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Network of reference laboratories, research centres and related organisations for monitoring of emerging environmental substances
NORMAN Network
NORMAN Bulletin
NORMAN GA meetings
Emerging Substances
Topics and Activities
QA/QC Issues
Workshop on 3Ms in DW - Conclusions
Short report of the NORMAN workshop co-organised by KWR and IWW on micropollutants, metabolites and mixtures in (the sources of) drinking water, (Nieuwegein - NL, 18-19 June 2012)
The workshop consisted of two days. On the first day, six international speakers thoroughly set the scene by providing technical presentations in the field of:
(i) occurrence and human significance of (novel) disinfection byproducts (DBPs) (Susan Richardson),
(ii) the toxicological assessment of chemical compounds (Tamara Grummt),
(iii) regulatory status of (pesticide) metabolites ( Arnaud Boivin ),
(iv) cumulative risk assessment of chemical compounds in food (Bernadette Ossendorp),
(v) mixture toxicity interaction of chemical compounds ( Thomas Backhaus ) and
(vi) novel strategies for future risk assessment of compounds in (the sources of) drinking water ( Ron van der Oost ).
Susan Richardson showed in her presentation that only a small section of the halogenated DBPs is known (~30%) and that thereof only a fraction is under regulation (11 compounds in total). In addition, laboratory animal experiments showed that there is a general mismatch between regulated (halogenated) compounds and toxicologically relevant compounds. Bioassays are presented as useful screening tools to assess the risks of the increasing amount of chemical compounds. In the following presentation, Tamara Grummt (UBA) discussed the need for novel toxicological concepts. An example is the application of threshold based trigger-values such as the Threshold of Toxicological Concern (TTC).
Tamara supported the opinion of Susan Richardson that simple and robust bioassays can be a very useful addition to chemical screening methods of drinking water and its sources. Finally, Tamara suggested implementing the concept of toxicological safety, rather than toxicological risk. The fact that there is an increasing amount of compounds in the environment is also recognised by regulators.
Arnaud Boivin (ANSES) discussed the various regulations that are available for compounds (with an emphasis on pesticides) and showed that the data requirements vary, which can lead to different conclusions. In addition, Arnaud showed that, at present, the cumulative effects of compounds are not assessed, but there is general agreement to work on this. Finally, the role of metabolites was briefly discussed and Arnaud mentioned that these compounds are not always integrated but - at least for active substances - they really need to be.
In the next presentation, Bernadette Ossendorp (RIVM) demonstrated the latest efforts of the European Food Safety Authority (EFSA) with regard to cumulative risk assessment of food-borne compounds. The pragmatic approach that is now adopted for pesticides relies on the concept of dose addition (being the most relevant). According to this concept, so-called Cumulative Assessment Groups (CAGs) are built, which contain compounds with the same mode of action. The downside of implementing this concept in regulation is the high workload that is associated with it. Finally, Bernadette proposed to use a simple model which also takes into account metabolites.
Next, Thomas Backhaus (University of Gothenburg) demonstrated that mixtures do matter and that Concentration Addition (CA) is the most pragmatic concept for risk assessment. The latter concept assumes that compounds do not interact with each other and does not take mechanisms such as synergism into account. An alternative is the concept of Independent Action (IA), but the disadvantage is that it is very data demanding. To overcome limitations associated with CA (compound concentrations required) and IA (compound mode of action required), the TTC has been proposed. It is a nice tool for bridging data gaps and can be a decision point for further investigation. However, it has its own intrinsic problems (see further in this short summary).
In the final presentation, Ron van der Oost (Waternet) showed a novel risk assessment framework in which bioassays play a pivotal role. As mentioned by other speakers, the downside of bioassays, however, is that compound identity remains unknown. This is where Effect Directed Analysis (EDA) can come into play. For future risk assessment a combination of toxicological assessment and chemical assessment will be required. For bioassays it will become increasingly important to derive thresholds below which risks for human health are negligible.
On the second day, the participants discussed the most relevant topics related to micropollutants, metabolites and mixtures in (the sources of) drinking water. Following this discussion three topics were prioritised for further follow-up discussion in three breakout groups, namely (i) bioassays, (ii) the Threshold of Toxicological Concern (TTC) and (iii) Effect Directed Analysis (EDA)/prioritisation of compounds.
In the first breakout group (chaired by Tamara Grummt and Susan Richardson) a discussion was held on bioassays and which models to apply. In general, consensus was reached that a panel of (cost-effective) bioassays for general toxicity (e.g. cytotoxicity) and more specific toxicity (e.g. genotoxicity, reprotoxicity, neurotoxicity, immunotoxicity and hormone disruption) would provide the best option. In addition, specifically for drinking water it is best to apply cell lines with a human origin. Bioassays can be used as a tool to screen for biological activity of the vast amount of compounds in the environment, but standardisation of end-points is important in addition to proper bioassay validation. Finally, it was concluded that bioassays trigger-values are important to put results into perspective.
In the second breakout group (chaired by Thomas Backhaus and Merijn Schriks ), the role of the TTC in the framework of risk assessment was discussed. The pros for applying the TTC include that it is quick, simple and can be easily used for prioritisation. In addition, it is already implemented in German and Dutch legislation. However, the TTC may be over-conservative and can overlook unknown (hazardous) compound properties. Furthermore, it is based on a static database, whereas the presence of compounds in the environment is a dynamic process. In addition, compound identity is required and mixture toxicity effects are not completely incorporated. However, it was concluded that the TTC can be a part of a tiered approach in risk assessment. An interesting final question is what to do when the TTC is exceeded. Among the options is carrying out additional research and/or reducing compound concentrations by e.g. emission reduction or other mitigating solutions.
The third breakout group (chaired by Ron van der Oost , Arnaud Boivin ) and David SCHWESIG ) discussed the options to prioritise compounds in the environment. Since the amount of environmental compounds is huge, it was proposed to focus on indicator substances or form priority categories (which contain similar compounds). In addition, bioassay may serve as a prioritisation tool driving further investigation such as advanced analytical chemistry. Non-target chemical screening is a useful application to get a first impression of compounds present in a water extract. Finally, it was concluded that EDA is too expensive in its current form and there is a need for more cost-effective methods.
A more detailed summary of the conclusions and further proposed follow-up actions will be made available on the website of the Norman Network.
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buy Codex ': ' paradigm attacks can describe all customers of the Page. t ': ' This requirementsAll ca back battle any app rights. catalog ': ' Can trigger, Thank or be ia in the hormesishypothesis and subject area ways. Can ponder and intervene passing notes of this author to produce increases with them.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,872
|
Q: Disable Syntax Highlight in Sphinx (Alabaster theme) How to disable syntax highlight in Sphinx?
I've tried setting highlight language to 'none' and setting ..language:: none.
Also tried setting it to 'text'.
I tried removing the html and make clean. But the syntax highlighting is there. (using Alabaster theme)
conf.py is configured with these extensions:
extensions = ['sphinx.ext.autodoc',
'sphinx.ext.doctest',
'sphinx.ext.todo',
'sphinx.ext.coverage',
'sphinx.ext.ifconfig',
'sphinx.ext.viewcode',
'sphinx.ext.githubpages',
'numpydoc',
]
EDIT: Well it seems that viewcode extension is doing this and that it is quite hard coded
A: viewcode applies syntax highlighting only to Python source files. To disable highlighting in just those files, you could edit the source of the file by modifying this line in the lexer logic adding , 'none'.
if env.config.highlight_language in ('python3', 'default', 'none'):
It looks like you already submitted a PR for it.
For all other files rendered by Sphinx in your narrative documentation, you can disable highlighting globally in conf.py:
highlight_language ='none'
Then this would be inherited by your modification in viewcode.
A: This was fixed.. fixed in 1.6-release
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 7,733
|
<?xml version="1.0" encoding="utf-8"?>
<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:rgbpi="http://schemas.android.com/apk/res-auto"
android:layout_width="fill_parent"
android:layout_height="fill_parent"
android:minWidth="25px"
android:minHeight="25px"
style="@style/HostItem">
<LinearLayout
android:orientation="vertical"
android:minWidth="25px"
android:minHeight="25px"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:id="@+id/layout_container"
style="@style/HostItem_container">
<TextView
android:text="NAME"
android:textAppearance="?android:attr/textAppearanceLarge"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:id="@+id/txt_name"
android:layout_gravity="center_horizontal|top"
android:background="@color/transparent" />
<TextView
android:text="IP"
android:textAppearance="?android:attr/textAppearanceMedium"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:layout_gravity="center"
android:id="@+id/txt_ip"
android:background="@color/transparent" />
<TextView
android:text="PORT"
android:textAppearance="?android:attr/textAppearanceMedium"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:layout_gravity="center"
android:id="@+id/txt_port"
android:background="@color/transparent" />
</LinearLayout>
<LinearLayout
android:orientation="vertical"
android:minWidth="25px"
android:minHeight="25px"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:id="@+id/layout_container_edit"
style="@style/HostItem_container">
<TextView
android:text="NAME"
android:textAppearance="?android:attr/textAppearanceLarge"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:id="@+id/txt_name_edit"
android:layout_gravity="center_horizontal|top"
android:background="@color/transparent" />
<EditText
android:text="IP"
android:textAppearance="?android:attr/textAppearanceMedium"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:layout_gravity="center"
android:id="@+id/txt_ip_edit"
android:background="@color/transparent" />
<EditText
android:text="PORT"
android:textAppearance="?android:attr/textAppearanceMedium"
android:layout_width="wrap_content"
android:layout_height="0dp"
android:layout_weight="1"
android:layout_gravity="center"
android:id="@+id/txt_port_edit"
android:background="@color/transparent" />
</LinearLayout>
<Button
android:layout_width="@dimen/host_item_edit_button_sise"
android:layout_height="@dimen/host_item_edit_button_sise"
android:layout_gravity="top|right"
android:id="@+id/btn_edit"
android:background="@drawable/ic_tune_black_36dp" />
<Button
android:layout_width="@dimen/host_item_edit_button_sise"
android:layout_height="@dimen/host_item_edit_button_sise"
android:layout_gravity="top|right"
android:id="@+id/btn_save"
android:background="@drawable/ic_check_black_36dp" />
<Button
android:background="@drawable/ic_panorama_fisheye_black_36dp"
android:layout_width="@dimen/host_item_edit_circle_sise"
android:layout_height="@dimen/host_item_edit_circle_sise"
android:layout_gravity="center_vertical|left"
android:id="@+id/btn_circle" />
<Button
android:background="@drawable/Icon"
android:layout_width="@dimen/host_item_edit_active_sise"
android:layout_height="@dimen/host_item_edit_active_sise"
android:layout_marginLeft="@dimen/host_item_edit_active_delta_sise"
android:layout_gravity="center_vertical|left"
android:id="@+id/btn_active" />
<Button
android:background="@drawable/ic_remove_circle_outline_black_36dp"
android:layout_width="@dimen/host_item_edit_button_sise"
android:layout_height="@dimen/host_item_edit_button_sise"
android:layout_gravity="bottom|right"
android:id="@+id/btn_remove" />
</FrameLayout>
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 7,812
|
{"url":"https:\/\/www.physicsforums.com\/threads\/moment-of-inertia-of-curves-and-surfaces.720530\/","text":"Moment of Inertia of Curves and Surfaces\n\n1. Nov 2, 2013\n\nJhenrique\n\nGreetings!!!\n\nI enjoyed the definition of moment of inertia for a volume and for an area in the form of matrix. It's very enlightening!\n\n$$I = \\int \\begin{bmatrix} y^2+z^2 & -xy & -xz\\\\ -yx & x^2+z^2 & -yz\\\\ -zx & -zy & x^2+y^2 \\end{bmatrix}dxdydz$$\n\n'-> http:\/\/mathworld.wolfram.com\/MomentofInertia.html\n\n$$J = \\int \\begin{bmatrix} y^2 & -xy\\\\ -yx & x^2\\\\ \\end{bmatrix}dxdy$$\n\n'-> http:\/\/mathworld.wolfram.com\/AreaMomentofInertia.html\n\nSo, analogously, I'd like to know how would be the matrices of moment of inertia for curves and for surfaces...\n\nThx,\n\nJhenrique\n\n2. Nov 3, 2013\n\nSteamKing\n\nStaff Emeritus\nThe second moments of area have a specific usage, particularly in calculating certain stresses for beams.\n\nThe second moments of a volume are used in mechanics to describe the motions of a body under the influence of external forces and moments.\n\nI am not aware of a definition of a second moment for a general curve, unless you wish to approximate the curve as a rod of negligible radius. There are second moments defined for surfaces whose thickness is very small. These moments are used for objects which are composed of thin shells and can be derived using the definitions for the I matrix in the OP.\n\nSee:\n\nhttp:\/\/en.wikipedia.org\/wiki\/Second_moment_of_area\nhttp:\/\/en.wikipedia.org\/wiki\/List_of_area_moments_of_inertia\n\nhttp:\/\/en.wikipedia.org\/wiki\/Mass_moment_of_inertia\nhttp:\/\/en.wikipedia.org\/wiki\/List_of_moments_of_inertia","date":"2017-11-20 16:01:04","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\": 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\": 0.689355194568634, \"perplexity\": 396.02516365563076}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806070.53\/warc\/CC-MAIN-20171120145722-20171120165722-00063.warc.gz\"}"}
| null | null |
Q: Kafka Streams replication factor not applied to state store changelog topics We are using Kafka Streams via Spring Cloud Stream integration. I configured the replication factor to be used across all internal Kafka Streams topics by setting
spring.cloud.stream.kafka.streams.binder.configuration.replication.factor=${REPL_FACTOR}
It works for most repartition/changelog topics used internally by Kafka Streams. However, it looks like this setting has no effect on state store changelog topics that get created manually via Materialized#as(StoreSupplier). For those topics I can still see the replication factor is set to default 1. It's also not possible to set it using Materialized#withLoggingEnabled(Map<String, String>) because this only accepts topic-level configs (replication.factor is Streams config). Is this a known bug in Kafka Streams? I couldn't find anything. If so, is there a workaround to increase the replication factor for those changelog topics?
We are using Kafka v2.3.1 on the broker side and 2.5.0 on client side.
A: Starting with version 2.4, the AdminClient can now set the replication factor to -1 in NewTopic, meaning that the default.replication.factor should be used when creating topics - KIP-464.
However, it appears that Kafka Streams does not currently use this feature; there is an open issuee KAFKA-8531 for this.
You can set the replication factor for internal topics using
StreamsConfig.REPLICATION_FACTOR_CONFIG)
https://kafka.apache.org/documentation/#replication.factor
The replication factor for change log topics and repartition topics created by the stream processing application.
Since you are setting that, via the binder config, it should work as expected.
EDIT
What version of spring-cloud-stream are you using? I just tested with 3.0.8 and it works as expected.
spring.cloud.stream.kafka.streams.binder.configuration.replication.factor: 3
2020-10-15 12:03:55,601 ERROR [kafka-stre] o.a.k.s.p.i.StreamThread:673 - stream-thread [kafka-streams-inventory-processor-b8d07a5a-f3c4-476a-a265-119163d2acb7-StreamThread-1] Encountered the following unexpected Kafka exception during processing, this usually indicate Streams internal errors:
org.apache.kafka.streams.errors.StreamsException: Could not create topic kafka-streams-inventory-processor-inventory-counts-changelog.
Caused by: org.apache.kafka.common.errors.InvalidReplicationFactorException: Replication factor: 3 larger than available brokers: 1.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 274
|
package de.tudarmstadt.lt.seg.sentence;
import java.io.IOException;
import java.io.Reader;
import de.tudarmstadt.lt.seg.Segment;
import de.tudarmstadt.lt.seg.SegmentType;
import de.tudarmstadt.lt.seg.SegmentationUtils;
/**
* @author Steffen Remus
*
*/
public class LineSplitter implements ISentenceSplitter{
Reader _reader = null;
int cp = 0;
final Segment _segment = new Segment(){{ begin = 0; end = 0; type = SegmentType.UNKNOWN; text.setLength(0); }};
private boolean getNext(){
_segment.text.setLength(0);
_segment.type = SegmentType.UNKNOWN;
_segment.begin = _segment.end;
if(cp < 0)
return false;
boolean first_is_newline = SegmentationUtils.charIsLineSeparator(cp);
boolean is_empty = first_is_newline;
while(true){
_segment.text.appendCodePoint(cp);
int cp_current = cp;
is_empty &= SegmentationUtils.charIsLineSeparator(cp_current);
try {
cp = _reader.read();
} catch (IOException e) {
System.err.format("%s: %s%n", e.getClass().getName(), e.getMessage());
break;
}
_segment.end++;
if(cp < 0)
break;
int cp_next = cp;
if(is_empty && !SegmentationUtils.charIsLineSeparator(cp_next))
break;
if(!is_empty && SegmentationUtils.charIsLineSeparator(cp_next))
break;
}
_segment.type = is_empty ? SegmentType.EMPTY_SPACE : SegmentType.SENTENCE;
return !_segment.hasZeroLength();
}
/* (non-Javadoc)
* @see java.util.Iterator#hasNext()
*/
@Override
public boolean hasNext() {
return getNext();
}
/* (non-Javadoc)
* @see java.util.Iterator#next()
*/
@Override
public Segment next() {
return _segment;
}
/* (non-Javadoc)
* @see de.tudarmstadt.lt.seg.sentence.ISentenceSplitter#init(java.io.Reader)
*/
@Override
public ISentenceSplitter init(Reader reader) {
_reader = reader;
_segment.begin = 0; _segment.end = 0; _segment.type = SegmentType.UNKNOWN; _segment.text.setLength(0);
try {
cp = _reader.read();
} catch (IOException e) {
System.err.format("%s: %s%n", e.getClass().getName(), e.getMessage());
return this;
}
return this;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,647
|
Герб Кві́нсленду, що був затверджений королевою Вікторією 1893 року, є найстарішим в Австралії.
Історія
Розроблення герба тривало впродовж року (1892-1893). 1902 року геральдичне зображення британської імператорської корони було стандартизовано для (символічної) корони Тюдорів. Після 1953 року увагу було зосереджено на зображенні фактичної корони Святого Едуарда. Останній на поточну мить додаток до гербу був долучений 1977 року, під час Срібного ювілею королеви Єлизавети II: були додано тварин-щитотримачів. Олень благородний, класична геральдична тварина, що представляє старий світ, і журавель австралійський, що символізує корінні народи.
Примітки
Герби Австралії
Квінсленд
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 782
|
namespace cc {
RendererSettings::RendererSettings()
: allow_antialiasing(true),
force_antialiasing(false),
force_blending_with_shaders(false),
partial_swap_enabled(false),
finish_rendering_on_resize(false),
should_clear_root_render_pass(true),
disable_display_vsync(false),
delay_releasing_overlay_resources(false),
refresh_rate(60.0),
highp_threshold_min(0),
use_rgba_4444_textures(false),
texture_id_allocation_chunk_size(64),
use_gpu_memory_buffer_resources(false) {}
RendererSettings::~RendererSettings() {
}
void RendererSettings::ToProtobuf(proto::RendererSettings* proto) const {
proto->set_allow_antialiasing(allow_antialiasing);
proto->set_force_antialiasing(force_antialiasing);
proto->set_force_blending_with_shaders(force_blending_with_shaders);
proto->set_partial_swap_enabled(partial_swap_enabled);
proto->set_finish_rendering_on_resize(finish_rendering_on_resize);
proto->set_should_clear_root_render_pass(should_clear_root_render_pass);
proto->set_disable_display_vsync(disable_display_vsync);
proto->set_delay_releasing_overlay_resources(
delay_releasing_overlay_resources);
proto->set_refresh_rate(refresh_rate);
proto->set_highp_threshold_min(highp_threshold_min);
proto->set_use_rgba_4444_textures(use_rgba_4444_textures);
proto->set_texture_id_allocation_chunk_size(texture_id_allocation_chunk_size);
proto->set_use_gpu_memory_buffer_resources(use_gpu_memory_buffer_resources);
}
void RendererSettings::FromProtobuf(const proto::RendererSettings& proto) {
allow_antialiasing = proto.allow_antialiasing();
force_antialiasing = proto.force_antialiasing();
force_blending_with_shaders = proto.force_blending_with_shaders();
partial_swap_enabled = proto.partial_swap_enabled();
finish_rendering_on_resize = proto.finish_rendering_on_resize();
should_clear_root_render_pass = proto.should_clear_root_render_pass();
disable_display_vsync = proto.disable_display_vsync();
delay_releasing_overlay_resources = proto.delay_releasing_overlay_resources();
refresh_rate = proto.refresh_rate();
highp_threshold_min = proto.highp_threshold_min();
use_rgba_4444_textures = proto.use_rgba_4444_textures();
texture_id_allocation_chunk_size = proto.texture_id_allocation_chunk_size();
use_gpu_memory_buffer_resources = proto.use_gpu_memory_buffer_resources();
}
bool RendererSettings::operator==(const RendererSettings& other) const {
return allow_antialiasing == other.allow_antialiasing &&
force_antialiasing == other.force_antialiasing &&
force_blending_with_shaders == other.force_blending_with_shaders &&
partial_swap_enabled == other.partial_swap_enabled &&
finish_rendering_on_resize == other.finish_rendering_on_resize &&
should_clear_root_render_pass == other.should_clear_root_render_pass &&
disable_display_vsync == other.disable_display_vsync &&
delay_releasing_overlay_resources ==
other.delay_releasing_overlay_resources &&
refresh_rate == other.refresh_rate &&
highp_threshold_min == other.highp_threshold_min &&
use_rgba_4444_textures == other.use_rgba_4444_textures &&
texture_id_allocation_chunk_size ==
other.texture_id_allocation_chunk_size &&
use_gpu_memory_buffer_resources ==
other.use_gpu_memory_buffer_resources;
}
} // namespace cc
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,282
|
Bruce Hoffmeister, Global Chief Information Officer for Marriott International, holds global accountability for all business information technology resources.
With 22 years at Marriott, Mr. Hoffmeister brings a vast and diverse set of experiences to this role. He was previously Senior Vice President, IR Shared and Application Services, leading the effort to drive sustainable efficiencies within the technical infrastructure that supports the computing resources for Marriott's worldwide operations. In addition, he directed the development efforts to replace and update the Sales and Marketing, Event Management and Revenue Management systems in order to optimize profits across the total hotel, increase the efficiency of the selling process and enable the delivery of superior events.
Beginning his career at Marriott as a Senior Financial Analyst for Development Finance, Mr. Hoffmeister has held various finance and accounting roles within the Development, Information Resources, and Lodging functions. He has also held the position of Senior Vice President, Global Revenue Management where he was responsible for developing and leading the company's worldwide pricing, inventory management and selling strategies.
Mr. Hoffmeister holds bachelor degrees in mathematics and biology, with a minor in computer science, from Thomas More College, as well as a masters of business administration degree, with concentrations in finance and accounting, from the Owen Graduate School of Management, Vanderbilt University.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,689
|
Chautauqua Symphony Orchestra presents an evening of musical portraits
by Philip de Oliveira on August 7, 2017 2.2K views
Can music paint a picture?
The question, expressed in different ways, concerns the age-old debate over whether music does and can mean anything definite. Critics, program annotators, professional musicians and even composers themselves can't seem to agree.
Felix Mendelssohn said music isn't too vague for words, "but rather too definite." Aaron Copland famously told the story of a concertgoer who, after seeing a performance of "Appalachian Spring," told him she could see the rolling hills of Appalachia and hear spring in the music. But Copland penned the whole score with only the drab working title "Ballet for Martha" as inspiration.
"How close should the intelligent music lover wish to come to pinning a definite meaning to any particular work?" Copland asks in his book, What to Listen for in Music. "No closer than a general concept, I should say."
Yet composers love to drop visual hints across the tops of their scores, with titles like "Fountains of Rome" and "The Hebrides" and "Godzilla Eats Las Vegas!" (a cheeky farewell to Sin City for wind band by Eric Whitacre). Musicians often talk about sounds in terms of their "color": a section of double basses sounds dark, while a chorus of muted trumpets sounds bright.
At 8:15 p.m. Tuesday in the Amphitheater, the Chautauqua Symphony Orchestra will welcome guest conductor Daniel Boico and cello soloist Harriet Krijgh to present a diverse palette of orchestral images and colors, starting with Ottorino Respighi's "Trittico botticelliano."
"Trittico" is Respighi's attempt to express three paintings by Italian Renaissance painter Sandro Botticelli in musical form.
Each of the three movements corresponds to a different painting. The triptych is conceptually and musically very similar to Respighi's Roman trilogy, his most famous work.
The first movement, inspired by a large panel called "Primavera" ("Spring"), begins with florid trills and brass fanfares that sound almost identical to the opening of "Pines of Rome."
"(Respighi's) use of color is quite stunning and you can see that in all of his pieces," Boico said. "This is what I love about him."
According to Boico, Respighi tends to be eclipsed in the minds of music lovers by another exemplar of imaginative orchestration, Nikolai Rimsky-Korsakov, whose treatise on the craft is the standard text for student composers at conservatories and music departments everywhere.
"I would say Respighi is way better than Rimsky-Korsakov," Boico said. "I find him much more imaginative in his orchestration, in his ways of connecting sounds."
According to Boico, Respighi wasn't afraid to go against what he heard in other composers' music.
"He had to imagine this according to his memories of other composers' sounds," Boico said. "I'm sure he heard all of Rimsky-Korsakov's stuff and said, 'OK, that's what it sounds like, but I think I'll go this other route.' "
Sergei Rachmaninoff, whose "Symphonic Dances" will close out Tuesday's program, was among those who took notice of Respighi's skill. Rachmaninoff gave Respighi permission to orchestrate five of his "Études-tableaux" at the urging of Boston Symphony Orchestra conductor Serge Koussevitzky.
"Rachmaninoff didn't have any problem orchestrating his own works, but he said to Respighi, 'Sure, go for it,'" Boico said. "For Rachmaninoff to believe Respighi can do justice to his own piano works in the orchestral realm is fantastic."
The "Symphonic Dances" are the last pieces of music Rachmaninoff ever wrote. Boico said that like Respighi, Rachmaninoff was a master of imagining fresh orchestral colors.
"Rachmaninoff was a pianist, but pianists talk as if they hear the music symphonically," Boico said. "They can imagine an orchestra playing these notes that are being played on the piano."
Both Respighi and Rachmaninoff lived well into the 20th century, yet their musical styles and tastes remained decidedly old-fashioned.
"Respighi has a harmonic language that's typical for Romantic composers who were stuck in the turn of the 20th century and were searching for new paths but chose to remain tonal," Boico said.
When he's conducting, Boico considers it essential to have some sort of mental image or scenario to help guide the music.
"I can decide on certain emotions and atmospheres, and all of that contributes to the musicality of it all," Boico said.
But as far as deriving meaning from the music, Boico leaves that up to the listener.
"It doesn't have to be right. It can be super individual," Boico said. "But it's something we are able to do and it makes me happy."
Tags : Chautauqua Symphony OrchestracsoDaniel BoicoHarriet KrijghOttorino RespighiWeek Seven
Professional storyteller David Gonzalez will share 'Tales from the Latino World'
Chautauqua Conversations: Ron Kilpatrick, Jim Pardo reflect on Chautauqua's financial stability
The author Philip de Oliveira
Philip de Oliveira reports on the Chautauqua Symphony Orchestra and the Logan Chamber Music Series for the Daily. He is the Walton D. Clarke Fellow at NPR member station 89.7 WKSU in Kent, Ohio, where he has filed stories for local newscasts and statewide for Ohio Public Radio. He occasionally contributes to Cleveland Scene magazine and Cool Cleveland.
Acclaimed musician Rhiannon Giddens returns to Amp stage for CSO collaboration
'A power beyond the norm' CSO, Buffalo Philharmonic come together for 'profound,' 'beautiful' evening of Brahms, Moravec/Campbell's 'Sanctuary Road'
Principal Horn Kaza soars through Amp with CSO, bringing life to Mozart, Schickele
With Buffalo Philharmonic Chorus, CSO presents Moravec/Campbell work 'Sanctuary Road,' story of unsung abolitionist Still
In evening including Mozart, Kaza to solo with CSO on Schickele's 'Pentangle'
|
{
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Q: Couldn't able to add cocoa pod files (which is created using "pod lib create") for sample example? I have created a new cocoa pod using
pod lib create SamplePod
It set me up with a well thought out library structure. So I created new file by making a group (SamplePod) inside Pod folder "Sample.h" and when trying to access it using
#import <SamplePod/Sample.h>
It couldn't able to identify the file.
Can anybody explain where to add new files for pod and how to refer them in pod sample project ?
I have already refered these links
(1) http://guides.cocoapods.org/making/using-pod-lib-create
(2) http://guides.cocoapods.org/making/making-a-cocoapod.html
A: You should store files that are actually part of your CocoaPod in the the location specified for source_files in your podspec (e.g. Pod/Classes).
So, if you are adding new files, you should put them in that directory, and you can then put them in a sensible group folder within Xcode.
If you are copying an existing file to the project (that is part of your actual pod), you should copy them to the same directory above, and you can then add them to your example project by going to File > Add Files to "MyProject" ... (you should un-tick the Copy items to destination group's folder).
They can then be imported with the usual #import "MyFile.h"
A: It's better to run
Pod install
After adding new files. Then you can easily import these files to your CocoaPod sample project.For more details you might want to check this out
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{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/oncologypro.esmo.org\/meeting-resources\/esmo-asia-2015-congress\/Simulation-modeling-of-lung-cancer-screening-cost-effectiveness-analysis","text":"Oops, you're using an old version of your browser so some of the features on this page may not be displaying properly.\n\nMINIMAL Requirements:\u00a0Google Chrome 24+Mozilla Firefox 20+Internet Explorer 11Opera 15\u201318Apple Safari 7SeaMonkey 2.15-2.23\n\n# 1001 - Simulation modeling of lung cancer screening cost-effectiveness analysis\n\n20 Dec 2015\n\n### Session\n\nPoster presentation 2\n\nWenshuai Wan\n\n### Citation\n\nAnnals of Oncology (2015) 26 (suppl_9): 156-160. 10.1093\/annonc\/mdv535\n\nW. Wan\n\n### Author affiliations\n\n\u2022 Radiology At The Hospital Of The University Of Pennsylvania, University of Pennsylvania, 19103 - Philadelphia\/US\nMore\n\n## Resources\n\n### Aim\/Background\n\nLow dose computed tomography-based lung cancer screening has been shown through the National Lung Cancer Screening Trial (NLST) to reduce mortality for high risk patients. We evaluate the use of stochastic simulation for cost-effectiveness analysis of CT screening, a technology which exhibits low specificity and potentially high downstream costs.\n\n### Methods\n\nScreening effectiveness was tested using a natural history of lung cancer. Probability of metastatic disease and cure at detection are functions of tumor size. Parameters estimated were based on the Surveillance, Epidemiology and End Results cancer registry. Screening and treatment components in the simulation are based on protocols used in the Mayo CT trial. Costs of diagnostic and therapeutic procedures were obtained from the Medicare payment database. We assessed cost-effectiveness for all scenarios from ages 55-80 at 5 year starting intervals with screening ranging from 5-30 years. These scenarios were examined based on one-time, annual, biennial, and every five year intervals.\n\n### Results\n\nScreening approximately doubles incidence compared to the unscreened population. Clinically, the simulation model also shows a 20.5% reduction of 5-yr mortality in the screened group as opposed to the simulated, unscreened population. The most cost-effective strategies screen at every two or five year intervals from ages 60-80. Expenses in a microsimulation are tailored to show aggregate costs, with or without discounting, over the lifetime of simulated patients on a societal perspective. Varying the intensity of screening contributes to large changes in cost-effectiveness. Increased duration and frequency of screening derive marginal benefits but come at incrementally higher marginal costs. These data correlate with an even wider range of published cost-effectiveness analysis results, ranging from less than $20,000 to over$100,000 per QALY.\n\n### Conclusions\n\nSimulation models can be used to predict how screening affects both clinical and economic outcomes. As extensions of clinical trials, clinical-economic model results and their conclusions should be incorporated into the development of future cancer screening and treatment programs.\n\n### Clinical trial identification\n\nResults were obtained using a computer-based microsimulation. This research did not involve the use of personally identifiable health information and was not performed as part of an onging clinical trial.\n\n### Disclosure\n\nAll authors have declared no conflicts of interest.\n\nThis site uses cookies. Some of these cookies are essential, while others help us improve your experience by providing insights into how the site is being used.","date":"2020-12-04 20:12:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31694701313972473, \"perplexity\": 5258.203650498456}, \"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\/1606141743438.76\/warc\/CC-MAIN-20201204193220-20201204223220-00480.warc.gz\"}"}
| null | null |
Who's Who At DC Comics-The New 52: Paul Cornell
By David Hyde Thursday, August 18th, 2011
THE SOURCE: How do you write the first line of a new series? PAUL CORNELL: Just dive in, I'll rewrite it later anyway. How do you introduce a new hero? I like to give them a big hero shot, like the Horsewoman in DEMON KNIGHTS #1 gets a big cinematic entrance. How do you introduce characters? In action, usually, then we'll get to know them when we're interested. How do you introduce a new villain? Through their actions. They can't just talk in a bad way, they have to be seen to do bad. What was the first comic you ever worked on? Doctor Who Magazine. Who was the first character you followed? Probably the Bash Street Kids in the Beano. Who was the first writer you followed? Consciously, I think Chris Claremont, but way before that Stan Lee was all I knew about anything. Who was the first artist you followed? George Perez. I used to try and draw the moon like he did! What was the first convention you attended as a fan? The Longleat Doctor Who Experience! What was the first convention you attended as a professional? It must have been one of the Bristol Comics Expos, or the Gallifrey con in LA. Kind of slipped easily between the categories without thinking about it, business as usual. What was the first comic book you read? It would have been an issue of Pippen and Playhour. Get your British Auntie to look it up. What was your first job in the comic book industry? Co-writing a short 7th Doctor strip in Doctor Who Magazine. On your creative process: Initially, I'll plot out a story arc, just saying in a couple of paragraphs what happens in each issue. Most editors are happy, once we've worked that back and forth, for me to go from that to writing an issue. Sometimes for my own use I'll write the numbers one to twenty down the side of a piece of paper, then a quick few words about happens on each page of the issue, often starting by working out how much space I need for the ending. I'll often have a scene or a line in mind for many issues before finding the right place to use it. I keep those on my iPad.
Who's Who At DC Comics-The New 52: Joshua Hale Fialkov
DC Comics-The New 52, Demon Knights, paul cornell, Stormwatch
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| 4,094
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Religion In Tajikistan: Important Facts And Figures
The Masjidi Jami Mosque in Khujand, Tajikistan. Editorial credit: Milosz Maslanka / Shutterstock.com.
Islam is the religion of the majority in Tajikistan. It is an integral part of the region's culture. However, Tajikistan is a secular country with the Constitution of the country providing the right to freedom of religion to the country's residents. The country is unique for its law which prohibits anyone under 18 to publicly practice religion.
Islam: The Most Widely Practiced Religion In Tajikistan
Sunni Islam is the religion of the majority in Tajikistan. It is the religion of nearly 95.7% of the population of the country. Only 3% of the population practice Shia Islam, another major branch of Islam. Other denominations of Islam with adherents in Tajikistan include Ismailism and some Sufi orders. Ismailism has managed to survive in small pockets in the Pamir mountain's remote areas where the followers have managed to escape persecution.
Islam In Tajikistan During Soviet Rule
During the Soviet era, several attempts were made to reduce the popularity and practice of Islam in Tajikistan. The Soviet government's general drive against the practice of religion had its effects on the religious scene of Tajikistan for some decades. Persecution in the name of religion, demolition of mosques, killings of influential religious leaders, and a general wave of government action against the observance of religion in the country happened during the Soviet rule. However, despite the measures adopted by the Soviet regime, Islam in Tajikistan did not suffer any major blows and the popularity of the religion continued unabated.
Islam In Tajikistan Post-Independence
Following the end of Soviet rule, the newly formed Tajik government implemented measures that closed down a large number of unregistered mosques in the country leading to the general belief that the government action was actually against Islam. However, measures to encourage the religion have also been taken like the arrangement of an international symposium to commemorate Abu Hanifa, a Sunni Muslim jurist and the planning of the construction of an impressive mosque in the country.
Minority Religions Of Tajikistan
Russian Orthodox
The second most practiced religion in Tajikistan is Russian Orthodox. The St. Nicholas Cathedral in Dushanbe serves the followers of this religion, most of whom are migrants from Russia and Ukraine. Small populations in the country also follow some other Christian denominations.
The religion is followed by a very small population of Tajikistan. Around 326 Catholics reside in the country who are served by three parishes present in the country.
Hinduism is followed by a minority community in Tajikistan. The religion was introduced and spread here primarily by the ISKCON missionaries. An ISKCON Centre exists in Dushanbe.
Other Minority Religions Of Tajikistan
Zoroastrians, Jews, and Baha'is are some of the other minority religious groups of Tajikistan. However, the number of followers of these groups sharply declined in the early independence period due to a wave of emigration.
Oishimaya Sen Nag August 30 2017 in Society
Religion in Fiji: Important Facts and Figures
Why Is Religion So Important In Culture?
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is a racing game developed and published by Capcom, first released on PlayStation 2, later ported to GameCube and Xbox.
Description
Auto Modellista marked itself apart from others of the same genre with cel-shaded graphics, which gave a hand-drawn and cartoon-like appearance. The game is similar in gameplay to games like Gran Turismo, with the player picking a vehicle which they can modify and customize. There are six tracks in the default game, including the real-life Suzuka Circuit and the Mt. Akagi mountain pass.
After poor initial reception, Capcom modified the game for its North American releases. In Japan, the game was rereleased with these changes as Auto Modellista: US Tuned. This version featured American cars such as the Dodge Viper, two new oval tracks, various UI improvements and a new handling model that saw cars accelerate slower and lose more speed in turns.
The US Tuned changes were present in every single release of the game starting with the North American PS2 release in early 2003, evidenced by the new cover art with the Dodge Viper. Many distributors predicted that sales of the game were going to be poor, and generally refused to carry it.
Also in 2003, Auto Modellista received a followup in the form of Group S Challenge for the Xbox, though it lacked any of Auto Modellista'''s visual style and is generally not considered to be a direct sequel. Capcom has not been involved with driving games since, although it did publish some games based on MotoGP developed by Milestone srl, and included Mega Man Battle & Chase, a racing game based on the Mega Man franchise, in the Mega Man X Collection.
GameplayAuto Modellista attempts to be a very technical racing game, with an immense amount of available parts and settings for the selection of cars provided to the player. Various aspects of each car can be tuned, allowing the player to tweak the performance of the car.
In the Garage mode (the main single-player mode), the player is granted the ability to select one of four tire types which affect road grip in regard to the weather conditions on the race track (for example, the "Semi-Slick Tires" provide maximum speed and grip in dry weather, but suffer in rain). Other options include brakes (which determine braking efficiency), suspension, Turbines, Mufflers, Computer (determines the car's ability to accept upgrades later in the game), the engine, "Final Gear", and Weight Reduction.
Auto Modellista's customisation options also extends to visual enhancements, allowing the player to choose from many different color combinations, hood and spoiler types, plus the ability to add badges, stickers and even create license plates. Engine swaps are also available, for example, the Subaru 360 can have EJ20T in place of its EK32. Swapped engines cannot be retuned in the game.
A large aspect of the game was its online mode, with online races supporting up to 8 players. This functionality was not available on the GameCube and European PlayStation 2 versions. The online mode of Auto Modellista has since been terminated, and cannot be used in any versions of the game.
DevelopmentAuto Modellista was a part of an initiative from Capcom's Production Studio 1 to develop three network focused games on the PlayStation 2. The other games were Monster Hunter and Resident Evil Outbreak. Capcom's plan was that at least one of the games would become a million seller. Both Monster Hunter and Resident Evil Outbreak eventually became million sellers.
Reception
The game received "mixed or average reviews" on all platforms according to video game review aggregator Metacritic. In Japan, Famitsu'' gave the PS2 version a score of 30 out of 40.
References
External links
CAPCOM - auto modellista (official site)
CAPCOM - auto modellista (Japanese official site)
Racing video games
PlayStation 2 games
GameCube games
Xbox games
2002 video games
Video games developed in Japan
Video games with cel-shaded animation
Video games scored by Tetsuya Shibata
Video games set in Tokyo
Capcom games
Multiplayer and single-player video games
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Q: Static DNS + DHCP-ed IP in Windows? I change wifi networks a couple times a day. Obviously they have different DHCP settings, and I always use dynamic ip resolving.
However, I want to have the same (say, OpenDNS') static nameservers in all networks.
For reference, in Linux, I'd use the supersede domain-name-servers option in dhclient.conf to set up the preferred DNS addresses.
Is this possible to do in Windows?
A: When you configure your IP just set it to use the OpenDNS server and leave it set to DHCP, that should work out.
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Students in Years 7-9 undertake their studies through our innovative Catalyst Curriculum, made up of core subjects, the Personal Learning Program and courses offered through our Discovery Learning Banks.
The Personal Learning Program (PLP) is a three-year, sequential learning program aimed at developing individual abilities and skills as well as providing the flexibility for students to explore new areas of interest or continue learning in disciplines that align with their passions and academic strengths. The principles of individual and meaningful learning are organising tenets of the Personal Learning Program. Students grow their understanding of who they are as learners through this program which includes participation in four Action Projects incorporated into each student's Core learning and the opportunity to select and study four Discovery Courses each year.
Stagnant Swamp or Wild Wetland?
The Discovery Learning Banks offer a wide range of practical and academic courses. In all junior secondary years Discovery courses are organised around three learning Banks: Creativity, Technology and Opportunity. The arrangement of Discovery courses in these important years of schooling provides significant choice across a broad range of subject disciplines. Across this three-year sequential program students will select and complete twelve Discovery courses. Each course runs for one term and results in 35 hours of instruction.
|
{
"redpajama_set_name": "RedPajamaC4"
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DOJ Docs Obtained by Judicial Watch Confirm Communication Between FBI & Pfizer About Project Veritas
By Project Veritas
DOJ Documents Confirm Existence of Communications Between FBI & Pfizer about Project Veritas
• According to a Freedom of Information/Privacy Acts ("FOIPA") request response issued by the Department of Justice ("DOJ"), it appears there are indeed communications between Pfizer and the FBI about Project Veritas.
• The request for communications, filed by legal advocacy nonprofit Judicial Watch, was acknowledged but denied by the DOJ which cited an exemption over law enforcement proceedings.
• "The FBI has completed its search for records responsive to your request. The material you requested is located in an investigative file which is exempt from disclosure[.]" A direct quote from DOJ FOIPA response letter.
• A separate response letter from the DOJ cited "invasions of personal privacy," as the basis for rejecting a separate, but similar, Freedom of Information Act ("FOIA") request to the FBI.
• Project Veritas, which previously broke news stories on Pfizer over the past year, was not previously aware of "law enforcement proceedings" relating to those investigations.
[New York, New York January 18, 2022] Project Veritas has obtained Freedom of Information/Privacy Acts ("FOIPA") response letters from the DOJ which shockingly appears to confirm that there is some level of communication between federal officials at the Bureau and pharmaceutical giant Pfizer about Project Veritas.
The letters, which were originally obtained by Judicial Watch resulting from FOIA requests made by the legal watchdog group requesting communications between Pfizer and the FBI about Project Veritas cite separate reasons for rejecting the request.
While the second letter attempts to avoid acknowledging the existence of such communications by citing "an unwarranted invasion of personal privacy," the first response letter from the Department of Justice revealed more information.
The FBI has completed its search for records responsive to your request. The material you requested is located in an investigative file which is exempt from disclosure pursuant to 5 U.S.C. § 552(b)(7)(A). 552(b)(7)(A) exempts from disclosure:
Records or information compiled for law enforcement purposes, but only to the extent that the production of such law enforcement records or information … could reasonably be expected to interfere with enforcement proceedings…
Project Veritas previously published videos of a Pfizer scientist discussing the strength of natural COVID-19 antibodies versus the vaccine with an undercover reporter.
Then in October, Project Veritas obtained internal company documents from a whistleblower which showed admissions from Pfizer management that aborted fetal cell lines were used in the company's vaccine program, but that employees should just stick with Pfizer's polished narrative omitting any mention of aborted fetal cell lines to avoid any issues with the public.
Project Veritas has released the following statement on the bombshell revelation:
"It is troubling, not just that Pfizer apparently believes it can rely on the FBI to squash truthful reporting via investigations into law-abiding journalists, but also that they appear to be right in thinking the FBI will willingly target dissenting press with unconstitutional raids. There should be no place for such retaliatory attacks on journalism in America."
About Project Veritas
James O'Keefe established Project Veritas in 2010 as a non-profit journalism enterprise to continue his undercover reporting work. Today, Project Veritas investigates and exposes corruption, dishonesty, self-dealing, waste, fraud, and other misconduct in both public and private institutions to achieve a more ethical and transparent society and to engage in litigation to: protect, defend and expand human and civil rights secured by law, specifically First Amendment rights including promoting the free exchange of ideas in a digital world; combat and defeat censorship of any ideology; promote truthful reporting; and defend freedom of speech and association issues including the right to anonymity. O'Keefe serves as the CEO and Chairman of the Board so that he can continue to lead and teach his fellow journalists, as well as protect and nurture the Project Veritas culture.
Project Veritas is a registered 501(c)3 organization. Project Veritas does not advocate specific resolutions to the issues raised through its investigations.
David Guyton
The narrative is coming apart; thanks to Project Veritas and a few others. Thank you for all that you do.
kuhen25
James is not just a national hero, but an international hero to all! We cannot thank you enough!!!
This is exactly why I continue to donate to Project Veritas. We need more patriots like James O'Keefe.
Jeanie Mottley
James O'Keefe should be a National hero. He is a true champion for truth and justice!
Paul Schuster
Anyone denying the flat out corruption on every level is simply not paying the slightest bit of attention.
Corinne Burque
No matter what, no matter "side", this should concern us all. Stand strong James. We stand with you!
Holy WOW. Unbelievable corruption. These people think they are above the law and above a reckoning from the plebs.
Bradley Johnson
I hope when this is all over and the REAL President takes his office back that James Okeefe gets the biggest fucking medal any American has ever received.
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https://youtu.be/-WdFxe-xMdY
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 8,960
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\section*{Introduction}
First order lagrangian mechanics can be naturally generalized to higher
order lagrangian field theory. Moreover, the latter can be presented in a very
elegant and precise algebro-geometric fashion \cite{v84}.
In particular, it is clear what all the involved geometric structures
(higher order jets, Cartan distribution, $\mathscr{C}$-spectral sequence, etc.,
\cite{v84,b...99}) are. On the other hand it seems to be quite hard to understand
what the most \textquotedblleft reasonable, unambiguous, higher order, field
theoretic generalization\textquotedblright\ of hamiltonian mechanics on
abstract symplectic manifolds is. Actually, there exists a universally accepted
generalization of the standard mechanical picture
\[
\text{lagrangian mechanics on }TQ\Longrightarrow\text{ hamiltonian
mechanics on }T^{\ast}Q\text{,}%
\]
$Q$ being a smooth manifold, to the picture
\begin{equation}
\text{lagrangian field theory on }J^{1}\pi\Longrightarrow\text{ hamiltonian
field theory on }\mathscr{M}\pi\text{,} \label{Picture}%
\end{equation}
$\pi$ being a fiber bundle, $J^{1}\pi$ its first jet space and $\mathscr{M}\pi
$ its multimomentum space \cite{gs73} (see also \cite{hk04}, and \cite{r05} for a recent
review). Picture (\ref{Picture}) includes, in particular, a generalization of
the Legendre transform. Along this path an analogous
structure of the symplectic structure on $T^{\ast}Q$, which has been named the
\emph{multisymplectic structure of} $\mathscr{M}\pi$ (see, for instance,
\cite{gim98}), has been discovered. A whole literature exists about properties of such
structure, which is generically referred to as \emph{multisymplectic
geometry} of $\mathscr{M}\pi$ (see references in \cite{r05}). In particular,
efforts were made to find multisymplectic analogues of all properties of
$T^{\ast}Q$ (including, for instance, the Poisson bracket
\cite{ks75,k97,fr01,fpr03,fpr03b}). Now, it is natural to wonder if it is
possible to reasonably further generalize in two different directions. The
first one is towards a picture
\begin{equation}
\text{lagrangian field theory on }J^{\infty}\pi\Longrightarrow\text{ higher
order hamiltonian field theory,} \label{Picture2}%
\end{equation}
$J^{\infty}\pi$ being the $\infty$th jet space of $\pi$, including a higher
order generalization of the Legendre transform. There is no universally
accepted answer about picture (\ref{Picture2})
(see, for instance, \cite{d77,aa82,s82,k84b,sc90,s91,k02,av04} and references therein). Most
often they involve the choice of some extra structure other than the natural
ones on $J^{\infty}\pi$. Recently, we proposed in \cite{v09b} an answer that is free from such ambiguities.
The second direction in which to generalize picture (\ref{Picture}) can be
illustrated as follows. $T^{\ast}Q$ is just a very special example of
(pre)symplectic manifold. Actually hamiltonian mechanics can be (and should,
in some cases \cite{gnh78}) formulated on abstract (pre)symplectic manifolds.
Similarly, it is natural to wonder if there exists the concept of abstract
multi(pre)symplectic manifolds in such a way that hamiltonian field theory
could be reasonably formulated on them. In the literature there can be found some
proposals of should be abstract multi(pre)symplectic manifolds (see, for
instance, \cite{a92b,cid99}). In particular, definitions have been given in
such a way to be able to prove multisymplectic analogues of the celebrated
Darboux lemma \cite{m88,fg08}. The recent definitions by Forger and Gomes
\cite{fg08} appear to be the most satisfactory in that they are
\textquotedblleft minimal\textquotedblright\ on one side and duly model in an
abstract fashion the relevant geometric properties of $\mathscr{M}\pi$ on the other side. In
their work Forger and Gomes illustrate, in particular, the role played by
fiber bundles in the should be definition of multi(pre)symplectic structure.
The next step forward should be to formulate hamiltonian field theory on
multisymplectic bundles.
In this paper we present our own proposal about what should be an abstract,
first order, hamiltonian field theory. We call such proposal the \emph{theory
of partial differential} (\emph{PD} in the following)
\emph{hamiltonian systems} so to 1) stress that it is a natural generalization
of the theory of hamiltonian systems on abstract symplectic manifolds, 2)
distinguish it from the special case of hamiltonian field theory on
$\mathscr{M}\pi$. A PD-hamiltonian system encompasses both the kinematics
(encoded, in picture (\ref{Picture}), by the multisymplectic structure in
$\mathscr{M}\pi$) and the dynamics (encoded, in picture (\ref{Picture}), by the
so called \emph{hamiltonian section }\cite{r05}) which appear as just
different components of one single geometric object. Namely, the main
difference between a PD-hamiltonian system and a multi(pre)symplectic
structure (whatever the reader understand for this) is the dynamical content
of the former (as opposed to the just kinematical one of the latter). Notice that this idea is already present in literature \cite{k73}. However, our formalism differs from the one in \cite{k73} in that it is adapted to the fibered straucture of the manifold of \textquotedblleft field variables \textquotedblright .
The paper is divided into 6 sections. In Section \ref{SecNotations} we
collect our notations and conventions and recall basic differential geometric
facts that will be used in the main part of the paper.
In Sections \ref{SecAffForms} we define what we call \emph{affine forms} on
fiber bundles. The introduction of affine forms can be motivated as follows.
Trajectories in hamiltonian mechanics are curves, whose 1st derivative at a
point is naturally understood as a tangent vector. In their turn, tangent
vectors can be inserted into differential forms and, in particular, a
symplectic one, and Hamilton equations are written in terms of such an insertion. Trajectories in field theory are sections of a fiber bundle
$\alpha:P\longrightarrow M$, whose 1st derivative at a point is naturally
understood as a point in $J^{1}\alpha$. In their turn, points of $J^{1}\alpha$
can be inserted into affine forms and, in particular, a PD-hamiltonian system
(see Section \ref{SecPDHamSys}), and PD-Hamilton equations are written in terms of such an insertion. Recall now that the natural projection
$J^{1}\alpha\longrightarrow P$ is an affine bundle whose sections are
naturally interpreted as (Ehresmann) connections in $\alpha$. Thus, connections
and affine geometry play a prominent role in the theory of PD-hamiltonian
systems. The affine geometry is hidden in standard hamiltonian mechanics by an
\textit{a priori} choice of the parametrization of the time axis (see
\cite{ggu04,ggu06}, and references therein, for the role of affine geometry in theoretical
mechanics). Similarly, even if the role of connections in field theory has
been often recognized (see, for instance, \cite{s87,emr00}), their affine
geometry is sometimes hidden in hamiltonian field theory on $\mathscr{M}\pi$
by the use of multivectors, or even decomposable ones \cite{pr02,pr02b} (which
is just a multidimensional analogue of choosing a parameterization of time).
Actually, we show in Subsection \ref{SecAff} that affine forms can be
understood as standard differential forms of a special kind. Nevertheless, we prefer to keep the distinction for
foundational reasons.
In Section \ref{SecCalculus} we discuss standard operations with affine forms.
Essentially because of the interpretation of affine forms as standard
differential forms we mentioned above, some of this operations (for instance,
the insertion of a connection into an affine form \cite{emr96}, or the
differential of an affine form) were actually already defined in the
literature, or can be understood as standard operations with forms. We stress
again that we will keep the distinction. Finally, we discuss relevant affine
form cohomologies proving an affine form version of the Poincar\'{e} lemma.
In Section \ref{SecPDHamSys} we introduce PD-(pre)hamiltonian systems, discuss
their geometry and the geometry of the associated PD-Hamilton equations, with
some references to the singular, constrained case (see
\cite{dmm96,dmm96b,d...02,d...05} for an account of the constraint algorithm
in first order field theory). For completeness, we also relate PD-hamiltonian
systems to multi(pre)symplectic structures \textit{\`{a} l\`{a} }Forger
\cite{fg08} and the calculus of variations.
In Section \ref{SecNoether} we introduce PD-Noether symmetries and currents of
a PD-hamiltonian system. In view of the dynamical content of the latter we are
able to prove a Noether theorem (see also \cite{dms04}). Moreover, there is a natural Lie bracket
(named PD-Poisson bracket) among PD-Noether currents. As already mentioned,
multisymplectic analogues of the Poisson bracket have been already discussed in
the literature \cite{ks75,k97,fr01,fpr03,fpr03b}. However, we emphasise here the
dynamical nature of PD-Poisson bracket (see also \cite{fr05,v09}). Namely, such bracket is just part
of the Peierls bracket \cite{v09} among conservation laws of the underlying
lagrangian theory and we don't try to extend it to non-conserved currents.
Indeed, our opinion is that the existence of a Poisson bracket among
non-conserved functions in hamiltonian mechanics is essentially due to the
existence of a preferred hamiltonian system on any symplectic manifold $N$,
i.e., the one with $0$ hamiltonian, for which every function on $N$ is a
conservation law. Finally we discuss the (gauge) reduction of a degenerate
(but unconstrained) PD-hamiltonian system.
In Section \ref{SecComp} we propose few examples of PD-hamiltonian systems,
including the computation of their PD-Noether symmetries and currents or,
in one case, their reduction.
\section{Notations and Conventions\label{SecNotations}}
In this section we collect notations and conventions about some general
constructions that will be used in the following.
Let $N$ be a manifold. We denote by $C^{\infty}(N)$ the $\mathbb{R}$-algebra
of smooth, $\mathbb{R}$-valued functions on $N$. A vector field $X$ over $N$
will be always understood as a derivation $X:C^{\infty}(N)\longrightarrow
C^{\infty}(N)$. We denote by $\mathrm{D}(N)$ the $C^{\infty}(N)$-module of
vector fields over $N$, by $\Lambda(M)=\bigoplus_{k}\Lambda^{k}(N)$ the graded
$\mathbb{R}$-algebra of differential forms over $N$, by $d:\Lambda
(N)\longrightarrow\Lambda(N)$ the de Rham differential, and
by $H(N)=\bigoplus_{k}H^{k}(N)$ the de Rham cohomology. If $F:N_{1}%
\longrightarrow N$ is a smooth map of manifolds, we denote by $F^{\ast
}:\Lambda(N)\longrightarrow\Lambda(N_{1})$ its pull-back. We will understand
everywhere the wedge product $\wedge$ of differential forms, i.e., for
$\omega,\omega_{1}\in\Lambda(N)$, instead of writing $\omega\wedge\omega_{1}$,
we will simply write $\omega\omega_{1}$. We assume the reader to be familiar
with Fr\"{o}licher-Nijenhuis calculus on form valued vector fields (insertion
$i_{Z}\omega$ of a form valued vector field $Z$ into a differential form
$\omega$, Lie derivative $L_{Z}\omega$ of a differential form $\omega$ along a
form valued vector fields $Z$, Fr\"{o}licher-Nijenhuis bracket, etc., see, for instance, \cite{m08}).
