajamaGithub"
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Q: Freelance web development vs. design Do web design freelancers (I'm thinking individuals, not companies) typically do all development and design work themselves? What are the options for those with a strong base in back-end web development but less with web design, color schemes, layouts, graphic design, etc.?
A: As a freelancer myself, this is what I do. I normally talk to the person who is giving me the project and get myself a designer. The designer designs the UI and provides me with all the necessary HTML, CSS etc etc. Using that as the base, I change plain vanilla HTML to server components (ASP.Net) and I start working on the design and then start development. My design methodology is mostly agile, as in freelance projects (depending upon the scale), you normally don't have separation of phases. So agile works well for me.
A: Some people do try and do both.. and in most cases I have seen they do one well and the other... not so much. Designing and Programming generally take two different types of people. Often I've seen designers subcontract the programming work and focus on the designing aspects.
Personally I consider myself formost a programmer. I understand the basic aspects of designing web pages but I would rather let somone else deal with those details.
A: I'm up front about not doing any design work. I handle only the programming side. However, I keep all HTML seperate from the rest of the code, in easy to identify template files (e.g header.php, footer.php, signup.php, etc), so its very easy for them to get a designer and have him change it without having to deal with much code.
A: I typically get all of my design work sub-contracted. That is, I'll hire a designer to build a photoshop version of the site, usually getting it 90% of the way there and then either get someone to cut up that design into HTML & CSS, or do that myself (depending on a number of factors).
There are many options for getting the initial design. I tend to use the same designer that I've used in the past, but you can use sites like 99designs.com or more generic freelance sites like elance.com to find designers. I would tend to prefer the designer-specific 99designs.com, they're more expensive but the quality is generally higher.
In the past, I've also sub-contracted out the HTML/CSS cutting, but I find that is more hit-and-miss than the designs themselves. I usually prefer to do the cutting myself, since I can tailor the HTML/CSS to fit better with the server-side generation of the HTML. I find that many companies which provide HTML/CSS cutting don't take tweaking or modifications into account and making changes to the layout or slight tweaks later on can be very painful. I've been burnt by that often enough - it's ended up costing me more time tweaking a badly cut design than if I had just done the whole thing from scratch - that I don't even bother much any more.
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| 7,750
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In Snowflake, Arizona at Northland Pioneer College a self-defense course is being offered on March 15, 2014 to develop your defensive marksmanship and self-defense skills.
"Self-defense is more than a class — it's a mind-set. Students will learn how to think defensively, assess threats and learn avoidance techniques," noted NRA-certified instructor Richard Harris, who will be teaching the "Basic Self Defense with a Handgun" course (reference STC 099x-73705). The $75 fee includes range time and a student manual containing all of the techniques and skills learned in the self-defense course. Students will need to bring their own firearm with holster, eye and ear protection and 100 rounds of ammunition.
Students under the age of 21 must be accompanied by a parent or a guardian. All weapons and ammunition should be left secured out of sight in a vehicle prior to range activities.
Sign up for the Basic Handgun Self-Defense workshop at least a week before the class starts to ensure it is not canceled due to insufficient enrollment.
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=Majorization&diff=cur&oldid=4677","text":"# Difference between revisions of \"Majorization\"\n\n## Definition\n\nWe say a nonincreasing sequence of real numbers $a_1, \\ldots ,a_n$ majorizes another nonincreasing sequence $b_1,b_2,\\ldots,b_n$, and write $\\{a_i\\}_{i=1}^n$$\\{b_i\\}_{i=1}^n$ if and only if all for all $1 \\le k \\le n$, $\\sum_{i=1}^{k}a_i \\ge \\sum_{i=1}^{k}b_i$, with equality when $\\displaystyle k = n$. If $\\displaystyle \\{a_i\\}$ and $\\displaystyle \\{b_i\\}$ are not necessarily nonincreasing, then we still write $\\displaystyle \\{a_i\\}$$\\displaystyle \\{b_i\\}$ if this is true after the sequences have been sorted in nonincreasing order.\n\n### Minorization\n\nWe will occasionally say that $b_1, \\ldots, b_n$ minorizes $a_1, \\ldots, a_n$, and write $\\displaystyle \\{b_i\\}$$\\displaystyle \\{a_i\\}$, if $\\displaystyle \\{a_i\\}$$\\displaystyle \\{b_i\\}$.\n\n## Alternative Criteria\n\nIt is also true that $\\{a_i\\}_{i=1}^n$$\\{b_i\\}_{i=1}^n$ if and only if for all $1\\le k \\le n$, $\\sum_{i=k}^n a_i \\le \\sum_{i=k}^n b_i$, with equality when $\\displaystyle k=1$. An interesting consequence of this is that the finite sequence $\\displaystyle \\{a_i\\}$ majorizes $\\displaystyle \\{b_i\\}$ if and only if $\\displaystyle \\{-a_i\\}$ majorizes $\\displaystyle \\{-b_i\\}$.\n\nWe can also say that this is the case if and only if for all $t \\in \\mathbb{R}$,\n\n$\\sum_{i=1}^{n}|t-a_i| \\ge \\sum_{i=1}^{n}|t-b_i|$.\n\nBoth of these conditions are equivalent to our original definition.","date":"2021-04-17 07:30:53","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\": 28, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9980432391166687, \"perplexity\": 9803.146531706578}, \"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-2021-17\/segments\/1618038118762.49\/warc\/CC-MAIN-20210417071833-20210417101833-00146.warc.gz\"}"}
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Table of Contents
Title Page
Copyright Page
Acknowledgements
ONE - Baku, Azerbaijan Sunday, 11:33 P.M.
TWO - Camp Springs, Maryland Sunday, 4:12 P.M.
THREE - Washington, D.C. Sunday, 6:32 P.M.
FOUR - Baku, Azerbaijan Monday, 2:47 A.M.
FIVE - Washington., D. C. Sunday, 8:00 P.M.
SIX - Hellspot Station, the Caspian Sea Monday, 3:01 A.M.
SEVEN - Camp Springs, Maryland Monday, 12:44 A.M.
EIGHT - Baku, Azerbaijan Monday, 4:00 P.M.
NINE - Washington, D.C. Monday, 11:55 A.M.
TEN - Baku, Azerbaijan Monday, 9:21 P.M.
ELEVEN - Washington, D. C. Monday, 3:00 P.M.
TWELVE - Camp Springs, Maryland Monday, 3:14 P.M.
THIRTEEN - Gobustan, Azerbaijan Monday, 11:56 P.M.
FOURTEEN - New York, New York Monday, 4:01 P.M.
FIFTEEN - Washington, D.C. Monday, 4:03 P.M.
SIXTEEN - Baku, Azerbaijan Tuesday, 12:07 A.M.
SEVENTEEN - Washington, D. C. Monday, 4:13 P.M.
EIGHTEEN - Gobustan, Azerbaijan Tuesday, 1:22 A.M.
NINETEEN - Baku, Azerbaijan Tuesday, 1:35 A.M.
TWENTY - Washington, D.C. Monday, 6:46 P.M.
TWENTY-ONE - Baku, Azerbaijan Tuesday, 3:58 A.M.
TWENTY-TWO - Saint Petersburg, Russia Tuesday, 4:01 A.M.
TWENTY-THREE - Washington, D.C. Monday, 7:51 P.M.
TWENTY-FOUR - Khachmas, Azerbaijan Tuesday, 4:44 A.M.
TWENTY-FIVE - Saint Petersburg, Russia Tuesday, 4:47 A.M.
TWENTY-SIX - Washington, D. C. Monday, 9:00 P.M.
TWENTY-SEVEN - Baku, Azerbaijan Tuesday, 5:01 A.M.
TWENTY-EIGHT - Washington, D.C. Monday, 10:03 P.M.
TWENTY-NINE - Baku, Azerbaijan Tuesday, 6:15 A.M.
THIRTY - Baku, Azerbaijan Tuesday, 6:16 A.M.
THIRTY-ONE - Washington, D.C. Monday, 11:11 P.M.
THIRTY-TWO - Washington, D.C. Monday, 11:24 P.M.
THIRTY-THREE - Baku, Azerbaijan Tuesday, 8:09 A.M.
THIRTY-FOUR - Washington, D.C. Tuesday, 12:10 A.M.
THIRTY-FIVE - Saint Petersburg, Russia Tuesday, 8:30 A.M.
THIRTY-SIX - Washington, D.C. Tuesday, 12:30 A.M.
THIRTY-SEVEN - Baku, Azerbaijan Tuesday, 9:01 A.M.
THIRTY-EIGHT - Saint Petersburg, Russia Tuesday, 9:31 A.M
THIRTY-NINE - Teheran, Iran Tuesday, 10:07 A.M.
FORTY - Washington, D.C. Tuesday, 1:33 A.M.
FORTY-ONE - Washington, D.C Tuesday, 1:34 A.M.
FORTY-TWO - Saint Petersburg, Russia Tuesday, 9:56 A.M.
FORTY-THREE - Baku, Azerbaijan Tuesday, 10:07 A.M.
FORTY-FOUR - Washington, D.C. Tuesday, 2:08 A.M.
FORTY-FIVE - Saint Petersburg, Russia Tuesday, 10:20 A.M.
FORTY-SIX - Baku, Azerbaijan Tuesday, 10:31 A.M.
FORTY-SEVEN - Washington, D.C. Tuesday, 2:32 A.M.
FORTY-EIGHT - Baku, Azerbaijan Tuesday, 10:47 A.M.
FORTY-NINE - Saint Petersburg, Russia Tuesday, 11:02 A.M.
FIFTY - Washington, D.C. Tuesday, 3:06 A.M.
FIFTY-ONE - Baku, Azerbaijan Tuesday, 11:09 A.M.
FIFTY-TWO - Washington, D.C. Tuesday, 3:13 A.M.
FIFTY-THREE - Baku, Azerbaijan Tuesday, 11:15 A.M.
FIFTY-FOUR - Washington, D.C. Tuesday, 3:17 A.M.
FIFTY-FIVE - Baku, Azerbaijan Tuesday, 11:22 A.M.
FIFTY-SIX - Washington, D.C. Tuesday, 4:27 A.M.
FIFTY-SEVEN - Washington, D.C. Tuesday, 4:41 A.M.
FIFTY-EIGHT - Saint Petersburg, Russia Tuesday, 12:53 P.M.
FIFTY-NINE - Washington, D.C. Tuesday, 5:04 A.M.
SIXTY - Washington, D.C. Tuesday, 6:46 A.M.
EPILOGUE
**THE BESTSELLING NOVELS OF Tom Clancy**
**RAINBOW SIX**
Clancy's shocking story of international terrorism—closer to reality than any government would care to admit.
"GRIPPING . . . BOLT-ACTION MAYHEM."
— _People_
**EXECUTIVE ORDERS**
Jack Ryan has always been a soldier. Now he's giving the orders.
"AN ENORMOUS, ACTION-PACKED, HEAT-SEEKING MISSILE OF A TOM CLANCY NOVEL."
— _Seattle Times_
**DEBT OF HONOR**
It begins with the murder of an American woman in the backstreets of Tokyo. It ends in war. . .
"A SHOCKER CLIMAX SO PLAUSIBLE YOU'LL WONDER WHY IT HASN'T YET HAPPENED!"
— _Entertainment Weekly_
**THE HUNT FOR RED OCTOBER**
The smash bestseller that launched Clancy's career—the incredible search for a Soviet defector and the nuclear submarine he commands . . .
"BREATHLESSLY EXCITING!"
_—Washington Post_
**RED STORM RISING**
The ultimate scenario for World War III—the final battle for global control . . .
"THE ULTIMATE WAR GAME . . . BRILLIANT!"
— _Newsweek_
**PATRIOT GAMES**
CIA analyst Jack Ryan stops an assassination—and incurs the wrath of Irish terrorists . . .
"A HIGH PITCH OF EXCITEMENT!"
— _Wall Street Journal_
**THE CARDINAL** **OF THE KREMLIN**
The superpowers race for the ultimate Star Wars missile defense system . . .
_"CARDINAL_ EXCITES, ILLUMINATES ... A REAL PAGE-TURNER!"
— _Los Angeles Daily News_
**CLEAR AND PRESENT DANGER**
The killing of three U.S. officials in Colombia ignites the American government's explosive, and top secret, response ...
"A CRACKLING GOOD YARN!"
— _Washington Post_
**THE SUM OF ALL FEARS**
The disappearance of an Israeli nuclear weapon threatens the balance of power in the Middle East—and around the world . . .
"CLANCY AT HIS BEST . . . NOT TO BE MISSED!"
_—Dallas Morning News_
**WITHOUT REMORSE**
The Clancy epic fans have been waiting for. His code name is Mr. Clark. And his work for the CIA is brilliant, cold-blooded, and efficient . . . but who is he really?
"HIGHLY ENTERTAINING!"
— _Wall Street Journal_
_Novels by Tom Clancy_
THE HUNT FOR RED OCTOBER
RED STORM RISING
PATRIOT GAMES
THE CARDINAL OF THE KREMLIN
CLEAR AND PRESENT DANGER
THE SUM OF ALL FEARS
WITHOUT REMORSE
DEBT OF HONOR
EXECUTIVE ORDERS
RAINBOW SIX
SSN: STRATEGIES OF SUBMARINE WARFARE
_Created by Tom Clancy and Steve Pieczenik_
TOM CLANCY'S OP-CENTER
TOM CLANCY'S OP-CENTER: MIRROR IMAGE
TOM CLANCY'S OP-CENTER: GAMES OF STATE
TOM CLANCY'S OP-CENTER: ACTS OF WAR
TOM CLANCY' S OP-CENTER: BALANCE OF POWER
TOM CLANCY'S OP-CENTER: STATE OF SIEGE
TOM CLANCY'S OP-CENTER: DIVIDE AND CONQUER
TOM CLANCY' NET FORCE
TOM CLANCY'S NET FORCE: HIDDEN AGENDAS
TOM CLANCY'S NET FORCE: NIGHT MOVES
_Created by Tom Clancy and Martin Greenberg_
TOM CLANCY'S POWER PLAYS: POLITIKA
TOM CLANCY'S POWER PLAYS: RUTHLESS.COM
TOM CLANCY'S POWER PLAYS: SHADOW WATCH
_Nonfiction_
SUBMARINE: A GUIDED TOUR INSIDE A NUCLEAR WARSHIP
ARMORED CAV: A GUIDED TOUR OF AN ARMORED CAVALRY REGIMENT
FIGHTER WING: A GUIDED TOUR OF AN AIR FORCE COMBAT WING
MARINE: A GUIDED TOUR OF A MARINE EXPEDITIONARY UNIT
AIRBORNE: A GUIDED TOUR OF AN AIRBORNE TASK FORCE
CARRIER: A GUIDED TOUR OF AN AIRCRAFT CARRIER
INTO THE STORM: A STUDY IN COMMAND
( _written with General Fred Franks)_
EVERY MAN A TIGER
_(written with General Charles Horner)_
This is a work of fiction. Names, characters, places, and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.
TOM CLANCY'S OP-CENTER: DIVIDE AND CONQUER
A Berkley Book / published by arrangement with
Jack Ryan Limited Partnership and S & R Literary, Inc.
PRINTING HISTORY
Berkley edition / June 2000
All rights reserved.
Copyright © 2000 by Jack Ryan Limited Partnership and
S & R Literary, Inc.
This book may not be reproduced in whole or in part, by mimeograph or any other means, without permission. For information address:
The Berkley Publishing Group, a division of Penguin Putnam Inc.,
375 Hudson Street, New York, New York 10014.
The Penguin Putnam Inc. World Wide Web site address is <http://www.penguinputnam.com>
eISBN : 978-1-101-00366-4
BERKLEY®
Berkley Books are published by The Berkley Publishing Group, a division of Penguin Putnam Inc., 375 Hudson Street, New York,
New York 10014.
BERKLEY and the "B" design are trademarks belonging to
Penguin Putnam Inc.
<http://us.penguingroup.com>
_**Acknowledgments**_
We would like to acknowledge the assistance of Martin H. Greenberg, Larry Segriff, Robert Youdelman, Esq., Tom Mallon, Esq., and the wonderful people at Penguin Putnam, including Phyllis Grann, David Shanks, and Tom Colgan. As always, we would like to thank Robert Gottlieb of The William Morris Agency, our agent and friend, without whom this book would never have been conceived. But most important, it is for you, our readers, to determine how successful our collective endeavor has been.
—Tom Clancy and Steve Pieczenik
**PROLOGUE**
**_Washington, D.C. Sunday, 1:55 P.M._**
The two middle-aged men sat in leather armchairs in a corner of the wood-paneled library. The room was in a quiet corner of a Massachusetts Avenue mansion. The blinds were drawn to protect the centuries-old art from the direct rays of the early-afternoon sun. The only light came from a dull fire that was smoldering in the fireplace. The fire gave the old, wood-paneled room a faintly smoky smell.
One of the men was tall, stout, and casually dressed with thinning gray hair and a lean face. He was drinking black coffee from a blue Camp David mug while he studied a single sheet of paper resting in a green folder. The other individual, seated across from him with his back to the bookcase, was a short bulldog of a man with a three-piece gray suit and buzz-cut red hair. He was holding an empty shot glass that, moments before, had been brimming with scotch. His legs were crossed, his foot was dancing nervously, and his cheek and chin bore the nicks of a quick, unsatisfactory shave.
The taller man shut the folder and smiled. "These are wonderful comments. Just perfect."
"Thank you," said the red-haired man. "Jen's a very good writer." He shifted slowly, uncrossing his legs. He leaned forward, causing the leather seat to groan. "Along with this afternoon's briefing, this is really going to accelerate matters. You know that, don't you?"
"Of course," the taller man said. He put his coffee mug on a small table, rose, and walked to the fireplace. He picked up a poker. "Does that scare you?"
"A little," the red-haired man admitted.
"Why?" the taller man asked as he threw the folder into the flames. It caught fire quickly. "Our tracks are covered."
"It's not us I'm worried about. There _will_ be a price," the red-haired man said sadly.
"We've discussed this before," the taller man said. "Wall Street will love it. The people will recover. And any foreign powers that try to take advantage of the situation will wish they hadn't." He jabbed the burning folder. "Jack ran the psychological profiles. We know where all the potential trouble spots are. The only one who's going to be hurt is the man who created the problem. And he'll recover. Hell, he'll do better than recover. He'll write books, give speeches, make millions."
The taller man's words sounded cold, though the red-haired man knew they weren't. He had known the other man for nearly thirty-five years, ever since they served together in Vietnam. They fought side by side in Hue during the Tet offensive, holding an ammunition depot after the rest of the platoon had been killed. They both loved their country passionately, and what they were doing was a measure of that deep, deep love.
"What's the news from Azerbaijan?" the taller man asked.
"Everyone's in place." The red-haired man looked at his watch. "They'll be eyeballing the target close-up, showing the man what he has to do. We don't expect the next report for another seven hours or so."
The taller man nodded. There was a short silence broken only by the crackling of the burning folder.
The red--haired man sighed, put his glass on the table, and rose. "You've got to get ready for the briefing. Is there anything else you need?"
The taller man stabbed the ashes, destroying them. Then he replaced the poker and faced the red-haired man.
"Yes," he said. "I need you to relax. There's only one thing we have to fear."
The red-haired man smiled knowingly. "Fear itself."
"No," said the other. "Panic and doubt. We know what we want, and we know how to get there. If we stay calm and sure, we've got it."
The red-haired man nodded. Then he picked up the leather briefcase from beside the chair. "What was it that Benjamin Franklin said? That revolution is always legal in the first person, as in 'our' revolution. It's only illegal in the third person, as in 'their' revolution."
"I never heard that," said the taller man. "It's nice."
The red-haired man smiled. "I keep telling myself that what we're doing is the same thing the founding fathers did. Trading a bad form of government for a better one."
"That's correct," the other man said. "Now, what I want you to do is go home, relax, and watch a football game. Stop worrying. It's all going to work out."
"I wish I could be as confident."
"Wasn't it Franklin who also said, 'In this world nothing can be said to be certain, except death and taxes'? We've done the best we can, and we've done everything we can. We have to put our trust in that."
The red-haired man nodded.
They shook hands, and the shorter man left.
A young aide was working at a large, mahogany desk outside the library. She smiled up at the red-haired man as he strode down the long, wide, carpeted corridor toward the outside door.
He believed that this would work out. He truly did. What he didn't believe was that the repercussions would be so easy to control.
_Not that it matters,_ he thought as a security guard opened the door for him and he stepped into the sunlight. He pulled sunglasses from his shirt pocket and slipped them on. _This has to be done, and it has to be done now_.
As he walked down the paved drive to his car, the red-haired man held tight to the notion that the founding fathers had committed what many considered to be treasonous acts when they forged this nation. He also thought of Jefferson Davis and the Southern leaders who formed the Confederacy to protest what they considered repression. What he and his people were doing now was neither unprecedented nor immoral.
But it _was_ dangerous, not just for themselves but for the nation. And that, more than anything, would continue to scare the hell out of him until the country was firmly under their control.
**ONE**
**_Baku, Azerbaijan Sunday, 11:33 P.M._**
David Battat looked impatiently at his watch. They were over three minutes late. _Which is nothing to be concerned about,_ the short, agile American told himself. A thousand things could have held them up, but they would be here. They would come by launch or motorboat, possibly from another boat, possibly from the wharf four hundred yards to his right. But they _would_ arrive.
_They had better_ , he thought. He couldn't afford to screw up twice. Not that the first mistake had been his fault.
The forty-three-year-old Battat was the director of the Central Intelligence Agency's small New York field office, which was located across the street from the United Nations building. Battat and his small team were responsible for electronic SOS activities: spying on spies. Keeping track of foreign "diplomats" who used their consulates as bases for surveillance and intelligence-gathering activities. Battat also had been responsible for overseeing the activities of junior agent Annabelle Hampton.
Ten days before, Battat had come to the American embassy in Moscow. The CIA was running tests in the communications center on an uplink with a new high-gain acoustic satellite. If the satellite worked on the Kremlin, the CIA planned on using it in New York to eavesdrop more efficiently on foreign consulates. While Battat was in Moscow, however, Annabelle helped a group of terrorists infiltrate the United Nations. What made it especially painful was that the young woman did it for pay, not principle. Battat could respect a misguided idealist. He could not respect a common hustler.
Though Battat had not been blamed officially for what Annabelle did, he was the one who had run the background check on her. He was the one who had hired her. And her "seconding action," as it was officially classified, had happened during his watch. Psychologically and also politically, Battat needed to atone for that mistake. Otherwise, chances were good that he would get back to the United States and discover that the field agent who had been brought in from Washington to operate the office in his absence was now the permanent New York field director. Battat might find himself reassigned to Moscow, and he didn't want that. The FBI had all the ins with the black marketeers who were running Russia and the Bureau didn't like to share information or contacts with the CIA. There wouldn't be anything to do in Moscow but debrief bored _aparatchiks_ who had nothing to say except that they missed the old days and could they please get a visa to anywhere west of the Danube?
Battat looked out over the tall grasses at the dark waters of the Bay of Baku, which led to the Caspian Sea. He raised his digital camera and studied the _Rachel_ through the telephoto lens. There was no activity on the deck of the sixty-one-foot motor yacht. A few lights were on below deck. They must be waiting. He lowered the camera. He wondered if the passengers were as impatient as he was.
_Probably_ , he decided. Terrorists were always edgy but focused. It was an unusual combination, and one way that security forces zeroed in on potential troublemakers in crowds.
Battat looked at his watch again. Now they were five minutes late. Maybe it was just as well. It gave him a chance to get a handle on the adrenaline, to concentrate on the job. It was difficult.
Battat had not been in the field for nearly fifteen years. In the closing days of the war in Afghanistan, he had been a CIA liaison with the Mujahideen guerrilla fighters. He had reported from the front on Soviet troop strength, arms, deployment, tactics, and other battlefield details. Anything the military might need to know if the United States ever fought Soviet or Soviet-trained soldiers. That was back when the United States still had people on the ground collecting solid, firsthand intelligence instead of satellites gathering pictures and audio transmissions, which teams of experts then had to interpret. Former operatives like Battat who had been trained in HUMINT—human intelligence—called those experts "educated lucky guessers," since they were wrong just as often as they were right.
Now, dressed in black boots, blue jeans, leather gloves, a black turtleneck, and a black baseball cap, Battat was watching for a possible new enemy. One of those satellites Battat hated had picked up a communication during a test run in Moscow. For reasons as yet unknown, a group known as "Dover Street" was meeting on the _Rachel,_ presumably a boat, to pick up "the Harpooner." If this was the same Harpooner the CIA had missed grabbing in Beirut and Saudi Arabia, they wanted him. Over the past twenty-five years, he had been responsible for the deaths of hundreds of Americans in terrorist bombings. After discussing the contents of the message with Washington, it was decided that Battat would photograph the individuals and return to the American consulate in Baku for positive ID. After that, the boat would be tracked by satellite, and a special ops team would be dispatched from Turkey to take him out. No extradition debate, no political hot potato, just a good, old-fashioned erasure. The kind the CIA used to do before Iran-Contra gave black ops a bad name. Before "do something" was replaced by "due process." Before good manners replaced good government.
Battat had flown to Baku. Clearing customs, he had taken the crowded but clean metro out to the Khatayi stop on the sea. The ride cost the equivalent of three cents, and everyone was exceedingly polite, helping one another on and off and holding the doors for late arrivals.
The United States embassy in Baku maintained a small CIA field office staffed by two agents. The agents were presumably known to the Azerbaijani police and rarely went into the field themselves. Instead, they brought in outside personnel whenever neccessary. The embassy would not be happy to be presented with the action as a fait accompli. But there were increasing tensions between the United States and Azerbaijan over Caspian oil. The republic was attempting to flood the market with inexpensive oil to bolster its weak economy. That represented enormous potential damage to American oil companies, who were only marginally represented here—a holdover from the days of the Soviet Union. The CIA in Moscow did not want to inflame those tensions.
Battat spent the late afternoon walking around a section of beach, looking for a particular boat. When he found it, anchored about three hundred yards offshore, he made himself comfortable on a low, flat rock among a thatch of high reeds. With his backpack, water bottle, and bag dinner at his side and the camera hanging around his neck, he waited.
The smell of salty air and oil from the offshore rigs was strong here, like nowhere else in the world. It almost burned his nostrils. But he loved it. He loved the sand under his rubber soles, the cool breeze on his cheek, the sweat on his palms, and the accelerated beat of his heart.
Battat wondered how many foreign invaders had stood on these shores, perhaps in this very spot. The Persians in the eleventh century. The Mongols in the thirteenth and fourteenth centuries. The Russians in the eighteenth century, then the Persians again, then the Soviets. He couldn't decide whether he was part of a dramatic historical pageant or an ugly, unending rape.
_Not that it matters_ , he told himself. He wasn't here to safeguard Azerbaijan. He was here to redeem himself and to protect American interests.
Crouched among the high reeds at this isolated section of beachfront, Battat felt as though he had never been away from the field. Danger did that. It was like a fond song or a familiar food smell, a bookmark in the soul. He loved that, too. He also felt good about what he was doing. Not just to atone for Annabelle but because it was right.
Battat had been here for nearly seven hours now. The cell phone communications they'd intercepted said that the pickup was scheduled for eleven-thirty P.M. The Harpooner was supposed to be there to examine the parcel, whatever it was, then pay for it and leave.
Just then, something happened on the boat. A hatch door opened, and a man climbed out onto the deck. Battat looked out at the water. The man turned on a radio. It was playing what sounded like local folk tunes. Maybe that was a signal. Battat's gaze swept across the water.
Suddenly, an elbow locked around Battat's throat from behind and yanked him to his feet. He gagged. He tried to tuck his chin into the elbow, to relieve the pressure on his throat so he could breathe, but the attacker was well trained. He had locked his right arm around his throat and was pushing Battat's head with his left hand so he couldn't turn it. Battat tried to drive an elbow back into the attacker's gut, but the man was standing to the side. Finally, he tried to reach back and grab the shoulder of the choking arm and pull the attacker over.
The attacker responded by tilting his own body back and lifting Battat from the ground. Although Battat was able to grab the man's shoulder, he couldn't throw the attacker. Battat's feet were in the air and he had no leverage.
The struggle lasted five seconds. The attacker's arm squeezed against the American's carotid arteries from the side, immediately cutting the blood supply to the head and causing Battat to black out. Taking no chances, the attacker kept pressing the arteries for another half minute. Then he dropped the unconscious body to the sand.
The Harpooner reached into the pocket of his windbreaker. He removed a syringe from his pocket, pulled off the plastic tip, and injected the man in the neck. After wiping away the small drop of blood, he took out a flashlight and flicked it on. He waved it back and forth several times. Another flashlight answered from the _Rachel_.
Then both lights went dark. Moments later, a motor dinghy lowered from the boat and headed toward shore.
**TWO**
**_Camp Springs, Maryland Sunday, 4:12 P.M._**
Paul Hood sat on an armchair in the corner of the small, TV-lit hotel room. The heavy shades were drawn and a football game was on, but Hood wasn't really watching it. He was watching reruns in his mind. Reruns of over sixteen years of married life.
_Old pictures in my new home,_ he thought.
Home was an anonymous fifth-floor suite at the Days Inn on Mercedes Boulevard, located a short distance from Andrews Air Force Base. Hood had moved in late Saturday night. Though he could have stayed at a motel right next to the base where Op-Center was located, he wanted the option of being able to get away from work. Which was ironic. It was Hood's dedication to Op-Center that had cost him his marriage.
Or so his wife maintained.
Over the past several years, Sharon Hood had become increasingly frustrated by the long hours her husband kept at Op-Center. She grew tense and angry each time an international crisis caused him to miss one of their daughter Harleigh's violin recitals or their son Alexander's ball games. She was bitter that virtually every vacation they planned had to be canceled because of a coup attempt or assassination that demanded his attention. She resented how he was on the phone, even when he was with his family, checking with Deputy Director Mike Rodgers on how the mobile Regional Op-Center was performing in field tests or discussing with Intelligence Chief Bob Herbert what they could do to strengthen the new relationship with Op-Center's Russian counterpart in Saint Petersburg.
But Hood had never believed that work itself was really the problem. It was something older and deeper than that.
Even when he had resigned his position as director of Op-Center and went to New York for Harleigh's performance at a United Nations reception, Sharon still wasn't happy. She was jealous of the attention that other mothers on the junket gave him. Sharon realized that the women were drawn to Hood because he had been a highly visible mayor of Los Angeles. After that, he had held a powerful job in Washington, where power was the coin of the realm. It didn't matter to Sharon that Hood put no stock in fame and power. It didn't matter to her that his replies to the women were always polite but short. All Sharon knew was that she had to share her husband again.
Then came the nightmare. Harleigh and the other young musicians were taken hostage in the Security Council chambers by renegade United Nations peacekeepers. Hood had left Sharon at the State Department's understaffed crisis center so that he could oversee Op-Center's successful covert effort to rescue the teenagers and the captive foreign delegates. In Sharon's eyes, he had not been there for her again. When they returned to Washington, she immediately took the children to her parents' house in Old Saybrook, Connecticut. Sharon had said she wanted to get Harleigh away from the media zoo that had pursued the children from New York.
Hood couldn't argue with that. Harleigh had seen one of her friends seriously wounded and several other people executed. She was almost killed herself. She had suffered the clinical consequences of classic stressor triggers for post-traumatic stress disorder: threats to the physical integrity of herself and others; fear and helplessness; and a guilt response to survival. After all that, to have been surrounded by TV lights and shouting members of the press corps would have been the worst thing for Harleigh.
But Hood knew that wasn't the only reason his wife had gone back to Old Saybrook. Sharon herself needed to get away. She needed the comfort and safety of her childhood home in order to think about her future.
About _their_ future.
Hood shut off the TV. He put the remote on the night table, lay back on the bunched pillows, and looked up at the white ceiling. Only he didn't see a ceiling. Hood saw Sharon's pale face and dark eyes. He saw how they had looked on Friday when she came home and told him she wanted a divorce.
That wasn't a surprise. It was actually a relief in some ways. After Hood had returned from New York, he met briefly with the president about repairing the rift between the United States and the UN. Being back at the White House, being plugged into the world, had made him want to withdraw his resignation from Op-Center. He _liked_ the work he was doing: the challenge, the implications, the risk. On Friday evening, after Sharon had told him of her decision, he was able to withdraw his resignation with a clear conscience.
By the time Hood and Sharon talked again on Saturday, the emotional distancing had already begun. They agreed that Sharon could use their family attorney. Paul would have Op-Center's legal officer, Lowell Coffey III, recommend someone for him. It was all very polite, mature, formal.
The big questions they still had to decide were whether to tell the kids and whether Hood should leave the house immediately. He had called Op-Center's staff psychologist Liz Gordon, who was counseling Harleigh before turning her over to a psychiatrist who specialized in treating PTSD. Liz told Hood that he should be extremely gentle whenever he was around Harleigh. He was the only family member who had been with her during the siege. Harleigh would associate his strength and calmness with security. That would help to speed her recovery. Liz added that whatever instability was introduced by his departure was less dangerous than the ongoing strife between him and his wife. That tension would not show Hood in the light Harleigh needed to see him. Liz also told him that intensive therapy for Harleigh should begin as soon as possible. They had to deal with the problem, or she ran the risk of being psychologically impaired for the rest of her life.
After having discussed the situation with Liz Gordon, Hood and Sharon decided to tell the kids calmly and openly what was happening. For the last time as a family, they sat in the den—the same room where they had set up their Christmas tree every year and taught the kids Monopoly and chess and had birthday parties. Alexander seemed to take it well after being assured that his life wouldn't change very much. Harleigh was initially upset, feeling that what had happened to her was the cause. Hood and his wife assured Harleigh that was not the case at all, and they would both be there for her.
When they were finished, Sharon had dinner with Harleigh at home, and Hood took Alexander out to their favorite greasy pit, the Corner Bistro—the "Coroner Bistro" as the health-conscious Sharon called it. Hood put on his best face, and they had a fun time. Then he came back to the house, quickly and quietly packed a few things, and left for his new home.
Hood looked around the hotel room. There was a glass-covered desk with a blotter, a lamp, and a folder full of postcards. A queen-sized bed. An industrial-strength carpet that matched the opaque drapes. A framed print of a painting of a harlequin whose outfit matched the carpet. A dresser with a built-in cabinet for a minirefrigerator and another cabinet for the TV. And, of course, a drawer with a Bible. There was also a night table with a lamp like the one on the desk, four wastebaskets, a clock, and a box of tissues he had moved from the bathroom.
_My new home_ , he thought again.
Except for the laptop on the desk and the pictures of the kids beside it—last year's school photos, still in their warping cardboard frames—there was nothing of home here. The stains on the carpet weren't apple juice Alexander had spilled as a boy. Harleigh hadn't painted the picture of the harlequin. The refrigerator wasn't stocked with rows of plastic containers filled with that wretched kiwi-strawberry-yogurt juice that Sharon liked. The television had never shown home videotapes of birthday parties, pool parties, and anniversaries, of relatives and coworkers who were gone. Hood had never watched the sun rise or set from this window. He had never had the flu or felt his unborn child kick in this bed. If he called out to the kids, they wouldn't come.
Tears pressed against the backs of his eyes. He turned to look at the clock, anything to break the steady succession of thoughts and pictures. He would have to get ready soon. Time—and government—stopped for no man. He still had professional obligations. But lord God, Hood thought, he didn't feel like going. Talking, putting on a happy face the way he did with his son, wondering who knew and who didn't in the instant message machine known as the Washington grapevine.
He looked up at the ceiling. Part of him had wanted this to happen. Hood wanted the freedom to do his job. He wanted an end to being judged and criticized by Sharon. He also wanted to stop constantly disappointing his wife.
But another part of him, by far the largest part, was bitterly sad that it had come to this. There would be no more shared experiences, and the children were going to suffer for their parents' shortcomings.
As the finality of the divorce hit him, hit him hard, Hood allowed the tears to flow.
**THREE**
**_Washington, D.C. Sunday, 6:32 P.M._**
Sixty-one-year-old First Lady Megan Catherine Lawrence paused before the late-seventeenth-century gilded pier mirror over a matching commode. She gave her short, straight, silver hair and ivory satin gown one last check before picking up her white gloves and leaving her third-floor salon. Satisfied, the tall, slender, elegant woman crossed the South American rug collected by President Herbert Hoover and entered the private presidential bedroom. The president's private dressing room was directly across from her. As she stepped out, she looked out at the lamp-lit white walls and light-blue Kennedy curtains, the bed that was first used by Grover and Frances Cleveland, the rocking chair where delicate, devoted Eliza Johnson awaited word of her husband Andrew's impeachment trial in 1868, and the bedside table where each night the seventh president, Andrew Jackson, would remove a miniature portrait of his dead wife from its place beside his heart, set it on the table next to her well-read Bible, and made certain that her face was the first thing he saw each morning.
As she looked out at the room, Megan smiled. When they first moved into the White House, friends and acquaintances would say to her, "It must be amazing having access to all the secret information about President Kennedy's missing brain and the Roswell aliens." She told them the secret was that there _was_ no secret information. The only amazing thing was that, after nearly seven years of living in the White House, Megan still felt a thrill to be here among the ghosts, the greatness, the art, and the history.
Her husband, former Governor Michael Lawrence, had been president of the United States for one term when a series of stock market tumbles helped the moderate conservative lose a close election to Washington outsiders Ronald Bozer and Jack Jordan. Pundits said it was as much the family lumber fortune of the Oregon redwood that had made the president a target, since he was largely unaffected by the downturn. Michael Lawrence didn't agree, and he was not a quitter. Rather than become a token partner in some law firm or join the board of directors of his family corporation, the former president stayed in Washington, set up a nonpartisan think tank, American Sense, and was a hands-on manager. He used the next eight years to find ways to fix or fine-tune what he perceived had been wrong with his first term, from the economy to foreign policy to social programs. His think tank members did the Sunday morning talk show circuit, wrote op-ed pieces, published books, and gave speeches. With a weak incumbent vice president to run against, and a new vice president on his own ticket—New York Senator Charles Cotten—Mi—chael Lawrence decisively won reelection. His popularity rating remained in the 60 percent region, and reelection was considered a fait accompli.
Megan crossed the room to the president's dressing room. The door was shut, which was the only way to keep the bathroom warm, since draftiness came with the old walls and history. That meant her husband was probably still in the shower, which was surprising. Selected guests would be arriving at the second-floor study for a small, private half-hour cocktail reception at seven. Her husband usually liked to be ready fifteen minutes before that to sit with his thick personnel folder and review the likes, dislikes, hobbies, and family data of foreign guests. Tonight, he had the newly appointed acting ambassadors from Sweden and Italy coming up before a state dinner for key United Nations delegates. Their predecessors had been assassinated during the recent siege, and the replacements had been named quickly to show the world that terrorism could not stop the pursuits of peace and diplomacy. The president wanted a chance to meet the two men privately. After that, they'd **go** down to the Blue Room for a formal predinner reception with other influential United Nations delegates. Then it was on to the dinner itself, which was designed to show unity and support after the attack the previous week.
The president had come up shortly before six o'clock, which should have given him plenty of time to shower and shave. Megan couldn't understand what was keeping him. Perhaps he was on the phone. His staff tried to keep calls to the private residence to a minimum, but he'd been getting more and more calls over the past few days, sometimes in the small hours of the morning. She did not want to sleep in one of the guest bedrooms, but she wasn't a youngster anymore. Years ago, when they first started campaigning for public office, she used to be able to get by on two or three hours of sleep. No more. It had to be even worse on her husband. He was looking more tired than usual and desperately needed rest. The crisis at the United Nations had forced them to cancel a planned vacation in the northwest, and they had not been able to reschedule it.
The First Lady stopped by the six-panel door and listened. The shower was not running. Neither was the water in the sink. And it didn't sound as if he was on the phone.
"Michael?"
Her husband did not answer. She turned the bright brass handle and opened the door.
There was a narrow anteroom before the bathroom. In an alcove to the right was a stand-alone cherry wood wardrobe where the president's valet left his clothes for the day. In an alcove to the left was a matching cherry wood dressing table with a large, brightly lit wall mirror above it. The president was dressed in a royal blue bathrobe. He was standing there, breathing heavily, a look of rage in his narrow blue eyes. His fists were white-knuckle tight at his sides.
"Michael, are you all right?"
He glared at her. She had never seen him look so angry and—disoriented was the word that came to mind. It frightened her deeply.
"Michael, what _is_ it?"
He looked back at the mirror. His eyes softened and his hands relaxed. His breathing came more easily. Then he slowly lowered himself into a walnut side chair in front of the dressing table.
"It's nothing," he said. "I'm fine."
"You don't look fine," she said.
"What do you mean?"
"A moment ago, you looked like you wanted to take a bite out of something," Megan told him.
He shook his head. "That was just leftover energy from my exercises," he said.
"Your exercises? I thought you were at a meeting before."
"I was just doing isometrics," he told her. "Senator Samuels does them for ten minutes every morning and evening. He says they're a great tension releaser when you can't get to the gym."
Megan did not believe him. Her husband perspired easily when he exercised. His forehead and upper lip were dry. Something else was happening here. He had seemed increasingly distant the past few days, and it was starting to scare her.
She stepped forward, coming to his side, and touched his face.
"Something's bothering you, hon," she said. "Talk to me."
The president looked at her. "It's nothing," he said. "These past couple of days have been rough, that's all."
"You mean the calls at night—"
"That, plus everything else that's going on," the president said.
"Is it worse than usual?"
"In some ways," he said.
"Do you want to talk about it?"
"Not right now," he said, forcing a little smile. His deep voice had regained some of its vigor and confidence, and his eyes had a little sparkle now. The president took her hands in his and rose. He stood just over six-foot-four. He looked down at her. "You look beautiful."
"Thank you," Megan said. "But you've still got me worried."
"Don't be," he said. He looked to his right. There was a shelf with a gold clock that had belonged to Thomas Jefferson. "It's late," the president said. "I'd better get ready."
"I'll wait for you," she told him. "And you'd better do something about your eyes."
"My eyes?" he said, glancing at the mirror. He'd gotten up even earlier than she had that morning, and his eyes were severely bloodshot. It was bad for an individual in a position of great responsibility to look weak or tired.
"I didn't sleep very well last night," he said, touching and tugging on the skin around them. "A few eyedrops will take care of that." The president turned back to his wife and kissed her gently on the forehead. "It's all right, I promise," he said, then smiled again and turned away.
Megan watched as her husband walked slowly toward the bathroom and shut the door. She heard him turn on the shower. She listened. Michael usually hummed rock and roll oldies when he showered. Sometimes he even sang. Tonight he was silent.
For the first time in a long time, Megan didn't believe what her husband had told her. No politician was entirely truthful on the outside. Sometimes they had to say what voters and political rivals wanted to hear. But Michael was an honest man on the inside, at least with Megan. When she looked into his eyes, she knew whether or not he was hiding something. When he was, Megan could usually coax him into telling her about it.
But not today, and that bothered her deeply. She was suddenly very scared for him.
Slowly, Megan walked back toward her own dressing room. She pulled on her gloves and tried to concentrate on what she had to do for the next four hours. She had to be an outgoing hostess. She had to be gracious and complimentary to the delegates' wives. At least she would be with people she didn't know. It was easier to hide her feelings when she was with strangers. They would not know that she was putting on an act.
But it _would_ be an act.
Megan went back into the bedroom. There was a small, early-nineteenth-century mahogany Tambour writing cabinet on her side of the bed. She picked up a folder from her executive secretary and went over the guest list, paying particular attention to the names of the foreign delegates and their wives. There was a phonetic guide beside each name, and she reviewed the pronunciation aloud. The names came easily to the First Lady. She had an affinity for language and had planned on becoming a translator when she met and married her husband. Ironically, she had wanted to work for the United Nations.
Megan closed the folder and set it down. She looked around the room. The magic was still here, the lurking spirits and the resonance of great drama. But she was also acutely aware of something she didn't often feel here. Here, in a house that was literally watched by every eye in the world.
She suddenly felt a great sense of isolation.
**FOUR**
_**Baku, Azerbaijan Monday, 2:47 A.M.**_
David Battat awoke slowly.
The sea air was chilly and becoming raw. David was lying on his belly, his face turned to the reeds in front of the water. There was cool moisture on his cheeks, condensation from the Caspian.
He tried to move, but his head felt as if it were made of concrete. His throat was raw, and his neck hurt. He touched it gently and winced. The skin was bruised and extremely sore. His camera was gone. The CIA team back in Moscow wouldn't be able to study the photographs he took to see who else might have been on the boat, or calculate how much weight it was carrying by where the waterline reached. Artillery and missiles weighed a lot more than explosives, currency, or drugs.
Battat tried to push himself off the ground. As he did, he felt as though a spike had been hammered through the back of his neck. He dropped, waited a few seconds, then tried again even more slowly. He managed to get his knees under him, then sat looking out across the dark water.
The _Rachel_ was gone. He'd blown this big time. Like it or not, he'd have to let Moscow know as soon as possible.
Battat's head throbbed, and he lowered himself back to the ground. He rested on his forearms, placed his forehead on the cool earth, and tried to get a handle on the pain. He also tried to make sense of what had happened.
Why was he still alive? Battat wondered. The Harpooner had never let anyone live. Why him?
Then it occurred to him that maybe he went down before the Harpooner even arrived. Maybe some waterfront thug had happened by, saw his camera and backpack, and decided to steal them. Battat couldn't decide which was worse: letting his target sneak up on him or being mugged. Not that it mattered. They were both bad.
The operative took a long breath, then rose slowly, first to his knees again and then to his feet. He stood unsteadily as his head pounded. He looked around for his backpack. That was gone, too. No flashlight, no chance to look around for footprints or other clues.
He looked at his watch. His wrist was trembling, and he used his free hand to steady it. It would be dawn in less than three hours. Fishermen would be setting out soon, and Battat didn't want to be seen here. Just in case he wasn't meant to survive, he didn't want anyone to know that he had. He walked slowly from the shore, his head drumming. Each swallow was painful, and the collar of his turtleneck chafed his bruised neck.
But the worst pain was none of those.
The worst pain was the knowledge that he'd failed.
**FIVE**
_**Washington., D. C. Sunday, 8:00 P.M.**_
As he entered the White House through the East Appointment Gate, Paul Hood remembered the first time he brought his children here. Hood had come to Washington for a conference of mayors. Harleigh was eight at the time, and Alexander was six. Alexander was not impressed by the imposing G. P. A. Healy painting of Abraham Lincoln or the magnificent Blue Room chairs bought by James Monroe or even the secret service officers. Alexander had seen paintings and chairs and police officers in Los Angeles. The spectacular chandelier in the State Dining Room was barely worth an upward glance, and the Rose Garden was just grass and flowers. But as they crossed the lawn toward E Street, the young boy finally saw something that impressed him.
Horse chestnuts.
The dark green chestnuts growing from the stout trees resembled nothing so much as little floating mines with Herz horns projecting from all sides. Alexander was convinced that they were little bombs to keep prowlers out. They'd bump their heads, and the chestnuts would explode. Alexander's father played along with the idea, even snatching a few of the chestnuts—carefully, of course—so they could plant them in the ground back at home. Harleigh finally busted her dad by stepping on one of the newly planted chestnuts and failing to blow up.
Sharon had never approved of the deception. She felt that it encouraged militarism. Hood felt that it was just a boy's imagination at work, nothing more.
It was rare that Paul Hood came to the White House without thinking of the horse chestnut trees. Tonight was no different, except that for the first time in years, Hood had the strong desire to go out back and pluck a few. Bring them to his son as a token, a memory of a good time shared. Besides, walking around the grounds would have been preferable to what he was doing.
He had dressed in his tuxedo, driven to the White House, and presented his calligraphic invitation at the East Appointment Gate. A junior secret service agent met Hood there and escorted him to the Red Room, which adjoined the State Dining Room. The president and First Lady were still in the Blue Room, which was the next room over. Though no one said so, the smaller Red Room—typically used for entertaining by the first ladies—was for the B-level guests.
Hood recognized but did not really know many of the people who were there. He knew some of them from conferences, some from briefings, and many from other dinners he attended here. The White House had two hundred fifty state dinners every year, and he was invited to at least fifteen of those. His background in Los Angeles government—which really meant knowing movie stars—finance, and espionage made him an ideal dinner guest. He could talk to generals, world leaders, diplomats, reporters, senators, and their spouses, informing and entertaining them and also not offending them. That was important.