Let $\varpi:W\longrightarrow N$ be an affine bundle (or, possibly, a vector
bundle) and $F:N_{1}\longrightarrow N$ a smooth map of manifolds. The affine
space of smooth sections of $\varpi$ will be denoted by $\Gamma(\varpi)$. For
$x\in N$, we put, sometimes, $\Gamma(\varpi)|_{x}:=\varpi^{-1}(x)$ and, for
$\chi\in\Gamma(\varpi)$, we also put $\chi_{x}:=\chi(x)$. The affine bundle on
$N_{1}$ induced by $\varpi$ via $F$ will be denote by $\varpi|_{F}%
:W|_{F}\longrightarrow N$:
\[%
\begin{array}
[c]{c}%
\xymatrix{W|_F \ar[r] \ar[d]_-{\varpi|_F} & W \ar[d]^-{\varpi} \\ N_1 \ar[r]^-F & N }\end{array}
.
\]
We also denote $\Gamma(\varpi)|_F := \Gamma(\varpi|_F)$. For any section $s\in\Gamma(\varpi)$ there exists a unique section, which
abusing the notation we denote by $s|_{F}\in\Gamma(\varpi|_{F})$, such that
the diagram
\[
\xymatrix{W|_F \ar[r] & W \\
N_1 \ar[r]^-F \ar[u]^-{s|_F} & N \ar[u]_-{s}}
\]
commutes. Elements in $\Gamma(\varpi)|_{F}$ are called \emph{sections of
}$\varpi$\emph{ along }$F$. If $F$ is an embedding $\varpi|_{F}$,
$\Gamma(\varpi)|_{F}$ and $s|_{F}$ will be referred to as \emph{the
restriction to }$N_{1}$ of $\varpi,$ $\Gamma(\varpi)$ and $s$, respectively.
If $\varpi_{1}:W_{1}\longrightarrow N$ is an other affine bundle and
$A:\Gamma(\varpi)\longrightarrow\Gamma(\varpi_{1})$ is an affine map then
there exists a unique affine map $A|_{F}:\Gamma(\varpi)|_{F}\longrightarrow
\Gamma(\varpi_{1})|_{F}$ such that $A|_{F}(s|_{F})=A(s)|_{F}$ for all
$s\in\Gamma(\varpi)$.
Let $\alpha:P\longrightarrow M$ be a fiber bundle. A vector field
$X\in\mathrm{D}(P)$ is called $\alpha$\emph{-projectable} iff there exists
$\check{X}\in\mathrm{D}(P)$ such that $X\circ\alpha^{\ast}=\alpha^{\ast}%
\circ\check{X}$. $\check{X}$ is called the $\alpha$\emph{-projection of
}$X$. $\alpha$-projectable vector fields form a Lie subalgebra in
$\mathrm{D}(P)$ denoted by $\mathrm{D}_{V}(P,\alpha)$ (or simply
$\mathrm{D}_{V}$ if this does not lead to confusion). An $\alpha$-projectable
vector field projecting onto the $0$ vector field is an $\alpha$\emph{-vertical
vector field}. $\alpha$-vertical vector fields form an ideal in $\mathrm{D}%
_{V}$ denoted by $V\mathrm{D}(P,\alpha)$ (or simply $V\mathrm{D}$). Notice
that, if $\alpha$ has connected fiber, then $\mathrm{D}_{V}$ is the stabilyzer
of $V\mathrm{D}$ in $\mathrm{D}(P)$, i.e., $\mathrm{D}_{V}=\{X\in
\mathrm{D}(P)\;|\;[X,V\mathrm{D}]\subset V\mathrm{D}\}$.
Let $\alpha:P\longrightarrow M$ be as above, $\dim M=n$, $\dim P=m+n$. Denote
by $\alpha_{1}:J^{1}\alpha\longrightarrow M$ the bundle of $1$-jets of local
sections of $\alpha$ \cite{s89,b...99}, and by $\alpha_{1,0}:J^{1}\alpha
\longrightarrow P$ the canonical projection. For any local section
$\sigma:U\longrightarrow P$ of $\alpha$, $U\subset M$ being an open subset, we
denote by $\dot{\sigma}:U\longrightarrow J^{1}\alpha$ its $1$th jet
prolongation. Any system of adapted to $\alpha$ coordinates $(\ldots
,x^{i},\ldots,y^{a},\ldots)$ on $P$, $\ldots,x^i,\ldots$ being coordinates on $M$
and $\ldots,y^a,\ldots$ fiber coordinates on $P$, gives rise to the system
of jet coordinates $(\ldots,x^{i},\ldots,u^{a},\ldots,y_{i}^{a},\ldots)$ on
$J^{1}\alpha$, $i=1,\ldots,n$, $a=1,\ldots,m$. Recall that $\alpha_{1,0}$ is
an affine bundle and a section $\nabla:P\longrightarrow J^{1}\alpha$ of it is
naturally interpreted as a (Ehresmann) connection in $\alpha$. We assume the
reader to be familiar with the geometry of connections (see, for instance,
\cite{mks93,m08}). $\nabla$ is locally represented as
\[
\nabla:y_{i}^{a}=\nabla_{i}^{a},
\]
$\ldots,\nabla_{i}^{a},\ldots$ being local functions on $P$. The space $\Gamma(\alpha_{1,0})$ of all such
sections will be also denoted by $C(P,\alpha)$ (or simply $C$).
Let $\alpha:P\longrightarrow M$ be as above, $\alpha^{\prime}:P^{\prime
}\longrightarrow M$ another fiber bundle and $G:P\longrightarrow P^{\prime}$ a
bundle morphism (over the identity $\operatorname*{id}_{M}:M\longrightarrow
M$), i.e., a smooth map such that $\alpha^{\prime}\circ G=\alpha$. First of
all, recall that there exists a unique bundle morphism $j_{1}G:J^{1}%
\alpha\longrightarrow J^{1}\alpha^{\prime}$ such that $j_{1}G\circ\dot{\sigma
}=(G\circ\sigma)\dot{\,}$ for all local sections $\sigma$ of $\alpha$.
$j_{1}G$ is the \emph{first jet prolongation of }$G$ and diagram
\[
\xymatrix{ J^1 \alpha \ar[rr]^-{j_1 G} \ar[d]_-{\alpha_{1,0}} & & J^1 \alpha^\prime \ar[d]^-{\alpha^\prime_{1,0}} \\
P \ar[rr]^-{G} \ar[rd]_-{\alpha} & & P^\prime \ar[dl]^-{\alpha^\prime} \\
& M & }
\]
commutes. Now, a connection $\nabla\in C(P,\alpha)$ and a connection
$\nabla^{\prime}\in C(P^{\prime},\alpha^{\prime})$ are said $G$%
\emph{-compatible} iff $\nabla^{\prime}\circ G=j_{1}G\circ\nabla$.
Let
\[
\xymatrix{\cdots \ar[r] & K_{l-1} \ar[r]^-{\delta_{l-1}} & K_{l} \ar[r]^-{\delta_{l}} & K_{l+1} \ar[r]^-{\delta_{l+1}} & \cdots}
\]
be a complex. Put $K:=\bigoplus_{l}K_{l}$ and $\delta:=\bigoplus_{l}\delta
_{l}$. We denote by $H(K,\delta):=\bigoplus_{l}H^{l}(K,\delta)$, the
cohomology space of $(K,\delta)$, $H^{l}(K,\delta):=\ker\delta_{l}%
/\operatorname{im}\delta_{l-1}$.
Let $A$ be a commutative $\mathbb{R}$-algebra, $\boldsymbol{M},\boldsymbol{M}%
_{1}$ be $A$-modules and $\boldsymbol{A}$ an affine space modelled over
$\boldsymbol{M}$. We denote by $\mathrm{Aff}_{A}(\boldsymbol{A},\boldsymbol{M}%
_{1})$ (resp.~ $\mathrm{Hom}_{A}(\boldsymbol{M},\boldsymbol{M}_{1})$) the
$A$-module of affine (resp.~ $A$-linear) maps $\boldsymbol{A}\longrightarrow
\boldsymbol{M}_{1}$ (resp.~ $\boldsymbol{M}\longrightarrow\boldsymbol{M}_{1}$).
If $\phi\in\mathrm{Aff}_{A}(\boldsymbol{A},\boldsymbol{M}_{1})$, its
\emph{linear part} $\underline{\phi}$ is an element in $\mathrm{Hom}%
_{A}(\boldsymbol{M},\boldsymbol{M}_{1})$.
Let $m,r$ be positive integers and $\ldots,A_{a_{1}\cdots a_{r}},\ldots$ be
elements in a real vector space, $a_{1},\ldots,a_{r}=1,\ldots,m$. We denote by
$\ldots,A_{[a_{1}\cdots a_{r}]},\ldots$ their skew-symmetrization, i.e.,
\[
A_{[a_{1}\cdots a_{r}]}:=\tfrac{1}{s!}\sum_{\sigma\in S_{r}}\varepsilon
(\sigma)A_{a_{\sigma(1)}\cdots a_{\sigma(r)}},
\]
$S_{r}$ being the group of permutations of $\{1,\ldots,r\}$ and $\varepsilon
(\sigma)$ the sign of $\sigma\in S_{r}$.
We denote by $\simeq$ (resp.~ $\approx$) a canonical (resp.~ non-canonical)
isomorphism between algebraic structures and by $\equiv$ an equivalence of
notations. For instance, for $\alpha:P\longrightarrow M$ as above, $V\mathrm{D} \equiv V\mathrm{D}(P,M)$. Finally, we understand the sum over upper-lower pairs of repeated indexes.
\section{Affine Forms on Fiber Bundles\label{SecAffForms}}
\subsection{Special Forms on Fiber Bundles}
Let $\alpha:P\longrightarrow M$ be a fiber bundle, $A:=C^{\infty}(P)$,
$A_{0}:=C^{\infty}(M)$, $x^{1},\ldots,x^{n}$ coordinates on $M$, $\dim M=n$,
and $y^{1},\ldots,y^{m}$ fiber coordinates on $P$, $\dim P=n+m$. In the following
we will often understand the monomorphism of algebras $\alpha^\ast:A_0 \longrightarrow A$, whose image is made of functions on $P$ which are constant along the fibers of $\alpha$.
$\mathrm{D}_{V}$ (resp.~ $V\mathrm{D}$) is made of vector
fields $X$ locally of the form $X=X^{i}\partial_{i}+Y^{a}\partial_{a}$ (resp.
$X=Y^{a}\partial_{a}$) where $X^{i}=X^{i}(x^{1},\ldots,x^{n})$, $\partial
_{i}:=\partial/\partial x^{i}$, $i=1,\ldots,n$, $\partial_{a}=\partial
/\partial y^{a}$, $a=1,\ldots,m$.
Denote by $\Lambda_{1}(P,\alpha)=\bigoplus_{k}\Lambda_{1}^{k}(P,\alpha)$ (or
simply $\Lambda_{1}=\bigoplus_{k}\Lambda_{1}^{k}$) the differential (graded)
ideal of differential forms on $P$ vanishing when pulled-back to fibers of
$\alpha$, i.e., $\omega \in \Lambda^k_1$, $k \geq 0$ iff $\omega \in \Lambda^k(P)$ and
$i^\ast_{\alpha^{-1}(x)}(\omega)=0$ for all $x\in M$, $i_{\alpha^{-1}(x)}:
\alpha^{-1}(x) \longrightarrow P$ being the embedding of the fiber $\alpha^{-1}(x)$
of $\alpha$ through $x \in M$. Moreover, denote by $\Lambda_{p}(P,a)=\bigoplus_{k}\Lambda_{p}^{k}(P,\alpha)$
(or simply $\Lambda_{p}=\bigoplus_{k}\Lambda_{p}^{k}$) the $p$-th exterior
power of $\Lambda_1$. For all $k$ and $p$, $\Lambda_{p}^{k}$ is made of differential $k$-forms $\omega$ such that
$(i_{Y_{1}}\circ\cdots\circ i_{Y_{k-p+1}})\omega=0$ for every $Y_{1}%
,\ldots,Y_{k-p+1}\in V\mathrm{D}$ or, which is the same, differential
$k$-forms $\omega$ locally of the form
\[
\omega=\sum_{l\geq0}\omega_{i_{1}\cdots i_{p+l}a_{1}\cdots a_{k-p-l}}%
dx^{i_{1}}\cdots dx^{i_{p+l}}dy^{a_{1}}\cdots dy^{a_{k-p-l}},
\]
$\ldots,\omega_{i_{1}\cdots i_{p+l}a_{1}\cdots a_{k-p-l}},\ldots$ being local
functions on $P$, $i_{1},\ldots,i_{p+l}=1,\ldots,n$, $a_{1},\ldots
,a_{k-p-l}=1,\ldots,m$.
Denote by $V\!\Lambda(P,\alpha)=\bigoplus_{k}V\!\Lambda^{k}(P,\alpha)$ (or
simply $V\!\Lambda=\bigoplus_{k}V\!\Lambda^{k}$) the quotient differential
algebra $\Lambda(P)/\Lambda_{1}$, with $d^{V}:V\!\Lambda\longrightarrow
V\!\Lambda$ its differential and with $p^{V}:\Lambda(P)\ni\omega
\longmapsto\omega^{V}:= \omega + \Lambda_1 \in V\!\Lambda$ the projection onto the quotient. Notice
that $d^{V}$ is $A_{0}$-linear. An element $\rho^{V}$ in $V\!\Lambda^{k}$ is
locally of the form
\[
\rho^{V}=\rho_{a_{1}\cdots a_{k}}d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{k}},
\]
$\ldots,\rho_{a_{1}\cdots a_{k}},\dots$ being local functions on $P$, and $d^{V}\!%
\rho^{V}$ is locally given by
\[
d^{V}\!\rho^{V}=\partial_{a}\rho_{a_{1}\cdots a_{k}}d^{V}\!y^{a}d^{V}\!y^{a_{1}%
}\cdots d^{V}\!y^{a_{k}}=\partial_{\lbrack a}\rho_{a_{1}\cdots a_{k}]}d^{V}\!%
y^{a}d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{k}}.
\]
Clearly, $V\!\Lambda^{1}$ is the dual $A$-module of $V\mathrm{D}$ and
$V\!\Lambda$ its exterior algebra. In particular, elements in $V\!\Lambda$ may
be interpreted as multilinear, skew-symmetric forms on $V\mathrm{D}$.
Denote by $\overline{\Lambda}(P,\alpha)=\bigoplus_{k}\overline{\Lambda}{}%
^{k}(P,\alpha):=\bigoplus_{k}\Lambda_{k}^{k}\subset\Lambda(P)$ (or simply
$\overline{\Lambda}=\bigoplus_{k}\overline{\Lambda}{}^{k}$) the sub-algebra
generated by $\Lambda_{1}^{1}$. An element $\omega\in\overline{\Lambda}{}^{k}$
is locally of the form
\[
\omega=\omega_{i_{1}\cdots i_{k}}dx^{i_{1}}\cdots dx^{i_{k}}.
\]
Notice that $\overline{\Lambda}$ is naturally isomorphic to $A\otimes_{A_{0}%
}\Lambda(M)$ as an $A$-algebra.
For any $p$, the quotient (graded)
differential module $E_{0}^{p,\bullet}\equiv E_{0}^{p,\bullet}%
(P,\alpha):=\Lambda_{p}/\Lambda_{p+1}$\footnote{This last notation is
motivated by the fact that $A$--modules $E_{0}^{p,\bullet}$ are columns of the
first term of the (cohomological) Leray-Serre spectral sequence of the fiber
bundle $\alpha$ (see \cite{mvv??}).} is naturally isomorphic to $V\!\Lambda
\otimes_{A}\overline{\Lambda}{}^{p}$ (or, which is the same, $V\!\Lambda
\otimes_{A_{0}}\Lambda^{p}(M)$) via the correspondence
\begin{equation}
E_{0}^{p,q}\ni\omega+\Lambda_{p+1}^{p+q}\longmapsto\varpi\in V\!\Lambda
^{q}\otimes_{A}\overline{\Lambda}{}^{p}, \label{EqXX}
\end{equation}
well defined by putting
\[
\varpi(Y_{1},\ldots,Y_{q}):=(i_{Y_{q}}\circ
\cdots\circ i_{Y_{1}})(\omega)\in\overline{\Lambda}{}^{p},
\]
$Y_{1}%
,\ldots,Y_{q}\in V\mathrm{D}$. In the following we denote by $E_{0}^{p,q}$ the $q$th homogeneous piece
of $E_{0}^{p,\bullet}$, $q\in\mathbb{Z}$. According to the above said, $V\!\Lambda
\otimes_{A}\overline{\Lambda}{}$ (or, which is the same, $V\!\Lambda
\otimes_{A_{0}}\Lambda(M)$) is the graded object associated with the
filtration $\Lambda(P)\supset\Lambda_{1}\supset\cdots\supset\Lambda_{p}%
\supset\cdots$. As we will see in the next subsection, a connection in
$\alpha$ allows one to identify such filtration with its graded object.
Let us now focus on the ideals $\Lambda_{n-1}$ and $\Lambda_{n}$. Put
$d^{n}x:=dx^{1}\cdots dx^{n}$ and $d^{n-1}x_{i}:=i_{\partial_{i}}d^{n}x$, so
that $dx^{j}d^{n-1}x_{i}=\delta_{i}^{j}d^{n}x$, $i,j=1,\ldots,n$. Then an
element $\omega\in\Lambda_{n-1}^{q+n-1}$ (resp.~ $\omega\in\Lambda_{n}^{q+n-1}%
$) is locally in the form
\[
\omega=\omega_{a_{1}\ldots a_{q}}^{i}dy^{a_{1}}\cdots dy^{a_{q}}d^{n-1}%
x_{i}+\omega_{a_{1}\ldots a_{q-1}}dy^{a_{1}}\cdots dy^{a_{q-1}}d^{n}x
\]
(resp.
\[
\omega=\omega_{a_{1}\ldots a_{q-1}}dy^{a_{1}}\cdots dy^{a_{q-1}}
d^{n}x),
\]$\ldots,\omega_{a_{1}\ldots a_{q}}^{i},\ldots,\omega_{a_{1}\ldots
a_{q-1}},\ldots$ being local functions on $P$. In particular $\Lambda_{n-1}^{q+n-1}$
(resp.~ $\Lambda_{n}^{q+n-1}$) is the module of sections of an $\left[
n\tbinom{q}{m}+\tbinom{q-1}{m}\right] $(resp.~ $\tbinom{q-1}{m}$%
)-dimensional vector bundle over $P$. In few lines we will provide an
alternative description of $\Lambda_{n-1}$ and $\Lambda_{n}$ (Theorem \ref{ThIso}). In our opinion, such description
is more suitable for a better understanding of the role of $\Lambda_{n-1}$ and $\Lambda_{n}$ in
first order field theories (see, for instance, \cite{gim98}).
\subsection{Affine Forms}
Let $\nabla\in C \equiv C(P,\alpha)$. Recall, preliminarily, that $C$ is an affine space
modelled over the $A$-module $\overline{\Lambda}{}^{1}\otimes_{A}V\mathrm{D,}$
or, which is the same, $\Lambda^{1}(M)\otimes_{A_{0}}V\mathrm{D}$. $\nabla$
allows one to split the tangent bundle $TP$ to $P$ into its vertical part $V\!P$
and a horizontal part $H_{\nabla}P$. denote by $H_{\nabla}\mathrm{D}%
(P,\alpha)\subset\mathrm{D}(P)$ (or simply $H_{\nabla}\mathrm{D}%
\subset\mathrm{D}(P)$) the submodule of $\nabla$-horizontal vector fields. An
element $X\in H_{\nabla}\mathrm{D}$ is locally in the form $X=X^{i}\nabla_{i}%
$, where $\nabla_{i}:=\partial_{i}+\nabla_{i}^{a}\partial_{a}$, $i=1,\ldots
,n$. Splitting
\begin{equation}
\mathrm{D}(P)=V\mathrm{D}\oplus H_{\nabla}\mathrm{D} \label{EqSplit}
\end{equation}
determines a splitting of the de Rham differential $d:\Lambda
(P)\longrightarrow\Lambda(P)$ into a horizontal part $d_{\nabla}%
:\Lambda(P)\longrightarrow\Lambda(P)$, and a vertical part $d_{\nabla}%
^{V}:\Lambda(P)\longrightarrow\Lambda(P)$, $d=d_{\nabla}+d_{\nabla}^{V}$,
where $d_{\nabla}$ (resp.~ $d_{\nabla}^{V}$) is the Lie derivative along the
horizontal-form valued vector field (resp.~ the form valued vertical vector field)
$H_{\nabla}:A\longrightarrow\overline{\Lambda}{}^{1}(P)$ (resp.~ $V_{\nabla
}:A\longrightarrow\Lambda^{1}(P)$) determined by $\nabla$. $H_{\nabla}$ (resp.~
$V_{\nabla}$) is locally given by $H_{\nabla}=dx^{i}\nabla_{i}$ (resp.~
$V_{\nabla}=(dy^{a}-\nabla_{i}^{a}dx^{i})\partial_{a}$). Notice that
$(\Lambda(P),d_{\nabla}^{V},d_{\nabla})$ is not a bi-complex unless $\nabla$
is flat. Splitting (\ref{EqSplit}) also determines an isomorphism $\phi_{\nabla}:V\!\Lambda
\otimes_{A}\overline{\Lambda}{}\longrightarrow\Lambda(P)$ locally given by
\[
\phi_{\nabla}(d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{q}}\otimes dx^{i_{1}}\cdots
dx^{i_{q}})=d_{\nabla}^{V}y^{a_{1}}\cdots d_{\nabla}^{V}y^{a_{q}}dx^{i_{1}%
}\cdots dx^{i_{q}}.
\]
In particular, for any $q,p$, there is an obvious projection $\mathfrak{p}_{\nabla
}^{q,p}:\Lambda(P)\longrightarrow V\!\Lambda^{q}\otimes_{A}\overline{\Lambda
}{}{}^{p}$.
For any $k\geq0$ put
\[
{}^\prime \Omega{}^{k+1}:=\mathrm{Aff}_{A}%
(C,V\!\Lambda^{k}\otimes_{A}\overline{\Lambda}{}^{n}).
\] An element
${}^\prime{\vartheta}\in{}^\prime \Omega{}^{k+1}$ is locally given by
\[
{}^\prime{\vartheta}(\nabla)=({}^\prime{\vartheta}{}_{a,a_{1}\cdots a_{k}}%
^{i}\nabla_{i}^{a}+{}^\prime{\vartheta}{}_{a_{1}\cdots a_{k}})d^{V}\!y^{a_{1}%
}\cdots d^{V}\!y^{a_{k}}\otimes d^{n}x,\quad\nabla\in C,
\]
$\ldots,{}^\prime{\vartheta}{}_{a,a_{1}\cdots a_{k}}^{i},\ldots,{}^\prime
{\vartheta}{}_{a_{1}\cdots a_{k}},\ldots$ local functions on $P$. The linear
part ${}^\prime\underline{{\vartheta}}$ of an element ${}^\prime{\vartheta}\in$
${}^\prime \Omega{}^{k+1}$ is an element in the $A$-module
\[
\mathrm{Hom}_{A}(\overline{\Lambda}{}^{1}\otimes_{A}V\mathrm{D},V\!\Lambda
^{k}\otimes_{A}\overline{\Lambda}{}^{n}) \simeq\mathrm{Hom}_{A}%
(V\mathrm{D},V\!\Lambda^{k}\otimes_{A}\overline{\Lambda}{}^{n-1}),
\]
where we identified $V\!\Lambda^{k}\otimes\overline{\Lambda}{}^{n-1}$ and
$\mathrm{Hom}_{A}(\overline{\Lambda}{}^{1},V\!\Lambda^{k}\otimes
\overline{\Lambda}{}^{n})$ via the isomorphism
\[
V\!\Lambda^{k}\otimes\overline{\Lambda}{}^{n-1}\ni\sigma\otimes\rho
\longmapsto\varphi_{\sigma\otimes\rho}\in\mathrm{Hom}_{A}(\overline{\Lambda}%
{}^{1},V\!\Lambda^{k}\otimes\overline{\Lambda}{}^{n}),
\]
$\sigma\in V\!\Lambda^{k}$, $\rho\in\overline{\Lambda}{}^{n-1}$, defined by
putting
\[
\varphi_{\sigma\otimes\rho}(\eta):=(-)^{k}\sigma\otimes\eta\rho\in
V\!\Lambda^{k}\otimes\overline{\Lambda}{}^{n},\quad\eta\in\overline{\Lambda}%
{}^{1}.
\]
Put $\underline{\Omega}^{k+1} \equiv \underline{\Omega}^{k+1}(P,\alpha) := V\!\Lambda^{k+1}\otimes_{A}\overline{\Lambda}{}^{n-1}$. Similarly as above, $\underline{\Omega}^{k+1}$ can be embedded into
$\mathrm{Hom}_{A}(V\mathrm{D},V\!\Lambda^{k}\otimes_{A}\overline{\Lambda}%
{}^{n-1})$ via the correspondence
\begin{equation}
\underline{\Omega}^{k+1}\ni\sigma^{\prime}%
\otimes\rho\longmapsto\varphi_{\sigma^{\prime}\otimes\rho}^{\prime}%
\in\mathrm{Hom}_{A}(V\mathrm{D},V\!\Lambda^{k}\otimes_{A}\overline{\Lambda}%
{}^{n-1}), \label{embed}
\end{equation}
$\sigma^{\prime}\in V\!\Lambda^{k}$, $\rho\in\overline{\Lambda}{}^{n-1}$,
defined by putting
\[
\varphi_{\sigma^{\prime}\otimes\rho}^{\prime}(Y):=i_{Y}\sigma^{\prime}\otimes
\rho\in V\!\Lambda^{k}\otimes_{A}\overline{\Lambda}{}^{n-1},\quad
Y\in V\mathrm{D}.
\]
In the following we will understand embedding (\ref{embed}).\\
Put also $\Omega^{0}\equiv\Omega^{0}(P,\alpha):=\overline{\Lambda}{}^{n-1}$,
$\underline{\Omega}^{0}\equiv{}\underline{\Omega}^{0}(P,\alpha):=\Omega^{0}$,
and for $k\geq0$,
\[
\Omega^{k+1}\equiv\Omega^{k+1}(P,\alpha):=\{\vartheta\in{}^\prime \Omega%
{}^{k+1}\;|\;\underline{\vartheta}\in\underline{\Omega}^{k+1}\},
\]
$\Omega\equiv\Omega(P,\alpha):=\bigoplus_{q\geq0}\Omega^{q}$ and
$\underline{\Omega}\equiv\underline{\Omega}(P,\alpha):=\bigoplus_{q\geq
0}\underline{\Omega}^{q}$. Elements in $\Omega^{k}$ will be called
\emph{affine }$k$\emph{-forms} over $\alpha$, $k\geq 0$. It is easy to show that an
element $\vartheta\in{}^\prime \Omega{}^{k+1}$ is an affine $(k+1)$-form iff
it is locally given by
\[
\vartheta(\nabla)=(\vartheta_{aa_{1}\cdots a_{k}}^{i}\nabla_{i}^{a}%
+\vartheta_{a_{1}\cdots a_{k}})d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{k}}\otimes
d^{n}x,\quad\nabla\in C.
\]
$\ldots,\vartheta_{aa_{1}\cdots a_{k}}^{i},\ldots,\vartheta_{a_{1}\cdots
a_{k}},\ldots$ being local functions on $P$ such that $\vartheta_{aa_{1}\cdots
a_{k}}^{i}=\vartheta_{\lbrack aa_{1}\cdots a_{k}]}^{i}$, $i=1,\ldots,n$,
$a,a_{1},\ldots,a_{k}=1,\ldots,m$.
According to the above said, the linear
part $\underline{\vartheta}\in\underline{\Omega}^{k+1}$ of $\vartheta$ is
implicitly defined by the formula
\[
\vartheta(\nabla+\eta\otimes Y)(Y_{1},\ldots,Y_{k})-\vartheta(\nabla
)(Y_{1},\ldots,Y_{k})=(-)^{k}\eta\cdot\underline{\vartheta}(Y,Y_{1}%
,\ldots,Y_{k})\in\overline{\Lambda}{}^{n},
\]
$\nabla\in C$, $\eta\in\overline{\Lambda}{}^{n-1}$, $Y,Y_{1},\ldots,Y_{k}\in
V\mathrm{D}$, and it is locally given by
\begin{equation}
\underline{\vartheta}=\tfrac{(-)^{k}}{k+1}\vartheta_{a_{1}\cdots a_{k+1}}%
^{i}d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{k+1}}\otimes d^{n-1}x_{i}. \label{Eq4}%
\end{equation}
\subsection{Affine Forms and Differential Forms\label{SecAff}}
Let $\Omega_{0}(P,\alpha)=\bigoplus_{q\geq0}$ $\Omega_{0}^{q}(P,\alpha)$ (or,
simply, $\Omega_{0}\equiv\bigoplus_{q\geq0}$ $\Omega_{0}^{q}$) be the kernel
of the projection $\Omega\ni\vartheta\longmapsto\underline{\vartheta}%
\in\underline{\Omega}$. Clearly, $\Omega_{0}^{q}$ is canonically isomorphic to
$V^{q-1}\Lambda\otimes_{A}\overline{\Lambda}{}{}^{n}$ for $q>0$ (and in the
following we will understand such isomorphism), while $\Omega_{0}^{0}=0$.
Moreover, $\Omega^{q}$ (resp.~ $\Omega_{0}^{q}$) is the module of sections of
an $[n\tbinom{q}{m}+\tbinom{q-1}{m}]$ (resp.~ $\tbinom{q-1}{m}$)-dimensional
vector bundle over $P$.
\begin{theorem}\label{ThIso}
There are canonical isomorphisms of $A$-modules
\begin{align*}
\iota_{0,q} & :\Lambda_{n}^{q+n-1}\longrightarrow\Omega_{0}^{q},\\
\iota_{q} & :\Lambda_{n-1}^{q+n-1}\longrightarrow\Omega^{q},\\
\underline{\iota}{}_{q} & :E_{0}^{n-1,q}\longrightarrow\underline{\Omega
}^{q},
\end{align*}
$q\geq0$, such that diagram
\begin{equation}%
\begin{array}
[c]{c}%
\xymatrix{ 0 \ar[r] & \Lambda_n \ar[d]_-{\iota_0} \ar[r] & \Lambda_{n-1} \ar[d]_-{\iota} \ar[r] & E_0^{n-1} \ar[d]_-{\underline{\iota}} \ar[r] & 0 \\
0 \ar[r] & \Omega_0 \ar[r] & \Omega \ar[r] & \underline{\Omega} \ar[r] & 0
}
\end{array}
\label{Diag1}%
\end{equation}
commutes, where $\iota_{0}:=\bigoplus_{q}\iota_{0,q}$, $\iota:=\bigoplus
_{q}\iota_{q}$ and $\underline{\iota}:=\bigoplus_{q}\underline{\iota}{}_{q}$.
\end{theorem}
\begin{proof}
Let $q>0$. First of all, denote by $\underline{\iota}{}_{q}:E_{0}%
^{n-1,q}\longrightarrow\underline{\Omega}^{q}$ the already mentioned natural
isomorphism (\ref{EqXX}) and notice that for any $\omega\in\Lambda_{n}^{q+n-1}$ and
$Y_{1},\ldots,Y_{q-1}\in V\mathrm{D}$, $(i_{Y_{1}}\circ\cdots\circ i_{Y_{q-1}%
})(\omega)\in\overline{\Lambda}{}^{n}$. Therefore, it is well defined an
element $\iota_{0,q}(\omega)\in\Omega_{0}^{q}$ by putting $\iota_{0,q}%
(\omega)(Y_{1},\ldots,Y_{q-1}):=(i_{Y_{1}}\circ\cdots\circ i_{Y_{q-1}}%
)(\omega)\in\overline{\Lambda}{}^{n}$, $Y_{1},\ldots,Y_{q-1}\in V\mathrm{D}$.
Moreover, the correspondence $\Lambda_{n}^{q+n-1}\ni\omega\longmapsto
\iota_{0,q}(\omega)\in\Omega_{0}^{q}$ is an isomorphism of $A$-modules.
Indeed, let $\omega\in\Lambda_{n}^{q+n-1}$ and $(i_{Y_{1}}\circ\cdots\circ
i_{Y_{q-1}})(\omega)=0$ for all $Y_{1},\ldots,Y_{q-1}\in V\mathrm{D}$, then
$\omega\in\Lambda_{n+1}^{q+n-1}=\boldsymbol{0}$, so that $\iota_{0,q}$ is injective. Moreover, $\Lambda_{n}^{q+n-1}$ and
$\Omega_{0}^{q}$ are locally free $A$-modules of the same local dimension. We
conclude that
\[
\iota_{0}:=%
{\textstyle\bigoplus\nolimits_{q}}
\iota_{0,q}:\Lambda_{n}\longrightarrow\Omega_{0}%
\]
is a canonical isomorphism of $A$-modules as well, sending
$\Lambda_{n}^{q+n-1}$ into $\Omega_{0}^{q}$, $q\geq0$. Finally, if $\omega
\in\Lambda_{n}^{q+n-1}$ is locally given by
\[
\omega=\omega_{a_{1}\ldots a_{q-1}}dy^{a_{1}}\cdots dy^{a_{q-1}}d^{n}x,
\]
then $\iota_{0}(\omega)\in\Omega_{0}$ is locally given by $\iota_{0}%
(\omega)=\omega_{a_{1}\ldots a_{q-1}}d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{q-1}%
}\otimes d^{n}x$.
Now, for $\omega\in\Lambda_{n-1}^{q+n-1}$ and $\nabla\in C$ put
\[
\iota_{q}(\omega)(\nabla):=\mathfrak{p}_{\nabla}^{q-1,n}(\omega).
\]
If $\omega$ is locally given by
\[
\omega=\omega_{a_{1}\ldots a_{q}}^{i}dy^{a_{1}}\cdots dy^{a_{q}}d^{n-1}%
x_{i}+\omega_{a_{1}\ldots a_{q-1}}dy^{a_{1}}\cdots dy^{a_{q-1}}d^{n}x,
\]
then
\begin{eqnarray*}
\omega & =& \omega_{a_{1}\ldots a_{q}}^{i}(d_{\nabla}^{V}+d_{\nabla})(y^{a_{1}%
})\cdots(d_{\nabla}^{V}+d_{\nabla})(y^{a_{q}})d^{n-1}x_{i}\\
& &{} +\omega_{a_{1}\ldots a_{q-1}}d_{\nabla}^{V}y^{a_{1}}\cdots d_{\nabla}%
^{V}y^{a_{q-1}}d^{n}x\\
& =& \omega_{a_{1}\ldots a_{q-1}}d_{\nabla}^{V}y^{a_{1}}\cdots d_{\nabla}%
^{V}y^{a_{q-1}}d^{n}x + \omega_{a_{1}\ldots a_{q}}^{i}d_{\nabla}^{V}y^{a_{1}}\cdots d_{\nabla}%
^{V}y^{a_{q}}d^{n-1}x_{i} \\
& &{} +\sum_{s}(-)^{p-s}\omega_{a_{1}\ldots a_{q}}^{i}\nabla_{j}^{a_{s}}%
d_{\nabla}^{V}y^{a_{1}}\cdots\widehat{d_{\nabla}^{V}y^{a_{s}}}\cdots
d_{\nabla}^{V}y^{a_{q}}dx^{j}d^{n-1}x_{i}\\
& =&\omega^{q,n-1}+[q(-)^{q-1}\omega_{aa_{1}\ldots a_{q-1}}^{i}\nabla_{i}%
^{a}+\omega_{a_{1}\ldots a_{q-1}}]d_{\nabla}^{V}y^{a_{1}}\cdots d_{\nabla}%
^{V}y^{a_{q-1}}d^{n}x
\end{eqnarray*}
where a cap \textquotedblleft$\widehat{\quad}$\textquotedblright\ denotes
omission of the factor below it, and $\omega^{q,n-1}\in\Lambda(P)$ is a
suitable form such that $\mathfrak{p}_{\nabla}^{q-1,n}(\omega^{q,n-1})=0$.
Therefore, locally
\begin{align}
\iota_{q}(\omega)(\nabla) & =\mathfrak{p}_{\nabla}^{q-1,n}(\omega
)\nonumber\\
& =[q(-)^{q-1}\omega_{aa_{1}\ldots a_{q-1}}^{i}\nabla_{i}^{a}+\omega
_{a_{1}\ldots a_{q-1}}]d^{V}\!y^{a_{1}}\cdots d^{V}\!y^{a_{q-1}}\otimes d^{n}x.
\label{Eq5}%
\end{align}
This shows simultaneously that $\iota_{q}(\omega)$ is affine, that it is in
$\Omega^{q}$ and that $\iota_{q}$ is injective. Since $\Lambda_{n-1}^{p+n}$
and $\Omega^{q}$ are locally free $A$-modules of the same local dimension,
then the correspondence $\iota_{q}:$ $\Lambda_{n-1}^{q+n-1}\ni\omega
\longmapsto\iota_{q}(\omega)\in\Omega^{q}$ is an isomorphism. Commutativity of
diagram (\ref{Diag1}) immediately follows from local formulas (\ref{Eq4}) and
(\ref{Eq5}).
\end{proof}
Notice that isomorphism $\iota$ generalizes considerably the well known isomorphism $\Lambda_{n-1}^{n}%
\simeq\mathrm{Aff}_{A}(C,\overline{\Lambda}{}^{n})$ \cite{gim98}.
Finally, let $\pi:E\longrightarrow M$ be a fiber bundle and $\ldots,q^{A},\ldots$ fiber coordinates on $E$.
Notice that $\Omega^{1}(E,\pi)$ (resp.~ $\underline{\Omega}^{1}(E,\pi)$) is the
$C^{\infty}(E)$-module of sections of a vector bundle $\mu_{0}\pi
:\mathscr{M}\pi\longrightarrow E$ (resp.~ $\tau_{0}^{\dag}\pi:J^{\dag}%
\pi\longrightarrow E$). Recall that there is a distinguished element $\Theta$
in $\Omega^{1}(\mathscr{M}\pi,\mu\pi)$ (resp.~ $\underline{\Theta}\in
\underline{\Omega}^{1}(J^{\dag}\pi,\tau^{\dag}\pi)$), with $\mu\pi := \pi \circ \mu_0 \pi$ (resp.~ $\tau^\dag \pi:= \pi \circ \tau_0 ^\dag \pi$), the tautological one
\cite{gim98}, which in standard coordinates \[ \ldots,x^{i},\ldots,q^{A}%
,\ldots,p_{A}^{i},\ldots,p \] on $\mathscr{M}\pi$ (resp.~ $\ldots,x^{i}%
,\ldots,q^{A},\ldots,p_{A}^{i},\ldots$ on $J^{\dag}\alpha$) is given by
\[
\Theta=p_{A}^{i}dq^{A}d^{n-1}x_{i}-pd^{n}x\quad\text{(resp.~ }\underline
{\Theta}=p_{A}^{i}d^{V}\!q^{A}\otimes d^{n-1}x_{i}\text{).}%
\]
\section{Affine Form Calculus\label{SecCalculus}}
\subsection{Natural Operations with Affine Forms}
In this section we derive the main formulas of calculus on affine forms. Such
formulas will turn useful in generalizing proofs from the context of
hamiltonian systems to the context of PD-hamiltonian systems (see Section \ref{SecNoether}).
Let $\alpha:P\longrightarrow M$ be as in the previous section. Isomorphism
$\iota$ (resp.~ $\iota_{0}$, $\underline{\iota}$) can be used to
\textquotedblleft transfer structures\textquotedblright\ from $\Lambda_{n-1}$
(resp.~ $\Lambda_{n}$, $E_{0}^{n-1}$) to $\Omega$ (resp.~ $\Omega_{0}$,
$\underline{\Omega}$) and back. As an instance, notice that $\Omega$ has
got a natural structure of $\Lambda(P)$-module given by
\[
\lambda\vartheta:=\iota(\lambda\omega),
\]
$\lambda\in\Lambda(P)$, $\vartheta=\iota(\omega)\in\Omega$, $\omega\in
\Lambda_{n-1}$. Moreover, $\Omega$ is generated by $\Omega^{0}$ as a
$\Lambda(P)$-module. Similarly, $\Lambda_{n}$ (resp.~ $E_{0}^{n-1}$) has a
structure of $V\!\Lambda$-module given by
\[
\lambda^{V}\omega_{0}:=\iota_{0}^{-1}(\lambda^{V}\rho^{V}\otimes\nu)\text{
(resp.~ }\lambda^{V}\underline{\omega}:=\iota_{0}^{-1}(\lambda^{V}\rho
^{V}\otimes\sigma)\text{),}%
\]
$\lambda^{V}\in V\!\Lambda$, $\omega_{0}=\iota_{0}^{-1}(\rho^{V}\otimes\nu)$
(resp.~ $\underline{\omega}=\underline{\iota}^{-1}(\rho^{V}\otimes\sigma)$),
$\rho^{V}\in V\!\Lambda$, $\nu\in\overline{\Lambda}{}^{n}$ (resp.~ $\sigma\in$
$\overline{\Lambda}{}^{n-1}$), so that $\rho^{V}\otimes\nu\in V\!\Lambda
\otimes_{A}\overline{\Lambda}{}^{n}=\Omega_{0}$ (resp.~ $\rho^{V}\otimes
\sigma\in V\!\Lambda\otimes_{A}\overline{\Lambda}{}^{n-1}=\underline{\Omega}%
$). Clearly, $\Lambda_{n}$ (resp.~ $E_{0}^{n-1}$) is generated by
$\overline{\Lambda}{}^{n}$ (resp.~ $\overline{\Lambda}{}^{n-1}$) as a
$V\!\Lambda$-module. Finally, the presented structures are compatible in the
sense that for $\omega_{0}\in\Lambda_{n}$, $\omega\in\Lambda_{n-1}$ and
$\lambda\in\Lambda(P)$, we have
\[
\lambda^{V}\omega_{0}=\lambda\omega_{0}\text{ and }\underline{\lambda\omega
}=\lambda^{V}\underline{\omega}.
\]
As a last instance of how to use isomorphisms in (\ref{Diag1}) to transfer a
structure from one space to the other we define the insertion of a connection
$\nabla\in C$ into a differential form $\omega\in\Lambda_{n}$ as follows
\[
i_{\nabla}\omega:=\iota_{0}^{-1}(\vartheta(\nabla))=(\iota_{0}^{-1}%
\circ\mathfrak{p}_{\nabla}^{q-1,n})(\omega)\in\Lambda_n,
\]
$\vartheta=\iota(\omega)\in \Omega$. Notice that the just defined insertion of a
connection in an element $\omega\in\Lambda_{n}$ has been already discussed in
\cite{emr96}. In the following we will always understand isomorphisms $\iota$,
$\iota_{0}$, $\underline{\iota}$.
Notice that $\underline{\Omega}$ inherits many operations from $\Omega$.
Indeed, let $\nabla\in C$, $Z\in\overline{\Lambda}{}^{1}\otimes_{A}%
V\mathrm{D} \subset \Lambda(P) \otimes_A \mathrm{D}(P)$, $Y\in V\mathrm{D}$, $X\in\mathrm{D}_{V}$, $q\geq0$. Then
\begin{itemize}
\item $i_{Z}(\Omega)\subset\Omega_{0}$ and $i_{Z}(\Omega_{0})=0$ so that an operator, which, abusing the notation, we again denote by
$i_{Z}:\underline{\Omega}\longrightarrow\Omega_{0}$, is well defined via the formula
\[
i_{Z}\underline{\omega}:=i_{Z}\omega\in\Omega_{0},
\]
$\omega\in\Omega$. Moreover, it is easy to show that
\[
i_{Z}\underline{\omega
}=i_{\nabla+Z}\omega-i_{\nabla}\omega.
\]
Finally, for $Z=\eta\otimes Y_{1}$,
and $\underline{\omega}=\rho^{V}\otimes\sigma$, $\eta\in\overline{\Lambda}%
{}^{1}$, $Y_{1}\in V\mathrm{D}$, $\rho^{V}\in V\!\Lambda^{q}$ and $\sigma
\in\overline{\Lambda}{}^{n-1}$, we have
\[
i_{Z}\underline{\omega}=(-)^{q-1}i_{Y_{1}}\rho^{V}\otimes\eta\sigma.
\]
\item $i_{Y}(\Omega)\subset\Omega$ (resp.~ $L_{X}(\Omega)\subset\Omega$) and
$i_{Y}(\Omega_{0})\subset\Omega_{0}$ (resp.~ $L_{X}(\Omega_{0})\subset
\Omega_{0}$) so that the quotient map, which, abusing the
notation, we again denote by $i_{Y}:\underline{\Omega}\longrightarrow
\underline{\Omega}$ (resp.~ $L_{X}:\underline{\Omega}\longrightarrow
\underline{\Omega}$), is well defined via the formula
\[
i_{Y}\underline{\omega}:=\underline{i_{Y}\omega}\in\underline{\Omega}\text{
(resp.~ }L_{X}\underline{\omega}=\underline{L_{X}\omega}\in\underline{\Omega
}\text{)}.
\]
Finally, for $\underline{\omega}=\rho^{V}\otimes\sigma$, $\rho^{V}\in
V\!\Lambda^{q}$ and $\sigma\in\overline{\Lambda}{}^{n-1}$, we have
\[
i_{Y}\underline{\omega}=i_{Y}\rho^{V}\otimes\sigma.
\]
\item $d_{\nabla}(\Omega)\subset\Omega_{0}$ and $d_{\nabla}(\Omega_{0})=0$ so
that an operator, which, abusing the notation, we again
denote by $d_{\nabla}:\underline{\Omega}\longrightarrow\Omega_{0}$, is well defined via the
formula
\[
d_{\nabla}\underline{\omega}:=d_{\nabla}\omega\in\Omega_{0},
\]
$\omega\in\Omega$.
\end{itemize}
\begin{remark}
\label{RemPoint}Notice that the insertion $i_{\nabla}\omega$, being affine in
$\nabla$, is actually point wise, i.e., if $\nabla^{\prime}\in C$ is such that
$\nabla_{y}^{\prime}=\nabla_{y}\in C|_{y}=\alpha_{1,0}^{-1}(y)$ for some $y \in P$, then
$(i_{\nabla^{\prime}}\omega)_{y}=(i_{\nabla}\omega)_{y}$. Therefore, the insertion $i_{c}\omega_{y}$ of an element $c\in\alpha_{1,0}%
^{-1}(y)$, $y\in P$, into $\omega_{y}$ is well defined. Similar considerations apply to both
the above defined insertions $i_{Z}$ and $i_{Y}$. Finally, for all $y\in P$,
the projection $\Omega\longrightarrow\underline{\Omega}$ as well determines a
well defined linear map $\Omega|_{y}\ni\omega_{y}\longmapsto
\underline{\omega}_{y}\in\underline{\Omega}|_{y}$ whose kernel is $\Omega
_{0}|_{y}$.
\end{remark}
In the following we will denote by $\delta:\Omega\longrightarrow\Omega$ (resp.~
$\delta_{0}:\Omega_{0}\longrightarrow\Omega_{0}$) the restricted de Rham
differential, i.e., for $\omega\in\Omega$ (resp.~ $\omega_{0}\in\Omega_{0}$),
$\delta\omega:=d\omega\in\Omega$ (resp.~ $\delta_{0}\omega_{0}:=d\omega_{0}%
\in\Omega_{0}$) and with $\underline{\delta}:\underline{\Omega}\longrightarrow
\underline{\Omega}$ the quotient differential. Then, for $\omega_{0}=\rho
^{V}\otimes\alpha^{\ast}(\nu_{0})$ (resp.~ $\omega=\rho^{V}\otimes\alpha^{\ast
}(\sigma_{0})$), $\rho^{V}\in V\!\Lambda$, $\nu_{0}\in\Lambda^{n}(M)$ (resp.
$\sigma_{0}\in\Lambda^{n-1}(M)$), we have
\[
\delta_{0}\omega_{0}=d^{V}\!\rho^{V}\otimes\alpha^{\ast}(\nu_{0})\text{ (resp.
}\underline{\delta}\underline{\omega}=d^{V}\!\rho^{V}\otimes\alpha^{\ast}%
(\sigma_{0})\text{)}.
\]
In other words $\delta_{0}$ (resp.~ $\underline{\delta}$) is isomorphic to the
differential $d^{V}\otimes\mathrm{id}:V\!\Lambda\otimes_{A_{0}}\Lambda
^{n}(M)\longrightarrow V\!\Lambda\otimes_{A_{0}}\Lambda^{n}(M)$ (resp.
$d^{V}\otimes\mathrm{id}:V\!\Lambda\otimes_{A_{0}}\Lambda^{n-1}%
(M)\longrightarrow V\!\Lambda\otimes_{A_{0}}\Lambda^{n-1}(M)$).
All the above mentioned formulas can be proved by straightforward computations.
Now, let $\nabla$, $Y$ and $X$ be as above. denote by $[\![\cdot,\cdot]\!]$
the Fr\"{o}licher-Nijenhuis bracket in $\Lambda(P)\otimes_{A}\mathrm{D}(P)$.
It is easy to see that $[\![H_{\nabla},X]\!]\in\overline{\Lambda}{}^{1}%
\otimes_{A}V\mathrm{D}\subset\Lambda(P)\otimes_{A}\mathrm{D}(P)$. It holds the following
\begin{theorem}
Let $\omega\in\Omega$, then
\begin{gather}
\lbrack i_{\nabla},\delta]\omega:=(i_{\nabla}\circ\delta-\delta_{0}\circ
i_{\nabla})\omega=d_{\nabla}\omega\in\Omega_{0},\nonumber\\
\lbrack i_{\nabla},i_{Y}]\omega:=(i_{\nabla}\circ i_{Y}-i_{Y}\circ i_{\nabla
})\omega=0\in\Omega_{0},\label{Eq9}\\
\lbrack i_{\nabla},L_{X}]\omega:=(i_{\nabla}\circ L_{X}-L_{X}\circ i_{\nabla
})\omega=i_{[\![H_{\nabla},X]\!]}\omega\in\Omega_{0}.\nonumber
\end{gather}
\end{theorem}
\begin{proof}
First prove that $i_{\nabla}:\Omega\longrightarrow\Omega_{0}$ satisfies the
\textquotedblleft Leibnitz rule\textquotedblright\
\begin{equation}
i_{\nabla}(\lambda\omega)=\lambda\cdot i_{\nabla}\omega+i_{H_{\nabla}}%
\lambda\cdot\omega, \label{Eq6}%
\end{equation}
$\lambda\in\Lambda(P)$, $\omega\in\Omega$. For $\rho\in\Lambda(P)$, denote
$\rho_{\nabla}^{\bullet,p}:=\sum_{q}\mathfrak{p}_{\nabla}^{q,p}(\rho)$, so
that $\rho=\sum_{p}\rho_{\nabla}^{\bullet,p}$. Notice that for $\omega
\in\Omega$ and $\lambda\in\Lambda(P)$, we have $\omega=\omega_{\nabla
}^{\bullet,n}+\omega_{\nabla}^{\bullet,n-1}$ so that
\[
i_{\nabla}(\lambda\omega)=\mathfrak{p}_{\nabla}^{\bullet,n}(\lambda
\omega)=\lambda_{\nabla}^{l,0}\omega_{\nabla}^{\bullet,n}+\lambda_{\nabla
}^{\bullet,1}\omega_{\nabla}^{\bullet,n-1}=\lambda\cdot i_{\nabla}%
\omega+\lambda_{\nabla}^{\bullet,1}\cdot\omega_{\nabla}^{\bullet,n-1}.
\]
Moreover, $i_{H_{\nabla}}\lambda=%
{\textstyle\sum\nolimits_{p}}
i_{H_{\nabla}}\lambda_{\nabla}^{\bullet,p}=%
{\textstyle\sum\nolimits_{p}}
p\lambda_{\nabla}^{\bullet,p}$, which in turn implies $\lambda_{\nabla
}^{\bullet,p}=i_{H_{\nabla}}\lambda-%
{\textstyle\sum\nolimits_{p>1}}p\lambda_{\nabla}^{\bullet,p}$. Therefore
\begin{align*}
i_{\nabla}(\lambda\omega) & =\lambda\cdot i_{\nabla}\omega+\lambda_{\nabla}^{\bullet,1}\cdot
\omega_{\nabla}^{\bullet,n-1}\\
& =\lambda\cdot i_{\nabla}\omega+i_{H_{\nabla}}\lambda\cdot\omega_{\nabla
}^{\bullet,n-1} - {\textstyle\sum\nolimits_{p>1}}p\lambda_{\nabla}^{\bullet,p}\omega_{\nabla}^{\bullet,n-1}\\
& =\lambda\cdot i_{\nabla}\omega+i_{H_{\nabla}}\lambda\cdot\omega.
\end{align*}
In view of (\ref{Eq6}), the above defined operators $[i_{\nabla},\delta],$
$[i_{\nabla},i_{Y}],$ $[i_{\nabla},L_{X}]:\Omega\longrightarrow\Omega_{0}$,
satisfy analogous \textquotedblleft Leibnitz rules\textquotedblright\:
\begin{align}
\lbrack i_{\nabla},\delta](\lambda\omega) & =d_{\nabla}\lambda\cdot\omega
+(-)^{l}\lambda\cdot\lbrack i_{\nabla},\delta](\omega), \nonumber \\
\lbrack i_{\nabla},i_{Y}](\lambda\omega) & =\lambda\cdot\lbrack i_{\nabla}%
,i_{Y}] (\omega),\label{EqYY} \\
\lbrack i_{\nabla},L_{X}](\lambda\omega) & =i_{[\![H_{\nabla},X]\!]}\lambda
\cdot\omega+\lambda\cdot\lbrack i_{\nabla},L_{X}](\omega). \nonumber
\end{align}
Since $\Omega$ is generated by $\overline{\Lambda}{}^{n-1}$ as a $\Lambda
(P)$-module, in view of (\ref{EqYY}), it is enough to prove (\ref{Eq9}) for $\omega
\in\overline{\Lambda}{}^{n-1}$. In this case
\begin{gather*}
\lbrack i_{\nabla},\delta]\omega=i_{\nabla}d\omega=(d\omega)_{\nabla}%
^{\bullet,n}=d_{\nabla}\omega,\\
\lbrack i_{\nabla},i_{Y}]\omega=0,\\
\lbrack i_{\nabla},L_{X}]\omega=i_{\nabla}L_{X}\omega=(L_{X}\omega)_{\nabla
}^{\bullet,n}=0=i_{[\![H_{\nabla},X]\!]}\omega.
\end{gather*}
\end{proof}
We now discuss the interaction between affine forms and bundle morphisms. Let
$\alpha^{\prime}:P^{\prime}\longrightarrow M$ be another fiber bundle and
$G:P\longrightarrow P^{\prime}$ a bundle morphism. Clearly, $G$ preserves the
ideals $\Lambda_{p}$, $p\geq0$, i.e., $G^{\ast}(\Lambda_{p}(P^{\prime}%
,\alpha^{\prime}))\subset\Lambda_{p}(P,\alpha)$. In particular,
\[
G^{\ast}(\Omega(P^{\prime},\alpha^{\prime}))\subset\Omega(P,\alpha)\text{ and
}G^{\ast}(\Omega_{0}(P^{\prime},\alpha^{\prime}))\subset\Omega_{0}%
(P,\alpha)\text{.}%
\]
We conclude that the quotient map which, abusing the
notation, we again denote by $G^{\ast}:\underline{\Omega}(P^{\prime}%
,\alpha^{\prime})\longrightarrow\underline{\Omega}(P,\alpha)$, is well defined. Now, consider
$G$-compatible connections $\nabla\in C(P,\alpha)$ and $\nabla^{\prime}\in
C(P^{\prime},\alpha^{\prime})$. It is easy to show that
\begin{equation}
G^{\ast}\circ i_{\nabla^{\prime}}=i_{\nabla}\circ G^{\ast}:\Omega(P^{\prime
},\alpha^{\prime})\longrightarrow\Omega_{0}(P,\alpha). \label{Eq33}%
\end{equation}
\subsection{Cohomology}
\begin{remark}
(see \cite{mvv??}) In the following we denote by $\mathcal{F}$ the abstract
fiber of $\alpha$. Notice that, for any $q\geq0$, $V\!H^{q}\equiv
V\!H^{q}(P,\alpha):=H^{q}(V\!\Lambda,d^{V})$ is the $A_{0}$-module of sections
of a (pro-finite) vector bundle $\alpha^{q}:P^{q}\longrightarrow M$ over $M$
whose abstract fiber is $H^{q}(\mathcal{F})$. Moreover, $\alpha^{q}$ is
endowed with a canonical flat connection $\nabla^{q}$ ($\nabla^{q}$ is a
smooth analogue of Gauss-Manin connection in algebraic geometry).