Sharon usually came with him to those dinners. Being in the health-food business, she was generally unhappy with the fare, though she always loved the settings, which were from different administrations, different centuries. When Sharon couldn't make it, Op-Center's press liaison Ann Farris went with Hood. She liked any food that was put in front of her and, unlike Sharon, enjoyed talking to whoever she was seated with.
This was the first time Hood had come stag. Regardless of how the White House might try to position it, Hood did not consider Mala Chatterjee as his date. The UN secretary-general was also coming alone and was assigned a seat at Hood's table, directly to his left.
Hood opened the door and looked into the long, chandelier-lit dining room. Fourteen round tables had been brought into the dining room. Each one was set for ten people. Hood's invitation had said that he was seated at table two, near the center of the room. That was good. He was rarely seated so close to the president. If things got tense between him and Chatterjee, Hood would be able to exchange knowing glances with the First Lady. Megan Lawrence had been raised in Santa Barbara, California. She had spent time with Hood when he was mayor of Los Angeles, and they got to know each other quite well. She was a smart, classy lady with a dry sense of humor.
While senior staff members watched, liveried White House waitstaff hurried around, making last-minute adjustments to the rose centerpieces. They were dressed in black jackets and were multiethnic, which was to be expected at an affair of this kind.
The White House selected from a large pool of security-cleared hourly employees. And though no one liked to admit it, the composition of the staff was determined by the nature of the dinner. The young and attractive personnel were filling crystal water glasses and making sure the flatware was spaced exactly alike from setting to setting.
Straight ahead was the towering 1869 portrait of Abraham Lincoln that hadn't impressed Alexander. It was the only painting in the dining room. Directly across from him, inscribed on the mantel, was a passage written by John Adams to his wife Abigail before they moved into the newly completed executive mansion. Franklin Roosevelt had read the lines and liked them so much that they became the official White House prayer. The inscription read:
I pray Heaven to bestow the best of blessings on this house and all that shall hereafter inhabit it. May none but honest and wise men ever rule under this roof.
_Sorry, Mr. Adams, Hood thought. We managed to blow that one_.
One of the senior attendants walked over. Dressed in white trousers and a white waistcoat with gold braid, he politely but insistently shut the door. Hood stepped back into the Red Room. It had grown noisier and more crowded as people began filing in from the Blue Room. He couldn't imagine what it was like in here before air-conditioning.
Hood happened to be facing the door to the Blue Room as Mala Chatterjee entered. She was on the arm of the president, who was followed by the First Lady and two delegates. The vice president and Mrs. Cotten came in next followed by California Senator Barbara Fox. Hood knew Fox well. She looked uncharacteristically confused. Hood didn't get to ask why. At almost exactly that moment, the door to the State Dining Room opened. There was no more rushing around inside the hall. The twenty members of the waitstaff were lined up along the northwest wall, while attendants stood in a row by the door to show guests to their tables.
Hood made no effort to link up with Chatterjee. She was an intense woman, and she seemed caught up in her conversation with the president. He turned and went back into the dining hall.
Hood watched as the glitterati entered beneath the golden light of the chandelier. There was something almost ghostly about the procession: people moving slowly, stiffly dignified, and without much expression; voices low and hollow in the echoing chamber, with only occasional polite laughter; chairs soundlessly lifted and moved by attendants so they didn't drag on the hardwood floor; and a sense that this scene had been repeated over and over throughout the years, throughout the centuries, with the same people: those who had power, those who wanted it, and people like Hood who were the buffers between them.
Hood took a sip of water. He wondered if divorce turned all men into cynics.
Chatterjee had left the president's side and was being shown to the table. Hood rose as the New Delhi native neared. The attendant pulled out her chair. The secretary-general thanked him and sat down. Without obviously ignoring Hood, the forty-three-year-old woman managed not to look at him. Hood had no patience for that.
"Good evening, Madam Secretary-General," Hood said.
"Good evening, Mr. Hood," she replied, still without looking at him.
Other people began arriving at the table. Chatterjee turned and smiled at Agriculture Secretary Richard Ortiz and his wife. That left Hood staring at the back of the secretary-general's head. He exited the awkward moment by reaching for his napkin, putting it on his lap, and looking the other way.
Hood tried to put himself in Chatterjee's position. The attorney-turned-diplomat had only been on the job for a short while when the terrorists struck. She had joined the United Nations as an avowed peacekeeper, and here were terrorists executing diplomats and threatening to shoot children. Chatterjee's negotiating tactics had failed, and Hood had embarrassed her publicly by infiltrating the Security Council and ending the crisis with quick, violent action. Chatterjee was further humiliated by the way many member nations loudly applauded Hood's attack.
But Hood and Secretary-General Chatterjee were supposed to be putting that ill will behind them, not nurturing it. She was an avowed advocate of first move detente, in which one party demonstrated trust by being the first to lay down arms or surrender land.
_Or maybe she only believes in that when she advocates others to make the first move,_ Hood thought.
Suddenly, someone appeared behind Hood and spoke his name. He turned and looked up. It was the First Lady.
"Good evening, Paul."
Hood rose. "Mrs. Lawrence. It's good to see you."
"It's been too long," she said, taking his hand in hers and holding it tight. "I miss those Los Angeles fund-raisers."
"We had fun," Hood said. "We made some history, and hopefully we did some good, too."
"I like to think so," the First Lady said. "How is Harleigh?"
"She took a very hard hit, and is having a rough time," Hood admitted.
"I can't even imagine," the First Lady said. "Who's working with her?"
"Right now, it's just Liz Gordon, our staff psych at Op-Center," Hood said. "Liz is getting a little trust going. Hopefully, in a week or two, we can bring in some specialists."
Megan Lawrence smiled warmly. "Paul, maybe there's something we can do to help each other. Are you free for lunch tomorrow?"
"Sure," he said.
"Good. I'll see you at twelve-thirty." The First Lady smiled, turned, and went back to her table.
_That was strange_ , Hood thought. " _Maybe there's_ _something we can do to help each other."_ What could she possibly need his help for? Whatever it was, it must be important. A First Lady's social calendar was usually well-booked months in advance. She would have had to move her engagements around to make room for him.
Hood sat back down. The table had been joined by Deputy Secretary of State Hal Jordan and his wife Barri Allen-Jordan as well as two diplomats and their spouses who Hood did not know. Mala Chatterjee did not introduce him, so he introduced himself. The secretary-general continued to ignore him, even after the president rose at his table to offer a toast and say a few words about how he hoped this dinner and its show of unity would send a message to terrorists that the civilized nations of the world would never yield to them. As the White House photographer took pictures and a C-SPAN camera unobtrusively recorded the event from the southwest corner of the hall, the president underscored his faith in the United Nations by announcing officially, and to great applause, that the United States was about to retire its nearly two billion dollar debt to the United Nations.
Hood knew that paying off the debt had very little to do with terrorists. The United Nations didn't scare them, and the president knew it, even if Mala Chatterjee didn't. What the two billion dollars did was get the United States out of the doghouse with poor countries like Nepal and Liberia. With thawed economic relations in the Third World, we could then convince them to take loans with the provision that they buy American goods, services, and military intelligence. That would become a self-perpetuating source of income for American companies, even when other nations started putting money into those countries. That was the great thing about a government budgetary surplus and a politically expedient moment. When they came together, an administration could look benevolent and score points on the stock exchange.
Hood was only half listening to the speech when the president said something that drew him back in.
"Finally," the president said, "I am happy to inform you that American intelligence leaders are presently earmarking personnel and resources for a vital new initiative. It is their intention to work closely with governments around the world and guarantee that attacks against the United Nations cannot, do not, and _will not_ happen again."
There was mild applause from tables where there were delegates. But the statement had caught Hood's attention because he knew something that the president apparently did not.
It wasn't true.
**SIX**
_**Hellspot**_ **_Station, the Caspian Sea_** _**Monday, 3:01 A.M**._
The white Cessna U206F flew low over the dark Caspian Sea, its single engine roaring loudly. Its only occupants were a Russian pilot and the man seated beside him, an Englishman of average build and average appearance.
This trip had started out off the coast of Baku. After taking off, the seaplane had headed northeast and had traveled nearly two hundred miles in the past ninety minutes. It had been a smooth, quiet ride. Neither the pilot nor his passenger spoke a word the entire time. Though forty-one-year-old Maurice Charles spoke Russian—along with nine other languages—he did not know the pilot well and did not trust even those people he did know well. That was one reason he'd managed to survive as a mercenary for nearly twenty years.
When they finally arrived, all the pilot said was, "Below, four o'clock."
Charles looked out his window. His pale blue eyes fixed on the target. It was a beautiful thing. Tall, brightly lit, majestic.
And alone.
The semisubmersible offshore oil drilling platform stood approximately 150 feet above the water and was surrounded by sea. There was a helipad on the north side of the platform, a 200-foot-tall derrick beside it on the northwest side, and a network of tanks, cranes, antennae, and other equipment in the oil processing area.
The rig was like a lady standing on a deserted avenue under a streetlamp late at night by the Mersey back home. Charles could do what he wanted with it. And he would.
Charles picked up a camera that was sitting in his lap. He popped the button on the tan leather carrying case and removed the top. The camera was the same thirty-five-millimeter reflex that he had used in his first assignment, back in Beirut in April 1983. He began snapping pictures. A second camera, the one he had taken from the CIA operative on the beach, lay on the floor of the cabin between his feet along with the man's backpack. There might be names or numbers in there that would prove useful. Just like the operative himself would be useful, which was why Charles had left him alive.
The airplane circled the oil platform twice, once at 600 feet and once at 300 feet. Charles exposed three rolls of film, then indicated to the pilot that it was all right to leave. The seaplane swung back to its cruising altitude of 2000 feet and headed to Baku. There, Charles would rejoin the crew of the _Rachel,_ which by now would have removed the white banner with the fake name. They had ferried him to the plane and would be his partners in the next part of the undertaking.
But that would only be the start. His employers in America had very specific goals, and the team Charles had put together were experts in achieving those goals: turning neighbor against neighbor, nation against nation, through acts of terrorism and assassination. Before they were finished, the region would be awash in fire and blood from around the world.
And though he had already made a lot of money in the terrorist game, he had spent a lot of that wealth buying weapons, passports, transportation, anonymity. With this job, he would be richer than he had ever dared to imagine. And he had a fertile imagination.
When he was growing up in Liverpool, Charles had often dreamed about wealth and how he might obtain it. He thought about it when he swept the train station where his father sold tickets. He thought about it when he slept with his two brothers and grandfather in the living room of their one-bedroom flat, a flat that always smelled of perspiration and trash from the adjoining alley. He thought about it when he helped his father coach the local men's football team. The elder Charles knew how to communicate, how to strategize, how to win. He was a natural leader. But Maurice's father, his family, his working-class people were held down by the upper class. They were not permitted to go to the better schools, even if they could have afforded them. They weren't allowed to work in the upper levels of banking, of communications, of politics. They had funny, common accents and brawny shoulders and weather-beaten faces and weren't taken seriously.
Charles grew up feeling bad that the only outlet, the only joy his father had was football. Charles also idolized the Beatles because they had made it out—the same reason, ironically, his father and so many of his contemporaries hated "those young punks." Charles realized that he could not escape poverty musically because he had no talent for that and it had already been done. He had to get out his way, make a mark that was uniquely his own. How could he have known that he would find his hidden skills by joining the Royal Marines, 29 Commando Regiment, Royal Artillery, and learning to work with explosives? By discovering the pleasure and genius involved in tearing things down?
It was a glorious feeling to put events like this in motion. It was the creation of art: living, breathing, powerful, bleeding, changing, utterly unforgettable art. There was nothing else like it in the world, the aesthetics of destruction. And what was most rewarding was that the CIA had inadvertently helped him by sending that man to watch for him. The agency would conclude that it couldn't be the Harpooner who had attacked their man. No one had ever survived an encounter with the Harpooner.
Charles settled comfortably into his seat as the Cessna left the lights of the rig behind.
That was the beauty about being an artist, he told himself.
It gave him the right and privilege to surprise.
**SEVEN**
_**Camp Springs, Maryland Monday, 12:44 A.M.**_
Throughout the Cold War, the nondescript two-story building located near the Naval Reserve flight line at Andrews Air Force Base was a staging area for pilots and their crews. In the event of a nuclear attack, their job would have been to evacuate key officials from the government and military to a safe compound in the Blue Ridge Mountains.
But the ivory-colored building with its neat, green lawn was not just a monument to the Cold War. The seventy-eight full-time employees who worked there now were employed by the National Crisis Management Center, familiarly known as Op-Center, an independent agency that was designed to collect, process, and analyze data on potential crisis points domestically and abroad. Once that was done, Op-Center then had to decide whether to defuse them preemptively through political, diplomatic, media, economic, legal, or psychological means or else—after gaining the approval of the Congressional Intelligence Oversight Committee—to terminate them through military means. To this end, Op-Center had at its disposal a twelve-person tactical strike team known as Striker. Led by Colonel Brett August, Striker was based at the nearby Quantico FBI Academy.
In addition to the offices upstairs, a secure basement had been built into the facility to house the more sensitive intelligence retrieval systems and personnel. It was here that Paul Hood and his top advisers worked.
Hood came directly from the White House affair. He was still dressed in his tuxedo, which earned him a "Good morning, Mr. Bond" greeting from the Naval officer at the gate. It made him smile. It was the only thing that had done that for days.
A strange uneasiness had settled over Hood after the president made his comments. He couldn't imagine why the president had said the United States would offer intelligence assistance to the United Nations. If there was one thing many member nations feared, it was that the United States was already using the international organization as a means of spying on them.
The president's short speech had pleased some people, most notably delegates who were targets for acts of terrorism. But it struck some other attendees as odd. Vice president Cotten appeared surprised, as did Secretary of State Dean Carr and America's United Nations Ambassador Meriwether. And Mala Chatterjee had been openly bothered by the comment. So much so that she'd actually turned to Hood and asked if she had understood the president correctly. He told her that he believed she had. What he didn't tell her was that Op-Center would almost certainly have been involved in or briefed about any such arrangement. Something might have been arranged during the time that he was away, but Hood doubted it. When he visited his office the day before to catch up on business he had missed, he saw no reference to a multinational intelligence effort.
Hood didn't bother talking to anyone after the dinner. He left promptly and went to Op-Center, where he did additional digging into the matter. This was the first time he had seen the weekend night crew since his return. They were glad to see him, especially weekend night director Nicholas Grillo. Grillo was a fifty-three-year-old former Navy SEAL intelligence expert who had moved over from the Pentagon around the same time Hood had first joined Op-Center. Grillo congratulated him on the fine job he and General Rodgers had done in New York and asked how his daughter was. Hood thanked him and told him that Harleigh would be all right.
Hood began by accessing the files of the DCI—the Director of Central Intelligence. This independent body was a clearinghouse of information for four other intelligence departments: the Central Intelligence Agency; Op-Center; the Department of Defense, which included the four branches of the military, the National Reconnaissance Office, the National Security Agency, and the National Imagery and Mapping Agency; and Department Intelligence, which consisted of the Federal Bureau of Investigation, the Department of State, the Department of Energy, and the Department of Treasury.
Once Hood was into the DCI database, he asked for recent agreements or initiatives pertaining to the United Nations. There were nearly five thousand listings. He eliminated those that did not involve intelligence-gathering for the United Nations and its members. That reduced the list to twenty-seven. Hood browsed those quickly. The last was filed a week before, a preliminary report about the failure of the CIA field office to catch Annabelle Hampton's terrorist-support activities in New York. Blame was placed on New York field office head David Battat and his supervisor in Washington, Deputy Assistant Director Wong. Wong was given a written warning, which was not entered into his record. Battat was given a sterner reprimand, which did not become part of his permanent dossier. But Battat would be hung out to dry for a while, doing what Bob Herbert had once described as "sewer rat-a-tat" jobs—dirty work in the line of fire. The kind of work that freshmen agents usually had to perform.
There was nothing about a United Nations operation involving any of the fourteen intelligence agencies. Given the new detente the president was trying to establish with the United Nations, it wasn't surprising that Lawrence would look for a way to help them. But presenting a desire or opportunity as a done deal was mystifying.
The president would have needed the cooperation of the head of at least one of these agencies just to undertake a _study_ for such a proposition, and that wasn't anywhere in the files. There wasn't even any correspondence, electronic or otherwise, requesting such a study. The only answer Hood could think of was a handshake deal between the president and the CIA, FBI, or one of the other groups. But then one of those persons would have been there at tonight's dinner, and the only representative from the intelligence community was Hood. Perhaps the president was trying to force the issue, the way John F. Kennedy did when he announced, publicly, that he wanted Congress to give NASA the funds to put a man on the moon. But United States involvement in international intelligence-gathering was an extremely sensitive area. A president would be reckless to attempt a wide-ranging operation like this without assurances from his own team that it was possible.
It could all be the result of a series of misunderstandings. Maybe the president thought he had the support of the intelligence community. Confusion was certainly not uncommon in government. The question was what to do now that the idea had been presented to the world body. The United States intelligence community was sure to be torn. Some experts would welcome the opportunity to plug directly into resources in nations like China, Colombia, and several former Soviet republics where they currently had very restricted access. Others—Hood included—would be afraid of joining forces with other nations and being fed false data, data that would then become part of U.S. intelligence gospel with potentially disastrous results. Herbert once told him about a situation in 1978, just before the overthrow of the shah of Iran, when antiextremist forces provided the CIA with a code used by supporters of the Ayatollah Khomeini to communicate via telefax. The code was accurate—then. Once the ayatollah assumed power, the shah's files were raided, and the code was found to be in American hands. The code remained in the CIA's system and was used to interpret secret communiques. It wasn't until the ayatollah's death in 1989—when the secret communiques said he was recovering—that the CIA went back and took a close look at the code and the disinformation they'd received. Ten years of data had to be reviewed and much of it purged.
Hood could just imagine what Teheran would say about joining this new antiterrorism network. _"Sure, sign us up. And don't forget to use this new code to monitor the Sunni terrorists working out of Azerbaijan. "_ It could be a real code for real transmissions, or the Iranians could use false transmissions to create deeper mistrust of the Sunnis. The United States could not refuse to help them, because the president had offered; we could not trust the code; and yet what if it turned out to be real and we ignored it?
The whole thing was a potential for disaster. For his part, Hood intended to contact Burton Gable, the president's chief of staff, to find out what he knew about the situation. Hood didn't know Gable well, but he had been one of Lawrence's think tank geniuses and was instrumental in getting the president reelected. Gable hadn't been at the dinner, but there was no policy undertaking in which he was not involved.
Hood went back to the motel, napped, then was back at Op-Center at five-thirty. He wanted to be there when his staff arrived.
Hood had spoken to psychologist Liz Gordon about Harleigh, and to attorney Lowell Coffey about the divorce, so both of them knew he was coming back. Hood had also informed General Rodgers, who had let intelligence chief Bob Herbert know.
Herbert rolled in first. He had lost his wife and the use of his legs in the American embassy bombing in Beirut in 1983. But he had turned that setback into an advantage: Herbert's customized wheelchair was a mini-communications center with phone, fax, and even a satellite uplink that helped to make him one of the most effective intelligence collectors and analysts in the world.
Rodgers followed him in. Though the gray-haired officer had played a key role in ending the terrorist standoff at the United Nations, he was still recovering emotionally from the torture he'd suffered at the hands of Kurdish terrorists in the Middle East. Since his return, there hadn't been quite the same fire in his eyes or bounce in his walk. Though he hadn't broken, some proud, vital part of him had died in that cave in the Bekaa Valley.
Rodgers and Herbert were happy to see him. The two men stayed long enough to welcome him back and for Hood to brief them on what had happened at the state dinner. Herbert was blown away by what the president had said.
"That's like the Goodyear Blimp saying it's going to watch the stands for rowdy fans instead of watching the Super Bowl," Herbert said. "No one would believe that. No one."
"I agree," Hood said. "Which is why we've got to find out why the president said it. If he has a plan that we don't know about, we need to be brought into the loop. Talk to the other intel people and find out."
"I'm on it," Herbert said as he wheeled out.
Rodgers told Hood that he would get in touch with the heads of Army, Navy, Air Force, and Marine intelligence to find out what their knowledge of the situation was.
When Herbert and Rodgers left, Hood was visited by the only key members of the team who hadn't known about Hood's return, FBI and Interpol liaison Darrell McCaskey and press liaison Ann Farris. McCaskey was just back from a stay in Europe, working with his Interpol associates and nurturing a romance with Maria Corneja, an operative he had worked with in Spain. Hood had a good sense about people, and his instincts told him that Darrell would be handing in his resignation before long to return to Maria. Since McCaskey was gone while Hood's retirement was briefly in effect, he had not missed his boss.
Ann Farris was a different story. The five-foot, seven-inch-tall divorcee had always been close to Hood and had hated to see him leave. Hood knew that she cared for him, though no one could have told that just by looking at her. The thirty-four-year-old woman had developed the perfect poker face for reporters. No question, no revelation, no announcement made her jump. But to Hood, her large, dark-rust eyes were more articulate than any speech-maker or television moderator he had ever heard. And right now, her eyes were telling Hood that she was happy, sad, and surprised all at once.
Ann walked toward the desk. She was dressed in what she called her "uniform," a black pantsuit and white blouse with a pearl necklace. Her brown hair was shoulder length and held back from her face with a pair of clips. Hood's office was stripped of his personal touches. He hadn't had time to put the photographs and mementos back. Yet after the struggles with Sharon and the coldness of his hotel room, Ann's arrival suddenly made this place seem like home.
"Mike just told me," she said.
"Told you what?"
"About Sharon," Ann replied. "About your coming back. Paul, are you all right?"
"I'm a little banged up, but I'll be okay."
Ann stopped in front of the desk. _Was it only just ten days ago that she had stood there while I packed?_ Hood thought. It seemed so much longer. Why did pain stretch time while happiness made it feel so short?
"What can I do, Paul?" Ann asked. "How are Sharon and the kids?"
"We're all reeling. Liz is helping Harleigh, Sharon and I are pretty civil, and Alexander is Alexander. He's okay." Hood dragged a hand through his wavy black hair. "As for what you can do, I just realized we're going to have to send out a press release about my return."
"I know." She smiled. "A head's-up _would_ have been a big help."
"I'm sorry," Hood said.
"That's all right," Ann replied. "You had other things on your mind. I'll write something up and show it to you."
Ann looked down at him, her shoulder-length brown hair framing her angular features. Hood had always felt the sexual tension between them. Hell, he thought. Everyone around them did. Bob Herbert and Lowell Coffey used to tease Hood about it. Hood's unwillingness to give in to that tension had always kept Ann at a distance. But he could feel that distance closing.
"I know you have a lot to do," Ann said, "but if you need anything, I'm here. If you want to talk or don't want to be by yourself, don't be shy. We go back quite a few years."
"Thanks," Hood said.
Ann's eyes held him for a long moment. "I'm sorry for what you and your family are going through, Paul. But you've done an amazing job here, and I'm glad you're back."
"It's good to be back," Paul admitted. "I think that frustrated me more than anything else."
"What did?" she asked.
"Not being able to finish the work I started," he said. "It may sound corny, but the teamwork of exceptional men and women built this nation. Op-Center is a part of that tradition. We have a great team here doing important work, and I hated leaving that."
Ann continued to look at him. She seemed to want to say something more but didn't. She stepped back from the desk.
"Well, I've got to get to work on the press release," she said. "Do you want me to say anything about the situation with Sharon?"
"No," Hood said. "If anyone wants to know, tell them. Otherwise, just say I had a change of heart."
"That's going to make you sound wishy-washy," she said.
"What the _Washington Post_ thinks isn't going to affect my job performance," he said.
"Maybe not now," Ann said. "But it might if you ever decide to run for public office again."
Hood looked at her. "Good point," he said.
"Why don't we tell them that the president asked you to return?" she said.
"Because he didn't," Hood said.
"You two had a private meeting when you came back from New York," she said. "He won't deny asking you to return. It shows loyalty on his part. Everyone benefits."
"But it isn't true," Hood said.
"Then let's just say this," Ann said. "After meeting with the president, you decided to reconsider your resignation. That's true."
"You really want to get the president in there."
"Whenever I can," Ann said. "It gives us weight."
"Weight?" Hood said. "You mean suction."
"Excuse me?"
"Nick Grillo said that the word-de-jour is _suction._ "
"Actually, that's not quite right," Ann informed him. " _Weight_ is when someone has credibility. _Suction_ is when they have considerable influence. There's a difference."
"I see," Hood said. They smiled at each other. Hood looked away. "I'd better get to work," he said. "There's a lot of catching up to do."
"I'm sure," Ann said. "I'll e-mail you a copy of the press release before it goes out."
"Thanks again," Hood said. "For everything."
"Sure." Ann hesitated. She looked at Hood for a long moment more and then left.
Hood turned to the computer monitor on his right. He did not want to watch Ann go. Ann Farris was a beautiful, intelligent, very sexual woman. For the five years they had known each other, they had flirted, she more openly than he. Now that Hood was going to be single, he felt uneasy about continuing the game. There was no longer someone between them. Flirting no longer felt like a game.
But Hood did not have time to think about that now. There was a lot to do. He had to review the daily briefings that had gone to Mike Rodgers during the past week, which included intelligence data collected from around the world as well as ongoing covert operations. He also had to look at reports from the rest of the staff and have a glance at the schedule for the upcoming week before he went to see the First Lady. He noticed that Rodgers was going to be interviewing the final candidates to replace Martha Mackall, the political liaison who had been assassinated in Spain, as well as candidates for the new post of economic adviser. With more and more nations linked together financially—"Siamese megatuplets," was how Lowell Coffey had put it—poli—tics was becoming a troublesome sideshow to the force that really drove the world.
Hood decided to let Mike make those hires. Not only had he started the process, but Hood was going to be too busy with everything else. But with all that was going on, one thing remained true.
Paul Hood loved this work, this place.
It was good to be back.
**EIGHT**
_**Baku, Azerbaijan Monday, 4:00 P.M.**_
Azerbaijan is a nation in flux.
Because of political conflict in the Nagorno-Karabakh region, twenty percent of the country—mostly in the southwest, along the borders with Armenia and Iran—are occupied by rebel forces. Though a cease-fire has been observed since 1994, firefights occur with some regularity. Privately, diplomats fear that the self-proclaimed Republic of Nagorno-Karabakh will become the next Kosovo. Protests, often violent, erupt in Baku and other cities without warning. Some of them pertain to politics, others to general unrest. Since the breakup of the Soviet Union, there has been an extreme shortage of staples such as medical supplies, produce, and new technology. Cash—preferably U.S. dollars—is the only form of exchange recognized in most areas of the country, including the capital.
The United States has managed to openly support the legitimate government of Azerbaijan without alienating the powerful insurgent forces. Loans have been granted to Baku, while goods have been sold directly to "the people"—primarily the rebels. In the event of widespread revolt, the United States wants to have open lines of communication on both sides.
Maintaining that balance is the primary task of the small American embassy. Since March 1993, the fifteen employees and ten marine guards have operated from a small stone building at 83 Azadlig Prospect. In the back of that building, in a windowless, wood-paneled room, is the Department of News Services. Unlike the small press department, which issues news releases and arranges for interviews and photo ops with U.S. congressmen, senators, and other government leaders, officially the job of the DNS is to collect news clippings from around Russia and keep them on file for reference.
Officially.
In fact, the DNS is staffed by one CIA operative who gathers intelligence from around the nation. Most of the information comes from electronic surveillance that is conducted both from the office via satellite and from vans. Some of it comes from personnel who are paid to watch, listen to, and photograph government officials—sometimes in compromising situations. Some of those situations are also arranged by the DNS.
Because he was hurt, David Battat did not want to attempt returning to Moscow. Instead, he made his way to the embassy on foot. He was taken to see Deputy Ambassador Dorothy Williamson, who brought in Senior Researcher Tom Moore. Williamson was a large woman with curly black hair. Battat guessed her to be about forty. Moore was a lean giant in his thirties with a long, gaunt face and a lugubrious expression. If Battat had to be stranded in Baku, his expression would be gloomy as well.
Williamson's aide was a smart veteran named Ron Friday. He was the only one who gave Battat an encouraging smile. Battat appreciated that.
While Battat gave Moore a quick rundown on what had happened, Williamson had the Marine medic take a look at Battat's wounds. There was swelling in his throat and traces of blood in his saliva, though the damage did not appear to be serious. When the medic was finished with him, Battat was taken to the DNS room. He was given privacy while he called Moscow. He spoke to Pat Thomas, the assistant director of public information at the embassy. Thomas was also an OTR—off the record—field director for the CIA. That meant there was no record of him at agency headquarters. His reports were delivered directly to Washington in the diplomatic pouch.
Thomas did not take the news well. If Battat had succeeded in identifying the Harpooner, Thomas would have been a hero. Instead, he would have to explain to his counterpart in Baku and his superior in Washington how they had managed to blow the relatively simple job of surveillance.
Thomas said that he would think about their next step and let him know. Food was brought in. Battat ate, even though he had left his appetite back at the beach, along with his self-esteem, his energy, the mission, and his career. Then he sat in a chair resting until Williamson and Moore arrived for a second, more thorough, conversation. Moore looked grim. This was going to be painful.
Acoustic devices planted in the walls caused conversations to sound like static to the electronic eavesdropping devices that the Azerbaijanis had placed on surrounding buildings.
Battat told them that Moscow had suspected the Harpooner was in Baku, and he had been sent to try and identify him. This news did not meet with the approval of the senior researcher.
"The field office in Moscow obviously didn't feel it was necessary to involve us in this operation," Moore complained. "Do you want to tell me why?"
"They were afraid that our target might have people watching the embassy," Battat said.
"Not all of our people are in the embassy," Moore pointed out. "We have external resources."
"I understand," Battat said. "But Moscow felt that the fewer people who were in the loop, the better our chances of surprising the target."
"Which didn't really help, did it?" Moore said.
"No,"
"Whoever attacked you obviously knew you were coming."
"Apparently, though I don't understand how," Battat said. "I was well hidden, and I wasn't using anything that gave out an electronic pulse. The camera was one of the digital seventies. No flash, no glass in front to reflect light, no moving parts that clicked."
"Couldn't this Harpooner or his people have done a routine sweep of the shore?" the deputy ambassador asked.
"I was watching for that," said Battat. "I got to the site early, at a spot we'd selected through satellite imaging. We chose it specifically so that I could see and hear people coming and going."
"Then why didn't you see or hear the goddamned assailant coming?" asked Moore.
"Because they hit me just when something started to happen out on the boat I was watching," he said. "Someone came from below and turned on a radio. It was a perfect distraction."
"Which suggests that someone knew you were in that spot, Mr. Battat," Moore said.
"Probably."
"Possibly even before you got there," Moore went on.
"I don't see how, but I can't rule it out," Battat agreed.
"What I really want to know, though, is whether this was even the Harpooner," Moore went on.
"What do you mean?" the deputy ambassador asked.
"The Harpooner has been a terrorist for over two decades," Moore told her. "He has personally run or been a part of at least fifteen terrorist strikes that we know of and probably many more that we don't know about. He's eluded countless efforts to trap him thanks, in large part, to his ability to stay mobile. He has no permanent address that we know of, hires whoever he needs, and rarely uses the same people twice. We only know what he looks like because one of his arms suppliers once snuck a photo to us. The supplier's body was found a few months later on a sailboat, slit from chin to belly with a fish-gutting knife— _after_ we'd relocated him and given him a new ID."
"I see," the deputy ambassador said.
"He left the knife behind," Moore said. "He always leaves his weapons behind, from spearguns to bowline stirrups."
"Sea-related things," said Williamson.
"Often," Moore said. "We suspect he was in the naval service somewhere—not a big leap of faith, though we haven't been able to trace him. But in all that time, the Harpooner never left a witness. Which means that either it wasn't the Harpooner who attacked Mr. Battat or the Harpooner wanted him alive."
The deputy ambassador regarded Battat. "For what reason?"
"I can't think of one," Battat admitted.
The three were silent for a moment. The only sound was the hum of the air vent.
"Mr. Battat, the presence of a man like the Harpooner in this region could have terrible ramifications for all of us," said the deputy ambassador.
"Which is another reason why we should have been in the loop on this!" Moore said angrily. "Hell, we know who the undercover guys are that are watching us, and they haven't been around for days. They're too busy trying to find a Russian spy who slipped out of jail two days ago."
"Again, I'm sorry," said Battat.
"Would you mind staying in Baku while we try to make sense of all this?" the deputy ambassador asked.
"Not at all," said Battat. "I want to help."
"Hopefully, it's not too late for that," Moore said.
They rose. "What about the _Rachel?"_ Battat asked.
"I've sent a small plane out to look for it," Moore told him. "But they've had several hours head start, and God knows which direction they went. I'm not optimistic."
"Can't you trace the name?" Battat asked. "Isn't there a local registry?"
"There is," Moore told him, "and the _Rachel_ isn't in it. We're checking records in Dagestan, Kalmyk, and other republics on the Caspian, but my guess is she's a rogue."
Moore showed Battat to a small guest room on the second floor of the building. There was a cot in the corner, and Battat lay down to think. The boat, the music they played, the brief glimpse he had of the man on deck—he replayed the sounds and images over and over, looking for more information. Something that might tell him who the crew of the _Rachel_ were, how they were dressed, or where they might have come from. In SD sessions—subconscious debriefing—trained interviewers would walk agents through experiences to help them remember lost details. The interviewers would ask about the color of the sky, the look of the water, the force of the wind and the smells riding it. Once the agent was reimmersed in the scene, the interviewer would move him around, ask him to describe distinctive markings on the hull of the boat or whether there were banners on the stern or mast or sounds coming from the deck or below. It always surprised Battat how much information the brain stored that was not always immediately accessible.
Though Battat closed his eyes and breathed slowly and deeply and went through the SD checklist, he could not remember anything that brought him closer to whoever was on the boat or from what direction his assailant might have come. He could not even remember the feel of the fabric on the arm that had been choking him or the smell of the man who had attacked him. He couldn't remember if the man's cheek had touched him and whether he was bearded or clean-shaven. Battat had been too focused on trying to survive.
Battat's eyes remained shut. They stopped looking into the past and gazed ahead. He would stay in Baku, but not just because the deputy ambassador had asked. Until Battat found whoever had attacked him, his confidence was broken and his life belonged to them.
Which, he realized, could be why he was left alive.
**NINE**
_**Washington, D.C. Monday, 11:55 A.M.**_
It had always amazed Hood how different Washington looked during the daytime. At night, the white facades were brightly lit and appeared to stand alone, shining with Olympian grandeur. In the day, situated between modern office buildings, vending carts, and glossy restaurant logos, beneath loud and ever-present jet traffic and security barricades of concrete and steel, the landmarks seemed almost antique instead of timeless.
Yet both were Washington. They represented an old, increasingly monolithic bureaucracy that had to be dealt with, and a vision of greatness that could not be ignored or diminished.
Hood parked in the Ellipse on the southern side of the grounds. He crossed E Street and walked up East Executive to the East Appointment Gate. He was buzzed through the iron gate and, after passing through a metal detector, waited inside the East Wing for one of the First Lady's aides.
Of all the landmarks in Washington, Hood had always been partial to the Capitol. For one thing, it was the guts of the government, the place where Congress put wheels on the president's vision. They were often square wheels or wheels of different sizes, but nothing could move without them. For another thing, the building itself was a vast museum of art and history, with treasures everywhere. Here a plaque indicating where the desk of Congressman Abraham Lincoln was located. There a statue of General Lew Wallace, the onetime governor of the territory of New Mexico and the author of _Ben-Hur._ Somewhere else a sign indicating the status of the search for the cornerstone of the building, which was laid over two hundred years before in a little-noted ceremony and was somehow buried and then lost under numerous modifications to the foundation.
The White House wasn't as imposing as the Capitol. It was a much smaller structure, with peeling paint and warping wood on the exterior. But its grounds and columns, its rooms and many familiar angles were intertwined in American memory with images of great leaders doing great things—or, sometimes, infamous, very human things. It would always be the symbolic heart of the United States.
A young male assistant to the First Lady arrived. He brought Hood to the elevator that led to the third floor. Hood was somewhat surprised that the First Lady wanted to see him upstairs. She had an office on the first floor and typically received visitors there.
Hood was taken to the First Lady's sitting room, which adjoined the presidential bedroom. It was a small room with a main door that led to the corridor and another, he assumed, that opened into the bedroom. There was a gold settee against the far wall, two matching wing chairs across from it, and a coffee table between them. A tall secretary with a laptop sat on the opposite wall. The Persian rug was white, red, and gold; the drapes were white, and they were drawn. A small chandelier threw bright shards of light around the room.
Hood looked at the two portraits on the wall. One was of Alice Roosevelt, daughter of Theodore. The other was a painting of Hannah Simpson, mother of Ulysses S. Grant. He was wondering why they were here when the First Lady entered. She was dressed casually in beige slacks and a matching sweater. Her aide shut the door behind her, leaving the two of them alone.
"Nancy Reagan found them in the basement," Megan said.
"I beg your pardon?"
"The portraits," she said. "She found them personally. She hated the idea of women being left to gather dust."
Hood smiled. They embraced lightly, and then Megan gestured toward the settee.
"There are still wonderful things down there," Megan said as they sat. "Furnishings, books, documents, _things_ like Tad Lincoln's writing slate and a diary that belonged to Florence Harding."
"I thought most of that memorabilia was in the Smith-sonian."
"A lot of it is. But many of the family-related things are still here. People have gotten jaded by all the scandals over the years," Megan said. "They forget how much the White House was and is a home. Children were born and raised here, there were weddings, birthdays, and holidays."
Coffee arrived, and Megan was silent as it was served. Hood watched her as the White House steward quietly and efficiently set out the silver service, poured the first cup, then left.
The passion in Megan's voice was exactly as Hood remembered. She never did anything she didn't care deeply about, whether it was addressing a crowd or advocating greater education spending on TV talk shows or discussing the White House with an old friend. But there was something in her expression he had never seen before. The old enthusiasm stopped short of her eyes. When he looked in them, they seemed frightened. Confused.
Hood picked up his cup, took a sip of coffee, then turned to Megan.
"I appreciate your coming," the First Lady said. Her cup and saucer were on her lap, and she was looking down. "I know you're busy and that you have problems of your own. But this isn't just about me or the president, Paul." She looked up. "It's about the nation."
"What's wrong?" Hood asked.
Megan breathed deeply. "My husband has been behaving strangely over the last few days."
Megan fell silent. Hood didn't push her. He waited while she drank some of her coffee.
"Over the past week or so, he's been more and more distracted," she said. "He hasn't asked about our grandson, which is very unusual. He says that it's work, and maybe it is. But things got very strange yesterday." She regarded Hood intently. "This remains between us."
"Of course."
Megan took a short, reinforcing breath. "Before the dinner last night, I found him sitting at his dressing table. He was running late. He wasn't showered or dressed. He was just staring at the mirror, flushed and looking as though he'd been crying. When I asked him about it, he said he'd been exercising. He told me that his eyes were bloodshot because he hadn't been sleeping. I didn't believe him, but I let it be. Then, at the predinner reception, he was flat. He smiled and was pleasant, but there was no enthusiasm in him at all. Until he received a phone call. He took it in his office and returned about two minutes later. When he came back, his manner was entirely different. He was outgoing and confident."
"That's certainly how he seemed at dinner," Hood said. "When you say the president was flat, what exactly do you mean?"
Megan thought for a moment. "Do you know how someone gets when they're really jet-lagged?" she asked. "There's a glassiness in their eyes and a kind of delayed reaction to whatever is said?"
Hood nodded.
"That's exactly how he was until the call," Megan said.
"Do you know _who_ called?" Hood asked.
"He told me it was Jack Fenwick."
Fenwick was a quiet, efficient man who had been the president's budget director in his first administration. Fenwick had joined Lawrence's American Sense think tank, where he added intelligence issues to his repertoire. When the president was reelected, Fenwick was named the head of the National Security Agency, which was a separate intelligence division of the Department of Defense. Unlike other divisions of military intelligence, the NSA was also chartered to provide support for nondefense activities of the Executive Branch.
"What did Fenwick tell the president?" Hood asked.
"That everything had come together," she told Hood. "That was all he would say."
"You have no idea who or what that is?"
Megan shook her head. "Mr. Fenwick left for New York this morning, and when I asked his assistant what the phone call was about, she said something very strange. She asked me, 'What call?' "
"Did you check the log?"
Megan nodded. "The only call that came into that line at that time was from the Hay-Adams Hotel."
The elegant old hotel was located on the other side of Lafayette Park, literally across the street from the White House.
"I had a staff member visit the hotel this morning," Megan went on. "He got the names of the night staff, went to their homes, and showed them pictures of Fenwick. They never saw him."
"He could have come in a back entrance," Hood said. "Did you run a check of the registry?"
"Yes," she said. "But that doesn't mean anything. There could have been any number of aliases. Congressmen often use the hotel for private meetings."
Hood knew that Megan wasn't just referring to political meetings.
"But that wasn't the only thing," Megan went on. "When we went downstairs to the Blue Room, Michael saw Senator Fox and went over to thank her. She seemed very surprised and asked _why_ he was thanking her. He said, 'For budgeting the initiative.' I could see that she had no idea what he was talking about."
Hood nodded. That would explain the confusion he had noticed when Senator Fox entered the room. Things were beginning to fall into place a little. Senator Fox was a member of the Congressional Intelligence Oversight Committee. If any kind of intelligence operation had been approved, she would have to have known about it. Apparently, she was as surprised to learn about the international intelligence-sharing operation as Hood had been. Yet the president either assumed or had been told, possibly by Jack Fenwick, that she had helped make it happen.
"How was the president after the dinner?" Hood asked.
"That's actually the worst of it," Megan said. Her composure began to break. She set her coffee cup aside and Hood did likewise. He moved closer. "As we were getting ready for bed, Michael received a call from Kirk Pike."
The former chief of Navy Intelligence, Pike was the newly appointed director of the CIA.
"He took the call in the bedroom," Megan went on. "The conversation was brief, and when Michael hung up, he just sat on the bed, staring. He looked shell-shocked."
"What did Pike tell him?"
"I don't know," Megan told him. "Michael didn't say. It may have been nothing, just an update that got his mind working. But I don't think he slept all night. He wasn't in bed when I got up this morning, and he's been in meetings all day. We usually talk around eleven o'clock, even if it's just a quick hello, but not today."
"Have you talked to the president's physician about this?" Hood asked.
Megan shook her head. "If Dr. Smith can't find anything wrong with my husband, he might recommend that Michael see Dr. Benn."
"The psychiatrist at Walter Reed," Hood said.
"Correct," Megan said. "Dr. Smith and he work closely together. Paul, you know what will happen if the president of the United States goes to see a psychiatrist. As much as we might try to keep something like that a secret, the risks are much too high."
"The risks are higher if the president isn't well," Hood said.
"I know," Megan said, "which is why I wanted to see you. Paul, there are too many things going on that don't make sense. If there's something wrong with my husband, I'll insist that he see Dr. Benn and to hell with the political fallout. But before I ask Michael to submit to that, I want to know whether something else is going on."
"Glitches in the communications system or a hacker playing tricks," Hood said. "Maybe more Chinese spies."
"Yes," Megan said. "Exactly."
He could see Megan's expression, her entire mood, lighten when he said that. If it were something from the outside, then it could be fixed without hurting the president.
"I'll see what I can find out," Hood promised.
"Quietly," Megan said. "Please, don't let this get out."
"I won't," Hood assured her. "In the meantime, try and talk to Michael. See if you can get him to open up somehow. Any information, any names other than what you've told me, will be a big help."
"I'll do that," Megan said. She smiled. "You're the only one I can trust with this, Paul. Thank you for being there."
He smiled back. "I get to help an old friend and my country. Not a lot of people get that chance."
Megan rose. Hood stood, and they shook hands. "I know this is not an easy time for you, either," the First Lady said. "Let me know if there's anything _you_ need."
"I will," Hood promised.
The First Lady left, and her aide returned to show Hood out.
**TEN**
_**Baku, Azerbaijan Monday, 9:21 P.M.**_
Pat Thomas experienced two miracles in one day.
First, the Aeroflot TU-154 that was scheduled to leave Moscow at six P.M. did so. On time. With the possible exception of Uganda Royal Airways, Aeroflot was the most notoriously late carrier Thomas had ever flown on. Second, the airplane landed in Baku at 8:45 P.M.—five minutes ahead of schedule. During his five years of service at the American embassy in Moscow, Thomas had never experienced either of those events. What was more, despite a relatively full aircraft, the airline had not double- or triple-booked his seat.
The slim, nearly six-foot-tall, forty-two-year-old Thomas was assistant director of public information at the embassy. What the title of ADPI really meant was that Thomas was a spy: a diplomatic private investigator was how he viewed the acronym. The Russians knew that, of course, which was the reason one or two Russian agents always shadowed Thomas in public. He was certain that someone in Baku would be waiting to tail him as well. Technically, of course, the KGB was finished. But the personnel and the infrastructure of the intelligence operation were still very much in place and very much in use as the Federal Security Service and other "services."
Thomas was dressed in a three-piece gray winter suit that would keep him warm in the heavy cold that always rolled in from the Bay of Baku. Thomas knew he would need more than that—strong Georgian coffee or even stronger Russian cognac—to warm him after the reception he expected to receive at the embassy. Unfortunately, keeping secrets from your own people was part of the spy business, too. Hopefully, they would vent a little, Thomas would act contrite, and everyone could move on.
Thomas was met by a staff car from the embassy. He didn't rush tossing his single bag in the trunk. He didn't want any Russian or Azerbaijani agents thinking he was in a hurry. He paused to pop a sucker into his mouth, stretched, then climbed into the car. Be boring. That was the key when you thought you were being watched. Then, if you had to speed up suddenly, chances were good you might surprise and lose whoever was trailing you.
It was a thirty-minute drive from Baku International Airport to the bay-side region that housed the embassies and the city's commercial district. Thomas never got to spend more than a day or two at a time here, though that was something he still meant to do. He had been to the local bazaars, to the Fire Worshipper's Temple, to the State Museum of Carpets—a museum with a name like that demanded to be seen—and to the most famous local landmark, the Maiden Tower. Located in the old Inner City on the bay and at least two thousand years old, the eight-story tower was built by a young girl who either wanted to lock herself inside or throw herself into the sea—no one knew for certain which version was true. Thomas knew how she felt.
Thomas was taken to see Deputy Ambassador Williamson, who had returned from dinner and was sitting behind her desk, waiting for him. They shook hands and exchanged a few banal words. Then she picked up a pen and noted the time on a legal pad. Moore and Battat came to her office moments later. The agent's neck was mottled black and gunmetal gray. In addition to the bruises, he looked exhausted.
Thomas offered Battat his hand. "Are you all right?"
"A little banged up," Battat said. "I'm sorry about all this, Pat."
Thomas made a face. "Nothing's guaranteed, David. Let's see how we can fix it."
Thomas looked at Moore, who was standing beside Battat. The men had met several times at various Asian embassy conferences and functions. Moore was a good man, what they called a twenty-four/seven-an agent who lived and ate his work twenty-four hours a day, seven days a week. Right now, Moore was making no attempt to conceal his dark, unforgiving mood.
Thomas extended his hand. Moore accepted it.
"How have you been?" Thomas asked.
"That isn't important," Moore said. "I'm not happy now. There was no reason for this to go down the way it did."
"Mr. Moore, you're correct," Thomas said as he released his hand. "In retrospect, we should have done this all differently. The question is, how do we fix it now?"
Moore sneered. "You don't get off that easily," he said. "Your team mounted a small operation here and didn't tell us. Your man says you were worried about security risks and other factors. What do you think, Mr. Thomas—that the Azerbaijani are wet-wired into the system? That we can't conduct a surveillance without them finding out?"
Thomas walked to an armchair across from Williamson. "Mr. Moore, Ms. Williamson, we had a short time to make a quick decision. We made a bad one, a wrong one. The question is, what do we do now? If the Harpooner is here, can we find him and stop him from getting away?"
"How do we bail you out, you mean?" Moore asked.
"If you like," Thomas conceded. Anything to get this out of reverse and moving ahead.
Moore relaxed. "It isn't going to be easy," he said. "We've found no trace of the boat Mr. Battat says he saw, and we have a man watching the airport. No one who fits the description of the Harpooner has left today."
"What about working backward?" Thomas said. "Why would the Harpooner be in Baku?"
"There are any number of targets a terrorist for hire could hit," Moore said. "Or he may just have been passing through on his way to another republic or to the Middle East. You know these people. They rarely take a direct route anywhere."
"If Baku was just a layover, the Harpooner is probably long gone," Thomas said. "Let's concentrate on possible targets in the region and reasons for hitting those targets."