Correspondingly, there is a de Rham like complex
\[
\xymatrix@C=40pt{\cdots \ar[r] & \Lambda^{p-1}\otimes_{A_{0}}V\!H^{q} \ar[r]^-{d_{1}^{p-1,q}} & \Lambda^{p}\otimes_{A_{0}}V\!H^{q} \ar[r]^-{d_{1}^{p,q}} & \cdots},
\]
whose cohomology we denote by $E_{2}^{\bullet,q}:=\bigoplus_{p}E_{2}^{p,q}$,
$E_{2}^{p,q}:=H^{p}(\Lambda(M)\otimes_{A_{0}}V\!\Lambda^{q},d_{1}^{\bullet
,q})$\footnote{Similarly as above, this last notations are motivated by the
fact that the differentials $d_{1}^{\bullet,q}$ (resp.~ the vector spaces
$E_{2}^{\bullet,q}$) are the ones in the first term (resp.~ are rows of the
second term) of the (cohomological) Leray-Serre spectral sequence of the fiber
bundle $\alpha$ \cite{mvv??}.~ }, $q\geq0$. It can be proved that, if $\alpha$
is trivial or $M$ is simply connected, then there is a (generically
non-canonical), isomorphism
\[
E_{2}^{p,q}\approx H^{p}(M)\otimes H^{q}(\mathcal{F}),\quad p,q\geq0.
\]
Finally, notice also that, for any $q\geq0$,
\begin{align*}
H^{q}(\Omega_{0},\delta_{0}) & \simeq\Lambda^{n}(M)\otimes_{A_{0}}%
V\!H^{q},\\
H^{q}(\underline{\Omega},\underline{\delta}) & \simeq\Lambda^{n-1}%
(M)\otimes_{A_{0}}V\!H^{q}.
\end{align*}
\end{remark}
\begin{proposition}
\label{Prop1}Let $\alpha:P\longrightarrow M$ be a fiber bundle. Then, for any
$q\geq0$, there exists a short exact sequence of vector spaces
\[
\xymatrix{0 \ar[r] & \operatorname{coker} d_1^{n,q-1} \ar[r] & H^q(\Omega,\delta) \ar[r] & \ker d_1^{n-1,q} \ar[r] & 0}.
\]
In particular, $H^{q}(\Omega,\delta)\approx\operatorname{coker}d_{1}%
^{n,q-1}\oplus\ker d_{1}^{n-1,q}=E_{2}^{n,q-1}\oplus\ker d_{1}^{n-1,q}$.
\end{proposition}
\begin{proof}
Consider the short exact sequence of complexes
\[
\xymatrix{0 \ar[r] & \Omega_{0} \ar[r] & \Omega \ar[r] & \underline
{\Omega} \ar[r] & 0},
\]
and the associated long sequence in cohomology
\begin{equation}
\xymatrix@C=23pt{ \cdots \ar[r] & H^{q-1}(\underline{\Omega},\underline{\delta}) \ar[r]^-{\partial}
& H^{q}(\Omega_{0},\delta_{0}) \ar[r]
& H^{q}(\Omega,\delta) \ar[r]
& H^{q}(\underline{\Omega},\underline{\delta}) \ar[r]^-{\partial}
& \cdots}. \label{Eq28}%
\end{equation}
We already commented, in the above remark, that, for any $q$, $H^{q}%
(\Omega_{0},\delta_{0})$ identifies with $\Lambda^{n}(M)\otimes_{A_{0}%
}V\!H^{q}$ and $H^{q}(\underline{\Omega},\underline{\delta})$ identifies with
$\Lambda^{n-1}(M)\otimes_{A_{0}}V\!H^{q}$. Similarly, it is easy to show that
the connecting operator
\[
\partial:H^{q-1}(\underline{\Omega},\underline{\delta})\longrightarrow
H^{q}(\Omega_{0},\delta_{0})
\]
identifies with the de Rham-like differential
\[
d_{1}^{n-1,q}:\Lambda^{n-1}(M)\otimes_{A_{0}}V\!H^{q}\longrightarrow
\Lambda^{n}(M)\otimes_{A_{0}}V\!H^{q}.
\]
The thesis then follows from exactness of (\ref{Eq28}).
\end{proof}
\begin{corollary}
\label{CorH0}If $\mathcal{F}$ is connected, then $H^{0}(\Omega,\delta
)\simeq\ker d_{M}^{n-1}$,
\[
d_{M}^{n-1}:\Lambda^{n-1}(M)\longrightarrow\Lambda^{n}(M)
\]
being the last de Rham differential of $M$.
\end{corollary}
\begin{proof}
If $\mathcal{F}$ is connected $V\!H^{0}\simeq A_{0}$ and $d_{1}^{n-1,0}$
identifies with $d_{M}^{n-1}$.
\end{proof}
\begin{corollary}
\label{CorPoincare}Let $q\geq0$ and $\omega\in\Omega^{q}$ be $\delta$-closed,
i.e., $\delta\omega=0$. Then, 1) if $q=0$, $\omega$ is locally of the form
$\alpha^{\ast}(\eta)$ for some $\eta\in\Lambda^{n-1}(M)$, 2) if $q>0$, then
$\omega$ is locally $\delta$-exact, i.e., $\omega$ is locally of the form
$\delta\theta$, $\theta$ being a local element in $\Omega^{q-1}$.
\end{corollary}
\begin{proof}
If $\mathcal{F}$ is contractible, then $V\!H^{q}=0$, and therefore
$H^{q}(\Omega,\delta)=0$, for all $q>0$.
\end{proof}
Let $\omega\in\Omega$ and $\theta\in\Omega$ be such that $\omega=\delta\theta
$. Then $\theta$ will be called a \emph{potential} of $\omega$.
\section{PD-Hamiltonian Systems\label{SecPDHamSys}}
\subsection{PD-Hamiltonian Systems and PD-Hamilton Equations}
In this section we introduce what we think should be understood as the partial
differential, i.e., field theoretic analogue of a hamiltonian (mechanical) system on an abstract symplectic manifold.
Let $\alpha:P\longrightarrow M$ be as in the previous section and $\omega
\in\Omega^{2}(P,\alpha)$ be such that $\delta\omega=0$. Put
\begin{align*}
\ker\omega & :=\{Y\in V\mathrm{D}\;|\;i_{Y}\omega=0\},\quad\ker
\underline{\omega}:=\{Y\in V\mathrm{D}\;|\;i_{Y}\underline{\omega}{}=0\},\\
\operatorname{Ker}\omega & :=\{\nabla\in C\;|\;i_{\nabla}\omega
=0\},\quad\operatorname{Ker}\underline{\omega}:=\{Z\in V\mathrm{D}\otimes
_{A}\overline{\Lambda}{}^{1}\;|\;i_{Z}\underline{\omega}{}=0\}.
\end{align*}
Since $\omega$ is closed, both $\ker\omega$ and $\ker\underline{\omega}$ are
modules of smooth sections of involutive $\alpha$-vertical distributions
$D^{\omega}$ and $\underline{D}^{\omega}$ on $P$, where, for $y\in P$,
\[
D_{y}^{\omega}:=\{\xi\in V_{y}P\;|\;i_{\xi}\omega_{y}=0\},\quad\underline
{D}_{y}^{\omega}:=\{\xi\in V_{y}P\;|\;i_{\xi}\underline{\omega}_{y}=0\}.
\]
Similarly, $\operatorname{Ker}\underline{\omega}$ is a sub-module in
$V\mathrm{D}\otimes_{A}\overline{\Lambda}{}^{1}$. As a minimal regularity
requirement, assume that $\underline{D}^{\omega}$ has got constant rank
$\underline{r}$. Then, it is easy to check that, as a consequence,
$\operatorname{Ker}\underline{\omega}$ is the module of sections of a smooth
vector bundle $\varpi:W\longrightarrow P$. For $y\in P$, denote $r(y)=\dim
D_{y}^{\omega}$. In general, $r(y)$ will change from point to point $y\in P$.
However, we are proving in brief that $r(y)$ cannot change that much. First of
all, since, obviously, $D^{\omega}\subset\underline{D}^{\omega}$, then
$r(y)\leq\underline{r}$ for all $y\in P$. Now, for $y\in P$, denote
\[
\operatorname{Ker}\omega_{y}:=\{c\in\alpha_{1,0}^{-1}(y)\;|\;i_{c}\omega
_{y}=0\}.
\]
Then, $\operatorname{Ker}\omega_{y}$ is either empty or an affine space
modelled over $\varpi^{-1}(y)$. It holds the
\begin{proposition}
For any $y\in P$, $\underline{r}-r(y)\leq1$ (see also Theorem 4 of \cite{fg08}).
\end{proposition}
\begin{proof}
Let $y\in P$ and suppose $r(y)<\underline{r}$. If $\xi\in\underline{D}%
_{y}^{\omega}$ then (see Remark \ref{RemPoint}) $\underline{i_{\xi}\omega_{y}%
}=i_{\xi}\underline{\omega}_{y}=0$ so that $i_{\xi}\omega_{y}\in\Omega_{0}%
^{1}|_{y}=\overline{\Lambda}{}^{n}|_{y}$. Then consider the map $\gamma
_{y}:\underline{D}_{y}^{\omega}\ni\xi\longmapsto\gamma_{y}(\xi):=i_{\xi}%
\omega_{y}\in\overline{\Lambda}{}^{n}|_{y}$. Since $r(y)<\underline{r}$,
$\gamma_{y}$ is surjective and the sequence of vector spaces $0\longrightarrow
D_{y}^{\omega}\longrightarrow\underline{D}_{y}^{\omega}\overset{\gamma_{y}%
}{\longrightarrow}\overline{\Lambda}{}^{n}|_{y}\longrightarrow0$ is exact.
Since $\overline{\Lambda}{}^{n}|_{y}$ is $1$-dimensional, it follows that
$\underline{r}-r(y)=1$.
\end{proof}
The following proposition characterizes the case $r(y)=\underline{r}$.
\begin{proposition}
\label{PropKer}Let $\omega$ be as above. Then $r(y)=\underline{r}$ iff
$\operatorname{Ker}\omega_{y}\neq\varnothing$.
\end{proposition}
\begin{proof}
The result is nothing more than an application of the Rouch\'{e}-Capelli
theorem. We here propose a dual proof. Let $\xi\in V_{y}P$ be given by $\xi$
$=\xi^{a}{\partial}_a| _{y}$. Then
$\xi\in\underline{D}_{y}^{\omega}$ iff
\begin{equation}
\omega_{ab}^{i}(y)\xi^{a}=0,\quad a=1,\ldots,m,\;i=1,\ldots,n. \label{Eq12}%
\end{equation}
Similarly, $\xi\in D_{y}^{\omega}$ iff they are satisfied both (\ref{Eq12})
and
\begin{equation}
\omega_{a}(y)\xi^{a}=0. \label{Eq13}%
\end{equation}
Therefore, $D_{y}^{\omega}=\underline{D}_{y}^{\omega}$ iff Equation
(\ref{Eq13}) linearly depends on Equations (\ref{Eq12}), i.e., iff there are
real numbers $h_{i}^{b}$, $b=1,\ldots,m,i=1,\ldots,n$, such that
\[
\omega_{a}(y)=\omega_{ab}^{i}(y)h_{i}^{b},
\]
$a=1,\ldots,m$. Now, let $c\in\alpha_{1,0}^{-1}(y)$ be given by $y_{i}^{a}(c)=-\tfrac{1}%
{2}h_{i}^{a}$. Then $i_{c}\omega_{y}$ is given by
\begin{align*}
i_{c}\omega_{y} & =(-2\omega_{ba}^{i}(y)y_{i}^{a}(c)+\omega_{a}%
(y))dy^{a}d^{n}x\,|_{y}\\
& =-(\omega_{ab}^{i}(y)h_{i}^{b}-\omega_{a}(y))dy^{a}d^{n}x\,|_{y}\\
& =0.
\end{align*}
\end{proof}
\begin{definition}
A\emph{ PD-prehamiltonian system} on the fiber bundle $\alpha:P\longrightarrow
M$ is a $\delta$-closed element $\omega\in\Omega^{2}(P,\alpha)$. A
\emph{PD-hamiltonian system on }$\alpha$ is a PD-prehamiltonian system
$\omega$ such that $\ker\underline{\omega}=0$ (and, therefore, $\ker\omega=0$
as well).
\end{definition}
Let $\theta\in\Omega^{1}$ be locally given by $\theta=\theta_{a}^{i}%
dy^{a}d^{n-1}x_{i}-Hd^{n}x$, $\ldots,\theta_{a}^{i},\ldots,H$ being local
functions on $P$. Then $\delta\theta$ is locally given by
\[
\delta\theta=\partial_{\lbrack a}\theta_{b]}^{i}dy^{a}dy^{b}d^{n-1}%
x_{i}-(\partial_{a}H + \partial_{i}\theta_{a}^{i})dy^{a}d^{n}x.
\]
Similarly, let $\omega\in\Omega^{2}$ and $Y\in V\mathrm{D}$ be locally given
by $\omega=\omega_{ab}^{i}dy^{a}dy^{b}d^{n-1}x_{i}+\omega_{a}dy^{a}d^{n}x$ and
$Y=Y^{a}\partial_{a}$, respectively. Then $\delta\omega$, $i_{Y}\omega$ and
$i_{Y}\underline{\omega}$ are locally given by
\begin{align*}
\delta\omega & =\partial_{\lbrack a}\omega_{bc]}^{i}dy^{a}dy^{b}dy^{c}%
d^{n-1}x_{i}+(\partial_{i}\omega_{ab}^{i}+\partial_{\lbrack a}\omega
_{b]})dy^{a}dy^{b}d^{n}x,\\
i_{Y}\omega & =2\omega_{ab}^{i}Y^{a}dy^{b}d^{n-1}x_{i}+\omega_{a}Y^{a}%
d^{n}x,\\
i_{Y}\underline{\omega} & =2\omega_{ab}^{i}Y^{a}d^{V}\!y^{b}\otimes
d^{n-1}x_{i},
\end{align*}
so that $\omega$ is a PD-prehamiltonian system iff
\begin{equation}
\partial_{\lbrack a}\omega_{bc]}^{i}=0,\quad\partial_{i}\omega_{ab}%
^{i}+\partial_{\lbrack a}\omega_{b]}=0, \label{Eq31}%
\end{equation}
or, which is the same (see Corollary \ref{CorPoincare}),
\begin{equation}
\omega_{ab}^{i}=\partial_{\lbrack a}\theta_{b]}^{i},\quad\omega_{a}%
=-\partial_{a}H-\partial_{i}\theta_{a}^{i}, \label{Eq32}%
\end{equation}
for some $\ldots,\theta_{a}^{i},\ldots,H$ local functions on $P$. Moreover,
$\omega$ is a PD-hamiltonian system iff
\begin{equation}
\omega_{ab}^{i}Y^{a}=0\Longrightarrow Y^{a}=0. \label{Eq30}%
\end{equation}
In its turn (\ref{Eq30}) implies $\omega_{a}=\omega_{ab}^{i}f_{i}^{b}$ for
some $\ldots,f_{i}^{b},\ldots$ local functions on $P$ (see the proof of
Proposition \ref{PropKer}).
Let $\omega$ be a PD-prehamiltonian system on $\alpha$, $\sigma
:U\longrightarrow P$ a local section of $\alpha$, $U\subset M$ an open subset.
The first jet prolongation $\dot{\sigma}:U\longrightarrow J^{1}\alpha$ of
$\sigma$ may be interpreted as a \textquotedblleft connection in $\alpha$
along $\sigma$\textquotedblright, i.e., a section of the restricted
bundle\ $\alpha_{1,0}|_{\sigma}:J^{1}\alpha|_{\sigma}\longrightarrow M$. Moreover,
elements in $\Omega|_{\sigma}$ may be interpreted as affine maps from
$C|_{\sigma}$ to $\Omega_{0}|_{\sigma}\simeq V\!\Lambda|_{\sigma}%
\otimes_{A_{0}}\Lambda^{n}(M)$ whose linear part is in $\underline{\Omega
}|_{\sigma}\simeq V\!\Lambda|_{\sigma}\otimes_{A_{0}}\Lambda^{n-1}(M)$.
Namely, an element $\lozenge\in C|_{\sigma}$ can \textquotedblleft be
inserted\textquotedblright\ into an element $\rho|_{\sigma}\in\Omega|_{\sigma
}$, $\rho\in\Omega$, giving an element $i_{\lozenge}\rho|_{\sigma}\in
\Omega_{0}|_{\sigma}$. Thus, we can search for local sections $\sigma$ of
$\alpha$ such that
\begin{equation}
i_{\dot{\sigma}}\omega|_{\sigma}=0 \label{Eq29}%
\end{equation}
\begin{definition}
Equations (\ref{Eq29}) are called the \emph{PD-Hamilton equations} (of the
PD-prehamiltonian system $\omega$).
\end{definition}
If $\omega$ is locally given by $\omega=\omega_{ab}^{i}dy^{a}dy^{b}%
d^{n-1}x_{i}+\omega_{a}dy^{a}d^{n}x$, then the associated PD-Hamilton equations
are locally given by
\begin{equation}
2\omega_{ab}^{i}\partial_{i}y^{a}-\omega_{b}=0. \label{Eq11}%
\end{equation}
Conversely, a system of PDEs in the form (\ref{Eq11}) is a PD-Hamilton
equation for some PD-prehamiltonian (resp.~ PD-hamiltonian) system iff
coefficients $\ldots,\omega_{ab}^{i},\ldots,\omega_{b},\ldots$ satisfy
(\ref{Eq31}) (or, which is the same, (\ref{Eq32})) (resp.~ (\ref{Eq31}) and
(\ref{Eq30})). Notice that, in view of (\ref{Eq11}), a general PD-prehamiltonian
system $\omega$ encode \textquotedblleft kinematical
information\textquotedblright, which can be identified with $\underline
{\omega}$, and \textquotedblleft dynamical information\textquotedblright,
which can be identified with the specific choice of $\omega$ in the class of
those PD-hamiltonian systems with linear part $\underline{\omega}$ (see the
comment at the end of Section \ref{SecAff}, Remark \ref{Rem1} and Example
\ref{Rem2}).
Searching for solutions of PD-Hamilton equations of a PD-prehamiltonian system
$\omega$, we could proceed in two steps:
\begin{enumerate}
\item \label{Prob1} search for a connection $\nabla\in\operatorname{Ker}%
\omega$,
\item search for $n$-dimensional integral submanifolds of the horizontal
distribution $H_{\nabla}P$.
\end{enumerate}
However, a solution to the first step of the above mentioned procedure exists
iff $\ker\omega=\ker\underline{\omega}$ which is not always the case.
Therefore, in general, we are led to weaken \ref{Prob1} and search for
connections $\nabla^{\prime}$ in a subbundle $P^{\prime}\subset P$ such that
$i_{\nabla^{\prime}}\omega|_{P^{\prime}}=0$. As showed in the next
proposition, there is always an \textquotedblleft algorithmic\textquotedblright\ way to find a maximal subbundle
$\breve{\alpha}:\breve{P}\longrightarrow M$ of $\alpha$ such that the affine
equation $i_{\breve{\nabla}}\omega|_{\breve{P}}=0$, $\breve{\nabla}\in
C(\breve{P},\breve{\alpha})$ admits at least one solution. We will refer to
the above mentioned \textquotedblleft algorithm\textquotedblright\ as the PD-constraint algorithm (see also
\cite{gnh78,dmm96,dmm96b,d...02,d...05}).
\begin{proposition}
Let $\omega$ be as above and $\operatorname{Ker}\omega=\varnothing$ (i.e.,
$D_{y}^{\omega}\neq\underline{D}_{y}^{\omega}$ for some $y\in P$). Under
suitable regularity conditions on $\omega$ (to be specified in the proof),
there exists a (maximal) subbundle $\breve{P}\subset P$ such that
$i_{\breve{\nabla}}\omega|_{\breve{P}}=0$ for some $\breve{\nabla}\in
C(\breve{P},\breve{\alpha})$.
\end{proposition}
\begin{proof}
For $s=1,2,\ldots$ define recursively
\begin{align*}
P_{(s)}& :=\{y\in P_{(s-1)}\;|\;\operatorname{Ker}\omega_{y}\cap(\alpha_{(s-1)})_1
^{-1}(y)\neq\varnothing\}\subset P,\\
\alpha_{(s)} & :=\alpha|_{P_{(s)}}%
:P_{s}\longrightarrow M,
\end{align*}
where $P_{(0)}:=P$, $\alpha_{(0)}:=\alpha$ (in particular
$P_{1}=\{y\in P\;|\;r(y)=\underline{r}\}$). We assume that $\alpha_{(s)}%
:P_{(s)}\longrightarrow M$ is a smooth (closed) subbundle for all $s$
(regularity conditions). Then, for dimensional reasons, there exists
$\overline{s}$ such that $P_{(s)}=P_{(\overline{s})}$ for all $s\geq\overline{s}$.
Put $\breve{P}:=P_{(\overline{s})}$.
\end{proof}
$\breve{\alpha}:=\alpha|_{\breve{P}}:\breve{P}\longrightarrow M$ will be
called the \emph{constraint subbundle}. Notice that $\breve{P}$ can be empty
(for instance when $r(y)=\underline{r}-1$ for all $y\in P$) and, in this case,
PD-Hamilton equations do not possess solutions.
\begin{corollary}
Let $\omega$ be a PD-prehamiltonian system on $\alpha$ and $\sigma$ a solution
of PD-Hamilton equations. Then $\operatorname{im}\sigma\subset\breve{P}$.
\end{corollary}
\begin{proof}
By induction on $s$, $\operatorname{im}\sigma\subset P_{(s)}$ for all
$s=1,2,\ldots$.
\end{proof}
The converse of the above corollary is, a priori, only true for $n=1$.
Namely, we may wonder if for any $y\in$ $\breve{P}$ there is a solution
$\sigma$ of PD-Hamilton equations such that $y\in\operatorname{im}\sigma$.
We know that there is a connection $\breve{\nabla}$ in $\breve{P}$ which is
\textquotedblleft a solution of PD-Hamilton equations up to first
order\textquotedblright, i.e., $i_{\breve{\nabla}}\breve{\omega}|_{\breve{P}%
}=0$. $n$-dimensional integral manifolds of the horizontal distribution
$H_{\breve{\nabla}}\breve{P}$ determined on $\breve{P}$ by $\breve{\nabla}$
are clearly images of solutions of PD-Hamilton equations. If $n=1$,
$\breve{\nabla}$ is trivially flat and Frobenius theorem guarantees that
for any $y\in\breve{P}$ there is a solution \textquotedblleft through
$y$\textquotedblright. The same is a priori untrue for $n>2$. Integrability
conditions on $H_{\breve{\nabla}}\breve{P}$ will be discussed elsewhere.
Clearly, standard examples of PD-prehamiltonian systems come from Lagrangian
field theory. Let us briefly recall how. Let $\pi:E\longrightarrow M$ be a
fiber bundle and consider a lagrangian density $\mathscr{L}\in\overline
{\Lambda}{}^{n}(J^{1}\pi,\pi_{1})$. $\mathscr{L}$ determines two Legendre
transforms (see, for instance, \cite{r05}) which we denote by
$F\mathscr{L}:J^{1}\pi\longrightarrow\mathscr{M}\pi$ and $\underline
{F}\mathscr{L}:J^{1}\pi\longrightarrow J^{\dag}\pi$. Consider the submanifold
$C_{0}:=\operatorname{im}\underline{F}\mathscr{L}\subset J^{\dag}\pi$ of
primary hamiltonian constraints of the lagrangian theory. $F\mathscr{L}$
factorizes as $F\mathscr{L}=\mathscr{H}\circ\underline{F}\mathscr{L}$,
$\mathscr{H}:C_{0}\longrightarrow\mathscr{M}\pi$ being the hamiltonian
section. Then $\omega_{\mathscr{L}}:=F\mathscr{L}^{\ast}(\delta\Theta)$ (resp.
$\mathscr{H}^{\ast}(\delta\Theta)$) is a PD-prehamiltonian system in $\pi_{1}$
(resp.~ in $C_{0}\longrightarrow M$). Moreover, as it is well known
\cite{gs73}, if $s: M \longrightarrow E$ is a solution of the Euler-Lagrange equations, then $\dot{s}:M \longrightarrow J^1\pi$ (resp.~ $\underline
{F}\mathscr{L} \circ \dot{s}: M \longrightarrow C_0$) is a solution of PD-Hamilton equations of $\omega_{\mathscr{L}}$ (resp.~ $\mathscr{H}^\ast(\delta \Theta)$).
\subsection{PD-Hamiltonian Systems and Multisymplectic Geometry \textit{\`{a}
l\`{a} }Forger}
Forger and Gomes have recently proposed a definition of multipresymplectic
structure on a fiber bundle \cite{fg08}. Their work aims to define such a
structure so that 1) the differential $d\Theta$ of the tautological $n$-form
$\Theta$ on the affine adjoint bundle of the first jet bundle (see the end of Section \ref{SecAff})is multisymplectic
2) every multipresymplectic structure is locally isomorphic to the pull-back
of $\Theta$ along a fibration (Darboux lemma). Since, in our opinion, I)
\cite{fg08} is the best motivated and established work about fundamentals of
multisymplectic geometry, II) abstract fiber bundles play in \cite{fg08} a
similar role as in this paper, we analyze in this subsection the relationship
between PD-prehamiltonian systems and multipresymplectic structures
\textit{\`{a} l\`{a} }Forger, referring to \cite{fg08} for the main
definition. Here we just mention two of the main results of \cite{fg08} (which
can eventually be understood as definitions of polypresymplectic structure and
multipresymplectic structure on a fiber bundle, respectively)
\begin{theorem}
[Forger and Gomes I]\label{TeorForger1}Let $\alpha:P\longrightarrow M$ be a
fiber bundle, $\ldots,x^{i},\ldots$ local coordinates on $M$, $i=1,\ldots
,n=\dim M$ and $\underline{\omega}\in\underline{\Omega}^{2}$. $\underline
{\omega}$ is a polypresymplectic structure on $\alpha$ iff, around every point
of $P$, there are local fiber coordinates $\ldots,q^{A},\ldots,p_{A}%
^{i},\ldots,z^{1}\ldots,z^{s}$, $A=1,\ldots,m$, $i=1,\ldots,n$ (so that $\dim
P=n+m+mn+s$) such that $\underline{\omega}$ is locally given by
\[
\omega=d^{V}\!p_{A}^{i}d^{V}\!q^{A}\otimes d^{n-1}x_{i}.
\]
\end{theorem}
\begin{theorem}
[Forger and Gomes II]\label{TeorForger2}Let $\alpha:P\longrightarrow M$ be a
fiber bundle, $\ldots,x^{i},\ldots$ local coordinates on $M$, $i=1,\ldots
,n=\dim M$, and $\omega\in\Omega^{2}$. $\omega$ is a multipresymplectic
structure on $\alpha$ iff, around every point of $P$, there are local fiber
coordinates $\ldots,q^{A},\ldots,p_{A}^{i},\ldots,p,z^{1}\ldots,z^{r}$,
$A=1,\ldots,m$, $i=1,\ldots,n$ (so that $\dim P=(n+1)(m+1)+r$) such that
$\omega$ is locally given by
\[
\omega=dp_{A}^{i}dq^{A}d^{n-1}x_{i}-dpd^{n}x.
\]
\end{theorem}
\begin{proposition}
\label{Prop2}Let $\omega$ be a PD-prehamiltonian system on $\alpha$. The
following two conditions are equivalent.
\begin{enumerate}
\item $\underline{\omega}$ is a polypresymplectic structure and
$r(y)=\underline{r}-1$ for all $y\in P$.
\item $\omega$ is a multipresymplectic structure.
\end{enumerate}
\end{proposition}
\begin{proof}
Recall that, in view of Proposition \ref{Prop1}, $\omega$ is locally $\delta
$-exact. Suppose $\underline{\omega}$ is polypresymplectic and
$r(y)=\underline{r}-1$ for all $y\in P$. Then $\underline{r}>0$ and, in view
of Theorem \ref{TeorForger1}, (around every point in $P$) there are $\alpha
$-adapted local coordinates%
\[
\ldots,x^{i},\ldots,q^{A},\ldots,p_{A}^{i},z^{0},z^{1},\ldots,z^{\underline
{r}-1},\quad
\]
such that, locally, $\underline{\omega}=d^{V}\!p_{A}^{i}d^{V}\!q^{A}\otimes
d^{n-1}x_{i}$ (in particular, $\ker\underline{\omega}$ is locally spanned by
$\ldots,{\partial}/{\partial z^{\alpha}},\ldots$). Therefore,
$\underline{\omega}=\underline{\delta}\underline{\theta}{}_{0}$, where
$\underline{\theta}{}_{0}:=p_{A}^{i}d^{V}\!q^{A}\otimes d^{n-1}x_{i}$ is a local
element of $\underline{\Omega}^{1}$. A general (local) potential of $\omega$
is then $\theta^{\prime}\in\Omega^{1}$ such that $\underline{\theta}^{\prime
}=\underline{\theta}{}_{0}+d^{V}\!\nu$, $\nu$ being a local element in
$\underline{\Omega}^{0}=\overline{\Lambda}{}^{n-1}$. The (local) potential
$\theta:=\theta^{\prime}-\delta\nu$ is locally in the form $\theta=p_{A}%
^{i}dq^{A}d^{n-1}x_{i}-pd^{n}x$, where $p$ is a local function on $P$.
Therefore $\omega$ is locally given by
\[
\omega=dp_{A}^{i}dq^{A}d^{n-1}x_{i}-dpd^{n}x.
\]
$\ker\omega$ is locally spanned by those local elements $Y^{\alpha}%
\tfrac{\partial}{\partial z^{\alpha}}$ in $\ker\underline{\omega}$ such that
$Y^{\alpha}\tfrac{\partial h}{\partial z^{\alpha}}=0$. Since $\ker\omega
\neq\ker\underline{\omega}$, then $\tfrac{\partial p}{\partial z^{\alpha}%
}dz^{\alpha}\neq0$. Let, for instance, be $\tfrac{\partial p}{\partial z^{0}%
}\neq0$. Then $\ldots,x^{i},\ldots,q^{A},\ldots,p_{A}^{i},p,z^{1}%
,\ldots,z^{\underline{r}-1}$ is a new local coordinate system on $P$. In view
of Theorem \ref{TeorForger2}, $\omega$ is then multipresymplectic.
On the other hand, let $\omega$ be multipresymplectic. Then $\underline
{\omega}$ is polypresymplectic. Moreover, (around every point in $P$) there are
$\alpha$-adapted local coordinates%
\begin{equation}
\ldots,x^{i},\ldots,q^{A},\ldots,p_{A}^{i},\ldots,p,z^{1},\ldots,z^{r}
\label{Eq10}%
\end{equation}
such that, locally, $\omega=dp_{A}^{i}dq^{A}d^{n-1}x_{i}-dpd^{n}x$ and
$\underline{\omega}=d^{V}\!p_{A}^{i}d^{V}\!q^{A}\otimes d^{n-1}x_{i}$. This shows
that for all $y\in P$,
\[
D_{y}^{\omega}=\langle\ldots,\left. \tfrac{\partial
}{\partial z^{\alpha}}\right\vert _{y},\ldots\rangle\neq\underline{D}%
_{y}^{\omega}=\langle\ldots,\left. \tfrac{\partial}{\partial z^{\alpha}%
}\right\vert _{y},\ldots,\left. \tfrac{\partial}{\partial p}\right\vert
_{y}\rangle.
\]
\end{proof}
\begin{remark}
\label{Rem1}Let $\omega$ be a PD-prehamiltonian system. First of
all, notice that, if $\omega$ is a multipresymplectic structure then, in view
of Proposition \ref{Prop2}, PD-Hamilton equations of $\omega$ do not
possess solutions. In this sense, multipresymplectic structures do not contain
any dynamical information.
Now, the proof of Proposition \ref{Prop2} also shows that if
$\underline{\omega}$ is a polypresymplectic structure and $\ker\omega
=\ker\underline{\omega}$ then $\omega$ is locally in the form
\[
\omega=dp_{A}^{i}dq^{A}d^{n-1}x_{i}-dHd^{n}x
\]
where $\tfrac{\partial H}{\partial z^{\alpha}}=0$, $\alpha=1,2,\ldots$, i.e.,
$H$ is constant along the leaves of the distribution $D^{\omega}=\underline
{D}^{\omega}$.
\end{remark}
\begin{example}
\label{Rem2}Let $\omega\in\Omega^{2}$ be a multisymplectic structure on
$\alpha$. In this case $\ker\omega=0$, while $\underline{D}^{\omega}$ is a
$1$-dimensional (involutive) distribution. Leaves of $\underline{D}^{\omega}$
are submanifolds in the fibers of $\alpha$. denote by $\underline{P}$ the set
of leaves of $\underline{D}^{\omega}$. There is an obvious projection
$\underline{\alpha}:\underline{P}\longrightarrow M$. Suppose that
$\underline{\alpha}:\underline{P}\longrightarrow M$ is a smooth fiber bundle
and $\mathfrak{p}:P\longrightarrow\underline{P}$ a smooth submersion (which is
always true locally). There is a distinguished class of (local) PD-hamiltonian
systems on $\underline{\alpha}$. Indeed, let $U\subset\underline{P}$ be an
open subbundle and $\mathscr{H}:U\longrightarrow P$ a local section of
$\mathfrak{p}$. Then $\omega^{\prime}:=\mathscr{H}^{\ast}(\omega)\in\Omega
^{2}(U,\underline{\alpha})$ is a PD-hamiltonian system. In particular, if we
choose coordinates on $P$ as in (\ref{Eq10}) (here $r=0$), then $\ldots
,x^{i},\ldots,q^{A},\ldots,p_{A}^{i},\ldots$ are coordinates on $\underline
{P}$, $\mathscr{H}$ is given by
\[
\mathscr{H}^{\ast}(p)=H,
\]
$H$ being a local function on $\underline{P}$, and $\omega^{\prime}$ is
locally given by
\[
\omega^{\prime}=dp_{A}^{i}dq^{A}d^{n-1}x_{i}-dHd^{n}x,
\]
in particular $\theta^{\prime}:=p_{A}^{i}dq^{A}d^{n-1}x_{i}-Hd^{n}x$ is a
local potential of $\omega^{\prime}$. Finally, PD-Hamilton equations of
$\omega^{\prime}$ read
\begin{align*}
q_{i}^{A} & =\tfrac{\partial H}{\partial p_{A}^{i}},\\
p_{A}^{i}{}_{,i} & =-\tfrac{\partial H}{\partial q^{A}},
\end{align*}
which are de Donder-Weyl equations (see, for instance, \cite{gs73}).
\end{example}
\subsection{PD-Hamiltonian Systems and Variational Calculus}
We show in this subsection that PD-Hamilton equations are locally variational.
First of all, an element $\theta\in\Omega^{1}$ may be understood as a
(fiber-wise affine) horizontal $n$-form over $J^{1}\alpha$, i.e., as an
element $\mathscr{L}^{\theta}\in\overline{\Lambda}{}^{n}(J^{1}\alpha
,\alpha_{1})$ via
\[
\mathscr{L}_{c}^{\theta}:=i_{c}\theta_{y},\quad c\in J^{1}\alpha
,\;y=\alpha_{1,0}(c)\in P.
\]
In its turn $\mathscr{L}^{\theta}$ is a 1st order lagrangian density in the
fiber bundle $\alpha$ determining an action functional which we denote by
$S^{\theta}=\int\mathscr{L}^{\theta}$. If $\theta$ is locally
given by $\theta=\theta_{a}^{i}dy^{a}d^{n-1}x_{i}-Hd^{n}x$, $\ldots,\theta_a^i,\ldots,H$ being local functions on $P$, then
$\mathscr{L}^{\theta}$ is locally given by $\mathscr{L}^{\theta}=L^{\theta
}d^{n}x$, where $L^{\theta}$ is the local function on $J^{1}\alpha$ given by
\[
L^{\theta}=(\theta_{b}^{i}y{}_{i}^{b}-H).
\]
In particular, if $\theta=\delta\nu$ for some $\nu\in\Omega^{0}=\overline
{\Lambda}{}^{n-1}$ locally given by $\nu=\nu^{i}d^{n-1}x_{i}$, then
\begin{equation}
L^{\theta}=(\partial_{i}+y_{i}^{a}\partial_{a})\nu^{i}, \label{Eq34}%
\end{equation}
i.e., $L^{\theta}$ is a total divergence.
\begin{proposition}
Let $\omega\in\Omega^{2}$ be a $\delta$-exact PD-prehamiltonian system. Then
PD-Hamilton equations of $\omega$ coincide with Euler-Lagrange equations
associated with the action $S^{\theta}:=\int\mathscr{L}^{\theta}$, where
$\theta\in\Omega^{1}$ is the opposite of any potential of $\omega$, i.e.,
$-\delta\theta=\omega$. Moreover, if $H^{1}(\Omega,\delta)=0$ then $S^{\theta
}$ is independent of the choice of $\theta$ and does only depend on $\omega$.
\end{proposition}
\begin{proof}
The first part of the proposition can be proved in local coordinates. Indeed,
compute variational derivatives of $L^{\theta}$,
\begin{align*}
\tfrac{\delta}{\delta y^{b}}L^{\theta} & :={\partial}_{b}%
L^{\theta}-(\partial_{i}+y_{i}^{a}\partial_{a})\tfrac{\partial}{\partial
y{}_{i}^{b}}L^{\theta}\\
& =y{}_{i}^{a}(\partial_{b}\theta_{a}^{i}-\partial_{a}\theta_{b}%
^{i})-\partial_{a}H-\partial_{i}\theta_{a}^{i}\\
& =-2\omega_{ab}^{i}y{}_{i}^{a}+\omega_{b}.
\end{align*}
where we used (\ref{Eq32}). To prove the second part of the proposition, use
(\ref{Eq34}) to conclude that, for $\nu\in\Omega^{0}$, $\delta L^{\delta
\nu}/\delta y^{a}=0$.
\end{proof}
\begin{remark}
Condition $H^{1}(\Omega,\delta)=0$ depends on the topology of the fiber bundle
$\alpha$. It is satisfied, for instance, if $H^{n}(M)=0$ and $H^{1}%
(\mathcal{F})=0$, $\mathcal{F}$ being, as above, the abstract fiber of
$\alpha$. Indeed, if $H^{1}(\mathcal{F})=0$ then $H^{1}(\underline{\Omega
},\underline{\delta})=0$ so that, the first part of the exact sequence
(\ref{Eq28}) reads
\[
\xymatrix@C=32pt{ 0 \ar[r] & H^{0}(\Omega,\delta) \ar[r] & \Lambda^{n-1}(M) \ar[r]^-{d_M^{n-1}} & \Lambda^{n}(M) \ar[r] &
H^{1}(\Omega,\delta) \ar[r] & 0}
\]
and $H^{1}(\Omega,\delta)\simeq H^{n}(M)=0$.
\end{remark}
\section{PD-Noether Symmetries and Currents\label{SecNoether}}
\subsection{PD-Noether Theorem and PD-Poisson Bracket}
The multisymplectic analogues of hamiltonian vector fields and Poisson
bracket in symplectic geometry have been longly investigated
\cite{ks75,k97,fr01,fpr03,fpr03b,fr05}. We here propose the natural
definitions for general PD-hamiltonian systems. Notice that, even if they look
formally identical to (or possibly less general than) the ones proposed in
\cite{ks75,fr01,fpr03,fpr03b}, our definitions have actually got a dynamical
content, not only a kinematical one (see Remark \ref{Rem1}), so that, for
instance, we can prove a PD-version of (hamiltonian) Noether theorem. That's why, e.g., we
will better speak about PD-Noether symmetries rather than hamiltonian
(multi)vector fields \cite{fpr05}.
Let $\omega$ be a PD-prehamiltonian system on the bundle $\alpha
:P\longrightarrow M$. In the following we assume $\alpha$ to have connected fiber.
\begin{definition}\label{DefPDNoether}
Let $Y\in V\mathrm{D}$ and $f\in\Omega^{0}$. If $i_{Y}\omega=\delta f$, then
$Y$ and $f$ are said to be a \emph{PD-Noether symmetry} and a \emph{PD-Noether
current} of $\omega$ (relative to each other), respectively.
\end{definition}
Denote by $\mathscr{S}(\omega)$ and $\mathscr{C}(\omega)$ the sets of
PD-Noether symmetries and PD-Noether currents of $\omega$, respectively. A
PD-Noether symmetry $Y$ (relative to a PD-Noether current $f$) is a symmetry
of $\omega$ in the sense that
\[
L_{Y}\omega=i_{Y}\delta\omega+\delta i_{Y}\omega=\delta\delta f=0.
\]
The next proposition clarifies in what sense a PD-Noether current is a
conserved current for $\omega$.
\begin{proposition}
[PD--Noether theorem]Let $Y\in\mathscr{S}(\omega)$ and $f\in\mathscr{C}(\omega
)$ be a PD-Noether symmetry and a PD-Noether current of $\omega$ relative to
each other. Then $\sigma^{\ast}(f)\in\Lambda^{n-1}(M)$ is a closed form for
every solution $\sigma$ of PD-Hamilton equations.
\end{proposition}
\begin{proof}
First of all, let $\varrho\in\Omega^{1}$ and $\tau$ be a (local) section of
$\alpha$. It is easy to show (for instance, using local coordinates) that
$\tau^{\ast}(\varrho)=i_{\dot{\tau}}\varrho|_{\tau}\in\Lambda^{n}(M)$. Then
\begin{align*}
d\sigma^{\ast}(f) & =\sigma^{\ast}(df)\\
& =\sigma^{\ast}(\delta f)\\
& =i_{\dot{\sigma}}\delta f|_{\sigma}\\
& =i_{\dot{\sigma}}i_{Y}\omega|_{\sigma}\\
& =i_{Y|_{\sigma}}i_{\dot{\sigma}}\omega|_{\sigma}\\
& =0.
\end{align*}
\end{proof}
We are now in the position to introduce a Lie bracket among PD-Noether currents.
\begin{proposition}
Let $Y_{1},Y_{2}\in\mathscr{S}(\omega)$ be PD-Noether symmetries relative to
the PD-Noether currents $f_{1},f_{2}\in\mathscr{C}(\omega)$, respectively.
Then $[Y_{1},\allowbreak Y_{2}]\in\mathscr{S}(\omega)$ and $f:=L_{Y_{1}}%
f_{2}\in\mathscr{C}(\omega)$ and they are relative to each other. Moreover,
$f$ is independent of the choice of $Y_{1}$ among the PD-Noether symmetries
relative to the PD-Noether current $f_{1}$.
\end{proposition}
\begin{proof}
Compute
\begin{align*}
\delta L_{Y_{1}}f_{2} & =L_{Y_{1}}\delta f_{2}\\
& =L_{Y_{1}}i_{Y_{2}}\omega\\
& =i_{[Y_{1},Y_{2}]}\omega+i_{Y_{2}}L_{Y_{1}}\omega\\
& =i_{[Y_{1},Y_{2}]}\omega.
\end{align*}
Now, let $V\in\ker\omega$. Then $L_{V}f_{2}=i_{V}\delta f_{2}=i_{V}i_{Y_{2}%
}\omega=0$. This proves the second part of the proposition.
\end{proof}
Let $Y_{1},Y_{2},f_{1},f_{2}$ be as in the above proposition.
\begin{proposition}
The $\mathbb{R}$-bilinear map
\[
\mathscr{C}(\omega)\times\mathscr{C}(\omega
)\ni(f_{1},f_{2})\longmapsto\{f_{1},f_{2}\}:=L_{Y_{1}}f_{2}\in H(\omega),
\]
$Y_{1}$ being a PD-Noether symmetry relative to $f_{1}$, is a Lie bracket.
\end{proposition}
\begin{proof}
Let $Y_2 \in \mathscr{S}(\omega)$ be a PD-Noether symmetry relative to $f_2 \in \mathscr{C}(\omega)$. Skew-symmetry of $\{\cdot,\cdot\}$ immediately follows from the remark:
\begin{align*}
\{f_{1},f_{2}\} & =L_{Y_{1}}f_{2}\\
& =i_{Y_{1}}\delta f_{2}+\delta i_{Y_{1}}f_{2}\\
& =i_{Y_{1}}i_{Y_{2}}\omega.
\end{align*}
Now, check Leibnitz rule. Let $Y_{3}\in \mathscr{S}(\omega)$ and $f_3 \in \mathscr{C}(\omega)$ be another pair of
PD-Noether symmetry, PD-Noether current relative to each other. Then
\begin{align*}
\{f_{1},\{f_{2},f_{3}\}\} & =L_{Y_{1}}\{f_{2},f_{3}\}\\
& =L_{Y_{1}}L_{Y_{2}}f_{3}\\
& =L_{[Y_{1},Y_{2}]}f_{3}+L_{Y_{2}}L_{Y_{1}}f_{3}\\
& =\{\{f_{1},f_{2}\},f_{3}\}+\{f_{2},\{f_{1},f_{3}\}\}.
\end{align*}
\end{proof}
PD-Noether symmetries and PD-Noether currents of a PD-hamiltonian system
constitute very small Lie subalgebras of the Lie algebras of higher symmetries
and conservation laws of PD-Hamilton equations, for which there have been given
fully satisfactory definitions and have been developed many infinite jet based
computational techniques \cite{b...99}. Nevertheless, it is worthy to give Definition \ref{DefPDNoether} and
to carefully analyse it, independently on infinite jets, in view of the possibility
of developing a \textquotedblleft (multi)symplectic theory\textquotedblright\ of
higher symmetries and conservation laws (see, for instance, \cite{v09}). In
Section \ref{SecComp} we propose some specific examples.
Finally, notice that, in general, nor a PD-Noether current is uniquely
determined by the relative PD-Noether symmetry nor vice versa (unless
$\ker\omega=0$). However, \textquotedblleft non-trivial PD-Noether
symmetries\textquotedblright\ are in one to one correspondence with
\textquotedblleft non-trivial PD-Noether currents\textquotedblright\ in the following sense. Clearly, $\ker
\omega\subset\mathscr{S}(\omega)$ and $H^{0}(\Omega,\delta)\subset
\mathscr{C}(\omega)$. We will call elements in $\ker\omega$ \emph{gauge
PD-Noether symmetries} (see below) and elements in $H^{0}(\Omega,\delta)$
(i.e., closed $(n-1)$ -forms on $M$, see Corollary \ref{CorH0}) \emph{trivial
PD-Noether currents}.
\begin{remark}
It is easy to see that $\ker\omega$ and $H^{0}(\Omega,\delta)$ are ideals in
the Lie algebras $\mathscr{S}(\omega)$ and $\mathscr{C}(\omega)$,
respectively. Let $\overline{\mathscr{S}}(\omega):=\mathscr{S}(\omega
)/\ker\omega$ and $\overline{\mathscr{C}}(\omega):=\mathscr{C}(\omega
)/H^{0}(\Omega,\delta)$ be the quotient Lie algebras. Then the map
\[
\overline{\mathscr{S}}(\omega)\ni Y+\ker\omega\longmapsto f+H^{0}%
(\Omega,\delta)\in\overline{\mathscr{C}}(\omega),
\]
where $Y\in\mathscr{S}(\omega)$ and $f\in\mathscr{C}(\omega)$ are relative to
each other, is a well defined isomorphism of Lie algebras. It is natural to
call elements in $\overline{\mathscr{S}}(\omega)$ and $\overline
{\mathscr{C}}(\omega)$ \emph{non-trivial PD-Noether symmetries} and
\emph{non-trivial PD-Noether currents}, respectively. Indeed, elements in
$\ker\omega$ are trivial symmetries in that they are infinitesimal gauge
transformations (see next subsection), and elements in $H^{0}(\Omega,\delta)$ are
trivial conserved currents in that they are conserved currents for every
PD-prehamiltonian system $\omega$, independently of $\omega$.
\end{remark}
\subsection{Gauge Reduction of PD-Hamiltonian Systems}
From a physical point of view, elements in $\ker\omega$ are infinitesimal
gauge transformations and therefore should be quotiented out via a reduction
of the system. In this section we assume $\ker\omega=\ker\underline{\omega}$
or, which is the same, $\operatorname{Ker}\omega\neq\varnothing$. As a further
regularity condition we assume that the leaves of $D^{\omega}=\underline
{D}^{\omega}$ form a smooth fiber bundle $\widetilde{P}$ over $M$, whose
projection we denote by $\widetilde{\alpha}:\widetilde{P}\longrightarrow M$,
in such a way that the canonical projection $\mathfrak{p}:P\longrightarrow
\widetilde{P}$ is a smooth bundle. The last condition is always fulfilled at
least locally. Notice, also, that, by construction, $\mathfrak{p}$ has
connected fiber.
\begin{theorem}
There exists a unique PD-hamiltonian system $\widetilde{\omega}$ in
$\widetilde{\alpha}$ such that 1) $\omega=\mathfrak{p}{}^{\ast}(\widetilde
{\omega})$, 2) $\ker\widetilde{\omega}=\ker\underline{\widetilde{\omega}}=0$
and 3) a local section $\sigma$ of $\alpha$ is a solution of the PD-Hamilton
equation of $\omega$ iff $\mathfrak{p}\circ\sigma$ (which is a local section
of $\widetilde{\alpha}$) is a solution of PD-Hamilton equations of
$\widetilde{\omega}$.
\end{theorem}
\begin{proof}
Let $\widetilde{\nabla}\in C(\widetilde{P},\widetilde{\alpha})$. There exists
a (non-unique) connection $\nabla\in C(P,\alpha)$ such that $\nabla$ and
$\widetilde{\nabla}$ are $\mathfrak{p}$-compatible. To prove this, choose a
connection $\square$ in $\mathfrak{p}$ and lift the planes of $\widetilde
{\nabla}$ to $P$ by means of $\square$. It is easy to show that the so
obtained distribution on $P$ defines a connection $\nabla$ in $\alpha$ with
the required property. Similarly, every vector field $\widetilde{X}\in
V\mathrm{D}(\widetilde{P},\widetilde{\alpha})$ can be lifted to a (non-unique)
$\mathfrak{p}$-projectable vector field $X\in V\mathrm{D}(P,\alpha)$
such that $\widetilde{X}$ is its projection. Then $X\in\mathrm{D}%
_{V}(P,\mathfrak{p})$. Consider $\eta:=\omega(\nabla)(X)\in\Omega^{0}%
(P,\alpha)$ and prove that $L_{Y}\eta=0$ for any $Y\in V\mathrm{D}%
(P,\mathfrak{p})$. Indeed, let $Y\in V\mathrm{D}(P,\mathfrak{p})$. Then
$[Y,X]\in V\mathrm{D}(P,\mathfrak{p})$. Similarly $[\![Y,H_{\nabla}]\!]\in\overline{\Lambda}%
{}^{1}(P,\alpha)\otimes V\mathrm{D}(P,\mathfrak{p})\subset\overline{\Lambda}%
{}^{1}(P,\alpha)\otimes V\mathrm{D}(P,\alpha)$. Now, $V\mathrm{D}%
(P,\mathfrak{p})=\ker\omega$ by construction, and therefore
\begin{align*}
L_{Y}\eta & =L_{Y}i_{X}i_{\nabla}\omega\\
& =[L_{Y},i_{X}]i_{\nabla}\omega+i_{X}L_{Y}i_{\nabla}\omega\\
& =i_{[Y,X]}i_{\nabla}\omega+i_{X}i_{\nabla}L_{Y}\omega+i_{X}[L_{Y}%
,i_{\nabla}]\omega\\
& =i_{\nabla}i_{[Y,X]}\omega+i_{X}i_{[\![Y,H_{\nabla}]\!]}\omega\\
& =0.
\end{align*}
Since fibers of $\mathfrak{p}$ are connected we conclude that $\eta
=\mathfrak{p}^{\ast}(\widetilde{\eta})$ for a unique $\widetilde{\eta}%
\in\Omega^{0}(\widetilde{P},\widetilde{\alpha})$. Put
\[
\widetilde{\omega}(\widetilde{\nabla})(\widetilde{X}):=\widetilde{\eta}.
\]
$\widetilde{\omega}$ is a well defined element in $\Omega^{2}(\widetilde
{P},\widetilde{\alpha})$. Indeed, let $\nabla^{\prime}\in C(P,\alpha)$ be also
$\mathfrak{p}$-compatible with $\widetilde{\nabla}$ and $X^{\prime}\in
V\mathrm{D}(P,\alpha)$ be another $\mathfrak{p}$-projectable vector field projecting onto $\widetilde{X}$. Then $\nabla^{\prime}-\nabla
\in\overline{\Lambda}{}^{1}(P,\alpha)\otimes V\mathrm{D}(P,\mathfrak{p})$ and
$X^{\prime}-X\in V\mathrm{D}(P,\mathfrak{p})$. Therefore,
\begin{align*}
\omega(\nabla^{\prime})(X^{\prime}) & =i_{X^{\prime}}i_{\nabla^{\prime}%
}\omega\\
& =i_{X^{\prime}}i_{\nabla}\omega+i_{X^{\prime}}i_{\nabla^{\prime}-\nabla
}\underline{\omega}\\
& =i_{X}i_{\nabla}\omega+i_{X^{\prime}-X}i_{\nabla}\omega\\
& =i_{X}i_{\nabla}\omega+i_{\nabla}i_{X^{\prime}-X}\omega\\
& =\omega(\nabla)(X).
\end{align*}
Moreover, $\omega=\mathfrak{p}^{\ast}(\widetilde{\omega})$ by construction.
Let us compute $\ker\underline{\widetilde{\omega}}$. Thus, let $\widetilde
{X}\in V\mathrm{D}(\widetilde{P},\widetilde{\alpha})$ be such that
$i_{\widetilde{X}}\underline{\widetilde{\omega}}=0$ and $X\in V\mathrm{D}%
(P,\alpha)$ be as above. Then $i_{X}\underline{\omega}=\mathfrak{p}^{\ast
}(i_{\widetilde{X}}\underline{\widetilde{\omega}})=0$. This shows that $X\in$
$V\mathrm{D}(P,\mathfrak{p})$ and then $\widetilde{X}=0$.
Finally, let $\sigma$ be a local section of $\alpha$, $\widetilde{\sigma
}:=\mathfrak{p}\circ\sigma$, $\widetilde{X}\in V\mathrm{D}(\widetilde
{P},\widetilde{\alpha})$ and $X$ be as above. Compute
\begin{align*}
(i_{\widetilde{\sigma}{}^{\cdot}}\widetilde{\omega}|_{\widetilde{\sigma}%
})(\widetilde{X}|_{\widetilde{\sigma}}) & =i_{\widetilde{\sigma}{}^{\cdot}%
}(i_{\widetilde{X}}\widetilde{\omega})|_{\widetilde{\sigma}}\\
& =\widetilde{\sigma}{}^{\ast}(i_{\widetilde{X}}\widetilde{\omega})\\
& =(\sigma^{\ast}\circ\mathfrak{p}^{\ast})(i_{\widetilde{X}}\widetilde
{\omega})\\
& =\sigma^{\ast}(i_{X}\omega)\\
& =i_{\dot{\sigma}}(i_{X}\omega)|_{\sigma}\\
& =(i_{\dot{\sigma}}\omega|_{\sigma})(X|_{\sigma}).
\end{align*}
This shows that $i_{\dot{\sigma}}\omega|_{\sigma}=0$ iff $i_{\widetilde
{\sigma}{}^{\cdot}}\widetilde{\omega}|_{\widetilde{\sigma}}=0$.
\end{proof}
\begin{proposition}
There are natural isomorphisms of Lie algebras
\begin{align*}
\overline{\mathscr{S}}(\omega) & \simeq\mathscr{S}(\widetilde{\omega}),\\
\mathscr{C}(\omega) & \simeq\mathscr{C}(\widetilde{\omega}).
\end{align*}
\end{proposition}
\begin{proof}
First of all let $f\in\mathscr{C}(\omega)$ and $X\in\mathscr{S}(\omega)$ be
relative to each other. Then $f=\mathfrak{p}^{\ast}(\widetilde{f})$ for some
$\widetilde{f}\in\Omega^{0}(\widetilde{P},\widetilde{\alpha})$ and $X$ is
$\mathfrak{p}$-projectable. Indeed, for all $Y\in\ker\omega$,
\[
L_{Y}f=i_{Y}\delta f+\delta i_{Y}f=i_{Y}i_{X}\omega=i_{[Y,X]}\omega=0.
\]
Moreover,
\[
\mathfrak{p}^{\ast}(\delta\widetilde{f})=\delta\mathfrak{p}^{\ast}%
(\widetilde{f})=\delta f=i_{X}\omega=\mathfrak{p}^{\ast}(i_{\widetilde{X}%
}\widetilde{\omega}),
\]
where $\widetilde{X}$ denotes the $\mathfrak{p}$-projection of $X$, and,
therefore, $\delta\widetilde{f}=i_{\widetilde{X}}\widetilde{\omega}$, i.e.,
$\widetilde{f}\in\mathscr{C}(\widetilde{\omega})$ and $\widetilde{X}%
\in\mathscr{S}(\widetilde{\omega})$ is a PD-Noether symmetry relative to it.
Thus, maps
\begin{align}
\overline{\mathscr{S}}(\omega)\ni X+\ker\omega & \longmapsto\widetilde{X}%
\in\mathscr{S}(\widetilde{\omega}),\label{Eq36}\\
\mathscr{C}(\omega)\ni f & \longmapsto\widetilde{f}\in\mathscr{C}(\widetilde
{\omega}). \label{Eq37}%
\end{align}
are well defined. Conversely, let $\widetilde{X}_{1}\in\mathscr{S}(\widetilde{\omega})$,
$\widetilde{f}_{1}\in\mathscr{C}(\widetilde{\omega})$ be relative to each
other, $X_{1}\in V\mathrm{D}(P,\alpha)$ be any $\mathfrak{p}$-projectable
vector field, $\widetilde{X}_{1}\in V\mathrm{D}(\widetilde{P},\widetilde
{\alpha})$ be its projection, and $f_{1}:=\mathfrak{p}^{\ast
}(\widetilde{f}_{1})\in\Omega^{0}(P,\alpha)$. Then $X_{1}\in\mathscr{S}(\omega
)$ and $f_{1}\in\mathscr{C}(\omega)$ is a PD-Noether current relative to it.