"The Nagorno-Karabakh and Iran are our biggest concerns," Williamson said. "The people in NK have voted themselves an independent republic, while Azerbaijan and Armenia are both fighting to claim it. The whole region will probably explode when Azerbaijan gets enough money to buy more advanced weapons for its military. That would be bad enough for both nations, but with Iran just fifteen miles to the south, it could end up being quite an explosion. As for Iran, even without the NK situation, Teheran and Baku have been gnawing at each other for years over access to everything from offshore oil to Caspian sturgeon and caviar. When the Soviet Union watched over the Caspian, they took what they wanted. And not only are there problems, but the problems overlap," Williamson added. "Sloppy drilling by Azerbaijan has caused a quarter-inch-thick oil film in parts of the sea where Iran fishes for sturgeon. The pollution is killing the fish."
"What is the oil situation, exactly?" Thomas asked.
"There are four major oil fields," Williamson said. "Azeri, Chirag, Guneshli, and Azerbaijan. Azerbaijan and the Western Consortium members that underwrite the drilling are convinced that international law protects their exclusive rights to the sites. But their claim is based on boundaries that are defined by fishing rights, which both Iran and Russia insist do not apply. So far, the arguments have all been diplomatic."
"But if someone perpetrated a new action somewhere," Thomas said, "such as an embassy explosion or an assassination—"
"There could be a disastrous chain reaction reaching into a half-dozen surrounding nations, affecting oil supplies worldwide, and drawing the United States into a major foreign war," Williamson said.
Moore added sarcastically, "That's why we like to be kept informed about covert actions in our backward little outpost."
Thomas shook his head. _"Mea culpa_. Now, can we all agree to look ahead instead of back?"
Moore regarded him for a moment, then nodded.
"So," Williamson said, looking down at her notes. "As I understand this, there are two possible scenarios. First, that the individual who attacked Mr. Battat was not the Harpooner, in which case we may have nothing more than a drug smuggler or gunrunner on our hands. One who managed to get the drop on Mr. Battat and then slip away."
"Correct," said Thomas.
"What are the chances of that?" Williamson asked.
"They're unlikely," Thomas said. "We know that the Harpooner is in the region. An official from the Department of State Bureau of Intelligence and Research was on a Turkish Airlines flight from London to Moscow and made a tentative ID of the Harpooner. He tried to follow the target but lost him."
"You're saying an INR guy and the world's most wanted terrorist just happened to be on the same flight?" Moore said.
"I can't speak for the Harpooner, only for the DOS official," Thomas replied. "But we're finding that more and more terrorists and spies take the diplomatic routes. They try to pick up intel from laptops and phone calls. DOS has issued several alerts about that. Maybe it was a coincidence; maybe there was a diskette or phone number the Harpooner wanted to try and steal when the official went to the rest room. I don't know."
"The official was able to identify the Harpooner based on what?" Williamson asked.
"The only known photograph," Thomas told him.
"It was a good picture, reliable," Moore assured her.
"We were notified and did some checking," Thomas went on. "It fit with some intel we had picked up independently. The passenger was traveling under an assumed name with a fake British passport. We checked taxi records, found that he had been picked up at the Kensington Hilton in London. He'd only been there for one night, where he met with several people who, according to the concierge, looked and sounded Middle Eastern. We tried to track the individual in Moscow, but no one saw him leave the terminal. So we checked flights to other areas. Someone matching his description had shown a Russian passport in the name of Gardner and flown to Baku."
"It is the Harpooner's boat," Deputy Ambassador Williamson said suddenly. "It has to be."
The others looked at her.
"You've heard of it?" Thomas asked.
"Yes. I went to college," Williamson said. "Gardner is the captain of the _Rachel_ in _Moby-Dick._ It's one of the ships that was chasing the elusive white whale. She failed to capture him, I might add."
Thomas regarded Battat unhappily. "The Harpooner," Thomas said. "Dammit. Of course. He planted that for us to find."
"Now, there's a smart terrorist," Moore said. "If you recognize the allusion, you would have thought it a joke and wouldn't have bothered to pursue. If you thought it was real, then the Harpooner knew just where you'd be looking for him. And he would be there, waiting to stop you."
"But the boat was real," Battat said. "I saw the name—"
"A name that was put there to hold your attention for a while," Thomas said. "Shit. We fell for that one, big time."
"Which brings us to the second and suddenly very likely scenario," Williamson said. "If the Harpooner has been in Baku, there are two things we need to find out pretty damn quick. First, what he wanted and second, where he is now. Is that about right?"
Thomas nodded.
Moore rose. "I'm betting he's no longer using the Russian passport. I'll get into the hotel computers and check the names of the guests against our passport registry database. See if any new names pop up."
"He may also be working with people here, in which case he may not be staying at a hotel," Thomas said.
"I'll give you a list of known or suspected foreign cells," Moore told him. "You and Mr. Battat can crosscheck those with people the Harpooner might have worked with before."
Battat said he would do that.
"There's one other thing we should try," Thomas said. "We pretty much tapped out our Moscow-based sources on this before Mr. Battat came down. It wasn't very productive, but that was all we had time for. What about other governments in the region?"
"We haven't made any significant intelligence inroads with any of them," the deputy ambassador admitted. "We don't have the personnel to nurse the relationships, and a lot of the republics, including Azerbaijan, have had their resources strained with internal problems. Everyone is busy spying on each other, especially on Chechnya."
"Why there?" Battat asked.
"Because despite the coalition government that exists on paper, Chechnya is really controlled by Islamic militias intent on destabilizing and bringing down the other republics, including Russia," she said. "I'm hoping that the initiative the president announced last night in Washington will remedy that."
"What initiative?" Battat asked.
"An intelligence cooperative with the United Nations," Moore told him. "He announced it last night in Washington."
Battat rolled his eyes.
"You know, there is one place we might be able to try," Thomas said. "A couple of years ago I remember hearing that the National Crisis Management Center was involved with a Russian group based in Saint Petersburg."
"A Russian crisis management group," Moore said. "Yeah, I remember hearing about that."
"I can call Washington and have them contact Op-Center," Moore said. "See if they still have a relationship with the Russians."
"When you do, have them contact Bob Herbert over there," Thomas suggested. "He's the head of intelligence—a really capable guy from what I hear. I understand that the new guy running the place, General Rodgers, is something of a hard-ass."
"He's not running Op-Center," the deputy ambassador said.
"Who is?" asked Thomas.
"Paul Hood," said the deputy ambassador. "We got a directory update this morning. He withdrew his resignation."
Moore snickered. "I'll bet he won't be involved in the UN intelligence program."
"Regardless," Thomas said, "have them contact Herbert. The Harpooner may try to slip out of the region by heading north, into Scandanavia. If he does, the Russians may be able to help us up there."
Thomas agreed. Everyone rose then, and Thomas offered his hand to the deputy ambassador. "Thank you for everything," Thomas said. "I'm truly sorry about all this."
"So far, no real harm has been done."
"We're going to see that it stays that way," Thomas said.
"I'll have a room prepared for the two of you," Williamson said. "It's not fancy, but it's a place to crash."
"Thanks," Thomas said. "But until we find our man, I have a feeling I won't be getting a lot of sleep."
"None of us will, Mr. Thomas," Williamson assured him. "If you'll excuse me, Ambassador Small is due back from Washington at ten P.M. He'll want to be briefed on this as soon as possible."
Thomas left and walked down the corridor to Moore's office. The ADPI hated having lost the Harpooner. But he also hated the fact that the bastard was probably laughing at them for taking the whale bait. He also wondered if the Harpooner might somehow have known that Battat had come from Moscow. Maybe that was why he'd let the agent live, to create conflict between the CIA office in Moscow and Baku. Or maybe he did it just to confuse them, have them waste time wondering why he hadn't killed Battat.
Thomas shook his head. _Your mind is all over the damn place_ , he chided himself. _Stop it. You've got to_ focus. But that was going to be tough, Thomas knew, because the Harpooner was obviously a man who liked to keep his trackers off balance by mixing games with reality.
And so far, he was doing a helluva job.
**ELEVEN**
_**Washington, D. C. Monday, 3:00 P.M.**_
The cell phone rang in the office of the red-haired man. He shooed out two young assistants who closed the door behind them. Then he swiveled his chair so the high leather back was facing the door. He looked out the window, drew the cell phone from his inside jacket pocket, and answered on the fifth ring. If the phone had been stolen or lost and someone answered before that, the caller had been instructed to hang up.
"Yes?" the red-haired man said softly.
"He's completed phase one," said the caller. "Everything is exactly on schedule."
"Thank you," said the red-haired man and clicked off. He immediately punched in a new number. The phone was answered on the fifth ring.
"Hello?" said a gravelly voice.
"We're on track," said the red-haired man.
"Very good," said the other.
"Anything from Benn?" asked the red-haired man.
"Nothing yet," said the other. "It will come."
The men hung up.
The red-haired man put the phone back in his jacket pocket. He looked out across his desk and the office beyond. The photographs with the president and foreign heads of state. The commendations. A seven-by-ten-inch American flag that had been given to him by his mother. The red-haired man had carried it, folded, in his back pocket during his tour of duty in Vietnam. It was framed on the wall, still creased and soiled with sweat and mud, the lubricants of combat.
As the red-haired man called his two aides back to the office, the ordinary nature of that act, the return of routine, underscored the extreme and complex nature of what he and his partners were undertaking. To remake the international political and economic map was one thing. But to do it quickly, in a stroke such as this, was unprecedented.
The work was daunting, and it was exciting. If the operation ever were to become publicly known, it would be considered monstrous by some. But to many, so were the American Revolution and the Civil War in their day. So was the involvement of the United States in World War II, before Pearl Harbor. The red-haired man only hoped that if their actions were ever revealed, people would understand _why_ they had been necessary. That the world in which the United States existed was radically different from the world into which the United States had been born. That in order to grow it was sometimes necessary to destroy. Sometimes rules, sometimes lives.
Sometimes both.
**TWELVE**
_**Camp Springs, Maryland Monday, 3:14 P.M.**_
Paul Hood called Senator Fox after returning from the White House. She admitted being totally confused by the president's remarks and had put in a call to him to talk about it. Hood asked her to hold off until after he had had a chance to review the situation. She agreed. Then Hood called Bob Herbert. Hood briefed the intelligence chief on his conversation with the First Lady, after which he asked Herbert to find out what he could about the phone call from the hotel and whether anyone else had noticed any odd behavior from the president. Because Herbert stayed in touch with so many people—never asking them for anything, just seeing how they were doing, what the family was up to—it was easy for him to call and slip in important questions among the chitchat without making it seem as though he were fishing.
Now the two men were back in Hood's office. But the Herbert who wheeled through the door was different than before.
"Is everything all right?" Hood asked.
The usually outgoing Mississippi native didn't answer immediately. He was extremely subdued and staring ahead at something only he could see.
"Bob?" Hood pressed.
"They thought they had him," Herbert said.
"What are you talking about?"
"A friend of mine at the CIA slipped me some news from the embassy in Moscow," Herbert said.
"Why?"
Herbert took a long breath. "Apparently, they had a solid lead that the Harpooner was in Baku."
"Jesus," Hood said. "What for?"
"They don't know," Herbert said. "And they lost him. They sent one freakin' guy to do the recon and—sur—prise!—he got clocked. I can't blame them for wanting to be low profile, but with a guy like the Harpooner, you have to have backup."
"Where is he now?" Hood asked. "Is there anything we can do?"
"They don't have a clue where he went," Herbert said. He shook his head slowly and swung the computer monitor up from the armrest. "For almost twenty years what I've wanted most out of life is to be able to hold the bastard's throat between my hands, squeeze real hard, and look into his eyes as he dies. If I can't have that, I want to know that he's decaying in a hole somewhere with no hope of ever seeing the sun. That's not a lot to ask for, is it?"
"Considering what he did, no," Hood said.
"Unfortunately, Santa's not listening," Herbert said bitterly. He angled the monitor so he could see it. "But enough about that son of a bitch. Let's talk about the president."
Herbert shifted in his seat. Hood could see the anger in his eyes, in the hard set of his mouth, in the tense movements of his fingers. "I had Matt Stoll check the Hay-Adams phone log."
Matt Stoll was Op-Center's computer wizard.
"He hacked into the Bell Atlantic records," Herbert said. "The call came from the hotel, all right, but it didn't originate in any of the rooms. It originated in the system itself."
"Meaning?"
"Meaning someone didn't want to be _in_ one of the rooms where they might have been seen coming or going," Herbert said. "So they got to the wires somewhere else."
"What do you mean 'got' to them?" Hood asked.
"They hooked in a modem to transfer a call from somewhere else," Herbert said. "It's called _dial-up_ _hacking_. It's the same technology phone scammers use to generate fake dial tones on public phones in order to collect credit card and bank account numbers. All you need to do is get access to the wiring at some point in the system. Matt and I brought up a blueprint of the hotel. The easiest place to do that would have been at the phone box in the basement. That's where all the wiring is. But there's only one entrance, and it's monitored by a security camera—too risky. Our guess is that whoever hacked the line went to one of the two public phones outside the Off the Record bar."
Hood knew the bar well. The phones were right beside the door that opened onto H Street. They were in closetlike booths and there were no security cameras at that spot. Someone could have slipped in and gotten away without being seen.
"So, with the help of a dial-up hacker," Hood said, "Jack Fenwick could have called the president from anywhere."
"Right," Herbert told him. "Now, as far as we can tell, the First Lady is correct. Fenwick's in New York right now, supposedly attending top-level meetings with UN ambassadors. I got his cell phone number and called several times, but his voice mail picked up. I left messages for him to call me, saying it was urgent. I left the same message at his home and office. So far, I haven't heard from him. Meanwhile, Mike and I checked with the other intel departments. The president's announcement was news to each of them. Only one of them was involved in this cooperative effort with the United Nations."
"The National Security Agency," Hood said.
Herbert nodded. "Which means Mr. Fenwick must have sold the president some bill of goods to convince him they could handle this operation solo."
Herbert was correct, though in one way the National Security Agency would have been the perfect agency to interface with new intelligence partners. The primary functions of the NSA are in the areas of cryptology and both protecting and collecting signals intelligence. Unlike the CIA and the State Department, the NSA is not authorized to maintain undercover personnel on foreign soil. Thus, they do not generate the kind of knee-jerk paranoia that would make foreign governments nervous about cooperating with them. If the White House was looking for an intel group to pair with the United Nations, the NSA was it. What _was_ surprising, though, was that the president didn't brief the other agencies. And he should have at least notified Senator Fox. The Congressional Intelligence Oversight Committee is directly responsible for approving programs of counter-proliferation, counterterrorism, counternarcotics, counterintelligence, and covert activites abroad. What the president had proposed certainly fell under their jurisdiction.
But because the NSA _does_ operate independently, and in very specific areas, it's also the least-equipped to organize and oversee a massive undertaking of the kind described by the president. That was the reason Hood didn't believe Lawrence when he announced the initiative at the dinner. It was why a large part of him still didn't believe it.
"Did you talk to Don Roedner about this?" Hood asked. Roedner was the Deputy National Security Adviser, second in command to Fenwick.
"He's with Fenwick, and I couldn't get him on the phone either," Herbert told him. "But I did talk to Assistant Deputy National Security Adviser Al Gibbons. And this is where things get a little weirder. Gibbons said that he was present at an NSA meeting on Sunday afternoon where Fenwick didn't mention a goddamn thing about a cooperative intelligence effort with other nations."
"Was the president at that meeting?"
"No," Herbert said.
"But just a few hours later, Fenwick called the president and apparently told him that they had an intelligence deal with several foreign governments," Hood said.
Herbert nodded.
Hood considered that. It was possible that the UN initiative was on a need-to-know basis and that Gibbons wasn't part of that loop. Or maybe there was a bureaucratic struggle between different divisions of the NSA. That wouldn't have been unprecedented. When Hood first came to Op-Center, he studied the pair of 1997 reports that had effectively authorized the creation of Op-Center. Report 105-24 issued by the Senate Select Committee on Intelligence and 105-135 published by the House Permanent Select Committee on Intelligence the two arms of the Congressional Intelligence Oversight Committee—both proclaimed that the intelligence community was extremely top-heavy with _"intramural struggles, waste, and uninformed personnel lacking depth, breadth, and expertise in political, military, and economic analysis,"_ as the SIC report summed it up. Congressional reports didn't get much rougher than that. When Op-Center was chartered by act of Congress, Hood's mandate had been to hire the best and the brightest while the CIA and other intelligence groups worked on cleaning house. But the current situation was unusual, even by intelligence community standards, if the NSA's senior staff didn't know what was going on.
"This whole thing just doesn't make sense," Herbert said. "Between Op-Center and the CIA, we already have official cooperative intelligence plans with twenty-seven different nations. We have intelligence relationships with eleven other governments unofficially, through connections with high-ranking officials. Military intelligence has their hands in seven other nations. Whoever talked the president into this wants their own discreet, dedicated intelligence line for a reason."
"Either that, or they wanted to embarrass him," Hood said.
"What do you mean?"
"Sell him a project, tell him it's been cleared with other agencies and foreign governments, and then have him make a big public stumble."
"Why?"
"I don't know," Hood said.
He didn't, but he didn't like where this was leading him. Op-Center had once run a psy-ops game called Alternate Reality on how to make Saddam Hussein so paranoid that he would turn on his most trusted advisers. What if a foreign government were doing something like that to the president?
It was a far-fetched idea, but so was the KGB killing a dissident by poking him with a poisoned umbrella, and the CIA attempting to slip Fidel Castro a poison cigar. Yet these things had happened.
Then there was another option he didn't want to consider : that it wasn't a foreign government but our own. It was possible.
It could also be less sinister than that. The First Lady said her husband wasn't himself. What if she was right? Lawrence had spent four tough years in the White House and then eight tough years winning it back. Now he was in the hot seat again. That was a lot of pressure.
Hood was aware of several presidents who had showed signs of breaking during extended periods of stress: Woodrow Wilson, Franklin Roosevelt, Richard Nixon, and Bill Clinton. In the case of Nixon, his closest advisers encouraged him to resign not just for the health of the nation but for his own mental well-being. With Clinton, the president's staff and friends decided not to bring in doctors or psychiatrists but to keep a careful watch and _hope_ he came through the impeachment crisis. He did.
But in at least two cases, allowing the president to carry the full burden of decision making and politicking was not the best policy. Wilson ended up with a stroke trying to push the League of Nations through Congress. And toward the end of World War II, burdened by the pressure of winning the war and drawing up plans for a postwar world, Roosevelt's closest advisers feared for his health. Had they impressed on him the absolute need to slow down, he might not have died of a cerebral hemorrhage.
Any of those scenarios could be correct, or they could all be dead wrong. But Hood had always believed that it was better to consider every option, even the least likely, rather than be surprised. Especially when the result of being right could be cataclysmic. He would have to proceed carefully. If he could get to see the president, he would have an opportunity to lay his few cards on the table and also observe Lawrence, see whether Megan's concerns had merit. The worst that could happen was the president would ask for his resignation. Fortunately, he still had his last one on file.
"What are you thinking?" Herbert asked.
Hood reached for the telephone. "I've got to see the president."
"Excellent," Herbert said. "Straight ahead has always been my favorite way, too."
Hood punched in the president's direct line. The phone beeped at the desk of his executive secretary, Jamie Leigh, instead of going through the switchboard. Hood asked Mrs. Leigh if she could please squeeze him in for a few minutes somewhere. She asked him for a log line for the calendar to let the president know what this was about. Hood said that it had to do with Op-Center having a role in the United Nations intelligence program.
Mrs. Leigh liked Hood, and she arranged for him to see the president for five minutes, from four-ten to four-fifteen.
Hood thanked her then looked at Herbert. "I've got to get going," Hood said. "My appointment's in forty minutes."
"You don't look happy," Herbert said.
"I'm not," Hood said. "Can we get someone to nail down who Fenwick is meeting in New York?"
"Mike was able to connect with someone at the State Department when you two were up there," Herbert said.
"Who?"
"Lisa Baroni," Herbert told him. "She was a liaison with the parents during the crisis."
"I didn't meet her," Hood said. "How did Mike find her?"
"He did what any good spymaster does," Herbert said. "When he's someplace new, he looks for the unhappy employee and promises them something better if they deliver. Let's see if she can deliver."
"Good," Hood said as he rose. "God. I feel like I do whenever I go to Christmas Eve Mass."
"And how is that?" Herbert asked. "Guilty that you don't go to church more often?"
"No," Hood replied. "I feel like there's something going on that's much bigger than me. And I'm afraid that when I figure out what that is, it's going to scare the hell out of me."
"Isn't that what church is supposed to be about?" Herbert asked.
Hood thought about that for a moment. Then he grinned as he left the office. "Touché," he said.
"Good luck," Herbert replied as he wheeled out after him.
**THIRTEEN**
_**Gobustan, Azerbaijan Monday, 11:56 P.M**._
Gobustan is a small, rustic village located forty-three miles south of Baku. The region was settled as far back as 8000 B.C. and is riddled by caves and towering out-croppings of rock. The caves boast prehistoric art as well as more recent forms of expression—graffiti left two thousand years ago by Roman legionnaires.
Situated low in the foothills, just beneath the caves, are several shepherds' shacks. Spread out over hundreds of acres of grazeable land, they were built early in the century and most of them remain in use, though not always by men tending their flocks. One large shack is hidden behind a rock that commands a view of the entire village. The only way up is along a rutted dirt road cut through the foothills by millennia of foot traffic and erosion.
Inside, five men sat around a rickety wooden table in the center of the small room. Another man sat on a chair by a window overlooking the road. There was an Uzi in his lap. A seventh man was still in Baku, watching the hospital. They weren't sure when the patient would arrive, but when he did, Maurice Charles wanted his **man** to be ready.
The window was open, and a cool breeze was blowing in. Except for the occasional hooting of an owl or rocks dislodged by prowling foxes in search of field mice, there was silence outside the shack—the kind of silence that the Harpooner rarely heard in his travels around the world.
Except for Charles, the men were stripped to their shorts. They were studying photographs that had been received through a satellite uplink. The portable six-inch dish had been mounted on the top of the shack, which had an unobstructed view of the southeastern sky and the GorizonT3. Located 35,736 kilometers above twenty-one degrees twenty-five minutes north, sixty degrees twenty-seven minutes east, that was the satellite the United States National Reconnaissance Office used to keep watch on the Caspian Sea. Charles's American contact had given him the restricted web site and access code, and he had downloaded images from the past twenty-four hours.
The decoder they used, a StellarPhoto Judge 7, had also been provided by Charles's contact through one of the embassies. It was a compact unit roughly the size and configuration of a fax machine. The SPJ 7 printed photographs on thick sublimation paper, a slick, oil-based sheet that could not be faxed or electronically transmitted. Any attempt to do so would be like pressing on a liquid crystal display. All the receiver would see was a smudge. The unit provided magnification with a resolution of ten meters. Combined with infrared lenses on the satellite, he was able to read the numbers on the wing of the plane.
Charles smiled. His plane was on the image. Or rather, the Azerbaijani plane that they had bought.
"Are you certain the Americans will find that when they go looking for clues?" asked one of the men. He was a short, husky, swarthy man with a shaved head and dark, deep-set eyes. A hand-rolled cigarette hung from his downturned lips. There was a tattoo of a coiled snake on his left forearm.
"Our friend will make sure of it," Charles said.
And they would. That was the reason for staging this attack on the Iranian oil rig. Once the incident occurred, the United States National Reconnaissance Office would search the satellite database of images from the Guneshli oil region of the Caspian. Surveillance experts would look back over the past few days to see who might have been reconnoitering near the rig. They would find the images of Charles's plane. Then they would find something else.
Shortly after the attack, a body would be dropped into the sea—the body of a Russian terrorist, Sergei Cherkassov. Cherkassov had been captured by Azerbaijan in the NK, freed from prison by Charles's men, and was presently being held on the _Rachel_. Cherkassov would be killed shortly before the attack, shot with a shell from an Iranian-made Gewehr 3 rifle. That was the same kind of bullet that would have been fired by security personnel on the rig. When the Russian's body was found—thanks to intelligence that would be leaked to the CIA—the Americans would find photographs in the terrorist's pockets: the photographs Charles had taken from the airplane. One of those photographs would show portions of the airplane's wing and the same numbers seen in the satellite view. Another of the photographs would have markings in grease pencil showing the spot that particular terrorist was supposed to have attacked.
With the satellite photographs and the body of the terrorist, Charles had no doubt that the United States and the rest of the world would draw the conclusion that he and his sponsors wanted them to draw.
The wrong one.
That Russia and Azerbaijan had united to try to force Iran from its lucrative rigs in Guneshli.
**FOURTEEN**
_**New York, New York Monday, 4:01 P.M.**_
The State Department maintains two offices in the vicinity of the United Nations Building on New York's East Side. One is the Office of Foreign Missions and the other is the Bureau of Diplomatic Security.
Forty-three-year-old attorney Lisa Baroni was the assistant director of diplomatic claims for the Diplomatic Liaison Office. That meant whenever a diplomat had a problem with the United States' legal system, she became involved. A legal problem could mean anything from an allegedly unlawful search of a diplomat's luggage at one of the local airports, or a hit-and-run accident involving a diplomat, to the recent seizure of the Security Council by terrorists.
Ten days before, Baroni had been on hand to provide counsel for diplomats but found herself giving comfort to parents of children who were held hostage during the attack. That was when she'd met General Mike Rodgers. The general talked with her briefly when the siege was over. He said he was impressed by the way she had remained calm, communicative, and responsible in the midst of the crisis. He explained that he was the new head of Op-Center in Washington and was looking for good people to work with. He asked if he could call her and arrange an interview. Rodgers had seemed like a nononsense officer, one who was more interested in her talent than her gender, in her abilities more than in the length of her skirt. That appealed to her. So did the prospect of going back to Washington, D.C. Baroni had grown up there, she had studied international law at Georgetown University, and all her friends and family still lived there. After three years in New York, Baroni could not wait to get back.
But when General Rodgers finally called, it was not quite the call Baroni had been expecting.
It came early in the afternoon. Baroni listened as Rodgers explained that his superior, Paul Hood, had withdrawn his resignation. But Rodgers was still looking for good people and offered her a proposition. He had checked her State Department records and thought she would be a good candidate to replace Martha Mackall, the political officer who had been assassinated in Spain. He would bring her to Washington for an interview if she would help him with a problem in New York.
Baroni asked if the help he needed was legal. Rodgers assured her it was. In that case, Baroni told him, she would be happy to help. That was how relationships were forged in Washington. Through back-scratching.
What Rodgers needed, he explained, was the itinerary of NSA Chief Jack Fenwick who was in New York for meetings with United Nations delegates. Rodgers said he didn't want the published itinerary. He wanted to know where Fenwick actually ended up.
That should have been relatively easy for Baroni to find. Fenwick had an office in her building, and he usually used it when he came to New York. It was on the seventh floor, along with the office for the secretary of state. However, Fenwick's New York deputy said that he wasn't coming to the office during this trip but was holding all of his meetings at different consulates.
Instead, Baroni checked the file of government-issued license plates. This listing was maintained in the event of a diplomatic kidnapping. The NSA chief always rode in the same town car when he came to New York. Baroni got the license number and asked her friend, Detective Steve Mitchell at Midtown South, to try to find the car on the street. Then she got the number of the car's windshield-mounted electronic security pass. The ESP enabled vehicles to enter embassy and government parking garages with a minimum of delay, giving potential assassins less time to stage ambushes.
The ESP didn't show up on any of the United States checkpoints, which were transmitted immediately to State Department security files. That meant that Fenwick was visiting foreign embassies. Over one hundred nations also transmitted that data to the DOS within minutes. Most of those were close U.S. allies, such as Great Britain, Japan, and Israel. Fenwick had not yet gone to visit any of them. She used secure e-mail to forward to Rodgers the information where Fenwick hadn't been.
Then, just after four P.M., Baroni got a call from Detective Mitchell. One of his squad cars spotted the chief of staff's car leaving a building at 622 Third Avenue. That was just below Forty-second Street. Baroni looked up the address in her guide to permanent missions.
The occupant surprised her.
**FIFTEEN**
_**Washington, D.C. Monday, 4:03 P.M.**_
Paul Hood arrived at the west wing of the White House at four o'clock. Even before he had finished passing through the security checkpoint, a presidential intern had arrived to show him to the Oval Office. Hood could tell he had been here at least several months. Like most seasoned interns, the freshly scrubbed young man had a slightly cocky air. Here he was, a kid in his early twenties, working at the White House. The ID badge around his neck was his trump card with women at bars, with chatty neighbors on airplanes, with brothers and cousins when he went home for the holidays. Whatever anyone else said or did, he was interacting with the president, the vice president, cabinet, and congressional leaders on a daily basis. He was exposed to real power, he was plugged into the world, and he was moving past the eyes and ears of all media where the expressions and casual utterances of even people like him could cause events that would ripple through history. Hood remembered feeling a lot of that when he was a kid working in the Los Angeles office of the governor of California. He could only imagine how much more extreme it was for this kid, the sense of being at the center of the universe.
The Oval Office is located at the far southeast corner of the West Wing. Hood followed the young man in silence as they made their way through the busy corridors, passed by people who did not seem at all self-important. They had the look and carriage of people who were very late for a plane. Hood walked past the office of the national security adviser and the vice president, then turned east at the vice president's office and walked past the office of the press secretary. Then they turned south past the cabinet room. They walked in silence all the while. Hood wondered if the young man wasn't speaking to him because the kid had a sense of propriety or because Hood wasn't enough of a celebrity to merit talking to. Hood decided to give him the benefit of the doubt.
The office past the cabinet room belonged to Mrs. Leigh. She was seated behind her desk. Behind it was the only door that led to the Oval Office. The intern excused himself. Hood and the president's tall, white-haired secretary greeted each other with smiles. Mrs. Leigh was from Texas, with the steel, poise, patience, and dry, self-effacing humor required for the guardian of the gate. Her husband was the late Senator Titus Leigh, a legendary cattleman.
"The president's running a few minutes late," Mrs. Leigh said. "But that's all right. You can tell me how you are."
"Coping," Hood said. "And you?"
"Fine," she replied flatly. "My strength is the strength of ten because my heart is pure."
"I've heard that somewhere," Hood said as he continued toward the secretary's desk.
"It's Lord Tennyson," she replied. "How is your daughter?"
"She's strong, too," Hood said. "And she has an awful lot of people pulling for her."
"I don't doubt that," Mrs. Leigh said, still smiling. "Let me know if there's anything I can do."
"I absolutely will," Hood said. He looked into her gray eyes. "There is something you can do for me, though."
"And that is?"
"Off the record?"
"Of course," she assured him.
"Mrs. Leigh, has the president seemed all right to you?" Hood asked.
The woman's smile wavered. She looked down. "Is that what this meeting is about?"
"No," Hood said.
"What makes you ask a question like that?"
"People close to him are worried," Hood said.
"And you're the one who's been asked to bell the cat?" she asked.
"Nothing that calculated," Hood said as his cell phone beeped. He reached into his jacket pocket and answered the phone.
"This is Paul."
"Paul, it's Mike."
"Mike, what's up?" If Rodgers was calling him here, now, it had to be important.
"The target was seen leaving the Iranian mission to the UN about three minutes ago."
"Any idea where he was the rest of the time?" Hood asked.
"Negative," said Rodgers. "We're working on that. But apparently, the car didn't show up at the embassies of any of our top allies."
"Thanks," Hood said. "Let me know if you find out anything else."
Hood hung up. He put the phone back in his pocket. That was strange. The president had announced an intelligence initiative involving the United Nations, and one of the first missions the national security adviser visits belongs to Iran. As a sponsor of the kind of terrorism the United Nations opposed, that did not make sense.
The door to the Oval Office opened.
"Mrs. Leigh, would you do me a favor?" Hood said.
"Yes."
"Would you get me Jack Fenwick's itinerary in New York?"
"Fenwick? Why?"
"He's one of the reasons I asked you the question I did," Hood replied.
Mrs. Leigh looked at Hood. "All right. Do you want it while you're with the president?"
"As soon as possible," Hood said. "And when you get the file number, let me know what else is in the file. I don't need specific documents, just dates when they were filed."
"All right," she said. "And Paul—what you asked before ? I have noticed a change."
He smiled at her. "Thanks. If there's a problem, we're going to try and fix it quickly and quietly, whatever it is."
She nodded and sat at her computer as the vice president emerged from the Oval Office. Charles Cotten was a tall, stout man with a thin face and thinning gray hair. He greeted Paul Hood with a warm handshake and a smile but didn't stop to talk. Mrs. Leigh punched the phone intercom. The president answered. She told him that Paul Hood was here, and the president asked her to send him in. Hood went around the desk and walked into the Oval Office.
**SIXTEEN**
**Baku,** _**Azerbaijan Tuesday, 12:07 A.M.**_
David Battat lay on the flimsy cot and stared at the dark ceiling of the damp basement storehouse. Pat Thomas slept on his back in a cot on the other side of the small room, breathing softly, regularly. But Battat couldn't sleep.
His neck still ached, and he was angry at himself for having gotten cold-cocked, but that wasn't what was keeping him awake. Before going to sleep, Battat had reviewed the original data the CIA had received about the Harpooner. He could not put it out of his mind. All signs, including a reliable eyewitness, pointed to it having been the terrorist that was being met by the _Rachel_. And if that were so, if the Harpooner had passed through Baku on his way to somewhere else, Battat was deeply troubled by one question: _Why am I still alive?_
Why would a terrorist with a reputation for scorched-earth attacks and homicidal behavior leave an enemy alive? To mislead them? To make them think it wasn't the Harpooner who was there? That had been his initial reaction. But maybe the terrorist had left him alive for another reason. And Battat lay there, trying to figure out what that reason could be.
The only reason he could think of would be to carry misinformation back to his superiors. But he had not carried any information back, other than what was already known: that the _Rachel_ was where it was supposed to be. And without knowing who got on or where it went, that information did them no good.
Battat's clothes had been gone over carefully for an electronic bug or a radioactive tracer of some kind. Nothing had been found, and the clothes were subsequently destroyed. If one had been located, it would have been used to spread disinformation or to misdirect the enemy. Moore had gone through Battat's hair, checked under his fingernails, looked in his mouth and elsewhere for a microtransmitter that could be used to locate Battat or eavesdrop on any conversations he might have. Nothing had been found.
There wasn't a damn thing, he thought. And it gnawed at him because he didn't think this was a screw-up. He was alive for a reason.
He shut his eyes and turned on his side. Thinking about this while he was dead tired would get him nowhere. He had to sleep. He forced himself to think about something pleasant: what he would do when he found the Harpooner.
The thought relaxed him. As he lay there, Battat began to feel warm. He attributed that to the poor ventilation in the room and the distress he was feeling over everything that had happened.
A few minutes later, he was asleep.
A few minutes after that, he began to perspire.
A few minutes after that, he was awake and gasping for breath.
**SEVENTEEN**
_**Washington, D. C. Monday,**_ _**4:13**_ _**P.M.**_
The president was writing on a white legal pad when Hood entered. The president told Hood to have a seat; he needed to make a few notes before they talked. Hood quietly shut the door behind him and walked toward a brown leather armchair in front of the desk. He turned off his cell phone and sat down.
The president was dressed in a black suit and silver and black striped tie. A rich yellow light gleamed off the panes of bulletproof glass behind the president. Beyond it, the Rose Garden looked rich and alive. Everything seemed so right here, so healthy and normal, that for a moment Hood doubted himself.
But only for a moment. Hood's instincts got him where he was; there was no reason to start doubting them now. Besides, the battle was always somewhere else, never in the command tent.
The president finished writing, put down his pen, and looked at Hood. His face was drawn and wan, but his eyes had their usual gleam.
"Talk to me, Paul," the president said.
Hood grew warm behind the ears. This wasn't going to be easy. Even if he were correct, it wasn't going to be easy convincing the president that members of his staff might be running an operation of their own. Hood did not have a lot to go on, and part of him wished that he had gone to the First Lady before coming here. It would have been better to let her talk to him in private. But if the intelligence Herbert had received was right, there might not be time for that. Ironically, Hood would have to keep Megan Lawrence out of this. He did not want the president to know that his wife had been talking about him behind his back.
Hood leaned forward. "Mr. President, I have some concerns about the United Nations intelligence operation."
"Jack Fenwick is setting it all up," the president said. "There'll be a comprehensive briefing when he returns from New York."
"Will the NSA be running the project?"
"Yes," the president informed him. "Jack will be reporting directly to me. Paul, I hope this visit isn't about some kind of territorial pissing contest between Op-Center and the NSA—"
"No, sir," Hood assured him.
The intercom beeped. The president answered. It was Mrs. Leigh. She said she had something for Paul Hood. The president frowned and asked her to bring it in. He looked at Hood.
"Paul, what's going on?"
"Hopefully, nothing," Hood said.
Mrs. Leigh walked in and handed Hood a single sheet of paper.
"Is this all?" Hood asked.
She nodded.
"What about the file itself?"
"Empty," she said.
Hood thanked Mrs. Leigh, and she left.
"What file is empty?" the president asked irritably. "Paul, what the hell is going on?"
"I'll tell you in a moment, Mr. President," Hood said. He looked down at the paper. "From eleven A.M. this morning until four P.M., Jack Fenwick was scheduled to meet with representatives of the government of Iran at their permanent mission in New York."
"Impossible," said the president.
"Sir, Mrs. Leigh obtained this from the NSA office," Hood said. He handed the president the paper. "It has their file number on top. And according to intel we received, Fenwick did spend a good part of the afternoon at the Iranian mission."
The president looked at the paper and was still for a long moment. Then he shook his head slowly. "Fenwick was supposed to be meeting with the Syrians, the Vietnamese, a half-dozen others," he said. "That's what he told me last night. Hell, we aren't even close to reaching an intelligence agreement with Iran."
"I know," Hood said. "But Fenwick was there. And except for this document, the file is empty. As far as the NSA is concerned, there is no such thing as the UN initiative."
"This has to be bullshit," the president said dismissively. "More bullshit." The president jabbed the intercom button on his phone. "Mrs. Leigh, get me Jack Fenwick—"
"Sir, I don't think you should talk to anyone at the NSA," Hood said.
"Excuse me?"
"Not yet, at least," Hood said.
"Hold on, Mrs. Leigh," the president said. "Paul, you just told me my national security adviser is way off the playbook. Now you're telling me not to bother finding out if that's true?"
"Before you do that, we need to talk," Hood said.
"About what?"
"I don't believe this situation with Fenwick is a miscommunication," Hood said.
"Neither do I," the president said. "My conversations with him were very explicit. That's why he and I need to talk."
"But what if something is very wrong?" Hood asked.
"Explain."
"What if this is a rogue operation of some kind?" Hood asked.
"You're out of your mind," the president said. He appeared stunned. "Christ, Paul, I've known most of these people for fifteen, twenty years—they're good friends!"
Hood understood. And all he could think to say was, "'Et tu, Brute?' "
The president looked at him. "Paul, what are you talking about?"
"When Julius Caesar was killed by republicans in the senate, it was his closest and oldest friend who organized the assassination," Hood said.
The president looked at him. A moment later, he told Mrs. Leigh to forget the call. Then he shook his head slowly. "I'm listening," the president said. "But this better be good."
Hood knew that. What he didn't know was where to begin. There was a possible conspiracy and possible mental illness. Perhaps both. He decided to start at the beginning and work his way through.
"Mr. President, why did Fenwick call you last night?" he asked.
"He had finished a day of meetings with ambassadors at the Hay-Adams," the president said. "There was strong opposition to the intelligence initiative from several key governments. He was supposed to let me know if and when he finally pulled it all together."
"Mr. President," Hood said, "we don't believe that Jack Fenwick was at the Hay-Adams Hotel last night. The call he made to you was apparently routed to the hotel from somewhere else."
"From where?" the president asked.
"I don't know," Hood admitted. "Perhaps he was already in New York. Was Fenwick also liaising with the CIOC?"
"No," the president said. "Getting approvals from the Oversight Committee was the responsibility of Fenwick's deputy, Don Roedner, and Red Gable on this end."
Hood didn't know Roedner any better than he knew Gable. He didn't even know Gable _had_ a nickname.
"Sir," Hood continued, "last night, when you thanked Senator Fox for budgeting Mr. Fenwick's initiative, that was the first she'd heard about it."
President Lawrence froze, but only for a moment. His expression changed slowly. He looked very strange for a moment, both twenty years older and like a lost boy. He sat back.
"Gable wouldn't go behind my back on something," the president said faintly. "He wouldn't. And if he did, I'd read it in his face."
"When was the last time you saw him?" Hood asked.
The president thought. "Friday, at the cabinet meeting."
"There were a lot of people there, a lot of issues on the table," Hood said. "You might have missed it. Or maybe he was snookered by the NSA."
"I can't believe that, either," the president said.
"I see," Hood said. "Well, if Fenwick and Gable aren't rogue, there's only one other option I can think of."
"Which is?"
Hood had to be careful how he said this. He was no longer floating ideas about the president's staff but about the president himself.
"Maybe none of this happened," Hood said. "The UN initiative, the meetings with foreign governments—none of it."
"You mean I imagined it all," the president said. Hood didn't answer.
"Do you believe that?" the president asked.
"I do not," Hood replied truthfully. If nothing else, there was the rerouted phone call from the Hay-Adams, and the president didn't imagine that. "But I won't lie to you, Mr. President," Hood went on. "You do seem tense, guarded, distracted. Definitely not yourself."
The president took a long breath. He started to say something and then stopped. "All right, Paul. You've got my attention. What do we do next?"
"I suggest we proceed under the assumption that we've got a serious problem," Hood said. "I'll continue the investigation from our end. We'll see what we can find out about the Iranian connection. Check on what else Fenwick has been doing, who he's been talking to."
"Sounds good," Lawrence said. "Fenwick is due back late tonight. I won't say anything to him or to Red until I hear from you. Let me know as soon as you learn anything else."
"I will, sir."
"Will you also bring Senator Fox up to speed?"
Hood said he would and then stood. So did the president. He seemed a little stronger now, more in command. But the things Megan had told Hood still troubled him.
"Mr. President," Hood said, "I do have one more question."
The president looked at Hood intently and nodded once.
"A few minutes ago, you said that this was 'more bullshit,' " Hood said. "What did you mean?"
The president continued to regard Hood. "Before I answer that, let me ask you a question."
"All right."
"Don't you already know the answer to that?" the president asked.
Hood said that he did not.
"You came to see me only because of what happened last night?" the president asked.
Hood hesitated. The president knew that he and the First Lady were old friends. It was not Hood's place to tell the president that his wife was worried about him. But Hood also did not want to be just one more person who was lying to the president.
"No," Hood answered truthfully. "That is not the only reason."
The president smiled faintly. "Fair enough, Paul. I won't press you."
"Thank you, sir."
"But I will tell you one thing about the bullshit," the president said. "This is not the only mix-up we've had here over the past few weeks. It's been frustrating." The president extended his hand across his desk. "Thanks for coming, Paul. And thanks for pushing me."
Hood smiled and shook the president's hand. Then he turned and left the Oval Office.
There was a group of eager-looking Boy Scouts waiting outside with a photographer. The young men were award-winners of some kind, judging by their sashes. Hood winked at them, taking a moment to savor their openmouthed awe and innocence. Then he thanked Mrs. Leigh as he passed her desk. She flashed a concerned look at Hood, and he indicated that he would call her. She mouthed a thank-you and then showed the Boy Scouts inside.
Hood walked briskly to his car. He started the engine, then took out his cell phone and checked his messages. There was only one. It was from Bob Herbert. As Hood headed toward Fifteenth Street, he called Herbert back.
"Bob, it's Paul," said Hood. "What's up?"
"Plenty," Herbert said. "First of all, Matt traced the call that came from the Hay-Adams."
"And?"
"The call originated on Fenwick's cell phone."
"Bingo!" Hood said.
"Maybe, maybe not," Herbert replied.
"Explain," Hood said.
"I got a call a few minutes ago, one I didn't expect to get," Herbert said.
"From?"
"Fenwick," Herbert replied. "He was open and sounded surprised by what I had to say. He told me he didn't speak to the president last night. He said his briefcase was stolen, which is why he didn't get the calls I left on his cell phone. He only got the one I left at his office."
"I'm not ready to buy that," Hood replied. "The president did receive a call, and it was routed through the hotel."
"True," Herbert said. "But do you remember Marta Streeb?"
"The woman who had the affair with Senator Lancaster?" Hood asked.
"Right."
"What about her?"
"Her calls were run through a phone bank at Union Station so they couldn't be traced," Herbert said.
"I remember," Hood said. "But the president isn't having an affair."
"Are you sure?" Herbert asked. "His wife said he was acting strange. That could be guilt—"
"It could be, but let's rule out the national security issues first," Hood snapped.
"Sure," Herbert replied.
Hood took a moment to calm down. His anger surprised him. Hood had never had an affair, but for some reason, Herbert's comment made him feel guilty about Sharon.
"What else did Fenwick have to say?" Hood asked.
"That he doesn't know a damn thing about any UN initiative," Herbert said. "He didn't get any calls about it and didn't read about it in the paper. He told me he was sent to New York to help the Iranians with the situation involving the Harpooner and possible Azerbaijani terrorists in the Caspian. And there could be some truth to that," Herbert pointed out. "If the CIA was compromised over there, the Iranians might need to turn to someone else for help. Someone that could get them signal intelligence capacity ASAP."
"Were the Iranians working with the CIA on this?"
"I'm trying to find that out," Herbert said. "You know those Company guys. They don't like to share. But think about it. Op-Center's worked with other governments, some of them hostile. We'd get in bed with Teheran if all we were going to do was snuggle a little."
That was true, Hood had to admit.
"And Fenwick was at the mission," Herbert continued. "That much is pretty clear."
"It's about the only thing that is," Hood replied. "Bob, you said that Fenwick was sent to New York. Did he say who sent him?"
"Yes," Herbert replied, "and I don't think you're going to like this. Fenwick says the president was the one who sent him."
"Triple-O?" Hood asked. Triple-0 was _oral orders only_. They were given when an official didn't want to leave a paper trail to or from a potentially explosive situation.
"Triple-O," Herbert told him.
"Jesus," Hood said. "Look—someone else would have to have been in this Iranian loop."
"Sure," Herbert agreed. "The veep, probably. The chief of staff—"
"Call Vice President Cotten's office," Hood said. "Find out what he has to say. I'll be there as soon as possible."
"I'll call out for pizza," Herbert told him.
Hood hung up and concentrated on getting himself through the maddening rush-hour traffic.
At the moment, it was a welcome diversion.
**EIGHTEEN**
_**Gobustan, Azerbaijan Tuesday, 1:22 A.M.**_
The other men had gone to sleep on threadbare bedrolls they had bought secondhand in Baku. But Maurice Charles was still awake, still sitting at the wooden table in the shepherd's shack. Though he never had trouble sleeping before a mission, he did have trouble waiting for other people to do things. Things on which the mission depended. Until then, he would not—could not—rest.
When the phone finally beeped, he felt a nearly electric shock. This was it. The last unfinished business before H-hour.
Charles went to the equipment table. Beside the StellarPhoto Judge 7 was a Zed-4 unit, which had been developed by the KGB in 1992. The secure phone system was the size and general shape of an ordinary hardcover book. The small, flat receiver fit neatly into the side. It was a remarkable improvement over the point-to-point radios Charles had used when he was first starting out. Those had a range of two and one-half miles. The Zed-4 utilized a series of satellite links to pick up cellular transmissions from around the world. A series of internal audio enhancers and boosters virtually eliminated breakup and lost signals.
The Zed-4 was also quite secure. Most secure-phone calls, including the United States Tac-Sat units, were encrypted with a 155-digit number. In order to crack the code, eavesdroppers had to factor that into its two-component prime numbers. Even using sophisticated computers like the Cray 916, that could take weeks. The CIA had managed to cut that time into days by stealing computer time from personal computers. In 1997, the agency began using Internet servers to piggyback the numbers into home computer systems. Small amounts of memory were appropriated to work on the problem without the user being aware of it. Networked throughout a system of millions of PCs, the CIA was able to add gigabytes of computation power to the problem. It also created a problem for counterprogrammers, since it was not possible to shut down the CIA's so-called Stealth Field System. Thus, the Zed-4 was created using a complex encryption code of 309 digits. Even the SFS lacked sufficient power to break that code in a timely fashion.
Charles answered on the third ring. "B-sharp," he said. That was the receiver code name.
"C-natural," said the caller.
"Go ahead," said Charles.
"I'm across the street from the target," said the caller. "They're bringing him out the side door."
"No ambulance?"
"No," said the caller.
"Who's with him?" Charles asked.
"Two men," said the caller. "Neither of them in uniform."
Charles smiled. Americans were so predictable. If there were more than one operative, they invariably went to the user's manual. "How to Be a Soldier or Spy," Rule Fifty-three: Put the man above the mission. That thinking went at least as far back as the United States cavalry out West. Whenever the more aggressive Native American tribes like the Apaches were being pursued, they would stop to attack homesteaders. The warriors would always rape one of the women, leaving her where the cavalry was certain to find her. Invariably, the soldiers would send the woman back to the fort with an escort. That would not only delay the pursuing column but leave them depleted.
"Is backup in place?"
"Yes, sir."
"Then take them," Charles said.
"It's done," the caller said confidently. "Out."
The phone went dead. Charles hung up.
That was it. The last piece. He'd allowed the one agent to live to draw the others out. An injection in the neck, a fast-acting bacterial pneumonia, and the entire local cast was out of commission. Now there would be no one to put pieces together, to stop him from completing the mission.
Charles had one more call to place before he went to bed. It was to a secure line in Washington, to one of the few men who knew of Charles's involvement in this operation.
To a man who didn't follow the rule book.
To a man who helped devise one of the most audacious schemes of modern times.
**NINETEEN**
**Baku,** _**Azerbaijan Tuesday, 1:35 A.M.**_
The ride to the VIP Hospital took just under ten minutes. The VIP was the only hospital the American embassy deemed to be up to the standards of western health care. They had an arrangement with Dr. Kanibov, one of the city's few English-speaking physicians. The fifty-seven-year-old Kanibov was paid off the books to be available for around-the-clock emergencies and to recommend qualified specialists when necessary.
Tom Moore didn't know if a specialist was going to be necessary. All he knew was that Pat Thomas had woken him twenty minutes earlier. Thomas had heard David Battat moaning on his cot. When Thomas went over to check on Battat, he found him soaked with perspiration and trembling. The embassy nurse had a look at him and took Battat's temperature. He had a fever of 105. The nurse suggested that Battat may have hit his head or suffered capillary damage when he was attacked. Rather than wait for an ambulance, Thomas and Moore loaded Battat into one of the embassy staff cars in the gated parking lot and brought him to the hospital themselves. The medic called ahead to let Dr. Kanibov know that they had a possible case of neurogenic shock.
_This is all we need, to be down a man,_ Thomas thought as he drove through the dark, deserted streets of the embassy and business district. It was bad enough to have too few people to deal with normal intelligence work. But to find the Harpooner, one of the world's most elusive terrorists, was going to take more. Thomas only hoped that his call to Washington would get them timely cooperation on a Saint Petersburg connection.
Dr. Kanibov lived just a block from the hospital. The tall, elderly, white-goateed physician was waiting when they arrived. Battat's teeth were chattering, and he was coughing. By the time a pair of orderlies put him on a gurney just inside the door, the American's lips and fingernail beds were rich blue.
"Very restricted blood flow," said Kanibov to one of the orderlies. "Oxygen." He looked in Battat's mouth. "Traces of mucus. Suction, then give me an oral temperature."
"What do you think is wrong?" Thomas asked.
"I don't know yet," Kanibov said.
"The nurse at the embassy said it could be neurogenic shock," Thomas said to the doctor.
"If it were, his face would be pale, not flushed," the doctor said with annoyance. He looked at Thomas and Moore. "You gentlemen can wait here or you can go back and wait—"
"We'll stay here," Thomas informed him. "At least until you know what's wrong."
"Very well," the doctor said as they wheeled Battat into the ward.
It seemed strangely quiet for an emergency room, Thomas thought. Whenever his three boys hurt themselves back in Washington or in Moscow, the ERs were like the West Wing of the White House: loud, purposeful chaos. He imagined that the clinics in the poorer sections of Baku must be more like that. Still, the silence was unnerving, deathlike.
Thomas looked at Moore. "There's no sense for both of us to be here," Thomas said. "One of us should get a little sleep."
"I wasn't sleeping," Moore said. "I was making those contacts we discussed and reviewing files."
"Did you find anything?" Thomas asked.
"Nothing," Moore said.
"All the more reason for you to go back to the embassy," Thomas said. "David is my responsibility. I'll wait here."
Moore considered that. "All right," he said. "You'll call as soon as you know something?"
"Of course," Thomas said.
Moore gave him a reassuring pat on the shoulder, then walked back through the lobby. He pushed the door open and walked around the front of the car to the driver's side.
A moment later, Tom Moore's head jerked to the right and he dropped to the asphalt.
**TWENTY**
_**Washington, D.C. Monday, 6:46 P.M.**_
Paul Hood arrived at Op-Center, where he was to meet with Bob Herbert and Mike Rodgers. He also telephoned Liz Gordon. He asked her to wait around so he could talk to her later. He wanted to get her input on what, if anything, might be happening with the president from a clinical standpoint.
Hood bumped into Ann Farris on the way to his office. She walked with him through the tight, winding maze of cubicles to the executive wing. As Herbert had joked when he first went to work at Op-Center, that was where the cubicles had ceilings.
"Anything interesting going on?" Ann asked.
"The usual confusion," Hood said. "Only this time, it's happening in Washington, not overseas."
"Is it something really bad?"
"I don't know yet," Hood said. "There seems to be a loose cannon somewhere in the NSA." Hood didn't want to say anything about the president possibly having mental lapses of some kind. It wasn't that he didn't trust Ann, but Megan Lawrence had told him something in confidence. For now, he wanted to keep the number of people with whom he shared that as small as possible. "What's going on in your department?"
"The usual efficiency and expert coordination," she said with a disarming smile.
"You mean nothing's going on."
"Exactly," Ann said. She waited a moment, then asked, "Do you expect to be here long?"
"A couple of hours," he said. "There's no reason to go back to the hotel. I'd just sit there and watch some bad sitcom."
"Can I interest you in dinner?" she asked.
"It may be a long night," Hood said.
"I don't have any plans, either," she said. "My son is staying with his dad this week. There's nothing for me to go home to but a spoiled cat and those same sitcoms."
Hood's heart began thumping a little faster than usual. He very much wanted to say yes to Ann. But he was still a married man, and going out with a divorced female coworker could cause trouble, legally as well as ethically. And Op-Center did not need this distraction. The intelligence team was brilliant at uncovering information. Hood having dinner with Farris would be common knowledge by morning. Besides, if dinner with Ann was in the back of his mind, he would not be focusing on a crisis in the executive branch.
"Ann, I wish I could," he said sincerely. "But I don't know when I'll be finished here. Some other time?"
"Sure," she said with a small, sad smile. She touched the back of his hand. "Have a good meeting."
"Thanks," Hood said.
Ann left, and Hood continued on his way.
Hood felt terrible now. He had not done what he really wanted to do, which was have dinner with Ann. And he had hurt her feelings.
He stopped. He wanted to go after her and tell her he would have the dinner. But once he started down that road, there was no turning back. Hood continued toward his office.
Hood buzzed Rodgers and Herbert when he arrived. Rodgers said he would be right over. Herbert was on the computer and said he would be with them in a few minutes.
Rodgers was alert and professional when he arrived. The general had always wanted to run Op-Center. If he harbored any resentment about having it handed to him and then abruptly pulled away, it did not show. Above all, Rodgers was a good man and a team player.
General Rodgers had spent most of the day overseeing the activities of Op-Center while Paul Hood was involved with the president and the UN initiative. As Hood briefed his deputy director about Herbert's talk with Fenwick, Herbert wheeled in. The intelligence chief was flushed and perspiring slightly. He had hurried to get here.
"How's your relationship with Sergei Orlov at the Russian Op-Center?" Herbert asked breathlessly.
The question surprised Hood. "I haven't spoken to him in about six months. Why?"
"I just received a message that was forwarded from the U.S. embassy in Baku," Herbert said. "One of the CIA's people over there, Tom Moore, is now convinced that Baku has had a visit from the Harpooner. Moore doesn't know why the bastard's there—"
"It could have something to do with what you were just telling me about," Rodgers said to Hood. "Bob's conversation with Fenwick—"
"About Iran fearing terrorist attacks from Azerbaijan," Hood said.
Rodgers nodded.
"I agree that that's a possibility," Herbert said. "Paul, if it is the Harpooner, Moore wants to catch him going into or keep him from getting out of the former USSR. He's hoping that the Russian Op-Center can help."
"How?" Hood asked. "Orlov and I shared our files years ago. There was nothing on the Harpooner."
"Orlov's facility was new then," Herbert said. "He or his people may have found something in the old KGB files since then. Something they might not have told us about."
"It's possible," Hood agreed. Op-Center was understaffed, and the situation at their Russian counterpart was even worse. Keeping up a regular flow of information was difficult.
"In addition to intel on the Harpooner," Herbert said, "More was hoping that Orlov's people might be able to watch the northern and northwestern sections of Russia. He was thinking that the Harpooner might try to leave the region through Scandinavia."
Hood looked at his watch. "It's about three in the morning over there," he said.
"Can you reach him at home?" Herbert asked. "This is important. You know it is."
Herbert was right. Regardless of the intelligence chief's desire to see the terrorist captured, tried, and executed, the Harpooner was a man who deserved to be out of circulation.
"I'll call," Hood said.
"Before you do, what about President Lawrence?" Rodgers asked. "How did things go over there?"
"I'll fill you in after I talk to Orlov," Hood said as he accessed his secure phone list on the computer. He found Orlov's number. "But from the look of it, we're facing a lose-lose situation. Either the president is suffering from some kind of mental fatigue, or we've got a group of top officials running a black ops action of some kind—"
"Or both," Herbert said.
"Or both," Hood agreed. "I've got Liz Gordon coming in later to talk about what the president might be experiencing."
Before punching in Orlov's home telephone number, Hood called Op-Center's linguistics office. He got Orly Turner on the line. Orly was one of Op-Center's four staff translators. Her area of expertise was Eastern Europe and Russia. Hood conferenced her in to the call. Though Orlov spoke English well enough, Hood wanted to make sure there were no misunderstandings, no delays if technical terms or acronyms had to be explained.
"You want to know what my gut tells me?" Herbert said.
"What?" Hood asked as he punched in Orlov's number.
"That all of this is related," Herbert said. "The president being out of the loop, Fenwick dealing secretly with Iran, the Harpooner showing up in Baku. It's all part of a big picture that we haven't figured out yet."
Herbert left the office. Hood didn't disagree with him. In fact, his own gut was willing to go one step further.
That the big picture was bigger than what they imagined.
**TWENTY-ONE**
_**Baku, Azerbaijan Tuesday, 3:58 A.M.**_
When Tom Moore went down, Pat Thomas ran toward the hospital door. He was halfway out when he saw blood pulsing from the side of Moore's head. Thomas stopped and jumped back just as a shot blew out the glass in the door. The bullet punched into his left thigh and knocked him down. He landed in a sitting position and continued to scuttle back. A second bullet chewed up the green tile inches in front of his foot. Thomas hurried backward along the floor, propelled by his palms and right heel. The wound burned viciously, and each move was agony. He left a long smear of blood behind him.
It was a few moments before the hospital staff realized what had happened. One of the nurses, a young woman, ran forward and helped pull Thomas back. Several orderlies followed. They dragged him behind the admissions desk. Another nurse called the police.
A bald-headed doctor knelt beside Thomas. He was wearing off-white surgical gloves and shouted instructions in Azerbaijani to other hospital workers who were in front of the counter. As he did, he took a pocket knife from his white coat and carefully cut away the fabric around the wound.
Thomas winced as the khaki fabric came away. He watched as the doctor exposed the wound.
"Will I live?" Thomas asked.
The doctor didn't answer. Suddenly, the bald man started to rise. But instead of getting up, he straddled the American's legs. He sat on the wound, sending fire up through his patient's waist. Thomas wanted to scream, but he could not. A moment later, the doctor slipped a hand behind the America's head, holding it in place, and pushed the knife blade through his throat. The metal entered the skin just behind Thomas's chin and pinned his mouth shut. The blade continued upward until Thomas could feel the point of the blade under his tongue.
Thomas choked as he coughed blood into his closed mouth. He raised his hands and tried to push the bald man back. But he was too weak. Calmly and quickly, the bald man angled the knife back. Then he drew the knife down until it reached Thomas's larynx. He cut swiftly to the left and right, following the line of the jaw all the way to the ears. Then he removed the blade, rose, and allowed Thomas to flop to the floor. The doctor pocketed the knife and walked away without a glance back.
The American lay there, his arms weak and his fingers moving aimlessly. He could feel the warm blood flowing from both sides of his throat as the flesh around it grew cold. He tried to call out, but his voice was a burbling whisper. Then he realized that his chest was moving but no air was going to it. There was blood in his throat.
Thomas's thoughts were confused. His vision swirled black. He thought about flying up to Baku, about meeting with Moore. He wondered how Moore was. And then he thought about his children. For a moment, he was back playing ball with them on the front lawn.
Then they were gone.
**TWENTY-TWO**
_**Saint Petersburg, Russia Tuesday, 4:01 A.M.**_
General Sergei Orlov was standing in the snow in the small town of Nar'yan Mar on the Arctic Ocean when a peeping bird caused him to start. He turned to look for it and found himself staring at his alarm clock.
He was back in his one-bedroom apartment in Saint Petersburg.
"Damn you," Orlov said as the phone rang again. The former cosmonaut did not often dream of the town where he grew up. He hated being taken away from it and from his loving parents.
"Sergei?" his wife Masha said groggily beside him.
"I have it," Orlov told her. He picked up the receiver of the cordless phone. He held it to his chest to stifle the ringing. "Go back to sleep."
"All right," she said.
Orlov listened enviously to the cozy rustle of the sheets as his wife curled up on her side. He got out of bed, pulled a bathrobe from the edge of the door, and pulled it on as he stepped into the living room. Even if this were a wrong number, Orlov would have trouble getting back to sleep.
He finally answered the telephone. "Hello," Orlov said with a trace of annoyance.
"General Orlov?" said the voice on the other end. It was a man.
"Yes?" Orlov said as he rubbed his eyes vigorously with his free hand. "Who is this?"
"General, it's Paul Hood," said the caller.
Orlov was suddenly very much awake. "Paul!" he practically shouted. "Paul Hood, my friend. How are you? I heard that you resigned. And I heard about what happened in New York. Are you all right?"
Orlov walked over to an armchair while the woman translated. The general had a decent command of English, the result of the years he spent as a goodwill ambassador for the Russian space program after his flying days were finished. But he let the woman translate to be sure he didn't miss anything.
Orlov sat down. Standing just under five-foot-seven, he had the narrow shoulders and compact build that had made him an ideal cosmonaut. Yet he had presence. His striking brown eyes, high cheekbones, and dark complexion were, like his adventurous spirit, a part of his Manchu heritage. He walked with a significant limp due to a left leg and hip badly broken when his parachute failed to deploy in what turned out to be his last space mission.
"I'm fine," Hood said in reply. "I withdrew my resignation."
While Turner translated, Orlov turned on the lamp beside the chair and sat down. He picked up a pen and pad he kept on the small end table.
"Good, good!" Orlov said.
"Listen, General," Hood went on, "I'm very sorry to be calling you so early and at home."
"It's no bother, Paul," Orlov replied. "What can I do for you?"
"The terrorist who calls himself the Harpooner," Hood said. "You and I once spoke about him."
"I remember," said Orlov. "We've been looking for him in connection with the terror bombings in Moscow several years ago."
"General, we believe he is in Azerbaijan."
Orlov's full lips tightened. "That would not surprise me," he said. "We thought we had him located in Moscow two days ago. A guard near Lenin's Tomb was very confident in his identification. He summoned police assistance, but by the time it had arrived, the suspect had disappeared."
"Do you mean the police lost him, or the suspect knew he was being watched and managed to get away?" Hood asked.
"The police are generally good at surveillance," Orlov replied. "The subject went around a corner and was gone. He could have changed clothes somehow—I don't know. The Kievskaya metro stop is near where he was last seen. It is possible he went down there."
"It's more than possible," Hood said. "That was where one of our embassy people spotted him."
"Explain, please," Orlov said.
"We had heard that he was in Moscow," Hood said. "The embassy person followed the man he thought was the Harpooner onto the metro. They went to a transfer station, and the Harpooner got off. He boarded another train, left it at the Paveletskaya stop, then he literally vanished."
Orlov was now very interested. "You're sure it was Paveletskaya?" he asked.
"Yes," Hood asked. "Is that significant?"
"Perhaps," Orlov said.
"General Orlov," Hood said, "however the Harpooner left Moscow, it's possible that he may be headed back there or toward Saint Petersburg. Do you think you could help us try and find him?"
"I would love to capture that monster," Orlov replied. "I will contact Moscow and see what they have. In the meantime, please send whatever information you have to my office. I will be there within the hour."
"Thank you, General," Hood said. "And again, I'm sorry to have wakened you. I didn't want to lose any time."
"You did the right thing," Orlov assured him. "It was good speaking with you. I will talk to you later in the day."
Orlov rose and went back to the bedroom. He hung up the phone, kissed his precious, sleeping Masha on the forehead, then quietly went to the closet and removed his uniform. He carried it into the living room. Then he went back for the rest of his clothes. He dressed quickly and quietly, then left his wife a note. After nearly thirty years, Masha was not unaccustomed to his comings and goings in the middle of the night. When he had been a fighter pilot, Orlov was often called for missions at odd hours. During his spacefaring years, it was common for him to suit up while it was still dark. Before his first orbital flight he had left her a note that read, "My dearest—I am leaving the earth for several days. Can you pick me up at the spaceport on Sunday morning? Your loving husband, Sergei. PS: I will try to catch you a shooting star."
Of course, Masha was there.
Orlov left the apartment and took the stairs to the basement garage. The government had finally given him a car after three years, since the buses were unreliable. And with everything that was going on in and around Russia, from restless republics to rampant gangsterism in major cities, it was often imperative for Orlov to be able to get to his Op-Center's headquarters.
And it was imperative now. The Harpooner was back in Russia.
**TWENTY-THREE**
_**Washington, D.C. Monday, 7:51 P.M.**_
Liz Gordon came to Hood's office after his conversation with Orlov. A husky woman with sparkling eyes and short, curly brown hair, Gordon was chewing nicotine gum and carrying her ever-present cup of coffee. Mike Rodgers remained for the talk.
Hood told Gordon how the president had seemed during their meeting. Hood also gave the woman a brief overview of the possible covert activities that might explain what appeared to be the president's delusions.
When Hood was finished, Gordon refilled her coffee cup from a pot in the corner of the office. Though Hood had been dubious of psychiatry when he had first come to Op-Center, Gordon's profiling work had impressed him. He had also been won over by her thoroughness. She brought a mathematician's prooflike manner to the process. That, coupled with her compassion, had made her an increasingly valuable and respected member of the team. Hood did not have any trouble entrusting his daughter to her.
"The president's behavior does not seem extreme," Gordon said, "so we can eliminate some very serious dementias, which would indicate a complete or nearcomplete loss of intellectual capacity. That leaves us with dangerous but more elusive delusions, of which there are basically six kinds. First there's organic, which is brought on by illness such as epilepsy or brain lesions. Second is substance-induced, meaning drugs. Third is somatic, which involves a kind of hyperawareness of the body—anorexia nervosa or hypochondria, for example. What you've described doesn't sound like any of those. Besides, they certainly would have been caught by the president's physician during one of his regular checkups. We can also rule out delusions of grandeur—megalo—mania—since that would show up in public. We haven't seen any of that.
"The only two possibilities are delusions of reference and delusions of persecution," she went on. "Delusions of reference is actually a mild form of delusions of persecution, in which innocent remarks are deemed to be critical. That doesn't seem to apply here. But I can't be as quick to rule out persecution delusions."
"Why not?" Hood asked.
"Because the sufferer will go to great pains to conceal them," she said. "He or she believes that others are trying to stop them or hurt them in some way. They often imagine a conspiracy of some kind. If the president fears that people are out to get him, he won't want to confide in anyone."
"But the stress might come out in little bursts," Rodgers said.
"Exactly," Gordon told him. "Crying, withdrawal, distraction, temper—all of the things Paul described."
"He seemed to want to trust me," Hood said.
"That's true and also characteristic of the illness," Gordon said. "Delusions of persecution is a form of paranoia. But as a sage once said, 'Sometimes even paranoids have enemies.' "
"Is there something we should do?" Hood asked. "The First Lady's feelings notwithstanding, we have to do something if the president can't continue to function under these circumstances."
"Whatever is going on sounds like it's in an advanced-early stage," Gordon said. "The effects are unlikely to be permanent."
Hood's phone beeped.
"If there is a conspiracy, and you can expose it quickly," Gordon went on, "there is every reason to believe the president can stay on the job after a short rest. Whatever has happened probably wouldn't have any effects, long-term or short."
Hood nodded as he answered the phone. "Yes?"
"Paul, it's Bob," said Herbert.
"What's up?"
"A major situation," he said. "I just got a call from the CIA suit who relayed Tom Moore's request to me from Baku. Moore and the CIA guy from Moscow, Pat Thomas, were just wasted. They were taking David Battat to the hospital—the guy the Harpooner attacked during the stakeout. Moore was tagged by a sniper outside the hospital, and Thomas had his throat cut in the lobby."
"By who?" Hood asked.
"We don't know."
"No one saw him?" Hood asked.
"Apparently not," Herbert replied. "Or if they did, they didn't see him again."
"Where is Battat?"
"He's still at the hospital, which is why the suit called me," Herbert said. "The embassy called for police protection, but we don't know whether they've been compromised or not. The CIA is out of people, and they're afraid Battat will be next, and soon. We don't have anyone in Baku, but I thought—"
"Orlov," Hood said urgently. "I'll call him now."
**TWENTY-FOUR**
_**Khachmas, Azerbaijan Tuesday, 4:44 A.M.**_
Maurice Charles did not like to repeat himself.
If he arrived someplace by car, he liked to leave by bus or rail. If he went west by air, he liked to go east by car or bus. If he wore a hat in the morning, he took it off in the afternoon. Or else he wore a different one or dyed his hair. If he destroyed a car with a pipe bomb, he attacked the next target with C-4. If he had done surveillance along a coastline, he retreated inland for a short time. Repetition was the means by which entrepreneurs in any field were undone. Patterns enabled lesser thinkers to anticipate you. The only exceptions were densely populated cities where he might be seen. If he found a relatively obscure route through a place like that, he would use it more than once. The risk of being spotted and identified was greater than the risk of reusing an out-of-the-way road or tunnel.
Because Charles had surveyed the Caspian oil drilling site by plane, he decided to return to it by boat. The American and possibly Russian satellites would be looking for an aircraft by now. He and his team would take the motor yacht, which would have a different name on its side than it had the day before. One of the team members had made those arrangements in Baku. It would be waiting for them in Khachmas, a coastal town some fifty miles north of Baku. A freelance crew had been hired in Baku and sailed up with one of Charles's Iranian sailors. Not only was Khachmas closer to their target, it was unlikely that anyone would recognize them or the vessel.
After a short sleep, which was all he needed, Charles and his comrades had climbed into a van that was parked behind the shack. Their gear was already on board, and they drove from Gobustan back toward Baku. They traveled along roads that were utterly deserted at this time of night. Though Charles did not drive, he did not sleep. He sat in the backseat with a .45 in his lap. If anyone approached the van for any reason, he wanted to be awake.
The van arrived in sleepy Khachmas shortly before 4:30. They had driven the seventy miles nonstop. No one had approached them.
The _Rachel_ —now the _Saint Elmo_ —was waiting in a slip at a ramshackle marina. The berth was close to shore. The hired crew had been dismissed. They had departed in their own boat, a fishing vessel, which had accompanied the motor yacht north.
Wearing night-vision goggles, Charles stood watch while the equipment was transferred from the van to the _Saint Elmo_. When all the gear was on board, one of the team members drove off in the van. The vehicle would be painted locally and driven to another city. Finally, the motor yacht set off.
The trip to the target would take fifty minutes. The sun would just be coming up when they arrived. That was important. Working at sea, Charles did not like to use artificial lights. They were too easy to spot in the dark and reflected on the water. He also didn't like to work during bright daylight when the wet suits glistened. Early dawn was best. There would be just enough time to get the job done and depart without being seen.
Then he would leave Azerbaijan and do nothing but enjoy life for a month or two. Savor the international ramifications of what he had accomplished. Cherish the fact, as he always did, that no world leader, no army, no business, had a greater impact on international events than he did.
**TWENTY-FIVE**
_**Saint Petersburg, Russia Tuesday, 4:47 A.M.**_
After the fall of the Soviet Union, many officials in Moscow were afraid of the Ministerstvo Bezopasnosti Ruskii, or MBR, the Security Ministry of Russia. They were even more afraid than when the intelligence agency had been known as the KGB and was routinely tapping their phone lines and opening their mail. The officials feared that leaders of the former Soviet intelligence group would either support ousted Communists in an effort to recapture power or attempt to seize power themselves. Because of this, the Kremlin's new regime had created an autonomous intelligence agency outside of Moscow, away from the immediate reach of the MBR. They based it in Saint Petersburg. And, following the adage of hiding in plain sight, they located the Op-Center in one of the most visited places in Russia: the Hermitage.
The Hermitage was built by Catherine the Great as a retreat. The towering, white, neoclassical building was formally known as the Winter Palace. It was a place where Catherine could enjoy the gems and great old masters paintings, drawings, and sculptures she had collected. She literally acquired them at a rate of one every other day from 1762 to 1772. When Catherine first opened her home to the patrician public, her only comments were that visitors should be joyful. However, she added, they "shall not try to damage, break, or gnaw at anything." The Hermitage remained a repository of the imperial collection until 1917. After the Russian Revolution, the Hermitage was opened to all the people. Its collection was expanded to include art from other schools as well as modern art. It currently houses over 8,000 paintings, 40,000 etchings, and 500,000 illustrations. Today, it is second only to the Louvre in Paris in terms of the size of its collection.
The Russian Op-Center was constructed underneath a fully operational television studio. Though the broadcast facility had been built as a cover for the construction of the intelligence center, satellite dishes beamed famed Hermitage programs around the world. Most of the time, however, the highly advanced uplinks allowed the Op-Center to interface with satellites for both domestic and international electronic communications. The comings and goings of museum staff and tourists helped to disguise the presence of Op-Center personnel. Also, the Kremlin had decided that in the event of war or revolution, no one would bomb the Hermitage. Even if an enemy had no use for art as an aesthetic possession, paintings and sculptures were always as negotiable as currency.
It was still dark when the fifty-three-year-old Orlov arrived at the museum. Because the Hermitage was still closed, he entered through an inconspicuous studio door on the northeastern side of the museum. As he did, he gazed north across the dark Neva River. Directly across the water were the stately Academy of Sciences and Museum of Anthropology. Nearby was the Frunze Naval College. In addition to training cadets, the college housed the dozen soldiers of the center's special operations force, Molot, which meant _Hammer_.
There was a guard seated behind a desk inside the TV studio. Orlov acknowledged him as he passed. The elderly guard stood and saluted. The general reached a door and used the keypad to enter. Once inside, he made his way through the dark reception area and down a short flight of stairs. At the far end, he punched the new day's four-digit code on a keypad, and the door popped open. The next day's number was always given to Orlov by the center's security chief at the end of each workday. When Orlov shut the door behind him, the overhead lighting snapped on automatically. There was another, longer set of stairs. He walked down where a second keypad gained him access to the Op-Center.
The facility consisted of a very long corridor with offices to the left and right. Orlov's office was at the end, literally at the shores of the Neva. There were times when he could hear barges passing overhead.
Ordinarily, Orlov did not arrive until nine o'clock. There was a skeletal night staff, and they were surprised to see the general. He greeted them without stopping. When he entered his small, wood-paneled office, he shut the door and walked over to his desk. The desk faced the door. On the walls were framed photographs Orlov had taken from space. There were no photographs of the general himself. Though he was proud of his accomplishments, he didn't enjoy looking at the past. All he saw was how short he fell of his goals. How he had hoped to walk on the moon and command a manned mission to Mars. How he had dreamed of seeing the cosmonaut corps grow and prosper. Perhaps if he had used his celebrity more constructively, more aggressively, he could have helped make that happen. Perhaps if he had spoken out against the war in Afghanistan. That struggle drained the nation's resources and pride and hastened the union's downfall.
There were no photographs of himself because General Orlov preferred to look ahead. The future held no regrets, only promise.
There was a voice mail from Paul Hood. The message did not say very much. Only that the matter was urgent. Orlov sat down and booted his computer. As he opened his secure phone list and auto-dialed Hood, he thought back to how the American Op-Center had helped him prevent a cabal of right-wing Russian officials from overthrowing the government. The counterattack had cost Hood one of his top field operatives, Lieutenant Colonel Charles Squires. Since then, the two Op-Centers had occasionally exchanged information. But they had never become fully integrated partners, which was something both Hood and Orlov had wanted. Unfortunately, like many of the progressive dreams Orlov had, the bureaucrats had not been ready for this. Distrust between the nations was still too deep.
The phone beeped once. Hood answered.
"Hello?" Hood said.
"Paul, it's Sergei," Orlov said.
Op-Center's translator was on standby. It only took her a moment to get on the line.
"General, I need your trust, and I need it fast," Hood said. His urgent tone left no room for discussion.
"Of course," Orlov said.
"Our team searching for the Harpooner suffered a catastrophic hit at a hospital in Baku," Hood informed him. "It happened a little over an hour ago. Two of our men were killed. The first was taken down by a sniper outside the hospital. The second had his throat cut inside the lobby. The last man is a patient. His name is David Battat, and he is ill with a fever of some kind."
Orlov took a moment to write the name down.
"The police are at the hospital, but we don't know who the killer is," Hood said. "He or she may still be in the hospital."
"The killer could be a police officer," Orlov pointed out.
"Exactly," Hood said. "General, do you have anyone in Baku?"
"Yes, we do," Orlov said without hesitation. "In what room is Mr. Battat located?"
"He's in one fifty-seven," Hood said.
"I will send someone at once," Orlov said. "Tell no one."
Hood gave him his word.
Orlov hung up.
The three most powerful Russian intelligence groups had their own personnel. These groups were the MBR; the military's Glavnoye Razvedyvatelnoye Upravlenie, or GRU, the Main Intelligence Directorate; and the Ministerstvo Vnutrennikh Del, or MVD, the Ministry of Internal Affairs. The Russian Op-Center did not have the financial resources to maintain its own network of intelligence and counterintelligence personnel, so it was necessary to share people with other relatively small
Russian agencies. These were administered by the Sisteme Objedinennovo Utschotya Dannych o Protivniki, or SOUD, the Interlinked System for Recognizing Enemies. SOUD also provided personnel for the Sluzhba Vneshney Razvedki, or SVR, the Foreign Intelligence Service; the Federal'naya Sluzhba Bezopasnosti, or FSB, the Federal Security Service; the Federal'naya Sluzhba Kontr-razvedky, or FSK, the Federal Counterintelligence Service; and the Federal'naya Sluzhba Okhrani, or FSO, the Federal Protective Service.
Orlov quickly accessed the SOUD files. He input the highest-priority code, Red Thirteen. This meant that the request was not only coming from a senior official—level thirteen—but involved a case of immediate national emergency: the apprehension of the Harpooner. The Red Thirteen code gave Orlov the names, locations, and telephone numbers of field personnel around the world. Even if the operatives were involved in other sit uations, he would be authorized to commandeer them.
Orlov went to the file for Baku, Azerbaijan.
He found what he was looking for.
He hesitated.
General Orlov was about to ask a deep-cover operative to try to help an American spy. If the Americans were planning an operation in Baku, this would be the quickest way to expose and neutralize Russian intelligence resources. But to believe that, Orlov would have to believe that Paul Hood would betray him.
Orlov made the call.
**TWENTY-SIX**
_**Washington, D. C. Monday, 9:00 P.M.**_
Paul Hood was angry when he hung up with Orlov.
Hood was angry at the system, at the intelligence community, and at himself. The dead men were not his people. The man at risk was not his operative. But they had failed, and the Harpooner had succeeded, partly because of the way spies did business. The Harpooner commanded a team. Most American agents worked as part of a team. Theoretically, that should give the operatives a support system. In practice, it forced them to operate within a bureaucracy. A bureaucracy with rules of conduct and accountability to directors who were nowhere near the battlegrounds. No one could fight a man like the Harpooner with baggage like that. And Hood was guilty of supporting that system. He was as guilty as his counterparts at CIA, NSA, or anywhere else.
The irony was that Jack Fenwick had apparently done something off the books. It was Hood's job to find out what that was.
_The bureaucrats are checking up on the bureaucrats,_ Hood thought bitterly. Of course, he probably should not be thinking at all right now. He was tired and frustrated about the situation with Battat. And he had not even called home to see how Harleigh was doing.
Rodgers had stayed with Hood between the time he first phoned Orlov and Orlov returned the call. While they waited for Bob Herbert to come back, Rodgers left to grab a soda. Hood decided to call home. It did not improve his mood.
He was doing just the thing that Sharon had always hated. Working late. Calling home as an afterthought. He could hear the anger in her throat, in the tightness of her mouth, in the brevity of her answers.
"I'm doing laundry," Sharon said. "Harleigh is in the den playing solitaire on the computer. Alexander is in his room doing homework and studying for a history test."
"How does Harleigh seem today?" Hood asked.
"How do you think?" Sharon said. "Your own psychologist said it's going to be a while before we see any kind of change. If we see any kind of change," Sharon added. "But don't worry, Paul. I'll handle whatever comes up."
"I'm not going anywhere, Sharon," Hood said. "I want to help."
"I'm glad. Do you want me to get Alexander?" she asked.
"Not if he's studying," Hood said. "Just tell him I called."
"Sure."
"Good night," Hood said.
He could feel Sharon hesitate. It was only a moment, but it felt much, much longer. " 'Night, Paul," she said, then hung up.
Hood sat there holding the phone for several moments. Now he was a bastard and a bureaucrat. He lay the phone in its cradle, folded his hands, and waited for Rodgers. As he sat there, something began to tick inside him. It wasn't a clock or a bomb. It was like a cam and rocker arm. And with each click of the arm, a spring grew tighter inside him. A desire to do something—and not just debate or call the Russians for help. Hood wanted to act. Something was not right, and he needed to know what it was.
Rodgers and Herbert arrived together. They found Hood staring at the back wall of his office where plaques and framed photographs once hung, the mementos of his years in government. Pictures with world leaders, with constituents. Photographs of Hood laying cornerstones or working in a Thanksgiving soup kitchen.
His life as a bloody goddamn bureaucrat. As part of the problem, not the solution.
"Are you all right?" Herbert asked.
"Fine," Hood said.
"Did you get news?" Herbert pressed.
"No," Hood said. "But I want to make some."
"You know where I stand on that," Herbert said. "What were you thinking of?"
"Battat," Hood said. That was not entirely true. He was thinking that he never should have withdrawn his resignation. He should have left Op-Center and never looked back. He wondered if resigning had actually been for him and not to spend more time with his family, as he had believed. But he was back, and he was not going to run away.
Battat was the next stop in his thought process. "This man was sent to the hospital with some kind of sickness where a pair of assassins were waiting," he said. "That doesn't sound like a coincidence."
"No, it doesn't," Herbert agreed. "My brain trust and I have been looking into that."
Herbert's brain trust consisted of four deputy intelligence directors who had been brought to Op-Center from military intelligence, the NSA, and the CIA. They were three men and one woman who ranged in age from twenty-nine to fifty-seven. With input from Darrell McCaskey, who liaised with the FBI and Interpol, Op-Center had the best per capita intelligence team in Washington.
"Here's what we've been thinking," Herbert said. "The CIA is ninety-nine percent certain the Harpooner passed through Moscow and went to Baku. A DOS agent thinks he saw him on a flight to Moscow, but that may have been intentional."
"Why?" Rodgers asked.
"It wouldn't be unprecedented for a terrorist to let himself be seen," Herbert said. "Back in 1959, the Soviet spy Igor Slavosk allowed himself to be seen at Grand Central Station in New York so he could draw police attention and bring FBI personnel to his apartment. When they got to the place down on Jane Street, it blew up. Slavosk came back, collected badges and IDs, and had perfect fakes made. He used them to get into FBI headquarters in Washington. So, yes, it's possible the Harpooner allowed his presence to be known through channels."
"Go on," Hood said quietly. He was getting impatient. Not at Bob Herbert; the intelligence chief was simply a convenient target. Hood wanted Orlov to call him back. He wanted to hear that everything was all right at the hospital. He wanted some good news for a change.
"Sorry," Herbert said. "So the Harpooner somehow lets it be known that he's going to Baku. He has some kind of operation planned. He knows there are CIA personnel attached to the embassy. He also knows that the CIA might not want to expose those people since police from the Azerbaijani Ministry of Internal Security are probably keeping an eye on embassy personnel, watching for foreign intelligence operations. So the CIA brings someone in from Moscow."
"Battat," said Hood.
"Yes," Herbert said. He seemed a little uneasy. "David Battat was the head of the CIA's New York City field office. He was the man who hired Annabelle Hampton."
"The junior officer we busted during the UN siege?" Rodgers said.
Herbert nodded. "Battat was in Moscow at the time. We checked him. He's clean. One of our CIA contacts told me he was sent to Baku to do penance for the New York screwup."
Hood nodded. "All right. You've got Battat in Baku." "Battat goes out to a target area to watch for the Harpooner and gets taken down," Herbert said. "Not taken out, which the Harpooner could have done with no problem. Battat was apparently infected with a virus or chemical designed to drop him at a specific time. Something serious enough so that he'd be taken to the hospital."
"Under guard from his fellow CIA operatives," Hood said.
"Exactly," Herbert replied. "Pretty maids all in a row."
"Which leaves the Harpooner free of CIA interference to do whatever he's planning," Hood said.
"That's what it looks like," Herbert said. "No one but the United States, Russia, and probably Iran has any kind of intelligence presence in Baku."
"Because of the Caspian oil?" Rodgers asked.
Herbert nodded. "If the Harpooner also hit operatives from Moscow and Teheran, we haven't heard about it."
Hood thought about that. "Iran," he said softly.
"Excuse me?" Herbert said.
"That's the second time we've been talking about Iran today," Hood said.
"But not for the same—" Herbert said, then stopped.
"Not for the same reason?" Hood asked.
"Aw, no," Herbert said after a moment. "No."
"Hold on," Rodgers said. "What am I missing?"
"You're thinking the game of telephone could go from the Harpooner to Teheran to Jack Fenwick to the NSA to the CIA," Herbert said.
"It's possible," Hood said.
"That would put Fenwick in bed with them on something involving the Harpooner," Herbert said.
"Something he would not want the president to know about," Hood pointed out.
Herbert was shaking his head. "I don't want this to be happening," he said. "I don't want us working with the sonofabitch who killed my wife."
"Bob, I need you to calm down," Hood said.
Herbert was glaring at Hood's desk.
"If the Harpooner is up to something in Baku, we might still be able to get him," Hood said. "But only if we stay focused."
Herbert did not respond.
"Bob"
"I hear you," Herbert said. "I'm focused."
Hood looked at Rodgers. A minute ago, Hood wanted to lash out. Now that one of his friends was hurting, the desire had subsided. All he wanted to do was help Herbert.
Why did he never feel that way about Sharon when she was angry?
"Mike," Hood said, "we really need to pin down what Fenwick's been up to and who, if anyone, he's been working with."
"I'll get that information," Rodgers said. "But I can tell you this much. I found two e-mails in my computer files from six months ago. They were written by Jack Fenwick and Burt Gable."
"What were the memos about?" Hood asked.
"They were responding to a Pentagon white paper," Rodgers said. "The paper was about the minimal threat of possible Russian military alliances with neighbors who were not part of the former Soviet Union. Fenwick and Gable took issue with that." "The head of the National Security Agency and the president's chief of staff both took issue to the report, independently," Hood said.
"Correct," said Rodgers. "The memos were sent to all the members of congress and various military leaders."
"I wonder if the two men met philosophically on-line," Hood said. "What was the time code on the memos?"
"A few hours apart," Rodgers said. "They didn't appear to be part of a concerted effort. But they both shared an aggressive disapproval of the report."
"I guess it doesn't matter whether Fenwick and Gable issued those memos independent of one another or whether they found out they had something in common when they read them," Hood said. "The question is whether they did something about it. Whether they got together and did some plotting."
"What makes you think they might have?" asked Herbert, easing back into the conversation.
"Gable's name came up today in my talk with the president," Hood said. "He and Fenwick's assistant Don Roedner were responsible for keeping the CIOC in the loop about that UN initiative."
"And didn't," Herbert said.
"No, they didn't." Hood tapped the desk slowly. "We've got two issues here," he said a moment later. "Fenwick's activities in New York and the Harpooner's activities in Baku."
"Assuming they are separate." Herbert said. "The two operations do have Iran in common. The Harpooner has worked for Teheran before."
Hood nodded. "What if he's working for them again?"
"Against Azerbaijan," said Herbert.
"It's possible," Rodgers said. "The Iranians have two potential areas of conflict with Azerbaijan. The Caspian oil reserves and the bordering Nagorno-Karabakh region."
"But why would Fenwick want to be involved in something like that?" Herbert said. "Just to prove the Pentagon wrong? Then what?"
"I don't know," Hood said. He looked at Rodgers. "Get to him and make him open up. Not only about Iran but about why he lied to the president."
"Tell him you've got information you can only tell him face-to-face," Herbert said.
"Right," Hood said. "Have Liz work out a psych profile of the president. One based on firsthand observations, including my own, that makes it look as though Lawrence is losing his grip. Bring that to Fenwick, ostensibly on the Q.T. Ask if he's heard anything about this."
Rodgers nodded and left.
Hood looked at Herbert. "If Iran has any military adventures on the drawing board, they may have moved troops or matériel. The NRO may have noticed something. Has Stephen Viens gone back to work there?"
"Last week," Herbert said.
The NRO was the National Reconnaissance Office, the top-secret facility that manages most of America's spy satellites. An agency of the Department of Defense, the NRO is staffed by personnel from the CIA, the military, and civilian DOD personnel. The existence of the NRO was declassified in September of 1992, twenty years after it was first established. Stephen Viens was an old college buddy of Op-Center's computer chief Matt Stoll. He had been extremely helpful getting information to Op-Center when more established groups like military intelligence, the CIA, and the NSA were fighting for satellite time. Viens had been accused of hiding money in a black ops situation but was later vindicated.
"Good," Hood said. "See if Viens can find anything. The NRO may have spotted activity in Iran without perceiving any immediate danger."
"I'm on it," Herbert said.
The intelligence chief wheeled his chair from the office. Hood sat back. He looked at the phone. He wanted to hear from Orlov. He wanted to hear that the Russian had someone in place and that Battat would be all right. He wanted to hear that they had managed to put the brakes on the bad news and could start turning this situation around.
_We have to_ , Hood thought. There was something out there. Something big and dangerous. He did not know what it was or who was behind it. He did not know if the pieces Op-Center had collected would fit together. He only knew one thing for certain: Whatever it was, it had to be stopped.
**TWENTY-SEVEN**
_**Baku, Azerbaijan Tuesday, 5:01 A.M.**_
David Battat felt frigidly cold and light-headed. He could hear his heart in his ears, feel it in his throat. He was aware of being wheeled somewhere. There were faces over him. Lights flashed by. Then he felt himself being lifted. He was placed on a bed, still experiencing a sense of forward motion. He was not strapped down, but there were raised metal gates on the side of the bed.
Battat shut his eyes. He did not know what had happened to him. He remembered waking up at the embassy, perspiring and shaking. Moore and Thomas brought him to the car, and then he must have slept. The next thing he knew, he woke up on a gurney.
He heard people moving around him. He coughed and opened his eyes. There was a white-haired man looking down at him.
"Mr. Battat, can you hear me?" the man shouted.
Battat nodded.
"We are going to undress you and put you in a gown," the man said to him. "Then we need to get an IV into you. Do you understand?"
Battat nodded. "What . . . happened?"
"You're ill," the doctor told him as a pair of male nurses came over. They began lifting and undressing him. "You have a very high fever. We have to bring it down."
"Okay," Battat said. What else could he say? He could not have resisted if he wanted to. But he did not understand how he could have gotten sick. He had felt fine before.
The medical team worked on him for several minutes. Battat was not entirely aware of what they were doing. He only knew that he was being shifted and turned and poked. He felt a pinch in his right arm, at the elbow, and then there was no further pain. He was also shivering, and he felt cold. Sweat had soaked into Battat's pillow. His fever warmed it quickly. His head sank into the down, muffling the sounds of the people and whatever it was they were doing. He shut his eyes again and allowed his mind to go wherever it wanted.
Soon it was quiet and dark. Battat began to feel a little warmer, more comfortable. He no longer heard drumming in his ears. He was awake, but his thoughts were dreamlike. His mind went back over the days. He saw short, blurry visions of the embassy in Moscow, the trip to Baku, the seashore, the sudden pain of the attack. A pinch in his neck. He was unaware of time passing or the hospital room. There was just a strange, not unpleasant sense of drifting. There must be something in the IV. Something that was relaxing him.
Then Battat heard something click. It sounded like a gun hammer cocking. He opened his eyes. There was a window to the left of the bed, but it was shut. He glanced toward the foot of the bed. The last time he had looked, the door was ajar. Now it was shut. A doctor or nurse must have closed it. The room was even quieter than before. It was nice. He shut his eyes again. There were no more visions, only darkness. Battat slipped quickly into a dreamless sleep.
There was another click. The sound woke Battat, and he opened his eyes. The door was still closed. But now there was someone in the room. He could see a dark figure standing in front of the door. The figure was black against the darkness behind it.
Battat was not sure he was awake.
"Hi," he said. He heard his own voice. He was definitely awake.
Slowly, the shadow moved toward him. Someone must have come to check on him.
"It's all bright," Battat said in a soft slur. "You can turn on the light. I'm awake."
The figure did not speak. Battat could not make out whether it was a man or a woman. It appeared to be wearing a medical robe of some kind. And it was holding something long and slender. Battat could see the silhouette low at its side. It looked like a knife.
"Do you speak English?" Battat asked.
There was a monitor on the wall behind Battat. The green glow threw a faint light on the figure as it stopped beside the bed. It was a man. And he was definitely holding a knife. The long blade gleamed in the dull light.
"What is this?" Battat asked. It was beginning to penetrate his foggy mind that the newcomer was not a doctor. Battat tried to move, but his arms felt like they were full of wet sand.
The man's arm went back.
"Someone!" Battat said, trying to raise his voice. "Help me—"
And then the man vanished.
A moment later, sounds came from the floor. There were low grunts, chattering, and then a long, slow groan. They were followed by silence.
Battat tried to raise himself on an elbow. His arm shook, and he fell back down.
Suddenly, someone rose beside the bed.
"There may be others," said the figure. "We have to leave."
The sharp, thickly accented voice belonged to a woman. There were an awful lot of people here.
"I thought this was a private room," Battat said.
With swift, sure movements, the woman lowered the gate beside the bed, unhooked the IV, and raised Battat to a sitting position. She kept her hand on his back.
"Can you walk?" she asked.
"If you let go ... I'm not sure I can sit," he replied.
The woman lay Battat back down and stepped away from the bed. She was a tall, lean woman with broad shoulders. He could see now that she was wearing a police uniform. The woman went to the window and pulled the curtains aside. She turned the latch and raised the window. A cool, salty breeze blew in. It made him shiver. The woman looked outside. Then she grabbed a bathrobe from a hook behind the door and returned to the bed. She sat Battat up again and pulled the robe around his shoulders.
"What are we doing?" he asked. Without the IV in his arm, he was feeling a little more focused. His head was also hurting from sitting up.
"No talk," she said.
"But wait," he said.
"They've killed your companions, and they're trying to kill you," she snapped. "I was sent to get you out."
"Killed them?"
"Quiet!" she hissed.
Battat stopped talking.
His head ached as the woman helped him stand. She grabbed Battat's clothes, then slipped his left arm around her shoulder and helped him to the window. As they hobbled over, Battat tried to focus on what she had just told him. Were Moore and Thomas dead? If so, it had to be the Harpooner. Maybe he thought they knew more than they did. But if they were dead, who had sent this woman to help him? And how did he know that she was not working for the Harpooner? She might be taking him somewhere so the killer could finish the job.
But Battat knew he might as well trust her. He was certainly in no condition to resist. Besides, the woman was being gentle with him. And if she had wanted him dead, she could have killed him in the bed. Or she could have let the other intruder kill him.