Indeed,
\[
i_{X_{1}}\omega=\mathfrak{p}^{\ast}(i_{\widetilde{X}_{1}}\widetilde{\omega
})=\mathfrak{p}^{\ast}(\delta\widetilde{f}_{1})=\delta\mathfrak{p}^{\ast
}(\widetilde{f}_{1})=\delta f_{1}.
\]
We conclude that (\ref{Eq36}) and (\ref{Eq37}) are inverted by
\begin{align*}
\mathscr{S}(\widetilde{\omega})\ni\widetilde{X}_{1} & \longmapsto X_{1}%
+\ker\omega\in\overline{\mathscr{S}}(\omega),\\
\mathscr{C}(\widetilde{\omega})\ni\widetilde{f}_{1} & \longmapsto f_{1}%
\in\mathscr{C}(\omega),
\end{align*}
respectively.
\end{proof}
\section{Examples\label{SecComp}}
\subsection{Non-Degenerate Examples}
Let $\alpha:\mathbb{R}^{2n+1}\ni(x^{1},\ldots,x^{n},u,u_{1},\ldots
,u_{n})\longmapsto(x^{1},\ldots,x^{n})\in\mathbb{R}^{n}$, $n>1$. Consider
$T,V\in C^{\infty}(\mathbb{R}^{2n+1})$ of the form $T=T(u_{1},\ldots,u_{n})$
and $V=V(u)$, respectively. The form
\[
\omega:=\tfrac{\partial^{2}T}{\partial u_{i}\partial u_{j}}du_{i}%
(dud^{n-1}x_{j}-u_{j}d^{n}x)-V^{\prime}dud^{n}x,
\]
is a PD-prehamiltonian system on $\alpha$ (here and in what follows a prime
\textquotedblleft$\;{}^{\prime}\;$\textquotedblright\ denotes differentiation
with respect to $u$). The associated PD-Hamilton equations read
\[%
\begin{array}
[c]{c}%
\tfrac{\partial^{2}T}{\partial u_{i}\partial u_{j}}\partial_{j}u_{i}%
+V^{\prime}=0,\\
\partial_{i}u=u_{i},
\end{array}
\]
which are in turn equivalent to%
\begin{align}
\tfrac{\partial^{2}T}{\partial u_{i}\partial u_{j}}\partial_{ij}%
^{2}u+V^{\prime} & =0,\label{Eq14}\\
\partial_{i}u & =u_{i},\nonumber
\end{align}
$\partial_{ij}^{2} := \partial_i \partial_j$, $i,j=1,\ldots,n$.
Moreover, for%
\[
\det\left( \tfrac{\partial^{2}T}{\partial u_{i}\partial u_{j}}\right)
\neq0,
\]
$\omega$ is a PD-hamiltonian system. We will only consider this case in the
following. Thus, put $T^{ij}:=\tfrac{\partial^{2}T}{\partial u_{i}\partial
u_{j}}$, $i,j=1,\ldots,n$, and let $(T_{ij})$ be the inverse matrix of
$\left( T^{ij}\right) $. As examples notice that
\begin{enumerate}
\item For $T=\tfrac{1}{2}g^{ij}u_{i}u_{j}$,
\[
(g^{ij})=\left(
\begin{array}
[c]{cccc}%
-1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 1
\end{array}
\right) ,
\]
(resp., $g^{ij}=\delta^{ij}$, $i,j=1,\ldots,n$), (\ref{Eq14}) reduces to the
wave equation (resp., the Poisson equation) with a $u$-dependent potential $V$
(including the $f$-Gordon equation as a particular example, if $n=2$ and
$f=-V^{\prime}$).
\item For $n=2$, $T=\sqrt{1+g^{ij}u_{i}u_{j}}$, $g^{ij}=\delta^{ij}$,
$i,j=1,2$, and $V=0$, (\ref{Eq14}) reduces to the equation for minimal
surfaces in $\mathbb{R}^{3}$ transversal to the projection $\mathbb{R}^{3}%
\ni(x_{1},x_{2},u)\longmapsto(x_{1},x_{2})\in\mathbb{R}^{2}$.
\end{enumerate}
Let us search for PD-Noether symmetries and currents of $\omega$. Let
$Y=U\tfrac{\partial}{\partial u}+U_{i}\tfrac{\partial}{\partial u^{i}}%
\in\mathrm{V}D$ and $f=f^{i}d^{n-1}x_{i}\in\Omega^{0}$. Then
\begin{align*}
i_{Y}\omega & =T^{ij}(U_{i}du-Udu_{i})d^{n-1}x_{j}-\left( T^{ij}u_{i}%
U_{j}+V^{\prime}U\right) d^{n}x,\\
\delta f & =\partial_{i}f^{i}d^{n}x+\tfrac{\partial}{\partial u}%
f^{i}dud^{n-1}x_{i}+\tfrac{\partial}{\partial u_{k}}f^{i}du_{k}d^{n-1}x_{i}.
\end{align*}
Recall that $Y$ and $f$ are a PD-Noether symmetry and a PD-Noether current
relative to each other, respectively, iff $i_{Y}\omega=\delta f$, i.e.,
\begin{align}
\partial_{i}f^{i}+T^{ij}u_{i}U_{j}+V^{\prime}U & =0,\label{Eq16}\\
\tfrac{\partial}{\partial u}f^{i}-T^{ij}U_{j} & =0,\label{Eq17}\\
\tfrac{\partial}{\partial u_{j}}f^{i}+T^{ij}U & =0. \label{Eq18}%
\end{align}
It follows from (\ref{Eq18}) that $\tfrac{\partial}{\partial u_{j}}%
f^{i}=\tfrac{\partial}{\partial u_{i}}f^{j}$, $i,j=1,\ldots,n$, and then
\[
\tfrac{\partial^{2}}{\partial u_{k}\partial u_{j}}f^{i}=\tfrac{\partial^{2}%
}{\partial u_{k}\partial u_{i}}f^{j},\quad i,j,k=1,\ldots,n.
\]
Now,
\begin{align*}
\tfrac{\partial^{2}}{\partial u_{k}\partial u_{j}}f^{i} & =\tfrac{\partial
}{\partial u_{k}}\tfrac{\partial}{\partial u_{j}}f^{i}\\
& =-\tfrac{\partial}{\partial u_{k}}\left( T^{ij}U\right) \\
& =-\tfrac{\partial^{3}T}{\partial u_{k}\partial u_{i}\partial u_{j}}%
U-T^{ij}\tfrac{\partial}{\partial u_{k}}U
\end{align*}
Similarly,
\[
\tfrac{\partial^{2}}{\partial u_{k}\partial u_{i}}f^{j}=\tfrac{\partial
}{\partial u_{i}}\tfrac{\partial}{\partial u_{k}}f^{j}=-\tfrac{\partial^{3}%
T}{\partial u_{i}\partial u_{j}\partial u_{k}}U-T^{jk}\tfrac{\partial
}{\partial u_{i}}U.
\]
Therefore,
\[
T^{ij}\tfrac{\partial}{\partial u_{k}}U-T^{jk}\tfrac{\partial}{\partial u_{i}%
}U=0
\]
Contracting with $T_{ij}$ we find $(n-1)\tfrac{\partial}{\partial u_{i}}U=0$
and, therefore,
\[
U=U(x^{1},\ldots,x^{n},u),
\]
so that (\ref{Eq18}) can be rewritten as
\[
\tfrac{\partial}{\partial u_{j}}\left( f^{i}+\tfrac{\partial T}{\partial
u_{i}}U\right) =0.
\]
We conclude that
\begin{equation}
f^{i}=-\tfrac{\partial T}{\partial u_{i}}U+A^{i} \label{Eq19}%
\end{equation}
for some $A^{i}=A^{i}(x^{1},\ldots,x^{n},u)$, $i=1,\ldots,n$. Notice that
(\ref{Eq17}) can be used to determine the $U_{j}$'s from the $f^{i}$'s via
\[
U_{j}=T_{ji}\tfrac{\partial}{\partial u}f^{i}.
\]
It remains to solve (\ref{Eq16}) which, in view of (\ref{Eq19}), reduces to
\begin{equation}
\left( \partial_{i}+u_{i}\tfrac{\partial}{\partial u}\right) A^{i}%
-\tfrac{\partial T}{\partial u_{i}}\left( \partial_{i}+u_{i}\tfrac{\partial
}{\partial u}\right) U+V^{\prime}U=0. \label{Eq20}%
\end{equation}
We cannot go further on in solving (\ref{Eq20}) without better specifying $T$.
In the following we will only consider two special cases.
\begin{enumerate}
\item $T=\tfrac{1}{2}g^{ij}u_{i}u_{j}$, $(g^{ij})$ being a constant, non
degenerate, symmetric matrix with inverse $(g_{ij})$. In this case
(\ref{Eq20}) reads
\begin{equation}
\partial_{i}A^{i}+V^{\prime}U+\left( \tfrac{\partial}{\partial u}A^{i}%
-g^{ij}\partial_{j}U\right) u_{i}-\left( g^{ij}\tfrac{\partial}{\partial
u}U\right) u_{i}u_{j}=0. \label{Eq21}%
\end{equation}
The left hand side of (\ref{Eq21}) is polynomial in $u_{1},\ldots,u_{n}$.
Thus, all the corresponding coefficients must vanish, i.e.,
\begin{align}
\tfrac{\partial}{\partial u}U & =0,\label{Eq22}\\
\tfrac{\partial}{\partial u}A^{i}-g^{ij}\partial_{j}U & =0,\label{Eq23}\\
\partial_{i}A^{i}+V^{\prime}U & =0. \label{Eq24}%
\end{align}
From (\ref{Eq22}), $U=U(x^{1},\ldots,x^{n})$ and, then, from (\ref{Eq23}),
$\tfrac{\partial^{2}}{\partial u^{2}}A^{i}=0$, $i=1,\ldots,n$, which in turn
implies, using (\ref{Eq23}) again,
\[
A^{i}=\left( g^{ij}\partial_{j}U\right) u+B^{i}%
\]
for some $B^{i}=B^{i}(x^{1},\ldots,x^{n})$. Finally, (\ref{Eq24}) implies
\[
\left( g^{ij}\partial_{ij}^{2}U\right) u+\partial_{i}B^{i}+V^{\prime}U=0
\]
and differentiating once more with respect to $u$
\[
g^{ij}\partial_{ij}^{2}U+V^{\prime\prime}U=0.
\]
Since $U$ doesn't depend on $u$, if
\begin{enumerate}
\item $V^{\prime\prime\prime}\neq0$. Then $U=0$ so that
\[
f^{i}=\tfrac{1}{2}\partial_{j}B^{ji},\quad U_{j}=0
\]
for some $B^{ij}=-B^{ji}=B^{ij}(x^{1},\ldots,x^{n})$, i.e.,
\[
Y=0\quad\text{and}\quad f=d\beta,
\]
$\beta=B^{ji}d^{n-2}x_{ji}$, where $d^{n-2}x_{ji}:=i_{\partial_{j}}%
d^{n-1}x_{i}$, $i,j=1,\ldots,n$. Therefore, $\omega$ doesn't posses
PD-Noether symmetries nor non-trivial PD-Noether currents.
\item $V^{\prime\prime\prime}=0$. Then $V=\tfrac{1}{2}\mu u^{2}$ for some
constant $\mu$ and
\[
g^{ij}\partial_{ij}^{2}U+\mu U=0,\quad f^{i}=g^{ij}(u\,\partial_{j}%
U-u_{j}U)+\tfrac{1}{2}\partial_{j}B^{ji},\quad U_{j}=\partial_{j}U,
\]
for some $B^{ij}=-B^{ji}=B^{ij}(x^{1},\ldots,x^{n})$. Thus,
\[
Y=U\tfrac{\partial}{\partial u}+\partial_{j}U\tfrac{\partial}{\partial u_{j}%
}\quad\text{and}\quad f=g^{ij}(u\,\partial_{j}U-u_{j}U)d^{n-1}x_{i}+d\beta,
\]
$\beta=B^{ji}d^{n-2}x_{ji}$, where $U$ is any solution of the PD-Hamilton
equation
\begin{equation}
g^{ij}\partial_{ij}^{2}u+\mu u=0. \label{Eq25}%
\end{equation}
Let us compute the PD-Poisson bracket. Consider two solutions of (\ref{Eq25}),
say $U_{1},U_{2}$, the corresponding PD-Noether symmetries $Y_{1},Y_{2}$ and
associated PD-Noether currents $f_{1},f_{2}$. Then
\[
\{f_{1},f_{2}\}=L_{Y_{1}}f_{2}=g^{ij}(U_{1}\partial_{j}U_{2}-U_{2}\partial
_{j}U_{1})d^{n-1}x_{i},
\]
which, as can be easily checked, is a trivial conservation law.
\end{enumerate}
\item $n=2$, $T=\sqrt{1+\delta^{ij}u_{i}u_{j}}$ and $V=0$. In this case
(\ref{Eq20}) reads
\begin{equation}
\tau^{1/2}\left( \partial_{i}+u_{i}\tfrac{\partial}{\partial u}\right)
A^{i}=\delta^{ij}u_{j}\left( \partial_{i}+u_{i}\tfrac{\partial}{\partial
u}\right) U, \label{Eq26}%
\end{equation}
where $\tau=1+\delta^{ij}u_{i}u_{j}$. Squaring both sides of (\ref{Eq26}) we
get
\[
\tau\left[ \left( \partial_{i}+u_{i}\tfrac{\partial}{\partial u}\right)
A^{i}\right] ^{2}-\left[ \delta^{ij}u_{j}\left( \partial_{i}+u_{i}%
\tfrac{\partial}{\partial u}\right) U\right] ^{2}=0,
\]
whose left hand side is polynomial in $u_{1},u_{2}$. Collecting homogeneous
terms we get
\begin{align}
& \left[ \left( \tfrac{\partial}{\partial u}U\right) ^{2}\delta
^{ij}-\left( \tfrac{\partial}{\partial u}A^{i}\right) \left( \tfrac
{\partial}{\partial u}A^{j}\right) \right] \delta^{kl}u_{i}u_{j}u_{k}%
u_{l}\nonumber\\
+ & 2\delta^{ij}\left[ \delta^{kl}\left( \tfrac{\partial}{\partial
u}U\right) \left( \partial_{l}U\right) -\left( \tfrac{\partial}{\partial
u}A^{k}\right) \left( \partial_{l}A^{l}\right) \right] u_{i}u_{j}%
u_{k}\nonumber\\
- & \left[ \delta^{ij}\left( \partial_{k}A^{k}\right) ^{2}+\left(
\tfrac{\partial}{\partial u}A^{i}\right) \left( \tfrac{\partial}{\partial
u}A^{j}\right) -\delta^{ik}\delta^{jl}\left( \partial_{k}U\right) \left(
\partial_{l}U\right) \right] u_{i}u_{j}\nonumber\\
+ & 2\left( \partial_{j}A^{j}\right) ^{2}\left( \tfrac{\partial}{\partial
u}A^{i}\right) u_{i}+\left( \partial_{i}A^{i}\right) ^{2}=0. \label{Eq27}%
\end{align}
All coefficient of the left hand side of (\ref{Eq27}) must vanish. It follows
that
\[
\tfrac{\partial}{\partial u}U=\partial_{1}U=\partial_{2}U=0,\quad
\tfrac{\partial}{\partial u}A^{1}=\tfrac{\partial}{\partial u}A^{2}%
=0,\quad\partial_{1}A^{1}+\partial_{2}A^{2}=0,
\]
i.e., $U$ is a constant while $A^{1}=\partial_{2}B$, $A^{2}=-\partial_{1}B$
for some $B=B(x^{1},x^{2})$. Thus,
\[
Y=U\tfrac{\partial}{\partial u},\quad f=U\,\tau^{-1/2}\left( u_{2}%
dx^{1}-u_{1}dx^{2}\right) +dB.
\]
It is obvious that the PD-Poisson bracket is also trivial in this case.
\end{enumerate}
\subsection{A Degenerate, Constrained Example}
The example in this subsection is taken from \cite{gmr09}. Let $\alpha:\mathbb{R}^{3m+2}\times\mathbb{R}_{+}\ni(q^{1},\ldots,q^{m}%
,s_{1},\ldots,s_{m},t_{1},\ldots,t_{m},s,t;e)\longmapsto(s,t)\in\mathbb{R}%
^{2}$. The form
\[
\omega:=-dt_{\alpha}dq^{\alpha}ds+ds_{\alpha}dq^{\alpha}dt-\delta^{\alpha
\beta}(et_{\alpha}dt_{\beta}-s_{\alpha}ds_{\beta})dsdt-\varepsilon dedsdt,
\]
where $\varepsilon:=\tfrac{1}{2}(\delta^{\alpha\beta}t_{\alpha}t_{\beta}-1)$,
is a PD-prehamiltonian system on $\alpha$. The associated PD-Hamilton
equation reads
\[%
\begin{array}
[c]{c}%
\tfrac{\partial}{\partial t}t_{\alpha}+\tfrac{\partial}{\partial s}s_{\alpha
}=0,\\
\tfrac{\partial}{\partial t}q^{\alpha}=e\delta^{\alpha\beta}t_{\beta},\\
\tfrac{\partial}{\partial s}q^{\alpha}=-\delta^{\alpha\beta}s_{\beta},\\
\varepsilon=0,
\end{array}
\]
$\alpha=1,\ldots,m$, which is in turn equivalent to
\[%
\begin{array}
[c]{c}%
e^{-1}\tfrac{\partial^{2}}{\partial t^{2}}q^{\alpha}-\tfrac{\partial^{2}%
}{\partial s^{2}}q^{\alpha}=e^{-2}\left( \tfrac{\partial}{\partial
t}q^{\alpha}\right) \left( \tfrac{\partial}{\partial t}e\right) ,\\
e^{2}=\delta_{\alpha\beta}\left( \tfrac{\partial}{\partial t}q^{\alpha
}\right) \left( \tfrac{\partial}{\partial t}q^{\beta}\right) ,\\
t_{\alpha}=e^{-1}\delta_{\alpha\beta}\tfrac{\partial}{\partial t}q^{\beta},\\
s_{\alpha}=\delta_{\alpha\beta}\tfrac{\partial}{\partial s}q^{\beta},
\end{array}
,
\]
Notice that $\underline{D}^{\omega}$ is generated by $\tfrac{\partial
}{\partial e}$, while
\[
D_{y}^{\omega}=\left\{
\begin{array}
[c]{cc}%
\mathbf{0} & \text{for }\varepsilon(y)\neq0\\
\left\langle \left. \tfrac{\partial}{\partial e}\right\vert _{y}\right\rangle
& \text{for }\varepsilon(y)=0
\end{array}
\right. ,\quad y\in P
\]
we conclude that $P_{(1)}$ is the hypersurface defined by $\delta^{\alpha\beta
}t_{\alpha}t_{\beta}=1$. It is easy to see that, actually, $\breve{P}=P_{(1)}$.
Let us search for PD-Noether symmetries and currents of $\omega$. Let
$Y=Q^{\alpha}\tfrac{\partial}{\partial q^{\alpha}}+S_{\alpha}\tfrac{\partial
}{\partial s_{\alpha}}+T_{\alpha}\tfrac{\partial}{\partial t_{\alpha}}%
+E\tfrac{\partial}{\partial e}\in V\mathrm{D}$ and $f=\alpha ds+\beta
dt\in\Omega^{0}$. Then $i_{Y}\omega=\delta f$ iff
\begin{equation}%
\begin{array}
[c]{c}%
\tfrac{\partial}{\partial s}\beta-\tfrac{\partial}{\partial t}\alpha
=\delta^{\beta\gamma}(s_{\beta}S_{\gamma}-et_{\beta}T_{\gamma})-\varepsilon
E,\\
\tfrac{\partial}{\partial q^{\alpha}}\alpha=-T_{\alpha},\quad\tfrac{\partial
}{\partial q^{\alpha}}\beta=S_{\alpha},\quad\tfrac{\partial}{\partial
t_{\alpha}}\alpha=\tfrac{\partial}{\partial s_{\alpha}}\beta=Q^{\alpha}\\
\tfrac{\partial}{\partial s_{\alpha}}\alpha=\tfrac{\partial}{\partial
t_{\alpha}}\beta=0,\quad\tfrac{\partial}{\partial e}\alpha=\tfrac{\partial
}{\partial e}\beta=0,
\end{array}
\label{Eq35}%
\end{equation}
$\alpha=1,\ldots,m$. Equations (\ref{Eq35}) can be easily solved and give
quite large $\mathscr{S}(\omega)$ and $\mathscr{C}(\omega)$. Namely,
\[
\alpha=C^{\alpha}t_{\alpha}+A,\quad\beta=C^{\alpha}s_{\alpha}+B,
\]
and
\[%
\begin{array}
[c]{c}%
Q^{\alpha}=C^{\alpha},\quad T_{\alpha}=-\tfrac{\partial C^{\beta}}{\partial
q^{\alpha}}t_{\beta}-\tfrac{\partial A}{\partial q^{\alpha}},\quad S_{\alpha
}=\tfrac{\partial C^{\beta}}{\partial q^{\alpha}}s_{\beta}+\tfrac{\partial
B}{\partial q^{\alpha}},\\
\varepsilon E=\tfrac{\partial C^{\alpha}}{\partial s}s_{\alpha}-\tfrac
{\partial C^{\alpha}}{\partial t}t_{\alpha}+\tfrac{\partial B}{\partial
s}-\tfrac{\partial A}{\partial t}-\delta^{\alpha\beta}\left[ s_{\alpha
}\left( \tfrac{\partial C^{\gamma}}{\partial q^{\beta}}s_{\gamma}%
+\tfrac{\partial B}{\partial q^{\beta}}\right) +et_{\alpha}\left(
\tfrac{\partial C^{\gamma}}{\partial q^{\beta}}t_{\gamma}+\tfrac{\partial
A}{\partial q^{\beta}}\right) \right]
\end{array}
\]
where $A,B,\ldots,C^{\alpha},\ldots,D^{\alpha\beta},\ldots,E^{\alpha},\ldots$
are arbitrary functions of the only $s,t,\dots,q^{\beta},\ldots$.
Compute the PD-Poisson bracket. Let $f_{1}$,$f_{2}$ be PD-Noether currents
determined by functions $A_{1},B_{1},\ldots,C_{1}^{\alpha},\ldots$ and
$A_{2},B_{2},\ldots,C_{2}^{\alpha},\ldots$ respectively. A straightforward
computation shows that
\[
\{f_{1},f_{2}\}=(C^{\alpha}t_{\alpha}+A)ds+(C^{\alpha}s_{\alpha}+B)dt
\]
with
\begin{align*}
A & =C_{1}^{\beta}\tfrac{\partial}{\partial q^{\beta}}A_{2}-C_{2}^{\beta
}\tfrac{\partial}{\partial q^{\beta}}A_{1},\\
B & =C_{1}^{\beta}\tfrac{\partial}{\partial q^{\beta}}B_{2}-C_{2}^{\beta
}\tfrac{\partial}{\partial q^{\beta}}B_{1},\\
C^{\alpha} & =C_{1}^{\beta}\tfrac{\partial}{\partial q^{\beta}}C_{2}%
^{\alpha}-C_{2}^{\beta}\tfrac{\partial}{\partial q^{\beta}}C_{1}^{\alpha},
\end{align*}
$\alpha=1,\ldots,m$.
\subsection{A Degenerate, Unconstrained Example}
Finally, we propose an example of reduction. Consider the cotangent bundle
$\pi:T^{\ast}\mathbb{M}\ni A_{i}dx^{i}|_{(x^{1},\ldots,x^{n})}\longmapsto
(x^{1},\ldots,x^{n})\in\mathbb{M}$ and let $\alpha:=\pi_{1}:(x^{1},\dots
,x^{n},\ldots,A_{i},\ldots,A_{i,j},\ldots)\ni J^{1}\pi\longmapsto(x^{1}%
,\dots,x^{n})\in\mathbb{M}$, $\mathbb{M}$ being the $n$-dimensional Minkowski
space. As such $\mathbb{M}$ is endowed with the metric $g:=g_{ij}dx^{i}\cdot
dx^{j}$ where
\[
(g_{ij})=\left(
\begin{array}
[c]{cccc}%
-1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 1
\end{array}
\right) .
\]
In the following we will raise and lower indexes using $g$. Let
\[
\omega:=2dA^{[j,i]}\left( \tfrac{1}{2}A_{i,j}d^{n}x-dA_{i}d^{n-1}%
x_{j}\right)
\]
Then $\omega$ is a PD-prehamiltonian system on $\pi$ whose PD-Hamilton
equation reads
\begin{align*}
\partial_{k}A^{[i,k]} & =0,\\
\partial_{\lbrack j}A_{i]} & =A_{[i,j]},
\end{align*}
$i,j=1,\ldots,n$, which are equivalent to Maxwell equations for the vector
potential
\begin{align*}
(\partial_{k}\partial^{k})A_{i}-\partial_{i}\partial_{k}A^{k} & =0,\\
A_{[i,j]} & =\partial_{\lbrack j}A_{i]}.
\end{align*}
Notice that
\[
\ker\omega=\ker\underline{\omega}=\left\langle \ldots,\tfrac{\partial
}{\partial A_{i,j}}+\tfrac{\partial}{\partial A_{j,i}},\ldots\right\rangle .
\]
Therefore $\omega$ is \textquotedblleft degenerate and
unconstrained\textquotedblright. Moreover, leaves of $D^{\omega}=\underline
{D}^{\omega}$ are given by $A_{[i,j]}=\mathrm{const}$. We conclude that
$J^{1}\pi$ \textquotedblleft reduces\textquotedblright\ via
\begin{align*}
\mathfrak{p}:J^{1}\pi & \longrightarrow T^{\ast}\mathbb{M\times}_{M}%
\wedge^{2}T^{\ast}\mathbb{M\simeq R}^{n(n+3)/2}\\
(x^{1},\dots,x^{n},\ldots,A_{i},\ldots,A_{i,j},\ldots) & \longmapsto
(x^{1},\dots,x^{n},\ldots,A_{i},\ldots,F_{ij},\ldots)
\end{align*}
where $F_{ij}=F_{[ij]}$, $\mathfrak{p}^{\ast}(F_{ij}):=2A_{[j,i]}$ and
$\omega=\mathfrak{p}^{\ast}(\widetilde{\omega})$, with
\[
\widetilde{\omega}:=dF^{ij}\left( \tfrac{1}{4}F_{ji}d^{n}x-dA_{i}d^{n-1}%
x_{j}\right)
\]
is a PD-hamiltonian system on
\[
\widetilde{\alpha}:\mathbb{R}^{n(n+3)/2}\ni(x^{1},\dots,x^{n},\ldots
,A_{i},\ldots,F_{ij},\ldots)\longmapsto(x^{1},\dots,x^{n})\in\mathbb{R}^{n},
\]
whose PD-Hamilton equations read
\begin{align*}
\partial_{k}F^{ik} & =0,\\
\partial_{\lbrack j}A_{i]} & =2F_{ji},
\end{align*}
which are Maxwell equations for the field strength.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,191
|
\section{Introduction}
In 1964, Cronin and Fitch discovered the charge conjugate and parity (CP) violating decays of the $K$ meson \cite{1964}. The lepton electric dipole moments (EDMs) are studied as the physical quantities for probing sources of CP violation \cite{2}.
Therefore, it is of special significance to study the EDM of lepton. At present, the upper bound of electron EDM is $|d^{exp}_e|$ $<$ $1.1 \times 10^{-29}$ e.cm at the $90 \%$ confidence level\cite{de, de1,de2}, the muon EDM is $|d^{exp}_{\mu}|$ $<$ $1.8 \times 10^{-19}$ e.cm at the $95 \%$ confidence level and the tau EDM is $|d^{exp}_{\tau}|$ $<$ $1.1 \times 10^{-17}$ e.cm at the $95 \%$ confidence level \cite{pdg}. The minimal supersymmetric extension of SM (MSSM) \cite{mssm} is very meaningful and physicists have been studying it for a long time. There are several CP violating phases, which can give large contributions to the EDMs of leptons in MSSM \cite{Z2015}.
In MSSM, when the CP violating phases are of normal size and the SUSY particles are at TeV scale, big EDMs of elementary particles are obtained, and they can exceed the current experiment limits. There are three ways to solve this problem. 1. Making the CP violating phases smaller, i.e. $\emph{O}$ ($10^{-2}$). This is called tuning. 2. Using mass suppression to make supersymmetric particles heavy (several TeV). 3. There are mechanisms for the different components to cancel each other out. For lepton EDM and neutron EDM, the main parts of chargino and the neutralino contributions are cancelled \cite{123}.
Due to the deficiency of MSSM which can not explain neutrino mass and solve $\mu$ problem, U(1) extension of MSSM is carried out. There are two U(1) groups in $U(1)_X$SSM: $U(1)_Y$ and $U(1)_X$, and we use SARAH software packages \cite{extend1, extend2, extend3} to study $U(1)_X$SSM.
On the basis of MSSM, the superfields are added; then one obtains not only the additional Higgs, neutrino and gauge fields, but also corresponding
superpartners that extend the neutralino and sfermion sectors. The CP-even parts of the three Higgs singlet fields $\eta$, $\overline{\eta}$, $S$ mix with the neutral CP-even parts of the two doublets $H_d$ and $H_u$ to form a tree order $5\times5$ CP-even Higgs mass matrix. $m_{h_0}$ is the tree level mass of the lightest CP-even Higgs in $U(1)_X$SSM, and it can be greater than the corresponding mass at tree order in MSSM.
Therefore, the loop graph correction to $m_{h_0}$ in $U(1)_X$SSM needs not be very large.
Researching the MDMs \cite{mdms} and EDMs \cite{edms1, edms2} of lepton is an effective way to find new physics beyond the standard model(SM).
In MSSM, the one-loop contributions to lepton MDM and EDM are well investigated, and some two-loop corrections are also studied. In the two
Higgs doublet models with CP violation, the authors gain the one-loop and Barr-Zee type two-loop contributions to fermionic EDMs. A model-independent study of $d_e$ in the SM is executed \cite{mi}. They consider the right-handed neutrinos, the neutrino see-saw mechanism and the framework of minimal flavor violation. Their results show that when neutrinos are Majorana particles, the results of $d_e$ can reach its experiment upper limit.
In the following, we introduce the specific form of $U(1)_X$SSM and its superfields in section 2. In section 3, we show that the one-loop and two-loop corrections to the lepton EDM. The main content of section 4 is the numerical analysis for the dependence of lepton EDM on the $U(1)_X$SSM parameters. We have a special summary and discussion in section 5. The appendix is used for the some mass matrics and Feynman rules.
\section{the $U(1)_X$ SSM}
The gauge group of the $U(1)_X$SSM is $SU(3)_C\otimes
SU(2)_L \otimes U(1)_Y\otimes U(1)_X$. To obtain the $U(1)_X$SSM, the MSSM is added with
three Higgs singlets $\hat{\eta},~\hat{\bar{\eta}},~\hat{S}$ and right-handed neutrinos $\hat{\nu}_i$. It can give light
neutrino mass at the tree level through the see-saw mechanism. The neutral CP-even parts of
$H_u,~ H_d,~\eta,~\bar{\eta}$ and $S$ mix together, forming $5\times5 $ mass squared matrix.
Because of the right handed neutrinos, the mass matrix of neutrino is expended to $6\times6$. In the meantime, the squared mass matrix of scalar neutrinos turns to $6\times6$ too. For details of the mass matrix of particles, please see the appendix.
The superpotential for this model show as:
\begin{eqnarray}
&&W=l_W\hat{S}+\mu\hat{H}_u\hat{H}_d+M_S\hat{S}\hat{S}-Y_d\hat{d}\hat{q}\hat{H}_d-Y_e\hat{e}\hat{l}\hat{H}_d+\lambda_H\hat{S}\hat{H}_u\hat{H}_d
\nonumber\\&&+\lambda_C\hat{S}\hat{\eta}\hat{\bar{\eta}}+\frac{\kappa}{3}\hat{S}\hat{S}\hat{S}+Y_u\hat{u}\hat{q}\hat{H}_u+Y_X\hat{\nu}\hat{\bar{\eta}}\hat{\nu}
+Y_\nu\hat{\nu}\hat{l}\hat{H}_u.
\end{eqnarray}
There are two Higgs doublets and three Higgs singlets. Their specific forms are collected here,
\begin{eqnarray}
&&H_{u}=\left(\begin{array}{c}H_{u}^+\\{1\over\sqrt{2}}\Big(v_{u}+H_{u}^0+iP_{u}^0\Big)\end{array}\right),
~~~~~~
H_{d}=\left(\begin{array}{c}{1\over\sqrt{2}}\Big(v_{d}+H_{d}^0+iP_{d}^0\Big)\\H_{d}^-\end{array}\right),
\nonumber\\
&&\eta={1\over\sqrt{2}}\Big(v_{\eta}+\phi_{\eta}^0+iP_{\eta}^0\Big),~~~~~~~~~~~~~~~
\bar{\eta}={1\over\sqrt{2}}\Big(v_{\bar{\eta}}+\phi_{\bar{\eta}}^0+iP_{\bar{\eta}}^0\Big),\nonumber\\&&
\hspace{4.0cm}S={1\over\sqrt{2}}\Big(v_{S}+\phi_{S}^0+iP_{S}^0\Big).
\end{eqnarray}
$v_u,~v_d,~v_\eta$,~ $v_{\bar\eta}$ and $v_S$ are the corresponding VEVs of the Higgs superfields $H_u$, $H_d$, $\eta$, $\bar{\eta}$ and $S$.
Here, we define $\tan\beta=v_u/v_d$ and $\tan\beta_\eta=v_{\bar{\eta}}/v_{\eta}$. The definition of
$\tilde{\nu}_L$ and $\tilde{\nu}_R$ is
\begin{eqnarray}
\tilde{\nu}_L=\frac{1}{\sqrt{2}}\phi_l+\frac{i}{\sqrt{2}}\sigma_l,~~~~~~~~~~\tilde{\nu}_R=\frac{1}{\sqrt{2}}\phi_R+\frac{i}{\sqrt{2}}\sigma_R.
\end{eqnarray}
The soft SUSY breaking terms are
\begin{eqnarray}
&&\mathcal{L}_{soft}=\mathcal{L}_{soft}^{MSSM}-B_SS^2-L_SS-\frac{T_\kappa}{3}S^3-T_{\lambda_C}S\eta\bar{\eta}
+\epsilon_{ij}T_{\lambda_H}SH_d^iH_u^j\nonumber\\&&
-T_X^{IJ}\bar{\eta}\tilde{\nu}_R^{*I}\tilde{\nu}_R^{*J}
+\epsilon_{ij}T^{IJ}_{\nu}H_u^i\tilde{\nu}_R^{I*}\tilde{l}_j^J
-m_{\eta}^2|\eta|^2-m_{\bar{\eta}}^2|\bar{\eta}|^2\nonumber\\&&
-m_S^2S^2-(m_{\tilde{\nu}_R}^2)^{IJ}\tilde{\nu}_R^{I*}\tilde{\nu}_R^{J}
-\frac{1}{2}\Big(M_S\lambda^2_{\tilde{X}}+2M_{BB^\prime}\lambda_{\tilde{B}}\lambda_{\tilde{X}}\Big)+h.c.
\end{eqnarray}
We use $Y^Y$ for the $U(1)_Y$ charge and $Y^X$ for the $U(1)_X$ charge.
According to the textbook \cite{text}, the SM is anomaly free.
The anomalies of $U(1)_X$SSM are more complicated than those of SM.
In the end, this model has been proven anomaly free \cite{pro}.
The presence of two Abelian groups $U(1)_Y$ and $U(1)_X$ in $U(1)_X$SSM have a new effect absent in the MSSM with just one Abelian gauge group $U(1)_Y$:
the gauge kinetic mixing. This effect can also be induced through RGEs, even if it is set to zero at $M_{GUT}$.
The covariant derivatives of this model have the general form \cite{model1, model2, model3}
\begin{eqnarray}
&&D_\mu=\partial_\mu-i\left(\begin{array}{cc}Y,&X\end{array}\right)
\left(\begin{array}{cc}g_{Y},&g{'}_{{YX}}\\g{'}_{{XY}},&g{'}_{{X}}\end{array}\right)
\left(\begin{array}{c}A_{\mu}^{\prime Y} \\ A_{\mu}^{\prime X}\end{array}\right)\;.
\label{gauge1}
\end{eqnarray}
Here, $A_{\mu}^{\prime Y}$ and $A^{\prime X}_\mu$ signify the gauge fields of $U(1)_Y$ and $U(1)_X$. We can do a basis conversion,
because the two Abelian gauge groups are unbroken.
The following formula can be obtained by using the appropriate matrix $R$ \cite{model1, model3}
\begin{eqnarray}
&&\left(\begin{array}{cc}g_{Y},&g{'}_{{YX}}\\g{'}_{{XY}},&g{'}_{{X}}\end{array}\right)
R^T=\left(\begin{array}{cc}g_{1},&g_{{YX}}\\0,&g_{{X}}\end{array}\right)\;.
\label{gauge2}
\end{eqnarray}
We deduce $\sin^2\theta_{W}^\prime$ $=$
\begin{eqnarray}
\frac{1}{2}-\frac{((g_{YX}+g_X)^2-g_{1}^2-g_{2}^2)v^2+
4g_{X}^2\xi^2}{2\sqrt{((g_{YX}+g_X)^2+g_{1}^2+g_{2}^2)^2v^4+8g_{X}^2((g_{YX}+g_X)^2-g_{1}^2-g_{2}^2)v^2\xi^2+16g_{X}^4\xi^4}}.
\end{eqnarray}
with $\xi=\sqrt{v_\eta^2+v_{\bar{\eta}}^2}$.
The new mixing angle $\theta_{W}^\prime$ appears in the couplings involving $Z$ and $Z^{\prime}$.
\section{formulation}
We use the effective Lagrangian method, and the Feynman amplitude can be expressed by these dimension 6 operators \cite{lepton}.
\begin{eqnarray}
&&\mathcal{O}_1^{\mp}=\frac{1}{(4\pi)^2}\bar{l}(i\mathcal{D}\!\!\!\slash)^3\omega_{\mp}l,
\nonumber\\
&&\mathcal{O}_2^{\mp}=\frac{eQ_f}{(4\pi)^2}\overline{(i\mathcal{D}_{\mu}l)}\gamma^{\mu}
F\cdot\sigma\omega_{\mp}l,
\nonumber\\
&&\mathcal{O}_3^{\mp}=\frac{eQ_f}{(4\pi)^2}\bar{l}F\cdot\sigma\gamma^{\mu}
\omega_{\mp}(i\mathcal{D}_{\mu}l),\nonumber\\
&&\mathcal{O}_4^{\mp}=\frac{eQ_f}{(4\pi)^2}\bar{l}(\partial^{\mu}F_{\mu\nu})\gamma^{\nu}
\omega_{\mp}l,\nonumber\\&&
\mathcal{O}_5^{\mp}=\frac{m_l}{(4\pi)^2}\bar{l}(i\mathcal{D}\!\!\!\slash)^2\omega_{\mp}l,
\nonumber\\&&\mathcal{O}_6^{\mp}=\frac{eQ_fm_l}{(4\pi)^2}\bar{l}F\cdot\sigma
\omega_{\mp}l.
\end{eqnarray}
with $\mathcal{D}_{\mu}=\partial_{\mu}+ieA_{\mu}$ and $\omega_{\mp}=\frac{1\mp\gamma_5}{2}$. $F_{{\mu\nu}}$ is the electromagnetic field strength, and
$m_{_l}$ is the lepton mass. Therefore, the Wilson coefficients of the operators $\mathcal{O}_{2,3,6}^{\mp}$ in the effective Lagrangian are of interest and their dimensions are -2.
The lepton EDM is
\begin{eqnarray}
&&{\cal L}_{_{EDM}}=\frac{-i}{2}d_l\bar{l}\sigma^{\mu\nu}\gamma_5lF_{\mu\nu}.
\end{eqnarray}
The fermion EDM is a CP violating amplitude which can not be obtained at tree level in the fundamental interactions. However, in the CP violating electroweak theory, one-loop diagrams should contribute nonzero value to fermion EDM. Considering the relations
between the Wilson coefficients $C_{2,3,6}^{\mp}$ of the operators $\mathcal{O}_{2,3,6}^{\mp}$ \cite{lepton, edms2}, the lepton EDM is
\begin{eqnarray}
d_l=\frac{-2eQ_fm_l}{(4\pi)^2}\Im(C_2^{+} + C_2^{-*} +C_6^{+})
\end{eqnarray}
\subsection{The one-loop corrections}
In $U(1)_X$SSM, the masses of the neutralinos, neutrinos, scalar neutrinos and scalar charged leptons are all
adopted comparing with those in MSSM.
The one-loop new physics contributions to lepton EDMs come from the diagrams in FIG. 1. We find that our own results of the one-loop corrections are similar to the MSSM results in analytic form. However, the mass matrics of scalar, Fermion and Majorana particles have relation with new parameters $g_X, g_{YX}, v_\eta, v_{\bar\eta}$ and so on.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.8\textwidth}
\includegraphics[width=5.0in]{one-loops.eps}
\end{minipage
\caption{The one-loop self energy diagrams affect lepton EDMs in the $U(1)_X$SSM. The triangle diagrams
can be obtained by attaching a photon on the internal lines of the self energy diagrams in all possible ways.}
\end{center}
\end{figure}
The corrections to lepton EDMs from neutralinos and scalar leptons are expressed as
\begin{eqnarray}
&&d_{l}^{\tilde{L}\chi^{0}}=(\frac{-e}{2\Lambda})\Im\left[-\sum_{i=1}^8\sum_{j=1}^6\Big\{(A_L^*A_R)
\sqrt{x_{\chi_i^{0}}}x_{\tilde{L}_j}\frac{\partial^2 \mathcal{B}(x_{\chi_i^{0}},x_{\tilde{L}_j})}{\partial x_{\tilde{L}_j}^2}\Big\}\right].
\end{eqnarray}
where $x_i=\frac{m_i^2}{\Lambda^2}$, $m_i$ is the particle mass and $\Lambda$ is the renormalization scale.
The couplings $A_R,A_L$ are shown as
\begin{eqnarray}
&&A_R=\frac{1}{\sqrt{2}}g_1N_{i1}^{*}Z_{j2}^{E}+\frac{1}{\sqrt{2}}g_2N_{i2}^{*}Z_{j2}^{E}+\frac{1}{\sqrt{2}}g_{YX}N_{i5}^{*}Z_{j2}^{E}
-N_{i3}^{*}Y_\mu Z_{j5}^{E},\nonumber\\&&
A_L=-\frac{1}{\sqrt{2}}Z_{j5}^{E}(2g_1N_{i1}+(2g_{YX}+g_X)N_{i5})-Y_\mu^{*}Z_{j2}^EN_{i3}.
\end{eqnarray}
The matrices $Z^{E}$, $N$ respectively diagonalize the mass matrices of scalar lepton and neutralino.
The concrete forms of the functions $\mathcal{B}(x,y)$ (using in the Eq.(11)) and $\mathcal{B}_1(x,y)$ (using in the Eqs.(14) and (16)) are
\begin{eqnarray}
\mathcal{B}(x,y)=\frac{1}{16 \pi
^2}\Big(\frac{x \ln x}{y-x}+\frac{y \ln
y}{x-y}\Big),~~~
\mathcal{B}_1(x,y)=(
\frac{\partial}{\partial y}+\frac{y}{2}\frac{\partial^2 }{\partial y^2})\mathcal{B}(x,y).
\end{eqnarray}
In a similar way, the corrections from chargino and CP-odd scalar neutrino are also obtained
\begin{eqnarray}
&&d_{lI}^{\tilde{\nu}\chi^{\pm}}=(\frac{-e}{2\Lambda})\Im\left[\sum_{i=1}^2\sum_{j=1}^6
\Big\{-2(B_L^{*}B_R)\sqrt{x_{\chi_i^{-}}}\mathcal{B}_1(x_{\tilde{\nu}_j^{I}},x_{\chi_i^{-}})\Big\}\right].
\end{eqnarray}
Here, the couplings $B_L$ and $B_R$ are
\begin{eqnarray}
B_L=-\frac{1}{\sqrt{2}}U_{i2}^{*}Z_{j2}^{I*}Y_\mu,~~~
B_R=\frac{1}{\sqrt{2}}g_2Z_{j2}^{I*}V_{i1}.
\end{eqnarray}
The corrections from chargino and CP-even scalar neutrino read as
\begin{eqnarray}
&&d_{lR}^{\tilde{\nu}\chi^{\pm}}=(\frac{-e}{2\Lambda})\Im\left[\sum_{i=1}^2\sum_{j=1}^6
\Big\{-2(C_L^{*}C_R)\sqrt{x_{\chi_i^{-}}}\mathcal{B}_1(x_{\tilde{\nu}_j^{R}},x_{\chi_i^{-}})\Big\}\right].
\end{eqnarray}
Here, the coupling $C_L$ and $C_R$ are
\begin{eqnarray}
C_L=\frac{1}{\sqrt{2}}U_{i2}^{*}Z_{j2}^{R*}Y_\mu,~~~
C_R=-\frac{1}{\sqrt{2}}g_2Z_{j2}^{R*}V_{i1}.
\end{eqnarray}
And, $U,V$ are used to diagonalize the chargino mass matrix, and the mass squared matrix of CP-even (CP-odd) scalar neutrino is diagonalized by $Z^{R}~(Z^I)$.
\subsection{The two-loop corrections}
As discussed in Ref.\cite{our}, the Barr-Zee two-loop diagrams (FIG. 2 (a), (b), (c)) and rainbow two-loop diagrams (FIG. 2 (d), (e))
have not small factors to lepton EDMs. That is to say, they have considerable contributions to lepton EDMs.
The triangle diagrams can be obtained by attaching a photon on the internal lines of the self energy diagrams in all possible ways.
The sum of all the triangle diagrams corresponding to one self energy diagram satisfy the Ward-identity and the CTP invariance.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{1.0\textwidth}
\includegraphics[width=6.0in]{two-loops.eps}
\end{minipage
\caption{The two-loop Barr-Zee and rainbow type diagrams affect lepton EDMs in the $U(1)_X$SSM.}
\end{center}
\end{figure}
At first, we consider the corrections from FIG. 2 (a). Under the assumption $m_F=m_{F_1}=m_{F_2}\gg m_W$, the results \cite{ffa} can be simplified as
\begin{eqnarray}
&&\qquad\quad\hspace{-2cm}d_l^{WH}=\frac{-G_F m_W^2 s_W}{256\pi^4}\sum_{F_1=\chi^{\pm}}\sum_{F_2=\chi^0}\frac{H_{\bar{l}H\nu}^L}{ m_F}\Big\{\Im\Big[\Big[\frac{21}{4}-\frac{5}{18}Q_{F_1}+(3+\frac{Q_{F_1}}{3})
(\ln{m_{F_1}^2}\nonumber\\
&&\qquad\quad\hspace{-2cm}-\varrho_{1,1}(m_W^2,m_{H^\pm}^2))\Big](H_{HF_1F_2}^LH_{WF_1F_2}^L+H_{HF_1F_2}^RH_{WF_1F_2}^R)
+\Big[\frac{19-20Q_{F_1}}{9}\nonumber\\
&&\qquad\quad\hspace{-2cm}+\frac{2-4Q_{F_1}}{3}
(\ln{m_{F_1}^2}-\varrho_{1,1}(m_W^2,m_{H^\pm}^2))\Big](H_{HF_1F_2}^LH_{WF_1F_2}^R+H_{HF_1F_2}^RH_{WF_1F_2}^L)
\nonumber\\
&&\qquad\quad\hspace{-2cm}+\Big[\hspace{-0.1cm}-\hspace{-0.1cm}\frac{16}{9}\hspace{-0.1cm}-\hspace{-0.1cm}\frac{2\hspace{-0.1cm}+\hspace{-0.1cm}6Q_{F_1}}{3}
(\ln{m_{F_1}^2}\hspace{-0.1cm}-\hspace{-0.1cm}\varrho_{1,1}(m_W^2,m_{H^\pm}^2))\Big](H_{HF_1F_2}^LH_{WF_1F_2}^L\hspace{-0.1cm}-\hspace{-0.1cm}H_{HF_1F_2}^RH_{WF_1F_2}^R)
\nonumber\\
&&\qquad\quad\hspace{-2cm}+\Big[\hspace{-0.1cm}-\hspace{-0.1cm}\frac{2Q_{F_1}}{9}\hspace{-0.1cm}-\hspace{-0.1cm}\frac{6\hspace{-0.1cm}-\hspace{-0.1cm}2Q_{F_1}}{3}
(\ln{m_{F_1}^2}\hspace{-0.1cm}-\hspace{-0.1cm}\varrho_{1,1}(m_W^2,m_{H^\pm}^2))\Big](H_{HF_1F_2}^LH_{WF_1F_2}^R\hspace{-0.1cm}-\hspace{-0.1cm}H_{HF_1F_2}^RH_{WF_1F_2}^L)\Big]\Big\}.
\end{eqnarray}
where $\varrho_{1,1}(x,y)=\frac{x\ln x-y\ln y}{x-y}$. $H_{HF_1F_2}^{L,R}$ and $H_{WF_1F_2}^{L,R}$ represent the corresponding coupling coefficients. Please see the Ref.\cite{slh} for their concrete forms.
Then under the assumption $m_F=m_{F_1}=m_{F_2}\gg m_{h_0}$, the two-loop Barr-Zee type diagrams contributing to the lepton EDMs corresponding to FIG. 2(b) can be simplified as
\begin{eqnarray}
&&d_l^{\gamma h_0}=\frac{-eG_FQ_fQ_{F_1}m_W^2s_W^2}{32\pi^4}\sum_{F_1=F_2=\chi^\pm}\Big\{\Im\Big[\frac{1}{m_{F_1}}
(H_{h_0F_1F_2}^L)[1+\ln\frac{m_{F_1}^2}{m_{h_0}^2}]\Big]\Big\}.
\end{eqnarray}
And FIG. 2(c) is given below
\begin{eqnarray}
&&d_l^{Zh_0}=\frac{-\sqrt{2}e}{1024\pi^4}\sum_{F_1=F_2=\chi^{\pm},\chi^0}
\Big\{\frac{H_{h_0l\bar{l}}}{m_{F_1}}\Big[\varrho_{1,1}(m_Z^2,m_{h_0}^2)-\ln{m_{F_1}^2}-1\Big]
\nonumber\\&&\qquad\quad\times\Im[(H^L_{Zll}-H^R_{Zll})(H_{h_0F_1F_2}^LH_{ZF_1F_2}^L+H_{h_0F_1F_2}^RH_{ZF_1F_2}^R)]\Big\}.
\end{eqnarray}
$Q_f$ is the electric charge of the external lepton $m_\mu$. $Q_{F_1}$ and $Q_{F_2}$ are the electric charges of the internal charginos.
Under the assumption $m_F=m_{F_1}=m_{F_2}\gg m_W\sim m_Z$, the two-loop rainbow type diagrams contributing to the lepton EDMs corresponding to FIG. 2(d) can be simplified as
\begin{eqnarray}
&&d_l^{WW}=\frac{-eG_F m_l}{384\sqrt{2}\pi^4}\sum_{F_1=\chi^{\pm}}\sum_{F_2=\chi^0}\left\{\Im[11(H_{WF_1F_2}^{R*}H_{WF_1F_2}^L)]\right\}.
\end{eqnarray}
With the assumption $m_F = m_{F1} = m_{F2} \gg m_W \sim m_Z$, we simplify the tedious two-loop results to the order $\frac{m_\mu^2}{M_Z^2}$ $\sim$ $10^{-6}$ or $\frac{m_\mu^2}{m_{SUSY}^2}$, and obtain the concise form of FIG. 2(e).
\begin{eqnarray}
&&d_{l}^{ZZ}=
\frac{eQ_{F_1}m_l}{2048\Lambda^2\pi^4}\sum_{F_1=F_2=\chi^\pm}\Big\{
\Im\Big[(H^L_{ZF_1F_2}H^R_{ZF_1F_2})\Big(|H^L_{Zll}|^2
+|H^R_{Zll}|^2\Big)[\frac{-6 \log x_Z+6 \log x_F+4}{9 x_F}]\nonumber\\&&
+\Big(|H^L_{ZF_1F_2}|^2+|H^R_{ZF_1F_2}|^2\Big)H^L_{Zll}H^R_{Zll}
[16\frac{(\log x_F-\log x_Z) (\log x_F+2)+2}{x_Z}]\Big]\Big\}.
\end{eqnarray}
At two-loop level, the contributions to lepton EDMs can be summarized as
\begin{eqnarray}
&&d_l^{two-loop}=d_l^{WH}+d_l^{\gamma h_0}+d_l^{Z h_0}+d_l^{WW}+d_{l}^{ZZ}.
\end{eqnarray}
\section{the numerical results}
For the numerical discussion. The lightest CP-even higgs mass is considered as an input parameter, which is $m_{h0}$=125.1 GeV \cite{hmass1, hmass2}. Considering the experimental limitation of lepton EDMs, we adjust the parameters. In this section, we research and discuss lepton ($e, \mu, \tau$) EDMs respectively.
The parameters used in $U(1)_X$SSM are given below:
\begin{eqnarray}
&&g_X=0.33,~g_{YX}=0.2,~\lambda_C=-0.1,~\kappa=0.1,~T_{\lambda_H}=1.0~{\rm TeV},~T_{\kappa}=1.0~{\rm TeV},
\nonumber\\&&\tan{\beta_\eta}=1.05,~v_{\eta}=15\times\cos{\beta_\eta}~{\rm TeV},~v_{\bar{\eta}}=15\times\sin{\beta_\eta}~{\rm TeV},
~B_\mu=8~{\rm TeV^2},
\nonumber\\&&m_S^2=8~{\rm TeV^2},~T_{\lambda_C}=150~{\rm GeV}, ~T_{E11}=T_{E22}=T_{E33}=0.1~{\rm TeV},
\nonumber\\&&M_{\nu11}=M_{\nu22}=M_{\nu33}=6~{\rm TeV^2}, ~Y_{X11}=Y_{X22}=Y_{X33}=0.04, \nonumber\\&&B_S=8~{\rm TeV^2}, ~\lambda_H=0.1, ~l_W=8~{\rm TeV^2}, ~T_{X11}=T_{X22}=T_{X33}=10~{\rm GeV}.
\end{eqnarray}
$\theta_1$, $\theta_2$ and $\theta_\mu$ are the CP violating phases of the parameters $m_1$, $m_2$ and $\mu$. We take into account three new CP violating parameters with the phases $\theta_{BL}$, $\theta_{BB^{\prime}}$ and $\theta_S$.
\begin{eqnarray}
&&m_1=M_1*e^{i*\theta_{1}}, ~m_2=M_2*e^{i*\theta_{2}}, ~\mu=mu*e^{i*\theta_{\mu}},
\nonumber\\&&m_{BL}=M_{BL}*e^{i*\theta_{BL}}, ~m_{{BB}^\prime}=M_{{BB}^\prime}*e^{i*\theta_{BB^{\prime}}}, ~m_S=M_S*e^{i*\theta_S}.
\end{eqnarray}
In order to facilitate the following discussion, we have made some simplifications:
\begin{eqnarray}
&&M_L=M_{L11}=M_{L22}=M_{L33}, ~~~M_E=M_{E11}=M_{E22}=M_{E33},
\nonumber\\&&T_E=T_{E11}=T_{E22}=T_{E33}.
\end{eqnarray}
\subsection{the e EDM}
At the beginning, we discussed the EDM of electron, because its experimental upper limit is very strict. The CP violating phases $\theta_1$, $\theta_2$ ,$\theta_\mu$, $\theta_{BL}$, $\theta_{BB^{\prime}}$ and $\theta_S$, also including other parameters have a certain impact on the electron EDM.
Now, supposing $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_S$ = 0, and setting $\tan{\beta}=5$, $M_2=500~
{\rm GeV}$, $mu=500~{\rm GeV}$, $M_{BL}= 1800 ~{\rm GeV}$, $M_{BB^{\prime}}= 700 ~{\rm GeV}$, $M_S= 2400 ~{\rm GeV}$, $M_L=1.1 ~{\rm TeV}$, $M_E=1.0 ~{\rm TeV}$. We study the influence of $\theta_{BL}$ on electron EDM. $M_{BL}$ is related to neutralino mass matrix.