When they reached the window, the woman told Battat to lean on the sill. He did, unsteadily. She kept a hand on him, helping to keep him upright as she slipped around him. She landed quietly among the hedges outside the window and then helped him down. She put his arm back around her shoulder and then crouched. They listened for several seconds.
Battat was shivering again, his teeth clattering. But at least he was more awake than before. After a moment, they were on the move again. He felt as if he was being carried through the night. They had emerged in back of the hospital and were making their way around to the north side. They stopped at a car. To Battat's surprise, it wasn't a police car but a small black Hyundai.
She probably was not a policewoman at all. Battat did not know if that were a good thing or a bad thing. But as she laid him across the backseat and climbed behind the wheel, he knew one thing for certain.
If he remained conscious, he would find out very soon.
**TWENTY-EIGHT**
_**Washington, D.C. Monday, 10:03 P.M.**_
The red-haired man sat behind his large desk. The office was dark, save for the glow of a green-shaded desk lamp and the red light on top of the phone. That meant the scrambler function was engaged.
"People are asking about Fenwick's trip," said the red-haired man.
"What people?" said the man on the other end of the line.
"The intelligence unit at Op-Center."
"Op-Center is well removed from the president," the other man said. "They don't have the same clout as the CIA—"
"I'm not so sure about that," the red-haired man interrupted.
"What do you mean?"
"I was told that Director Hood asked for and received a private meeting with the president a few hours ago," said the red-haired man.
"I know."
"Do you know what they discussed?" asked the red-haired man.
"No. More fallout from the United Nations affair. I'd guess. Do you have reason to believe otherwise?" the man asked.
"Paul Hood spoke briefly with the First Lady last night." the red-haired man said. "I checked his file. They knew each other in the past."
"Knew each other in a way we can use?"
"No," said the red-haired man. "It was platonic. Anyway, she might have seen a change in the president. Maybe she said something to Hood. I just don't know."
"I see," said the other.
There was a long silence. The red-haired man waited. He was concerned about the unexpected presence of Op-Center. The other agencies had all been covered. He and his partners had been counting on the transition period between Paul Hood and General Rodgers to keep Op-Center's eyes looking inward. Unfortunately, that had not happened. But with H-hour approaching on the foreign operation, they could not afford to have anyone watching. Harpooner had seen to it on his end. They must see to it on their end.
"Is the other documentation ready?" the other man finally asked.
The red-haired man looked at his watch. He really needed glasses to read this close, but he was fighting that. He was fighting a lot of things. He moved his wrist back slightly. "In another hour or so," he replied.
"All right," said the other man. "I don't want to move against Op-Center directly. There isn't time. And without careful planning, we might do more harm than good."
"I agree," said the red-haired man.
"Let's continue with the plan," said the other man. "If Op-Center is watching Fenwick or the president without any real idea what we're up to, that should keep them busy enough. Just make sure Fenwick. doesn't do or say anything that might give them more information."
"Understood," said the red-haired man. "I'll let Fenwick know."
The other man thanked him and hung up.
The red-haired man placed the receiver in the cradle. He would call Fenwick in a minute. This was serious, unprecedented business. He needed a moment to remind himself that this was all being done for a good reason: to make sure that the United States survived the new millennium.
Despite this small setback, everything was still working the way they had planned. Reporters had been calling his office to find out about the new UN initiative, an initiatve that only the president seemed to be aware of. Members of the CIOC and even people at the UN apparently had not known about it. One very dogged TV reporter had called this evening to ask if the president had imagined "this whole thing, too." And Red Gable, the president's chief of staff, had answered off the record, "I honestly don't know, Sam. I do not know what is wrong with the president."
Though the quote would be off the record, Gable knew that his sentiment would be mentioned in the broadcast. The reporter reminded Red that this was the third time in a week the president had gotten something seriously wrong. The first time was at a breakfast with reporters. The president commented about farm subsidy legislation that was supposedly before congress. It was not. The second time, just two days ago, was at a press conference. The president's opening remarks included comments about a civil rights case that was supposedly before the Supreme Court. No such case existed. What Gable did not tell the reporter, of course, was that the set of documents the president had been given during his daily briefings was different from the set of documents that he should have seen. The real ones. Gable had slipped those documents into the president's files after he made the public misstatements. When the president had the files brought to him, he did not understand where the misinformation had come from. Investigations by Gable and his assistants failed to turn up any suspicious activity.
Gable did not smile. He could not. The situation was too serious. But he was gratified. The reporter and many of his colleagues were very concerned about the president's state of mind. By tomorrow afternoon, the rest of the nation would also be concerned. Events that were about to unfold a world away and in Washington had been very carefully orchestrated. Events that would be misinterpreted by everyone except the third and most important leader of their team: the vice president. The president would insist that Azerbaijan had attacked an Iranian oil rig. He would recommend staying out of the conflict because it was a local issue. As Iran built up its forces in the region, the vice president would publicly urge a different tack. He would say that he did not trust Iran and would strongly advise building up an American military presence in the Caspian. Fenwick would back up the vice president. He would report that during his meetings with the Iranians, they had spoken vaguely of events that were on the horizon. He would say that they asked the United States to do nothing while they strengthened their hold on oil reserves in the region.
The Iranians would deny that, of course. But no one in America would believe them.
The disagreement between the president and vice president would cause a very public rift.
And when the Harpooner's Iranian cohorts were found dead with photographs and other evidence of sabotage on their bodies—murdered by the Harpooner himself—the vice president and Fenwick would be vindicated.
Reporters would then openly discuss the president's questionable judgment. Washington would be abuzz with rumors that the president was unstable. Senators like Barbara Fox would have no choice but to support a motion to impeachment. Sex scandals were one thing. Mental illness was something much different. There would be calls for Lawrence to step down. For the good of the nation, Lawrence would have no choice but to resign.
Vice president Cotten would become president. He would ask Jack Fenwick to become his new vice president. Congress would quickly endorse his selection. Meanwhile, the American military would move into the Caspian. They would help the Azerbaijanis protect their rigs.
In the heat of rising tensions, President Cotten would remain strong.
And then something else would happen. Something that would demand an American response so firm, so devastating, that religious fanatics would never again attack a target under American protection.
In the end, Gable told himself, the career of a president was worth that sacrifice.
**TWENTY-NINE**
_**Baku, Azerbaijan Tuesday, 6:15 A.M.**_
When forty-seven-year-old Ron Friday first arrived in Baku, he felt as though he had been dropped into medieval times.
It was not a question of architecture. Embassy row was in a very modern section of the city. The modern buildings could have been lifted whole from Washington, D.C., or London, or Tokyo, or any other modern metropolis. But Baku was not like those cities where he had spent so much time. Once you moved past the embassies and business center of Baku, there was a pronounced sense of age. Many of the buildings had been standing when Columbus reached the Americas.
No, the architecture was not what made Baku seem so old, so feudal. It was a sense of entropy among the people. Azerbaijan had been ruled from the outside for so long, now that the people were free and independent, they seemed unmotivated, directionless. If it were not for petrodollars, they would probably slip deep into the Third World.
At least, that was Friday's impression. Fortunately, when the former Army Ranger and his people were finished with what they were doing here, Azerbaijan would not be quite so independent.
Friday entered his seven-story apartment building. The ten-year-old brick building was located two blocks from the embassy. He made his way up the marble stairs. Friday lived on the top floor, but he did not like being in elevators. Even when he was with the other embassy workers who lived here, he took the stairs. Elevators were too confining, and they left him vulnerable.
Friday walked toward his apartment. He could not believe that he had been here nearly six months. It seemed much longer, and he was glad his tenure was coming to an end. Not because Deputy Ambassador Williamson didn't need him. To the contrary, Friday had proven valuable to the diplomat, especially in her efforts to moderate Azerbaijani claims on Caspian oil. Friday's years as an attorney for a large international oil company served him well in that capacity. But Friday's real boss would need him elsewhere, in some other trouble spot. He would see to it that Friday was transferred.
To India or Pakistan, perhaps. That was where Friday really wanted to go. There were oil issues to be dealt with there, in the Arabian Sea and on the border between the Great Indian Desert in the Rajasthan province of India and the Thar Desert in Pakistan. But more than that, the Indian subcontinent was the place where the next big war would begin, perhaps triggered by a nuclear exchange. Friday wanted to be in there, helping to manipulate the politics of the region. It had been a dream of his ever since he was in college. Since the day when he had first gone to work for the National Security Agency.
Friday put the key in the door and listened. He heard the cat cry. Her mewing was a normal welcome. That was a very good indication that no one was waiting for him inside.
Friday had been recruited by the NSA when he was in law school. One of his professors, Vincent Van Heusen, had been an OSS operative during World War II. After the war, Van Heusen had helped draft the National Security Act of 1947, the legislation that led to the founding of the Central Intelligence Agency.
Professor Van Heusen saw in Friday some of the same qualities he himself had possessed as a young man. Among those was independence. Friday had learned that growing up in the Michigan woods where he attended a one-room schoolhouse and went hunting with his father every weekend—not only with a rifle but with a longbow. After graduating from NYU, Friday spent time at the NSA as a trainee. When he went to work for the oil industry a year later, he was also working as a spy. In addition to making contacts in Europe, the Middle East, and the Caspian, Friday was given the names of CIA operatives working in those countries. From time to time, he was asked to watch them—to spy on the spies, to make certain that they were working only for the United States.
Friday finally left the private sector five years ago, bored with working for the oil industry. They had become more concerned with international profits than with the vitality of America and its economy. But that was not why he quit. He left the private sector out of patriotism. He wanted to work for the NSA full-time. He had watched as intelligence operations went to hell overseas. Electronic espionage had replaced hands-on human surveillance. The result was much less efficient mass intelligence gathering. To Friday, that was like getting meat from a slaughterhouse instead of hunting it down. The food didn't taste as good when it was mass-produced. The experience was less satisfying. And over time, the hunter grew soft.
Friday had no intention of growing soft. So when his Washington contact told him that Jack Fenwick wanted to talk to him, Friday was eager to meet. Friday went to see him at the Off the Record bar at the Hay-Adams Hotel. It was during the week of the president's inauguration, so the bar was jammed, and the men were barely noticed. It was then that Fenwick suggested a plan so bold that Friday thought it was a joke. Or a test of some kind.
Then Friday agreed to meet with some of the other members of the group. And he believed.
Oh, how he believed. They sent him here and, through contacts in Iran, he was put in touch with the Harpooner. Iran did not realize they were going to be double-crossed. That once they had an excuse to move into the Caspian Sea, a new American president would move against them.
And the Harpooner? He did not care. Friday and the Harpooner had worked closely organizing the attack against Battat and the program of disinformation to the CIA.
Friday was still dressed in yesterday's clothes. In case anyone saw him, that would support the story he would tell them. It was just one of the many stories he had perfected over the years to cover meetings he had to make with operatives.
Or targets.
Friday was glad the Harpooner had put one of his other men inside the hospital as backup. They had hoped that Friday would be able to get both Moore and Thomas while they were outside. But the way the ambulance was parked he did not have a clear shot at Thomas. Friday hoped the Iranian assassin had been able to get the other man. It would have been easier, of course, if Friday could have taken all three men out in the embassy. But that might have exposed him. The embassy was not that large, and someone might have seen them. And there were security cameras everywhere. This way had been cleaner, easier.
After firing the shot, Friday had dropped the rifle the Harpooner had given to him. It was a G3, a Heckler & Koch model, Iranian manufacture. He had others at his disposal if he needed them. Friday had tossed the weapon in a shallow pond near the hospital. He knew the local police would search the area for clues and would probably find it. He wanted it to be traced back to Teheran. Friday and his people wanted to make very sure that the world knew Iran had assassinated two officials of the United States embassy. The Iranians would disavow that, of course, but America would not believe the Iranians. The NSA would see to that.
The Iranians who were working with the Harpooner had made cell phone calls to one another during the past few days. They had discussed the attack on the oil rig and described the two pylons that had to be destroyed: "target one" and "target two." The Iranians did not know that the Harpooner made certain those calls were monitored by the NSA. That the conversations were recorded and then digitally altered. Now, on those tapes, the targets the Iranians were discussing were embassy employees, not pylons.
In a phone call of his own, the Harpooner had added that the deaths would be a warning, designed to discourage Americans from pursuing any action against Iran in the coming oil wars. The Harpooner pointed out in the call that if Washington insisted on becoming involved, American officials would be assassinated worldwide.
Of course, that threat would backfire. After President Lawrence resigned, the new president of the United States would use the brutal murders as a rallying cry. He was not a live-and-let-live leader like the incumbent. Someone who was willing to cooperate with the United Nations to the detriment of his own nation. The assassinations, like the attacks on the oil rigs, would underscore that the United States had unfinished business from the previous century: the need to strike a decisive, fullscale blow against terrorist regimes and terrorist groups that were being protected by those regimes.
Friday entered his apartment. He saw the red light on his answering machine flashing. He walked over and played the message. There was only one, from Deputy Ambassador Williamson. She needed him to come to the embassy right away. She said that she had tried his cell phone but could not reach him.
Well, of course she could not. His cell phone had been in his jacket, and his jacket had been slung over a chair in another room. He had not heard the phone because he was in the bedroom of a woman he had met at the International Bar.
Friday called her back at the embassy. Williamson did not bother to ask where he had been. She just told him the bad news. Tom Moore had been shot and killed by a sniper outside the hospital. Pat Thomas's throat had been cut by an assassin inside the hospital.
Friday allowed himself a small, contented smile. The Harpooner's assassin had succeeded.
"Fortunately," Williamson went on, "David Battat was able to stop the man who tried to kill him."
Friday's expression darkened. "How?"
"His throat was cut with his own knife," she said.
"But Battat was ill—"
"I know," said the deputy ambassador. "And either Battat was delirious or afraid. After he stopped the killer, he left the hospital by the window. The police are out looking for him now. So far, all they've found was the rifle used to kill Mr. Moore. Metal detectors picked it up in a pond."
"I see," Friday said. The assassin did not speak English. Even if Battat were lucid, he could not have learned anything from the killer. But Fenwick and the Harpooner would be furious if Battat were still alive. "I'd better go out and join the search," Friday said.
"No," Williamson said. "I need you here at the embassy. Someone has to liaise between the Baku police and Washington. I've got to deal with the political ramifications."
"What political ramifications?" Friday asked innocently. This was going to be sweet. It was going to be very sweet.
"The police found the rifle they think was used in the attack on Moore," she said. "I don't want to talk about this on an open line. I'll tell you more when you get here."
That was good news, at least. The deputy ambassador had concluded that the killings were political and not random.
"I'm on my way," Friday said.
"Watch yourself," Williamson said.
"I always do," he replied. Friday hung up, turned around, and left the apartment. "I always do."
**THIRTY**
_**Baku, Azerbaijan Tuesday, 6:16 A.M.**_
The Harpooner and his team reached the oil rig just before dawn. The boat cut its engines one thousand feet from the nearest of the four columns. Then the Harpooner and four members of his Iranian team slipped into the water. They were all wearing wet suits and compressed air cylinders. Slipping beneath the dark surface of the sea, the men swam toward the rig.
Two of them carried waterproof pouches containing watergel high-energy explosives. The Harpooner had carefully injected the blue sticks with heat-sensitive pentanitroaniline. As the sun rose, the heat would cause the foil packet to warm. The sunlight itself would detonate the explosion.
Two other men carried an inflatable raft. This would allow them some stability underneath the platform. Many rigs had sensors on the columns and motion detectors along the sea line. Avoiding the columns and going under the motion detectors was the safest way to get inside the perimeter. Once the explosives were placed, it would be virtually impossible for the crew of the rig to get to them in time.
The Harpooner carried a spear gun and night-vision glasses. He would use the gun to fire the watergel packets around the support struts beneath the platform. The Harpooner had brought along only a dozen of the seven-eighths-inch sticks of explosive. He had learned long ago that the trick to destroying something big is not necessarily to hit it with something big. In hand-to-hand combat, a foe could be driven back with a powerful roundhouse punch. He can be debilitated faster, more efficiently, and with more control, with a finger pressed against his throat, just below the larynx and above the clavicle. Hooking the top of a foot behind the knee and then stepping down with the side of the foot will drop someone faster than hitting them with a baseball bat. Besides, all it takes to neutralize a bat attack is to move in close to the attacker.
The Iranian oil rigs in the Caspian Sea are mostly semisubmersible platforms. They rest on four thick legs with massive pontoons that sink below the waterline. There is a platform on top of the legs. The riser system—the underwater component, which includes the drill—descends from the derrick, which is mounted on the platform. The key to destroying a platform like that is not to take out the columns but to weaken the center of the platform. Once that has happened, the weight of the structures on top will do the rest. The Harpooner's team had been able to get copies of the oil rig blueprints. He knew just where to place the watergel.
The men reached the underbelly of the rig without incident. Though it was dark in the water, the higher struts of the rig caught the first glint of dawn. As the Harpooner eyeballed the target, two men inflated the raft while the other two attached a pair of watergel sticks beneath the tip of three spears. The twelve-inch-long sticks were carefully taped belly-to-belly. This configuration allowed the spear to be fitted into the tube muzzle. It also made sure that the sticks of watergel would not upset the balance of the spear. Though it would have been easier to assemble the package on the boat, the Harpooner had wanted to keep the watergel packets as dry as possible. Though moisture would not harm the explosives, wet foil would take longer for the sun to warm. These packets would only be exposed to direct sunlight for a half hour. He had to make certain they were dry enough—and thus hot enough—to explode within that time.
The raft was a six-man hexagonal platform. The Harpooner did not need it to hold six men. He wanted the larger size for stability. Larger rafts tended to ignore the smaller waves. That was important when he lay on his back to fire. He had removed the canopy to make it lighter. The large case in which it had been carried was discarded. The Harpooner climbed on board while the other men hung onto the sides to steady the raft even more.
The speargun was made of stainless steel. It was painted matte black to minimize reflected sunlight. The spears were also black. The weapon was comprised of a forty-inch-long black tube and a yellow grip and trigger at the end. Only a foot of spear protruded from the end. Normally, a rope was attached to the spears so that prey could be hauled back to the spearman. The Harpooner had removed these back on the boat.
There were six-inch-thick acoustic dampeners beneath the platform. They were located fifty feet above the sea. The hard rubber pads had been placed there to muffle the sounds of activity. This was done so that people who lived on the rig would suffer as little noise pollution as possible. The Harpooner had chosen his targets from the blueprints. He would fire two harpoons. The first would go into the padded area below and to the northeast of the derrick. The derrick was in the southwest corner of the platform. When the detonation occurred, the derrick would fall toward the center of the platform. A second harpoon would be fired into the platform at the point where the heavy center of the derrick would land. The second explosion, plus the impact of the derrick, would shatter the platform and cause it to collapse inward. Everything would slide to the center and tumble into the sea.
The Harpooner would not need the third harpoon to destroy the rig, though he did not tell his people that.
The terrorist donned night-vision glasses and lay on his back. The speargun had terrific recoil, equivalent to a twelve-gauge shotgun. That would give him quite a bump. But his shoulder could take it. He aimed the weapon and fired. There was a sound like a metallic cough and the spear flew through the dark.
It hit the target with a faint thunk. The Harpooner quickly repositioned himself to fire the second shaft. It, too, struck its target. He motioned the men to start back. As soon as the others ducked underwater, the Harpooner pulled the tape from the spear, grabbed one of the equipment bags, and slipped the watergel sticks inside. Then he slid into the water and followed his men back to the boat.
Upon boarding the vessel, the men dropped the remains of Sergei Cherkassov into the sea. On the way over, they had burned the body. It would look as though he had been killed in the blast. The photographs that had been taken from the airplane were already in his pocket. As far as the Iranians on board knew, the Russians and the Azerbaijanis would be blamed for the attack.
The Harpooner knew differently.
When Cherkassov was in the water, the boat departed. They were nearly out of visual range when the oil rig exploded.
The Harpooner was watching through high-powered binoculars. He saw the puff of yellow red smoke under the platform. He saw the tower shudder and then do a slow pirouette drop toward the center. A moment later, the muted pop of the first explosion reached the boat.
The Iranians on the deck all cheered. Which was odd, the Harpooner thought. Even though they thought they were doing this for the national good, they were happy about the deaths of at least one hundred of their countrymen.
A moment before the derrick hit, the second watergel packet exploded. The Harpooner had positioned the two to go off nearly at the same time. It would not have done for the derrick to crash, knock the spear from the rubber padding, and drop it into the sea. A second cloud of red and yellow smoke began to form, but it was flattened and disbursed when the derrick struck the platform. It hit with a small-sounding crunch. Debris flew into the morning sky, chasing away the distant gulls.
The entire rig shuddered. The whole thing reminded the Harpooner of a vignette he had seen as a child. A poplar tree had been split during a storm and fell across power lines. It hit them, bounced, then hit them again. The lines hung there for a moment before sagging and then ripping from the poles on the left and right. That was what happened here. The platform stood for a moment after the derrick struck. Then, slowly, the steel and concrete sagged where the second blast had weakened them. The platform bent inward. Sheds, cranes, tanks, and even the helicopter began sliding toward the crease. Their weight caused additional strain. The Harpooner could hear the ugly collisions in the distance, see the smoke and shattered pieces of wood and metal fly into the air.
And then it happened. The added weight was too much for the platform to bear. It cracked and dumped everything into the sea. The boat was now too far away for the Harpooner to make everything out. The collapse looked like a waterfall from this distance, especially when the cascade of white and silver debris hit the sea, sending up waves and spray.
As the rig disappeared beyond the horizon, all the Harpooner could see was a large ball of mist hanging in the new day.
He turned away, accepting the congratulations of the team. They were treating him like a football hero, but he felt more like an artist. Using the medium of explosives and a canvas of steel and concrete, the Harpooner had created a perfect destruction.
He went below to wash up. He always needed to wash after creation. It was a symbolic act of completion and of getting ready for the next work. Which would be soon. Very soon.
When the boat reached the docks, the Harpooner told the crew he wanted to go ashore. He told the Iranians he wanted to make certain that the Azerbaijani police had not already learned of the blast. If they had, the police might be checking incoming vessels. They might be looking for possible terrorists and also for eyewitnesses to the explosion.
The men thought that was a good idea.
The Harpooner told them that if he did not come back in five minutes, they should leave the dock and head to the open sea. The Harpooner said that if the police were talking to people, stopping them from leaving the area, he would figure out a way to elude them.
The men agreed. The Harpooner went ashore.
Six minutes later, there was a massive explosion in the harbor. The Harpooner had stuck a timed detonator into one of the sticks of watergel. He had set it and then left it below, under one of the bunks. Evidence from the attack was still on board. It would take a while, but eventually the authorities would find traces of the watergel on the boat and on the rig and realize that the Iranians, aided by a Russian terrorist, had attacked their own operation. The Iranians would dispute that, of course, and tensions would rise even higher. The United States would suspect that the Russians and Iranians were working together to seize the Caspian oil wells. There would be no way to avoid what was coming.
The Harpooner got in the repainted van and drove it from the harbor. There were no police there. Not yet. At this hour, the Baku police force was involved primarily in traffic management and accident investigation. Besides, there was no indication that a boat had attacked the rig or that it had come to Baku. That would come later, when they found the Russian and the Americans had sent over satellite photographs of the region.
The Harpooner headed toward the Old City. There, he drove up Inshaatchilar Prospekti toward the hotels on Bakihanov Kuchasi. Two days before, he had taken a hotel room under an assumed name. Here he was Ivan Ganiev, a telecommunications consultant. It was a name and profession he had chosen with care. If he were ever stopped by customs agents or police, he could explain why he was traveling with high-tech equipment. And being Russian had another advantage, especially here. One that would help him get out of the country when the time came.
He had left clothing, gear, and cash in the room and a do not disturb sign on the door. He would clean himself up, dye his hair, and then take a long nap. When he woke, he would apply a fake mustache, slip colored contact lenses into his eyes, and call a cab to take him to the train station. A cabdriver was always a good hostage in case he was discovered and surrounded. He would use his fake passport to leave the city.
He parked the van in an alley near the hospital. Then he pulled a packet of dental floss from his pocket. He rubbed it deeply between two teeth until his mouth filled with blood. Then he spat on the floor, dashboard, and seat cushion. It was the fastest way to draw blood. It also left no scars, in case anyone decided to stop him and check for wounds. He did not need a lot of blood. Just traces for the forensics people to find. When he was finished with that, he slipped a plastic mircochip in the gas tank. Then he replaced the cap.
When he was finished dressing the van, the Harpooner took the backpack containing the Zed-4 phone and left. When the authorities found the vehicle, they would also find evidence inside tying it to the Iranians in the boat. That would include their fingerprints on the wheel, glove compartment, and handles. They would assume that one or more of the men got away. The blood would suggest that he was injured. The police would waste time looking through hospital records for a possible perpetrator.
The Harpooner would return to Moscow. Then he would leave Russia and permit himself a rest. Possibly a vacation in some country where he had never committed terrorism. Some place where they would not be looking out for him.
Some place where he could sit back and read the newspapers.
Enjoy once again the impact his art had had on the world.
**THIRTY-ONE**
_**Washington, D.C. Monday, 11:11 P.M.**_
Paul Hood was concerned, confused, and tired.
Bob Herbert had just spoken with Stephen Viens of the National Reconnaissance Office. Viens was working late to catch up on paperwork that had collected during his absence. While Viens was there, an NRO satellite had recorded an explosion in the Caspian Sea. He had called Herbert, who wanted to know if anything unusual had happened in the region. Then Herbert called Paul Hood.
"According to our files, the coordinates of the explosion match those of Iran's Majidi-2 oil rig," Herbert said.
"Could it have been an accident?" Hood asked.
"We're checking that now," Herbert said. "We've got some faint radio signals coming from the rig, which means there may be survivors."
"May be?"
"A lot of those rigs have automatic beacons to signal rescue craft in the area," Herbert said. "That may be what we're hearing. The audio keeps breaking up, so we can't tell if it's a recording."
"Understood," Hood said. "Bob, I've got a bad feeling about this. Fenwick goes to the Iranian mission, and then an Iranian rig is attacked."
"I know," Herbert said. "I tried to call him, but there was no answer. I'm wondering if the NSA knew about this attack, and Fenwick took intelligence to the mission in New York."
"If Fenwick had intel, wouldn't Iran have tried to prevent the attack?" Hood asked.
"Not necessarily," Herbert told Hood. "Teheran has been itching for a reason to establish a stronger military presence in the Caspian Sea. An attack by Azerbaijan could give them that reason. It's no different than historians who say that Franklin Roosevelt allowed Pearl Harbor to be attacked so we'd have a reason to get into World War Two."
"But then why all the deception with the president?" Hood asked.
"Plausible deniability?" Herbert replied. "The president has been getting misinformation."
"Yes, but Jack Fenwick would not undertake something of this magnitude on his own," Hood said.
"Why not?" Herbert asked. "Ollie North ran an uberoperation during Iran-Centra—"
"A military officer might have the balls for that but not Jack Fenwick," Hood said. "I had a look at his dossier. The guy is Mr. Support Systems. He's instituted backup systems for backup systems at the NSA. Got congress to jack up the budget fifteen percent for next year. The CIA only got an eight percent bump and we got six."
"Impressive."
"Yeah," Hood said. "And he just doesn't strike me as the kind of guy to take this kind of chance. Not without backup."
"So?" Herbert said. "Maybe he's got it."
_Shit_ , Hood thought. _Maybe he does._
"Think about it," Herbert went on. "He got double the increases everyone else got. Who has that kind of sway with congress? Not President Lawrence, that's for sure. He's not conservative enough for the budget group."
"No, he's not," Hood agreed. "Bob, find out if Matt can get into Fenwick's phone records and calendar. See who he might have talked to and met with over the past few days and weeks."
"Sure," he said. "But it's going to be tough to draw any conclusions from that. The NSA head meets with practically everyone."
"Exactly," Hood said.
"I don't follow."
"If Fenwick were part of a black-ops situation, he would probably meet with his team away from the office. Maybe by seeing who he stopped meeting with, officially, we can figure out who he's been seeing on the sly."
"Nice one, Paul," Herbert said. "I wouldn't have thought of that."
"But that isn't what has me worried," Hood went on. The phone beeped. "Excuse me, Bob. Would you bring Mike up to date on this?"
"Will do," Herbert said.
Hood switched lines. Sergei Orlov was on the other end.
"Paul," Orlov said, "good news. We have your man."
"What do you mean you have him?" Hood asked. The Russian operative was only supposed to keep an eye on him.
"Our operative arrived in time to save him from joining his comrades," Orlov said. "The assassin was dispatched and left in the hospital room. Your man was taken from the hospital to another location. He is there now."
"General, I don't know what to say," Hood told him. "Thank you."
"Thank you is good enough," Orlov said. "But what do we do now? Can he help us get the Harpooner?"
"I hope so," Hood told him. "The Harpooner must still be there. Otherwise, he would not have had to draw these people out and assassinate them. General, did you hear what happened in the Caspian?"
"Yes," Orlov said. "An Iranian oil rig was destroyed. The Azerbaijanis are probably going to be blamed, whether they did it or not. Do you know anything more about it?"
"Not yet," Hood said. "But the operative you saved might. If the Harpooner's behind this attack, we need to know. Can you arrange for the American agent to call me here?"
"Yes," Orlov said.
Hood thanked him and said he would wait by the phone.
Orlov was correct. Suspicion would fall on Azerbaijan. They were the ones who disputed Iran's presence in that region of the sea. They were the ones who had the most to gain. But the Harpooner had done most of his work for Middle Eastern nations. What if Azerbaijan wasn't behind the attack? What if another nation was trying to make it seem that way?
Hood got back on the phone with Herbert. He also patched in Mike Rodgers and briefed them both. When he was finished, there was a short silence.
"Frankly, I'm stumped," Herbert said. "We need more intel."
"I agree," Hood said. "But we may have more intel than we think."
"What do you mean?" Herbert asked.
"I mean we've got the NSA working with Iran," Hood said. "We have a president who was kept out of the loop by the NSA. We have a terrorist who works with Iran taking out CIA agents in Azerbaijan. We have an attack on an Iranian oil installation off the coast of Azerbaijan. There's a lot of information there. Maybe we're not putting it together in the right way."
"Paul, do we know who in the CIA first found out the Harpooner was in Baku?" Rodgers asked.
"No," Hood said. "Good point."
"I'll get someone to find that out ASAP," Herbert said.
Hood and Rodgers waited while Herbert made the call. Hood sat there trying to make sense of the facts, but it still was not coming together. _Concerned, confused, and tired._ It was a bad combination, especially for a man in his forties. He used to be able to pull all-nighters without a problem. Not anymore.
Herbert got back on. "I've got someone calling the director's office, Code Red-One," he said. "We'll have the information soon."
Code Red-One signified an imminent emergency to national interest. Despite the competitiveness between the agencies, CR1s were generally not denied.
"Thanks," Hood said.
"Paul, do you know the story about the Man Who Never Was?" Rodgers asked.
"The World War Two story? I read the book in high school," Hood said. "He was part of the disinformation campaign during World War Two."
"Correct," Rodgers said. "A British intelligence group took the body of a homeless man, created a false identity for it, and planted papers on the body that said the Allies would invade Greece, not Sicily. The body was left where the Germans would find it. This helped divert Axis forces from Sicily. I mention this because a key player in the operation was a British general named Howard Tower. He was key in the sense that he was also fed misinformation."
"For what reason?" Hood asked.
"General Tower's communiques were intercepted by the Germans," Rodgers said. "British Intelligence saw to that."
"I'm missing something here," Herbert said. "Why are we talking about World War Two?"
"When Tower learned what had happened, he put a gun barrel in his ear and pulled the trigger," Rodgers said.
"Because he was used?" Hood asked.
"No," Rodgers said, "because he thought he'd screwed up."
"I'm still not getting this," Herbert admitted.
"Paul, you said the president was pretty upset when you spoke with him," Rodgers went on. "And when you met with the First Lady, she described a man who sounded like he was having a breakdown."
"Right," Hood said.
"That may not mean anything," Herbert said. "He's president of the United States. The job has a way of aging people."
"Hold on, Bob. Mike may be onto something," Hood said. There was something gnawing at Hood's stomach. Something that was getting worse the more he thought about it. "The president did not look tired when I saw him. He looked disturbed."
"I'm not surprised," Herbert said. "He was being kept out of the loop and made an apparent faux pas about the UN. He was embarrassed."
"But there's another component to this," Hood told him. "There's the cumulative psychological impact of disinformation. What if plausible deniability and bureaucratic confusion aren't the reasons the president was misled? What if there's another reason?"
"Such as?" Herbert asked.
"What if disinformation isn't the end but the means?" Hood said. "What if someone is trying to convince Lawrence that he's losing his grip?"
"You mean, what if someone is trying to gaslight the president of the United States?" Herbert declared.
"Yes," Hood replied.
"Well, it's going to take a lot of convincing before I buy that," Herbert said. "For one thing, anyone who tried that would never get away with it. There are too many people around the president—"
"Bob, we already decided that this is something Jack Fenwick would not, probably could not, do on his own," Hood said.
"Yes, but to make it work, he'd need a small army of people who were very close to the president," Herbert said.
"Who?" Hood asked. "The chief of staff?"
"For one," Herbert said. "He's privy to most of the same briefings the president receives."
"Okay," Hood said. "Gable's already on my list of unreliables. Who else? Who would be absolutely necessary for a plan like this to work?"
Before Herbert could answer, his phone beeped. He answered the call and was back in less than a minute.
"Don't tell me, 'I told you so,' " Herbert said.
"Why?" Hood asked.
"A high-level official at the CIA in Washington got the intel about the Harpooner from the NSA," Herbert told them. "The NSA didn't have anyone in Baku, so they notified the CIA. The CIA sent David Battat."
"Whom the Harpooner knew just where to find," Rodgers said. "Instead of killing him, the Harpooner poisoned him somehow. And then Battat was used to bring out Moore and Thomas at the hospital."
"Apparently," Herbert said.
"Paul, you asked a question a moment ago," Rodgers said. "You wanted to know who else would be necessary for a psy-ops maneuver to work against the president. That's a good question, but it's not the first one we need to answer."
"No?" Hood said. "What is?"
"Who would benefit the most from the mental incapacitation of the president?" Rodgers asked. "And at the same time, who would be in a perfect position to help make some of the disinformation happen?"
Hood's stomach was growling now. The answer was obvious.
The vice president of the United States.
**THIRTY-TWO**
_**Washington, D.C. Monday, 11:24 P.M.**_
Vice President Charles Cotten was in the ground-floor sitting room of the vice presidential residence. The mansion was located on the sprawling Massachusetts Avenue grounds of the United States Naval Observatory. It was a twenty-minute drive from here to the vice president's two offices: one in the White House and the other in the neighboring Old Executive Office Building. It was just a short walk from the mansion to the National Cathedral. Lately, Cotten had been spending more time than usual at the cathedral.
Praying.
An aide knocked and entered. The woman told the vice president that his car was ready. The vice president thanked her and rose from the leather armchair. He entered the dark, wood-paneled hallway and headed toward the front door. Upstairs, Cotten's wife and children were asleep.
_My wife and children_. They were words Cotten never thought would be part of his life. When he was a senator from New York, Cotten had been the ultimate lady's man. A new, gorgeous date to every function. The press referred to these younger women as "Cotten candy." There were regular jokes about what went on below the Cotten belt. Then he met Marsha Arnell at a Museum of Modern Art fund-raiser in Manhattan, and everything changed. Marsha was twenty-seven, eleven years his junior. She was a painter and an art historian. She was telling a group of guests about late—twentieth-century art and how the work of commercial artists like Frank Frazetta, James Bama, and Rich Corben defined a new American vision: the power of the human form and face blended with landscapes from dream and fantasy. Cotten was hypnotized by the young woman's voice, her ideas, and her vital and optimistic view of America.
They were married four months later.
For nearly ten years, Marsha and their twin girls had been the foundation of Charles Cotten's life. They were his focus, his heart, and their future was never far from his thoughts.
They were the reason the vice president had conceived of this plan. To preserve America for his family.
The fact was, the United States was at risk. Not just from terrorist attacks, though more and more those were becoming a very real threat. The danger facing the United States was that it was on the verge of becoming irrelevant. Our military could destroy the world many times over. But other nations knew that we would never do that, so they did not fear us. Our economy was relatively strong. But so were the economies of many other nations and alliances. The Eurodollar was strong, and the new South American League and their SAL currency was growing in power and influence. Central America and Mexico were talking about a new confederacy. Canada was being tempted to join the European economy. Those unions, those nations, did not face the kind of suspicion and resentment that greeted America the world over. The reason? America was a giant everyone wanted to see brought down. Not destroyed; they needed us too much for international policing. They simply wanted us humbled and humiliated. We were a meddling thug to our enemies and an overbearing big brother to our supposed allies.
These were not concerns that bothered other nations during times of international depression or world war. It was all right to invade France to free the French of Hitler. But it was not okay to fly over France to bomb Libya, the home of a different despot. It was all right to maintain a military presence in Saudi Arabia to protect the nation from Saddam Hussein. But it was not all right to fly jets from Riyadh to protect American troops in the region.
We were not respected, and we were not feared. That had to change. And it had to change long before Michael Lawrence was scheduled to leave the White House in three years. That would be too late to act.
The problem had not been caused by Michael Lawrence. He was simply the latest bearer of the torch of arrogant isolationism. When he was in the Senate, Cotten had felt that there needed to be a United States that was better integrated with the world. The one that Teddy Roosevelt had described. The one that carried a big stick and was not afraid to use it. But also one that knew how to speak softly. An America that knew how to use and exert diplomacy and economic pressure. One that had the resolve to use quiet assassination and blackmail instead of mounting very public and unpopular miniwars.
When the senator was tapped to share a ticket with presidential candidate Michael Lawrence, Cotten accepted. The public liked Lawrence's "I'm for the people" slogan and style, his perception as a man who had come back from the political wilderness to serve them. But he had wanted to balance his relatively up-front and independent manner with someone who knew how to work the back rooms of Congress and the corridors of power abroad.
Cotten left the mansion and slid into the car. The driver shut the door for him. They rolled into the dark, still night. Cotten's soul was on fire. He was not going to enjoy what he and his allies were about to do. He remembered when he had first approached them and others individually. Seemingly casual remarks were dropped. If they were ignored, he let the subject drop. If not, he pursued it with more pointed remarks. Cotten realized that was what it must be like for a married man to ask a woman to have an affair. Go too far with the wrong individual, and everything could be lost.
Each man had become involved for the same reason: patriotism. The creation of an America that led the world community rather than reacted to it. An America that rewarded peace with prosperity and punished warmongers not with a public pummeling and credibility but with quiet, lonely death. Lawrence was not willing to cross the line from legal war to illegal murder, even though lives would be saved. But the dawn of the twenty-first century was not a time for warfare. It bred short-term misery and long-term hatred. The world was becoming too small, too crowded for bombs. As distasteful as this was, a change had to come. For the nation and for the sake of its children. For the sake of his children.
The car moved swiftly through the empty streets. Washington was always so deserted at night. Only the spies and plotters were afoot. It seemed strange to think of himself in that capacity. He had always been a straight shooter. If you felt passionately about something, you spoke your mind. If you didn't feel passionately, then it probably was not worth doing. But this was different. This operation had to be kept very quiet. Kept only among those who were actively involved in its planning and execution.
Now this was it, Cotten thought. The last leg of the operation. According to the president's staff, announcing a UN intelligence initiative that did not exist had seriously rattled Lawrence. It had shaken him more than the other canards Fenwick and Gable had fed him and subsequently denied—usually during a cabinet session or meeting in the Oval Office.
"No, Mr. President," Cotten would say softly, seemingly embarrassed for the confusion of the president, "there was never a Pentagon report that Russia and China exchanged artillery fire over the Amur River. Sir, we had not heard that the FBI director had threatened to resign. When did this happen? Mr. President, don't you recall? We had agreed that Mr. Fenwick would share this new intelligence with Iran."
The question of sharing intelligence with Iran had been important to the final stage of the operation. Jack Fenwick had told the Iranian ambassador that according to United States intelligence sources, an attack would come from Azerbaijan. They weren't sure what the target would be, but it would probably be a terrorist attack in the heart of Teheran. Fenwick had assured Iran that if they retaliated, the United States would stay out of it. This nation wanted to nurture closer ties with the Islamic Republic of Iran, not stand in the way of its self-defense.
Lawrence, of course, would be pushed to behave in a less accommodating manner. And when he realized where his confused perceptions had taken the nation, he would be forced to resign.
The fact that Lawrence had known nothing about the meeting was irrelevant. At tonight's meeting with the so-called "Eyes Only Group"—Gable, Fenwick, and the vice president—the men would convince the president that he had been kept informed. They would show him memos that he had seen and signed. They would show him the calendar his secretary kept on the computer. The appointment had been added after she left for the day. Then they would jump right into the current crisis. They would trust and the president would lead. By morning, Michael Lawrence would be publicly committed to a path of confrontation with two of the most volatile nations on earth.
The following morning, with the help of unnamed NSA sources, the _Washington Post_ would run a front-page, above-the-fold article about the president's mental health. Though the newspaper piece would be hooked to the UN fiasco, it would also contain exclusive details about some of the president's increasingly dramatic and fully documented lapses. The nation would not tolerate instability from the commander-in-chief. Especially as he was about to send the nation to war.
Things would happen very quickly after that. There was no constitutional provision for the president to take a leave of absence. And there was no short-term cure for mental illness. Lawrence would be forced to resign, if not by public pressure then by act of congress. Cotten would become president. The United States military would immediately back down in the Caspian Sea to avoid a confrontation with Iran and Russia. Instead, through intelligence operations, they would prove that Iran had masterminded the entire operation in the first place. Teheran would protest, but the government's credibility would be seriously compromised. Then, through diplomacy, the United States would find ways to encourage moderates in Iran to seize more power. Meanwhile, spared a pounding from Iran and Russia, Azerbaijan would be in America's debt.
After the clouds of war drifted away, President Cotten would make certain of something else. That Azerbaijan and America shared in the oil reserves of the Caspian Sea. The Middle East would never again hold the United States hostage. Not in their embassies nor at the gas pump.
With order restored and American influence and credibility at its peak, President Charles Cotten would reach out to the nations of the world. They would be invited to join us in a permanent peace and prosperity. When their people experienced freedom and economic reward for the first time, they would cast those governments out. Eventually, even China would follow suit. They had to. People were greedy, and the old-line Communists would not live forever. If the United States stopped provoking them, providing the government with a public enemy, Beijing would weaken and evolve.
This was the world that Charles Cotten wanted for America. It was the world he wanted for his own children. He had thought about it for years. He had worked to achieve it. He had prayed for it.
And very soon, he would have it.
**THIRTY-THREE**
_**Baku, Azerbaijan Tuesday, 8:09 A.M.**_
David Battat was lying on a hard twin bed in the small, sparsely furnished studio apartment. There was a window to his left. Though the blinds were drawn, the room brightened as light leaked through the slats.
Battat was shivering but alert. His abductor, hostess, or savior—he had not yet decided which—was in the kitchenette off to the right. She had been making eggs, sausage, and tea when the phone rang.
Battat hoped the call was brief. The food smelled good, but the thought of tea was even better. He needed to warm himself inside. Do something to stop the trembling. He felt as though he had the flu. He was weak and everything he saw or heard seemed dreamlike. But his head and chest were also very tight. More than from any sickness he could remember. Hopefully, once he had tea and something to eat, he would be able to focus a little better, try to understand what had happened back at the hospital.
The woman walked over to the bed. She was carrying the phone. She stood about five-foot-nine and had a lean, dark face framed by thick, black, shoulder-length hair. Her cheekbones were pronounced, and her eyes were blue. Battat was willing to bet there was Lithuanian blood in her. She handed the receiver to Battat.
"There is someone who wishes to speak with you," she said in thickly accented English.
"Thank you," said Battat. His own voice was a weak croak. He accepted the cordless phone. He did not bother to ask her who it was. He would find out soon enough. "Hello?"
"David Battat?" said the caller.
"Yes—"
"David, this is Paul Hood, the director of Op-Center."
"Paul Hood?" Battat was confused. Op-Center found him here and was calling him now to ask about—that? "Sir, I'm sorry about what happened," Battat said, "but I didn't know that Annabelle Hampton was working with—"
"This isn't about the United Nations siege," Hood interrupted. "David, listen to me. We have reason to believe that the NSA set you and your colleagues up."
It took a moment for Battat to process what Hood had said. "They set us up to be murdered? Why?"
"I can't tell you that now," Hood replied. "What's important is that for the present, you're out of danger."
The young woman walked over with a cup of tea. She set it on the night table beside the bed. Battat used an elbow to drag himself into a sitting position. She helped him by putting strong hands under his arm and literally lifting him from the bed.
"What I need to know is this," Hood went on. "If we can locate the Harpooner, do you feel up to helping us take him down?"
"If there's a way for me to get the Harpooner, I'm up for it," Battat said. Just the thought of that energized him.
"Good," Hood told him. "We're working with a Russian intelligence group on this. I don't know when we'll have additional information. But when we do, I'll let you and your new partner know."
Battat looked over at the young woman. She was standing in the kitchenette spooning eggs onto two plates. The last time he was in the field, Russians were the enemy. It was a strange business they were in.
"Before I go, is there anything else you can tell us about the Harpooner?" Hood asked. "Anything you might have seen or heard while you were looking for him? Anything Moore or Thomas might have said?"
"No," Battat said. He took a sip of tea. It was stronger than he was used to. It was like a shot of adrenaline. "All I know is that someone put me in a choke hold from behind. The next thing I knew, I was on the ground. As for Moore and Thomas, they were as mystified as I was."
"Because—?"
"The Harpooner had let me live," Battat said.
"Assuming it was the Harpooner," Hood said. "Listen. Use the time you have to rest. We don't know where the Harpooner may turn up or how much time you may have to get to him. But we need you to be ready to move out."
"I'll be ready," Battat said.
Hood thanked him and hung up. Battat placed the phone on the night table. Then he took another swallow of tea. He still felt weak, but he was trembling a little less than before.
The young woman walked over with a plate for him. Battat watched her as she set the plate on his legs and placed a cloth napkin and utensils on the night table. She looked tired.
"My name is David Battat," he said.
"I know," she said.
"And you are—?" he pressed.
"In Baku, I am Odette Kolker," she said. There was finality in the young woman's voice. It told him two things. First, that she was definitely not an Azerbaijani recruited by the Russians. And second, that Battat would not be getting her real name. Not from her, anyway.
"I'm pleased to meet you," Battat said, extending his hand. "I'm also extremely grateful for everything you've done."
"You're welcome," she said.
The young woman shook Battat's hand firmly but perfunctorily. As she did, Battat noticed several small bloodstains on the sleeve of her off-white police blouse. There were no lacerations on her hand or forearm. The blood did not appear to be hers.
"Are you really a policewoman?" Battat asked.
"Yes," she replied.
"Were you working the night shift?" he asked.
"No," she replied. "I was called in to do this." She smiled slightly. "And I cannot collect overtime for it."
Battat sipped more tea and smiled back. "I'm sorry they had to wake you." He moved the plate to the night table and started to throw off the cover. "I probably shouldn't be taking your bed—"
"No, it's all right," she said. "I'm expected on duty in less than an hour. Besides, I'm accustomed to having unexpected guests."
"A hazard of the business," he said.
"Yes," Odette observed. "Now, if you'll excuse me, I'm going to eat. You should do the same. Eat and then rest."
"I will," Battat promised.
"Do you need salt or anything else?"
"No thank you," he said.
Odette turned and walked slowly toward the kitchenette.
Less than an hour ago, she had killed a man. Now she was serving Battat breakfast. This was a strange business. A very strange business indeed.
**THIRTY-FOUR**
_**Washington, D.C. Tuesday, 12:10 A.M.**_
"Hello, Paul."
Sharon's voice was thick and cold on the other end of the phone. Hood glanced at the clock on his computer. "Hi," he said warily. "Is everything okay?"
"Not really," she replied.
"I just got back from the hospital."
"What happened?"
"The short version," she said, "is that Harleigh freaked out about ninety minutes ago. I called an ambulance—I didn't know what else to do."
"You did the right thing," Hood said. "How is she?"
"Dr. Basralian sedated her, and she's sleeping now," Sharon went on.
"What does he think is wrong?" Hood asked. "Is it physical—?"
"He isn't sure," she said. "They're going to run tests in the morning. The doctor said that sometimes a traumatic event can have physical repercussions. It can affect the thyroid, cause it to get hyper, or create a surplus of adrenaline. Anyway, I didn't call so you'd drop what you're doing and go to see her. I just wanted you to know."
"Thank you," Hood said. "I'll still get over as soon as I can."
"No need for that," Sharon told him. "Everything's quiet. I'll let you know if there's a change."
"All right," Hood said. "If that's what you want."
"I do. Just some down-time. Tell me, Paul. Is there a problem?" Sharon asked.
"With what?"
"The world," Sharon said.
"Always," Hood replied.
"I tried the motel first," Sharon told him. "When you weren't there, I figured you must be putting out a fire somewhere."
Hood was not exactly sure how to take that remark. He tried not to read anything into it.
"There's a problem in the Middle East," Hood said. "Could be a bad one."
"Then I won't keep you," Sharon said. "Just don't kill yourself, Paul. You're not a kid anymore. You need sleep. And the kids need you."
"I'll take care of myself," he promised.
Sharon hung up. When Hood and his wife were together, Sharon used to be frustrated and angry whenever he worked long hours. Now that the two of them were apart, she was calm and concerned. Or maybe she was holding it all together for Harleigh's sake. Whatever the reason, it was a sad, sad joke being played on the Hood family.
But Hood did not have time to consider the injustice of it all or even the condition of his daughter. The phone rang a moment after he hung up. The call was from another concerned wife.
The president's.
**THIRTY-FIVE**
_**Saint Petersburg, Russia Tuesday, 8:30 A.M.**_
General Orlov was proud that his operative had been able to save the American. Proud, but not surprised.
Odette—Natalia Basov—had been working with him for three years. The thirty-two-year-old was a former decryption expert who had begun her career with the GRU, Soviet military intelligence. Her husband Viktor was an officer in the Spetsnaz, the Russian special forces. When Viktor was killed on a mission in Chechnya, Basov became deeply depressed. She wanted to get out from behind a desk. Because the GRU was being dismantled and its components downsized, Basov was sent to see Orlov. Orlov was happy to put her in the field. Not only was Basov skilled in electronic intelligence, her husband had taught her the self-defense techniques of the _systema,_ the lethal martial arts style of the Spetsnaz. Orlov himself had studied the basics as a way of staying in shape. The _systema_ did not rely on practiced moves or on physical strength. It taught that during an assault, your own defensive motion dictated what the counterattack should be. If you were struck on the right side of the chest, you instinctively turned the right side away to avoid the blow. As a result, your left side automatically came forward. Thus, your attack would be with the left arm. And it would not be a single blow. It would be a trinity. Perhaps a fist to the chin, an elbow to the jaw, and a swipe with the back of the hand, all in quick succession. While that was going on, you were positioning yourself to unleash the next trinity. Typically, an opponent did not get more than a first chance to strike. Multiple opponents were too busy avoiding their falling comrades to move in.
Basov had mastered the form well. And she had proven to be a valuable asset in Azerbaijan. Orlov's people had created a false identity for her, and she had obtained a job with the police force. That put her in a job to watch and question people, other officers, guards, and night watchmen at plants and military bases. To learn what was happening in Baku's corridors of power and in the military. Being a beautiful woman made men more inclined to talk to her, especially in bars. And underestimate her.
Basov said that she and her guest were safe, but they were not what bothered Orlov right now. What concerned him was finding the Harpooner. Basov had told Orlov that the Baku police radio was reporting an explosion in the harbor. A boat had blown up, killing everyone on board. Orlov was willing to bet that the boat had belonged to the Harpooner. That was his way—to destroy all the evidence along with some or all of his coworkers. The dead men would probably be blamed for the rig attack. Orlov wondered who they were. Azerbaijanis? Iraqis? Russians? There were any number of people he could have recruited for a job like that. Just as long as they did not know what usually happened to his employees.
Most of Orlov's staff began arriving at half-past eight. The general had left e-mail for the two key members of his intelligence team, Boris and Piotr, to come and see him as soon as possible. If the Harpooner had been responsible for the attack in the Caspian, he probably would not attempt to leave Baku immediately. In the past, the Harpooner apparently waited a day or two after an attack. And when he finally moved, he often passed through Moscow. No one knew why. Unfortunately, by the time authorities learned he was in the city, he had vanished. General Orlov did not want that to happen again. The question was how to find him. And Paul Hood might have unwittingly given them a clue.
Boris Grosky was a sullen, gray-haired intelligence veteran who missed the Cold War. Piotr Korsov was an eager newcomer who had studied at Technion in Haifa, Israel. He was openly thrilled to be working in a field he loved and for a man who had helped pioneer space travel. The men entered the windowless office within a minute of one another. They sat on the couch across from Orlov's desk, Boris drinking tea and Korsov sitting with a laptop on his knees.
Orlov briefed the men. Grosky became noticeably more interested when the general mentioned that the NSA and CIA might somehow be involved in the Caspian operation.
"What I want to know is this," Orlov said. "We have eavesdropped on cell phone communications between American intelligence operatives before. We've gotten through many of their secure lines."
"We've gotten through most of them," Grosky pointed out.
"They try to keep you out by altering the signal from second to second," Korsov said. "The shifts are all within just a few megahertz in the superhigh frequency. We've learned how to ride most of the shifts."
"The difficult part is decoding the messages, which are scrambled electronically," Grosky added. "The American agencies use very complex codes. Our computers aren't always up to the task of decrypting the calls."
"Do the same callers usually use the same signals, the same patterns?" Orlov asked Korsov.
"Usually," Korsov told him. "Otherwise, there would be audio crossover. Callers would keep bumping into one another."
"Do we keep records of the calls?" Orlov asked.
"The conversations?" Grosky asked. "Yes. We keep working on them, trying to decode—"
"I mean the signals," Orlov interrupted.
"Absolutely," said Grosky. "We send them up to the Laika so it can keep a lookout for those signals."
The Laika was the Russian Op-Center's sentry satellite. Named for the pioneering Soviet space dog, the Laika was in a high geostationary orbit over Washington, D.C. It could intercept signals from the United States, all of Europe, and parts of Asia.
"So, if the Harpooner spoke with an intelligence unit in Washington, we might have picked up the signal if not the content," Orlov said.
"That's right," said Kosov.
"Very good," said Orlov. "Go to the computer records for the past two weeks. Look up communiques between Azerbaijan and the National Security Agency in Washington. Get me all the information you have."
"Even if we haven't decrypted them," said Kosov.
"Yes," Orlov replied. "I want to know exactly where the Harpooner or his people might have been calling from."
"When you know that, what will you do?" Grosky asked.
"I'll call the American Op-Center and ask them to go through any satellite imaging they have for the region," Orlov said. "The Harpooner had to move explosives and personnel into position. If we can pinpoint his location, there may be a photographic record of it—"
"And clues to where he might be," Grosky said. Orlov nodded.
"We'll have that information for you as soon as possible," Kosov said eagerly. "It would be a coup if we could catch that monster."
"It would be," Orlov agreed.
The men left. Orlov put in a call to Paul Hood to bring him up to date.
Catching the Harpooner would be a highlight of his career. But more than that, he wondered if this close cooperation between Op-Centers could become increasingly routine. If the trust and sharing could lead to less suspicion and greater international security.
That would be the real coup.
**THIRTY-SIX**
_**Washington, D.C. Tuesday, 12:30 A.M.**_
"Paul, I'm glad I found you," Megan Lawrence said. "I think you should come here. There's something going on."
The First Lady's voice was steady when she got on the line, but Hood knew her well enough to know that it was Megan's "I have to be strong" voice. He had heard that voice during the campaign when there were hard questions from the press about an abortion she had had before she met the president. As she had years before, Megan was pulling this strength from deep inside. She would crash only when it was safe to do so.
"Talk to me," Hood said. He was drawing on his own emotional and psychological reserves to deal with the First Lady's problem. The call from Sharon had shaken him.
"We were just getting into bed when Michael received a call from Jack Fenwick," Megan said. "Whatever Fenwick said rattled my husband very much. His voice was calm while they talked and then afterward, but I watched this look come over him."
"What kind of look?" Hood asked.
"It's difficult to describe," she said.
"Was it guarded, startled, doubtful?" Hood asked.
"All of that," Megan replied.
Hood understood. That was what he saw in the Oval Office. "Where is the president now?" he asked.
"He went down to meet with Fenwick, the vice president, and Red Gable," Megan said.
"Did he say what the meeting was about?" Hood asked.
"No. But he told me not to wait up," she said.
It was probably about the Caspian situation. A small, nonconspiratorial part of Hood said that this might not be anything to worry about. On the other hand, the president was meeting with people who had fed him misinformation before. Perhaps that was what Megan had seen in her husband's expression. The fear that it might be happening again.
"Paul, whatever is going on, I think Michael needs to have friends around him," Megan said. "He should be with people he knows well and can trust. Not just policy advisers."
Hood's aide Stef Van Cleef beeped. She said there was a call from General Orlov. Hood told her to apologize to the general for the delay. He would take it in just a moment.
"Megan, I don't disagree," Hood said. "But I can't just invite myself to a meeting in the Oval Office—"
"You have the security clearance," she said.
"To get into the West Wing, not the Oval Office," he reminded her. Hood stopped. His eyes were on the beeping light on the phone. Maybe he would not have to get himself invited.
"Paul?"
"I'm here," Hood said. "Megan, listen to me. I'm going to take a call, and then I'm going to the White House. I'll call your private line later and let you know how things are going."
"All right," Megan said. "Thank you."
Hood hung up and took the call from Orlov. The Russian general briefed him on the plan to try to locate the Harpooner. Orlov also told him about the destruction of the boat in the harbor. He suspected that Azerbaijani officials would find bodies in the water, either the Harpooner's hirelings or people who were abducted to impersonate hirelings.
Hood thanked Orlov and informed the general that he would have Op-Center's full cooperation. Hood indicated that he would be away from the office for a while and that he should contact Mike Rodgers with any new information. When Hood hung up, he conferenced Herbert and Rodgers on his cell phone. He updated them as he hurried to the parking lot.
"Do you want me to let the president know you're coming?" Rodgers asked him.
"No," Hood said. "I don't want to give Fenwick a reason to end the meeting early."
"But you're also giving Fenwick and his people more time to act," Rodgers pointed out.
"We have to take that chance," Hood said. "If Fenwick and Gable are launching some kind of endgame, I want to give them time to expose it. Maybe we can catch them in the act."
"I still think it's risky," Rodgers said. "Fenwick will press the president to act before other advisers can be consulted."
"That could be why this was timed the way it was," Herbert pointed out. "If there's a plot of some kind, it was designed to happen when it was the middle of the night here."
"If this is tied to the Caspian situation, the president will have to act quickly," Rodgers went on.
"Mike, Bob, I don't disagree with what you're saying," Hood told them. "I also don't want to give these bastards a chance to discredit anything I may have to say before I get there."
"That's a tough call," Herbert said. "Real tough. You don't have a lot of information on the situation overseas."
"I know," Hood said. "Hopefully, we'll have more intel before too long."
"I'll be praying for you," Herbert said. "And if that doesn't work, I'll be checking other sources."
"Thanks," Hood said. "I'll be in touch."
Hood sped through the deserted streets toward the nation's capital. There was a can of Coke in the glove compartment. Hood kept it there for emergencies. He grabbed the can and popped the tab. He really needed the caffeine. Even warm, the cola felt good going down.
Rodgers was correct. Hood was taking a chance. But Hood had warned the president about Fenwick. The rerouted phone call, the visit to the Iranian mission, failure to communicate with Senator Fox and the COIC. Hopefully, Lawrence would look very carefully at whatever data was being presented to him. The president might also take the time to run the information through Op-Center, just to make sure it was valid.
But Hood's hopes did not change the fact that the president was under an unusual amount of stress. There was only one way to be certain what Michael Lawrence would do. That was for Hood to get there with new intelligence. And while Hood was there, to help the president sift through whatever information Fenwick was presenting to him.
And there was one more thing Hood had to do. Pray that Mike Rodgers was not right.
That there was still time.
**THIRTY-SEVEN**
_**Baku, Azerbaijan Tuesday, 9:01 A.M.**_
Maurice Charles settled into his small room at the Hyatt. The room had a queen-sized bed and a tall cabinet that held the TV and minibar. There was a desk to the left of them and a night table on either side of the bed. An armchair was tucked into a corner opposite the desk. There was very little room, which was fine with Charles. He did not like suites. There was too much open space. Too many places for people to hide.
The first thing Charles did was to tie a nylon rope to one of the legs of the desk. It was located near the window. The room was on the third floor of the ten-story hotel. If Charles were cornered there for any reason, the police would find it difficult to climb from the ground or rappel from the roof without making noise. That left only the door as a means of getting in. And he was prepared to deal with that. He carried cans of shaving cream that were actually filled with highly flammable liquid methanol. Spilled under the doorway and set aflame, it burned hot and fast and drove people back. That would give Charles time to shoot anyone who was waiting for him outside the window, then use the rope to climb out. Methanol was also a fatal poison. The liquid's fumes were so potent that even brief exposure to the vapors could cause blindness.
Charles turned on the light beside the bed and drew the heavy drapes. Next, he picked the locks between his room and the adjoining room. That was another route of escape in case he needed it. Then he pulled over the desk chair. He braced the back of the wooden chair under the knob of the door between his room and the next. He would be able to remove the chair quickly to escape. But if anyone on the other side tried the door, they would think it was locked.
The security arrangements took under a half hour. When they were finished, Charles sat on the bed. He went to his luggage and took out his .45. He placed it on the floor beside the bed. He pulled a Swiss army knife from his pocket and lay it on the night table. He also brought over a bag of several stuffed animals he had bought when he first came to Baku. All of the animals had costumes. If Charles were ever questioned, the plush toys were for his daughter. There were photos of a young girl in his wallet. It was not his daughter, but that did not matter. Then he opened the Zed-4. There was one last call to make.
The call was to the abandoned van. The microchip he had placed in the gas tank was a remote detonator. It had been nicknamed a Kamikaze Cell Phone by its Taiwanese inventor. The KCP had no function other than to pick up the signal, do its job, and then die. This particular KCP had been programmed to heat to 145 degrees Fahrenheit when triggered. Some chips could be programmed to emit high-pitched sounds to interfere with electronic signals or even confuse bloodhounds. Other chips could be used to create magnetic bursts that would cause radar or navigational tools to go haywire.
This chip would melt and leave no trace of itself. It would also set the gas tank afire. The police and fire department would be forced to respond at once to calls about a burning van. They would arrive in time to save some of the vehicle along with what little evidence Charles had left for them to find. That included the traces of Charles's blood. The heat of the fire would cause the water content of the blood to evaporate, leaving clear stains on the metal door handle, glove compartment knob, and other sections of the van that had not burned. The police would conclude that the wounded terrorist had tried to destroy the van and the evidence before leaving. They would assume that their quick response had enabled them to save what they were not supposed to see.
Charles punched in the number of the KCP. He waited while his signal traveled twenty-five miles into space and bounced back to a street three blocks away. There were two short clicks and then the dial tone returned. That meant the call had been completed. The chip had been designed to disconnect from the Zed-4 as it began to heat up.
Charles hung up. He put everything into his backpack except for the .45. As he did, he heard sirens. They stopped exactly where they were supposed to.
By the burning van.
Comforted by the unparalleled feeling of a job well done, Maurice Charles made the final preparations for his stay. He removed one of the pillows from the bed and put it on the floor between the bed and the window, directly in front of the nightstand. Then he lay down and looked to his right, toward the bed. The hem of the bedspread reached nearly to the floor. Beneath and beyond the bed, he could see the front door. If for some reason anyone came in, Charles would see their feet. That was all he had to see to stop them.
Charles kept his clothes and shoes on in case he had to leave in a hurry, but they did not distract him. Nothing did now. This was the time he enjoyed most. When he had earned his rest and his pay.
Soon, even the sound of the police and fire sirens did not penetrate his deep, rewarding sleep.
**THIRTY-EIGHT**
_**Saint Petersburg, Russia Tuesday, 9:31 A.M**_
At 9:22 A.M. Piotr Korsov e-mailed General Orlov a brief data file. The file contained a list of the secure calls that had been intercepted between Azerbaijan and Washington during the past few weeks. Most of those calls had been between the American embassy and either the CIA or the NSA. The Russian Op-Center had been unable to decrypt any of the conversations, but Orlov was able to scratch them off his list. Those calls were pretty much routine and not likely suspects for calls made by the Harpooner.
Over the past few days, there had also been calls to the NSA from Gobustan, a village to the south of Baku. They were all made before the attack on the oil rig. The calls from the embassy to the United States had a slightly different bandwith from the Gobustan calls. That meant the calls were made from different secure phones. In a note attached to the file, Korsov said he was watching for new calls made from either line.
Orlov was not very hopeful. The Harpooner probably would not signal his allies to tell them he had been successful. Whoever he was in league with would hear about that from their own intelligence sources.
The very fact that a secure satellite uplink had played any part in this business was personally disturbing to Orlov. That was the kind of technology his space flights had helped to pioneer—satellite communications. The fact that they were being so expertly abused by terrorists like the Harpooner made him wonder if the technology should have been developed at all. It was the same argument people had made for and against splitting the atom. It had produced plentiful and relatively clean atomic power, but it had also bred the atomic bomb. But Orlov had not had a hand in that work. Just in this.
Then again, Orlov thought, as Boris Pasternak wrote in one of his favorite novels, _Doctor Zhivago_ , "I don't like people who have never fallen or stumbled. Their virtue is lifeless and it isn't of much value. Life hasn't revealed its beauty to them." Progress had to allow monsters like the Harpooner to surface. That was how it showed the creators where the flaws were.
Orlov had just finished reviewing the material when his private internal line beeped. It was Korsov.
"We picked up a ping," Korsov said excitedly.
"What kind of ping?" Orlov asked. A _ping_ was how his intelligence officers described any kind of electronic communication.
"The same one we recorded as having been sent from Gobustan," Korsov replied.
"Was the call made from Gobustan?"
"No," Korsov replied. "It was made from Baku to a site very close by. A site that was also in Baku."
"How close?" Orlov asked.
"The caller and receiver were less than a quarter mile," Korsov told him. "We can't measure distances less than that."
"Maybe the Harpooner was calling accomplices who have another secure line," Orlov suggested.
"I don't think so," Korsov told him. "The phone call only lasted three seconds. As far as we can tell there was no verbal communication."
"What was sent?"
"Just an empty signal," Krosov said. "We've fed cartographical data into the computer. Grosky is overlaying the signal and trying to pinpoint the exact location now."
"Very good," Orlov said. "Let me know as soon as you have it."
As soon as Orlov hung up, he put in a call to Mike Rodgers to let him know about the apparent NSA-Harpooner connection and the possible location of the Harpooner. Then he called Odette. He hoped that the American she had saved was ready to move out. Orlov did not want to send Odette against the Harpooner unassisted, but he would if he had to. Because more than that, he did not want to lose the Harpooner.
As Orlov punched in Odette's number, he began to feel hopeful and upbeat. The technology that he had helped put into space was actually a two-edged sword. The Harpooner had been using a secure satellite uplink to help destroy lives. Now, with luck, that uplink would have an unexpected use.
To pinpoint the Harpooner and help destroy him.
**THIRTY-NINE**
_**Teheran, Iran Tuesday, 10:07 A.M.**_
The chief of the Supreme Command Council of the Armed Forces of the Islamic Republic of Iran had been called at home shortly after dawn. Teheran maintained listening posts on many of their oil rigs in the Caspian Sea. From there, they eavesdropped electronically on foreign shipping and on military sites along the Caspian coast. Each post sent a pulse every five minutes to indicate that the electronics were still on-line. The sudden silence of Post Four was the first indication anyone in Teheran had that something was wrong in the Caspian.
An F-14 Tomcat was immediately dispatched from the Doshan Tapeh Air Base outside of Teheran. The Tomcat was one of ten that remained of the seventy-seven that had been a part of the shah's state-of-the-art air force. The fighter confirmed that the oil rig had been destroyed. Salvage experts and military engineers were immediately parachuted into the region by a Kawasaki C-1 transport. While rescue patrol boats hurried to the site from Caspian fleet headquarters in Bandar-e Anzelli, the engineers found burn marks on the platform that were consistent with powerful high explosives. The fact that the underside had been struck suggested a submarine attack that had somehow eluded sonar detection. At nine-thirty A.M., the salvage experts found something more. The body of Russian terrorist Sergei Cherkassov.
The report galvanized the often fractious officers of the SCCAF as well as the minister of the Islamic Revolutions Guards Corps, the minster of foreign affairs, the minister of the interior, and the minister of intelligence. The moderates had joined the extremists, and by ten A.M., the order had been given: the IRI military was ordered to defend Iranian interests in the Caspian at any and all cost.
On the sea, the initial thrust was to be an antisubmarine defense. That was spearheaded by antisubmarine aircraft and helicopters. Marine battalions in the region were also mobilized. The second wave would consist of destroyers and frigates, which were to be stationed around the remaining rigs. Chinese-made Silkworm missiles were rushed to the forces defending the Caspian.
In the air, Chinese-made Shenyang F-6s began regular patrols from both the Doshan Tapeh Air Base and the Mehrabad Air Base. Three surface-to-air missile battalions in the region were also put on high alert.
At the same time, Iranian embassies in Moscow and Baku were ordered to notify the Russian and Azerbaijani governments that while the attack was under investigation, any further moves against Iranian interests would be regarded as a declaration of war by those governments. Iranian diplomats were informed by both governments that they had had no hand in the attack on the Iranian oil facility. Representatives of Moscow and Baku added that Iran's increased military presence was unwelcome. Both nations indicated that their own navies and air forces would be placed on alert and would increase patrols in the region.
By late morning, waters that had given lives to fishermen and oilmen the night before were rich with something else.
The promise of death.
**FORTY**
_**Washington, D.C. Tuesday, 1:33 A.M.**_
Mike Rodgers was in his office when General Orlov called. After hearing what the Russian had to say, Rodgers immediately called Paul Hood in his car and gave him the new information about the Harpooner.
"How certain is General Orlov. about the NSA-Harpooner connection?" Hood asked.
"I asked him that," Rodgers told Hood. "Orlov answered that he is very certain. Though I'm not sure the president is going to put a lot of credence in what a Russian general thinks."
"Especially if several of the president's top advisers refute that information," Hood said.
"Paul, if Orlov is correct, we're going to have to do more than tell the president," Rodgers said. "There's going to have to be a massive housecleaning in the NSA. We can't have American intelligence agencies hiring terrorists who have attacked American interests, taken American lives."
"Didn't we do that with the German rocket scientists after World War Two?" Hood asked.
"The operative phrase is, ' _after_ World War Two,' " Rodgers said. "We didn't hire German scientists to work for us while they were still building missiles to attack Great Britain."
"Good point," Hood said.
"Paul, this is the guy that helped kill Bob Herbert's wife," Rodgers said. "If Orlov's intel is true, the NSA has to be held accountable for this."
"I hear you," Hood said. "Look, I'll be at the White House soon. Work on trying to get me any kind of backup you can. See if Bob can dig up signal intelligence that backs up Orlov's claims."
"He's working on that now," Rodgers said.
Hood hung up, and Rodgers got up. He poured coffee from the pot that sat on a cart in the back of his room. It was an aluminum cart from the 1950s. He'd picked it up at a Pentagon garage sale ten years before. He wondered if the sounds of crisis still resonated somewhere deep in its molecular structure. Arguments and decisions about Korea, the Cold War, Vietnam.
_Or were they arguments about whose turn it was to treat for coffee and Danish?_ Rodgers wondered. That was part of war, too, of course. The moments of down-time that let decision makers catch their breath. Do something real instead of theoretical. Remind themselves that they were talking about people's lives and not just statistics.
When he sat back down, Rodgers started going through the files of the NSA's top officials. He was looking for people who had previous ties with Jack Fenwick or had ever investigated Middle Eastern terrorist groups. The NSA could not have contacted the Harpooner unless someone in one of those groups had helped. If it turned out that Orlov was right, Rodgers wanted to be ready to help with the purge. A purge of Americans who had collaborated with a man who had murdered American men and women, soldiers, and civilians.
He wanted to be ready with a vengeance.
**FORTY-ONE**
_**Washington, D.C Tuesday, 1:34 A.M.**_
The White House is an aging monument in constant need of repair. There is peeling paint on the southern columns and splitting wood on the third-floor terraces.
But in the West Wing, especially in the Oval Office, there is a sense of constant renewal. To outsiders, power is a large part of the appeal of the Oval Office. To insiders, it is the idea that an intense new drama presents itself every hour of every day. Whether it's small, cautious maneuvering against a political rival or the mobilization of the military for a massive offensive and possible casualties, each situation starts, builds, and ends. For someone who thrives on outthinking an adversary or on extrapolating short- and long-term results from quiet decisions, the Oval Office is the ultimate challenge. It clears the game board every few minutes and offers new contests with new rules. Some presidents are aged and drained by the process. Other presidents thrive on it.
There was a time until very recently when Michael Lawrence was invigorated by the problems that crossed his desk. He was undaunted by crises, even those that required quick military action and possible casualties. That was part of the job description. A president's task was to minimize the damage caused by inevitable aggression.
But something had changed over the past few days. Lawrence had always felt that however stressful situations got, he was at least in control of the process. He could chair meetings with confidence. Lately, that was no longer the case. It was difficult for him even to focus.
Lawrence had worked with Jack Fenwick and Red Gable for many years. They were old friends of the vice president, and Lawrence trusted Jack Cotten. He trusted his judgment. Lawrence would not have selected him as a running mate otherwise. As vice president, Cotten had been more closely involved in the activities of the NSA than any previous vice president. Lawrence had wanted it that way. For years, the CIA, the FBI, and military intelligence had had their own agendas. The Executive Branch needed its own eyes and ears abroad. Lawrence and Cotten had more or less appropriated the NSA for that task. The military could still utilize the NSA's chartered assets, which were the centralized coordination and direction of U.S. government intelligence technical functions and communications. Under Cotten, its role had quietly been expanded to increase the breadth and detail of intelligence that was coming directly to the president. Or, rather, to Fenwick and the vice president and then to the president.
The president stared at the open laptop on his desk. Jack Fenwick was talking about Iran. Data was downloading quickly from the NSA. Fenwick had some facts and a good deal of supposition. He also had an edge. He appeared to be going somewhere, though he had not yet indicated where.
Meanwhile, Lawrence's eyes stung, and his vision was foggy. It was difficult to concentrate. He was tired, but he was also distracted. He did not know who to believe or even what to believe. Was the data from the NSA real or falsified? Was Fenwick's intelligence accurate or fabricated?
Paul Hood suspected Fenwick of deception. Hood appeared to have the evidence for it. But what if it were Hood's evidence that wasn't trustworthy? Hood was going through an extremely stressful time. He had resigned his post at Op-Center, then returned. He had been at ground zero of the explosive UN hostage crisis. His daughter was suffering from an extreme case of post-traumatic stress disorder. Hood was in the process of getting a divorce.
What if it were Hood who had the agenda, not Fenwick, the president wondered. When Fenwick had arrived at the White House before, he admitted that he had been to the Iranian mission. He admitted it openly. But he insisted that the president had been informed. The vice president corroborated that fact. So did the calendar on the president's computer. As for the call regarding the United Nations initiative, Fenwick insisted that was not placed by him. He said the NSA would investigate. Could it have been placed by Hood?
"Mr. President?" Fenwick said.
The president looked at Fenwick. The national security adviser was seated in an armchair to the left of the desk. Gable was to the right, and the vice president was in the center.
"Yes, Jack?" the president replied.
"Are you all right, sir?" Fenwick asked.
"Yes," Lawrence replied. "Go on."
Fenwick smiled and nodded and continued.
The president sat up taller. He had to focus on the issue at hand. When he got through this crisis, he would schedule a short vacation. Very soon. And he would invite his childhood friend and golfing buddy, Dr. Edmond Leidesdorf, and his wife. Leidesdorf was a psychiatrist attached to Walter Reed. The president had not wanted to see him officially with this problem because the press would find out about it. Once that happened, his political career would be over. But they had played golf and gone sailing before. They could talk on a golf course or boat without raising suspicion.
"The latest intelligence puts the Russian terrorist Sergei Cherkassov at the scene of the explosion," Fenwick continued. "He had escaped from prison three days before the attack on the rig. His body was found at sea. There were burn marks consistent with flash explosives. There was also very little bloating. Cherkassov had not been in the water for very long."
"Do the Azerbaijanis have that information?" the president asked.
"We suspect they do," Fenwick replied. "The Iranian naval patrol that found Cherkassov radioed shore on an open channel. Those channels are routinely monitored by the Azerbaijanis."
"Maybe Teheran wanted the rest of the world to have the information," the president suggested. "It might turn them against Russia."
"That's possible," Fenwick agreed. "It's also possible that Cherkassov was working for Azerbaijan."
"He was being held in an Azerbaijani prison," the vice president said. "They might have allowed him to escape so that he could be blamed for the attack."
"How likely is that?" the president asked.
"We're checking with sources at the prison now," Fenwick said. "But it's looking very likely."
"Which means that instead of the attack turning Iran against Russia, Azerbaijan may have succeeded in uniting both nations against them," the vice president said.
Fenwick leaned forward. "Mr. President, there's one thing more. We suspect that creating a union between Russia and Iran may actually have been the ultimate goal of the Azerbaijani government."
"Why in hell would they do that?" the president asked.
"Because they are practically at war with Iran in the Nagorno-Karabakh region," Fenwick said. "And both Russia and Iran have been pressing claims on some of their oil fields in the Caspian."
"Azerbaijan wouldn't stand a chance against either nation individually," the president pointed out. "Why unite them?"
Even as he said it, the president knew why.
To win allies.
"How much of our oil do we get from that region?" the president asked.
"We're up to seventeen percent this year with a projection of twenty percent next year," Gable informed him. "We're getting much better prices from Baku than we are from the Middle East. That was guaranteed by the trade agreement we signed with Baku in March 1993. And they've been very good about upholding their end of the agreement."
"Shit," the president said. "What about the other members of the Commonwealth of Independent States?" he asked. "Where will they stand if two of their members go to war?"
"I took the liberty of having my staff put in calls to all of our ambassadors before I came over here," the vice president said. "We're in the process of ascertaining exactly where everyone stands. But a preliminary guess is that it will pretty much be split. Five or six of the poorer, smaller republics will side with Azerbaijan in the hopes of forming a new union with a share of the oil money. The other half will go with Russia for pretty much the same reason."
"So we risk a wider war as well," the president said.
"But this is more than just the possibility of us losing oil and watching a war erupt," Fenwick pointed out. "It's Iran and the Russian black market getting their hands on petrodollars that scares me."
The president shook his head. "I'm going to have to bring the joint chiefs in on this."
The vice president nodded. "We're going to have to move quickly. It's midmorning in the region. Things are going to happen very quickly. If they get ahead of us—"
"I know," the president said. He was suddenly energized, ready to deal with the situation. He looked at his watch and then at Gable. "Red, would you notify the joint chiefs to be here at three? Also, get the press secretary out of bed. I want him here as well." He looked at the vice president. "We'll need to alert the thirty-ninth Wing at Incirlik and the naval resources in the region."
"That would be the _Constellation_ in the North Arabian Sea and the _Ronald Reagan_ in the Persian Gulf, sir," Fenwick said.
"I'll put them on alert," the vice president said. He excused himself and went to the president's private study. It was a small room that adjoined the Oval Office on the western side. That was also where the president's private lavatory and dining parlor were located.
"We'll also have to brief NATO command," the president told Gable. "I don't want them holding us up if we decide to act. And we're going to need a complete chemical and biological workup of the Azerbaijani military. See how far they'll go if we don't join in."
"I already have that, sir," Fenwick said. "They've got deep reserves of anthrax as well as methyl cyanide and acetonitrile on the chemical side. All have surface-to-surface missile delivery systems. Most of the reserves are stored in or near the NK. We're watching to see if any of them are moved."
The president nodded as his intercom beeped. It was his deputy executive secretary Charlotte Parker.
"Mr. President," said Parker, "Paul Hood would like to see you. He says it's very important."
Fenwick did not appear to react. He turned to Gable and began talking softly as he pointed to data on his notepad.
Are they talking about the Caspian or about Hood? the president wondered. Lawrence thought for a moment. If Hood were the one who had lost his way—either intentionally or because of external pressures—this would be the time and the place to find out.
"Tell him to come in," said the president.
**FORTY-TWO**
_**Saint Petersburg, Russia Tuesday, 9:56 A.M.**_
"We have the Harpooner's location!" Korsov shouted. Orlov looked up as Korsov rushed into his office. The young intelligence officer was followed by Boris Grosky, who looked less glum than Orlov had ever seen him. He did not look happy, but he did not look miserable. Korsov was holding several papers in his hands.
"Where is he?" Orlov asked.
Korsov slapped a computer printout on Orlov's desk. There was a map and an arrow pointing to a building. Another arrow pointed to a street several blocks away.
"The signal originated at a hotel in Baku," Korsov said. "From there it went to Suleyman Ragimov Kuchasi. It's an avenue that runs parallel to Bakihanov Kuchasi, the location of the hotel."
"Was he calling someone with a cell phone?" Orlov asked.
"We don't believe so," Grosky said. "We've been monitoring police broadcasts from the area to find out more about the oil rig explosion. While we were listening, we heard about a van explosion on Suleyman Ragimov. The blast is being investigated now."
"It doesn't sound like a coincidence," Korsov added.
"No, it doesn't," Orlov agreed.
"Let's assume the Harpooner was behind that," Korsov said. "He might want to see it from his hotel room—"
"That might not be necessary, as long as he could hear it," Orlov said. "No. The Harpooner would be worried about security if he were staying in a hotel room. Do we have any way of fine-tuning the location of the signal?"
"No," Korsov said. "It was too brief, and our equipment is not sensitive enough to determine height in increments under two hundred feet."
"Can we get a diagram of the hotel?" Orlov asked.
"I have that," Korsov said. He pulled a page from the pile he was holding and laid it beside the map. It showed a ten-story hotel.
"Natasha is trying to break into the reservations list," Grosky said. He was referring to the Op-Center's twenty-three-year-old computer genius Natasha Revsky. "If she can get in, she will give us the names of all single male occupants."
"Get single females as well," Orlov said. "The Harpooner has been known to adopt a variety of disguises."
Grosky nodded.
"You feel very confident about this?" Orlov asked. Korsov had been leaning over the desk. Now he stood like a soldier, his chest puffed. "Completely," he replied.
"All right," Orlov said. "Leave the hotel diagram with me. This was very good work. Thank you both."
As Grosky and Korsov left, Orlov picked up the phone. He wanted to talk to Odette about the hotel and then get her on site. Hopefully, the American would be strong enough to go with her.
The Harpooner was not a man to tackle alone.
**FORTY-THREE**
_**Baku, Azerbaijan Tuesday, 10:07 A.M.**_
Odette Kolker was cleaning up the breakfast plates when the phone beeped. It was the apartment phone, not her cell phone. That meant it was not General Orlov who was calling.
She allowed her answering machine to pick up. It was Captain Kilar. The commander of her police unit had not been in when she phoned the duty sergeant to let him know that she would be out sick. Kilar was calling to tell her that she was a good and hardworking officer, and he wanted her to get well. He said that she should take whatever time she needed to recuperate.
Odette felt bad about that. She was hardworking. And though the Baku Municipal Police Department paid relatively well—twenty thousand manats, the equivalent of eight thousand American dollars—they did not pay overtime. However, the work Odette did was not always for the BMP and the people of Baku. The time she spent at her computer or on the street was often for General Orlov. Baku was a staging area for many of the arms dealers and terrorists who worked in Russia and the former Soviet republics. Checking on visa applications, customs activity, and passenger lists for boats, planes, and trains enabled her to keep track of many of these people.
After putting away the few dishes, Odette turned and looked back at her guest. The American had fallen asleep and was breathing evenly. She had placed a cool washcloth on his head and he was perspiring less than when she had brought him home. She had seen the bruises on his throat. They were consistent with choke marks. Obviously, the incident in the hospital was not the first time someone had tried to kill him. There was also a tiny red spot on his neck. A puncture wound, it looked like. She wondered if this illness were the result of his having been injected with a virus. The KGB and other Eastern European intelligence services used to do that quite a bit, typically with lethal viruses or poison. The toxin would be placed inside microscopic pellets. The pellets were sugar-coated metal spheres with numerous holes in their surface. These would be injected by an umbrella tip, pen point, or some other sharp object. It would take the body anywhere from several minutes to an hour or two to eat through the sugar coating. That would give the assassin time to get away. If this man had been injected, he probably was not supposed to die by the virus. He had been used to draw his colleagues out into the open. The hospital ambush had been well organized.
Just like the ambush that killed her husband in Chechnya, she thought. Her husband, her lover, her mentor, her dearest friend. They all perished when Viktor died on a cold, dark, and lonely mountainside.
Viktor had successfully infiltrated the Chechan mujihadin forces. For seven months, Viktor was able to obtain the ever-changing radio frequencies with which different rebel factions communicated. He would write this information down and leave it for a member of the KGB field force to collect and radio to Moscow. Then the idiot KGB officer got sloppy. He confused the frequency he was supposed to use with the one he was reporting about. Instead of communicating with his superiors, he broadcast directly to one of the rebel camps. The KGB officer was captured, tortured for information, and killed. He had not known Viktor's name but he knew which unit her husband had infiltrated and when he had arrived. The rebel leaders had no trouble figuring out who the Russian agent was. Viktor would always leave his information under a rock which he would chip in a distinctive fashion. While he was out one night, supposedly standing watch, Viktor was brought down by ten men, then taken into the mountains. There, his Achilles tendons were severed and his wrists were slashed. Viktor bled to death before he could crawl to help. His last message to her was painted on a tree trunk with his own blood. It was a small heart with his wife's initials inside.
Odette's cell phone beeped softly. She picked it up from the kitchen counter and turned her back toward her guest. The woman spoke softly so she would not wake him.
"Yes?"
"We believe we've found the Harpooner."
That got Odette's attention. "Where?"
"At a hotel not far from you," Orlov said. "We're trying to pinpoint his room now."
Odette moved quietly toward the bed. She was required to check her service revolver when she left police headquarters every night. But she kept a spare weapon in the nightstand. It was always loaded. A woman living alone had to be careful. A spy at home or abroad had to be even more careful.
"What's the mission?" Odette asked.
"Termination," Orlov said. "We can't take a chance that he'll get away."
"Understood," Odette said calmly. The woman believed in the work she was doing, protecting the interests of her country. Killing did not bother her when doing it would save lives. The man she had terminated just a few hours before meant little more to her than someone she might have passed in the street.
"Once we've narrowed down the guests who might be the Harpooner, you're going to have to make the final call," Orlov said. "The rest depends on what he does, how he acts. What you see in his eyes. He's probably going to have showered but still look tired."
"He's been a busy bastard," Odette said. "I can read that in a man."
"The chances are he won't open the door to the hotel staff," Orlov went on. "And if you pretend to be a housekeeper or security officer, that will only put him on guard."
"I agree," she said. "I'll find a way to get in and take him by surprise."
"I spoke to our profiler," Orlov said. "If you do get to him, he'll probably be cool and even pleasant and will appear to cooperate. He might attempt to bribe you or get you to be overconfident. Try to get your guard down so he can attack. Don't even listen. Make your assessment and do your job. I wouldn't be surprised if he also has several traps at the ready. A gas canister in an air duct, an explosive device, or maybe just a magnesium flash to blind you. He might have rigged it to a light switch or a remote control in his heel, something he can activate when he ties his shoe. We just don't know enough about him to say for certain how he secures a room."
"It's all right," Odette assured him. "I'll make the ID and neutralize him."
"I wish I could tell you to go in with a squad of police," Orlov said apologetically. "But that isn't advisable. A shout, rerouted traffic, anything out of the ordinary can alert him. Or the Harpooner may sense their presence. If he does, he may get away before you can even get to him. I'm sure he has carefully planned his escape routes. Or he may try to take hostages."
"I understand," Odette said. "All right. Where is the Harpooner registered?"
"Before I tell you that, how is your guest?" Orlov asked.
"He's sleeping," Odette replied. She looked down at the man on the bed. He was lying on his back, his arms at his side. His breathing was slow and heavy. "Whatever he's suffering from was probably artificially induced," she said. "Possibly by injection."
"How is his fever?"
"Down a bit, I think," she said. "He'll be okay."
"Good," Orlov said. "Wake him."
"Sir?" The order took her completely by surprise.
"I want you to wake him," Orlov told her. "You're bringing him with you."
"But that's not possible!" Odette protested. "I don't even know if the American can stand."
"He'll stand," Orlov said. "He has to."
"Sir, this is not going to help me—"
"I'm not going to have you face the Harpooner without experienced backup," Orlov said. "Now, you know the drill. Do it."
Odetted shook her head. She knew the drill. Viktor had taught it to her. Lit matches were applied to the soles of the feet. It not only woke up the ill or people who had been tortured into unconsciousness, but the pain kept them awake and alert as they walked.
Odette shook her head. By definition, field work was a solo pursuit. What had happened to Viktor underscored the danger of working with someone even briefly. Even if the American were well, she was not sure she wanted a partner. Ill, he would be more of a burden than an asset.
"All right," Odette said. She turned her back on the American and walked toward the kitchenette. "Where is he?"
"We believe the Harpooner is in the Hyatt," Orlov told her. "We're trying to have a look at their computer records now. I'll let you know if we learn anything from the files."
"I'll be there in ten minutes," Odette promised. "Is there anything else, General?"
"Just this," Orlov said. "I have grave reservations about sending you after this man. I want you both to be careful."
"We will," Odette said. "And thank you."
She hung up and hooked the cell phone on her belt. She removed the gun and ankle holster from the night table and slipped them on. Her long police skirt would cover the weapon. She slipped a silencer in her right pocket. She had brought a switchblade to the hospital. That was still tucked in her left skirt pocket. If she did not need it for self-defense, she would need it as a throwaway. If she were stopped for any reason, perhaps by hotel security, Odette could say that she was visiting a friend—the checkout who, of course, would no longer be there. Odette would be able to say that she knocked on the wrong door and the Harpooner attacked her. With her help—using information provided by Orlov and the Americans—the police would connect the dead man with the terrorist attack.
Hopefully, though, it would not be necessary to explain anything to anyone. With surprise on her side, Odette might be able to catch the Harpooner relatively unprepared.
Odette walked on slightly bent knees and tiptoed to the front door of the apartment. The hardwood floors creaked loudly underfoot. It was strange, Odette thought. It had never been necessary for her to be quiet here before. Until today, there had never been anyone but her in this bed. Not that she regretted that. Viktor had been all she ever wanted.
Odette opened the door. Before leaving, she looked back at the sleeping American.
The woman felt bad about lying to General Orlov. Though the coin of her profession was subterfuge and deceit, she had never lied to Orlov. Fortunately, this was a win-win situation for her. If she succeeded in bringing down the Harpooner, Orlov would be angry with her—but not very. And if she failed, she would not be around to hear Orlov complain.
Odette stepped into the corridor and quietly shut the door behind her. If she blew this assignment, she would probably have to listen to Viktor complain. Listen for all eternity.
She smiled. That, too, was a win situation.
**FORTY-FOUR**
_**Washington, D.C. Tuesday, 2:08 A.M.**_
A stoic secret service agent opened the door to the Oval Office and admitted Paul Hood. The large, white door closed with a small click. The sound seemed very loud to Hood as he crossed the carpet toward the president's desk. So did the sound of Hood's heart. He had no way of knowing for certain whether Fenwick was a rogue figure or working as part of a team. Either way, convincing others about possible involvement in an international conspiracy of some kind was going to be extremely difficult.
The mood in the room was hostile. Hood could feel that even before he saw the faces of the vice president, Fenwick, and Gable. None of the men looked back at him, and the president's expression was severe. Mike Rodgers once said that when he first joined the military, he had a commanding officer with a very singular expression of disapproval. He looked at you as though he wanted to tear heads off and use them for punting practice.
The president had that look.
Hood quickly made his way between the armchairs to the president's desk. The Washington Monument was visible through the windows behind the president. The tower was brightly moonlit in the flat, black night. Seeing it then gave Hood the flash of courage he needed.
"I'm sorry to intrude, Mr. President, gentlemen," Hood announced. "This couldn't wait."
"Things never can wait with you, can they?" Fenwick asked. He glanced back at the green folder in his lap.
_A preemptive strike_ , Hood thought. _The bastard was good._ Hood turned and looked at the NSA chief. The short, slender man had deep-set eyes beneath a head of thick, curly white hair. The whiteness of his hair emphasized the darkness of his eyes.
"Your team has a history of rushing blindly into evolving crises, Mr. Hood. North Korea, the Bekaa Valley, the United Nations. You're a lighted match waiting for the wrong tinderbox."
"We haven't blown one yet," Hood pointed out.
"Yet," Fenwick agreed. He looked at Lawrence. "Mr. President, we need to finish reviewing our data so that you can make a decision about the Caspian situation."
"What does Maurice Charles have to do with the Caspian situation?" Hood demanded. He was still looking at Fenwick. He was not going to let the man wriggle away.
"Charles? The terrorist?" Fenwick asked.
"That's right," Hood said. Hood said nothing else. He wanted to see where this went.
The president looked at Fenwick. "Did the NSA know that Charles was involved with this?"
"Yes, Mr. President, we did," Fenwick admitted. "But we don't know what his involvement was. We've been looking into that."
"Maybe I can point you in the right direction, Mr. Fenwick," Hood said. "Maurice Charles was in touch with the NSA both before and after the attack on the Iranian oil rig."
"That's bullshit!" Fenwick charged.
"You seem sure of that," Hood said.
"I am!" Fenwick said. "No one in my organization would have anything to do with that man!"
Hood had expected Fenwick to 3D the charge: disavow, deny, and delay. But neither the vice president nor Gable had jumped in to defend him. Perhaps because they knew it was true?
Hood turned to the president. "Sir, we have every reason to believe that Charles, the Harpooner, was involved in the destruction of that rig."
"Evidence from whom?" Fenwick demanded.
"Unimpeachable sources," Hood replied.
"Who?" Vice President Cotten asked.
Hood faced him. The vice president was a calm and reasonable man. Hood was going to have to bite the bullet on this one. "General Sergei Orlov, commander of the Russian Op-Center."
Gable shook his head. Fenwick rolled his eyes.
"The Russians," the vice president said dismissively. "They may have been the ones who sent Cherkassov into the region to attack the rig. His body was found in the water nearby."
"Moscow has every reason not to want us involved in the region," Gable said. "If Azerbaijan is chased out of the Caspian, Moscow can lay claim to more of the oil reserves. Mr. President, I suggest we table this side of the problem until we've dealt with the larger issue of the Iranian mobilization."
"We've reviewed the data Orlov provided, and we believe it's accurate," Hood stated.
"I'd like to see that data," Fenwick said.
"You will," Hood promised.
"You wouldn't also have given General Orlov any secure codes to help him listen in on alleged NSA conversations, would you?"
Hood ignored that. "Mr. President, the Harpooner is an expert at creating and executing complex cover stories. If he's involved in this operation, we have to look carefully at any evidence that comes in. We should also inform Teheran that this action may have nothing to do with Baku."
"Nothing?" Fenwick said. "For all we know, they may have hired the Harpooner."
"You may be right," Hood said. "What I'm saying is that we have no evidence of anything except the fact that the Harpooner is in the region and was probably involved in the attack."
"Secondhand evidence," Fenwick said. "Besides, I spent a day trying to open a dialogue with Teheran about an intelligence exchange. The bottom line is that they don't trust us, and we can't trust them."
"That is not the bottom line!" Hood snapped. He stopped. He had to watch that—showing anger. He was frustrated, and he was extremely tired. But if he lost control, he would also lose credibility. "The bottom line," Hood continued evenly, "is that misinformation has been passed regularly between the NSA, the CIOC, and the Oval Office—"
"Mr. President, we need to move on," Fenwick said calmly. "Iran is moving warships into the Caspian region. That is a fact, and it must be dealt with immediately."
"I agree," said the vice president. Cotten looked at Hood. There was condescension in the vice president's eyes. "Paul, if you have concerns about the actions of personnel at the NSA, you should bring your proof to the CIOC, not to us. They will deal with it."
"When it's too late," Hood said.
"Too late for what?" the president asked.
Hood turned to the president. "I don't know the answer to that, sir," Hood admitted. "But I do believe you should hold off making any decisions about the Caspian right now."
Fenwick shook his head. "Based on hearsay from Russians who may themselves be moving planes and ships into the region."
"Mr. Fenwick has a point," the president said.
"The Russians may indeed have designs on the Caspian oil," Hood agreed. "That in itself doesn't repudiate General Orlov's intelligence."