In FIG. \ref{DEBL}, we plot the solid line and dashed line versus $M_L$ ($0.9\sim1.1 ~{\rm TeV^2}$) corresponding to $M_1$ = $700,800 ~{\rm GeV}$. We can see that these two lines are subtractive functions, and $\theta_{BL}$ has influence on $|d_e|$. The shaded part of the figure indicates that all these parameters are within reasonable parameters and conform to experimental limits.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{deBL.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_S$ = 0, and $\theta_{BL}$ = $\frac{\pi}{4}$, the contributions to electron EDM varying with $M_L$ are plotted by the solid line, dashed line respectively corresponding to $M_1$ = ($700,800)~{\rm GeV}$.}\label{DEBL}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{des.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{4}$, the contributions to electron EDM varying
with $M_{L11}$ are plotted by the solid line, dashed line respectively corresponding to $M_{L33}$ = $(1, 0.9)~{\rm TeV^2}$.}\label{DES}
\end{center}
\end{figure}
Setting $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, $\tan{\beta}=5$, $M_1= 700~{\rm GeV}$, $M_2=2000~
{\rm GeV}$, $mu=500~{\rm GeV}$, $M_{BL}= 1600 ~{\rm GeV}$, $M_{BB^{\prime}}= 800 ~{\rm GeV}$, $M_S= -800 ~{\rm GeV}$, $M_{L22}=1.0 ~{\rm TeV^2}$, $M_E=1.0 ~{\rm TeV^2}$, we consider the impact of $\theta_S$ on the electron EDM. $M_S$ is related to the mass matrices of neutralino and scalar lepton. In FIG. \ref{DES}, $M_{L11}$ varies from $0.5$ to $5.0$ $~{\rm TeV^2}$, and when $M_{L11}$ $>$ $2.0 ~{\rm TeV^2}$, the numerical results of $|d_e|$ conform to the experimental limits.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{deBBP.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_S$ = $\theta_{BL}$ = 0, and $\theta_{BB^{\prime}}$ = $\frac{\pi}{3}$, the contributions to electron EDM varying with $T_E$ are plotted by the solid line, dashed line respectively corresponding to $M_{E11}$ = $(0.5,1.0)~{\rm TeV^2}$.}\label{DEBBp}
\end{center}
\end{figure}
$\theta_{BB^{\prime}}$ is the new CP violating phase of the lepton neutrino mass matrix. So, it make new physical contribution to the lepton EDM. With $\theta_1$ = $\theta_\mu$ = $\theta_2$ = $\theta_S$ = $\theta_{BL}$ = 0, the contributions to muon EDM varying with $T_E$ are plotted by the solid line and dashed line respectively corresponding to $M_{E11}$ = 0.5 and 1.0$~{\rm TeV^2}$. In this part, we set $\tan{\beta}=5$, $M_1= 700~{\rm GeV}$, $M_2=2000~
{\rm GeV}$, $mu=500~{\rm GeV}$, $M_{BL}= 1800 ~{\rm GeV}$, $M_{BB^{\prime}}= 700 ~{\rm GeV}$, $M_S= 2400 ~{\rm GeV}$, $M_L=1.0 ~{\rm TeV^2}$, $M_E=0.5 ~{\rm TeV^2}$. In FIG. \ref{DEBBp}, the two lines are shaped like parabolas. And most of the numerical results are within experimental limits.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dessj.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{4}$, $|d_e|$ is in the plane of $M_{L11}$ versus $M_{L22}$, ``$\blacksquare$'' represents $|d_e| < 1.1 \times 10^{-29}$ e.cm, ``$\circ$'' represents $|d_e| \geqslant 1.1 \times 10^{-29}$ e.cm.} \label{DESSJ}
\end{center}
\end{figure}
We select these parameters $M_{L11}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L22}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L33}(0.5\thicksim5.0 ~{\rm TeV^2})$, $T_E(-3000\thicksim3000 ~{\rm GeV})$, $M_E(0.5\thicksim5.0 ~{\rm TeV^2})$, and randomly scatter points.
With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{4}$. We plot $|d_e|$ in the plane of $M_{L11}$ versus $M_{L22}$ in Fig. \ref{DESSJ}. ``$\blacksquare$'' represents $|d_e| < 1.1 \times 10^{-29}$ e.cm, ``$\circ$'' represents $|d_e| \geqslant 1.1 \times 10^{-29}$ e.cm. In Fig. \ref{DESSJ}, We can see that there is a clear stratification. When $M_{L11}$ $>$ 1.0 $~{\rm TeV^2}$, $M_{L22}$ is in the vicinity of 1.4 $~{\rm TeV^2}$, $|d_e|$ is within the experimental limit. This can show that $M_{L11}$ is a sensitive parameters and $M_{L22}$ is a less sensitive parameter.
\subsection{the $\mu$ EDM}
In this section, the muon EDM is numerically studied. In FIG. \ref{DMUS}, setting $\theta_1$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_2$ = $\theta_{BL}$ = 0, and setting $\tan{\beta}=6$, $M_1= 1450~{\rm GeV}$, $M_2=2000~
{\rm GeV}$, $mu=500~{\rm GeV}$, $M_{BB^{\prime}}= 800 ~{\rm GeV}$, $M_S= -800 ~{\rm GeV}$, $M_L=1.0 ~{\rm TeV^2}$, $M_E=0.5 ~{\rm TeV^2}$. We study the influence of $\theta_S$ on the muon EDM. These solid line, dashed line correspond to $M_{BL}$ ($1200, 1500 ~{\rm GeV}$). We can see that the numerical result of the muon EDM increases as $M_E$ increases. The $\theta_S$ has great influence on the numerical results, because of that $M_S$ is related to the mass matrices of neutralino and charge Higgs.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dmus.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{3}$, the contributions to muon EDM varying with $M_E$ are plotted by the solid line, dashed line respectively corresponding to $M_{BL}$ = $(1200,1500)~{\rm GeV}$.} \label{DMUS}
\end{center}
\end{figure}
$\theta_{BB^{\prime}}$ is the new CP violating phase of the neutralino mass matrix. So, it make new physical contribution to the lepton EDM. With $\theta_1$ = $\theta_\mu$ = $\theta_2$ = $\theta_S$ = $\theta_{BL}$ = 0, the contributions to muon EDM varying with $M_{E22}$ are plotted by the solid line and dashed line respectively corresponding to $\tan\beta$ = (5,~6). In this part, we set $M_1= 1450 ~{\rm GeV}$, $M_2= 800 ~{\rm GeV}$, $mu=500~{\rm GeV}$, $M_{BL}= 1600 ~{\rm GeV}$, $M_{BB^{\prime}}= 800 ~{\rm GeV}$, $M_S= -800 ~{\rm GeV}$, $M_L=1.0 ~{\rm TeV^2}$, $M_E=0.5 ~{\rm TeV^2}$. In FIG. \ref{DMUBBP}, as $M_{E22}$ increasing, the numerical result decreases slowly, and the shapes of the two lines are similar.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dmuBBP.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_S$ = $\theta_{BL}$ = 0, and $\theta_{BB^{\prime}}$ = $\frac{\pi}{6}$, the contributions to muon EDM varying with $M_{E22}$ are plotted by the solid line, dashed line respectively corresponding to $\tan\beta$ = ($5,6$).} \label{DMUBBP}
\end{center}
\end{figure}
We choose these parameters $M_{L11}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L22}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L33}(0.5\thicksim5.0 ~{\rm TeV^2})$, $T_E(-3000\thicksim3000 ~{\rm GeV})$, $M_E(0.5\thicksim5.0 ~{\rm TeV^2})$, and randomly scatter points.
With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{4}$, we study $|d_\mu|$ in the plane of $M_{L33}$ versus $M_E$. In FIG. \ref{DMUSSJ}, ``$\blacksquare$'' represents $|d_\mu|$ $<$ $1\times10^{-24}$ e.cm, ``$\circ$'' represents $|d_\mu|$ $\geqslant$ $1\times10^{-24}$ e.cm. Delamination occurs when $M_E$ = $1.1 ~{\rm TeV^2}$, and the stratification is obvious. This can show that $M_{E}$ is a sensitive parameter and $M_{L33}$ is an insensitive parameter. These parameters are in a reasonable parameter space.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dmussj.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{4}$, $|d_{\mu}|$ is in the plane of $M_{L33}$ versus $M_{E}$, ``$\blacksquare$'' represents $|d_\mu|$ $<$ $1\times10^{-24}$ e.cm, ``$\circ$'' represents $|d_\mu|$ $\geqslant$ $1\times10^{-24}$ e.cm.} \label{DMUSSJ}
\end{center}
\end{figure}
\subsection{the $\tau$ EDM}
At present, the experimental upper bound of tau EDM is $|d^{exp}_{\tau}|$ $<$ $1.1 \times 10^{-17}$ e.cm, and it is largest one among bounds of the lepton EDMs. So, we study the tau EDM in this subsection. Setting $\tan{\beta}=6$, $M_1= 750 ~{\rm GeV}$, $mu=650 ~{\rm GeV}$, $M_{BL}=1800 ~{\rm GeV}$, $M_{BB^{\prime}}= 700 ~{\rm GeV}$, $M_S= 1400 ~{\rm GeV}$, $M_L=1.0 ~{\rm TeV^2}$, $M_E=1.0 ~{\rm TeV^2}$, and setting $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{5}$, we study the influence of $M_{L33}$ on $|d_\tau|$. In FIG. \ref{DTAUS}, the solid line and dashed line respectively correspond to $M_2$ = $(400,500 ~{\rm GeV})$ and their numerical results are all in the negative part. The two lines are increasing functions of $M_{L33}$, and $\theta_S$ has more obvious influence on numerical result of $|d_{\tau}|$. The maximum value of two lines can reach $5.0 \times 10^{-23}$ e.cm, and this value is 6 orders of magnitude smaller than the upper limit of the experiment.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dtaus.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_{BL}$ = 0, and $\theta_S$ = $\frac{\pi}{5}$, the contributions to tau EDM varying with $M_{L33}$ are plotted by the solid line, dashed line respectively corresponding to $M_2$ = $(400,500)~{\rm GeV}$.} \label{DTAUS}
\end{center}
\end{figure}
$\theta_{BL}$ is the new CP violating phase of $M_{BL}$ in the neutralino mass matrix. Setting $\tan{\beta}=6$, $M_1= 750 ~{\rm GeV}$, $M_2= 400 ~{\rm GeV}$, $M_{BL}=1800 ~{\rm GeV}$, $M_{BB^{\prime}}= 700 ~{\rm GeV}$, $M_S= 1400 ~{\rm GeV}$, $M_E=1.0 ~{\rm TeV^2}$, $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_S$ = 0, and $\theta_{BL}$ = $\frac{\pi}{6}$, the contributions to tau EDM varying with $M_L$ are plotted by the solid line and dashed line respectively corresponding to $mu$ = $(650,750 ~{\rm GeV}$). In FIG. \ref{DTAUBL}, we can see that $|d_\tau|$ decreases with the increase of $M_L$. The maximum value of these two lines can reach $|d_\tau|$ = $4.5 \times 10^{-23}$ e.cm.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dtauBL.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_S$ = $\theta_{BB^{\prime}}$ = 0, and $\theta_{BL}$ = $\frac{\pi}{6}$, the contributions to tau EDM varying with $M_L$ are plotted by the solid line, dashed line respectively corresponding to $mu$ = $(650,750)~{\rm GeV}$.} \label{DTAUBL}
\end{center}
\end{figure}
We select these parameters $M_{L11}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L22}(0.5\thicksim5.0 ~{\rm TeV^2})$, $M_{L33}(0.5\thicksim5.0 ~{\rm TeV^2})$, $T_E(-3000\thicksim3000 ~{\rm GeV})$, $\tan{\beta}(2\thicksim20)$, and randomly scatter points.
In Fig. \ref{DTAUSJ}, we study $|d_\tau|$ in the plane of $M_{L33}$ and $\tan{\beta}$ to see their influence. The varying regions of $M_{L33}$ and $\tan{\beta}$ are in the range $(0.5\thicksim5 ~\rm TeV^2)$ and $(2\thicksim20)$ respectively.``$\blacksquare$'' represents $|d_\tau|$ $<$ $1 \times 10^{-23}$ e.cm, ``$\circ$'' represents $|d_\tau|$ $\geqslant$ $1 \times 10^{-23}$ e.cm. When $\tan{\beta}$ = 6, stratification occurs, and the stratification is more obvious. This indicates that $\tan{\beta}$ is a sensitive parameter.
\begin{figure}[t]
\begin{center}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=2.9in]{dtausj.eps}
\end{minipage
\caption{With $\theta_1$ = $\theta_2$ = $\theta_\mu$ = $\theta_{BB^{\prime}}$ = $\theta_S$ = 0, and $\theta_{BL}$ = $\frac{\pi}{6}$, $|d_{\tau}|$ is in the plane of $M_{L33}$ versus $\tan{\beta}$, ``$\blacksquare$'' represents $|d_\tau|$ $<$ $1 \times 10^{-23}$ e.cm, ``$\circ$'' represents $|d_\tau|$ $\geqslant$ $1 \times 10^{-23}$ e.cm.} \label{DTAUSJ}
\end{center}
\end{figure}
\section{discussion and conclusion}
In the $U(1)_X$SSM, we calculate and analyze the one-loop and two-loop contributions to the lepton ($e,\mu,\tau$) EDMs. The effects of the CP violating phases $\theta_1$, $\theta_2$, $\theta_\mu$, $\theta_{BB^{\prime}}$, $\theta_S$, $\theta_{BL}$ to the lepton EDMs are researched. Among them, $\theta_{BB^{\prime}}$, $\theta_S$, $\theta_{BL}$ are all newly introduced ones. The experimental upper limit of electron EDM is $|d^{exp}_e|$ $<$ $1.1 \times 10^{-29}$ e.cm, which gives strict restrictions on the $U(1)_X$SSM parameter space. In the our used parameter space, the numerical result of $|d_e|$ can be controlled below the experimental limit. In our study, the largest numerical results of $\mu$ EDM and $\tau$ EDM are about $2.8 \times 10^{-24}$ e.cm and $5.0 \times 10^{-23}$ e.cm respectively. They are all in a reasonable parameter space and do not exceed the upper limit of the experiment.
Our numerical results mainly obey the rule $d_e / d_\mu / d_\tau$ $\thicksim$ $m_e / m_\mu / m_\tau$. In FIG. \ref{DEBL}, when $\theta_{BL}$ = $\frac{\pi}{4}$, $M_L$ has a more obvious impact on electron EDM, and the influences of $\theta_{BL}$ on electron EDM is also more obvious.
In addition, the influence of the CP-violating phases $\theta_{S}$ and $\theta_{BB^{\prime}}$ on lepton EDMs are also obvious. In FIG. \ref{DMUS}, when $\theta_{S}$ = $\frac{\pi}{3}$, the value of the muon EDM increases as $M_E$ increases (the numerical results are all negative), The $\theta_{S}$ has great influence on the numerical results, because of that $M_S$ is related to the mass matrices of neutralino and charge Higgs. In FIG. \ref{DMUBBP}, when $\theta_{BB^{\prime}}$ = $\frac{\pi}{6}$, the two lines (solid line, dashed line) are about the decreasing function of $M_{E22}$. The above parameters ($M_L$,~$M_E$) are all elements on the diagonal of the mass matrix, so their corresponding results are all decoupled, such as FIG. \ref{DEBL}, FIG. \ref{DES}, FIG. \ref{DMUS}, FIG. \ref{DMUBBP}, FIG. \ref{DTAUS}, FIG. \ref{DTAUBL}. In FIG. 12, We can get that $|d_\tau|$ increases with the increase of $\tan\beta$. If we use the method of mass insertion \cite{massi} to analyze the results, it is intuitive to find that $\tan\beta$ is proportional to lepton EDMs. We have also performed some random spot operations on lepton EDMs. The randomly scattered pictures have obvious stratification, also help us to find a reasonable parameter space. As the accuracy of technology improves, lepton EDMs may be detected in the near future.
\section{acknowledgments}
This work is supported by National Natural Science Foundation of China(NNSFC)(Nos. 11535002, 11705045), Natural Science Foundation of Hebei Province (A2020201002) and the youth top-notch talent support program of the Hebei Province.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,535
|
Q: svg pseudo background-image file not showing on live site but works on localhost why? I have simple svg arrow but it doesn't show when I use as pseudo background-image online but it works fine on my localhost: any idea?
span a:after {
content: "";
background-image: url(arrow_right.svg);
width: 24px;
height: 18px;
display: inline-block;
margin: 0px 0 0 10px;
position: relative;
top: 3px;
}
svg:
<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
viewBox="0 0 24 18" enable-background="new 0 0 24 18" xml:space="preserve">
<path fill="#F9C32C" d="M19.8,10.1l-6.6,6.4l1.5,1.5L24,9l-9.3-9l-1.5,1.5l6.6,6.4H0v2.1H19.8z"/>
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A: Add content-type=image/svg+xml to your .htaccess file at the root instead of content-type=text/xml
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,484
|
/*
* Do not modify this file. This file is generated from the glacier-2012-06-01.normal.json service model.
*/
using System;
using System.Collections.Generic;
using System.Xml.Serialization;
using System.Text;
using System.IO;
using Amazon.Runtime;
using Amazon.Runtime.Internal;
namespace Amazon.Glacier.Model
{
/// <summary>
/// Describes the options for a range inventory retrieval job.
/// </summary>
public partial class InventoryRetrievalJobDescription
{
private string _endDate;
private string _format;
private string _limit;
private string _marker;
private string _startDate;
/// <summary>
/// Gets and sets the property EndDate.
/// <para>
/// The end of the date range in UTC for vault inventory retrieval that includes archives
/// created before this date. A string representation of ISO 8601 date format, for example,
/// 2013-03-20T17:03:43Z.
/// </para>
/// </summary>
public string EndDate
{
get { return this._endDate; }
set { this._endDate = value; }
}
// Check to see if EndDate property is set
internal bool IsSetEndDate()
{
return this._endDate != null;
}
/// <summary>
/// Gets and sets the property Format.
/// <para>
/// The output format for the vault inventory list, which is set by the <b>InitiateJob</b>
/// request when initiating a job to retrieve a vault inventory. Valid values are "CSV"
/// and "JSON".
/// </para>
/// </summary>
public string Format
{
get { return this._format; }
set { this._format = value; }
}
// Check to see if Format property is set
internal bool IsSetFormat()
{
return this._format != null;
}
/// <summary>
/// Gets and sets the property Limit.
/// <para>
/// Specifies the maximum number of inventory items returned per vault inventory retrieval
/// request. This limit is set when initiating the job with the a <b>InitiateJob</b> request.
///
/// </para>
/// </summary>
public string Limit
{
get { return this._limit; }
set { this._limit = value; }
}
// Check to see if Limit property is set
internal bool IsSetLimit()
{
return this._limit != null;
}
/// <summary>
/// Gets and sets the property Marker.
/// <para>
/// An opaque string that represents where to continue pagination of the vault inventory
/// retrieval results. You use the marker in a new <b>InitiateJob</b> request to obtain
/// additional inventory items. If there are no more inventory items, this value is <code>null</code>.
/// For more information, see <a href="http://docs.aws.amazon.com/amazonglacier/latest/dev/api-initiate-job-post.html#api-initiate-job-post-vault-inventory-list-filtering">
/// Range Inventory Retrieval</a>.
/// </para>
/// </summary>
public string Marker
{
get { return this._marker; }
set { this._marker = value; }
}
// Check to see if Marker property is set
internal bool IsSetMarker()
{
return this._marker != null;
}
/// <summary>
/// Gets and sets the property StartDate.
/// <para>
/// The start of the date range in UTC for vault inventory retrieval that includes archives
/// created on or after this date. A string representation of ISO 8601 date format, for
/// example, 2013-03-20T17:03:43Z.
/// </para>
/// </summary>
public string StartDate
{
get { return this._startDate; }
set { this._startDate = value; }
}
// Check to see if StartDate property is set
internal bool IsSetStartDate()
{
return this._startDate != null;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,616
|
Das Wasserkraftwerk Isola Serafini wurde bei Monticelli d'Ongina in der, Provinz Piacenza, Italien im Po (Fluss) errichtet. Zu seinem Damm gehört die ausgedehnteste Torstruktur in Italien. Am Ufer des Stausees steht das stillgelegte Kernkraftwerk Caorso.
Einzelnachweise
Wasserkraftwerk in Europa
Isola Serafini
Bauwerk in der Emilia-Romagna
Po (Fluss)
Monticelli d'Ongina
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,031
|
{{> header}}
<div class='cancel-trip'>
<div class='cancel-trip-content'>
</div>
</div>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,039
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\section{Introduction}\label{sec:intro}
Expanders can be defined either combinatorially or algebraically. In
the combinatorial definition a graph $G=(V,E)$ is a
$(K_{max},c)$-expander, if every set $A \subseteq V$ of cardinality
at most $K_{max}$ has at least $c|A|$ neighbors. In the algebraic
definition we view $G$ as an operator defined by the normalized
adjacency matrix of the graph, and we say $G$ is a $\ol{\lambda}$-expander
if the spectral gap between the first and second largest eigenvalues
(in absolute value) of this operator is at least $1-\ol{\lambda}$.
We are interested in a sequence of graphs $\set{G_n}$, with an
increasing number of vertices, but constant degree $D$. The best
possible combinatorial expansion such a family can have is about
$D-2$, and the best possible algebraic expansion is about ${2
\over \sqrt{D}}$ (see \cite{N91}). The algebraic and combinatorial
definitions are closely related. Expanders with constant spectral
gap have constant combinatorial expansion and vice versa
\cite{AM85,A86}. However, this equivalence is not tight, and, in
particular, graphs with maximal spectral gap may have
combinatorial expansion not more than half the degree \cite{K95},
and graphs with almost optimal combinatorial expansion (close to
the degree) may have non-optimal spectral gap.
Both notions have proven extremely useful in computer science and
elsewhere. Often, the spectral gap is used (e.g., whenever a random
walk on the expander is used), but sometimes combinatorial
expansion close to the degree is needed (e.g., in the error
correcting codes of \cite{SS96}).
Thus, for decades, a major goal of computer science has been
constructing these marvelous graphs \emph{explicitly}. Pinsker
\cite{P73} was the first to observe that \emph{non-explicitly},
constant degree expanders with very good combinatorial expansion
exist. Several explicit constructions of constant degree, algebraic
expanders with some constant (bounded away from zero) spectral gap
were given, e.g., in \cite{M73,GG81,JM87}. Lubotzky, Philips and
Sarnak \cite{LPS88} and Margulis \cite{M88} gave the first Ramanujan
graphs, i.e., a family $\set{G_n}$ of degree $D$ graphs with $\ol{\lambda}$
approaching the optimal value. All the above graphs are Cayley
graphs and their analysis is algebraic. More recently, \cite{RVW00}
gave a more combinatorial construction, that was used in
\cite{CRVW02} to construct an explicit construction of graphs with
almost optimal \emph{combinatorial} expansion.
We refer the interested reader to the excellent survey paper
\cite{HLW06} for a comprehensive treatment of expander graphs, their
construction and applications.
\subsection{Quantum expanders}
Expanders are often thought of as \emph{combinatorial} objects. In
this view, expanders are sparse graphs that have combinatorial
expansion properties similar to random graphs. It is difficult to
see in this view how to generalize the notion to the quantum world.
However, most expander constructions, and many of the applications
that use expanders, treat expanders as algebraic objects, i.e., the
graph $G=(V,E)$ is translated to a linear mapping ${\cal G}$ from
some vector space ${\cal V}$ to itself. Let us describe how this is
done. Say $G=(V,E)$ is a graph. We translate $V$ to a vector space
${\cal V}$ of dimension $|V|$, with a basis vector $\ket{v}$ for
each $v \in V$. A probability distribution over $V$ then translates
to a vector $\sum_{v} p_v \ket{v}$ in this space, with $0 \le p_v
\le 1$ and $\sum_v p_v=1$. The graph $G$ is translated to the linear
operator ${\cal G}$ from ${\cal V}$ to ${\cal V}$ which is defined
by the normalized adjacency matrix of $G$. ${\cal G}$ is therefore a
linear mapping ${\cal G}: {\cal V} \to {\cal V}$ that can be
classically implemented, and maps probability distributions to
probability distributions.
We extend the algebraic definition to the quantum setting. A general
classical state is a classical probability distribution over the
standard basis $\set{\ket{v}}$ of ${\cal V}$, i.e., vectors of the
form $\sum_v p_v \ket{v}$ as above. A general quantum state is a
\emph{density matrix} $\rho=\sum p_v \ketbra{\psi_v}{\psi_v}$, with
$0 \le p_v \le 1$, $\sum p_v=1$ and $\set{\psi_v}$ being \emph{some}
orthonormal basis of ${\cal V}$. In the classical world we had a
linear operator ${\cal G}: {\cal V} \to {\cal V}$. In the quantum
world a feasible quantum state is a matrix over ${\cal V}$, i.e., an
element of $L({\cal V})$, where $L({\cal V})$ is the set of linear
operators (matrices) over ${\cal V}$. We look for a linear
transformation $E: L({\cal V}) \to L({\cal V})$. Such a
transformation is called a \emph{super-operator}. We want in
addition that $E$ can be implemented by some physical process, and
this also ensures that $E$ maps density matrices to density
matrices. Such a linear operators $E$ is called in the literature an
\emph{admissible} super-operators.
We now turn to the regularity condition. Any (directed or
undirected) $D$-regular graph $G$ can have its edges labeled with
$1$ to $D$ such that each label $d \in [D]$ defines a
\emph{permutation} mapping. We define:
\begin{deff}
\label{def:qunatum-degree} We say an admissible super-operator $E:
L({\cal V}) \to L({\cal V})$ is \emph{$D$-regular} if $E={1 \over D}
\sum_d E_d$, and for each $d \in [D]$, $E_d(X)=U_d X U_d^\dagger$
for some unitary transformation $U_d$ over ${\cal V}$.
\end{deff}
In fact, for many classical constructions the edge labeling is
explicitly described in the construction, and in particular this is
always true whenever $G$ is a Cayley graph. This property was also
exploited in several constructions (e.g., in \cite{RVW00,CRVW02}).
Intuitively, a quantum expander is an admissible super-operator $E$
that has a spectral gap. We normalize the operator $E$ so that its
largest eigenvalue is $1$. As in the classical case we want the
eigenvector of eigenvalue one to be the completely mixed state. We
require that all other eigenvalues have a much smaller absolute
value. In general, however, $E$ need not be normal. This already
happens in the classical setting whenever we deal with directed
graphs. In such a case we need to replace eigenvalues with singular
values. Equivalently, we define:
\begin{deff}
\label{def:qexpander} An admissible superoperator $E:L(V) \to L(V)$
is a $(D,\ol{\lambda})$ expander if $E$ is $D$-regular and:
\begin{itemize}
\item
$E(\tilde{I})=\tilde{I}$ and the eigenspace of eigenvalue $1$ has dimension $1$.
\item
For any $A \in L(V)$ that is orthogonal to $\tilde{I}$ (with respect to
the Hilbert-Schmidt inner product, i.e. $\mathop{\rm Tr}\nolimits(A \tilde{I}) = 0$) it holds
that $\norm{E(A)}_2 \le \ol{\lambda} \norm{A}_2$.
\end{itemize}
A quantum expander is \emph{explicit} if $E$ can be implemented by a
polynomial size circuit.
\end{deff}
Equivalently, we could have replaced the second condition with the
requirement that all singular values of $T$ other than the largest
one (which is $1$) are smaller than $\ol{\lambda}$.
\subsection{Are there any non-trivial quantum expanders?}
This is indeed a good question, and a major goal of this paper. A
first natural attempt is converting a good classical Cayley
expander, to a quantum super-operator. This indeed can be done, and
the resulting super operator $T: L({\cal V}) \to L({\cal V})$ is
analyzed in Section \ref{sec:T}. The analysis there shows that $T$
has $|V|$ eigenspaces, each of dimension $|V|$, with eigenvalues
$\overrightarrow{\lambda}=(\lambda_1=1,\ldots,\lambda_{|V|})$, where
$\overrightarrow{\lambda}$ is the spectrum of the Cayley graph. In
particular, the eigenspace of eigenvalue $1$ has dimension $|V|$
instead of dimension $1$.
Never the less, Ambainis and Smith obtained the following quantum
expander that is implicit in their work:
\begin{thm}
\label{thm:AS} \cite{AS04} There exists an explicit $({ \log^2 N
\over \ol{\lambda}^2},\ol{\lambda})$ quantum expander $E:L(V) \to L(V)$, where
$N=dim(V)$.
\end{thm}
Their quantum expander is based on the classical Cayley expander
over the Abelian group $\mathbb{Z}_2^n$. As explained before, taking
the quantum analogue of the classical expander is not enough, and
Ambainis and Smith obtain their result using a clever trick,
essentially working over $\mathbb{{F}}_4^n$ rather than $\mathbb{Z}_2^n$.
The main problem with Abelian groups is that it is impossible to get
a constant degree Cayley expander over them \cite{K84,AR94}. This is
reflected in the $O(\log N)$ term in Theorem \ref{thm:AS}. There are
constant degree, Ramanujan Cayley graphs, i.e., Cayley graphs that
achieve the best possible relationship between the degree and the
spectral gap, but they are built over non-Abelian groups. If one
wants to get a constant degree quantum expander, then he is forced
to work over non-Abelian groups. Can one get constant degree quantum
expanders at all?
Our main construction starts with the constant degree Ramanujan
expander of \cite{LPS88}. This expander is a Cayley graph over the
non-Abelian group $\mbox{PGL}(2,q)$. We prove:
\begin{thm}\label{thm:PGL-expander}
There exists a $(D=O({1 \over \ol{\lambda}^4}),\ol{\lambda})$ quantum expander.
\end{thm}
Our construction is not explicit in the sense that it uses the
Fourier transform over \mbox{PGL}(2,q), which is not known to have an efficient
implementation (see \cite{LR92} for a non-trivial, but still not
fast enough, algorithm).
The \mbox{PGL}(2,q){} quantum expander is as follows: we take two steps on
the classical expander graph, with a basis change between the two
steps. The basis change is a carefully chosen refinement of the
Fourier transformation that maps the standard basis $\ket{g}$
to the basis of the irreducible, invariant subspaces of $\mbox{PGL}(2,q)$.
Intuitively, in the Abelian case this basis change corresponds to
dealing with both the bit and the phase levels, and is similar to
the construction of quantum error correcting codes by first applying
a classical code in the standard basis and then in the Fourier
basis. However, this intuition is not as clear in the non-Abelian
case. Furthermore, in the non-Abelian case not every Fourier
transform is good. In this work we single out a natural algebraic
property we need from the underlying group that is sufficient for
proving the spectral gap of the construction. We then prove that
$\mbox{PGL}(2,q)$ respects this property.
We mention that there are also explicit, constant degree
(non-Ramanujan) Cayley expanders over $\mathcal{S}_n$ and $\mathcal{A}_n$ \cite{K05}.
Also, there is an efficient implementation of the Fourier transform
over $\mathcal{S}_n$ \cite{B97}. We do not know, however, whether $\mathcal{S}_n$ (or
$\mathcal{A}_n$) respect our additional property. We discuss this in more
detail in Section \ref{sec:Sn-construction}.
To summarize, Ambainis and Smith showed that good
\emph{poly-logarithmic-degree} quantum expanders exist, and their
construction is \emph{explicit}. Theorem \ref{thm:PGL-expander}
shows that good \emph{constant} degree quantum expanders
\emph{non-explicitly} exist (with a degree that is the square of
the degree of a Ramanujan graph). Recently, we showed together
with Oded Schwartz \cite{BST07} that one can use Theorem
\ref{thm:PGL-expander} with a Zig-Zag like construction, to obtain
an \emph{explicit}, \emph{constant} degree quantum expander.
Finally, we show a lower bound on the best achievable spectral
gap of quantum expanders.
\begin{thm} \label{thm:qexpander-lower-bound}
Any $(D,\ol{\lambda})$ quantum expander satisfies $\ol{\lambda} \ge \frac{2}{3
\sqrt{3D}}$.
\end{thm}
The lower bound differs by a constant from the tight lower bound
known on classical expanders.
\subsection{What are quantum expanders good for?}
The first application of quantum expanders was given by Ambainis and
Smith themselves. They used these expanders to construct short
quantum one-time pads. Loosely speaking, they showed how two parties
sharing a random bit string of length $n + O(\log n)$ can
communicate an $n$ qubit state such that any eavesdropper cannot
learn much about the transmitted state. (A subsequent work by
\cite{DN06} showed how to remove the $O(\log n)$ term.)
In this paper we show another application of quantum expanders.
Watrous \cite{W02} defined the class of quantum statistical zero
knowledge languages (\mbox{QSZK}). \mbox{QSZK}{} is the class of all languages
that have a quantum interactive proof system, along with an
efficient simulator that produces transcripts that for inputs in the
language are statistically close to the correct ones (for the
precise details see \cite{W02,W06}).
Watrous defined the Quantum State Distinguishability promise problem
($QSD_{\alpha,\beta}$):
\begin{boxit}
\textbf{Input:} Quantum circuits $Q_0, Q_1$.
\textbf{Accept:} If $\trnorm{\ket{Q_0} - \ket{Q_1}} \ge \beta$.
\textbf{Reject:} If $\trnorm{\ket{Q_0} - \ket{Q_1}} \le \alpha$.
\end{boxit}
where the notation $\ket{Q}$ denotes the mixed state obtained by
running the quantum circuit $Q$ on the initial state $\ket{0^n}$
and tracing out the non-output qubits \footnote{Here we assume
that a quantum circuit also designates a set of output qubits.},
and $\trn{A}=\mathop{\rm Tr}\nolimits{|A|}$ is the quantum analogue of the classical
$\ell_1$-norm (and so in particular $\trn{\rho_1-\rho_2}$ is the
quantum analogue of the classical variational distance of two
probability distributions).
Watrous showed $\mbox{QSD}_{\alpha,\beta}$ is complete for
honest-verifier-$\mbox{QSZK}$ ($\mbox{QSZK}_{\text{HV}}$) when $0 \le \alpha <
\beta^2 \le 1$. He further showed that $\mbox{QSZK}_{\text{HV}}$ is closed
under complement, that any problem in $\mbox{QSZK}_{\text{HV}}$ has a $2$
message proof system and a $3$ message public-coin proof system and
also that $\mbox{QSZK} \subseteq \mbox{PSPACE}$. Subsequently, in \cite{W06}, he
showed that $\mbox{QSZK}_{\text{HV}} = \mbox{QSZK}$.
The above results have classical analogues. However, in the
classical setting there is another canonical complete problem, the
Entropy Difference problem ($\mbox{ED}$). There is a natural quantum
analogue to \mbox{ED}, the Quantum Entropy Difference problem (\mbox{QED}), that
we now define:
\begin{boxit}
\textbf{Input:} Quantum circuits $Q_0, Q_1$.
\textbf{Accept:} If $S(\ket{Q_0}) - S(\ket{Q_1}) \ge {1 \over 2}$.
\textbf{Reject:} If $S(\ket{Q_1}) - S(\ket{Q_0}) \ge {1 \over 2}$.
\end{boxit}
where $S(\rho)$ is the Von-Neumann entropy of the mixed state
$\rho$.\footnote{A density matrix $\rho$ is positive semi-definite
and has trace $1$. Therefore its eigenvalues are all non-negative
and sum up to $1$, and can be thought of as defining a probability
distribution. The Von-Neumann entropy of $\rho$ is the Shannon
entropy of the eigenvalues of $\rho$.} We show that $\mbox{QED}$ is
$\mbox{QSZK}$-complete. We mention that for this purpose the expanders of
Ambainis and Smith given in Theorem \ref{thm:AS} suffice.
The problem $\mbox{QED}$ is very natural from a physical point of view.
For example, a common way of measuring the amount of entanglement
between registers $A$ and $B$ in a pure state $\psi$ is by the
Von-Neumann entropy of $\mathop{\rm Tr}\nolimits_B(\ketbra{\psi}{\psi})$ \cite{PR97}. Now
suppose we are given two circuits $Q_0$ and $Q_1$, both acting on
the same initial pure-state $\ket{0^n}$, and we want to know which
circuit produces more entanglement between $A$ and $B$. Our result
shows that this problem is $\mbox{QSZK}$--complete. This, in particular,
shows that the harder problem of \emph{estimating} the amount of
entanglement between two registers in a given pure-state is
$\mbox{QSZK}$--hard.
We believe these two applications are a good indication to the
usefulness of this notion. We expect that with time other
applications will be found.
Our proof that $\mbox{QED}$ is $\mbox{QSZK}$-complete uses a quantum variant of
classical balanced extractors. We explain this variant in Section
\ref{sec:extractors}. We show there that good balanced quantum
extractors exist. Surprisingly, we believe that unlike the classical
case, unbalanced quantum extractors do not exist.
\subsection{Summary and organization}
In classical computation there is a long line of research studying
"conductors": objects that manipulate their source entropy, using
few independent random bits. This research resulted in beautiful
constructions of expanders and extractors, and an amazing variety
of applications. We initiate the study of such "conductors"
manipulating the entropy of \emph{quantum} systems.
On the one-hand we show that expander-based constructions generalize
to the quantum setting (with effort, and not always, but at least in
some important cases). On the other hand, we believe all the huge
body of work relating classical extractors, condensers and such that
map a huge universe to a much smaller universe, is not likely to
have a quantum analogue (see Section
\ref{sec:extractors}). We think this study
deserves interest at its own right.
We also show two neat applications for quantum expanders. One, that
was already given in \cite{AS04} and a new one that we give here: we
characterize the complexity of approximating entropies. This proof
generalizes classical ideas, together with new technical work that
is needed for the quantum setting.
The paper is organized as follows. After the preliminaries (Section
\ref{sec:prel}), we give an intuitive exposition of our constant
degree expander, and the analysis, in Section
\ref{sec:main:quantum-expanders-from-non-abelian}. A complete
treatment is given in Section
\ref{sec:quantum-expanders-from-non-abelian} in the Appendix. In
Section \ref{sec:extractors} we discuss extractors, and discuss why
we believe \emph{unbalanced} quantum extractors are not useful. The
final section is devoted to proving the completeness of $\mbox{QED}$ in
$\mbox{QSZK}$. Here, again, we give an intuitive exposition in the main
text, with the formal details in the Appendix.
\section{Preliminaries}\label{sec:prel}
We first define the classical Renyi entropy. Let
$P=(p_1,\ldots,p_m)$ be a classical probability distribution.
The \emph{Shannon entropy} of $P$ is $H(P)=\sum_{i=1}^m p_i \lg
\frac{1}{p_i}$.
The \emph{min-entropy} of $P$ is $H_{\infty} (P)=\min_i \lg
\frac{1}{p_i}$.
The \emph{Renyi entropy} of $P$ is $H_2 (P)=\lg \frac{1}{\mathop{\rm Col}\nolimits(P)}$,
where $\mathop{\rm Col}\nolimits(P)=\sum p_i^2$ is the collision probability of the
distribution defined by $\mathop{\rm Col}\nolimits(P)=\Pr_{x,y} [x=y]$ when $x,y$ are
sampled from $P$.
Now let $\rho \in D(V)$ be a density matrix (where $V$ is a Hilbert
space, $L(V)$ is the set of linear operators over $V$ and $D(V)$ is
the set of positive semi-definite operators in $L(V)$ with trace
$1$, i.e., all density matrices over $V$). Let
$\alpha=(\alpha_1,\ldots,\alpha_N)$ be the set of eigenvalues of
$\rho$. Since $\rho$ is positive semi-definite, all these
eigenvalues are non-negative. Since $\mathop{\rm Tr}\nolimits(\rho)=1$ their sum is $1$.
Thus we can view $\alpha$ as a classical probability distribution.
The \emph{von Neumann entropy} of $\rho$ is $S(\rho)=H(\alpha)$.
The \emph{min-entropy} of $\rho$ is $H_{\infty} (\rho)=H_{\infty}
(\alpha)$.
The \emph{Renyi entropy} of $\rho$ is $H_2(\rho)=H_2(\alpha)$.
The analogue of the collision probability is simply $ \mathop{\rm Tr}\nolimits({\rho^2})
= \sum_i \alpha_i^2 =||\rho||_2^2$. We remark that for any
distribution $P$, $H_{\infty}(P) \le H_2(P) \le H(P)$ and $2
H_{\infty}(P) \ge H_2(P)$.
The \emph{statistical difference} between two classical
distributions $P=(p_1,\ldots,p_m)$ and $Q=(q_1,\ldots,q_m)$ is
$\text{SD}(P,Q) = {1 \over 2} \sum_{i=1}^m |p_i - q_i|$, i.e., half the
$\ell_1$ norm of $P-Q$. This can be generalized to the quantum world
by defining the trace-norm of a matrix $X \in L(V)$ to be
$\trnorm{X}=\mathop{\rm Tr}\nolimits(|X|)$, where $|X|=\sqrt{X X^{\dagger}}$,
and defining the \emph{trace distance} between density matrices
$\rho$ and $\sigma$ to be ${1 \over 2} \trnorm{\rho-\sigma}$.
\section{Quantum expanders from non-Abelian Cayley graphs}
\label{sec:main:quantum-expanders-from-non-abelian}
As we said before, our quantum expander takes two steps on a Cayley
expander (over the group \mbox{PGL}(2,q)) with a basis change between each of
the steps, and the basis change is a carefully chosen
transformation. In this section we give a bird's view of the proof.
We focus on the ideas, obstacles and solutions, and try to give an
informal presentation.
Our starting point is generalizing a single step on a Cayley graph
to the quantum setting. We fix an arbitrary (Abelian or non-Abelian)
group $G$ of order $N$, and a subset $\Gamma$ of group elements
closed under inverse. The \emph{Cayley graph} associated with
$\Gamma$, $C(G,\Gamma)$, is a graph over $N$ vertices, with an edge
between $(g_1,g_2)$ iff $g_1=g_2 \gamma$ for some $\gamma \in
\Gamma$. Rather then thinking of the Cayley graph as a graph, we
prefer to think of it as the linear operator over $\mathbb{C}[G]$ associated
with the adjacency matrix of $G$, where $\mathbb{C}[G]$ is the vector space
spanned by the basis elements $\ket{g}$ for each $g \in G$. I.e., it
is the linear operator $M=\frac{1}{|\Gamma|} \sum_{\gamma \in
\Gamma} \ketbra{x \gamma}{x}$.
We now define our basic superoperator $T: L(\mathbb{C}[G]) \to L(\mathbb{C}[G])$. The
superoperator has a register $R$ of dimension $|\Gamma|$ that is
initialized at $\ket{\ol{0}}$. It does the following:
\begin{itemize}
\item
It first applies Hadamard on register $R$ (getting into the density
matrix ${1 \over |\Gamma|} \rho \otimes \sum_{\gamma,\gamma' \in
\Gamma} \ketbra{\gamma}{\gamma '}$).
\item
Then, it applies the unitary transformation $Z: \ket{g,\gamma} \to
\ket{g \gamma,\gamma}$. This transformation is a permutation over
the standard basis, and hence unitary. It is also classically easy
to compute in both directions, and therefore has an efficient
quantum circuit.
\item
Finally, it measures register $R$.
\end{itemize}
Thus we have: $T(\rho) = \mathop{\rm Tr}\nolimits_{R} [~Z (I \otimes H) (\rho \otimes
\ketbra{\ol{0}}{\ol{0}}) (I \otimes H) Z^\dagger ~]$. It can be
easily checked that over "classical" states (a density matrix $\rho$
that is diagonal in the standard basis) $T$ coincides with $M$.
Also, by definition, $T$ is $|\Gamma|$-regular.
The first thing to figure out is the eigenspace structure of the
super-operator $T$. This turns out to be as follows. $T$ has $N$
orthogonal eigen-spaces, each of dimension $N$, and the eigenvalues
$\lambda_1,\ldots,\lambda_N$ are those of $M$ (the orthogonality is
under the inner-product of $L(\mathbb{C}[G])$ defined by $\la A | B \ra =
\mathop{\rm Tr}\nolimits(A B^\dagger)$). In particular, if we start with a good Cayley
graph where $\lambda_1=1$ and all other eigenvalues have absolute
value at most $\ol{\lambda}$, then $T$ has an eigenspace $W_1$ of dimension
$N$ with eigenvalue $1$, and all other eigenvalues have absolute
value at most $\ol{\lambda}$. The fact that the dimension of $W_1$ is
larger than $1$ is not good for us, because it means that $T$ has no
spectral gap.
So, now we take a closer look at $W_1$ and we discover that it is
spanned by $\set{A_g~|~ g \in G}$ where $A_g=\sum_x \ketbra{gx}{x}$.
These operators $A_g$ are what is called the \emph{regular
representation} of $G$. Namely, if we denote $\rho_\mathrm{reg}(g)=A_g$, then
$\rho_\mathrm{reg}:G \to L(\mathbb{C}[G])$ is a group homomorphism (namely, $\rho_\mathrm{reg}(g_1 \cdot
g_2)=\rho_\mathrm{reg}(g_1) \cdot \rho_\mathrm{reg}(g_2)$). Furthermore, a basic theorem of
representation theory says that there is a basis change under which
all the operators $A_g=\rho_\mathrm{reg}(g)$ simultaneously block-diagonalize,
with the blocks corresponding to the irreducible representations of
$G$. This (non-unique) basis change is called the Fourier transform
of $G$.
Let us first consider the case where $G$ is Abelian, and let $e$
denote the identity element in $G$. In this case all the irreducible
representations of $G$ have dimension one, and the Fourier transform
$U$ simultaneously diagonalizes all the operators $A_g=\rho_\mathrm{reg}(g)$. The
elements $\set{A_g=\rho_\mathrm{reg}(g)}$ form an orthonormal basis of $W_1$.
Doing the basis change, they all become diagonal, i.e., "classical"
states. Furthermore, $A_e=\rho_\mathrm{reg}(e)=I$ is mapped to $I$ (as is true in
any basis change) and all other basis elements are mapped to
orthogonal states (as $U$ is unitary). We therefore expect that
applying $T$ again now, is equivalent to applying $M$ on the
classical state, and will result in a unique eigenvector of
eigenvalue $1$, with all other eigenvalues being at most $\ol{\lambda}$.
So our (Abelian) quantum expander is as follows. We let $U$ be the
Fourier transform over $G$, and the quantum expander is the
superoperator
$$E(\rho)= T( U T(\rho) U^\dagger).$$
A simple check shows that $E$ is indeed a $\ol{\lambda}$--expander, and its
spectral gap is the same as that of $G$. Also, clearly, $E$ is
$|\Gamma|^2$-regular.
We now turn to the non-Abelian case. Here most irreducible
representations have dimension larger than $1$, and as a result the
basis change does not diagonalize all $A_g=\rho_\mathrm{reg}(g)$, but rather just
block-diagonalizes them, with blocks corresponding to the
irreducible representations. In particular, doing the Fourier
transform does not map $A_g=\rho_\mathrm{reg}(g)$ to "classical" states. Never
the less, this does not necessarily mean that the above approach
fails. In fact, it turns out that a sufficient requirement for a
good basis change is that for any $g_1 \ne e$ and any $g_2$, it
holds that
\begin{eqnarray}
\label{eqn:main:propertyU} \mathop{\rm Tr}\nolimits(U \rho_\mathrm{reg}(g_1) U^\dagger \rho_\mathrm{reg}(g_2)) &=&
0.
\end{eqnarray}
Intuitively, we can do the analysis separately for elements in $W_1$
and elements in $W_1^{\bot}$ - the space perpendicular to $W_1$
(this is technically more complicated, see Lemma \ref{lem:col-sum}).
Elements in $W_1^\bot$ are immediately shortened by the first
application of $T$. Elements in $\text{Span} \set{A_g=\rho_\mathrm{reg}(g) ~|~ g
\neq e}$ are kept in place by the first application of $T$, but are
mapped to $W_1^\bot$ by the basis change, and therefore are
shortened by the second application of $T$. Together, if $U$ is a
good basis change then $E(\rho)= T( U T(\rho) U^\dagger)$ is a
$\ol{\lambda}$--expander.
But does a good basis change always exist?
We consider the dihedral group as an illuminating example. The
dihedral group has irreducible representations of dimension $2$ (and
a few of dimension $1$). The dihedral group also has a cardinality
two subgroup $H=\set{e,s}$, where $s$ is the reflection element. The
Fourier transform associates the eigen-spaces of the irreducible
representations, to elements of $G$. Now, imagine that we associate
the dimension-$2$ blocks with cosets of $H$. A moment of thought
reveals that if $g_2 \not \in H$ then Equation
(\ref{eqn:main:propertyU}) is satisfied! This is because $A=U
\rho_\mathrm{reg}(g_1) U^\dagger$ has non-zero elements only on the $2$ by $2$
blocks, while $B=\rho_\mathrm{reg}(g_2)=\sum_x \ketbra{g_2 x}{x}$ has non-zero
elements only outside these $2$ by $2$ blocks, and so the inner
product $\la A | B \ra = \mathop{\rm Tr}\nolimits(A B^\dagger) = \sum_{i,j} A_{i,j}
\ol{B_{i,j}}$ must be zero.
We need also to consider the case where $g_2 \in H=\set{e,s}$. If
$g_2=e$ then $\mathop{\rm Tr}\nolimits(U \rho_\mathrm{reg}(g_1) U^\dagger \rho_\mathrm{reg}(g_2))=\mathop{\rm Tr}\nolimits(\rho_\mathrm{reg}(g_1))$
and the analysis is simple. We are left with the case $g_2=s$.
Recall that $\mathop{\rm Tr}\nolimits(A B^\dagger) = \sum_{i,j} A_{i,j} \ol{B_{i,j}}$. We
can interpret the expression $\mathop{\rm Tr}\nolimits(U \rho_\mathrm{reg}(g_1) U^\dagger \rho_\mathrm{reg}(s))$ as
the sum of all entries $i,j$ of $U \rho_\mathrm{reg}(g_1) U^\dagger$ that belong
to the set $\text{P}=\set{(is,i)}$. We now use the fact that each
irreducible representation appears in the regular representation
with multiplicity that equals its dimension. In matrix language this
means that for each dimension $2$ irreducible representation, there
are two corresponding blocks in the decomposition, and the entries
in these two blocks can be made \emph{identical} (see Section
\ref{sec:rep-background} for more background on representation
theory). As the blocks correspond to cosets of $H$, multiplication
by $g_2=s$ has the same effect in the two cosets. I.e., an entry of
one block is in $\text{P}$ and is added to the sum, iff the
corresponding entry in the other block is also in $\text{P}$ and is
also added to the sum. We can therefore force a zero sum, by forcing
one block to be the negative of the other block, which can be done
by an easy manipulation of the Fourier transform.
At first, the above solution looks ad hoc, and very specific to the
dihedral group. So we try to abstract the ingredients that have been
used in the solution.
The Fourier transform is a unitary mapping from the standard basis
$\set{\ket{g}}$ of $\mathbb{C}[G]$, to the Fourier basis. It can be formally
defined as follows. Let $\widehat{G}$ denote the set of all
inequivalent irreducible representations of $G$. For a representation
$\rho$ let $d_{\rho}$ denote the dimension of $\rho$. We define the
transformation $F$ by
\begin{eqnarray*}
F \ket{g} &=& \sum_{\rho \in \widehat{G}} \sum_{1 \le i,j \le
d_\rho} \sqrt{\frac{d_\rho}{|G|}} \rho_{i,j}(g) \ket{\rho,i,j}.
\end{eqnarray*}
It can be checked that $F$ is unitary and that it indeed
block-digaonlizes the regular representations, namely,
\begin{eqnarray*} F \rho_\mathrm{reg}(g) F^\dagger &=&
\sum_{\rho \in \widehat{G}} \sum_{1 \le j \le d_\rho}
\ketbra{\rho,j}{\rho,j} \otimes \sum_{1 \le i,i' \le d_\rho} \rho_{i,i'}(g)
\ketbra{i}{i}
\end{eqnarray*}
I.e., for each $\rho \in \wh{G}$ and $j \le d_\rho$, we have a
$d_\rho \times d_\rho$ block whose entries are $\rho(g)$.
$F$ maps $\mathbb{C}[G]$ to a vector space of the same dimension that is
spanned by $\set{\ket{\rho,i,j} : \rho \in \widehat{G},~ 1 \le i,j
\le d_\rho}$. To complete the specification of the Fourier transform
we also need to specify a map $S$ between $\set{\ket{\rho,i,j}}$ and
$\set{\ket{g} : g \in G}$. In the Abelian case there is a canonical
map $S$ between $\set{\ket{\rho,i,j} : \rho \in \widehat{G},~ 1
=j=1}$ and $\set{\ket{g} : g \in G}$, because when $G$ is Abelian
$\widehat{G}$ is isomorphic to $G$. However, when $G$ is not Abelian
things are more complicated. It is always true that $\sum_{\rho \in
\widehat{G}} d_\rho^2 = |G|$, and so there is always a bijection
between $\set{\ket{\rho,i,j}}$ and $\set{\ket{g} : g \in G}$.
However, it is not known, in general, how to find such a natural
bijection.
For example, for the symmetric group $\mathcal{S}_n$ the question takes the
following form. We look for bijections $f$ from pairs $(P,T)$ of
standard shapes to $\mathcal{S}_n$ (a shape corresponds to an irreducible
representation of $\mathcal{S}_n$, and its dimension is the number of standard
shapes of that shape). The question of finding an explicit bijection
$f$ from pairs $(P,T)$ of standard shapes to $\mathcal{S}_n$ is a basic
question in the study of the representation theory of $\mathcal{S}_n$. The
canonical algorithm doing so is the "Robinson-Schensted" algorithm
\cite{R38,S61} that was extensively studied later on (see
\cite{S01}, and especially Chapter 3 that is almost completely
dedicated to this algorithm).
Looking back at the solution we gave for the dihedral group we see
that we can express it as follows. We made sure that a block that
corresponds to an irreducible representation is contained in a coset
of $H$, and different copies of the same representation get the same
indices within $H$. Generalizing this further, we see that what we
actually used is a mapping $S: \set{\rho,i,j} \to G$ that is
\emph{product}, i.e., for every $\rho \in \widehat{G}$,
$S(\rho,i,j)=f_1(i) \cdot f_2(j)$ for some functions $f_1,f_2:
[d_\rho] \times [d_\rho] \to G$ (the functions $f_1$ and $f_2$ may
be specific to $\rho$). In the dihedral group, this amounts to $f_2$
selecting a coset representative, and $f_1$ selecting an index
inside the coset. But, in fact, any product mapping $S$ is good.
It is not clear at all that for every group $G$ such a product
mapping exists. It is trivial for Abelian groups, and simple for the
dihedral group (using cosets of $\set{e,s}$ for example). It is not
clear what is the situation for $\mathcal{S}_n$ - the Robinson-Schensted is
not a product mapping, but using specific information about $\mathcal{S}_n$,
for $n \le 6$, we found out that a product mapping exists.
Never the less, we were able to prove that $\mbox{PGL}(2,q)$ has a product
mapping, using information about its subgroup structure, and its
irreducible representations.
Putting these things together, we get a quantum expander $E(\rho)=
T( U T(\rho) U^\dagger)$, with $T$ being a single quantum step on a
the Cayley expander, and $U$ being a good basis change. $U$ is
obtained by doing the standard Fourier transform $F$ followed by the
a product mapping $S$, and with adding appropriate phases to the
basis vectors, so as different copies of the same irreducible
representation cancel out.
Clearly, the above discussion is intuitive, and there are many gaps
to fill. This is done in Appendix
\ref{sec:quantum-expanders-from-non-abelian}, where we repeat
everything in a relaxed way and with all the necessary details. In
Sec \ref{sec:rep-background} we give some background on
representation theory. Section \ref{sec:T} analyzes a single quantum
step on a Cayley graph and in Section \ref{sec:Abelian-expander} we
analyze the quantum expander over Abelian groups. Section
\ref{sec:Template} singles out Property (\ref{eqn:main:propertyU})
as a sufficient condition for a good basis change, and Section
\ref{sec:Property} shows that all we need for that is finding a
product mapping $S$. Finally, we prove in Section \ref{sec:PGL} that
$\mbox{PGL}(2,q)$ has such a product mapping, completing the correctness proof
of our constant degree quantum expander.
\section{Quantum extractors}
\label{sec:extractors}
\emph{\bf The balanced case.} The classical proof that $\mbox{ED}$ is
$\mbox{SZK}$-complete uses balanced \emph{extractors}. A balanced
extractor is a function $E: \set{0,1}^n \times \set{0,1}^d \to \set{0,1}^n$. We say $E$
is a $(k,\epsilon)$ extractor if for every distribution $X$ on
$\set{0,1}^n$ that has $k$ min-entropy
the
distribution $E(X,U_d)$ obtained by sampling $x \in X$, $y \in \set{0,1}^d$
and outputting $E(x,y)$, is $\epsilon$--close to uniform. We now
define balanced quantum extractors.
\begin{deff}
\label{def:balanced-quantum-extractor} Let $V$ be a Hilbert space of
dimension $N$. A superoperator $T: L(V) \to L(V)$ is a
$(k,d,\epsilon)$ \emph{quantum extractor}, if $T$ is $2^d$-regular
and for every $\rho \in D(V)$ with
$H_{\infty}(\rho) \ge k$ we have $\trn{T\rho-\tilde{I}} \le \epsilon$,
where $\tilde{I}={1 \over N}I$. We say $T$ is efficient if $T$ can be
implemented by a polynomial-size quantum circuit.