"How long do you need, Paul?"
"Give me another twelve hours," Hood said.
"Twelve hours will give Iran and Russia time to position ships in the Azerbaijani oil regions," Gable said.
The president looked at his watch. He thought for a moment. "I'll give you five hours," he said.
That was not what Hood wanted, but it was obviously all he was going to get. He took it.
"I'll need an office," Hood said. He did not want to waste time running back to Op-Center.
"Take the Cabinet Room," the president said. "That way I know you'll be done by seven. We'll be moving in then."
"Thank you, sir," Hood said.
Hood turned. He ignored the other men as he left the Oval Office. The hostility was much greater now than when he had come in. Hood was certain he had hit a bull's-eye. Just not with enough firepower.
It would have been too much to expect the president to buy everything he was telling him. Even after their earlier conversation, Lawrence was still obviously struggling with the idea that Jack Fenwick could be a traitor. But at least the president had not dismissed the idea entirely. Hood had been able to buy himself some time.
Hood walked down the quiet, green-carpeted hallway of the West Wing. He made his way past two silent secret service officers. One was posted outside the Oval Office. The other was standing down the hall between the doorway that led to the press secretary's office on the northwest end of the corridor and door to the Cabinet Room on the northeast side.
Hood entered the oblong room. There was a large conference table in the center of the room. Beyond it, in the northern end of the room, was a desk with a computer and a telephone. Hood went over and sat down.
The first thing Hood would do was contact Herbert. He had to try to get more information about the Harpooner's contacts with the NSA. Yet even having the exact time and location of the calls would probably not persuade the president that there was a conspiracy.
Hood needed proof. And right now, he did not know how he was going to get it.
**FORTY-FIVE**
_**Saint Petersburg, Russia Tuesday, 10:20 A.M.**_
When he was a cosmonaut, General Orlov had learned to read voices. Often, that was the only way he learned whether there was a problem with a flight. Ground control had once told him that all was well with his Salyut space station mission. In fact, pitting from micrometeoroid dust and a chemical cloud dumped by the spacecraft's own thrusters had corroded the solar array. The panels had been so seriously compromised that the station was going to lose power before a Kosmos ship from Earth was due to ferry them home.
The first hint of trouble came from the voice of the liaison in ground control. His cadence was a little different from usual. Orlov already had an ear for voices from the years he spent as a test pilot. Orlov insisted on being told what the problem was with the Salyut. The entire world heard the conversation, embarrassing the Kremlin. But Orlov was able to shut down noncritical systems and conserve power rather than wait for scientists to figure out how to realign the remaining panels while also shielding them from further corrosion.
Orlov trusted Natalia Basov. Completely. But he did not always believe her, which was not the same thing. There was something in her tone of voice that worried him. It was as if she had been concealing something. Just like the liaison at ground control.
Several minutes after they spoke on her cell phone, Orlov called the phone registered to Odette Kolker at her apartment. It rang a dozen times and no one answered. Orlov hoped that meant she had taken the American with her. Twenty minutes later, he called back again.
This time a man with a slurred voice answered. In English.
Orlov looked at the readout on the telephone to make sure he had the correct number. He did. The woman had left without the American.
"This is General Sergei Orlov," he said to the man. "Is this Mr. Battat?"
"Yes," Battat replied groggily.
"Mr. Battat, the woman who rescued you is my subordinate," Orlov went on. "She has gone out to try and apprehend the man who attacked you on the beach. You know who I am talking about?"
"Yes," Battat replied. "I do."
"She has no backup, and I'm worried about her and about the mission," Orlov said. "Are you well enough to get around the city?"
There was a short delay. Orlov heard grunts and moans.
"I'm on my feet, and I see my clothes hanging behind the door," Battat replied. "I'll take one step at a time. Where did she go?"
Orlov told the American he had no idea what Odette's plan was, or if she even had one. Orlov added that his team was still trying to get into the hotel computer to find out which rooms were occupied by single males.
Battat asked Orlov to call him a taxi, since he did not really speak the language.
Orlov said he would do that and thanked him. He gave Battat his telephone number at the Op-Center and then hung up.
Orlov sat still. Save for the faint buzz of the fluorescent light on his desk, his underground office was dead silent. Even space was not this quiet. There were always creaks as metal warmed and cooled or bumps as loose objects struck equipment. There were sounds of coolant moving through pipes and air rushing through vents. And every now and then there was someone talking in his headphone, either from Earth or somewhere else in the ship.
Not here. This was a lonelier-feeling place by far.
By now, Odette had probably reached the hotel and gone inside. He could phone her and order her back, but he did not think she would listen. And if she was intent on going through with this, he did not want to rattle her. She needed to know she had his support.
Orlov was angry at Odette for having disobeyed orders and lying to him. His anger was tempered by an understanding of what had driven the woman. Her husband had been a loner as well. A loner who had died because of someone else's carelessness.
Still, she would not stand in the way of Orlov's job. And that job was not just to capture or kill the Harpooner.
It was to make certain that Odette did not end up like Viktor.
**FORTY-SIX**
_**Baku, Azerbaijan Tuesday, 10:31 A.M.**_
There was a great deal of traffic, and it took Odette twice as long as she expected to reach the Hyatt Hotel. She parked on a side street less than a block from the employees' entrance. She did not want to park out front. There was still a sniper out there somewhere, the person who had shot the American diplomat outside the hospital. The killer might be bird-dogging the hotel for the Harpooner. He might have seen her car at the hospital and could recognize it again.
It was a sunny morning, and Odette enjoyed the brief walk to the front of the hotel. The air tasted richer and seemed to fill her lungs more than usual. She wondered if Viktor had felt this way while he was in Chechnya. If simple moments had seemed more rewarding when there was a real risk of losing it all.
Odette had been to the rear entrance of the hotel twice before. Once was to help a cook who had burned himself in a skillet fire. Another time was to quiet a man who was complaining about charges on his dinner bill. She knew her way around the back. Unfortunately, she didn't think she would find the Harpooner here. Odette assumed that when the Harpooner came and went, he used the front entrance. Sneaking out a delivery door or first-floor window might call attention to himself. Smart terrorists hid in plain sight.
And smart counterterrorists waited for them rather than charging into their lair, she thought.
But Odette had no idea when the Harpooner would be leaving. It could be the middle of the night. It could be early afternoon. It could be three days from now. She could not be here the entire time. She also had no idea whether or not he would be disguised. And for all she knew, he might even hire a prostitute to pose as his daughter, wife, or even his mother. There were some old prostitutes in Baku. Some very young ones, too. Odette had arrested a number of them.
There were many possibilities, all of which made it imperative that Odette get to the Harpooner before he left. The question was how to find him. She had no idea what his name was or what name he might be using.
Except for the Harpooner, Odette thought. She laughed to herself. Maybe she should run down the halls shouting that name. Watch to see which doors did not open. Anyone who did not need to see what the uproar was about had to be the Harpooner.
Odette rounded the corner and walked toward the front of the hotel. There was a kiosk around the corner. A newspaper extra was already announcing the Iranian buildup in the Caspian Sea. There were aerial reconnaissance photos of Iranian ships setting sail. Baku had always been relatively insulated from military action. This was something new for the nation's capital. That would help to explain the traffic. Most people lived in the suburbs. Many of them probably came to work, heard the news, and were getting out of town in the event of attack.
There was just one person standing beneath the gold and green awning. A doorman in a green blazer and matching cap. There were no tour buses, though that was not surprising. They usually left by nine A.M. Tourists who had entered the country as part of a group probably could not opt for early departure and had almost certainly gone ahead with their plans. In any case, checkout was not until noon. People who did want to leave were probably on the phones trying to book plane, train, or car reservations—
_Of course,_ she thought. _The phone._
Orlov had said that the Harpooner made a call using a secure phone. That would mean he probably had not made any calls using the hotel phone. She would look for a single male occupant with no phone charges on his bill.
Odette entered the hotel. She looked away from the front desk as she crossed the lobby. She did not want to risk being seen by the manager or any of the clerks who might recognize her. The first thing she did was turn to the right, toward the corridor that led to housekeeping. The long, simple office was located in the back of the hotel. There was a desk with a supervisor in the front of the office. Behind her was an array of cleaning carts. To her right was a Peg-Board with keys for all the rooms. A row of master keys was located on the bottom. These were given out to the cleaning staff each morning. Two keys remained.
Odette asked the elderly clerk if she could have more shampoo. Smiling pleasantly, the clerk rose and went to one of the carts. While the woman's back was turned, Odette took one of the master keys from the wall. The clerk returned with three small bottles of shampoo. The woman asked if she needed anything else. Odette said that she did not. Thanking her, Odette returned to the lobby and walked to the bank of telephone booths that lined an alcove in the back.
As she was walking, her phone beeped. She tucked herself into one of the booths, shut the door, then answered it.
Orlov said his team had broken into the hotel computer and they had five possibilities. Odette wrote down the names and room numbers.
"We might be able to narrow it down a little more," Orlov told her. "If someone wanted to get out of the country quickly, he would assume a nationality the Azerbaijani would not want around."
"Iranian," Odette said.
"No," Orlov countered. "Iranians might be detained. Russian is more likely. And there are two Russians at the hotel."
Odette said she might be able to narrow it down even further by checking the room telephone records.
"Good thinking," Orlov said. "Hold on while we're checking. Also, Odette, there's one thing more."
Odette felt her lower belly tighten. There was something about the general's voice.
"I spoke with Mr. Battat a few minutes ago," Orlov said.
Odette felt as if she'd run into a thick, low-lying tree branch. Her momentum died and her head began to throb. She did not think she had done wrong, leaving a sick man at home. But she had disobeyed an order and could think of nothing to say in her defense.
"The American is on his way to the hotel," General Orlov continued evenly. "I told him to look for you in the lobby. You're to wait until he arrives before you try to take down your man. Do you understand, Odette?"
"Yes, sir," she replied.
"Good," Orlov said.
The woman held on as Orlov's staff checked the records. Her palms were damp. That was less from nervousness than from having been caught. She was an honest woman by nature, and Orlov's trust was important to her. She hoped he understood why she had lied. It was not just to protect Battat. It was to allow herself to concentrate on the mission instead of on a sick man.
According to the hotel's records, two of the five men staying there had not made any calls from the room. One of them, Ivan Ganiev, was Russian. Orlov told her they were also checking the computer's housekeeping records. According to the last report, filed the day before, Ganiev's room, number 310, had not been cleaned in the three days he had been there.
Meanwhile, Orlov went to his computer and asked for a background check on the name. It came up quickly.
"Ganiev is a telecommunications consultant who lives in Moscow. We're checking the address now to make sure it's valid. He doesn't appear to work for any one company," Orlov said.
"So there's no personnel file we can check for his education or background," she said.
"Exactly," Orlov said. "He's registered with the Central Technology Licensing Bureau, but all it takes to get a license is a bribe. Ganiev does not have family in Moscow, does not appear to belong to any organizations, and receives his mail at a post office box."
That made sense, Odette thought. No mail collecting in the postbox, no newspapers piling up on the stoop. None of the neighbors would be certain whether he was there or not.
"Hold on, we have his address," Orlov added. He was silent for a moment. Then he said, "It's him. It has to be."
"Why do you say that?"
"Ganiev's residence is a block from the Kievskaya metro stop," Orlov told her.
"Which means—?"
"That's where we've lost the Harpooner on at least two other occasions," Orlov said.
Battat walked into the lobby just then. He looked like Viktor did after ten rounds of boxing in the military amateurs. Wobbly. Battat saw Odette and walked toward her.
"So it looks as though he's our man," Odette said. "Do we proceed as planned?"
This was the most difficult part of intelligence work. Making a determination about life and death based on an educated guess. If General Orlov were wrong, then an innocent man would die. Not the first and certainly not the last. National security was never error-free. But if he were correct, hundreds of lives might be spared. Then there was the option of attempting to capture the Harpooner and turn him over to Azerbaijani authorities. Even if it could be done, there were two problems with that. First, the Azerbaijanis would find out who Odette really was. Worse, they might not want to try to extradite the Harpooner. It was an Iranian rig he had attacked. And Russian buildings. And American embassies. The Azerbaijanis might want to make some kind of arrangement with him. Release him in exchange for his cooperation, for help in covert actions of their own. That was something Moscow could not risk.
"You're going to wait for the American to arrive?" Orlov asked.
"He's here now," Odette said. "Do you want to speak with him?"
"That won't be necessary," Orlov said. "The Harpooner will probably be traveling with high-tech equipment to go with his cover story. I want you to take some of it and any money he's carrying. Pull out drawers and empty the luggage. Make it look like a robbery. And work out an escape route before you go in."
"All right," she said.
There was nothing patronizing about Orlov's tone. He was giving instructions and also reviewing a checklist out loud. He was making sure that both he and Odette understood what must be done before she closed in.
Orlov was quiet again. Odette imagined him reviewing the data on his computer. He would be looking for additional confirmation that this was their quarry. Or a reason to suspect it was not.
"I'm arranging for airline tickets out of the country in case you need them when you're finished," Orlov said. He waited another moment and then decided as Odette knew he must. "Go and get him."
Odette acknowledged the order and hung up.
**FORTY-SEVEN**
_**Washington, D.C. Tuesday, 2:32 A.M.**_
Hood shut the door of the Cabinet Room behind him. There was a coffee machine on a small table in the far corner. The first thing Paul did upon entering was brew a pot using bottled water. He felt guilty doing that in the midst of a crisis, but he needed the caffeine kick. Desperately. Though his mind was speeding, his eyes and body from the shoulders down were crashing. Even the smell of the coffee helped as it began to brew. As he stood watching the steam, he thought back to the meeting he had just left. The shortest way of defusing the crisis on this end was to break Fenwick and whatever cabal he had put together. He hoped he could go back there with information, something to rattle Fenwick or Gable.
"I need time to think," he muttered to himself. Time to figure out how best to attack them if he had nothing more than he did now.
Hood turned from the coffeemaker. He sat on the edge of the large conference table and pulled over one of the telephones. He called Bob Herbert to see if his intelligence chief had any news or sources he could hit up for information about the Harpooner and possible contact with the NSA.
He did not.
"Unless no news is news," Herbert added.
Herbert had already woken several acquaintances who either worked for or were familiar with the activities of the NSA. Calling them in the middle of the night had the advantage of catching them off guard. If they knew anything, they would probably blurt it out. Herbert asked if any of them had heard about U.S. intelligence overtures to Iran.
None of them had.
"Which isn't surprising," Herbert said. "Something of that magnitude and delicacy would only be conducted at the highest executive levels. But it's also true that if more than one person knows about an operation over there, then everyone has heard at least a piece of the story. Not so here."
"Maybe more than one person at the NSA doesn't know about this," Hood said.
"That could very well be," Herbert agreed.
Herbert said he was still waiting to hear from HUM-INT sources in Teheran. They might know something about this.
"The only solid news we have is from Mike's people at the Pentagon," Herbert said. "Military Intelligence has picked up signs of Russian mobilization in the Caspian region. Stephen Viens at the NRO has confirmed that. The Slava-class cruiser _Admiral Lobov_ is apparently aleady heading south and the Udaloy II-class destroyer _Admiral Chebanenko_ is joining it along with several corvettes and small missile craft. Mike expects air cover over the Russian oil installations to commence within a few hours."
"All from something that started with the Harpooner—or whoever first hired him," Hood said.
"Eisenhower was the first to use the metaphor in 1954," Herbert said. "He said, 'You have a row of dominoes set up; you knock over the first one and what will happen to the last one is that it will go over very quickly.' He was talking about Vietnam, but it applies to this."
Herbert was right. You could count on the fact that dominoes not only fell, but they dropped quickly. And the only way to stop dominoes falling was to get far enough ahead of the chain and remove a few tiles.
After hanging up, Hood poured himself coffee, sat down in one of the leather seats, and called Sergei Orlov. The fresh, black coffee was a lifesaver. In the midst of chaos even a small respite seemed enormous.
The general brought Hood up to date on the situation with the Harpooner. Hood could hear the tension in the Russian's voice as he explained what the overall plan was. Hood related to Orlov's concern completely. There was worry for his operative Odette and a desperate desire to end the career of a notorious terrorist. Hood had been in that place. And he had both won there and lost there. This was not like a film or novel where the hero necessarily won.
Hood was still on the phone with General Orlov when the door opened. He glanced up.
It was Jack Fenwick. The time to think was over.
The NSA head entered the room and shut the door behind him. The Cabinet Room was a large room, but it suddenly seemed small and very close.
Fenwick walked over to the coffee and helped himself. Hood was nearly finished with the call. He ended the conversation as quickly as possible without seeming to hurry. He did not want Fenwick to hear anything. But he also did not want to show the NSA chief a hint of desperation.
Hood hung up. He took a swallow of coffee and glanced over at Fenwick. The man's dark eyes were on Hood.
"I hope you don't mind," Fenwick said. He indicated the coffee.
"Why should I?" Hood asked.
"I don't know, Paul," Fenwick shrugged. "People can get protective about things. Good coffee, by the way."
"Thanks."
Fenwick perched himself on the edge of the table. He was just a few feet from Hood. "We've taken a little break," Fenwick told him. "The president is waiting for the joint chiefs and secretary of state before making any decisions about the Caspian situation."
"Thanks for the update."
"You're welcome," Fenwick said. "I can give you more than an update," he went on. "I can give you a prediction."
"Oh?"
Fenwick nodded confidently. "The president is going to respond militarily. Emphatically. He has to."
Both Op-Center and the NSA had access to photographic reconnaissance from the NRO. No doubt Fenwick knew about the Russians as well.
Hood got up to freshen his coffee. As he did, he remembered what he had been thinking just a few minutes before.
The only way to stop the dominoes falling was to get far enough ahead of the chain and remove a few tiles.
"The question is not what the president will do, what the nation will do. The question is what are you going to do?" Fenwick said.
"Is that why you came here? To pick my brains?"
"I came here to stretch my legs," Fenwick said. "But now that we've gone there, I am curious. What are you going to do?"
"About what?" Hood asked as he poured more coffee. The dance was on. They were each watching their words.
"About the current crisis," Fenwick replied. "What part are you going to play?"
"I'm going to do my job," Hood said. He was either being interviewed or threatened. He had not yet decided which. Nor did he care.
"And how do you see that?" Fenwick asked.
"The job description says 'crisis management,' " Hood said. He looked back at Fenwick. "But at the moment, I see it as more than that. I see it as learning the truth behind this crisis and presenting the facts to the president."
"What truth is that?" Fenwick asked. Though his expression did not change, there was condescension in his voice. "You obviously don't agree with what Mr. Gable, the vice president, and I were telling him."
"No, I don't," Hood said. He had to be cautious. Part of what he was about to say was real, part of it was bluff. If he were wrong it would be the equivalent of crying wolf. Fenwick would not be concerned about anything Hood had to say. And Fenwick could use this to undermine Hood's credibility with the president.
But that was only if he were wrong.
"I've just been informed that we captured the Harpooner at the Hyatt Hotel in Baku," Hood said. He had to present it as a fait accompli. He did not want Fenwick calling the hotel and warning the terrorist.
"Then it's definitely the Harpooner?" Fenwick said. Fenwick took a sip of coffee and held it in his mouth. Hood let the silence hang there. After a long moment, Fenwick swallowed.
"I'm glad," Fenwick said without much enthusiasm.
"That's one less terrorist Americans have to worry about. How did you get him? Interpol, the CIA, the FBI—they've all been trying for over twenty years."
"We've been following him for several days," Hood went on. "We were observing him and listening to his phone calls."
"Who are _we_?"
"A group comprised of Op-Center, CIA, and foreign resources," Hood replied. "We pulled it together when we heard the Harpooner was in the region. We managed to lure him out using a CIA agent as bait."
Hood felt safe revealing the CIA's role since it was probably Fenwick who had given the information about Battat to the Harpooner.
Fenwick continued to regard Hood. "So you've got the Harpooner," Fenwick said. "What does all this have to do with the truth about what's going on? Do you know something that I don't?"
"The Harpooner apparently had a hand in what happened in the Caspian," Hood said.
"That doesn't surprise me," Fenwick said. "The Harpooner will work for anyone."
"Even us," Hood said.
Fenwick started when he heard that. Just a little, but enough so that Hood noticed. "I'm tired, and I don't have time for guessing games," Fenwick complained. "What do you mean?"
"We're talking to him now," Hood went on. "He seems willing to tell us who hired him in exchange for limited amnesty."
"Of course he does," Fenwick said dismissively.
"That bastard would probably say anything to save his hide."
"He might," Hood agreed. "But why lie when only the truth can save his life?"
"Because he's a twisted bastard," Fenwick said angrily. The NSA chief threw his cup into the wastebasket beneath the coffeemaker and got up from the table. "I'm not going to let you advise the president based on the testimony of a terrorist. I suggest you go home. Your work here is finished."
Before Hood could say anything else, Fenwick left the Cabinet Room. He pulled the door shut behind him. The room seemed to return to its former size.
Hood did not believe that Fenwick was concerned about the president getting misinformation. Nor did he believe that Fenwick was overworked and simply venting. Hood believed that he had come very close to exposing a relationship that Fenwick had worked hard to conceal.
A relationship between a high-ranking adviser to the president and the terrorist who had helped him to engineer a war.
**FORTY-EIGHT**
_**Baku, Azerbaijan Tuesday, 10:47 A.M.**_
When David Battat was six years old, he came down with the mumps and was extremely sick. He could barely swallow and his belly and thighs ached whenever he moved. Which was not so much of a problem because David had been too weak to move.
Battat felt too weak to move now. And it hurt when he did move. Not just in his throat and abdomen but in his legs, arms, shoulders, and chest. Whatever that bastard Harpooner had injected him with was debilitating. But it was also helpful, in a way. The pain kept him awake and alert. It was like a dull toothache all over his body. Whatever energy Battat had now was coming from anger. Anger at having been ambushed and debilitated by the Harpooner. And now anger at having been indirectly responsible for the deaths of Thomas and Moore.
Battat's hearing was muffled and he had to blink to see clearly. Yet he was extremely aware of his surroundings. The elevator was polished brass with green carpet. There were rows of small bright lightbulbs in the ceiling. There was a trapdoor in the back, and a fish-eye video lens beside it.
The elevator was empty except for Battat and Odette. When they reached the third floor, they stepped out. Odette took Battat's hand, like they were a young couple looking for their room. They checked the room numbers posted on the wall in front of them: 300 to 320 were to the right. That put 310 in the center of a long, brightly lit corridor. They started toward it.
"What are we doing?" Battat asked.
"Checking the stairwell first," Odette said. "I want to make sure the other killer isn't watching the room from there."
"And after that?" Battat asked.
"How would you feel about being married?" she asked.
"I tried it once and didn't like it," Battat said.
"Then you'll probably like this less," she replied. "I'll tell you what I'm thinking when we reach the stairwell."
They headed toward the stairwell, which was located at the opposite end of the corridor. As they neared 310, Battat felt his heart speed up. The "Do Not Disturb" sign was hanging from the door handle. There was something dangerous about the place. Battat felt it as they passed. It was not a physical sensation but a spiritual one. Battat was not prepared to go so far as to say it was palpable evil, but the room definitely had the feel of an animal's lair.
Odette released his hand when they reached the stairwell. She removed the gun from her holster and screwed on the silencer. Then she stepped ahead of Battat and cautiously peered through the window at the top of the door. No one was there. Odette turned the knob and stepped inside. Battat followed. He backed toward the concrete steps and leaned on the iron banister with one arm. It felt good not to have to move. Odette kept a heel in the door so it would not close and lock them out. She faced Battat.
"I'm sure the Harpooner has his room heavily protected from the inside," she said. "Since we probably won't be able to break in, we're going to have to try and draw him out."
"Agreed," Battat said. He was tired and dizzy and had to force himself to focus. "What do you propose?"
"You and I are going to have a lovers' quarrel," she said.
That got his attention. "About what?" he asked.
"It doesn't matter," she said. "As long as we end up arguing about which room is ours."
"One of us will say it's 312 and the other will insist it's 310," Battat said.
"Exactly," Odette replied. "Then we'll open the door to 310."
"How?"
Odette reached into her pocket.
"With this," she said as she pulled out the master key she had taken from the housekeeper. "If we're lucky, the Harpooner will only want to chase us away."
"What if someone else comes from their room or calls hotel security?" Battat asked.
"Then we argue more quickly," Odette said as she took off her jacket and slipped it over her forearm, concealing the gun.
The woman seemed to be growing impatient, a little anxious. Not that Battat blamed her. They were facing both the Harpooner and the unknown. If it were not for the dullness caused by whatever was afflicting him, he would have been experiencing fear on top of his lingering anger.
"This is not a science," she added. "The point of what we're doing is to distract the Harpooner long enough to kill him."
"I understand," Battat said. "What do you want me to do?"
"When I open the door, I want you to push it back hard," she said. "That should startle the Harpooner and also give me a moment to aim and fire. When we're finished, we come back to the stairwell and leave."
"All right," Battat said.
"Are you sure you feel up to this?" Odette asked.
"I'll be able to do what you want me to," he said. She nodded and gave him a reassuring half smile. Or maybe she was trying to reassure herself.
A moment later, they headed down the hall.
**FORTY-NINE**
_**Saint Petersburg, Russia Tuesday, 11:02 A.M.**_
Josef Norivsky was the Russian Op-Center's liaison between the country's other intelligence and investigative agencies as well as Interpol. He was a young, broad-shouldered man with short black hair and a long, pale face. He strode into General Orlov's office wearing an expression that was somewhere between fury and disbelief.
"Something is wrong," he said. Norivsky did not disseminate information unless he was sure of it. As a result, when he spoke, he had a way of making any statement seem like a pronouncement.
The intelligence liaison handed Orlov a set of eight-by-ten photographs. Orlov looked quickly at the eleven blurry black-and-white pictures. The shots showed five men in ski masks moving a sixth, unmasked man through a corridor made of cinder blocks.
"These photographs were taken by security cameras at the Lenkoran high-security prison in Azerbaijan," Norivsky explained. "We received them two days ago. The man without the mask is Sergei Cherkassov. The SIS was hoping we could help to identify the others."
The SIS was Azerbaijan's State Intelligence Service. They still maintained relatively close, cooperative relations with Russian intelligence groups.
"What have you come up with?" Orlov asked as he finished going through the photographs.
"The weapons they're carrying are IMI Uzis," said Norivsky. "They're based on the submachine guns Iran bought from Israel before the Islamic revolution. In and of themselves, they don't necessarily mean anything. Iranian arms dealers could have sold them to anyone. But look how the men are moving."
Orlov went back through the pictures. "I don't follow," he said.
Norivsky leaned over the desk and pointed to the fourth picture. "The men in the ski masks have formed a diamond shape around the Cherkassov. The point man covers the package, the escapee, the man in the rear watches their flank, and the men on the sides cover right and left. The fifth man, the only one who appears in pictures one and two, is ahead of the group, securing the escape route. Probably with a rocket launcher, according to reports." Norivsky stood. "This is the standard evacuation procedure used by VEVAK."
VEVAK was Vezarat-e Etella'at va Amniat-e Keshvar. The Iranian Ministry of Intelligence and Security.
"Why would Iran want to free a Russian terrorist from Azerbaijan?" Norivsky asked. The intelligence chief answered the question himself. "To use his talents? It's possible. But another possibility is that they wanted to dump his body at the attack site. How many bodies were found in the harbor at Baku? Four to six, depending on how the pieces eventually fit together."
"The same number of people who helped him to escape," Orlov said.
"Yes," Nirovsky replied.
"Which may mean they were all working together," Orlov said. "Nothing more than that."
"Except for the presence of the Harpooner," Norivsky pointed out. "We know that he has worked for Iran on many occasions. We know that he can usually be contacted through a series of associates in Teheran. What I'm saying, General, is, what if Iran organized the attack on its own oil rig as an excuse to move warships into the area?"
"That wouldn't explain the involvement of the American National Security Agency," Orlov said.
"But Cherkassov's presence might," Norivsky insisted. "Consider, sir. Iran threatens Azerbaijan. The United States becomes involved in that conflict. It has to. American oil supplies are being threatened. If the foe is only Iran, Americans are not opposed to an air and sea war. They have wanted to strike back at Teheran for decades, ever since the hostage crisis in 1979. But imagine that Russia is brought into the situation. At his trial, Cherkassov admitted working for the Kremlin. That was how he avoided execution. Suppose Azerbaijan or Iran retaliates by attacking Russian oil platforms in the Caspian. Are the people of the United States going to stand for a world war erupting in the region?"
"I don't think they would," Orlov said. He thought for a moment. "And maybe they wouldn't have to stand for it."
"What do you mean?" Norivsky asked.
"The Harpooner was working with the NSA, apparently to orchestrate this showdown," Orlov said. "What if someone in the American government made a deal with Iran before it happened?"
"Does the NSA have that kind of authority?" Norivsky asked.
"I don't believe so," Orlov said. "They would probably need higher-ranking officials working with them. Paul Hood at Op-Center indicated that contacts of that type may have taken place. What if the Americans agreed they would back down at a certain point? Allow Iran to have more of the oil-rich regions in exchange for American access to that oil?"
"A normalization of relations?" Norivsky suggested.
"Possibly," Orlov said. "The American military pushed to brinkmanship then pulled back for some reason. But what reason? That had to have been arranged as well."
Orlov did not know the answer, but he knew who might. Thanking Norivsky, Orlov rang his translator and put in a call to Paul Hood.
**FIFTY**
_**Washington, D.C. Tuesday, 3:06 A.M.**_
After Fenwick left the Cabinet Room, Hood sat alone at the long conference table. He was trying to figure out what he could tell the president to convince him that something was wrong with the intelligence he was receiving. That was going to be difficult without new information. Hood thought he had convinced him of Fenwick's duplicity earlier. But in the press of developing crises, crisis managers often took the advice of trusted and especially passionate friends. Fenwick was passionate, and Cotten was an old ally. Without hard facts, Hood would not be able to combat that. But what troubled him nearly as much was something the NSA head had said to Hood before leaving the Cabinet Room.
"I'm not going to let you advise the president." This was not just an international showdown. It was also a territorial fight in the Oval Office. But for what, exactly? It was not just about access to the president of the United States. Fenwick had tried to confuse Lawrence, to embarrass him, to mislead him. Why?
Hood shook his head and rose. Even though he had nothing to add to what he said before, Hood wanted to hear what the joint chiefs had to say. And Fenwick could not bar him from the Oval Office.
As Hood was leaving the Cabinet Room, his phone beeped. It was General Orlov.
"Paul, we have some disturbing information," Orlov said.
"Talk to me," Hood replied.
Orlov briefed him. When he was finished, Orlov said, "We have reason to believe that the Harpooner and Iranian nationals carried out the attack on the Iranian oil rig. We believe the attack may have been the same Iranians who freed the Russian terrorist Sergei Cherkassov from prison. This would make it seem as if Moscow was involved."
"Compelling the United States to lend its support to Azerbaijan as a counterbalance," Hood said. "Do you know if Teheran sanctioned the attack?"
"Very possibly," Orlov replied. "The Iranians appear to have been working for or were trained by VEVAK."
"In order to precipitate a crisis that would allow them to move in militarily," Hood said.
"Yes," Orlov agreed. "And the presence of Cherkassov, we think, was designed to give Iran a reason to threaten our oil facilities. To draw Russia into the crisis. Cherkassov may have had nothing to do with the attack itself."
"That makes sense," Hood agreed.
"Paul, you said before that members of your own government, of the NSA, were in contact with the Iranian mission in New York. That it was a member of the NSA that was in communication with the Harpooner in Baku. Could that agency be involved in this?"
"I don't know," Hood admitted.
"Perhaps the mission put them in contact with the Harpooner," Orlov suggested.
That was possible. Hood thought about it for a moment. Why would Fenwick help Iran to blow up its own rig and then encourage the president to attack Iran? Was this a plot to sucker Iran into a showdown? Was that why Fenwick had concealed his whereabouts from the president?
But Fenwick would have known about Cherkassov, Hood thought. He had to know that Russia would be drawn in as well.
And that still did not explain why Fenwick had made a point of calling the president right before the United Nations dinner. That was a move designed to humiliate Lawrence. To erode confidence in the president's—
_Mental state,_ Hood thought suddenly.
Hood followed the thread. Wasn't that what Megan Lawrence was concerned about? Mental instability, apparent or real, created by a careful pattern of deception and confusion? The president becomes deeply shaken. The United States finds itself on the precipice of war, led there by Fenwick. Lawrence tries to manage the crisis. What happens next? Does Fenwick undermine him somehow? Make him doubt his abilities—
_Or does he make the public doubt his abilities?_ Hood wondered.
Senator Fox was already concerned about the president. Mala Chatterjee had no love for him. The secretary-general would certainly give interviews stating that the president had been completely mistaken about the United Nations initiative. What if Gable or Fenwick were also to leak information about bad judgment the president had shown over the past few weeks?
Reporters would swallow it whole, Hood knew. It would be easy to manipulate the press with a story like that. Especially if it came from a reliable source like Jack Fenwick.
And it wasn't just Fenwick and Gable who were involved in this, Hood now knew for certain.
The vice president had been on the same page as Fenwick and Gable back in the Oval Office. Who stood to benefit most if the president himself and possibly the electorate were convinced that he was unfit to lead the nation in a time of crisis? The man who would succeed him, of course.
"General Orlov, have we heard from our people tracking the Harpooner?" Hood asked.
"They're both at the hotel where he is staying," Orlov reported. "They're moving in on him now."
"To terminate, not capture."
"We don't have the manpower to capture him," Orlov stated. "The truth is, we may not even have the manpower to complete the mission at hand. It's a great risk, Paul."
"I understand," Hood said. "General, are you solid about this information? That the men who attacked the Iranian rig are Iranian?"
"Until their body parts are collected and identified, an educated guess is the best I can do," Orlov said.
"All right," Hood said. "I'm going to take that information to the president. His advisers are pushing him to a military response. Obviously, we have to get him to postpone that."
"I agree," Orlov said. "We're mobilizing as well."
"Call me with any other news," Hood said. "And thank you, General. Thank you very much."
Hood hung up the phone. He ran from the Cabinet Room and jogged down the carpeted hallway toward the Oval Office. Canvas portraits of Woodrow Wilson and First Lady Edith Bolling Wilson looked down from the wall. She had effectively run the country in 1919 when her husband suffered a stroke. But she was protecting his health while looking out for the country's best interests. Not her own advancement. Had we become more corrupt since then? Or had the line between right and wrong become entirely erased? Did presumably virtuous ends justify corrupt means?
This was maddening. Hood had information, and he had a strong, plausible scenario. He had Fenwick turning pale when he said that the Harpooner had been captured. But Hood did not have proof. And without that, he did not see how he was going to convince the president to proceed slowly, carefully, regardless of what Iran did. Nor were the joint chiefs likely to be much help. The military had been itching for a legitimate reason to strike back at Teheran for over twenty years.
He turned the corner and reached the Oval Office. The secret service officer stationed at the door stopped him.
"I have to see the president," Hood told him.
"I'm sorry, sir, you'll have to leave," the young man insisted.
Hood wagged the badge that hung around his neck. "I have blue-level access," he said. "I can stand here. Please. Just knock on the door and tell the president I'm here."
"Sir, my doing that won't help you to see the president," the secret service agent told him. "They've moved the meeting downstairs."
"Where?" Hood asked. But he already knew.
"To the Situation Room."
Hood turned and swore. Fenwick was correct. He was going to keep him from seeing the president. The only way to get down there was with the next-level access badge, which was red level. Everyone who had that level would be down there. Being seduced and controlled by Jack Fenwick.
Hood walked back toward the Cabinet Room. He was still holding his cell phone and tapping it against his open palm. He felt like throwing the damn thing. He could not phone the president. Calls to the Situation Room went through a different switchboard than the rest of the White House. He did not have clearance for direct dial, and Fenwick would certainly have arranged it so that any calls Hood made would be refused or delayed.
Hood was accustomed to challenges, to delays. But he always had access to the people he needed to talk to and persuade. Even when terrorists had seized the United Nations Security Council, there had been ways to get in. All he needed was the resolve and manpower to do it. He was not accustomed to being utterly stonewalled like this. It was miserably frustrating.
He stopped walking. He looked up at the portrait of Woodrow Wilson, then looked at the painting of Mrs. Wilson.
"Shit," he said.
He glanced down at the phone. Maybe he wasn't as stonewalled as he thought.
Jogging again, Hood returned to the Cabinet Room. He was willing to bet there was one avenue Jack Fenwick hadn't closed down.
He couldn't have, even if he wanted to.
A queen always beat a Jack.
**FIFTY-ONE**
_**Baku, Azerbaijan Tuesday, 11:09 A.M.**_
As Odette walked down the hall, she had two concerns.
One worry was that she might be making a mistake about the identity of the man in room 310. That he was not, in fact, the Harpooner. Orlov had given Odette a general idea what the Harpooner looked like. But he had added that the Harpooner probably wore disguises. She had a mental picture of someone tall and aquiline with pale, hateful eyes and long fingers. Would she hesitate to shoot if someone not-so-tall and heavyset with blue, welcoming eyes and stubby fingers opened the door? Would that give him a chance to strike first?
An innocent man would come over and say "Hello," she told herself. The Harpooner might do that to throw off her guard. She _had_ to strike first, whoever was in there.
Her other concern was a question of confidence. She had been thinking about the reluctance she heard in General Orlov's voice. Odette wondered what concerned him most. That something would happen to her or that the Harpooner might escape? Probably both. Though she tried to rev up an "I'll show him" mentality, General Orlov's lack of confidence did not boost her own.
_It doesn_ ' _t matter,_ she told herself. _Focus on the goal and on nothing else._ The mission was all that mattered. The target was just a few doors down.
Odette and David Battat had agreed that she would start their spat. She was the one who had to open the door and go in. She should control the timing. The couple passed room 314. Odette was holding the key in her left hand. She still had the gun in her right hand, under the jacket, which was draped over her forearm. Battat was holding the switchblade at his side. He seemed to be somewhat more focused than he had been when he arrived. Odette was not surprised.
She was, too.
They passed room 312.
Odette turned to Battat. "Why are you stopping?" she asked him. Odette made sure not to shout just so the Harpooner could hear. Her tone was normal, conversational.
"What do you mean, 'Why am I stopping?'" he asked right back.
Odette moved ahead several steps. She stopped in front of room 310. Her heart was speeding. "Aren't we going inside?"
"Yes," he replied impatiently.
"That's not our room," Odette said.
"Yes it is," Battat said.
"No," Odette said. "This is our room."
"We're in 312," Battat said confidently.
She put the key in the slot of 310. That was the signal for Battat to step over to the room. He walked over and stopped directly behind her. His right shoulder was practically touching the door.
Odette's fingers were damp with sweat. She could actually smell the brass of the key. She hesitated. _This is what you've been waiting for_ , she reminded herself. An opportunity to prove herself and to make Viktor proud. She turned the key to the right. The bolt went with it. The door opened.
"I told you this was our room," she said to Battat. Odette swallowed hard. The words had caught in her throat and she did not want to show her fear. The Harpooner might hear it in her voice.
With the door open a sliver, Odette withdrew the key. She slipped it in her pocket and used that moment to listen. The TV was off and the Harpooner was not in the shower. Odette was half hoping he had been in the bathroom, cornered. But she heard nothing. She opened the door a little more.
There was a short, narrow hallway inside. It was cavedark and utterly still. They had assumed the Harpooner would be hiding in the room, but what if he were not? He could be out for a late breakfast. Or he might have left Baku. Perhaps he kept the room as a safe house in case he needed it.
_But what if he_ ' _s waiting for us?_ she thought then. And she answered her own question. _Then we_ ' _ll have to handle the situation._ Viktor used to say that nothing was guaranteed.
"What's wrong, honey?" Battat asked.
The words startled her. Odette looked back at her companion. The American's brow was pinched. He was obviously concerned. She realized that she was probably waiting too long to go in.
"Nothing's wrong," she said. She opened the door a little farther and reached in with her left hand. "I'm just looking for the light."
Odette pushed the door until it was halfway open. She could see the glowing red numbers of the alarm clock on the night table. There was a jagged line of white light in the center of the drapes. Its brilliance only made the rest of the room seem darker.
Odette's gun was still hidden under her jacket, still behind the half-closed door. She found the light switch with her left hand. She flicked it on. The hall light came on as did the lamps on the night tables. The walls and furniture brightened with a dull yellow orange glow.
Odette did not breathe as she stepped into the hallway. The bathroom was to her right. She turned and looked in. There were toiletries on the counter beside the sink. The soap was opened.
She looked at the bed. It had not been slept in, though the pillows had been moved around. She saw a suitcase on the luggage stand, but she did not see the Harpooner's shoes. Maybe he was out.
"Something's wrong here," Odette said.
"What do you mean?"
"That's not our bag on the luggage rack," she replied. Battat stepped in behind her. He looked around. "So I was right," he said. "This isn't our room."
"Then why did the key work?" she asked.
"Let's go back downstairs and find out," Battat urged. He was still looking around.
"Maybe the bellman made a mistake and put someone else in here," Odette suggested.
Battat suddenly grabbed Odette's left shoulder. He roughly shoved her into the bathroom and followed her in.
Odette turned and glared at Battat. He put a finger to his lips and moved very close.
"What's wrong?" she whispered.
"He's in there," Battat said quietly.
"Where?"
"Behind the bed, on the floor," Battat told her. "I saw his reflection in the brass headboard."
"Is he armed?" she asked.
"I couldn't tell," Battat said. "I'm betting he is." Odette put her jacket on the floor. There was no longer any reason to conceal the gun. Battat was standing a few steps in front of her, near the door. Just then she saw a small round mirror and extender arm attached to the wall to his right. She had an idea.
"Hold this," she whispered and handed Battat the gun. Then she walked around him, popped the mirror from its holder, and moved toward the door. Crouching, she carefully poked the mirror into the corridor. She angled it so that she could see under the bed.
No one was there.
"He's gone," she said quietly.
Odette extended the mirror arm a little farther so she could see more of the room. She angled it slowly from side to side. There was no one in the corners, and she could not see a bulge behind the drapes.
"He's definitely not here," she said.
Battat squatted behind her and looked into the mirror. Odette wondered if the feverish man had really seen anyone or if he had been hallucinating.
"Wait a second," Battat said. "Move the mirror so we can see the head of the bed."
Odette did as he asked. The drapes were moving there. It looked as if they were being stirred by a gentle wind.
"The window's open," Odette said.
Battat rose. He entered the room cautiously and looked around. "Damn."
"What?" Odette asked as she stood.
"There's a rope under the drape," he said and started toward it. "The bastard climbed—"
Suddenly, Battat turned and hurried back into the bathroom.
"Down!" he shouted and shoved Odette roughly to the floor. He dove down beside her, next to the fiberglass bathtub. Quickly, he pulled her jacket over their heads and lay beside her, his arm across her back.
A moment later, the hotel room was lit by a yellow red flare. There was a whooshing sound as the air became superheated. The flare died after a moment, leaving a sickly sweet smell mixed with the stench of burning fabric and carpet. The room smoke detector was squealing.
Odette whipped her jacket from them and knelt. "What happened?" she shouted.
"There was a TIC on the desk!" Battat yelled.
"A what?"
"A TIC," Battat said as he jumped to his feet. "Terrorist in a can. Come on—we've got to get out of here!"
Battat helped Odette up. She grabbed her jacket and the two of them swung into the hallway. Battat shut the door and staggered over to room 312. He was obviously having difficulty staying on his feet.
"What's a terrorist in a can?" Odette asked.
"Napalm with a benzene chaser," Battat said. "It looks like shaving cream and doesn't register on airport X-ray machines. All you have to do is twist the cap to set the timer, and _blam_." The main fire alarm began to clang behind them. "Give me the master key," he said as they reached 312.
Odette handed it over.
Battat opened the door. Smoke was already spilling through the door that connected the room to 310. Battat hurried past it and ran to the window. The heavy drapes were open. He edged toward the window, standing back just enough so that he could see out but not be seen from below. Odette stepped up behind him. Battat had to lean against the wall to keep from falling. They looked out at the empty parking lot.
"There," Battat said, pointing.
Odette moved closer. She looked out.
"Do you see him?" Battat asked. "In the white shirt, blue jeans, carrying a black backpack."
"I see him," Odette replied.
"That's the man I saw in the room," Battat said.
_So that_ ' _s the Harpooner,_ she thought. The monster cut an unimposing figure as he walked unhurriedly from the hotel. But his easygoing manner only made him seem even more noxious. People might be dying in the fire he set to cover his escape. Yet he did not care. Odette wished she could shoot him from here.
"He's probably going to keep moving slowly so he won't attract attention," Battat told her. He gave the gun back to her. He was panting, having trouble standing. "You've got enough time to catch up to him and take him out."
"What about you?"
"I'd only slow you down," he said.
She hesitated. An hour ago, she had not wanted him to be part of this. Now she felt as if she was deserting him.
"You're wasting time," Battat said. He gave her a gentle push and started toward the door. "Just go. I'll get to the stairwell and make my way back to the embassy. I'll see if I can do anything from there."
"All right," she said, then turned and hurried toward the door.
"He'll be armed!" Battat yelled after her. "Don't hesitate!"
She acknowledged with a wave as she left the room. The hallway was filling with smoke. The few guests who had been in their rooms were filing into the hallway to see what was happening. Housekeeping staff and security personnel were beginning to arrive. They were helping everyone toward the stairwell.
Odette told one of the security men that someone needed help in 312. Then she rushed ahead to the stairwell.
In less than a minute, she was in the street. The parking lot was on the other side of the building. She ran toward it.
The Harpooner was gone.
**FIFTY-TWO**
_**Washington, D.C. Tuesday, 3:13 A.M.**_
Paul Hood returned to the Cabinet Room and shut the door. He took a calming breath. The room smelled of coffee. He was glad. It covered the stink of treason. Then he took out his Palm Pilot, looked up a number, and went to the phone to enter it. This was not something that Hood wanted to do. It was something he had to do. It was the only way he could think of to prevent what was effectively shaping up as a coup d'état.
The phone was answered right after the second ring. "Hello?" said the voice on the other end.
"Megan, it's Paul Hood."
"Paul, where are you?" asked the First Lady. "I've been worried—"
"I'm in the Cabinet Room," he said. "Megan, listen. Fenwick is definitely involved in a conspiracy of some kind. My feeling is that he, Gable, and whoever else is in this have been trying to gaslight the president."
"Why would anyone want to make my husband think he's lost his mind?" she asked.
"Because they've also set in motion a confrontation with Iran and Russia in the Caspian Sea," Hood told her. "If they can convince the president or the public that he's not equipped to handle the showdown, he'll have to resign. Then the new president will either escalate the war or, more likely, he'll end it. That will win him points with the people and with Iran. Maybe then we'll all divide up the oil wells that used to belong to Azerbaijan."
"Paul, that's monstrous," Megan said. "Is the vice president involved with this?"
"Possibly," Hood said.
"And they expect to get away with it?"
"Megan, they are very close to getting away with it," Hood informed her. "The Caspian situation is revving up, and they've moved the strategy sessions from the Oval Office to the Situation Room. I don't have security clearance to go down there."
"I'll phone Michael on the private number and ask him to see you," Megan told him.
"That won't be enough," Hood said. "I need you to do something else."
Megan asked him what that was. Hood told her. "I'll do it," she said when he was finished. "Give me five minutes."
Hood thanked her and hung up.
What Hood had proposed was a potentially dangerous tactic for him and for the First Lady. And under the best of circumstances, it was not going to be pleasant. But it was necessary.
Hood looked around the room.
This was not like rescuing his daughter. That had been instinctive. He had to act if she were to survive. There had been no choice.
This was different.
Hood tried to imagine the decisions that had been made in this room over the centuries. Decisions about war, about depressions, about human rights, about foreign policy. Every one of them had affected history in some way, large or small. But more important than that, whether they were right or wrong, all of them had required a commitment. Someone had to believe they were making the proper decision. They had to risk anything from a career or national security to the lives of millions on that belief.
Hood was about to do that. He was about to do both, in fact. But there was a proverb that used to hang in the high school classroom where Hood's father taught civics. It was appropriate now:
"The first faults are theirs that commit them. The second theirs that permit them."
As Hood turned and left the Cabinet Room, he did not feel the weight of the decision he made. Nor did he feel the danger it represented.
He felt only the privilege of being able to serve his country.