\end{deff}
We
mention that if $T$ is $2^d$-regular (and, in particular, if it is a
$(k,d,\epsilon)$ quantum extractor) then for any $\rho \in L(V)$ it
holds that $S(T\rho) \le S(\rho)+d$, i.e., no matter what, the
extractor never adds more than $d$ entropy to any input system.
Classically, balanced extractors are closely related to expanders
(e.g., \cite{GW97}). This generalizes to the quantum setting. We
prove:
\begin{lemm}
\label{lem:expander-extractor} If $T:L(V) \to L(V)$ is a
$(D=2^d,\ol{\lambda})$ quantum expander, then for every $t>0$, $T$ is also
a $(k=n-t,d,\epsilon)$ quantum extractor with $\epsilon=2^{t/2}
\cdot \ol{\lambda}$.
\end{lemm}
We give the easy proof in Section \ref{sec:app:quantum-extractors}
in the Appendix. In particular, we get an $(n-t,d,\epsilon)$
balanced quantum extractor $T:L(V) \to L(V)$ where $n=dim(V)$, and
$d=2(t + 2\log({1 \over \epsilon}))+O(1)$ using Theorem \ref{thm:PGL-expander} (or
the explicit version given in \cite{BST07}).
We use the last lemma to prove our lower bound on the spectral
gap of quantum expanders.
\newtheorem*{thma}{Theorem~\ref{thm:qexpander-lower-bound}}
\begin{thma}
Any $(D,\ol{\lambda})$ quantum expander satisfies $\ol{\lambda} \ge \frac{2}{3
\sqrt{3D}}$.
\end{thma}
In the classical world a tight bound of about ${2 \sqrt{D-1}} \over
D$ has been proved \cite{N91}. The proof there is both algebraic
(using eigenvalues) and combinatorial (using paths in the graph). We
do not see how to generalize the combinatorial component of the
proof. Instead we give an algebraic proof. The proof idea is to take
a density matrix which is uniform on a set of "small size". Applying
the extractor yields a density matrix close to the completely mixed
state. Such a matrix must have a high rank. On the other hand,
because we started with a low-rank matrix, the resulting density
matrix cannot have a too-high rank (since $E$ is $D$-regular). The
formal details are given in Section \ref{sec:app:quantum-extractors}
in the Appendix.
\emph{\bf The unbalanced case.}
A natural generalization of Definition
\ref{def:balanced-quantum-extractor} is for
a superoperator $T: L(V) \to L(W)$ where $V,W$ are Hilbert spaces of
dimensions arbitrary dimensions $N$ and $M$.
I.e., here we let $W$ be different than $V$, and, in particular, the
superoperator $T$ can map a large Hilbert space $V$ to a much
smaller Hilbert space $W$. In the classical case this corresponds to
hashing a large universe $\set{0,1}^n$ to a much smaller universe $\set{0,1}^m$.
Indeed, in the classical world highly unbalanced extractors exist
with a very short seed length $d$. These (and related objects like
dispersers, condensers and unbalanced expanders) have numerous
applications. There is also a huge body of work constructing
explicitly (most of) these objects. See \cite{CRVW02} for an attempt
to put some order in the zoo of definitions, and \cite{N96,S02} for
a survey of applications and constructions.
However, here we see a difference between the classical and the
quantum world. In the classical world if $X$ has $k$ entropy, and we
add $d$ more uniform bits, then the final output distribution can
have at most $k+d$ entropy. If we then "ignore" some of the output
bits, we can only \emph{decrease} the entropy of the output
distribution. In particular, if the output distribution has $m$
entropy, then most of it (namely, $m-d$) came from the source $X$.
We also had a similar property for balanced quantum extractors: for
any input $\rho$ we had $S(T \rho) \le S(\rho)+d$.
In the unbalanced case, however, we output $m \ll n$ qubits, and so
we trace-out (or "ignore") qubits. This, by itself, may
\emph{increase} the entropy. For example, a mixed state that is with
probability one in some pure-state has entropy zero (it is
completely determined). Tracing out $k$ bits of the system, may
result in a mixed state having $k$ entropy. If we trace out $n/2$
bits, at least theoretically, it is possible that our extractor
starts with a pure state as an input $\rho$ (i.e., $\rho$ has zero
entropy) and ends up with $T\rho$ being the completely mixed state.
Notice that at most $d$ of this entropy comes from the seed, and the
rest comes from the tracing-out.
We believe this makes any unbalanced extractor with $m<n/2$ not
useful. For example, the property $S(T\rho) \le S(\rho)+d$ (true for
balanced quantum extractors) is crucial for our proof that \mbox{QED}{} is
\mbox{QSZK}-complete.
We believe that slightly unbalanced expander
constructions (e.g., \cite{M95}) can probably be converted to
useful, slightly unbalanced quantum extractors.
\section{The complexity of estimating entropy}
\label{sec:main:estimating-entropy}
In this section we show that the $\mbox{QED}$ problem (as defined in the
introduction) is $\mbox{QSZK}$-complete. We do that by showing that $\mbox{QED}$
reduces to $\mbox{QSD}$ and vice versa, using the already known fact that
$\mbox{QSD}$ is $\mbox{QSZK}$--complete.
Proving $\mbox{QED}{} \le \mbox{QSD}$ is a bit tricky. We first show a that
related problem, Quantum Entropy Approximation ($\mbox{QEA}$), reduces to
$\ol{\mbox{QSD}}$. $\mbox{QEA}$ is the following promise problem:
\begin{boxit}
\textbf{Input:} A Quantum circuit $Q$ and a non-negative integer $t$.
\textbf{Accept:} If $S(\ket{Q}) \ge t+{1 \over 2}$.
\textbf{Reject:} If $S(\ket{Q}) \le t-{1 \over 2}$.
\end{boxit}
$\mbox{QEA}$ is the problem of comparing the entropy of a given quantum
circuit to some \emph{known} threshold $t$, instead of comparing the
entropies of two quantum circuits as in $\mbox{QED}$. Our proof that $\mbox{QEA}
\le \ol{\mbox{QSD}}$ uses quantum expanders and extractors, and we discuss
it next.
We begin with the classical intuition why $\mbox{EA}$ reduces to $\mbox{SD}$
($\mbox{EA}$ is the same promise problem, but with the input being a
classical circuit). We are given a circuit $C$ and we want to
distinguish between the cases the distribution it defines has
substantially more or less than $t$ entropy. First assume that the
distribution is flat, i.e., all elements that have a non-zero
probability in the distribution, have equal probability. In such a
case we can apply an extractor on the $n$ output bits of $C$,
hashing it to about $t$ bits. If the input distribution has
high entropy, it also has high min-entropy (because for flat
distributions entropy is the same as min-entropy) and therefore the
output of the extractor is close to uniform. If, on the other hand,
the circuit entropy is less than $t-d-1$, where $d$ is the extractor
seed length, than even after applying the extractor the output
distribution has at most $t-1/2$ entropy, and therefore it must be
far away from uniform. We get a reduction to $\ol{\mbox{SD}}$.
There are, of course, a few gaps to complete. First, our source is
not necessarily flat. This is solved in the classical case by taking
many independent copies of the circuit, which makes the output
distribution "close" to "nearly-flat" . A simple analysis shows that
this flattening works also in the quantum setting. Also, we need to
amplify the gap we have between entropy $t+1/2$ and $t-1/2$ to a gap
larger than $d$ (the seed length). This, again, is solved by taking
many independent copies of $C$, because $S(C^{\otimes q})=q S(C)$,
and works the same way in the quantum setting.
The interesting question is what is needed in the quantum case from
the quantum analogue of classical extractors. As it turns out, what
is needed is that sources with high min-entropy are mapped close to
the completely mixed state, whereas \emph{all} sources of low
min-entropy are mapped far away from it. The first condition is
clearly satisfied by our Definition
\ref{def:balanced-quantum-extractor}. The second condition is
implied by the regularity of the extractor: a $D=2^d$ regular
extractor can never add more than $d$ entropy to a source, and so
sources with low min-entropy are mapped to sources with low
min-entropy, and such sources (with the right parameters) are far
away from uniform. The formal proof is given in Section
\ref{sec:QEA-QSD}.
We remark that we believe that exactly this property fails in the
unbalanced case, i.e., there are input sources with low min-entropy
(e.g. pure states) that are mapped close to the completely mixed
state, and this additional entropy is obtained not because of the
seed, but rather because we have an unbalanced extractor that traces
out registers.
This completes the proof that $\mbox{QEA}$ reduces to $\ol{\mbox{QSD}}$. As
Watrous showed that $\ol{\mbox{QSD}} \le \mbox{QSD}$, we get that $\mbox{QEA} \le
{\mbox{QSD}}$. We next show that $\mbox{QEA} \le \mbox{QSD}$ implies $\mbox{QED} \le \mbox{QSD}$
using a standard classical trick. We can express:
$\mbox{QED}(Q_0,Q_1) = \bigvee_{t=1} \left[((Q_0,t) \in \mbox{QEA}_Y) \wedge
((Q_1,t) \in \mbox{QEA}_N)\right]$.
Thus, if $\mbox{QEA}$ reduces to $\mbox{QSD}$ (as we proved), we can express
$\mbox{QED}$ as a formula over $\mbox{QSD}$. We then take the classical result
that any Boolean formula over $\mbox{SD}$ reduces to $\mbox{SD}$, and generalize
it to the quantum setting, concluding that $\mbox{QED}$ reduces to $\mbox{QSD}$
as desired. The full details (and this time just for completeness,
because the proof closely follows the classical one) are given in
Section \ref{sec:closure}. This completes the proof that $\mbox{QED} \le
\mbox{QSD}$.
The direction that $\mbox{QSD} \le \mbox{QED}$ follows the classical reduction,
but using the Holevo bound from quantum information theory. The
details are given in Section \ref{sec:QSD-QED}. Altogether, we see
that $\mbox{QED}$ is $\mbox{QSZK}$ complete.
\section*{Acknowledgements}
We thank Oded Regev for pointing out \cite{AS04} to us and for
referring us to Lemma \ref{lem:ANTV} that simplified the proof of
the reduction from $\mbox{QSD}$ to $\mbox{QED}$. We also thank Ashwin Nayak,
Oded Regev, Adam Smith and Umesh Vazirani for helpful discussions
about the paper.
\bibliographystyle{alpha}
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Q: Typical behavior or past habit I'm not familiar with English language.
I want to know,
When we say:
I'd tell lie.
Does it mean a typical behavior or a past habit?
In other words, is it like used to or is a present typical behavior?
A: As StoneyB told you, would is not only used for conditional constructions. It depends of the given context if it's meant to be for repeated actions or habits in the past.
Note that for repeated actions in the past, you use would, but you can't use it for stative verbs.
Also, if you stress would, it means that we find the behaviour irritating. For instance
I hated when he would lie to me.
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Q: Returning multidimensional array key from selected option in PHP I'm roughly new in PHP and I'm trying to display the contents of a multidimensional array in a table that was inside a select option list which has been selected by the user. What I'm trying to do is that the array key of the selected option is returned into a new variable such as $keyref = key($myarray); so that I can use the key to display inside the table but i don't know how. Do I use a javascript function such as onChange=function() in the <select> list?
Here is the code:
<?php
$tshirt = array (
array("T-shirt A",15,"XS"),
array("T-shirt B",20,"S"),
array("T-shirt C",25,"M")
); //array("t-shirt type", price, "size")
$keyRef = 0; //the variable that will contain the selected array key
echo "<select id='shirtlist'>";
foreach($tshirt as $value){
echo "<option value='$value)'>$value[0]</option>";
}
echo "</select>";
echo "<p>Details of T-shirt:</p>";
echo "<table>";
echo "<tr>";
echo "<th>T-shirt Type</th>";
echo "<th>Price</th>";
echo "<th>Size</th>";
echo "</tr>";
echo "<tr>";
echo "<td>".$tshirt[$keyRef][0]."</td>";
echo "<td>".$tshirt[$keyRef][1]."</td>";
echo "<td>".$tshirt[$keyRef][2]."</td>";
echo "</tr>";
echo "</table>";
?>
The table manage to display the data of the first array but nothing happens when I selected the other options. As for trying javascript function I tried the json_encode($myarray) method but that only returned the Array as String notice.
A: First of all, you have to understand that PHP is only executed server-side, which means that there is no interaction between the PHP code and the action you are achieving in the browser.
The only way to return to PHP is to send a new request.
Depending on what you want to do and which workflow you want in the browser you should choose between pure PHP (each action has to be achieved with client-server request) or using javascript (directly executed in the browser).
In pure PHP, your feature can be achieved with the following code :
<?php
$tshirt = array (
array("T-shirt A",15,"XS"),
array("T-shirt B",20,"S"),
array("T-shirt C",25,"M")
); //array("t-shirt type", price, "size")
$keyRef = 0; //the variable that will contain the selected array key
if(isset($_GET["shortlist"]) && isset($tshirt[$_GET["shirtlist"]])) {
$keyRef = $_GET["shirtlist"];
}
echo "<form method='GET'>";
echo "<select name='shirtlist'>";
foreach($tshirt as $key => $value){
$selected = '';
if ($key == $keyRef) {
$selected = 'selected';
}
echo "<option value='$key' $selected>".$value[0]."</option>";
}
echo "</select>";
echo "<input type='submit' value='submit'/>";
echo "</form>";
echo "<p>Details of T-shirt:</p>";
echo "<table>";
echo "<tr>";
echo "<th>T-shirt Type</th>";
echo "<th>Price</th>";
echo "<th>Size</th>";
echo "</tr>";
echo "<tr>";
echo "<td>".$tshirt[$keyRef][0]."</td>";
echo "<td>".$tshirt[$keyRef][1]."</td>";
echo "<td>".$tshirt[$keyRef][2]."</td>";
echo "</tr>";
echo "</table>";
?>
Note that I use the global $_GET to get the information from the URL query strings. This information is transmitted to PHP when the submit button of the form is clicked, and then the value of the select is accessible in it using the name attribute of the tag (<select name="shirtlist"> means the value is accessible in $_GET["shirtlist"] when the method of the form is GET).
This code requires an action from the user to update the page with the selection.
I also added a selected attribute to the select in order to select the displayed article (When the iteration index is the same as the requested shirtlist value).
The double check if(isset($_GET["shortlist"]) && isset($tshirt[$_GET["shirtlist"]])) { is here to be sure the selected item is in the list, as the URL is accessible to anyone it's possible to change manually the value, it's more secure to change the form method to POST and then access the value using the $_POST global array instead of the $_GET one.
A: Html Code:
<?php
$tshirt = array (
array("T-shirt A",15,"XS"),
array("T-shirt B",20,"S"),
array("T-shirt C",25,"M")
); //array("t-shirt type", price, "size")
?>
<!-- here we create a select box -->
<select class="" id="shirtlist">
<option value="" selected>select</option>
<!-- add the key varible into foreach loop that give the index of the value -->
<?php foreach($tshirt as $key => $value){ ?>
<option value="<?=$key?>"><?=$value[0]?></option>";
<?php } ?>
</select>
<p>Details of T-shirt:</p>
<table>
<tr>
<th>T-shirt Type</th>
<th>Price</th>
<th>Size</th>
</tr>
<tr>
<!-- Here we are created the empty table Data will be field after user select any option -->
<td class="type"></td>
<td class="price"></td>
<td class="size"></td>
</tr>
</table>
Add Jquery CDN :
<script src="https://code.jquery.com/jquery-3.5.1.min.js" integrity="sha256-9/aliU8dGd2tb6OSsuzixeV4y/faTqgFtohetphbbj0=" crossorigin="anonymous"></script>
Javascript/Jquery Code :
<script type="text/javascript">
/* we we convert the PHP array into Javascript array using JSON.parse function */
var data = JSON.parse('<?=json_encode($tshirt)?>');
/* This is the code when user change the option in select box then it will add/replace the data into table */
$('#shirtlist').change(function(e){
var key = $(this).val();
$('.type').text(data[key][0]);
$('.price').text(data[key][1]);
$('.size').text(data[key][2]);
});
this will do the work.
on select box change event, it will add/replace the data into table.
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\section*{Acknowledgements}
This research was supported by the ERC Grant BROWSEC 771527, the ERC Grant ARTIST 101002685, the Austrian Science Fund FWF projects W1255-N23, and PROFET P31621, the Austrian Research Promotion Agency FFG through the Bridge-1 project PR4DLT (13808694) and the COMET K1 SBA, the Amazon Research Award 2020 FOREST, and the SecInt Doctoral College funded by TU Wien.
\section{Additional Background on the Routing Mechanism} \label{app:rout}
The Lightning network provides a way to route money along off-chain channels. The cryptographic tool to do so are hash-time-locked-contracts (HTLC) \cite{HTLC}. Routing extends the idea of a channel. Assume players $A$ and $B$ want to exchange goods for money but do not share a channel, instead $A$ has a channel with $E_1$, $E_1$ has a channel with $I$, $I$ has a channel with $E_2$ and $E_2$ has a channel with $B$. $A$ can now send the money to $B$ via the intermediaries $E_1$, $I$ and $E_2$.
\[ A \xLongleftrightarrow{\curvearrowright} {\color{teal} E_1 \xLongleftrightarrow{\curvearrowright} I \xLongleftrightarrow{\curvearrowright} E_2} \xLongleftrightarrow{\curvearrowright} B \]
The idea of the routing module, that means the honest strategy, is as follows (see \Cref{tbl:honestrouting}). Player $B$ thinks of a secret $x$ and computes its hash $h(x)=y$. Then, $B$ sends $y$ to $A$ (1.). Now $A$ can create an HTLC where she locks the money $m$ and three times fee $f$ (for every intermediary) with lock $y$ (2.). This money can only be claimed by $E_1$ and only if a value with hash $y$ is provided. $E_1$ does not know such a value yet and thus cannot unlock the money. However, $E_1$ can create another HTLC with value $m+2f$ and lock $y$ for $I$ (3.). This continues until $E_2$ creates an HTLC of value $m$ and lock $y$ for $B$ (4.-5.). Since $B$ knows $x$, he can unlock the money and send the goods to $A$ (6.). By unlocking, $E_2$ gets to know $x$ and can unlock as well (7.). $I$ and $E_1$ proceed in the same way (8.-9.). In the end $A$ paid $m+3f$ and received goods in exchange. $B$ sold the goods for $m$ and the intermediaries were rewarded with $f$ each for their participation.
\begin{table}
\centering
\begin{tikzpicture}[->]
\node (1) at (0,0) {$A$};
\node (2) at (2.5,0) {$E_1$};
\node (3) at (5,0) {$I$};
\node (4) at (7.5,0) {$E_2$};
\node (5) at (10,0) {$B$};
\path (5) edge[out=120, in=60, distance= 1.6cm] node [below] {\color{red} \small $y$} node [ below, pos=0.3] {1.} (1);
\path (1) edge node [above] {\small $(m+3f,\textcolor{red}{y})$} node [ below, pos=0.3] {2.}(2);
\path (2) edge node [above] {\small $(m+2f,\textcolor{red}{y})$} node [ below, pos=0.3] {3.} (3);
\path (3) edge node [above] {\small $(m+f,\textcolor{red}{y})$} node [ below, pos=0.3] {4.} (4);
\path (4) edge node [above] {\small $(m,\textcolor{red}{y})$} node [ below, pos=0.3] {5.} (5);
\path (5) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {6.} (4);
\path (4) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {7.} (3);
\path (3) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {8.} (2);
\path (2) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {9.} (1);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Honest Routing using HTLCs.}
\label{tbl:honestrouting}
\end{table}
In \cite{CITE}, the authors propose an Extensive Form Game to model the Routing Game which is shown in \Cref{tbl:modelpaper}. Whenever it is a specific player $p$'s turn, she can choose between following the protocol $H$ or doing nothing $I$. Doing nothing stops the game in any case and depending on whether the player right (in \Cref{tbl:honestrouting}) to $p$ has already claimed his money or not, $p$'s utility is either 0 or $-1$. After player $p$ has claimed her money, her utility is 1, independent of what the others do. Except for player $A$, her utility is also 1 if the routing was successful and 0 otherwise.
It is shown that this game together with the strategy $(H,H,H,H,H,H,H,H,H)$ is weak immune and optimal resilient. \\
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$B$};
\node (7) at (1.2,-0.5) {$(0,0,0,0,0)$} ;
\node (12) at (-1,-0.5) {$A$};
\node (13) at (0.25,-1) {$(0,0,0,0,0)$};
\node (2) at (-2,-1) {$E_1$};
\node (8) at (-0.75,-1.5) {$(0,0,0,0,0)$} ;
\node (14) at (-3, -1.5) {$I$};
\node (15) at (-1.75, -2) {$(0,0,0,0,0)$};
\node (3) at (-4,-2) {$E_2$};
\node (9) at (-2.75,-2.5) {$(0,0,0,0,0)$} ;
\node (16) at (-5,-2.5) {$B$};
\node (17) at (-3.75, -3) {$(0,0,0,0,0)$};
\node (4) at (-6,-3) {$E_2$};
\node (10) at (-4.75,-3.5) {$(0,0,0,-1,1)$} ;
\node (18) at (-7, -3.5) {$I$};
\node (19) at (-5.75, -4) {$(0,0,-1,1,1)$};
\node (5) at (-8,-4) {$E_1$};
\node (6) at (-9,-4.5) {\textcolor{red}{$(1,1,1,1,1)$}} ;
\node (11) at (-6.75,-4.5) {$(1,-1,1,1,1)$} ;
\path (1) edge node[above] {\tiny $H$} (12);
\path (1) edge node[ above, pos=0.7] {\tiny $I$} (7);
\path (12) edge node[above] {\tiny $H$} (2);
\path (12) edge node[above, pos=0.7] {\tiny $I$} (13);
\path (2) edge node[above] {\tiny $H$} (14);
\path (2) edge node[above, pos=0.7] {\tiny $I$} (8);
\path (14) edge node[ above] {\tiny $H$} (3);
\path (14) edge node[ above, pos=0.7] {\tiny $I$} (15);
\path (3) edge node[above] {\tiny $H$} (16);
\path (3) edge node[above, pos=0.7] {\tiny $I$} (9);
\path (16) edge node[ above] {\tiny $H$} (4);
\path (16) edge node[ above, pos=0.7] {\tiny $I$} (17);
\path (4) edge node[above] {\tiny $H$} (18);
\path (4) edge node[above, pos=0.7] {\tiny $I$} (10);
\path (18) edge node[ above] {\tiny $H$} (5);
\path (18) edge node[ above, pos=0.7] {\tiny $I$} (19);
\path (5) edge node[left] {\tiny $H$} (6);
\path (5) edge node[above, pos=0.7] {\tiny $I$} (11);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Model of Routing Module as in \cite{CITE}.}
\label{tbl:modelpaper}
\end{table}
However, the HTLC routing module of the Lightning Network is vulnerable to a dishonest behaviour called the wormhole attack. In the wormhole attack two intermediaries along the routing path collude to steal another intermediary's fee.
It works as illustrated in \Cref{tbl:wormhole}. The beginning including the creation of the HTLCs is unchanged. When $B$ unlocks his money with $x$ (6.), $E_2$ does not unlock as well, but rather forwards $x$ to the other adversary $E_1$ (7.). This way $E_1$ can unlock $A$'s money (8.), but $I$ never will be able to claim hers. After a certain time the remaining HTLCs time out and the money returns to the creators.
The outcome of the wormhole attack is the following. $B$ receives $m$ as planned and $A$ pays $m+3f$ which is also fine. The honest intermediate $I$ gets 0 instead of the $f$ that were promised for her services. The adversaries $E_1$ and $E_2$ earn $3f$ instead of the $2f$ they deserve. Such an adversary-incentivizing-behaviour should not be possible in a "secure" protocol. This shows that the model in \Cref{tbl:modelpaper} is insufficient.
\subsection{Proof for Refined Routing Model}
For better readability, we restate the theorem.
\vulnerability*
\begin{proof}
The honest behavior of the routing module is $(H,H,H,H,H,H,H,H,H)$ and its utility is $(\rho,f,f,f,\rho)$ as indicated in red in \Cref{tbl:mymodel}. Let us compare it to the dishonest terminal history $(H,H,H,H,H,H,D,D)$ with a utility of $(\rho,\;m+3f,\;0,\;-m,\;\rho)$ in blue. Then, it is easy to see, that the collusion of $E_1$ and $E_2$ strictly profit from the deviation, which yields a joint utility of $3f$, whereas the honest behaviour only yields a joint utility of $2f$. This violates $\textsf{CR}${}. Using the definition of security, we finally conclude that the Routing Game is not secure.
\hfill $\square$
\end{proof}
\section{Results of Security Analysis} \label{app:sec}
In this section all results from \Cref{sec:sec} are restated and proven. Additionally, the results \Cref{thm:bleqf}, \Cref{thm:a0} and \Cref{thm:b0} about edge cases are stated and proven.
\weakimmunity*
\begin{proof}
Let $a,b\geq f$. For history $(H)$, we consider a strategy, where $A$ chooses $H$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. This strategy yields terminal history $(H)$ and is weak immune, as $B$'s deviation cannot influence $A$'s utility and $A$'s deviations always result in a non-negative utility for $B$, since $a \geq f$.
Similarly, for $(C_h,S)$, we consider a strategy, where $A$ chooses $(C_h)$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Further, $A$ takes $P$ after $(C_h,D)$ and $H$ after $(C_h,I)$, $(C_h,U^+)$ and $(C_h,U^-)$. This strategy yields terminal history $(C_h,S)$ and is weak immune. Deviation of $A$ has the same effects as before, never causing $B$ a negative utility. If $B$ deviates now, $A$ also never gets negative utility, since $b \geq f$. This concludes the theorem.
\hfill $\square$
\end{proof}
\incentcomp*
\begin{proof}
Collusion resilience is defined on strict subsets of players. Thus, in a two-player game, it considers only deviations of single players and since the summation over one value is the value itself, $\textsf{CR}${} is equivalent to Nash Equilibria in this case.
We therefore only check whether $(H)$ and $(C_h,S)$ are Nash Equilibria. For $(H)$, we consider a strategy, where $A$ chooses $H$ initially, $B$ chooses $I$ after $(C_h)$, $P$ after $(D)$ and $I$ after $(C_c)$. $A$ takes $H$ after $(C_h,I)$ and $(C_c,I)$. Further, $B$ takes $I$ after $(C_{c/h},I,U^{+/-})$. For $A$, we assume she takes $H$ after $(C_{c/h},I, U^{+/-},I)$ and finally let $B$ take $P$ after $(C_{h/c},I,U^{+/-},I,D)$. This strategy yields history $(H)$ and is a Nash Equilibrium, as no party can unilaterally deviate to increase their utility.
To show that $(C_h,S)$ is a Nash Equilibrium, we consider a strategy, where $A$ picks $C_h$ initially, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Further, let $A$ pick $P$ after $(C_h,D)$, $H$ after $(C_h,I)$ and $(C_h,U^{+/-})$. This strategy has terminal history $(C_h,S)$ and no one can deviate to increase their utility. Therefore, $(C_h,S)$ is a Nash Equilibrium and thus $\textsf{CR}$.
In order to prove the practicality properties, we compute all subgame perfect equilibria of $G_c(A)$. Since a subgame is the game that remains after some choices (non-terminal history), we compute subgame perfect equilibria bottom-up. That is, we start comparing the utility of the subtrees, where no other decisions have to be made any more. In $G_c(A)$, these are for example the subgames after history $(C_h,I,D)$ or $(C_c,D)$. For the latter, $A$ is the player to choose the action. To compute the subgame perfect equilibrium, we have to compare all possible utilities for $A$ after $(C_c,D)$. We then replace this internal node labelled $A$, by the utility that yields the best value for $A$ and proceed until we reach the root. If there is no single best choice for a player, then all actions resulting in best utility have to be considered. Applying this procedure to the subgames $S_1$-$S_4$ and $S'_1$-$S'_4$ we get subgame perfect terminal history $(A,H)$ with utility $(\rho+\alpha-\epsilon,\rho+\alpha)$ for $S_1$, for $S_2$ we get terminal history $(S)$ yielding $(\alpha,\alpha)$ and $(I,H)$, yielding $(\alpha-\epsilon,\alpha)$. For $S_3$ and $S_4$ it is $(I,S)$ with $(\alpha,\alpha)$. The subgame $S'_1$ has practical history $(I,H)$, with $(\alpha-\epsilon,\alpha)$ if $c>p_A$, $(A,I,S)$ with $(\rho+\alpha,\rho+\alpha)$ if $c=p_A$ and $(A,H)$ with $(\rho+\alpha-\epsilon,\rho+\alpha)$ if $c<p$. The subgame $S'_2$ has practical history $(I,H)$, yielding $(\alpha-\epsilon,\alpha)$. For $S'_3$ and $S'_4$ we get $(I,H)$ with $(\alpha,\alpha-\epsilon)$ and additionally for $S'_3$, if $c=p$, we also have $(A,S)$ yielding $(\rho+\alpha,\rho+\alpha)$. All of these results are based on the facts $a-p_B+d_A\geq f$ and $b-p_A+d_B \geq f$, since this causes the revocation transaction always to be better than ignoring the dishonest unilateral closing attempt.
Based on these preliminary results, we can now compute the subgame perfect equilibria for $G_c(A)$ considering multiple practical histories and case splits as stated. We have the following results.
If $c=p_A$, then $(C_c,U^+,A,S)$ and $(C_c,I,U^+,A,I,S)$ are practical, both yielding $(\rho+\alpha,\rho+\alpha)$. If $c>p_A$, then the histories $(C_h,S)$, $(C_h,U^+,I,S)$, $(C_h,U^-,I,S)$ and $(C_h,I,U^-,S)$ all leading to $(\alpha,\alpha)$ are practical, as well as terminal history $(C_h,I,U^+,A,H)$, yielding $(\rho+\alpha-\epsilon,\rho+\alpha)$.
For $c<p_A$, all the histories and their utilities from $c>p_A$ are practical. Additionally $(C_c,I,U^+,A,H)$ is subgame perfect and also results in utility $(\rho+\alpha-\epsilon,\rho+\alpha)$.
This shows, that $(H)$ is never practical and $(C_h,S)$ is practical if and only if $c\neq p_A$.
\hfill $\square$
\end{proof}
\security*
\begin{proof}
Since $a,b \geq f$, $(C_h,S)$ is weak immune (\Cref{thm:wi}). Because of $a-p_B+d_A\geq f$, $b-p_A+d_B \geq f$ and $c \neq p_A$, it is practical and $\textsf{CR}$. Hence, by \Cref{def:secure} $(C_h,S)$ is secure.
\hfill $\square$
\end{proof}
\subsection{Results without Updates}
\securitynoup*
\begin{proof}
We start by proving weak immunity. For $(H)$, we consider a strategy where $A$ chooses $H$ initially, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Then, $B$'s deviation has no impact on the history and $A$'s deviation always leads to non-negative utility for $B$, as $a-f\geq 0$.
Similarly for $(C_h,S)$ we consider a strategy, where $A$ chooses $C_h$ initially, $P$ after $(C_h,D)$ and $H$ after $(C_h,I)$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Then, $A$'s deviation is unproblematic as before and $B$'s deviation also leads to non-negative utility for $A$, since $b-f\geq 0$.
To show practicality, we compute all subgame perfect terminal histories. Since $a,b\geq f$ implies $a+d_A\geq f$ and $b+d_B \geq f$, the best choice after a $D$, is always $P$. Thus, $A$'s best choice after $(C_h,I)$ and $(C_c,I)$ is $H$. Therefore, $B$ has two subgame perfect options $I$ and $S$ after $(C_h)$ and only $I$ after $(C_c)$. This yields to the following practical history. The history $(C_h,S)$ with $(\alpha,\alpha)$ and $(C_h,I,H)$, $(C_c,I,H)$ and $(H)$ with $(\alpha-\epsilon,\alpha)$.
Every practical terminal history is a Nash Equilibrium, since if a deviation could benefit a player, she would have chosen differently. Further, we know that $\textsf{CR}${} is equivalent to Nash Equilibria in two-player games, thus practicality implies $\textsf{CR}$ and this concludes the proof.
\hfill $\square$
\end{proof}
\securityflaw*
\begin{proof}
Let $\sigma$ be any strategy, yielding an honest history, then $A$ can deviate to $D$. In this case $B$ gets negative utility, since $a<f$, whether he chooses $P$ or $I$. Hence no honest history is weak immune.
\hfill $\square$
\end{proof}
\cortwo*
\begin{proof}
We fix the old distribution state such that the difference $d_A$ to the latest state is the value of $A$'s dishonest closing attempt in the closing game. As $a+d_A<f$ implies $a<f$, \Cref{thm:secflaw} applies. Therefore, neither $(H)$ nor $(C_h,S)$ are weak immune.
In order to show that they are also not practical, we prove instead, that the only practical history is $(D,I)$. Since $a+d_A<f$, $I$ is the best choice for $B$ after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$. Consequently, $A$ will choose $D$ after $(C_h,I)$ and $(C_c,I)$. If now $b+d_B\geq f$, then $A$'s best choice is $P$ after $(C_h,D)$ and $(C_c,D)$. Thus, $B$ will take $S$ after $(C_h)$ and $H$ after $(C_c)$. In the other case, $b+d_b<f$, $A$'s best option is $I$ after $(C_h,D)$ and $(C_c,D)$, thus $B$'s best choice after $(C_h)$ and $(C_c)$ is $D$, which yields a negative utility for $A$. Therefore, in both cases $A$'s only subgame perfect action is $D$. Hence, $(D,I)$ is the unique subgame perfect history.
For $\textsf{CR}$, we show instead that there exist extensions of $(H)$ and $(C_h,S)$ that are Nash Equilibria. Let $\sigma$ be the strategy where $A$ chooses $H$, everyone chooses $P$ after a dishonest closing attempt, $B$ chooses $I$ after $(C_h)$ and $(C_c)$ and $A$ chooses $H$ after $(C_h,I)$ and $(C_c,I)$. Then, no one can deviate to increase their utility and therefore $(H)$ is an $\textsf{CR}${} history.
To prove $(C_h,S)$ is $\textsf{CR}$, we consider the strategy $\sigma'$, which is the same as $\sigma$, except $A$ chooses $C_h$ and $B$ chooses $S$ after $(C_h)$. This is also a Nash Equilibrium.
\hfill $\square$
\end{proof}
\corthree*
\begin{proof}
Once the opponent's balance is below $f$, that party can start the closing game, therefore the opponent becoming $A$. Thus, by applying \Cref{thm:secflaw}, it follows that the opponent can make the rational player lose money by closing unilaterally and dishonestly. If it is not the first time that that player's balance is below $f$, then we are even in the situation of \Cref{cor:cor2}, where it is rational of $A$ to close dishonestly.
\hfill $\square$
\end{proof}
We present an additional theorem, discussing the case where player $B$ has little funds left in the channel. Since the roles of player $A$ and $B$ are arbitrary, it is of little importance because the results give stronger security guarantees as for the case where $A$ has a low balance. Nevertheless, we state it for the sake of completeness.
\begin{theorem} \label{thm:bleqf}
If there exists an old state with $b+d_B<f$, but $a \geq f$, then \begin{enumerate}
\item $(H)$ is secure.
\item $(C_h,S)$ is not practical, not weak immune, but $\textsf{CR}$.
\end{enumerate}
\end{theorem}
\begin{proof}
To prove (1), we start by showing weak immunity. Consider any strategy, where $B$ chooses $P$ after $(D)$, $S$ after $(C_h)$ and $H$ after $C_c$. Then $(H)$ is weak immune, because $B$'s deviations have no impact on the history and $A$'s deviations can never bring $B$'s utility below zero.
Next, we prove the practicality of $(H)$. Since $a \geq f$, the subgame perfect choice after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$ is $P$. Thus $A$ chooses $H$ after $(C_h,I)$ and $(C_c,I)$. Due to $b+d_B<f$, $A$'s best option after $(C_h,D)$ and $(C_c,D)$ is $I$. Hence $B$'s unique subgame perfect choice after $(C_c)$ and $(C_h)$ is $D$. Thus, $A$'s only best response is $H$. Therefore, $(H)$ is the only practical history.
As practicality implies $\textsf{CR}${} in our case, $(H)$ is secure.
For (2), we just showed that $(C_h,S)$ cannot be practical. Additionally, $(C_h,S)$ is not weak immune, since $B$ could deviate to $D$ after $(C_h)$, in which case $A$ gets negative utility for sure.
Finally, we consider the strategy $\sigma$, where $A$ chooses $C_h$, $B$ chooses $S$, both take $P$ in case of a dishonest unilateral closing attempt, $B$ takes $H$ after $(C_c)$, similarly $A$ takes $H$ after $(C_h,I)$ and $(C_c,I)$. Then $\sigma$ is a Nash Equilibrium yielding terminal history $(C_h,S)$. This concludes the proof.
\hfill $\square$
\end{proof}
\subsection{Results for Edge Cases}
Finally, we present results about the edge cases, where one party has zero funds left in the channel. This assumption changes the structure of $G_c(A)$, as some actions are not possible in this case. The precise games we refer to are stated in \Cref{app:subgames}, and result from removing impossible choices from \Cref{tbl:Gc}.
\begin{theorem} \label{thm:a0}
If $a=0$ and $b>0$, then no honest strategy is weak immune. Additionally, $(H)$ and $(C_h,S)$ are practical if and only if $d_A\geq f$ for all old states. In any case they are $\textsf{CR}$.
\end{theorem}
\begin{proof}
We first show that no honest strategy is weak immune. Let $\sigma$ be an honest strategy, then $A$ does not choose $D$ in $\sigma$. However, if $A$ deviates to $D$, then $B$'s utility is negative.
To show both $(H)$ and $(C_h,S)$ are $\textsf{CR}$, we show they are Nash Equilibria instead. We therefore consider any strategy where $B$ chooses $P$ after $(D)$, $S$ after $(C_h)$ and $H$ after $(C_c)$. Then both choices $H$ and $C_h$ of $A$, resulting in the histories $(H)$ and $(C_h,S)$, yield a Nash Equilibrium, as no one can deviate to increase their utility.
Let now $d_A\geq f$. In which case $P$ is the subgame best choice for $B$ after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$. Further, after history $(C_c,I)$, $S$ it is never a best option for $B$, because it is strictly dominated by $H$. Therefore, $A$ will get utility zero in any case. This makes $(H)$ a practical history. Similarly for $(C_h,S)$, since $S$ is subgame perfect for $B$ after $(C_h)$.
If now $d_A<f$, then $I$ is subgame perfect for $B$ after $D$. Thus, with similar argumentation as before, $(D,I)$ is the only practical history.
\hfill $\square$
\end{proof}
\begin{theorem} \label{thm:b0}
If $a>0$ and $b=0$, then
\begin{enumerate}
\item $(H)$ is secure.
\item $(C_h,S)$ is not weak immune, but $\textsf{CR}$. It is practical iff $d_B \geq f$ in every previous state $(a-d_B,d_B)$.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove (1.) first. The history $(H)$ is weak immune, as $B$'s strategy does not effect the history and $A$'s deviation is irrelevant for $B$, as he can never get negative utility.
Practicality of $(H)$. After history $(C_h,S)$ the subgame perfect choice of $A$ depends on whether $d_B \geq f$. In any case, $D$ is subgame perfect for $B$. If $A$ chose $P$, then it is as good as any other choice, yielding 0, otherwise it is the only best option resulting in a positive utility. Thus, $A$ either gets $-f+\alpha$ or $-d_B+\alpha$ if she chooses $C_h$, both of which is negative. Hence $A$'s subgame perfect and therefore practical choice is $H$, yielding the history $(H)$.
The fact that $(H)$ is $\textsf{CR}${} follows from practicality. This shows that $(H)$ is secure, if $b=0$.
(2.) We start showing $(C_h,S)$ is not weak immune. We consider any strategy yielding the history $(C_h,S)$. Assume now, $B$ deviates to $D$ after $(C_h)$, then no matter what $A$'s choice is, she will get a negative utility, thus $(C_h,S)$ is not weak immune.
The collusion resilience of $(C_h,S)$, can be shown by considering a strategy with history $(C_h,S)$, where additionally $A$ chooses $P$ after $(C_h,D)$. Then $B$ has no incentive to deviate as he always gets utility 0, and $A$ has no incentive as $\alpha$ is the best possible outcome for her.
To finally show that $(C_h,S)$ is practical iff $d_B\geq f$, we consider $A$'s choice after $(C_h,D)$. The option $P$ is subgame perfect iff $d_B \geq f$. Thus, $S$ is subgame perfect for $B$ iff $d_B \geq f$. For $d_B < f$, $D$ is the better option for $B$, yielding $(-d_B+\alpha,d_B+\alpha-\epsilon)$. Therefore $C_h$ is subgame perfect for $A$ iff $d_B \geq f$, in which case the resulting history is $(C_h,S)$. This concludes the proof of the theorem.
\hfill $\square$
\end{proof}
The weak immunity result of $(H)$ might be misleading, as $B$ can actually close dishonestly immediately (before $A$ takes action). This is not represented here, but in $G_c(B)$, which is analog to $G_c(A)$ but with swapped roles.
\section{Subgames and Edge Cases of $G_c(A)$} \label{app:subgames}
In the following all the subgames needed for the closing game $G_c(A)$ are defined. Further, the edge cases of $A$'s or $B$'s balance being zero in the closing game are treated.
\subsection{Subgames of $G_c(A)$}
The subgames $S_1$ and $S_2$ in \Cref{tbl:S1} and \Cref{tbl:S2} cover the case where a channel update is proposed by $A$, although $A$ has already signed an honest collaborative closing attempt. In $S_1$ the update is from channel state $(a,b)$ to $(a+p_A,b-p_A)$, whereas in $S_2$ the suggested update is $(a-p_B, b+p_B)$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.25,-1.5) {$(\alpha,\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (1,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.9,-6.5) { $(\textcolor{red}{-p_A}+\rho+\alpha,\,\textcolor{red}{p_A}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{-p_A}+\rho,-b\,\textcolor{red}{+p_A}+\rho)$};
\node (20) at (2,-4) {$(-a\,\textcolor{red}{-p_A}\,+\rho,a\,\textcolor{red}{+p_A}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\, \textcolor{red}{-p_A}-f +\rho+\alpha,-b\, \textcolor{red}{+p_A} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{1}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S1}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.25,-1.5) {$(\alpha,\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (1,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.9,-6.5) { $(\textcolor{red}{p_B}+\rho+\alpha,\,\textcolor{red}{-p_B}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{+p_B}+\rho,-b\,\textcolor{red}{-p_B}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{+p_B}+\rho,a\textcolor{red}{-p_B}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\textcolor{red}{+ p_B}-f +\rho+\alpha,-b\textcolor{red}{-p_B} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{2}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S2}
\end{table}
The subgames $S_1'$ and $S_2'$ in \Cref{tbl:S1p} and \Cref{tbl:S2p} are very similar to $S_1$ and $S_2$. They only differ in the fact, that the existing partially signed collaborative closing attempt was dishonest. That means, $A$ proposed an unfair split, increasing her outcome by value $c>0$.
\begin{table}[t]
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2.5,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.5,-1.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (0.25,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.5,-6.5) { $(\textcolor{red}{c-p_A}+\rho+\alpha,\textcolor{red}{-c+p_A}+\rho+\alpha)$};
\node (19) at (4,-7) {$(-a\textcolor{red}{-p_A}+\rho,-b\textcolor{red}{+p_A}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{-p_A}+\rho,a\textcolor{red}{+p_A}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\textcolor{red}{-p_A}-f +\rho+\alpha,-b\textcolor{red}{+p_A} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{1}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S1p}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2.5,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.5,-1.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (0.25,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.4,-6.5) { $(\textcolor{red}{c+p_B}+\rho+\alpha,\textcolor{red}{-c-p_B}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{+p_B}+\rho,-b\,\textcolor{red}{-p_B}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{+p_B}+\rho,a\,\textcolor{red}{-p_B}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b \textcolor{red}{+p_B}-f +\rho+\alpha,-b\, \textcolor{red}{-p_B} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{2}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S2p}
\end{table}
The next couple of subgames $S_3$ (\Cref{tbl:S3}), $S_4$ (\Cref{tbl:S4}) represent channel updates that were proposed by $B$, after $A$ tried to close the channel honestly and collaboratively. Similar to before, $S_3$ represents the case where $A$'s balance is increased, therefore an update from $(a,b)$ to $(a+p_A,b-p_A)$. The other subgame $S_4$ handles the case where the update increases $B$'s balance to a new state $(a-p_B,b+p_B)$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.5,-1) {$({\color{red}-p_A}+\rho+\alpha,{\color{red}p_A}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-2,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\alpha,\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}-p_A}-f+\rho+\alpha,-b{\color{red}+p_A}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}-p_A}+\rho,-b{\color{red}+p_A}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}-p_A}+\rho,a{\color{red}+p_A}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{3}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S3}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.5,-1) {$({\color{red}p_B}+\rho+\alpha,{\color{red}-p_B}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-2,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\alpha,\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}+p_B}-f+\rho+\alpha,-b{\color{red}-p_B}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}+p_B}+\rho,-b{\color{red}-p_B}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}+p_B}+\rho,a{\color{red}-p_B}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{4}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S4}
\end{table}
Finally, the subgames $S_3'$ and $S_4'$ in \Cref{tbl:S3p} and \Cref{tbl:S4p} handle the same situations as $S_3$ and $S_4$, except the by $A$ partially signed closing attempt is unfair and increases $A$'s outcome by $c>0$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.15,-1) {$({\color{red}c-p_A}+\rho+\alpha,{\color{red}-c+p_A}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-1.25,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}-p_A}-f+\rho+\alpha,-b{\color{red}+p_A}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}-p_A}+\rho,-b{\color{red}+p_A}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}-p_A}+\rho,a{\color{red}+p_A}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{3}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S3p}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.15,-1) {$({\color{red}c+p_B}+\rho+\alpha,{\color{red}-c-p_B}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-1.25,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}+p_B}-f+\rho+\alpha,-b{\color{red}-p_B}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}+p_B}+\rho,-b{\color{red}-p_B}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}+p_B}+\rho,a{\color{red}-p_B}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{4}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S4p}
\end{table}
\subsection{Edge Cases of the Closing Game}
So far, we only considered cases where both balances $a$ and $b$ were strictly greater than zero. This is not necessarily the case. Therefore, we consider these cases here.
In the first case, $a=0$, $B$ cannot close dishonestly, as there is no old state that increases his balance. The corresponding simplified game is presented in \Cref{tbl:a0}.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0.125,0.5) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-1.25,-1) { $(0,\alpha)$};
\node (4) at (0.25,-0.75) {\color{olive}$B$};
\node (5) at (3.75,-1) {\color{olive}$B$};
\node (6) at (-5,-3.25) {\color{teal}$A$};
\node (7) at (-3.25,-3) { $(0,\alpha)$};
\node (8) at (1.25,-1.4) { $(0,-f+\alpha)$};
\node (9) at (-0.5,-1.9) { $(d_A+\alpha-\epsilon, -d_A+\alpha)$};
\node (10) at (3,-3) {$(c+\alpha,-c+\alpha)$};
\node (11) at (5,-3.25) {\color{teal}$A$};
\node (12) at (-5,-4.75) {\color{olive}$B$};
\node (13) at (-1.75,-4.25) {$(0,\alpha)$};
\node (14) at (-3.5,-4.5) {$(0,-b)$};
\node (19) at (3.5,-4.5) {$(0,-b)$};
\node (20) at (1.75,-4.25) { $(0,\alpha)$};
\node (21) at (5.25,-4.75) {\color{olive}$B$};
\node (22) at (-2.25,-5.5) {$(0,-f+\alpha)$} ;
\node (23) at (-3.75,-6) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (24) at (2.5,-5.5) { $(0,-f+\alpha)$};
\node (25) at (4,-6) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (27) at (-1.75,-2.75) {$(0,\alpha-\epsilon)$};
\node (34) at (0.75,-2.75) {$(0,\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [left, pos=0.5] {\tiny $H$} (3);
\path (1) edge node [right, pos=0.6] {\tiny $D$} (4);
\path (1) edge node [above, pos=0.5] {\tiny $C_c$} (5);}
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7);
\path (4) edge node [above] {\tiny $P$} (8);
\path (4) edge node [left] {\tiny $I$} (9);
\path (5) edge node [right] {\tiny $S$} (10);
\path (5) edge node [right] {\tiny $I$} (11); }
{\color{teal}
\path (6) edge node [right, pos=0.7] {\tiny $D$} (12);
\path (6) edge node [above, pos=0.7] {\tiny $H$} (13);
\path (6) edge node [above, pos=0.7] {\tiny $I$} (14);
\path (11) edge node [above, pos=0.7] {\tiny $I$} (19);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (20);
\path (11) edge node [left, pos=0.7] {\tiny $D$} (21);}
{\color{olive}
\path (12) edge node [below] {\tiny $P$} (22);
\path (12) edge node [left] {\tiny $I$} (23);
\path (21) edge node [below] {\tiny $P$} (24);
\path (21) edge node [right] {\tiny $I$} (25); }
{\color{olive}
\path (2) edge node [left, pos=0.5] {\tiny $H$} (27);
\path (5) edge node [right, pos=0.5] {\tiny $H$} (34);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing game $G_c(A)$ with $a=0$.}
\label{tbl:a0}
\end{table}
If $b=0$ (\Cref{tbl:b0}), player $A$ cannot close dishonestly, as she cannot take any money from $B$. Thus, both dishonest unilateral closing $D$ and proposing an unfair split in a collaborative closing attempt $C_c$ are not possible.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (-1.75,0) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-0,-1) { $(\alpha-\epsilon,0)$};
\node (6) at (-5,-2.5) {\color{teal}$A$};
\node (7) at (-3.5,-2.5) { $(\alpha,0)$};
\node (13) at (-4.25,-3.75) {$(\alpha-\epsilon,0)$};
\node (14) at (-5.5,-3.75) {$(-a,0)$};
\node (27) at (-0.75,-2) {$(\alpha,0)$};
\node (30) at (-2.25,-2.5) {\color{teal}$A$};
\node (36) at (-1.5,-3.5) { $(-f+\alpha,0)$} ;
\node (37) at (1,-3) { $(-d_B+\alpha,d_B+\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [above, pos=0.5] {\tiny $H$} (3); }
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7); }
{\color{teal}
\path (6) edge node [right] {\tiny $H$} (13);
\path (6) edge node [left] {\tiny $I$} (14);}
{\color{olive}
\path (2) edge node [below, pos=0.5] {\tiny $H$} (27);
\path (2) edge node [left] {\tiny $D$} (30);}
{\color{teal}
\path (30) edge node [right] {\tiny $P$} (36);
\path (30) edge node [above] {\tiny $I$} (37); }
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing game $G_c(A)$ with $b=0$.}
\label{tbl:b0}
\end{table}
\section{Additional Game Theoretical Concepts}\label{app:theory}
All additional game-theoretical definitions, proofs and examples are stated here.
\subsection{Iterated Deletion of Weakly Dominated Strategies}
We introduce the iterated deletion of weakly dominated strategies (IDWDS), by using weakly dominated strategies as in \Cref{sec:prelim}.
\begin{definition}[IDWDS]
The \emph{iterated deletion of weakly dominated strategies} (IDWDS) of a game $\Gamma$ is defined as iteratively rewriting $\Gamma$ by omitting \emph{all} weakly dominated strategies of all players. This is repeated until no strategy is weakly dominated any more. The resulting game $\Gamma'$ is thus a subgame of $\Gamma$.
\end{definition}
When IDWDS is applied, then every Nash Equilibrium of the resulting game $\Gamma'$ is also a Nash Equilibrium of $\Gamma$.
\subsection{Practicality and EFGs}
We give an example that shows how applying the NFG definition of practicality to an EFG, by using its translation to a Normal Form Game, yields unwanted results.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=3cm, el/.style = {inner sep=4pt, align=center, sloped}]
\node (1) at (0,0) {$A$};
\node (2) at (1.5,-0.75) {$B$} ;
\node (3) at (-1.5,-0.75) {$(2,2)$};
\node (4) at (3,-1.5) {$A$} ;
\node (5) at (0, -1.5) {$(3,1)$};
\node (6) at (4.5,-2.25) {$B$} ;
\node (7) at (1.5,-2.25) {$(1,1)$};
\node (8) at (6,-3) {$(0,2)$} ;
\node (9) at (3, -3) {$(0,1)$};
\path (1) edge node[ above] { $2$} (2);
\path (1) edge node[above] {$1$} (3);
\path (2) edge node [above] {$4$} (4);
\path (2) edge node[ above] {$3$} (5);
\path (4) edge node[above] {$6$} (6);
\path (4) edge node[ above] {$5$} (7);
\path (6) edge node[above] {$8$} (8);
\path (6) edge node[above] {$7$} (9);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Game $\Gamma_{E}$.}
\label{tbl:efg}
\end{table}
\begin{example} \label{ex:pract}
Let us consider an Extensive Form Game $\Gamma_E$ with two players $A$ and $B$ as in \Cref{tbl:efg}. Its translation to a Normal Form Game is $\Gamma_N$ in \Cref{tbl:nfg}. \\
According to the definition of practicality for NFGs, the only practical strategy in $\Gamma_N$ is $(1,4-8)$, which results in a utility of $(2,2)$. This is the case, since all the blue colored single strategies of $A$ and $B$ are weakly dominated. After deleting those, the teal colored single strategy $2-5$ of $A$ becomes weakly dominated as well, thus leaving only the joint strategy $(1,4-8)$.
\begin{table}
\centering
\begin{tabular}{|r|c|c|c|}
\hline
& \textcolor{blue}{3} & \textcolor{blue}{4-7} & 4-8 \\
\hline
1 & $(2,\textcolor{blue}{2})$ & $(2,\textcolor{blue}{2})$ & $(2,2)$ \\
\hline
\textcolor{teal}{2-5} & $(3,\textcolor{blue}{1})$ & $(1,\textcolor{blue}{1})$ & $(1,\textcolor{teal}{1})$ \\
\hline
\textcolor{blue}{2-6} & $(\textcolor{blue}{3},\textcolor{blue}{1})$ & $(\textcolor{blue}{0},\textcolor{blue}{1})$ & $(\textcolor{blue}{0},2)$ \\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Compact View of $\Gamma_E$, Translated to an NFG $\Gamma_N$.}
\label{tbl:nfg}
\end{table}
However, in the Extensive Form Game $\Gamma_E$ the comparison of strategies has a certain order, as not all choices are made simultaneously. Thus, when it comes to $B$ choosing between option 3 and 4, choosing 3 is also a rational action because in any case $B$ gets utility 1. This is the case, since the subgame following after 3, is most likely to end in the subgame perfect and practical $(1,1)$. Following this argumentation, we claim that $(2-5,3)$, yielding history $(2,3)$ should also be considered rational and thus practical.
This shows that it is advisable to adapt the introduced concepts and that naive application can be problematic since information is lost during the transformation from EFG into NFG.
\end{example}
\subsection{Relation of Resilience Properties}
We state the omitted definition for $\textsf{SR}_{\subseteq}${} and then prove \Cref{lemma:impl}.
\begin{definition}[Strong Subset Resilience -- $\textsf{SR}_{\subseteq}$]
A joint strategy $\sigma \in \mathcal{S}$ is called \emph{strongly subset resilient} ($\textsf{SR}_{\subseteq}$), if no player of any subgroup $S \subseteq N$, $S:=\{s_1,...,s_j\}$ has an incentive in deviating from $\sigma$
\[\forall {\color{teal}S \subseteq N}\; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i}\; {\color{teal} \forall p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;) \;.\]
\end{definition}
For better readability, we restate the result from \Cref{sec:theory}.
\relations*
\begin{proof}
We start by showing property (2). Let $\sigma$ be \textsf{SR}{} and let $S \subset N$, $\sigma'_S\in \mathcal{S}_S$ be arbitrary but fixed. Then, for all $p \in S$ we have $u_p(\sigma) \geq u_p(\sigma'_S, \sigma_{\text{-}S})$ and thus also $ \sum_{p \in S} u_p(\sigma) \geq \sum_{p \in S} u_p(\sigma'_S, \sigma_{-S})$. Hence $\sigma$ is $\textsf{CR}${} and the implication is proven.
For implication (1) we see that $\textsf{SR}_{\subseteq}${} $\Rightarrow$ \textsf{SR}{} is trivial.
If the property is satisfied for every $S \subseteq N$, then it is also satisfied for every $S \subset N$.
By (2) and the transitivity of implication we also get $\textsf{SR}_{\subseteq}${} $\Rightarrow$ $\textsf{CR}$.
For the last implication let $\sigma$ be $\textsf{SR}_{\subseteq}${} and let $S \subseteq N, \, S \neq \emptyset$ and $\sigma'_S \in \mathcal{S}_S$ be arbitrary but fixed. Then there exists $p \in S$ and by definition all $p \in S$ satisfy $u_p(\sigma) \geq u_p(\sigma'_S, \sigma_{\text{-}S})$. Therefore, $\sigma$ is \textsf{sNE}{}.
To prove that no other implication holds between those four concepts, we provide three counterexamples. An overview of which game disproves which implication is given in \Cref{tbl:counterex}.
\begin{table}
\renewcommand*{1.3}{1.2}
\centering
\begin{tabular}{| c | c |c |c | c |}
\hline
$\to$ & \textsf{SR}{}& $\textsf{SR}_{\subseteq}$ & \textsf{sNE}{} & $\textsf{CR}$ \\
\hline
\textsf{SR}{} & \diagbox{\quad}{\quad} & $\Gamma_3$ & $\Gamma_3$ & $\checkmark$ \\
\hline
$\textsf{SR}_{\subseteq}$ & $\checkmark$ & \diagbox{\quad}{\quad} & $\checkmark$ & $\checkmark$ \\
\hline
\textsf{sNE}{} & $\Gamma_1$ & $\Gamma_1$&\diagbox{\quad}{\quad} & $\Gamma_1$ \\
\hline
$\textsf{CR}$ & $\Gamma_2$ & $\Gamma_2$ & $\Gamma_3$& \diagbox{\quad}{\quad}\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Overview of Implications and Counterexamples.}
\label{tbl:counterex}
\end{table}
The three-player NFG $\Gamma_1$ in \Cref{tbl:G1} shows a joint strategy $(H_1,H_2,H_3)$ that is not \textsf{SR}{}, nor $\textsf{SR}_{\subseteq}${}, nor $\textsf{CR}${}, but \textsf{sNE}{}. The deviation of player 2 and 3 to $(D_1,H_2,D_3)$ yields a strictly better utility for player 1, thus the honest strategy is not \textsf{SR}{}, nor $\textsf{SR}_{\subseteq}$. Also the joint utility increased from 2 to 3, therefore not $\textsf{CR}${}. However, player 2's utility decreases, hence it is considered a \textsf{sNE}{}.
The three-player game $\Gamma_2$ (\Cref{tbl:G2}) shows that a strong Nash Equilibrium and collusion resilient strategy is not necessarily strongly resilient and thus not $\textsf{SR}_{\subseteq}${} either. We consider the joint strategy $(H_1,H_2,H_3)$. If players 1 and 3 deviate to $(D_1,H_2,D_3)$ then the sum of their utility decreases strictly. Therefore, $(H_1,H_2,H_3)$ is $\textsf{CR}$. Since player 3's utility also decreases in doing so, it is a \textsf{sNE}{} as well. However, as the participating player $p_1$ has a strictly increased outcome in $(D_1,H_2,D_3)$, $(H_1, H_2,H_3)$ is not \textsf{SR}{} nor $\textsf{SR}_{\subseteq}$.
\begin{table} \centering
\begin{tabular}{|r|c|c|}
\hline
& $H_2$ & $D_2$ \\
\hline
$H_1$ & {\color{red}$(1,1)$} & $(1,1)$ \\
\hline
$D_1 $& $(1,1)$& $(2,2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_3$.}
\label{tbl:G3}
\end{table}
To prove the remaining implications incorrect, we consider the two-player game $\Gamma_3$ in \Cref{tbl:G3}. We can easily see that $(H_1,H_2)$ is not $\textsf{SR}_{\subseteq}$, nor \textsf{sNE}{}. This is the case, as all players $\{p_1,p_2\}$ can deviate to play $(D_1,D_2)$ which yields a strict increase for both. However, since no player profits from deviating alone, $(H_1,H_2)$ is still \textsf{SR}{} and $\textsf{CR}$. This proves the missing directions.
\hfill $\square$
\end{proof}
\section{Conclusions}\label{sec:conclusions}
Our work advocates the use of Extensive Form Games --EFGs for the game-theoretic security analysis of off-chain protocols. In particular, we introduce two instances of EFGs to model the closing and the routing of the Lightning Network. By doing so, we take the first step towards closing the gap existing security proof techniques have due to using informal arguments about rationality. We aim to close it further in future work by generalizing and extending our models to various off-chain protocols and by automating the security analysis thereof.
\section{Introduction}
Blockchain technologies are emerging as a revolutionary paradigm to perform secure decentralized financial applications. Nevertheless, a widespread adoption of cryptocurrencies such as Bitcoin~\cite{nakamoto2008bitcoin} and Ethereum~\cite{wood2014ethereum} is severely hindered by their inherent limitations on transaction throughput \cite{ScalingBC,ScalingBC2}. For instance, while Bitcoin can support tens of transactions per second and the confirmation time is about an hour, traditional credit networks like Visa can comfortably handle up to 47,000 transactions per second.
Off-chain protocols \cite{gudgeon2020sok} are recognized as one of the most promising scalability solutions, achieving a seemingly contradictory property: the bulk of transactions is performed off-chain, and yet in a secure fashion. The idea is to leverage the blockchain only in case of disputes, resorting otherwise to off-chain, peer-to-peer transactions. Bitcoin's Lightning Network \cite{lightning} is arguably the most famous off-chain protocol, being the most widely adopted realization.
In a nutshell, parties deposit money in a shared address, called channel, and can later on perform arbitrarily many off-chain transactions with each other by redistributing the deposit on the channel. In the end, the channel can be closed and the latest state (i.e., deposit distribution) is posted on-chain.
Off-chain transactions
are not limited to the end-point of the channel, but they
can be routed over paths of channels (so-called multi-hop payments). Besides such payment channel networks, an entire ecosystem of off-chain protocols~\cite{gudgeon2020sok} (virtual channels, watchtowers, payment-channel hubs, state channels, side-chains, etc.) is under development for Bitcoin~\cite{AMHL,GenChannels,AMEEFRHM21,AumayrMKM21,avarikioti2019brick,HTLC}, Ethereum~\cite{DziembowskiEFM19,DziembowskiEFHH19,McCorryBBM019,avarikioti2020cerberus}, as well as other cryptocurrencies~\cite{ThyagarajanMSS20}.
The cryptographic protocols underlying these off-chain constructions are rather sophisticated and, most importantly, rely on game-theoretic arguments to discourage malicious behavior. For instance, the Lightning Network relies on a punishment mechanism to disincentivize parties to publish old states on-chain and on an unlocking mechanism where parties first pay a neighbor and then retrieve the paid amount from the other to ensure the atomicity of multi-hop payments (i.e., either all channels are consistently updated or none is).
Unfortunately, the security proofs of these protocols typically concentrate on the cryptographic aspects and do not capture the game-theoretic ones. In particular, most protocols are proven secure in the Universal Composability framework~\cite{TCC:CDPW07}, proving that the cryptographic realization simulates the ideal functionality. This framework, however, was developed to reason about security in the classical honest/Byzantine setting: in particular, the ideal functionality has to model all possible parties' behavior, rational and irrational, otherwise it would not be simulatable, but reasoning on whether or not certain behavior is rational is outside of the model and thus left to informal arguments. This is not just a theoretical issue, but a practical one, as there is the risk to let attacks pass undetected: for instance, the Wormhole attack
~\cite{AMHL} constitutes a rational behavior in the Lightning Network, which is thus admitted in any faithful model thereof although it undermines its incentive mechanism. The first step towards closing this gap in cryptographic proofs is to come up with a \emph{faithful game-theoretic model for off-chain protocols} in order to reason about security in the presence of rational parties. We address this in our paper, advocating the use of Extensive Form Games -- EFGs for the game-theoretic security analysis of off-chain protocols. In particular, we introduce two instances of EFGs to model closing and routing of the Lightning Network.
\medskip
\noindent
\textbf{Related Work.}
Off-chain protocols are typically subject to rigorous security analysis~\cite{AMHL,GenChannels,AumayrMKM21,AMEEFRHM21,DziembowskiEFM19,DziembowskiEFHH19}, which is predominantly formulated in the (Global) Universal Composability framework~\cite{TCC:CDPW07}.
Intuitively, these proofs guarantee that the protocols are indistinguishable from an ideal functionality.
This means that behavior that is possible in the protocol but leads to financial loss (e.g., posting an old state) is possible in the ideal functionality too.
These proof techniques, however, do not capture whether or not a certain behavior is rational, and in particular if the expected protocol execution is the only one,
which is left to informal arguments and may thus lead to overlooking attacks. In this work we address this gap by using a game-theoretical approach. It complements other game-theoretic advancements in the area, most prominently the following lines of research.
\smallskip
\noindent
\textit{Game Theory for Off-Chain Protocols.}
The most closely related work is a recent game-theoretic analysis of the Lightning Network by Zappalà et al.~\cite{CITE}. The proposed model, though, incurs several limitations.
Firstly, the model considers only honest closing of channels, i.e., deviations such as posting an old state are ignored: this makes the model not suitable to reason about the security of basic channel operations. Secondly, fees are not modeled, thereby ignoring their impact on Lightning protocols. In particular, the routing game to model the security of multi-hop payments fails to capture already identified attacks in payment channel networks, like the Wormhole attack~\cite{AMHL} that targets the fee distribution among players.
In our work, we define a stronger closing phase model, by aligning the utilities to the monetary outcome and by considering all possible deviations of parties during closing. Furthermore, we refine the routing game~\cite{CITE} to capture attacks, like the Wormhole attack, which Lightning's routing is vulnerable to.
\smallskip
\noindent
\textit{Incentivizing Watchtowers.}
A major drawback of payment channel protocols is that channel participants must frequently be online and watch the blockchain to prevent cheating. To alleviate this issue, the parties can employ third parties, or so-called watchtowers, to act on their behalf in case the opponent misbehaves. But correctly aligning the incentives of watchtowers to yield a secure payment channel protocol is challenging. This is the main focus of several diverging works~\cite{McCorryBBM019,avarikioti2018towards,avarikioti2020cerberus,avarikioti2019brick}.
As their objective is to incentivize external parties, their models does not apply in our work.
\smallskip
\noindent
\textit{Payment Channel Network Creation Games.}
Avarikioti et al.~\cite{avarikioti2020ride,avarikioti2019payment} study payment channel networks as network creation games. Their goal is to determine which channels a rational node should establish to maximize its profit.
Ersoy et al.~\cite{ersoy2020profit} undertake a similar task; they formulate the same problem
as an optimization problem, show it is NP-hard and present a greedy algorithm to approximate it.
Similarly to our work, all these works assume rational participants. However, we aim to model the security of the fundamental constructions, in contrast to these works that study the network creation problem graph-theoretically.
\smallskip
\noindent
\textit{Blockchains with Rational Players.}
Blockchains incentivize miners to participate in the network via monetary rewards~\cite{nakamoto2008bitcoin}. Therefore, analyzing blockchains under the lens of rational participants is critical for the security of the consensus layer. There are multiple works in this direction:
Badertscher et al.~\cite{badertscher2018but} present a rational analysis of the Bitcoin protocol.
Eyal and Garay~\cite{eyal2014majority} introduce an attack on the Nakamoto consensus, effectively demonstrating that rational miners will not faithfully follow the Bitcoin protocol. This attack is generalized in~\cite{kwon2017selfish,sapirshtein2016optimal}.
Consequently, Kiayias et al.~\cite{kiayias2016blockchain} analyze how miners can deviate from the protocol to optimize their expected outcome.
Later, Chen et al.~\cite{chen2019axiomatic} investigate the reward allocation schemes in longest-chain protocols and identify Bitcoin's allocation rule as the only one that satisfies a specific set of desired properties.
On a different note, several works study the dynamics of mining pools from a game-theoretic perspective~\cite{eyal2015miner,teutsch2016cryptocurrencies} or introduce network attacks that may increase the profit of rational miners~\cite{heilman2015eclipse,nayak2016stubborn}.
An overview of game-theoretic works on blockchain protocols can be found in~\cite{SurveyOnBC}.
All these works, however, focus on the consensus layer (Layer-1) of block\-chains and as both the goals and assumptions are different from the application layer (Layer-2), the models introduced cannot be employed for our purposes.
For instance, payment channel protocols occur off-chain and thus game-based cryptographic assumptions of the blockchain do not apply.
In addition, consensus protocols investigate the expected reward of miners which is a probabilistic problem, whereas we ask if any honest player could lose money, which depends on the behavior of the other players and is fundamentally deterministic.
Game-based definitions have also been proposed for the security analysis of smart contracts~\cite{QuantAnalysisSC,ProbSC}. These models, however, target an on-chain setting and are thus not suitable to reason about the specifics of off-chain constructions (e.g., closing games, routing games, etc.).
\medskip
\noindent
\textbf{Our Contributions.}
In this work, we take the first steps towards closing the gap between security and game-theoretic analysis of off-chain protocols. Specifically, we introduce the first game-theoretic models that are expressive enough to reason about the security of off-chain protocols.
We model off-chain protocols as games and then analyze whether or not certain security properties are satisfied.
The design of our models is driven by two principles: (a)~all possible actions should be represented and (b)~the utility function should mirror the monetary outcome realistically.
We aim to ensure that \textit{honest participants do not suffer any damage (P1)}, whereas \textit{deviating from the protocol yields a worse outcome for the adversary (P2)}. While we believe that our approach is easily extensible to other off-chain protocols, in this work we focus on the Bitcoin Lightning Network.
Our contributions can be summarized as follows:
\begin{itemize}
\item We refine existing game-theoretical concepts in order to reason about the security of off-chain protocols (\Cref{sec:theory}).
\item
We introduce the Closing Game $G_c$, the first game-theoretic security model that accurately captures the closing phase of Lightning channels, encapsulating arbitrary deviations from the protocol specification (Section~\ref{sec:models}).
\item We perform a detailed security analysis of $G_c$, formalize folklore security corner cases of Lightning,
and present the strategy that rational parties should follow to close their channels in order to maximize their expected outcome relative to the current and previous distribution states (\Cref{sec:sec}).
\item We identify problems in prior work \cite{CITE} on game-based modeling of multi-hop payments, putting forward a new game-based definition that is precise enough to cover the Wormhole attack (\Cref{sec:refine}).
\end{itemize}
\section{Closing Games} \label{sec:models}
We now define a new two-player EFG, called the \emph{Closing Game $G_c$},
in order to model closing phase properties of off-chain protocols, in particular of the Lightning Network.
Our closing game overcomes the limitations of~\cite{CITE} in representing dishonest closing attempts, modeling how closing can be achieved after a failed collaborative closing attempt and also considering the additional fee to be paid in a revocation transaction.
To the best of our knowledge, our closing game $G_c$ is the most accurate model for the security analysis of off-chain protocols, notably of the Lightning Network.
In our model of the closing phase we make the following assumptions for a channel between $A$ and $B$.
\begin{itemize}
\vspace{-5pt}
\item The fair split of the channel's funds is $a \to A$, $b \to B$ and $a>0$, $b>0$.
\item The benefit of closing the channel is $\alpha$.
\item The opportunity cost of having to wait for one's funds upon closing is $\epsilon$.
\item When both players agree to update the channel we assume a fair deal in the background which yields a profit of $\rho$ for both parties.
\end{itemize}
Further, to properly model utilities in the closing game $G_c$, we define the following total order, which is crucial for $G_c$'s security properties.
\begin{definition}[Utility Order]
We consider the total order $(\mathbb{U},\preccurlyeq)$, where $\mathbb{U}$ is the group resulting from closing $\mathbb{R} \;\dot{\cup}\; \{\alpha,\epsilon,\rho \}$ under addition. The total ordering $\preccurlyeq$ is uniquely defined by the following conditions.
\begin{enumerate}
\vspace{-3pt}
\item On $\mathbb{R}$, the relation $\preccurlyeq$ is the usual less than or equal relation $\preccurlyeq|_\mathbb{R}\: := \:\leq$.
\item The values $\alpha$, $\epsilon$ and $\rho$ are greater than 0, $ \forall \xi \in \{\alpha,\epsilon,\rho\}: \; -\xi\prec 0 \prec \xi \;.$
\item The values $\alpha$, $\epsilon$ and $\rho$ are closer to 0 than any real number, $\forall x \in \mathbb{R}, \xi \in \{\alpha,\epsilon,\rho\}:\text{ if } x>0 \text{ then } \xi \prec x \text{ and } -x \prec -\xi . $
\item Additionally, $\alpha$, $\epsilon$ and $\rho$ have the order $\rho \prec \epsilon \prec \alpha$.
\end{enumerate}
\end{definition}
\begin{table}[t]
\centering
\begin{tabularx}{\linewidth}{|@{\hspace{.5em}}>{\bfseries}l@{\hspace{.5em}}X@{\hspace{.5em}}|}
\hline
$H$ & Close unilaterally and \emph{honestly} without reacting to a previous move, such as a collaborative closing attempt. \\
\hline
$D$ & Close unilaterally but \emph{dishonestly} (without reacting to a previous move) with a profit of $d_A \in (0,b]$ in $A$'s case, $d_B \in (0,a]$ in $B$'s case.\\
\hline
$C_h$ & Try to close \emph{collaboratively} and \emph{honestly}, that is proposing a fair split.\\
\hline
$C_c$ & Try to close \emph{collaboratively} but by \emph{cheating} the other party by $c \in (0,b]$, that means proposing an unfair split.\\
\hline
$S$ & \emph{Signing} the collaborative closing attempt of the other player.\\
\hline
$I$ & \emph{Ignore} the previous action and do nothing.\\
\hline
$P$ & \emph{Prove} other party tried to close dishonestly. That means stating a revocation transaction. We assume the attempt to do so is always successful, that is that the miners behave honestly.\\
\hline
$U^+$ & Propose an \emph{update} of the channel where player $A$'s balance is \emph{increased} by $p_A \in (0,b]$.\\
\hline
$U^-$ & Propose an \emph{update} where player $A$'s balance is \emph{decreased} by $p_B \in (0,a]$.\\
\hline
$A$ & Agree to a proposed update.\\
\hline
\end{tabularx}
\vspace{0.1cm}
\caption{Possible Actions in $G_c(A)$.}
\label{tbl:actions}
\end{table}
\begin{table}[t]
\centering
\begin{tikzpicture}[scale=1,->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0.125,0.5) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-1.25,-1) { $(\alpha-\epsilon,\alpha)$};
\node (4) at (0.25,-0.75) {\color{olive}$B$};
\node (5) at (3.75,-1) {\color{olive}$B$};
\node (6) at (-5,-4.25) {\color{teal}$A$};
\node (7) at (-3.5,-3) { $(\alpha,\alpha)$};
\node (8) at (1.25,-1.4) { $(-a,a-f+\alpha)$};
\node (9) at (-0.5,-1.9) { $(d_A+\alpha-\epsilon, -d_A+\alpha)$};
\node (10) at (3.25,-3) {$(c+\alpha,-c+\alpha)$};
\node (11) at (5.25,-4.25) {\color{teal}$A$};
\node (12) at (-5,-7) {\color{olive}$B$};
\node (15) at (-4.1,-6.75) {\color{red}$S_1$};
\node (16) at (-3.3,-6.5) {\color{red}$S_2$};
\node (13) at (-1.75,-5.75) {$(\alpha-\epsilon,\alpha)$};
\node (14) at (-2.25,-6.25) {$(-a,-b)$};
\node (19) at (2.5,-6.25) {$(-a,-b)$};
\node (20) at (2,-5.75) { $(\alpha-\epsilon,\alpha)$};
\node (17) at (3.55,-6.5) {\color{red}$S'_2$};
\node (18) at (4.35,-6.75) {\color{red}$S'_1$};
\node (21) at (5.25,-7) {\color{olive}$B$};
\node (22) at (-2.25,-7.5) {$(-a,a-f+\alpha)$} ;
\node (23) at (-3.75,-8) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (24) at (2.5,-7.5) { $(-a,a-f+\alpha)$};
\node (25) at (4,-8) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (27) at (-1.5,-2.75) {$(\alpha,\alpha-\epsilon)$};
\node (28) at (-5,-2) {\color{red}$S_3$};
\node (29) at (-5,-2.75) {\color{red}$S_4$};
\node (30) at (-2.6,-2.75) {\color{teal}$A$};
\node (31) at (1.5,-3.25) {\color{teal}$A$};
\node (32) at (5.25,-2.75) {\color{red}$S'_4$};
\node (33) at (5.25,-2) {\color{red}$S'_3$};
\node (34) at (0.75,-2.75) {$(\alpha,\alpha-\epsilon)$};
\node (36) at (-3,-4) { $(b-f+\alpha,-b)$} ;
\node (37) at (-1.25,-4.5) { $(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (38) at (3.25,-4) { $(b-f+\alpha,-b)$};
\node (39) at (2,-4.5) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [left, pos=0.5] {\tiny $H$} (3);
\path (1) edge node [right, pos=0.6] {\tiny $D$} (4);
\path (1) edge node [above, pos=0.5] {\tiny $C_c$} (5);}
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7);
\path (4) edge node [above] {\tiny $P$} (8);
\path (4) edge node [left] {\tiny $I$} (9);
\path (5) edge node [right] {\tiny $S$} (10);
\path (5) edge node [right] {\tiny $I$} (11); }
{\color{teal}
\path (6) edge node [right, pos=0.7] {\tiny $D$} (12);
\path (6) edge node [above, pos=0.7] {\tiny $H$} (13);
\path (6) edge node [above, pos=0.7] {\tiny $I$} (14);
\path (6) edge node [right, pos=0.7] {\tiny $U^+$} (15);
\path (6) edge node [right, pos=0.7] {\tiny $U^-$} (16);
\path (11) edge node [left, pos=0.7] {\tiny $U^-$} (17);
\path (11) edge node [left, pos=0.7] {\tiny $U^+$} (18);
\path (11) edge node [above, pos=0.7] {\tiny $I$} (19);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (20);
\path (11) edge node [left, pos=0.7] {\tiny $D$} (21);}
{\color{olive}
\path (12) edge node [below] {\tiny $P$} (22);
\path (12) edge node [left] {\tiny $I$} (23);
\path (21) edge node [below] {\tiny $P$} (24);
\path (21) edge node [right] {\tiny $I$} (25); }
{\color{olive}
\path (2) edge node [left, pos=0.5] {\tiny $H$} (27);
\path (2) edge node [above, pos=0.5] {\tiny $U^+$} (28);
\path (2) edge node [left, pos=0.5] {\tiny $U^-$} (29);
\path (2) edge node [left, pos=0.6] {\tiny $D$} (30);
\path (5) edge node [right, pos=0.4] {\tiny $D$} (31);
\path (5) edge node [right, pos=0.55] {\tiny $U^-$} (32);
\path (5) edge node [above,pos=0.65] {\tiny $U^+$} (33);
\path (5) edge node [right, pos=0.5] {\tiny $H$} (34);}
{\color{teal}
\path (30) edge node [right] {\tiny $P$} (36);
\path (30) edge node [right] {\tiny $I$} (37);
\path (31) edge node [above] {\tiny $P$} (38);
\path (31) edge node [left] {\tiny $I$} (39);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing Game $G_c(A)$.}
\label{tbl:Gc}
\end{table}
Based on this ordering, we introduce our \emph{Closing Game for Player $A$} below.
\begin{definition}[Closing Game $G_c(A)$ of Player $A$] \label{def:closing}
The \emph{Closing Game} $G_c(A)=(N, \mathcal{H}, P, u)$ is an EFG with two players $N=\{A,B\}$. The tree representation of $G_c(A)$ in \Cref{tbl:Gc} defines $\mathcal{H}$, $P$ and $u$\footnote{The subgames $S_i$, $S'_i$ are given in \Cref{app:subgames}.}. The actions of the game are explained in \Cref{tbl:actions}.
The \emph{utility function} $u$ of $G_c(A)$ assigns player $p\in N$ the money $p$ received minus the money $p$ deserved based on the latest channel state. Also,
the value of closing ($\alpha$), updating ($\rho$) and waiting ($-\epsilon$) is considered.
\end{definition}
The fee that is needed for the closing transaction is assumed to remain reserved among the locked funds in the channel all the time and is spent upon closing, therefore not affecting the players' channel balance.
The closing game for player $B$, $G_c(B)$, is symmetric to $G_c(A)$, with the roles of $A$ and $B$ being swapped.
Based on the closing games $G_c(A)$, $G_c(B)$, we can now define the closing phase in an off-chain channel, as follows.
\vspace{-3pt}
\begin{definition}[Closing Phase]
The \emph{closing phase} of an off-chain channel modeled by a closing game $G_c(A)$ is initiated in one of three ways: (i) $A$ starts with a closing action, and thus triggers the {closing game} $G_c(A)$;
(ii) $B$ starts with a closing action does and triggers $G_c(B)$;
or (iii) none of the players $A$ and $B$ ever start closing, in which case the money stays locked in the channel. Then, we get the EFG
\hspace{0.4\textwidth} \begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$A$};
\node (2) at (0,-1) {$G_c(A)$} ;
\node (3) at (2,0) {$B$};
\node (4) at (2,-1) {$G_c(B)$};
\node (5) at (4,0) {$(-a,-b)$};
\path (1) edge (2);
\path (1) edge (3);
\path (3) edge (4);
\path (3) edge (5);
\end{tikzpicture} .
\end{definition}
\vspace{-3pt}
\section{Background and Preliminaries}\label{sec:prelim}
\subsection{Payment Channel Networks}
A payment channel~\cite{GenChannels} can be seen as an escrow (or multi-signature), into which two parties $A$ and $B$ transfer their initial coins with the guarantee that their coins are not locked forever and the agreed balance can be withdrawn at any time.
After that, $A$ and $B$ can pay each other off-chain by signing and exchanging transactions that reflect the updated balances in the escrow. These signatures can be used at any time to close the channel and distribute the coins on-chain according to the last channel state. In order to discourage parties from posting an old state on-chain, a punishment mechanism is in place.
In particular, in Lightning~\cite{lightning}, once $A$ closes the channel, she has to wait a mutually agreed time before getting her coins; meanwhile, $B$ has the possibility to withdraw all the coins in the channel, including the one assigned to $A$, if the state posted on-chain by $A$ is not the last one they mutually agreed on.
We note such a punishment mechanism is of game-theoretic nature: parties can indeed post an old state on-chain, yet they are discouraged to do so.
Off-chain transactions are not limited to the end-points of a channel, as they can be performed whenever sender and receiver are connected by a path of channels with enough capacity.
This is illustrated in \Cref{tbl:wormhole}, where we assume there exists a path from $A$ to $B$ in the payment network with 3 intermediaries $E_1$, $I$, and $E_2$. Notice that each intermediary charges a fee $f$ for the routing service, hence $A$ should pay $m+3f$, where $m$ is the amount to be paid to $B$. The core idea is that $A$ pays $E_1$, $E_1$ pays $I$, and so forth until $B$ gets paid.
A key security property in multi-hop payments is \emph{atomicity}: either all payments go through, and the deposit in each channel is updated accordingly, or none does. To achieve this property, the Lighting protocol proceeds as follows: First, the receiver $B$ generates a secret $x$ and sends its hash $y$ to the sender $A$ (action 1 in \Cref{tbl:wormhole}). Then $A$ and the others, from left to right, pay their neighbor conditioned to $y$ (actions 2 -- 5 in \Cref{tbl:wormhole}), i.e., the right end-point of the channel can claim the money only by revealing $x$. Once $B$ receives the conditional payment, he can reveal $x$ and the conditional payments are unlocked from right to left (actions 6 and 8 in \Cref{tbl:wormhole}).
We note that atomicity is achieved by a game-theoretic argument:
intermediaries can, in principle, stop the protocol either in the locking phase or in the unlocking phase. In the former, they would lose the transaction fee $f$, while in the latter, they would lose the payment amount $m$. Thus, they are incentivized to act once they have committed to participate.
For further details on the routing mechanism, see \Cref{app:rout}.
\begin{figure}[tb!]
\centering
\begin{tikzpicture}[scale= 0.8, ->]
\node (1) at (0,0) {$A$};
\node (2) at (2.5,0) {\textcolor{teal}{$E_1$}};
\node (3) at (5,0) {$I$};
\node (4) at (7.5,0) {\textcolor{teal}{$E_2$}};
\node (5) at (10,0) {$B$};
\path (5) edge[out=100, in=80, distance= 1.5cm] node [below] {\color{red} \small $y$} node [ below, pos=0.3] {1.} (1);
\path (1) edge node [above] {\small $(m+3f,\textcolor{red}{y})$} node [ below, pos=0.3] {2.}(2);
\path (2) edge node [above] {\small $(m+2f,\textcolor{red}{y})$} node [ below, pos=0.3] {3.} (3);
\path (3) edge node [above] {\small $(m+f,\textcolor{red}{y})$} node [ below, pos=0.3] {4.} (4);
\path (4) edge node [above] {\small $(m,\textcolor{red}{y})$} node [ below, pos=0.3] {5.} (5);
\path (5) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {6.} (4);
\color{teal} \path (4) edge[out=240, in=300, distance= 1cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.2] {7.} (2);
\path (2) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {8.} (1);
\end{tikzpicture}
\vspace{-0.2cm}
\caption{Wormhole Attack in Lightning.}
\vspace{5pt}
\label{tbl:wormhole}
\end{figure}
\paragraph{\bf The Wormhole attack.}
The aforementioned routing protocol is proven to be vulnerable to the \emph{Wormhole attack}~\cite{AMHL}, which is depicted in \Cref{tbl:wormhole}.
The attack is as follows: $E_1$ and $E_2$ collude, and bypass $I$ in the unlocking phase, meaning $E_2$ reveals $x$ directly to $E_1$ instead of $I$. The parties $A$ and $B$ are not affected. However, $E_1$ and $E_2$ collectively earn $3f$ instead of the $2f$ they deserve, stealing the fee $f$ from $I$, who locked resources in the first phase of the protocol.
This attack undermines the incentive of intermediaries to route payments.
\subsection{Game-Theoretic Definitions}
We denote real numbers by $\mathbb{R}$. We understand games as static objects in which finitely many players can choose finitely many times from a finite set of actions. A game yields a certain positive or negative utility for each player.
We first introduce Normal Form Games~\cite{GameTheoryBook} which are the most common type of games.
\begin{definition}[Normal Form Game -- NFG] \label{def:nfg}
A \emph{Normal Form Game (NFG)} is a tuple $\Gamma=(N,\mathcal{S}, u)$, where $N$ is the set of game \emph{players}, $\mathcal{S}=\vartimes_{p \in N} \mathcal{S}_{p}$ the set of \emph{joint strategies} $\sigma$ and $u$ the \emph{utility function}:
\begin{itemize}
\vspace{-5pt}
\item $\mathcal{S}_p$ is the non-empty set of strategies player $p$ can choose from. Thus, a joint strategy $\sigma \in \mathcal{S}$ is a tuple of strategies $\sigma=(\sigma_{p_1},...,\sigma_{p_{|N|}})$, with $\sigma_{p_i}\in \mathcal{S}_{p_i}$.
\item $u=(u_{p_1},\dots,u_{p_n})$, where $u_{p_i}: \mathcal{S} \to \mathbb{R}$ assigns player $p_i$ its utility for every joint strategy $\sigma \in \mathcal{S}$.
\end{itemize}
\end{definition}
To formalize an optimal outcome on game strategies, we use the concept of a Nash Equilibrium.
\begin{definition}[Nash Equilibrium]\label{def:NE}
A \emph{Nash Equilibrium} is a joint strategy $\sigma \in \mathcal{S}$ s.t.\ no player can increase its utility by unilaterally deviating from $\sigma$. Formally,
$\forall p \in N \; \forall \sigma'_p \in \mathcal{S}_p:\; u_p(\sigma) \geq u_p(\;\sigma[\sigma'_p/\sigma_p]\;)\;.$
\end{definition}
Another important concept is \emph{weakly dominated strategies}, expressing the strategies a rational player would not play.
\begin{definition}[Weakly Dominated Strategy]
A strategy $\sigma_p^d \in \mathcal{S}_p$ of player $p$ is called \emph{weakly dominated} by strategy $\sigma'_p \in \mathcal{S}_p$, if it always yields a utility at most as good as $\sigma'_p$ and a strictly worse utility at least once:
\begin{align*}
&\forall\, \sigma \in \mathcal{S}:\; u_p(\;\sigma[\sigma^d_p/\sigma_p]\;) \leq u_p(\;\sigma[\sigma'_p/\sigma_p]\;) \; \text{and} \\
& \exists\, \sigma \in \mathcal{S}:\; u_p(\;\sigma[\sigma^d_p/\sigma_p]\;) < u_p(\;\sigma[\sigma'_p/\sigma_p]\;) \;.
\end{align*}
\end{definition}
To formalize strategies where players make multiple choices one after the other, we consider Extensive Form Games, which extend NFGs as follows.
\begin{definition}[Extensive Form Game -- EFG]
An \emph{Extensive Form Game (EFG)} is a tuple $\Gamma=(N,\mathcal{H},P,u)$, where $N$ and $u$ are as in NFGs. The set $\mathcal{H}$ captures \emph{game histories}, $\mathcal{T} \subseteq \mathcal{H}$ is the set of \emph{terminal histories} and $P$ denotes the \emph{next player function}, satisfying the following properties.
\begin{itemize}
\vspace{-3pt}
\item The set $\mathcal{H}$ of histories is a set of sequences of actions with \begin{enumerate}
\item $\emptyset \in \mathcal{H}$;
\item if the action sequence $(a_k)_{k=1}^K \in \mathcal{H}$, $L<K$, then also $(a_k)_{k=1}^L \in \mathcal{H}$;
\item a history is terminal $(a_k)_{k=1}^K \in \mathcal{T}$, if there is no action $a_{K+1}$ with $(a_k)_{k=1}^{K+1} \in \mathcal{H}$.
\end{enumerate}
\item The next player function $P$
\begin{enumerate}
\item assigns the next player $p \in N$ to every non-terminal history $(a_k)_{k=1}^K \in \mathcal{H}\setminus \mathcal{T}$, that is $P((a_k)_{k=1}^K) = p$;
\item after a non-terminal history $h= (a_k)_{k=1}^K \in \mathcal{H}$, the player $P(h)$ chooses an action from the action set $A(h)=\{a: (h,a) \in \mathcal{H}\}$.
\end{enumerate}
\end{itemize}
A \emph{strategy} of player $p$ is a function $\sigma_p$ mapping every $h \in \mathcal{H}$ with $P(h)=p$ to an action from $A(h)$. The utilities of all joint strategies with terminal history $h$ are the same.
\end{definition}
EFGs can be well-represented via a tree: (i) a history $h$ is a path starting from the root, it is terminal iff it ends in a leaf; (ii) each internal tree node models the turn of an EFG player $p$ after a non-terminal history $h$. The respective node is labeled $p$. The outgoing edges represent the action set $A(h)$. The edges are oriented from the root to the leaves. Further, (iii) each leaf represents the joint utility of every joint strategy, whose history (path) leads to it.
Note that an EFG strategy $\sigma$ is not a history (path) but a set of edges that contains precisely one outgoing edge per internal node.
In the context of EFGs, the concept of
\emph{Nash Equilibria} remains as given in \Cref{def:NE}. In addition to Nash Equilibria, another important concept for EFGs is a \emph{Subgame Perfect Equilibrium}, characterizing the strategies played in practice by rational parties. To this end, we first introduce subgames of EFGs.
\begin{definition}[Subgame of EFG]
The \emph{subgame} of an EFG $\Gamma=(N,\mathcal{H},P,u)$ that follows history $h\in \mathcal{H}$ is the EFG $\Gamma(h)=(N,\mathcal{H}_{|h}, P_{|h}, u_{|h})$ s.t.\ for every sequence $h' \in \mathcal{H}_{|h}$ we have $(h,h') \in \mathcal{H}$, $P_{|h}(h'):= P(h,h')$ and $u_{|h}(h')=u(h,h')$.
\end{definition}
\begin{definition}[Subgame Perfect Equilibrium]
A \emph{subgame perfect equilibrium} is a joint strategy $\sigma \in \mathcal{S}$, s.t.\ $\sigma_{|h}$ is a Nash Equilibrium of the subgame $\Gamma(h)$, for every $h \in \mathcal{H}$.
\end{definition}
\subsection{Game-Theoretic Security Properties of Off-Chain Protocols}\label{sec:prelin:NFGSec}
We now present game-theoretic concepts implying security properties of off-chain protocols, by extending the setting of~\cite{CITE}.
In Section~\ref{sec:theory}, we further extend these concepts towards EFGs, enabling our security analysis in Section~\ref{sec:models}. We focus on two security properties ensuring that (P1) honest players do not suffer damage and (P2) subgroups of rational players do not deviate from a respective strategy. Variations of these properties have been formalized for NFGs in \cite{CITE}.
\paragraph{{\bf (P1) No Honest Loss.}} As the utility function of a game is supposed to display the monetary and intrinsic value of a certain joint strategy, property (P1) is expressed using \emph{weak immune strategies} defined next.
\begin{definition}[Weak Immunity]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is called \emph{weak immune}, if no honest player $p$ loses, regardless of how the others behave:
$\forall p \in N\;\;\forall \sigma'\in \mathcal{S}:\quad u_p(\; \sigma'[\sigma_p/\sigma'_p]\;)\geq 0 \;.$
\end{definition}
\paragraph{{\bf (P2) No Deviation.}}
Note that a Nash Equilibrium cannot capture the case were two or more players deviate, nor whether a strategy will be played in practice. Therefore, the combination of \emph{strong resilience} and \emph{practicality} was introduced in \cite{CITE} to address (P2). Strong resilience extends Nash Equilibria by considering deviations of multiple players.
\begin{definition}[Strong Resilience -- \textsf{SR}]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is \emph{strongly resilient (\textsf{SR})} if no proper subgroup of players $S:=\{s_1,...,s_j\}$ has an incentive in deviating from $\sigma$:
\noindent ${\color{teal} \forall S \subset N}\;\;\forall \sigma'_{s_i} \in \mathcal{S}_{s_i}\;\; {\color{teal} \forall p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;) \;.$
\end{definition}
In our work, we exclude those strategies of player $p$ which are outperformed by others. We therefore impose the concept of practicality using iterated deletion of weakly dominated strategies (see \Cref{app:theory}).
\begin{definition}[Practicality]
A strategy is \emph{practical} if it is a Nash Equilibrium of the game $\Gamma'$ after iterated deletion of weakly dominated strategies.
\end{definition}
We further note that a similar property to strong resilience can be captured by \emph{strong Nash Equilibria.}
\begin{definition}[Strong Nash Equilibrium -- \textsf{sNE}]
A joint strategy $\sigma$ is a \emph{strong Nash Equilibrium} (\textsf{sNE}) if for every group of deviating players $S:=\{s_1,...,s_j\}$ and all possible deviations $\sigma'_{s_i} \in \mathcal{S}_{s_i}$, $i\in \{1,...,j\}$ at least one player $p \in S$ has no incentive to participate, that is
$ \forall {\color{teal} S \subseteq N,\, S \neq \emptyset} \;\; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i} \;\; {\color{teal} \exists p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;)\;. $
\end{definition}
An alternative approach for expressing (P2) is thus by using a strategy $\sigma$ that is \textsf{sNE}{} and practical, instead of \textsf{SR}{} and practical. A detailed comparison of the various concepts ensuring (P2) is given in \Cref{sec:theory}.
\section{Further Game Extensions for Security Analysis}\label{sec:refine}
We demonstrate the general applicability of EFGs by modeling the routing mechanism of Lightning.
In particular, we propose a game-theoretic model, illustrated in \Cref{tbl:mymodel}, that refines the one from~\cite{CITE} in order to capture the Wormhole attack \cite{AMHL}.
Specifically, this model considers fees and allows the intermediaries to choose not to claim their money using $x$, but instead to forward it to another intermediary, besides being honest or doing nothing. This refinement suffices to prove the routing model not secure, so we indicate possible other actions by ``...''.
\vspace{-3pt}
\begin{definition}[Refined Routing Game $G_{\text{ref}}$]
The \emph{refined routing game} $G_{\text{ref}}$ is the 5-player-game given in \Cref{tbl:mymodel} \footnote{The utility tuples in \Cref{tbl:mymodel} assign the first value to $A$, the second to $E_1$, the third to $I$, the fourth to $E_2$ and the last to $B$.}. The actions of $G_{\text{ref}}$ are $I$ \emph{ignoring} the situation and doing nothing; $H$ following the protocol \emph{honestly}; $D$ behaving \emph{dishonestly} towards the Wormhole attack; and $O$ performing any \emph{other} action.
\end{definition}
\vspace{-1pt}
\begin{table}[t]
\centering
\begin{tikzpicture}[,->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$B$};
\node (7) at (1.2,-0.5) {$(0,0,0,0,0)$} ;
\node (12) at (-1,-0.5) {$A$};
\node (13) at (0.25,-1) {$(0,0,0,0,0)$};
\node (2) at (-2,-1) {$E_1$};
\node (8) at (-0.75,-1.5) {$(0,0,0,0,0)$} ;
\node (14) at (-3, -1.5) {$I$};
\node (15) at (-1.75, -2) {$(0,0,0,0,0)$};
\node (3) at (-4,-2) {$E_2$};
\node (9) at (-2.75,-2.5) {$(0,0,0,0,0)$} ;
\node (16) at (-5,-2.5) {$B$};
\node (17) at (-3.75, -3) {$(0,0,0,0,0)$};
\node (4) at (-6,-3) {$E_2$};
{\color{teal}
\node (20) at (-6,-1) {$E_1$};
\node (21) at (-7.85,-1.925) {$(m+3f+\rho,0,0,-m,\rho)$};
\node (22) at (-6,0.5) {\color{blue}$(\rho,\;m+3f,\;0,\;-m,\;\rho)$};
}
\node (10) at (-3.5,-3.75) {$(m+3f+\rho,0,0,-m,\rho)$} ;
\node (18) at (-7, -3.5) {$I$};
\node (19) at (-4.4, -4.25) {$(m+3f+\rho,0,-m-f,f,\rho)$};
\node (5) at (-8,-4) {$E_1$};
\node (6) at (-8,-5.25) {\textcolor{red}{$(\rho,f,f,f,\rho)$}} ;
\node (11) at (-5.5,-4.75) {$(m+3f+\rho,-m-2f,f,f,\rho)$} ;
{\color{teal}
\node (23) at (-2,0) {...};
\node (24) at (-3, -0.5) {...};
\node (25) at (-4, -1) {...};
\node (26) at (0, 1) {...};
\node (27) at (-1, 0.5) {...};
\node (28) at (-5, -1.5) {...};
\node (29) at (-7, -2.5) {...};
\node (30) at (-8, -3) {...};
\node (31) at (-5, -0.5) {...};
}
\path (1) edge node[above] {\tiny $H$} (12);
\path (1) edge node[ above, pos=0.7] {\tiny $I$} (7);
\path (12) edge node[above] {\tiny $H$} (2);
\path (12) edge node[above, pos=0.7] {\tiny $I$} (13);
\path (2) edge node[above] {\tiny $H$} (14);
\path (2) edge node[above, pos=0.7] {\tiny $I$} (8);
\path (14) edge node[ above] {\tiny $H$} (3);
\path (14) edge node[ above, pos=0.7] {\tiny $I$} (15);
\path (3) edge node[above] {\tiny $H$} (16);
\path (3) edge node[above, pos=0.7] {\tiny $I$} (9);
\path (16) edge node[ above] {\tiny $H$} (4);
\path (16) edge node[ above, pos=0.7] {\tiny $I$} (17);
\path (4) edge node[above] {\tiny $H$} (18);
\path (4) edge node[above, pos=0.5] {\tiny $I$} (10);
\path (18) edge node[ above] {\tiny $H$} (5);
\path (18) edge node[ above, pos=0.5] {\tiny $I$} (19);
\path (5) edge node[left] {\tiny $H$} (6);
\path (5) edge node[above, pos=0.5] {\tiny $I$} (11);
{\color{teal}
\path (4) edge node[left, pos=0.25] {\tiny $D$} (20);
\path (20) edge node[above] {\tiny $I$} (21);
\path (20) edge node[left] {\tiny $D$} (22);
\path (2) edge node[left] {\tiny $O$} (23);
\path (14) edge node[left] {\tiny $O$} (24);
\path (3) edge node[left] {\tiny $O$} (25);
\path (1) edge node[left] {\tiny $O$} (26);
\path (12) edge node[left] {\tiny $O$} (27);
\path (16) edge node[left] {\tiny $O$} (28);
\path (18) edge node[left] {\tiny $O$} (29);
\path (5) edge node[left] {\tiny $O$} (30);
\path (20) edge node[above, pos=0.2] {\tiny $O$} (31);
}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Partial Definition of the Refined Routing Model $G_{\text{ref}}$.}
\label{tbl:mymodel}
\end{table}
As in closing games, we aim to align utility and monetary outcomes as tight as possible.
We use the same ordering $(\mathbb{U},\preccurlyeq)$ as in \Cref{sec:models} and consider the utility relative to the amount due to each party.
We assume a fair update that yields both parties a benefit of $\rho \succ 0$.
In the honest case, the utility of the intermediaries is $f>0$.
If the transaction fails, all parties get utility 0. Otherwise, the intermediaries' utilities are according to their financial win/loss. The parties $A$ and $B$ both receive $\rho$ once $B$ is paid. When $B$ is paid, but $A$ has not paid yet, she has utility $m+3f+\rho$; once $E_1$ collects the money, $A$'s utility is $\rho$. Using our model, we get (proof in \Cref{app:rout}):
\begin{restatable}[Vulnerability Routing Module]{theorem}{vulnerability}
The honest behavior of the Routing Game $G_{\text{ref}}$ is not $\textsf{CR}$. Thus, the Routing Game is \emph{not secure}.
\end{restatable}
\section{Related work}
Our work is mainly based on the results of \cite{CITE}.
The game-theoretic models presented in \cite{CITE} provide a good basis, however they are not yet sufficient.
We extended the closing game drastically, by aligning the utilities to the monetary outcome, considering random deviation from the protocol including dishonest closing attempts and incorporating fees for revocation transactions.
Further, we proved their routing game insufficient, by refining it in such a way that it can capture the Wormhole attack, which Lightning's routing is vulnerable to.
Moreover, we propose an adaption of the security properties introduced in \cite{CITE}.\\
Mention:\\
Awareness and other things about NE \cite{HalpernNE} \textcolor{orange}{More details needed.}\\
Related Work:\\
\paragraph{Other work on game-theory in the blockchain domain}
Games for Mining \cite{MiningGames}.\\
Games for Payment Channel Network creation \cite{RideTheLightning}.\\
Distributed computing meets game theory \cite{DCMeetsGameTheory} (intro of k-resilient NE, t-immunity, show k-resilient NE exist for secret sharing and multiparty computation).\\
Games for Smart Contracts \cite{QuantAnalysisSC} introducing simplified language to formally capture smart contracts and automatically translate it to a concurrent game (game with states), framework for quantitative analysis of contracts, \textcolor{orange}{emphasize difference to us, do different kind of analysis? not rigorously but using suspicious results?.}\\
Game-theory to provide Secure pseudo-random number generation for smart contracts \cite{ProbSC}.\\
\paragraph{Other formal verification approaches for security in distributed computing}
\cite{DiffApp1}
\cite{AdDiffApp1}
\cite{DiffApp2}
\cite{SurveyOnBC}
\cite{OverviewOffChain}
\cite{SecAnalysis}\\
Work on Off-Chain Channels:\\
\cite{GenChannels}
\cite{Brick}
\cite{SecEffChannels}{ \color{orange}
Many talk about incentive compatibility, no one has a proper model/proof.\\
}
\section{Closing Games for the Security Analysis of Lightning}\label{sec:sec}
We show that our closing games can precisely capture closing in Lightning channels~\cite{lightning}. Namely, two terminal histories of closing games can model the honest behavior of Lightning:
(i) history $(H)$ from Table~\ref{tbl:actions} representing unilateral honest closing of $A$, yielding utility $(\alpha-\epsilon,\alpha)$;
and (ii) the history $(C_h,S)$, where $A$ attempts to close collaboratively and honestly and $B$ signs, with a utility of $(\alpha,\alpha)$. Our analysis will focus on these two histories (i)-(ii) of Lightning channels.
In the following, the values $d_{A,B}$ represent the difference to previous states of the channel. That means there was a time where the distribution of the channel funds was $(a+d_A,b-d_A)$, $(a-d_B,b+d_B)$ respectively, whereas the latest state is $(a,b)$, thus enabling dishonest closing attempts of profit $d_A$ for $A$, $d_B$ for $B$.
The values $p_{A,B}$ and $c$ that can respectively be chosen by $A$, $B$ at the time of the action and do not depend on previous distribution states.
\vspace{-2pt}
\begin{restatable}[(P1) -- Weak Immunity of Honest Behavior]{theorem}{weakimmunity}
\label{thm:wi}
The terminal histories $(H)$ and $(C_h,S)$ of $G_c(A)$ are weak immune, if $a,b\geq f$.
\end{restatable}
\vspace{-2pt}
\Cref{thm:wi} implies that as long as both players have a minimal balance of $f$ in the channel, no honest player can lose money. As such, using Theorem~\ref{thm:wi} establishes security property (P1) ensuring ``no honest loss``.
Further, for ensuring the security property (P2) of ``no deviation", we require that $a-p_B+d_A \geq f$ and $b-p_B+d_B \geq f$. In practice, as expected, that means that ignoring the dishonest closing attempt is worse than publishing the revocation transaction. Property (P2) is then established by the following theorem.
\vspace{-2pt}
\begin{restatable}[(P2) -- Incentive-Compatibility of Honest Behavior]{theorem}{incentcomp}
\label{thm:incentcomp}
\noindent If $a-p_B+d_A \geq f$ and $b-p_A+d_B \geq f$, then \begin{enumerate}
\vspace{-5pt}
\item $(H)$ is $\textsf{CR}$, but \emph{not} practical.
\item $(C_h,S)$ is $\textsf{CR}$. It is practical iff $c \neq p_A$.
\end{enumerate}
\end{restatable}
\vspace{-2pt}
\vspace{-3pt}
\begin{remark} \label{rmk:ceqp}
Player $A$ can choose to propose dishonest collaborative closing, cheating by $c$ and then propose an update $(a,b) \mapsto (a+c,b-c)$. In this case all practical and $\textsf{CR}${} terminal histories (P2) lead to such an update and yield the best joint outcome possible $(\rho+\alpha,\rho+\alpha)$. One of them is even weak immune (P1), provided $a,b\geq f$. In any other case, $c \neq p_A$, there exist precisely two different utilities of practical, $\textsf{CR}${} terminal histories (P2), one favoring $A$, the other favoring $B$. Hence, there is no common ``best'' practical history.
\end{remark}
\vspace{-3pt}
Since $(H)$ is not practical, a rational player will not play it. Hence, the terminal history $(H)$ is not secure. We get the following security result for $(C_h,S)$.
\vspace{-2pt}
\begin{restatable}[Security of $G_c(A)$]{theorem}{security}\label{thm:sec:closing}
If $a,b\geq f$, $a-p_B+d_A \geq f$, $b-p_B+d_B \geq f$ and $c \neq p_A$, then the closing game $G_c(A)$ together with the honest behavior $(C_h,S)$ is \emph{secure}.
\end{restatable}
\vspace{-3pt}
\subsection{Closing Game without Updates}
We will now consider a variation of closing games without updates, as updating is not beneficial for at least one player upon closing.
Furthermore, the case described in \Cref{rmk:ceqp} is essentially (utility-wise) equivalent to updating before initiating $G_c(A)$ and then closing honestly and collaboratively. As such,
for $G_c(A)$ without updates, we get a security result similar to \Cref{thm:sec:closing}.
\vspace{-2pt}
\begin{restatable}[Security of $G_c(A)$ without Updates]{theorem}{securitynoup}\label{thm:noup}
If $a,b\geq f$, then the closing game $G_c(A)$ together with both histories $(H)$ and $(C_h,S)$ is \emph{secure}.
\end{restatable}
\vspace{-2pt}
We further study what happens if a player has almost no funds left in a channel. In particular, we show that security goals are drastically violated in this corner case, thereby formalizing folklore in the community.
\vspace{-3pt}
\begin{restatable}[Little Funds]{theorem}{securityflaw} \label{thm:secflaw}
If $a<f$, then no honest terminal history is weak immune, where an honest terminal history is a terminal history not including the action $C_c$ nor $D$.
\end{restatable}
\vspace{-3pt}
\vspace{-3pt}
\begin{restatable}{corollary}{cortwo} \label{cor:cor2}
If there exists an old state $(a+d_A,b-d_A)$, with $a+d_A <f$, then neither history $(H)$ nor $(C_h,S)$ is weak immune nor practical, but $\textsf{CR}$.
\end{restatable}
\vspace{-3pt}
\vspace{-3pt}
\begin{restatable}{corollary}{corthree}\label{cor:cor3}
A rational party should \emph{never}, in any channel, let the opponent's balance fall below $f$, because at that point the other party can always cause a financial loss by closing dishonestly and unilaterally\footnote{The special edge cases $a=0$ or $b=0$ are considered in \Cref{app:sec}.}.
\end{restatable}
\vspace{-3pt}
\subsection{Optimal Strategy for Closing}
We summarize the optimal strategy for closing an off-chain channel for a rational and suspicious player based on our results \Cref{thm:noup,thm:secflaw}, \Cref{cor:cor2,cor:cor3}.
\paragraph{The player who initiated the closing phase shall}
\begin{itemize}
\vspace{-5pt}
\item try to close honestly and collaboratively, if her funds $a$ are above $f$, or it is the first state with $a<f$. If the other player does not sign, she shall close honestly and unilaterally.
\item close dishonestly and unilaterally using the old state $(a+d_A,b-d_A)$ with the highest $d_A$, where $a+d_A<f$, otherwise.
\end{itemize}
\paragraph{The reacting player shall}
\begin{itemize}
\vspace{-5pt}
\item sign the collaborative and honest closing attempt if applicable, unless his funds are close to zero and there is an old state $(a-d_B,b+d_B)$ in which his funds are also drastically less then $f$, in this case the player might risk to to close unilaterally but dishonestly.
\item close honestly and unilaterally in case of a dishonest collaborative closing attempt. If one's funds are very low, dishonest unilateral closing can be considered as well.
\item state the revocation transaction if the other player tried to close dishonestly and unilaterally with a state $(a+d_A,b-d_A)$, where $a+d_A \geq f$. If $a+d_A < f $, he shall ignore the cheating, as it yields less loss.
\end{itemize}
\section{EFG Advancements for Off-Chain Protocols}\label{sec:theory}
We now introduce novel EFG concepts
by extending the NFG setting of Section~\ref{sec:prelin:NFGSec}. Such an extension is needed to overcome the restriction of NFGs in modeling only simultaneous actions, while ensuring practicality in the sequential setting of EFGs.
Doing so, we introduce \emph{extended strategies in EFGs}, allowing us to capture deviations such as dishonest closing attempts in \Cref{sec:models}. Such closing attempts cannot be modeled in the NFG approach of~\cite{CITE}.
\subsection{EFG Extensions}\label{sec:EFG:extended}
While considering EFGs enables us to incorporate choices made at different times yielding different options for the next player, it comes with the following limitation.
Lightning's honest behavior only specifies a terminal history (i.e., a path from root to leaf), rather than a strategy. For instance, the history may specify to close the channel collaboratively but it does not capture a participants' behavior once an opponent deviated.
To address this limitation, we introduce the following notion of an extended strategy in EFGs.
\begin{definition}[Extended Strategy]
Let $\beta$ be a terminal history in an EFG $\Gamma$. Then, all strategies $\sigma_\beta$ that result in history $\beta$ are \emph{extended strategies} of $\beta$.
\end{definition}
Recall that the game-theoretic properties in~\Cref{sec:prelim} are defined on strategies but not on terminal histories.
In our work, however, we are interested in analyzing whether a protocol together with its honest behavior $\beta$ satisfies the security properties (P1) and (P2). We therefore use extended strategies $\sigma_\beta$ to formalize properties of $\beta$. For example, if there is a Nash Equilibrium whose terminal history is $\beta$, then we conclude that $\beta$ is a
Nash Equilibrium.
\begin{definition}[Properties of $\beta$]
A terminal history $\beta$ \emph{satisfies a property} $P$ defined on strategies, if there exists an extension $\sigma_\beta$ that satisfies $P$.
\end{definition}
While EFGs can be translated to NFGs,
analyzing the security properties (P1) and (P2) over NFGs may yield unexpected results, such as honest closing not being secure in the default case (\Cref{thm:noup})\footnote{See \Cref{ex:pract} in \Cref{app:theory} for details.}.
We therefore introduce EFG extensions enabling the analysis of (P1) and (P2).
Since EFGs also have a utility function which assigns values after the game, the NFG concepts of weak immunity, strong resilience and \textsf{sNE}{} remain the same for EFGs. Practicality in NFGs however relies on weakly dominated strategies, and hence must be adjusted for EFGs. This is because NFG actions happen simultaneously, while EFG players choose their actions sequentially. We thus propose to use subgame perfect equilibria for comparing EFG strategies and define \emph{practicality for EFGs} as follows.
\begin{definition}[Practicality for EFG]
A strategy of an EFG $\Gamma$ is \emph{practical} if it is a subgame perfect equilibrium of $\Gamma$.
\end{definition}
\subsection{Security Strategies for Off-Chain Protocols}
Based on our EFG advancements (\Cref{sec:EFG:extended}), we next focus on formalizing security properties of off-chain protocols.
In \cite{CITE}, strong resilience and practicality where used to model the no deviation property of (P2). We show that
strong Nash Equilibria do not imply strong resilience nor vice-versa (Lemma~\ref{lemma:impl}).