**FIFTY-THREE**
_**Baku, Azerbaijan Tuesday, 11:15 A.M.**_
It had been a long time since Maurice Charles had to make a sudden retreat from a safe site. It infuriated him to run from a place he had carefully prepared. But it infuriated him even more to run from anyone or anything. It did not even matter to him at the moment how someone had found out where he was. From their accents, the intruders were Russian and American. Perhaps Moscow and Washington had been tracking him without him knowing it. Perhaps he had slipped up somewhere. Or maybe one of his associates had made a mistake.
But Charles did not believe the couple had been there by accident. For one thing, he had taken both of the keys to room 310 when he checked in. The front desk did not have a third key to give out. When the click of the bolt being opened woke him up, he knew something was not right. For another thing, Charles had watched the woman's feet, listened to her speak as she came in. Everything about her entrance was tentative. If she truly thought this were her room, she would have strode in and turned on the light. Women were always eager to prove things when they believed they were correct.
Yet, as angry as Charles was, he refused to give in to his rage. The immediate task was to cover his tracks so he could get away. That meant eliminating the couple who had come to his room. He had not considered calling the assassins he had used the night before. He did not want it to be known that he had run into trouble. That would be bad for his reputation and bad for business.
He had gotten a good look at the couple's feet and pants. That would be enough to identify them. He had his gun and his knife. They would not survive the morning.
Charles had walked halfway into the parking lot before turning around. If the couple were looking out a window to find him, he wanted them to see him. He wanted them to come rushing downstairs to stop him from getting away. That would make them easier to spot. It would also tell him whether or not they had backup. If they had called for help, cars or other personnel would converge on the parking lot within moments. If that did not happen, he could dispatch them and then get out of the city by train as he had planned.
After giving the couple a chance to see him, Charles doubled back to the hotel. He entered by the side door, which led past a row of shops. There were fire sirens approaching the hotel but no police sirens. No other cars came speeding into the lot. That did not mean Charles was home free. But it did suggest that the man and woman had been acting without immediate backup near or on site. Losing himself in a crowd that was fleeing a fire should be easy. First, however, he had to finish his business with the intruders.
**FIFTY-FOUR**
_**Washington, D.C. Tuesday, 3:17 A.M.**_
During the administration of Harry Truman, the White House was virtually gutted and rebuilt due to the weakened condition of its centuries-old wooden beams and interior walls. The Trumans moved across the street to Blair House and, from 1948 to 1952, new foundations were laid and the decaying wooden struts were replaced by steel girders. A basement was also excavated, ostensibly to provide more storage space. In fact, it was created to provide safe areas for the president and members of his staff and family in the event of nuclear attack. Over the years, the basement was secretly expanded to include offices, command headquarters, medical facilities, surveillance posts, and recreational areas. It is now comprised of four levels that go down over two hundred feet.
All four basement levels are only accessible by a pair of elevators. These are located in both the East and West Wings. The West Wing elevator is located a short distance west of the president's private dining room, in a corner that is halfway between the Oval Office and the vice president's office. The carriage is small and wood-paneled and holds six people comfortably. Access to the elevator is gained by thumbprint identification. There is a small green monitor to the right of the door for this purpose. Since the White House recreation areas are down there, all the members of the First Family have access to the elevator.
Hood went to the vice president's office and waited outside. Because the vice president was at the White House, there was a secret service agent standing a little farther along the corridor. The vice president's office was close to the State Dining Room, where the original White House meets the newer, century-old West Wing.
Hood was there less than a minute when Megan Lawrence arrived. The First Lady was dressed in a medium-length white skirt and a red blouse with a blue scarf. She was wearing very little makeup. Her fair skin made her silver hair seem darker.
The secret service agent wished the First Lady a good morning as she passed. Megan smiled back at the young man and then continued on. She embraced Hood warmly.
"Thank you for coming down," Hood said.
Megan put her arm through his and turned toward the elevator. That gave her a reason to stand close to Hood and talk quietly. The secret service man was behind them.
"How are you going to handle this?" she asked.
"It's going to be a tough, uphill fight," Hood admitted. "Back in the Oval Office, the president was very focused. If your husband has had doubts about his ability to function, then what Fenwick and the others have given him is the perfect remedy. A crisis. They couldn't have planned it better. The president seemed to be putting a lot of trust in what Fenwick was telling him. He needed to. It was helping him get his confidence back."
"So you said," the First Lady remarked. "And they're all lies."
"I'm certain of it," Hood assured her. "The problem is, I don't have hard evidence."
"Then what makes you so sure they are lies?" the First Lady asked.
"I called Fenwick's bluff when we were alone in the Cabinet Room," Hood said. "I told him we had the terrorist who orchestrated the situation overseas. I told him the terrorist is going to tell us who he was working for. Meaning Fenwick. Fenwick told me I'll never get the information to the president."
They reached the elevator. Megan gently put her thumb on the screen. There was a faint hum behind it.
"Fenwick will deny he ever threatened you," she pointed out.
"Of course he will," Hood said. "That's why I need you to get the president away from the meeting. Tell him you need to see him for five minutes. If I did that, Fenwick and his people would chew me up. But they'll be very reluctant to attack you. That would turn the president against them."
"All right," Megan replied. The door slid open. The First Lady and Hood stepped in. She pressed button S1—Sublevel One. The door closed, and the elevator began to move.
"There's a guard downstairs," Megan said. "He's going to have to call ahead. I don't have access to the Situation Room."
"I know," Hood replied. "Hopefully, someone other than Fenwick or Gable will answer the phone."
"What if I can only get my husband alone? Just the two of us," Megan asked. "I get his attention. Then what?"
"Tell him what you've noticed over the past few weeks," Hood said. "Talk to him honestly about what we're afraid of, that Fenwick has been manipulating him. Buy me time, even if it's only two or three hours. I need that to get the evidence to stop a war."
The elevator stopped. The door opened. Outside was a brightly lit corridor. The walls were white and lined with paintings of American military officers and famous battles from the Revolution to the present. The Situation Room was located at the end of the corridor behind two black double doors.
A young, blond, fresh-faced marine guard was seated at a desk to the right of the elevator. There was a telephone, a computer, and a lamp on the desk. On a metal stand to his left were several security monitors.
The guard rose and looked from Hood to Megan. "Good morning, Mrs. Lawrence," he said. "Up kind of early for a swim," he added with a smile.
"Up kind of late, Corporal Cain," she smiled back. "This is my guest, Mr. Hood. And I'm not going for a swim."
"I didn't think so, ma'am," he replied. The guard's eyes shifted to Hood. "Good morning, sir."
"Good morning," Hood said.
"Corporal, would you please phone the president?" Megan said. "Tell him I need to speak with him. Privately, in person."
"Certainly," the guard said.
Cain sat and picked up the phone. He punched in the extension of the Situation Room.
Hood did not often pray, but he found himself praying that someone other than one of Fenwick's people was there to answer the phone.
A moment later, the guard said, "The First Lady is here to see the president."
The guard fell silent then. Hood and Megan stood still in the quiet corridor. The only sound was a high faint whine that came from the security monitors.
After a moment, the guard looked up. "No, sir," he said. "She's with a gentleman. A Mr. Hood." The guard fell silent again.
That wasn't a good sign. Only one of Fenwick's people would have thought to ask that question.
After several seconds the guard said, "Yes, sir," and hung up. He rose and looked at the First Lady. "I'm sorry, ma'am. I've been told that the meeting can't be interrupted."
"Told by whom?" she asked.
"Mr. Gable, ma'am."
"Mr. Gable is trying to keep Mr. Hood from delivering an important message to the president," Megan said. "A message that may prevent a war. I need to see my husband."
"Corporal," Hood said. "You're a military man. You don't have to take orders from a civilian. I'm going to ask you to place the call again. Ask to speak to an officer, and repeat the First Lady's message."
"If Mr. Gable gives you trouble, I will take responsibility," Megan said.
Corporal Cain hesitated, but only for a moment. He picked up the phone and remained standing as he punched in the extension.
"Mr. Gable?" he said. "I would like to speak with General Burg."
General Otis Burg was the chairman of the Joint Chiefs of Staff.
"No, sir," Cain said after a moment. "This is a military matter, sir. A security issue."
There was another pause. Hood tasted something tart in the back of his throat. He realized, after a moment, that it was blood. He was biting his tongue. He relaxed.
A few seconds later, Corporal Cain's voice and demeanor changed. His posture was stiffer, his tone formal. He was speaking with General Burg.
Cain repeated the request. Several seconds after that, the young Corporal hung up. He looked at the First Lady.
"Your husband will see you both," he said proudly.
Megan smiled and thanked him.
Hood and Megan turned and hurried down the corridor to the Situation Room.
**FIFTY-FIVE**
_**Baku, Azerbaijan Tuesday, 11:22 A.M.**_
Unsteadily, David Battat made his way down the stairwell.
Because of the late morning hour, not many people were exiting the hotel. Several of the people who did pass Battat asked if he needed help. The American told them that he had inhaled some smoke but would be all right. Hugging the iron banister, he made his way slowly down the concrete stairs. When Battat reached the lobby, he leaned against a wall near the house phones. He did not want to sit down. He was weak and dizzy and afraid he would not get back up. One of the hotel staff members, an assistant manager, asked him who he was and what room he was staying in. He said he was not a guest but had been visiting a friend. The young woman told him that firefighters wanted everyone to go outside. Battat said he would go out as soon as he caught his breath.
Battat looked across the lobby. It was crowded with people, mostly hotel staff, along with about fifty or sixty guests. The guests were concerned about their belongings and asking questions about security. They did not seem in a hurry to leave. There was no smoke in the lobby, and firefighters were just pulling into the circular drive in front of the hotel.
Battat was concerned about how Odette was making out. He had been proud of her when she left the hotel. If she had been afraid, she did not show it. He wished he were a little steadier. He did not like the idea of her having to face the Harpooner alone.
There was a side exit down the corridor to Battat's right. The parking lot was to the right, the front of the hotel to the left. Since the fire trucks were out front, he felt he stood a better chance of catching a taxi in the parking lot. If not, there was a major thoroughfare beyond the parking lot. He had seen it from the upstairs window. He could probably catch a bus there.
Pushing himself off the wall, Battat shuffled down the carpeted hallway. He felt feverish again, though he did not feel worse than he had before. His body was fighting whatever he had been injected with. That probably meant it was viral rather than chemical. He could finally get medical attention and start to shake this.
Battat's vision was misty as he moved past the bank of telephones. There were several shops beyond, their picture windows reflecting each other. There was no one inside, either customers or employees. The displays of shirts and trinkets, of luggage and toys, all seemed to merge as Battat neared. He tried to blink them clear. He could not. The sickness plus the exertion had worn him down much more than he thought. Battat gave serious thought to going back to the lobby and asking the fire department medics for a ride to the hospital. He had been afraid to go there lest someone recognize him from the night before and ask about the dead man in his room. But he was beginning to doubt that he could make it from the hotel, let alone reach the embassy.
Suddenly, someone appeared in Battat's line of vision. The American stopped and squinted. It was a man wearing jeans and a white shirt. There were straps around his shoulder.
A black backpack.
_Oh Christ,_ Battat thought as the man approached. He knew who it was. And he had no doubt that the man recognized him. And knew why he was in such a weakened condition. After all, it was probably this same man who had injected him with the toxin on the beach.
The Harpooner.
The assassin had just walked in through the side door. He was about twenty feet away. He was holding what looked like a knife in his right hand. Battat would not be able to fight him. He had to try and get back to the lobby.
Battat turned, but he moved too fast. His vision blurred and he stumbled against one of the shop windows. He quickly pushed off with his shoulder. He staggered ahead. If he could just get to the lobby, even if he fell square on his face, someone might get to him before the Harpooner could.
Battat reached the bank of phones. He extended his left arm, used it to move himself along the wall. Push, step, push, step.
He was halfway along the bank when he felt starched fabric slide along the front of his throat. A sleeve. A strong arm pulled back, putting Battat into a choke hold.
"The last time we met, I needed you alive," the assassin whispered harshly. "Not this time. Unless you tell me who you're working with."
"Up yours," Battat gasped.
Battat felt a knee against the small of his back. If the Harpooner intended to kill him standing up, he was going to be disappointed. Battat's legs gave out and he dropped to the floor. The Harpooner immediately released Battat and swung around in front of him. He straddled Battat and dropped a knee on his chest. Battat felt a sharp jab in his side and exhaled painfully. One or more of his ribs had been broken. The Harpooner brought the knife to the left side of the American's throat. He pressed the sharp tip just below the ear.
"No," the Harpooner hissed as he glared down at Battat. "This is going up yours."
Battat was too weak to fight. He was aware that he was going to be cut from ear to ear and then left to drown in his own blood. But there was nothing he could do about it. Nothing.
Battat felt a pinch in his throat. A moment later, he heard a soft pop and blood sprayed into his eyes. He thought it would hurt more, having his throat pierced. But there was no pain after the initial pinch. He did not feel the blade moving through his skin. And he was still able to breathe.
An instant later, Battat heard a second pop. He blinked hard to clear the blood from his eyes. He watched as the Harpooner just hovered there, crouched on his chest. Blood was pumping from a wound in his throat. There was no drama in his face, no great gesture befitting the size of his crimes. Just a momentary look of confusion and surprise. Then the killer's eyes shut, the knife fell from his hand, and the Harpooner tumbled to the floor between Battat and the phone bank.
Battat lay there. He did not know exactly what had happened until Odette appeared from behind. She was holding her silenced pistol in front of her and looking down at the Harpooner.
"Are you all right?" she asked Battat.
He reached up and felt his throat. Except for a trickle of blood on the left side, it felt intact.
"I think I'm okay," Battat said. "Thank you."
Battat managed to half wriggle, half crawl away as Odette bent and examined the Harpooner. The woman kept the gun pointed at the Harpooner's head as she felt his wrist for a pulse. Then she held her fingers under his nose, feeling for breath. But she had struck him once in the throat and once in the chest. His white shirt was already thick and dripping with blood.
"I'm glad you followed him," Battat said. He pulled a handkerchief from his pocket and pressed it to his own wound.
"I didn't," Odette said as she rose. "I lost him. But then I thought he might come back to try to cover his tracks. And I knew which one of us he would recognize."
Just then, a housekeeper in the lobby saw the body and screamed. Battat looked back. She was pointing at them and shouting for help.
Odette stepped around the corpse to help Battat to his feet. "We've got to get out of here," she said urgently. "Come on. My car isn't far—"
"Wait," Battat said. He bent over the Harpooner's body and began working on the straps of the backpack. "Help me get this off. There may be evidence we can use to identify his partners."
"You just get on your feet," Odette said as she pulled out her knife. "I'll do that."
Battat pulled himself up, using the ledge under the phones while Odette cut the backpack free. Then, lending Battat her shoulder, Odette led the American down the hall.
They were nearly at the door when someone yelled at them from behind.
"Stop!" a man yelled.
Battat and Odette turned. An elderly hotel security officer was standing just beyond the phone bank. Odette let Battat lean against one of the shop windows while she pulled her badge from her back pocket. She held it toward the security officer.
"I'm Odette Kolker of Metropolitan Squad Three," she said. "The man on the floor is a wanted terrorist. He started the fire in 310. Make sure the room is sealed off. I'm taking my partner to the hospital to see that he gets proper care. Then I'll be back."
Odette did not wait for the man to answer or for other security personnel to arrive. She turned and helped Battat from the building.
She did that well, Battat thought. Gave the man a mission, made him feel important, so he would not interfere with them.
The brisk, clear air and sharp sunshine helped give Battat yet another fresh start. This was the last one, though. He knew that for certain. The American's legs were rubbery, and he was having trouble holding his head up. At least his neck was not bleeding badly. And the handkerchief was keeping most of that inside, where it belonged.
Only after they had made their way through the parking lot to the rear of the hotel did it hit Battat. Odette had done it. She had not only saved his life but she had stopped the Harpooner. She had killed a terrorist who had eluded all of Europe's top security agencies. He was proud to have had a small hand in this. The only downside was that Odette probably would not be able to remain in Baku after this. It was going to be tough to explain this to her police superiors. And if the Harpooner had allies, they might come looking for her. It was probably a good time for Odette to assume another identity.
Five minutes later, Battat was seated in the passenger's seat of Odette's car. They pulled from the curb and headed toward the American embassy. It would be a short ride, but there was something that could not wait. The Harpooner's backpack was in Battat's lap. There was a small padlock on the flap. He borrowed Odette's knife and cut the flap away. He looked inside.
There were some documents as well as a Zed-4 phone. He had worked one of those when he was in Moscow. They were more compact and sophisticated than the American Tac-Sats.
Battat removed the phone from the case. There was an alphanumeric keypad along with several other buttons. Above them was a liquid crystal display on top. He pushed the menu button to the right of the display. For the Harpooner's sake, the instructions were in English.
And for the first time since David Battat arrived in Baku, he did something he had missed.
He smiled.
**FIFTY-SIX**
_**Washington, D.C. Tuesday, 4:27 A.M.**_
The Situation Room was a brightly lit chamber with a low ceiling, white walls, and soft, fluorescent lighting. There was a conference table in the center of the room and chairs along three of the four walls. Computer monitors were attached to the arms of the chairs. They provided aides with up-to-the-minute information. The fourth wall was fitted with a ten-foot-long high-definition TV monitor. The screen was linked to the National Reconnaissance Office. Real-time satellite images could be displayed there with magnification of objects up to three feet long. Most of these high-tech improvements were made within the last four years using over two billion dollars that had been allocated to fixing the White House recreation facilities, including the pool and tennis court.
Hood and the First Lady entered through the door that was under the high-definition monitor. The chiefs of the army, navy, and air force and the commandant of the marine corps were sitting along one side of the table with their chairman, General Otis Burg, in the center. Burg was a big, barrel-chested man in his late fifties. He had a shaved head and steel gray eyes that had been hardened by war and political bureaucracy. The joint chiefs' aides were seated behind them. Along the other side of the table were the president, the vice president, NSA head Fenwick, Chief of Staff Gable, and Deputy National Security adviser Don Roedner. Judging by their tense expressions, either it was a difficult meeting or they did not appreciate the interruption. Or both.
Several members of the Joint Chiefs of Staff registered surprise to see Hood with the First Lady. So did the president. He had been in the process of rising to go into an adjoining study and talk with her. The president froze and looked from Megan to Hood, then back to Megan. The new arrivals stopped at the head of the conference table.
"What's going on?" the president asked.
Hood glanced at the joint chiefs, who were a wall of impatience. He still did not know whether the frustration was with him or with the issue at hand. All he knew was that he would not have much time to present his case.
"Sir," Hood said, "there is increasing evidence that the attack on the Iranian oil rig was executed not by Azerbaijanis but by Iranians under the direction of the terrorist known as the Harpooner."
The president sat back down. "Why?" he asked. "So that Iran could justify moving ships into the region and seize as many oil resources as possible," Hood told him.
"And risk a military showdown with the United States?" Lawrence asked.
"No, sir," Hood replied. He looked at Fenwick. "I believe there is an agreement in place to make sure the United States does not interfere. Then, when the tensions are defused, we simply buy our oil from Teheran."
"And when was this agreement made?" the president asked.
"Yesterday, in New York," Hood said. "Probably after many months of negotiations."
"You're referring to Jack's visit to the Iranian mission," the president said.
"Yes, sir," Hood replied.
"Mr. Fenwick was not empowered to make such a promise," the president pointed out. "If he did make one, it would not be valid."
"It might be if you were not in office," Hood said.
"This is ridiculous!" Fenwick declared. "I was at the Iranian mission to try and expand our intelligence resources in the Middle East. I've explained that, and I can document it. I can tell you who I met with and when."
"All part of the big lie," Hood said.
"Mr. Roedner was with me," Fenwick said. "I have the notes I made, and I'll be happy to name my contacts. What do you have, Mr. Hood?"
"The truth," he replied without hesitation. "It's the same thing I had when you vowed to keep me from seeing the president."
"What I vowed was to keep you from bothering the president," Fenwick insisted. "Secret deals with Iran. The president being out of office. This isn't the truth, Mr. Hood. It's paranoia!"
The vice president looked at his watch. "Mr. President, forgive me, but we're wasting time. We need to get on with this meeting."
"I agree," said General Burg. "I'm not up to speed on any of this back-and-forth, and it isn't my job to say which of these gentlemen is full of gravy. But whether we play offense or defense, we have to make some quick decisions if we're going to match Iran's deployment."
The president nodded.
"Then get on with the meeting, Mr. President, General Burg," Hood said. "But please delay taking military action for as long as possible. Give me time to finish the investigation we've begun."
"I asked for evidence to back your claims," the president said, his voice extremely calm. "You don't have that."
"Not yet," Hood said.
"And we don't have the extra time I thought there'd be to investigate. We've got to proceed as if the Caspian threat is real," the president said with finality.
"Which is exactly what they want you to do!" Hood said. He was growing agitated and had to pull himself back. An outburst would undermine his own credibility. "We believe a crisis is being engineered, one that will call into question your ability to govern."
"People have argued about that for years," the president said. "They voted me out of office once. But I don't make decisions based on polls."
"I'm not talking about a policy debate," Hood said. "I'm talking about your mental and emotional state. That will be the issue."
Fenwick shook his head sadly. "Sir, mental health is the issue. Mr. Hood has been under a great deal of stress these past two weeks. His teenage daughter is mentally ill. He's going through a divorce. He needs a long vacation."
"I don't think Mr. Hood is the one who needs a leave of absence," the First Lady said. Her voice was clear and edged with anger. It quieted the room. "Mr. Fenwick, I have watched my husband being misled and misinformed for several weeks now. Mr. Hood looked into the situation at my personal request. His investigation has been methodical, and I believe his findings have merit." She glared at Fenwick. "Or do you intend to call me a liar as well?"
Fenwick said nothing.
The president looked at his wife. Megan was standing straight and stoic at Hood's side. There was nothing apologetic in her expression. The president looked tired, but Hood thought he also seemed sad. He could not tell whether it was because Megan had run an operation behind his back or because he felt he had let her down. The couple was silent. It was clearly an issue they would settle some other time, in private.
After a moment, the president's eyes returned to Hood. The sadness remained. "Your concern is noted and appreciated," the president said. "But I won't jeopardize the nation's interests to protect my own. Especially when you have no evidence that they're at risk."
"All I want is a few hours," Hood said.
"Unfortunately, we don't have a few hours," the president replied.
For a moment, Megan looked as though she was going to hug her husband. She did not. She looked at Fenwick and then at the joint chiefs. "Thank you for hearing us out," she said. "I'm sorry to have interrupted." She turned and started toward the door.
Hood did not know what else to say. He would have to go back to the Cabinet Room and work with Herbert and Orlov. Try to get the proof the president needed and get it quickly.
He turned to follow the First Lady from the Situation Room. As he did, there was a gentle beep from somewhere in the room. A cell phone. The sound had come from the inside pocket of Fenwick's suit.
_He shouldn't be able to get a signal in here,_ Hood thought. The walls of the Situation Room were lined with chips that generated random electrical impulses or impedence webs. The IWs were designed to block bugs from broadcasting to anyone on the White House grounds. They also blocked cell phone calls with one exception: transmissions relayed by the government's Hephaestus satellite array.
Hood turned back as the NSA chief had slipped a hand into his jacket. Fenwick took out the phone and shut off the ringer.
Bingo.
If it got through IW security, it had to be a Hephaestus call. Highest security. Who wouldn't Fenwick want to talk to right now?
Hood leaned over the NSA chief and pulled the phone from his hand. Fenwick reached for it, but Hood stepped away.
"What the hell are you doing?" Fenwick demanded. He pushed the chair back and rose. He walked toward Hood.
"I'm betting my career on a hunch," Hood said. He flipped open the cover and answered the call. "Yes?"
"Who is this?" asked the caller.
"This is Jack Fenwick's line at the NSA," Hood said. He walked toward the president. "Who's calling?"
"My name is David Battat," said the clear voice on the other end.
Hood felt the world slide off his shoulders. He held the cell phone so the president could listen as well. Fenwick stopped beside them. The NSA head did not reach for the phone. He just stood there. Hood saw just where the weight of the world had shifted.
"Mr. Battat, this is Paul Hood of Op-Center," said Hood.
"Paul Hood?" Battat said. "Why are you answering this line?"
"It's a long story," Hood said. "What is your situation?"
"A helluva lot better than Mr. Fenwick's," Battat said.
"We just took down the Harpooner and recovered his secure phone. This number was the first one that came up on the Harpooner's instant-dial menu."
**FIFTY-SEVEN**
_**Washington, D.C. Tuesday, 4:41 A.M.**_
Paul Hood stepped to a corner of the room to finish speaking with Battat. It was important that he get all the information he could about the Harpooner and what had happened.
While Hood did that, President Lawrence stood. He glanced over at his wife, who was standing by the door. He gave her a little smile. Just a small one to show that he was okay and that she had done the right thing. Then Lawrence turned to Fenwick. The NSA chief was still standing beside him. His arms were stiff at his side and his expression was defiant. The other men remained seated around the table. Everyone was watching Lawrence and Fenwick.
"Why did the Harpooner have your direct number and the Hephaestus access code?" the president asked. There was a new confidence in his voice.
"I can't answer that," Fenwick said.
"Were you working with Iran to orchestrate a takeover of Azerbaijani oil deposits?" the president asked.
"I was not."
"Were you working with anyone to organize a takeover of the Oval Office?" the president asked.
"No, sir," Fenwick replied. "I'm as puzzled as you are."
"Do you still believe that Mr. Hood is a liar?"
"I believe that he's misinformed. I have no explanation for what is going on," Fenwick said.
The president sat back down. "None at all."
"No, Mr. President."
The president looked across the table. "General Burg, I'm going to get the secretary of state and our UN ambassador working on this right away. How would you feel about coordinating a midlevel alert for the region?"
Burg looked at his colleagues in turn. No one voiced a protest. The general looked at the president. "Given the confusion about just who we should be fighting, I'm very comfortable with yellow status."
The president nodded. He looked at his watch. "We'll reconvene in the Oval Office at six-thirty. That will give me time to work with the press secretary to get something on the morning news shows. I want to be able to put people at ease about our troops and about the status of our oil supply." He regarded vice president Cotten and Gable. "I'm going to ask the attorney general to look into the rest of this situation as quietly as possible. I want him to ascertain whether treasonable acts have been committed. Do any of you have any thoughts?"
There was something challenging in the president's voice. Hood had just finished up with Battat and turned back to the table. He remained in the corner, however. Everyone else was still.
The vice president leaned forward and folded his hands on the table. He said nothing. Gable did not move. Fenwick's deputy, Don Roedner, was staring at the conference table.
"No suggestions at all?" the president pressed.
The heavy silence lasted a moment longer. Then the vice president said, "There will not be an investigation."
"Why not?" asked the president.
"Because you will have three letters of resignation on your desk by the end of the morning," Cotten replied.
"Mr. Fenwick's, Mr. Gable's, and Mr. Roedner's. In exchange for those resignations, there will be no charges, no prosecution, and no explanation other than that members of the administration had a difference of policy opinion."
Fenwick's forehead flushed. "Three letters, Mr. Vice President?"
"That's correct, Mr. Fenwick," Cotten replied. The vice president did not look at the NSA chief. "In exchange for complete amnesty."
Hood did not miss the subtext. Nor, he was sure, did the president. The vice president was in on this, too. He was asking the others to take a fall for him—though not a big one. Quitting an administration, high-ranking officials often tumbled upward in the private sector.
The president shook his head. "I have here a group of administration officials who apparently conspired with an international terrorist to steal oil from one nation, give it to another, reap foreign policy benefits, and in the process steal the office of president of the United States. And you sit there arrogantly declaring that these men will be given de facto amnesty. And that one of them, it appears, will remain in office, in line for the presidency."
Cotten regarded Lawrence. "I do declare that, yes," he said. "The alternative is an international incident in which the United States will be seen as having betrayed Azerbaijan. A series of investigations and trials that will ghost this administration and become its sole legacy. Plus a president who was unaware of what was going on among his closest advisers. A president who his own wife thought might be suffering from a mental or emotional breakdown. That will not boost public confidence in his abilities."
"Everyone gets off," the president said angrily. "I'm supposed to agree to that?"
"Everyone gets off," the vice president repeated calmly.
"Mr. Vice President, sir?" General Burg said. "I just want to say if I had my weapon here, I would shoot you in the ass."
"General Burg," the vice president replied, "given the pitiful state of our military, I'm confident you'd miss." He regarded the president. "There was never going to be a war. No one was going to shoot at anyone or be shot at. Peace would have been reached with Iran, relations would have been normalized, and Americans would have had a guaranteed fuel supply. Whatever one may think of the methods, this was all done for the good of the nation."
"Any time laws are broken, it is not for the good of the nation," the president said. "You endangered a small, industrious country trying to get its footing in a post-Soviet world. You sought to undo the will of the American electorate. And you betrayed my faith in you."
Cotten rose. "I did none of those things, Mr. President," he replied. "Otherwise, I would be resigning. I'll see you all at the six-thirty meeting."
"You will not be needed there," the president said.
"Ah," said the vice president. "You would prefer I go on the _Today Show_ to discuss administration policy in the Caspian region."
"No," the president replied. "I would prefer that you draft your letter of resignation to submit with the others."
The vice president shook his head. "I won't do that." "You will," the president replied. "And attribute your resignation to mental exhaustion. I won't make you a martyr to an anticonstitutional fringe. Find some other line of work, Mr. Cotten."
"Mr. President, you are pushing the wrong man," Cotten warned.
"I don't think so," the president replied. His eyes and voice grew steely. "You're correct, Mr. Cotten. I don't want a national or international scandal. But I'll suffer those before I leave a traitor in the line of succession to the office of president. Either you resign or, in exchange for that amnesty, I will urge Mr. Fenwick and his associates to tell the attorney general what they know about your involvement in this operation."
Cotten was silent. Red and silent.
The president reached for the phone in front of him. He pushed a button. "Corporal Cain?"
"Yes, Mr. President?"
"Please have an unarmed detail report to the Situation Room at once," Lawrence told him. "There are some gentlemen who need to be escorted to their offices and then from the grounds."
"Unarmed, sir?" Cain repeated.
"That's right," Lawrence said. "There won't be any trouble."
"Right away, sir."
"Wait outside the door when you're finished," the president added. "The men will be joining you in just a moment."
"Yes, sir."
The president hung up. He regarded the four men. "One more thing. Information about your participation in these events must not leave this room. Amnesty will not be based on anything I intend to do for you. Pardoning you would be a sin. It will be based solely on the absence of news."
The men turned and walked toward the door. Megan Lawrence stepped aside.
Hood's eyes met hers. The First Lady was glowing with pride. They were obviously thinking the same thing.
She was the only Lawrence who would be stepping aside this day.
**FIFTY-EIGHT**
_**Saint Petersburg, Russia Tuesday, 12:53 P.M.**_
In most intelligence agencies it's often difficult to tell night from day. That's because conspiracy and espionage never rest, so the counterterrorists and spybusters also work around the clock. Most are usually fully staffed. The distinction is even less noticeable in the Russian Op-Center because the facility is below ground. There are no windows anywhere.
But General Orlov always knew when it was afternoon. He knew because that was when his devoted wife called. She always rang shortly after lunchtime to see how her Sergei's sandwich was. She phoned even today, when she had not had time to prepare a bag lunch before he left.
Unfortunately, the call was brief. It often was. They usually had longer conversations when he was in space than they did at the Op-Center. Two minutes after Masha called, Orlov received a call from Odette. He told Masha he would have to call her back. She understood. Masha always understood.
Orlov switched lines. "Odette, how are you?" the general asked eagerly.
"I'm very well," the woman replied. "We accomplished our mission."
Orlov was unable to speak for a moment. He had been worried about Odette and concerned about the mission. The fact that she was safe and triumphant left him choked with pride.
"We terminated with complications," Odette went on, "but we got away. There were no other injuries."
"Where are you now?" Orlov asked.
"At the U.S. embassy," she said. "Mr. Battat is getting medical care. Then I'll be going to the police station. I had to show my badge to a hotel worker, but I think I'll be able to work it out with my superior. The Harpooner set a fire. I can tell the captain that I went there to see if I could help."
"So you don't want to leave, then?" Orlov. asked.
"I think there will be some interesting problems because of all this," she said. "I'd like to stay for a while."
"We'll talk about it," Orlov said. "I'm proud of you, Odette. And I know someone else would be, too."
"Thank you," she said. "I think Viktor was looking out for me today. So was David Battat. I'm glad you asked him to come along."
Odette gave Orlov additional information about what had happened. They arranged to talk again in six hours. If it became necessary for Odette to leave Baku, there was an Aeroflot flight she could catch at eight P.M.
Orlov took a moment to savor the victory's many rewards. First, having won the battle against a tenacious enemy. Second, having made the right decision to send Odette and Battat into the field together. And finally, having been able to help Paul Hood. Not only did it repay an old debt, but it hopefully opened the door to future close collaborations.
Odette said that Battat had spoken with Paul Hood. There was nothing Orlov could add to that. Orlov would call him in a few minutes. First, however, he wanted to brief the staff members who had been involved in the hunt.
He was about to send for Grosky and Kosov when the men came to his office door. Kosov was carrying a rolled-up blueprint.
"General," said the outgoing Kosov, "we have some news."
"Good news?" Orlov asked.
"Yes, sir," Kosov said. "That information the Americans gave us about the Harpooner's Russian identity has proved very useful."
"In what way?" Orlov asked.
"It suggested to us how he has been able to come to Moscow and disappear without ever being seen," Kosov said. He stepped forward and unrolled the blueprint on Orlov's desk. "This is a map of the old Soviet army railroad routes," he said. "As you know, they go underground well outside of Moscow and stop at various points beneath the city."
"It was designed that way so troops could be moved into place clandestinely, to put down riots or even foreign attacks," Grosky added.
"I know about these," Orlov said. "I've traveled in them."
"But what you may not know about is this one," Kosov said.
The intelligence analyst used a pen to point to a faint red line. It led from Kievskaya metro stop to several other stations around the city. Kosov was right. Orlov did not know what it was.
"This is unmarked, as you can see, even though it links up to the main trunk," Kosov continued. "We thought it might be a service tunnel of some kind, but we looked at an older map from the GRU files just to make certain. It was the old Stalin tunnel. If the German army had ever reached Moscow during World War II, Stalin would have been evacuated through this system. Only his closest military advisers know that it existed." Kosov stepped back and folded his arms. "We believe, sir, that all we need to do to catch our rat is to put video cameras at the entrance and exit. Sooner or later, the Harpooner is certain to show up there."
Orlov looked at the map for a moment, then sat back. "You may have solved a very perplexing riddle," he said. "Excellent work."
"Thank you, sir," Kosov beamed.
"Fortunately," Orlov went on, "the Harpooner was killed earlier today. The only rats that will be using the tunnel are the four-legged kind."
Grosky's mouth twisted slightly at one end. Kosov's expression seemed to fall entirely.
"But we could not have taken him without you, and I will say so in my report to the president's director of intelligence review," Orlov promised. He rose and extended his hand to each man in turn. "I am proud of you both and deeply grateful."
Kosov's disappointment evaporated quickly. Grosky's mouth remained bent. But even Grosky's perpetual sourness couldn't spoil the moment. An inexperienced woman, a sick man, and two former enemies had joined forces to win a big one.
It was an extraordinary feeling.
**FIFTY-NINE**
_**Washington, D.C. Tuesday, 5:04 A.M.**_
After the vice president and his team had been ushered away, the president asked Hood to wait for him. Hood stepped outside the Situation Room as the president and Megan stood alone behind the conference table, talking. The president took his wife's hands in his. He seemed composed, once again in control.
The Joint Chiefs of Staff filed out quickly after Cotten's group had been led off. They headed quickly toward the elevator. Before leaving, General Burg paused and turned to Hood. He shook the intelligence leader's hand.
"What you did in there was good work, smart work," the general said. "It was also ballsy. My congratulations, Mr. Hood. I'm proud to be associated with you. Proud to be an American."
Coming from anyone else under almost any other circumstance, that sentiment might have sounded corny. But the system had worked, despite the formidable forces and pressures rallied against it. General Burg had every reason to feel proud. Hood did.
"Thank you, General," Hood said sincerely.
After the Joint Chiefs left, the hall was quiet, save for the whispered conversation of the president and First Lady. Hood was relieved but still a little shell-shocked by everything that had just happened. He did not believe that the press would accept the given explanations for a mass resignation of the vice president and top administration officials. But that was a battle for other warriors and another day. Hood and his team had saved the presidency and defeated the Harpooner. Right now, all he wanted to do was hear what the president wanted to say, get back to the hotel, and go to sleep.
The president and First Lady emerged a few minutes later. They looked tired but content.
"Did your man in Baku have anything else to say?" the president asked as he walked toward Hood.
"Not really, sir," Hood said. "He's at the American embassy now. We'll talk again. If there's any other intel, I'll let you know at once."
The president nodded as he stopped next to Hood. Megan was standing beside him.
"I'm sorry to have kept you waiting, but Mrs. Lawrence and I wanted to thank you together," the president said. "She told me you've been working on this nonstop since Sunday night."
"It's been a long day and a half," Hood admitted.
"You're more than welcome to sleep upstairs, if you'd like," the president said. "Or a driver will take you home."
"Thank you, sir," Hood said. He looked at his watch. "Rush hour doesn't start until six, so I should be all right. I'll just roll down the window and enjoy the fresh air."
"If you're certain," the president said. He offered his hand. "I've got work to do. Megan will make sure you get back upstairs. And thank you again. For everything."
Hood accepted the president's hand. "It's been an honor, sir."
After the president left, Megan faced Hood. There were tears in her eyes. "You saved him, Paul. While I stood there, I watched him pull back from wherever they had taken him."
"He did that by himself," Hood said. "And without your heads-up, I wouldn't have acted on any of this."
"For once in your life, Paul, give the self-effacement a rest," Megan said. "You took all the risks in there. If things had gone the other way, you would have been ruined."
Hood shrugged.
Megan grimaced. "You're exasperating. Michael is right about one thing, though. You're tired. Are you sure you won't rest awhile before you head back?"
"I'm sure," Hood said. "There are still a few things we have to tie up, and I want to call Sharon."
"How's that going?" Megan asked.
"As good as could be expected," Hood said. "Harleigh's in the hospital so we're focused on that."
Megan touched his arm. "If you want to talk, I'm here."
Hood thanked her with a smile. They left together, and then Hood headed for his car. A plane rumbled in the distance. Hood looked up as he unlocked his car door. The first hint of daylight was appearing on the other side of the White House grounds.
Somehow, that seemed fitting.
**SIXTY**
_**Washington, D.C. Tuesday, 6:46 A.M.**_
Hood was surprisingly alert when he reached his office.
Mike Rodgers was gone. He had left a voice mail message two hours before about a military situation that was developing along the Pakistan-India border. Rodgers said he had gone home to get some rest before going off to a meeting at the Pentagon. Although General Rodgers was officially attached to Op-Center, he was called upon to assess flashpoints in different corners of the world.
Bob Herbert was still awake and "at the switch," as he described it. He came to Hood's office and quickly brought Hood up to speed on the little additional intelligence that Orlov had on the Harpooner and his movements. Then Herbert asked Hood how things had gone at the White House.
Herbert listened intently to his chief's matter-of-fact recitation of the facts. When Hood was finished, the intelligence head sighed. "I've been sitting here collecting intelligence while you were out there, in the field, saving America and the Constitution from a demagogue."
"Some guys have all the luck," Hood said dryly.
"Yeah," Herbert said. "But you're not the one I envy."
"Oh?"
Hood thought for a moment. Then, just before Herbert said it, Hood knew what was coming.
"I wish I had been the one who pulled the plug on the Harpooner," Herbert said. His voice was a low monotone. His eyes were staring. His mind was somewhere else. "I'd have done it slowly. Very slowly. I would have made him suffer the way I've suffered without my wife."
Hood did not know what to say, so he said nothing. Herbert looked at him. "I've got a lot of vacation time coming, Paul. I'm going to take it."
"You should," Hood said.
"I want to go to Baku and meet this woman Odette," Herbert said. "I want to see where it happened."
"I understand," Hood told him.
Herbert smiled. His eyes were damp. "I knew you would." His voice cracked. "Look at me. You're the one who's had his ass on the firing line twice in the past two weeks. But I'm the one cracking up."
"You've been carrying this pain and frustration for nearly twenty years," Hood said. "It's got to come out." He snickered humorlessly. "I'll break, too, Bob. One day the UN thing, the White House—it's all going to hit me and I'll come apart big time."
Herbert smiled. "Just hold on till I'm back from vacation so I can pick up all the cogs and wheels."
"It's a deal," Hood said.
Herbert wheeled around the desk and hugged Hood warmly. Then he turned his chair around and left the office.
Hood put in a quick call to General Orlov, thanking him for everything he had done and suggesting that they work out a way to integrate their two systems on some level. Create an Interpol for crisis management. Orlov was all for the idea. They agreed to talk about it the following day.
After hanging up with Orlov, Hood looked at the computer clock. It was still too early to call home. He decided to go to the hotel and phone Sharon and the kids from his room. There would be no other calls, no distractions.
Hood left his office and headed back upstairs. He greeted members of the day team as they arrived: Darrell McCaskey, Matt Stoll, and Liz Gordon. He told them each to go see Bob Herbert for an update. Hood said he would brief them more fully later in the day.
By the time he reached the parking lot, he was starting to crash. The caffeine had made its way through his system. Hood's body was definitely winding down. As he neared his car, he saw Ann Farris. She was just pulling through the gate. The press liaison saw him, waved, and drove over.
She rolled down the window. "Is everything all right?" she asked.
Hood nodded. "Just tired," he said. "Bob is still there. He'll brief you. There's nothing we have to press release, though. Not yet."
"Where are you headed?" she asked.
"Back to the hotel," he said. "I've got to get some rest."
"Hop in and I'll run you over," she said. "You don't look like you should be driving."
"I don't know when I'll be coming back," Hood told her. "I need the car."
"You'll be coming back this afternoon," Ann said. "I know you. A two- or three-hour power nap, and then you'll be back. Just call when you wake up, and I'll come and get you."
The offer sounded inviting. He did not feel like driving anymore.
"All right," Hood said.
Hood went to the passenger's side and slid in. He shut his eyes and had to be nudged awake when they arrived. He was groggy. Ann left her car out front and walked him to his room.
She returned a few minutes later, climbed behind the wheel, and sat there for a moment.
"Screw this," she said. Instead of driving off, she moved the car to the main lot. Then she went back inside.
Hood had just finished his short chat with Sharon. His wife had said that there had been no change in anything.
Hood removed his shoes and tie and was unbuttoning his shirt when there was a knock on the door. It had to be a bellboy with a fax from the office or his attorney. No one else knew he was here. He fished a dollar from his wallet and opened the door. He was surprised to see Ann.
"Thanks," she said, "but I didn't come back for my tip."
He smiled and let her in.
Ann was still wearing her jacket, but she looked different. There was something more accessible about her. It was in the eyes, he decided.
Hood shut the door behind her. As he did, he was surprised by something else. He was glad that she had come back.
**EPILOGUE**
_**Baku, Azerbaijan Tuesday, 3:00 P.M.**_
Throughout the late morning and early afternoon, the surprises kept coming for Ron Friday, each one more startling than the last.
First, Friday was surprised to find David Battat at the embassy. The CIA operative was being nursed to health by the embassy medic. He looked in remarkably good health and even better spirits.
Next, Friday was even more surprised to hear that a local policewoman had been responsible for killing the Harpooner. Friday himself would not have known how to find him or what he looked like. He could not imagine how a policewoman had gotten to him. Maybe it was an accident or they were mistaken. Perhaps someone else had been mistaken for the Harpooner. In any case, authorities were speculating that he had been the man behind the attack on the Iranian oil rig. Prodded by the United States, military mobilization was being delayed while an investigation was under way.
But the biggest surprise was the call from Jack Fenwick's executive secretary, Dori. Her boss, Don Roedner, Red Gable, and the vice president were all resigning later that morning. Dori did not know anything about the operation Fenwick had been running and was stunned by the announcement. Friday was stunned, too. He could not imagine how everything had come unraveled. He could not imagine what his old mentor must be feeling. He wished he could speak with him, say something reassuring.
But Friday had not been able to reach Fenwick on his cell phone. Someone else answered, and he quickly hung up. He did not know whether the NSA chief would be investigated and whether that investigation would ever get to him. Friday did not generally report to Fenwick directly. He reported to T. Perry Gord, assistant deputy director of South Asian affairs. There was no reason it should reach him. Gord knew nothing about Fenwick's other activities.
Still, after weighing whether or not to remain in Baku, Friday decided it would be best to leave. He would go somewhere that was a little bit off the radar. Someplace the international press would not be paying so much attention to over the next few weeks.
Fortunately, there was a situation developing on the India-Pakistan border that fell within Gord's jurisdiction. Rather than send someone over from Washington, Friday arranged to have himself transferred to the embassy in Islamabad in order to do on-site intelligence gathering. There was a Pakistan International Airlines flight leaving Moscow the following morning. He would fly from Baku tonight and make certain that he was on it.
_It would have been nice_ , he thought, _if it had all worked out for Fenwick._ With Cotten in the White House, Fenwick would have had unprecedented access and power. And any one of the few people who had taken part in the changeover would have been rewarded. Not just for their contribution but for their silence. On the other hand, one of the reasons Friday had gone into intelligence work was for the challenge. The danger. He had done his job. And he had enjoyed doing it, taking out a CIA operative who had CIA swagger. The kind that had helped to keep Friday back his whole life. That swagger did not prevent Thomas Moore from walking into a neat little NSA trap.
All right, Friday thought. Things had not worked out. It was on to the next project.
That, too, was one of the things Ron Friday enjoyed about intelligence work. It was never the same. He never knew who he might be working with—or against. In Islamabad, for example, it was not just a question of getting a good man to the flashpoint. It was getting the right man there quickly. Gord had heard through the grapevine that someone from Op-Center was being brought in to consult on the India-Pakistan situation and was probably going to be sent to the region. Over the past few years, Op-Center had taken over a great deal of the work Fenwick's team used to handle. That had resulted in ongoing budget and personnel battles at the NSA. Fenwick got the monies he wanted but it had turned a heated rivalry into a ferocious one.
Friday carefully disassembled and packed a rifle. He took along two boxes of shells. Because he was going to Islamabad with diplomatic credentials, his luggage would not be checked.
Showing up Op-Center was important. But as Friday had demonstrated in Baku and elsewhere, outperforming a rival was not the only way to bring them down.
Whoever this man Mike Rodgers was, he would learn that the hard way.
_Novels by Tom Clancy_
THE HUNT FOR RED OCTOBER
RED STORM RISING
PATRIOT GAMES
THE CARDINAL OF THE KREMLIN
CLEAR AND PRESENT DANGER
THE SUM OF ALL FEARS
WITHOUT REMORSE
DEBT OF HONOR
EXECUTIVE ORDERS
RAINBOW SIX
SSN: STRATEGIES OF SUBMARINE WARFARE
_Created by Tom Clancy and Steve Pieczenik_
TOM CLANCY'S OP-CENTER
TOM CLANCY'S OP-CENTER: MIRROR IMAGE
TOM CLANCY'S OP-CENTER: GAMES OF STATE
TOM CLANCY'S OP-CENTER: ACTS OF WAR
TOM CLANCY' S OP-CENTER: BALANCE OF POWER
TOM CLANCY'S OP-CENTER: STATE OF SIEGE
TOM CLANCY'S OP-CENTER: DIVIDE AND CONQUER
TOM CLANCY' NET FORCE
TOM CLANCY'S NET FORCE: HIDDEN AGENDAS
TOM CLANCY'S NET FORCE: NIGHT MOVES
_Created by Tom Clancy and Martin Greenberg_
TOM CLANCY'S POWER PLAYS: POLITIKA
TOM CLANCY'S POWER PLAYS: RUTHLESS.COM
TOM CLANCY'S POWER PLAYS: SHADOW WATCH
_Nonfiction_
SUBMARINE: A GUIDED TOUR INSIDE A NUCLEAR WARSHIP
ARMORED CAV: A GUIDED TOUR OF AN ARMORED CAVALRY REGIMENT
FIGHTER WING: A GUIDED TOUR OF AN AIR FORCE COMBAT WING
MARINE: A GUIDED TOUR OF A MARINE EXPEDITIONARY UNIT
AIRBORNE: A GUIDED TOUR OF AN AIRBORNE TASK FORCE
CARRIER: A GUIDED TOUR OF AN AIRCRAFT CARRIER
INTO THE STORM: A STUDY IN COMMAND
( _written with General Fred Franks)_
EVERY MAN A TIGER
_(written with General Charles Horner)_
|
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