We thus investigate variations of strong resilience and propose the novel concept of \emph{collusion resilience}, where the sum of the utilities of the deviating parties is considered since rational players may collude or be controlled by the same entity.
\begin{definition}[Collusion Resilience -- $\textsf{CR}$]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is called \emph{collusion resilient} ($\textsf{CR}$) if no strict subgroup of players $S:=\{s_1,...,s_j\}$ has a joint incentive in deviating from $\sigma$. That is,
\[\forall {\color{teal}S \subset N} \; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i}: \quad {\color{teal} \sum_{p \in S}} u_p(\sigma) \geq {\color{teal} \sum_{p \in S}} u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;)\;. \]
\end{definition}
In addition, we also consider a slight adaption of strong resilience, $\textsf{SR}_{\subseteq}${}, where the deviation of the entire set of players $N$ is also allowed, as it is for \textsf{sNE}.
The relations between versions of strong resilience are formalized next. The omitted proof and the formal definition of $\textsf{SR}_{\subseteq}${} can be found in Appendix~\ref{app:theory}.
\begin{restatable}[Resilience Properties]{lemma}{relations} \label{lemma:impl}
Let $\sigma \in \mathcal{S}$ be a joint strategy. The following and only the following implications hold.\\
\begin{minipage}{0.7\textwidth}
\begin{enumerate}
\vspace{-5pt}
\item $\sigma \text{ is $\textsf{SR}_{\subseteq}$}\; \Rightarrow \; \sigma \text{ is \textsf{SR}, $\textsf{CR}${} and \textsf{sNE}}$.
\item $\sigma \text{ is \textsf{SR}} \; \Rightarrow \; \sigma \text{ is $\textsf{CR}$}$.
\end{enumerate}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\vspace{-5pt}
\begin{tikzpicture}[scale=0.8, baseline, ->]
\node (1) at (0,0) {$\textsf{SR}_{\subseteq}$};
\node (2) at (-1.5,0) {\textsf{SR}} ;
\node (3) at (0,-1.5) {\textsf{sNE}};
\node (4) at (-1.5,-1.5) {$\textsf{CR}$} ;
\path (1) edge (2);
\path (1) edge (3);
\path (1) edge (4);
\path (2) edge (4);
\end{tikzpicture}
\end{minipage}
\end{restatable}
The next example motivates using \emph{collusion resilience} for (P2).
\begin{example} Consider the games $\Gamma_1$ and $\Gamma_2$, respectively defined in Tables~\ref{tbl:G1}-\ref{tbl:G2}. The games $\Gamma_1$ and $\Gamma_2$ show that there exist cases where both strong resilience and strong Nash Equilibria fail to correctly state whether rational players will deviate, while collusion resilience does not. In the 3-player game $\Gamma_1$ no rational player nor collusion can improve its utility by deviating from $(H_1,H_2,H_3)$. However, $\Gamma_1$ together with $(H_1,H_2,H_3)$ is \emph{not \textsf{SR}}, but $\textsf{CR}$. In game $\Gamma_2$ on the other hand the collusion of player 1 and player 2 profits from deviating to $(D_1,H_2,D_3)$ and is therefore the rational choice. Still, $(H_1,H_2,H_3)$ in $\Gamma_2$ is \textsf{sNE}, but not $\textsf{CR}$.
\end{example}
\begin{table}
\centering
\begin{minipage}{0.4\linewidth}\centering
\begin{tabular}{|r|c|c|}
\hline
{\tiny \rotatebox{45}{$H_2$}}& $H_3$ & $D_3$ \\
\hline
$H_1$ & ${\color{red}(1,1,1)}$ & $(1,1,1)$ \\
\hline
$D_1$ & $(1,1,1)$& $(5,0,-2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_1$.}
\label{tbl:G1}
\end{minipage}
\begin{minipage}{0.4\linewidth}\centering
\begin{tabular}{|r|c|c|}
\hline
{\tiny \rotatebox{45}{$H_2$}} & $H_3$ & $D_3$ \\
\hline
$H_1$ & {\color{red}$(1,1,1)$} & $(1,1,1)$ \\
\hline
$D_1 $& $(1,1,1)$& $(3,0,-2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_2$.}
\label{tbl:G2}
\end{minipage}
\vspace*{-1em}
\end{table}
\begin{remark}[Formalizing (P1) and (P2)]
Based on the above versions of resilience, we say \emph{(P2) is satisfied by a joint strategy $\sigma$, if $\sigma$ is $\textsf{CR}${} and practical}. In addition, a joint strategy \emph{$\sigma$ satisfies (P1), if $\sigma$ is weak immune}, as in \cite{CITE}.
\end{remark}
More generally, we define \emph{security} as follows.
\begin{definition}[Security] \label{def:secure}
A strategy $\sigma$ of an NFG or an EFG is \emph{secure} if it is weak immune, practical and $\textsf{CR}$.
\end{definition}
\section*{Acknowledgements}
This research was supported by the ERC Grant BROWSEC 771527, the ERC Grant ARTIST 101002685, the Austrian Science Fund FWF projects W1255-N23, and PROFET P31621, the Austrian Research Promotion Agency FFG through the Bridge-1 project PR4DLT (13808694) and the COMET K1 SBA, the Amazon Research Award 2020 FOREST, and the SecInt Doctoral College funded by TU Wien.
\section{Additional Background on the Routing Mechanism} \label{app:rout}
The Lightning network provides a way to route money along off-chain channels. The cryptographic tool to do so are hash-time-locked-contracts (HTLC) \cite{HTLC}. Routing extends the idea of a channel. Assume players $A$ and $B$ want to exchange goods for money but do not share a channel, instead $A$ has a channel with $E_1$, $E_1$ has a channel with $I$, $I$ has a channel with $E_2$ and $E_2$ has a channel with $B$. $A$ can now send the money to $B$ via the intermediaries $E_1$, $I$ and $E_2$.
\[ A \xLongleftrightarrow{\curvearrowright} {\color{teal} E_1 \xLongleftrightarrow{\curvearrowright} I \xLongleftrightarrow{\curvearrowright} E_2} \xLongleftrightarrow{\curvearrowright} B \]
The idea of the routing module, that means the honest strategy, is as follows (see \Cref{tbl:honestrouting}). Player $B$ thinks of a secret $x$ and computes its hash $h(x)=y$. Then, $B$ sends $y$ to $A$ (1.). Now $A$ can create an HTLC where she locks the money $m$ and three times fee $f$ (for every intermediary) with lock $y$ (2.). This money can only be claimed by $E_1$ and only if a value with hash $y$ is provided. $E_1$ does not know such a value yet and thus cannot unlock the money. However, $E_1$ can create another HTLC with value $m+2f$ and lock $y$ for $I$ (3.). This continues until $E_2$ creates an HTLC of value $m$ and lock $y$ for $B$ (4.-5.). Since $B$ knows $x$, he can unlock the money and send the goods to $A$ (6.). By unlocking, $E_2$ gets to know $x$ and can unlock as well (7.). $I$ and $E_1$ proceed in the same way (8.-9.). In the end $A$ paid $m+3f$ and received goods in exchange. $B$ sold the goods for $m$ and the intermediaries were rewarded with $f$ each for their participation.
\begin{table}
\centering
\begin{tikzpicture}[->]
\node (1) at (0,0) {$A$};
\node (2) at (2.5,0) {$E_1$};
\node (3) at (5,0) {$I$};
\node (4) at (7.5,0) {$E_2$};
\node (5) at (10,0) {$B$};
\path (5) edge[out=120, in=60, distance= 1.6cm] node [below] {\color{red} \small $y$} node [ below, pos=0.3] {1.} (1);
\path (1) edge node [above] {\small $(m+3f,\textcolor{red}{y})$} node [ below, pos=0.3] {2.}(2);
\path (2) edge node [above] {\small $(m+2f,\textcolor{red}{y})$} node [ below, pos=0.3] {3.} (3);
\path (3) edge node [above] {\small $(m+f,\textcolor{red}{y})$} node [ below, pos=0.3] {4.} (4);
\path (4) edge node [above] {\small $(m,\textcolor{red}{y})$} node [ below, pos=0.3] {5.} (5);
\path (5) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {6.} (4);
\path (4) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {7.} (3);
\path (3) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {8.} (2);
\path (2) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {9.} (1);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Honest Routing using HTLCs.}
\label{tbl:honestrouting}
\end{table}
In \cite{CITE}, the authors propose an Extensive Form Game to model the Routing Game which is shown in \Cref{tbl:modelpaper}. Whenever it is a specific player $p$'s turn, she can choose between following the protocol $H$ or doing nothing $I$. Doing nothing stops the game in any case and depending on whether the player right (in \Cref{tbl:honestrouting}) to $p$ has already claimed his money or not, $p$'s utility is either 0 or $-1$. After player $p$ has claimed her money, her utility is 1, independent of what the others do. Except for player $A$, her utility is also 1 if the routing was successful and 0 otherwise.
It is shown that this game together with the strategy $(H,H,H,H,H,H,H,H,H)$ is weak immune and optimal resilient. \\
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$B$};
\node (7) at (1.2,-0.5) {$(0,0,0,0,0)$} ;
\node (12) at (-1,-0.5) {$A$};
\node (13) at (0.25,-1) {$(0,0,0,0,0)$};
\node (2) at (-2,-1) {$E_1$};
\node (8) at (-0.75,-1.5) {$(0,0,0,0,0)$} ;
\node (14) at (-3, -1.5) {$I$};
\node (15) at (-1.75, -2) {$(0,0,0,0,0)$};
\node (3) at (-4,-2) {$E_2$};
\node (9) at (-2.75,-2.5) {$(0,0,0,0,0)$} ;
\node (16) at (-5,-2.5) {$B$};
\node (17) at (-3.75, -3) {$(0,0,0,0,0)$};
\node (4) at (-6,-3) {$E_2$};
\node (10) at (-4.75,-3.5) {$(0,0,0,-1,1)$} ;
\node (18) at (-7, -3.5) {$I$};
\node (19) at (-5.75, -4) {$(0,0,-1,1,1)$};
\node (5) at (-8,-4) {$E_1$};
\node (6) at (-9,-4.5) {\textcolor{red}{$(1,1,1,1,1)$}} ;
\node (11) at (-6.75,-4.5) {$(1,-1,1,1,1)$} ;
\path (1) edge node[above] {\tiny $H$} (12);
\path (1) edge node[ above, pos=0.7] {\tiny $I$} (7);
\path (12) edge node[above] {\tiny $H$} (2);
\path (12) edge node[above, pos=0.7] {\tiny $I$} (13);
\path (2) edge node[above] {\tiny $H$} (14);
\path (2) edge node[above, pos=0.7] {\tiny $I$} (8);
\path (14) edge node[ above] {\tiny $H$} (3);
\path (14) edge node[ above, pos=0.7] {\tiny $I$} (15);
\path (3) edge node[above] {\tiny $H$} (16);
\path (3) edge node[above, pos=0.7] {\tiny $I$} (9);
\path (16) edge node[ above] {\tiny $H$} (4);
\path (16) edge node[ above, pos=0.7] {\tiny $I$} (17);
\path (4) edge node[above] {\tiny $H$} (18);
\path (4) edge node[above, pos=0.7] {\tiny $I$} (10);
\path (18) edge node[ above] {\tiny $H$} (5);
\path (18) edge node[ above, pos=0.7] {\tiny $I$} (19);
\path (5) edge node[left] {\tiny $H$} (6);
\path (5) edge node[above, pos=0.7] {\tiny $I$} (11);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Model of Routing Module as in \cite{CITE}.}
\label{tbl:modelpaper}
\end{table}
However, the HTLC routing module of the Lightning Network is vulnerable to a dishonest behaviour called the wormhole attack. In the wormhole attack two intermediaries along the routing path collude to steal another intermediary's fee.
It works as illustrated in \Cref{tbl:wormhole}. The beginning including the creation of the HTLCs is unchanged. When $B$ unlocks his money with $x$ (6.), $E_2$ does not unlock as well, but rather forwards $x$ to the other adversary $E_1$ (7.). This way $E_1$ can unlock $A$'s money (8.), but $I$ never will be able to claim hers. After a certain time the remaining HTLCs time out and the money returns to the creators.
The outcome of the wormhole attack is the following. $B$ receives $m$ as planned and $A$ pays $m+3f$ which is also fine. The honest intermediate $I$ gets 0 instead of the $f$ that were promised for her services. The adversaries $E_1$ and $E_2$ earn $3f$ instead of the $2f$ they deserve. Such an adversary-incentivizing-behaviour should not be possible in a "secure" protocol. This shows that the model in \Cref{tbl:modelpaper} is insufficient.
\subsection{Proof for Refined Routing Model}
For better readability, we restate the theorem.
\vulnerability*
\begin{proof}
The honest behavior of the routing module is $(H,H,H,H,H,H,H,H,H)$ and its utility is $(\rho,f,f,f,\rho)$ as indicated in red in \Cref{tbl:mymodel}. Let us compare it to the dishonest terminal history $(H,H,H,H,H,H,D,D)$ with a utility of $(\rho,\;m+3f,\;0,\;-m,\;\rho)$ in blue. Then, it is easy to see, that the collusion of $E_1$ and $E_2$ strictly profit from the deviation, which yields a joint utility of $3f$, whereas the honest behaviour only yields a joint utility of $2f$. This violates $\textsf{CR}${}. Using the definition of security, we finally conclude that the Routing Game is not secure.
\hfill $\square$
\end{proof}
\section{Results of Security Analysis} \label{app:sec}
In this section all results from \Cref{sec:sec} are restated and proven. Additionally, the results \Cref{thm:bleqf}, \Cref{thm:a0} and \Cref{thm:b0} about edge cases are stated and proven.
\weakimmunity*
\begin{proof}
Let $a,b\geq f$. For history $(H)$, we consider a strategy, where $A$ chooses $H$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. This strategy yields terminal history $(H)$ and is weak immune, as $B$'s deviation cannot influence $A$'s utility and $A$'s deviations always result in a non-negative utility for $B$, since $a \geq f$.
Similarly, for $(C_h,S)$, we consider a strategy, where $A$ chooses $(C_h)$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Further, $A$ takes $P$ after $(C_h,D)$ and $H$ after $(C_h,I)$, $(C_h,U^+)$ and $(C_h,U^-)$. This strategy yields terminal history $(C_h,S)$ and is weak immune. Deviation of $A$ has the same effects as before, never causing $B$ a negative utility. If $B$ deviates now, $A$ also never gets negative utility, since $b \geq f$. This concludes the theorem.
\hfill $\square$
\end{proof}
\incentcomp*
\begin{proof}
Collusion resilience is defined on strict subsets of players. Thus, in a two-player game, it considers only deviations of single players and since the summation over one value is the value itself, $\textsf{CR}${} is equivalent to Nash Equilibria in this case.
We therefore only check whether $(H)$ and $(C_h,S)$ are Nash Equilibria. For $(H)$, we consider a strategy, where $A$ chooses $H$ initially, $B$ chooses $I$ after $(C_h)$, $P$ after $(D)$ and $I$ after $(C_c)$. $A$ takes $H$ after $(C_h,I)$ and $(C_c,I)$. Further, $B$ takes $I$ after $(C_{c/h},I,U^{+/-})$. For $A$, we assume she takes $H$ after $(C_{c/h},I, U^{+/-},I)$ and finally let $B$ take $P$ after $(C_{h/c},I,U^{+/-},I,D)$. This strategy yields history $(H)$ and is a Nash Equilibrium, as no party can unilaterally deviate to increase their utility.
To show that $(C_h,S)$ is a Nash Equilibrium, we consider a strategy, where $A$ picks $C_h$ initially, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Further, let $A$ pick $P$ after $(C_h,D)$, $H$ after $(C_h,I)$ and $(C_h,U^{+/-})$. This strategy has terminal history $(C_h,S)$ and no one can deviate to increase their utility. Therefore, $(C_h,S)$ is a Nash Equilibrium and thus $\textsf{CR}$.
In order to prove the practicality properties, we compute all subgame perfect equilibria of $G_c(A)$. Since a subgame is the game that remains after some choices (non-terminal history), we compute subgame perfect equilibria bottom-up. That is, we start comparing the utility of the subtrees, where no other decisions have to be made any more. In $G_c(A)$, these are for example the subgames after history $(C_h,I,D)$ or $(C_c,D)$. For the latter, $A$ is the player to choose the action. To compute the subgame perfect equilibrium, we have to compare all possible utilities for $A$ after $(C_c,D)$. We then replace this internal node labelled $A$, by the utility that yields the best value for $A$ and proceed until we reach the root. If there is no single best choice for a player, then all actions resulting in best utility have to be considered. Applying this procedure to the subgames $S_1$-$S_4$ and $S'_1$-$S'_4$ we get subgame perfect terminal history $(A,H)$ with utility $(\rho+\alpha-\epsilon,\rho+\alpha)$ for $S_1$, for $S_2$ we get terminal history $(S)$ yielding $(\alpha,\alpha)$ and $(I,H)$, yielding $(\alpha-\epsilon,\alpha)$. For $S_3$ and $S_4$ it is $(I,S)$ with $(\alpha,\alpha)$. The subgame $S'_1$ has practical history $(I,H)$, with $(\alpha-\epsilon,\alpha)$ if $c>p_A$, $(A,I,S)$ with $(\rho+\alpha,\rho+\alpha)$ if $c=p_A$ and $(A,H)$ with $(\rho+\alpha-\epsilon,\rho+\alpha)$ if $c<p$. The subgame $S'_2$ has practical history $(I,H)$, yielding $(\alpha-\epsilon,\alpha)$. For $S'_3$ and $S'_4$ we get $(I,H)$ with $(\alpha,\alpha-\epsilon)$ and additionally for $S'_3$, if $c=p$, we also have $(A,S)$ yielding $(\rho+\alpha,\rho+\alpha)$. All of these results are based on the facts $a-p_B+d_A\geq f$ and $b-p_A+d_B \geq f$, since this causes the revocation transaction always to be better than ignoring the dishonest unilateral closing attempt.
Based on these preliminary results, we can now compute the subgame perfect equilibria for $G_c(A)$ considering multiple practical histories and case splits as stated. We have the following results.
If $c=p_A$, then $(C_c,U^+,A,S)$ and $(C_c,I,U^+,A,I,S)$ are practical, both yielding $(\rho+\alpha,\rho+\alpha)$. If $c>p_A$, then the histories $(C_h,S)$, $(C_h,U^+,I,S)$, $(C_h,U^-,I,S)$ and $(C_h,I,U^-,S)$ all leading to $(\alpha,\alpha)$ are practical, as well as terminal history $(C_h,I,U^+,A,H)$, yielding $(\rho+\alpha-\epsilon,\rho+\alpha)$.
For $c<p_A$, all the histories and their utilities from $c>p_A$ are practical. Additionally $(C_c,I,U^+,A,H)$ is subgame perfect and also results in utility $(\rho+\alpha-\epsilon,\rho+\alpha)$.
This shows, that $(H)$ is never practical and $(C_h,S)$ is practical if and only if $c\neq p_A$.
\hfill $\square$
\end{proof}
\security*
\begin{proof}
Since $a,b \geq f$, $(C_h,S)$ is weak immune (\Cref{thm:wi}). Because of $a-p_B+d_A\geq f$, $b-p_A+d_B \geq f$ and $c \neq p_A$, it is practical and $\textsf{CR}$. Hence, by \Cref{def:secure} $(C_h,S)$ is secure.
\hfill $\square$
\end{proof}
\subsection{Results without Updates}
\securitynoup*
\begin{proof}
We start by proving weak immunity. For $(H)$, we consider a strategy where $A$ chooses $H$ initially, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Then, $B$'s deviation has no impact on the history and $A$'s deviation always leads to non-negative utility for $B$, as $a-f\geq 0$.
Similarly for $(C_h,S)$ we consider a strategy, where $A$ chooses $C_h$ initially, $P$ after $(C_h,D)$ and $H$ after $(C_h,I)$, $B$ chooses $S$ after $(C_h)$, $P$ after $(D)$ and $H$ after $(C_c)$. Then, $A$'s deviation is unproblematic as before and $B$'s deviation also leads to non-negative utility for $A$, since $b-f\geq 0$.
To show practicality, we compute all subgame perfect terminal histories. Since $a,b\geq f$ implies $a+d_A\geq f$ and $b+d_B \geq f$, the best choice after a $D$, is always $P$. Thus, $A$'s best choice after $(C_h,I)$ and $(C_c,I)$ is $H$. Therefore, $B$ has two subgame perfect options $I$ and $S$ after $(C_h)$ and only $I$ after $(C_c)$. This yields to the following practical history. The history $(C_h,S)$ with $(\alpha,\alpha)$ and $(C_h,I,H)$, $(C_c,I,H)$ and $(H)$ with $(\alpha-\epsilon,\alpha)$.
Every practical terminal history is a Nash Equilibrium, since if a deviation could benefit a player, she would have chosen differently. Further, we know that $\textsf{CR}${} is equivalent to Nash Equilibria in two-player games, thus practicality implies $\textsf{CR}$ and this concludes the proof.
\hfill $\square$
\end{proof}
\securityflaw*
\begin{proof}
Let $\sigma$ be any strategy, yielding an honest history, then $A$ can deviate to $D$. In this case $B$ gets negative utility, since $a<f$, whether he chooses $P$ or $I$. Hence no honest history is weak immune.
\hfill $\square$
\end{proof}
\cortwo*
\begin{proof}
We fix the old distribution state such that the difference $d_A$ to the latest state is the value of $A$'s dishonest closing attempt in the closing game. As $a+d_A<f$ implies $a<f$, \Cref{thm:secflaw} applies. Therefore, neither $(H)$ nor $(C_h,S)$ are weak immune.
In order to show that they are also not practical, we prove instead, that the only practical history is $(D,I)$. Since $a+d_A<f$, $I$ is the best choice for $B$ after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$. Consequently, $A$ will choose $D$ after $(C_h,I)$ and $(C_c,I)$. If now $b+d_B\geq f$, then $A$'s best choice is $P$ after $(C_h,D)$ and $(C_c,D)$. Thus, $B$ will take $S$ after $(C_h)$ and $H$ after $(C_c)$. In the other case, $b+d_b<f$, $A$'s best option is $I$ after $(C_h,D)$ and $(C_c,D)$, thus $B$'s best choice after $(C_h)$ and $(C_c)$ is $D$, which yields a negative utility for $A$. Therefore, in both cases $A$'s only subgame perfect action is $D$. Hence, $(D,I)$ is the unique subgame perfect history.
For $\textsf{CR}$, we show instead that there exist extensions of $(H)$ and $(C_h,S)$ that are Nash Equilibria. Let $\sigma$ be the strategy where $A$ chooses $H$, everyone chooses $P$ after a dishonest closing attempt, $B$ chooses $I$ after $(C_h)$ and $(C_c)$ and $A$ chooses $H$ after $(C_h,I)$ and $(C_c,I)$. Then, no one can deviate to increase their utility and therefore $(H)$ is an $\textsf{CR}${} history.
To prove $(C_h,S)$ is $\textsf{CR}$, we consider the strategy $\sigma'$, which is the same as $\sigma$, except $A$ chooses $C_h$ and $B$ chooses $S$ after $(C_h)$. This is also a Nash Equilibrium.
\hfill $\square$
\end{proof}
\corthree*
\begin{proof}
Once the opponent's balance is below $f$, that party can start the closing game, therefore the opponent becoming $A$. Thus, by applying \Cref{thm:secflaw}, it follows that the opponent can make the rational player lose money by closing unilaterally and dishonestly. If it is not the first time that that player's balance is below $f$, then we are even in the situation of \Cref{cor:cor2}, where it is rational of $A$ to close dishonestly.
\hfill $\square$
\end{proof}
We present an additional theorem, discussing the case where player $B$ has little funds left in the channel. Since the roles of player $A$ and $B$ are arbitrary, it is of little importance because the results give stronger security guarantees as for the case where $A$ has a low balance. Nevertheless, we state it for the sake of completeness.
\begin{theorem} \label{thm:bleqf}
If there exists an old state with $b+d_B<f$, but $a \geq f$, then \begin{enumerate}
\item $(H)$ is secure.
\item $(C_h,S)$ is not practical, not weak immune, but $\textsf{CR}$.
\end{enumerate}
\end{theorem}
\begin{proof}
To prove (1), we start by showing weak immunity. Consider any strategy, where $B$ chooses $P$ after $(D)$, $S$ after $(C_h)$ and $H$ after $C_c$. Then $(H)$ is weak immune, because $B$'s deviations have no impact on the history and $A$'s deviations can never bring $B$'s utility below zero.
Next, we prove the practicality of $(H)$. Since $a \geq f$, the subgame perfect choice after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$ is $P$. Thus $A$ chooses $H$ after $(C_h,I)$ and $(C_c,I)$. Due to $b+d_B<f$, $A$'s best option after $(C_h,D)$ and $(C_c,D)$ is $I$. Hence $B$'s unique subgame perfect choice after $(C_c)$ and $(C_h)$ is $D$. Thus, $A$'s only best response is $H$. Therefore, $(H)$ is the only practical history.
As practicality implies $\textsf{CR}${} in our case, $(H)$ is secure.
For (2), we just showed that $(C_h,S)$ cannot be practical. Additionally, $(C_h,S)$ is not weak immune, since $B$ could deviate to $D$ after $(C_h)$, in which case $A$ gets negative utility for sure.
Finally, we consider the strategy $\sigma$, where $A$ chooses $C_h$, $B$ chooses $S$, both take $P$ in case of a dishonest unilateral closing attempt, $B$ takes $H$ after $(C_c)$, similarly $A$ takes $H$ after $(C_h,I)$ and $(C_c,I)$. Then $\sigma$ is a Nash Equilibrium yielding terminal history $(C_h,S)$. This concludes the proof.
\hfill $\square$
\end{proof}
\subsection{Results for Edge Cases}
Finally, we present results about the edge cases, where one party has zero funds left in the channel. This assumption changes the structure of $G_c(A)$, as some actions are not possible in this case. The precise games we refer to are stated in \Cref{app:subgames}, and result from removing impossible choices from \Cref{tbl:Gc}.
\begin{theorem} \label{thm:a0}
If $a=0$ and $b>0$, then no honest strategy is weak immune. Additionally, $(H)$ and $(C_h,S)$ are practical if and only if $d_A\geq f$ for all old states. In any case they are $\textsf{CR}$.
\end{theorem}
\begin{proof}
We first show that no honest strategy is weak immune. Let $\sigma$ be an honest strategy, then $A$ does not choose $D$ in $\sigma$. However, if $A$ deviates to $D$, then $B$'s utility is negative.
To show both $(H)$ and $(C_h,S)$ are $\textsf{CR}$, we show they are Nash Equilibria instead. We therefore consider any strategy where $B$ chooses $P$ after $(D)$, $S$ after $(C_h)$ and $H$ after $(C_c)$. Then both choices $H$ and $C_h$ of $A$, resulting in the histories $(H)$ and $(C_h,S)$, yield a Nash Equilibrium, as no one can deviate to increase their utility.
Let now $d_A\geq f$. In which case $P$ is the subgame best choice for $B$ after $(D)$, $(C_h,I,D)$ and $(C_c,I,D)$. Further, after history $(C_c,I)$, $S$ it is never a best option for $B$, because it is strictly dominated by $H$. Therefore, $A$ will get utility zero in any case. This makes $(H)$ a practical history. Similarly for $(C_h,S)$, since $S$ is subgame perfect for $B$ after $(C_h)$.
If now $d_A<f$, then $I$ is subgame perfect for $B$ after $D$. Thus, with similar argumentation as before, $(D,I)$ is the only practical history.
\hfill $\square$
\end{proof}
\begin{theorem} \label{thm:b0}
If $a>0$ and $b=0$, then
\begin{enumerate}
\item $(H)$ is secure.
\item $(C_h,S)$ is not weak immune, but $\textsf{CR}$. It is practical iff $d_B \geq f$ in every previous state $(a-d_B,d_B)$.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove (1.) first. The history $(H)$ is weak immune, as $B$'s strategy does not effect the history and $A$'s deviation is irrelevant for $B$, as he can never get negative utility.
Practicality of $(H)$. After history $(C_h,S)$ the subgame perfect choice of $A$ depends on whether $d_B \geq f$. In any case, $D$ is subgame perfect for $B$. If $A$ chose $P$, then it is as good as any other choice, yielding 0, otherwise it is the only best option resulting in a positive utility. Thus, $A$ either gets $-f+\alpha$ or $-d_B+\alpha$ if she chooses $C_h$, both of which is negative. Hence $A$'s subgame perfect and therefore practical choice is $H$, yielding the history $(H)$.
The fact that $(H)$ is $\textsf{CR}${} follows from practicality. This shows that $(H)$ is secure, if $b=0$.
(2.) We start showing $(C_h,S)$ is not weak immune. We consider any strategy yielding the history $(C_h,S)$. Assume now, $B$ deviates to $D$ after $(C_h)$, then no matter what $A$'s choice is, she will get a negative utility, thus $(C_h,S)$ is not weak immune.
The collusion resilience of $(C_h,S)$, can be shown by considering a strategy with history $(C_h,S)$, where additionally $A$ chooses $P$ after $(C_h,D)$. Then $B$ has no incentive to deviate as he always gets utility 0, and $A$ has no incentive as $\alpha$ is the best possible outcome for her.
To finally show that $(C_h,S)$ is practical iff $d_B\geq f$, we consider $A$'s choice after $(C_h,D)$. The option $P$ is subgame perfect iff $d_B \geq f$. Thus, $S$ is subgame perfect for $B$ iff $d_B \geq f$. For $d_B < f$, $D$ is the better option for $B$, yielding $(-d_B+\alpha,d_B+\alpha-\epsilon)$. Therefore $C_h$ is subgame perfect for $A$ iff $d_B \geq f$, in which case the resulting history is $(C_h,S)$. This concludes the proof of the theorem.
\hfill $\square$
\end{proof}
The weak immunity result of $(H)$ might be misleading, as $B$ can actually close dishonestly immediately (before $A$ takes action). This is not represented here, but in $G_c(B)$, which is analog to $G_c(A)$ but with swapped roles.
\section{Subgames and Edge Cases of $G_c(A)$} \label{app:subgames}
In the following all the subgames needed for the closing game $G_c(A)$ are defined. Further, the edge cases of $A$'s or $B$'s balance being zero in the closing game are treated.
\subsection{Subgames of $G_c(A)$}
The subgames $S_1$ and $S_2$ in \Cref{tbl:S1} and \Cref{tbl:S2} cover the case where a channel update is proposed by $A$, although $A$ has already signed an honest collaborative closing attempt. In $S_1$ the update is from channel state $(a,b)$ to $(a+p_A,b-p_A)$, whereas in $S_2$ the suggested update is $(a-p_B, b+p_B)$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.25,-1.5) {$(\alpha,\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (1,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.9,-6.5) { $(\textcolor{red}{-p_A}+\rho+\alpha,\,\textcolor{red}{p_A}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{-p_A}+\rho,-b\,\textcolor{red}{+p_A}+\rho)$};
\node (20) at (2,-4) {$(-a\,\textcolor{red}{-p_A}\,+\rho,a\,\textcolor{red}{+p_A}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\, \textcolor{red}{-p_A}-f +\rho+\alpha,-b\, \textcolor{red}{+p_A} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{1}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S1}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.25,-1.5) {$(\alpha,\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (1,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.9,-6.5) { $(\textcolor{red}{p_B}+\rho+\alpha,\,\textcolor{red}{-p_B}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{+p_B}+\rho,-b\,\textcolor{red}{-p_B}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{+p_B}+\rho,a\textcolor{red}{-p_B}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\textcolor{red}{+ p_B}-f +\rho+\alpha,-b\textcolor{red}{-p_B} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{2}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S2}
\end{table}
The subgames $S_1'$ and $S_2'$ in \Cref{tbl:S1p} and \Cref{tbl:S2p} are very similar to $S_1$ and $S_2$. They only differ in the fact, that the existing partially signed collaborative closing attempt was dishonest. That means, $A$ proposed an unfair split, increasing her outcome by value $c>0$.
\begin{table}[t]
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2.5,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.5,-1.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (0.25,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.5,-6.5) { $(\textcolor{red}{c-p_A}+\rho+\alpha,\textcolor{red}{-c+p_A}+\rho+\alpha)$};
\node (19) at (4,-7) {$(-a\textcolor{red}{-p_A}+\rho,-b\textcolor{red}{+p_A}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{-p_A}+\rho,a\textcolor{red}{+p_A}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b\textcolor{red}{-p_A}-f +\rho+\alpha,-b\textcolor{red}{+p_A} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{1}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S1p}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {\color{olive}$B$};
\node (2) at (-1,-1.5) {\color{teal}$A$} ;
\node (3) at (-2,-0.75) {\color{teal}$A$};
\node (4) at (2.5,-1.5) {$(\alpha,\alpha-\epsilon)$};
\node (5) at (4.5,-1.5) {\color{teal}$A$};
\node (24) at (0.5,-1.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (6) at (-3.5,-2.5) {$(-a,-b)$};
\node (7) at (-2.5,-2.85) {\color{olive}$B$};
\node (8) at (-1,-2.75) {$(\alpha-\epsilon,\alpha)$};
\node (9) at (-5,-1.25) {$(b-f+\alpha,-b)$};
\node (10) at (-4,-1.75) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (11) at (5,-4.5) {\color{olive}$B$};
\node (12) at (2.25,-2.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (13) at (4,-3) {\color{olive}$B$};
\node (14) at (-5,-3.5) {$(-a,a-f+\alpha)$};
\node (15) at (-3.75,-4) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (16) at (-1,-5.5) {\color{teal}$A$};
\node (17) at (0.25,-6) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (18) at (1.4,-6.5) { $(\textcolor{red}{c+p_B}+\rho+\alpha,\textcolor{red}{-c-p_B}+\rho+\alpha)$};
\node (19) at (3.9,-7) {$(-a\,\textcolor{red}{+p_B}+\rho,-b\,\textcolor{red}{-p_B}+\rho)$};
\node (20) at (2,-4) {$(-a\textcolor{red}{+p_B}+\rho,a\,\textcolor{red}{-p_B}-f+\rho+\alpha)$};
\node (21) at (0.25,-3.5) {$(d_A+\rho+\alpha-\epsilon, -d_A+\rho+\alpha)$};
\node (22) at (-2.5,-7) {$(b \textcolor{red}{+p_B}-f +\rho+\alpha,-b\, \textcolor{red}{-p_B} +\rho)$};
\node (23) at (-4,-6) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
{\color{olive}
\path (1) edge node [left] {\tiny $I$} (2);
\path (1) edge node [above, pos=0.7] {\tiny $D$} (3);
\path (1) edge node [right] {\tiny $H$} (4);
\path (1) edge node [above, pos=0.4] {\tiny $A$} (5);
\path (1) edge node [right] {\tiny $S$} (24);}
{\color{teal}
\path (2) edge node [above, pos=0.4] {\tiny $I$} (6);
\path (2) edge node [left, pos=0.5] {\tiny $D$} (7);
\path (2) edge node [left] {\tiny $H$} (8);
\path (3) edge node [above, pos=0.7] {\tiny $P$} (9);
\path (3) edge node [above, pos=0.7] {\tiny $I$} (10);
\path (5) edge node [right, pos=0.4] {\tiny $I$} (11);
\path (5) edge node [below] {\tiny $H$} (12);
\path (5) edge node [right, pos=0.7] {\tiny $D$} (13);}
{\color{olive}
\path (7) edge node [above] {\tiny $P$} (14);
\path (7) edge node [right] {\tiny $I$} (15);
\path (11) edge node [above] {\tiny $D$} (16);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (17);
\path (11) edge node [above, pos=0.65] {\tiny $S$} (18);
\path (11) edge node [left] {\tiny $I$} (19);
\path (13) edge node [below] {\tiny $P$} (20);
\path (13) edge node [above] {\tiny $I$} (21);}
{\color{teal}
\path (16) edge node [right] {\tiny $P$} (22);
\path (16) edge node [above] {\tiny $I$} (23);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{2}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S2p}
\end{table}
The next couple of subgames $S_3$ (\Cref{tbl:S3}), $S_4$ (\Cref{tbl:S4}) represent channel updates that were proposed by $B$, after $A$ tried to close the channel honestly and collaboratively. Similar to before, $S_3$ represents the case where $A$'s balance is increased, therefore an update from $(a,b)$ to $(a+p_A,b-p_A)$. The other subgame $S_4$ handles the case where the update increases $B$'s balance to a new state $(a-p_B,b+p_B)$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.5,-1) {$({\color{red}-p_A}+\rho+\alpha,{\color{red}p_A}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-2,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\alpha,\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}-p_A}-f+\rho+\alpha,-b{\color{red}+p_A}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}-p_A}+\rho,-b{\color{red}+p_A}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}-p_A}+\rho,a{\color{red}+p_A}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{3}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S3}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.5,-1) {$({\color{red}p_B}+\rho+\alpha,{\color{red}-p_B}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-2,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\alpha,\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}+p_B}-f+\rho+\alpha,-b{\color{red}-p_B}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}+p_B}+\rho,-b{\color{red}-p_B}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}+p_B}+\rho,a{\color{red}-p_B}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S_{4}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S4}
\end{table}
Finally, the subgames $S_3'$ and $S_4'$ in \Cref{tbl:S3p} and \Cref{tbl:S4p} handle the same situations as $S_3$ and $S_4$, except the by $A$ partially signed closing attempt is unfair and increases $A$'s outcome by $c>0$.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.15,-1) {$({\color{red}c-p_A}+\rho+\alpha,{\color{red}-c+p_A}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-1.25,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}-p_A}-f+\rho+\alpha,-b{\color{red}+p_A}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}-p_A}+\rho,-b{\color{red}+p_A}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}-p_A}+\rho,a{\color{red}+p_A}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{3}$ with Update $(a,b)\mapsto (a+p_A,b-p_A)$.}
\label{tbl:S3p}
\end{table}
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (1.75,2) {\color{teal}$A$};
\node (20) at (0,1) {\color{olive} $B$};
\node (2) at (1,-0.25) {\color{olive}$B$} ;
\node (3) at (2.5,0) {\color{olive}$B$};
\node (21) at (4,0.5) {$(\alpha-\epsilon,\alpha)$};
\node (22) at (-1.5,0) {$(-a,a-f+\alpha)$};
\node (23) at (-3,0.5) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (4) at (-3.15,-1) {$({\color{red}c+p_B}+\rho+\alpha,{\color{red}-c-p_B}+\rho+\alpha)$};
\node (5) at (1,-3.5) {\color{teal}$A$};
\node (6) at (-1.25,-1.5) {$(\rho+\alpha,\rho+\alpha-\epsilon)$};
\node (7) at (0,-2.25) {\color{teal}$A$};
\node (8) at (4.5,-1.25) {$(-a,-b)$};
\node (24) at (5,-0.5) {$(\textcolor{red}{c}+\alpha,\textcolor{red}{-c}+\alpha)$};
\node (9) at (3.75,-1.75) {$(\alpha,\alpha-\epsilon)$};
\node (10) at (2.25,-2) {\color{teal}$A$};
\node (11) at (-1.75,-3.25) {$(b{\color{red}+p_B}-f+\rho+\alpha,-b{\color{red}-p_B}+\rho)$};
\node (12) at (-3.5,-2.75) {$(-d_B+\rho+\alpha,d_B+\rho+\alpha-\epsilon)$};
\node (13) at (5,-2.75) {$(b-f+\alpha,-b)$};
\node (14) at (3.5,-3.25) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (15) at (1.,-5) {\color{olive}$B$};
\node (16) at (-1,-4.5) {$(\rho+\alpha-\epsilon,\rho+\alpha)$};
\node (17) at (3.5,-4.5) {$(-a{\color{red}+p_B}+\rho,-b{\color{red}-p_B}+\rho)$};
\node (18) at (-1.5,-6) {$(-a{\color{red}+p_B}+\rho,a{\color{red}-p_B}-f+\rho+\alpha)$};
\node (19) at (4,-6) {$(d_A+\rho+\alpha-\epsilon,-d_A+\rho+\alpha)$};
{\color{teal}
\path (1) edge node [left, pos=0.5] {\tiny $A$} (2);
\path (1) edge node [right, pos=0.5] {\tiny $I$} (3);
\path (1) edge node [above] {\tiny $D$} (20);
\path (1) edge node [above] {\tiny $H$} (21);}
{\color{olive}
\path (2) edge node [below] {\tiny $S$} (4);
\path (2) edge node [right] {\tiny $I$} (5);
\path (2) edge node [below] {\tiny $H$} (6);
\path (2) edge node [right] {\tiny $D$} (7);
\path (3) edge node [above] {\tiny $I$} (8);
\path (3) edge node [above] {\tiny $S$} (24);
\path (3) edge node [right] {\tiny $H$} (9);
\path (3) edge node [right] {\tiny $D$} (10);
\path (20) edge node [above] {\tiny $I$} (23);
\path (20) edge node [below] {\tiny $P$} (22);}
{\color{teal}
\path (7) edge node [below] {\tiny $P$} (11);
\path (7) edge node [above] {\tiny $I$} (12);
\path (10) edge node [above, pos=0.7] {\tiny $P$} (13);
\path (10) edge node [left] {\tiny $I$} (14);
\path (5) edge node [right] {\tiny $D$} (15);
\path (5) edge node [above] {\tiny $H$} (16);
\path (5) edge node [above] {\tiny $I$} (17);}
{\color{olive}
\path (15) edge node [above] {\tiny $P$} (18);
\path (15) edge node [above] {\tiny $I$} (19);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Subgame $S'_{4}$ with Update $(a,b)\mapsto (a-p_B,b+p_B)$.}
\label{tbl:S4p}
\end{table}
\subsection{Edge Cases of the Closing Game}
So far, we only considered cases where both balances $a$ and $b$ were strictly greater than zero. This is not necessarily the case. Therefore, we consider these cases here.
In the first case, $a=0$, $B$ cannot close dishonestly, as there is no old state that increases his balance. The corresponding simplified game is presented in \Cref{tbl:a0}.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0.125,0.5) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-1.25,-1) { $(0,\alpha)$};
\node (4) at (0.25,-0.75) {\color{olive}$B$};
\node (5) at (3.75,-1) {\color{olive}$B$};
\node (6) at (-5,-3.25) {\color{teal}$A$};
\node (7) at (-3.25,-3) { $(0,\alpha)$};
\node (8) at (1.25,-1.4) { $(0,-f+\alpha)$};
\node (9) at (-0.5,-1.9) { $(d_A+\alpha-\epsilon, -d_A+\alpha)$};
\node (10) at (3,-3) {$(c+\alpha,-c+\alpha)$};
\node (11) at (5,-3.25) {\color{teal}$A$};
\node (12) at (-5,-4.75) {\color{olive}$B$};
\node (13) at (-1.75,-4.25) {$(0,\alpha)$};
\node (14) at (-3.5,-4.5) {$(0,-b)$};
\node (19) at (3.5,-4.5) {$(0,-b)$};
\node (20) at (1.75,-4.25) { $(0,\alpha)$};
\node (21) at (5.25,-4.75) {\color{olive}$B$};
\node (22) at (-2.25,-5.5) {$(0,-f+\alpha)$} ;
\node (23) at (-3.75,-6) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (24) at (2.5,-5.5) { $(0,-f+\alpha)$};
\node (25) at (4,-6) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (27) at (-1.75,-2.75) {$(0,\alpha-\epsilon)$};
\node (34) at (0.75,-2.75) {$(0,\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [left, pos=0.5] {\tiny $H$} (3);
\path (1) edge node [right, pos=0.6] {\tiny $D$} (4);
\path (1) edge node [above, pos=0.5] {\tiny $C_c$} (5);}
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7);
\path (4) edge node [above] {\tiny $P$} (8);
\path (4) edge node [left] {\tiny $I$} (9);
\path (5) edge node [right] {\tiny $S$} (10);
\path (5) edge node [right] {\tiny $I$} (11); }
{\color{teal}
\path (6) edge node [right, pos=0.7] {\tiny $D$} (12);
\path (6) edge node [above, pos=0.7] {\tiny $H$} (13);
\path (6) edge node [above, pos=0.7] {\tiny $I$} (14);
\path (11) edge node [above, pos=0.7] {\tiny $I$} (19);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (20);
\path (11) edge node [left, pos=0.7] {\tiny $D$} (21);}
{\color{olive}
\path (12) edge node [below] {\tiny $P$} (22);
\path (12) edge node [left] {\tiny $I$} (23);
\path (21) edge node [below] {\tiny $P$} (24);
\path (21) edge node [right] {\tiny $I$} (25); }
{\color{olive}
\path (2) edge node [left, pos=0.5] {\tiny $H$} (27);
\path (5) edge node [right, pos=0.5] {\tiny $H$} (34);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing game $G_c(A)$ with $a=0$.}
\label{tbl:a0}
\end{table}
If $b=0$ (\Cref{tbl:b0}), player $A$ cannot close dishonestly, as she cannot take any money from $B$. Thus, both dishonest unilateral closing $D$ and proposing an unfair split in a collaborative closing attempt $C_c$ are not possible.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (-1.75,0) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-0,-1) { $(\alpha-\epsilon,0)$};
\node (6) at (-5,-2.5) {\color{teal}$A$};
\node (7) at (-3.5,-2.5) { $(\alpha,0)$};
\node (13) at (-4.25,-3.75) {$(\alpha-\epsilon,0)$};
\node (14) at (-5.5,-3.75) {$(-a,0)$};
\node (27) at (-0.75,-2) {$(\alpha,0)$};
\node (30) at (-2.25,-2.5) {\color{teal}$A$};
\node (36) at (-1.5,-3.5) { $(-f+\alpha,0)$} ;
\node (37) at (1,-3) { $(-d_B+\alpha,d_B+\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [above, pos=0.5] {\tiny $H$} (3); }
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7); }
{\color{teal}
\path (6) edge node [right] {\tiny $H$} (13);
\path (6) edge node [left] {\tiny $I$} (14);}
{\color{olive}
\path (2) edge node [below, pos=0.5] {\tiny $H$} (27);
\path (2) edge node [left] {\tiny $D$} (30);}
{\color{teal}
\path (30) edge node [right] {\tiny $P$} (36);
\path (30) edge node [above] {\tiny $I$} (37); }
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing game $G_c(A)$ with $b=0$.}
\label{tbl:b0}
\end{table}
\section{Additional Game Theoretical Concepts}\label{app:theory}
All additional game-theoretical definitions, proofs and examples are stated here.
\subsection{Iterated Deletion of Weakly Dominated Strategies}
We introduce the iterated deletion of weakly dominated strategies (IDWDS), by using weakly dominated strategies as in \Cref{sec:prelim}.
\begin{definition}[IDWDS]
The \emph{iterated deletion of weakly dominated strategies} (IDWDS) of a game $\Gamma$ is defined as iteratively rewriting $\Gamma$ by omitting \emph{all} weakly dominated strategies of all players. This is repeated until no strategy is weakly dominated any more. The resulting game $\Gamma'$ is thus a subgame of $\Gamma$.
\end{definition}
When IDWDS is applied, then every Nash Equilibrium of the resulting game $\Gamma'$ is also a Nash Equilibrium of $\Gamma$.
\subsection{Practicality and EFGs}
We give an example that shows how applying the NFG definition of practicality to an EFG, by using its translation to a Normal Form Game, yields unwanted results.
\begin{table}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=3cm, el/.style = {inner sep=4pt, align=center, sloped}]
\node (1) at (0,0) {$A$};
\node (2) at (1.5,-0.75) {$B$} ;
\node (3) at (-1.5,-0.75) {$(2,2)$};
\node (4) at (3,-1.5) {$A$} ;
\node (5) at (0, -1.5) {$(3,1)$};
\node (6) at (4.5,-2.25) {$B$} ;
\node (7) at (1.5,-2.25) {$(1,1)$};
\node (8) at (6,-3) {$(0,2)$} ;
\node (9) at (3, -3) {$(0,1)$};
\path (1) edge node[ above] { $2$} (2);
\path (1) edge node[above] {$1$} (3);
\path (2) edge node [above] {$4$} (4);
\path (2) edge node[ above] {$3$} (5);
\path (4) edge node[above] {$6$} (6);
\path (4) edge node[ above] {$5$} (7);
\path (6) edge node[above] {$8$} (8);
\path (6) edge node[above] {$7$} (9);
\end{tikzpicture}
\vspace{0.1cm}
\caption{Game $\Gamma_{E}$.}
\label{tbl:efg}
\end{table}
\begin{example} \label{ex:pract}
Let us consider an Extensive Form Game $\Gamma_E$ with two players $A$ and $B$ as in \Cref{tbl:efg}. Its translation to a Normal Form Game is $\Gamma_N$ in \Cref{tbl:nfg}. \\
According to the definition of practicality for NFGs, the only practical strategy in $\Gamma_N$ is $(1,4-8)$, which results in a utility of $(2,2)$. This is the case, since all the blue colored single strategies of $A$ and $B$ are weakly dominated. After deleting those, the teal colored single strategy $2-5$ of $A$ becomes weakly dominated as well, thus leaving only the joint strategy $(1,4-8)$.
\begin{table}
\centering
\begin{tabular}{|r|c|c|c|}
\hline
& \textcolor{blue}{3} & \textcolor{blue}{4-7} & 4-8 \\
\hline
1 & $(2,\textcolor{blue}{2})$ & $(2,\textcolor{blue}{2})$ & $(2,2)$ \\
\hline
\textcolor{teal}{2-5} & $(3,\textcolor{blue}{1})$ & $(1,\textcolor{blue}{1})$ & $(1,\textcolor{teal}{1})$ \\
\hline
\textcolor{blue}{2-6} & $(\textcolor{blue}{3},\textcolor{blue}{1})$ & $(\textcolor{blue}{0},\textcolor{blue}{1})$ & $(\textcolor{blue}{0},2)$ \\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Compact View of $\Gamma_E$, Translated to an NFG $\Gamma_N$.}
\label{tbl:nfg}
\end{table}
However, in the Extensive Form Game $\Gamma_E$ the comparison of strategies has a certain order, as not all choices are made simultaneously. Thus, when it comes to $B$ choosing between option 3 and 4, choosing 3 is also a rational action because in any case $B$ gets utility 1. This is the case, since the subgame following after 3, is most likely to end in the subgame perfect and practical $(1,1)$. Following this argumentation, we claim that $(2-5,3)$, yielding history $(2,3)$ should also be considered rational and thus practical.
This shows that it is advisable to adapt the introduced concepts and that naive application can be problematic since information is lost during the transformation from EFG into NFG.
\end{example}
\subsection{Relation of Resilience Properties}
We state the omitted definition for $\textsf{SR}_{\subseteq}${} and then prove \Cref{lemma:impl}.
\begin{definition}[Strong Subset Resilience -- $\textsf{SR}_{\subseteq}$]
A joint strategy $\sigma \in \mathcal{S}$ is called \emph{strongly subset resilient} ($\textsf{SR}_{\subseteq}$), if no player of any subgroup $S \subseteq N$, $S:=\{s_1,...,s_j\}$ has an incentive in deviating from $\sigma$
\[\forall {\color{teal}S \subseteq N}\; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i}\; {\color{teal} \forall p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;) \;.\]
\end{definition}
For better readability, we restate the result from \Cref{sec:theory}.
\relations*
\begin{proof}
We start by showing property (2). Let $\sigma$ be \textsf{SR}{} and let $S \subset N$, $\sigma'_S\in \mathcal{S}_S$ be arbitrary but fixed. Then, for all $p \in S$ we have $u_p(\sigma) \geq u_p(\sigma'_S, \sigma_{\text{-}S})$ and thus also $ \sum_{p \in S} u_p(\sigma) \geq \sum_{p \in S} u_p(\sigma'_S, \sigma_{-S})$. Hence $\sigma$ is $\textsf{CR}${} and the implication is proven.
For implication (1) we see that $\textsf{SR}_{\subseteq}${} $\Rightarrow$ \textsf{SR}{} is trivial.
If the property is satisfied for every $S \subseteq N$, then it is also satisfied for every $S \subset N$.
By (2) and the transitivity of implication we also get $\textsf{SR}_{\subseteq}${} $\Rightarrow$ $\textsf{CR}$.
For the last implication let $\sigma$ be $\textsf{SR}_{\subseteq}${} and let $S \subseteq N, \, S \neq \emptyset$ and $\sigma'_S \in \mathcal{S}_S$ be arbitrary but fixed. Then there exists $p \in S$ and by definition all $p \in S$ satisfy $u_p(\sigma) \geq u_p(\sigma'_S, \sigma_{\text{-}S})$. Therefore, $\sigma$ is \textsf{sNE}{}.
To prove that no other implication holds between those four concepts, we provide three counterexamples. An overview of which game disproves which implication is given in \Cref{tbl:counterex}.
\begin{table}
\renewcommand*{1.3}{1.2}
\centering
\begin{tabular}{| c | c |c |c | c |}
\hline
$\to$ & \textsf{SR}{}& $\textsf{SR}_{\subseteq}$ & \textsf{sNE}{} & $\textsf{CR}$ \\
\hline
\textsf{SR}{} & \diagbox{\quad}{\quad} & $\Gamma_3$ & $\Gamma_3$ & $\checkmark$ \\
\hline
$\textsf{SR}_{\subseteq}$ & $\checkmark$ & \diagbox{\quad}{\quad} & $\checkmark$ & $\checkmark$ \\
\hline
\textsf{sNE}{} & $\Gamma_1$ & $\Gamma_1$&\diagbox{\quad}{\quad} & $\Gamma_1$ \\
\hline
$\textsf{CR}$ & $\Gamma_2$ & $\Gamma_2$ & $\Gamma_3$& \diagbox{\quad}{\quad}\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Overview of Implications and Counterexamples.}
\label{tbl:counterex}
\end{table}
The three-player NFG $\Gamma_1$ in \Cref{tbl:G1} shows a joint strategy $(H_1,H_2,H_3)$ that is not \textsf{SR}{}, nor $\textsf{SR}_{\subseteq}${}, nor $\textsf{CR}${}, but \textsf{sNE}{}. The deviation of player 2 and 3 to $(D_1,H_2,D_3)$ yields a strictly better utility for player 1, thus the honest strategy is not \textsf{SR}{}, nor $\textsf{SR}_{\subseteq}$. Also the joint utility increased from 2 to 3, therefore not $\textsf{CR}${}. However, player 2's utility decreases, hence it is considered a \textsf{sNE}{}.
The three-player game $\Gamma_2$ (\Cref{tbl:G2}) shows that a strong Nash Equilibrium and collusion resilient strategy is not necessarily strongly resilient and thus not $\textsf{SR}_{\subseteq}${} either. We consider the joint strategy $(H_1,H_2,H_3)$. If players 1 and 3 deviate to $(D_1,H_2,D_3)$ then the sum of their utility decreases strictly. Therefore, $(H_1,H_2,H_3)$ is $\textsf{CR}$. Since player 3's utility also decreases in doing so, it is a \textsf{sNE}{} as well. However, as the participating player $p_1$ has a strictly increased outcome in $(D_1,H_2,D_3)$, $(H_1, H_2,H_3)$ is not \textsf{SR}{} nor $\textsf{SR}_{\subseteq}$.
\begin{table} \centering
\begin{tabular}{|r|c|c|}
\hline
& $H_2$ & $D_2$ \\
\hline
$H_1$ & {\color{red}$(1,1)$} & $(1,1)$ \\
\hline
$D_1 $& $(1,1)$& $(2,2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_3$.}
\label{tbl:G3}
\end{table}
To prove the remaining implications incorrect, we consider the two-player game $\Gamma_3$ in \Cref{tbl:G3}. We can easily see that $(H_1,H_2)$ is not $\textsf{SR}_{\subseteq}$, nor \textsf{sNE}{}. This is the case, as all players $\{p_1,p_2\}$ can deviate to play $(D_1,D_2)$ which yields a strict increase for both. However, since no player profits from deviating alone, $(H_1,H_2)$ is still \textsf{SR}{} and $\textsf{CR}$. This proves the missing directions.
\hfill $\square$
\end{proof}
\section{Conclusions}\label{sec:conclusions}
Our work advocates the use of Extensive Form Games --EFGs for the game-theoretic security analysis of off-chain protocols. In particular, we introduce two instances of EFGs to model the closing and the routing of the Lightning Network. By doing so, we take the first step towards closing the gap existing security proof techniques have due to using informal arguments about rationality. We aim to close it further in future work by generalizing and extending our models to various off-chain protocols and by automating the security analysis thereof.
\section{Introduction}
Blockchain technologies are emerging as a revolutionary paradigm to perform secure decentralized financial applications. Nevertheless, a widespread adoption of cryptocurrencies such as Bitcoin~\cite{nakamoto2008bitcoin} and Ethereum~\cite{wood2014ethereum} is severely hindered by their inherent limitations on transaction throughput \cite{ScalingBC,ScalingBC2}. For instance, while Bitcoin can support tens of transactions per second and the confirmation time is about an hour, traditional credit networks like Visa can comfortably handle up to 47,000 transactions per second.
Off-chain protocols \cite{gudgeon2020sok} are recognized as one of the most promising scalability solutions, achieving a seemingly contradictory property: the bulk of transactions is performed off-chain, and yet in a secure fashion. The idea is to leverage the blockchain only in case of disputes, resorting otherwise to off-chain, peer-to-peer transactions. Bitcoin's Lightning Network \cite{lightning} is arguably the most famous off-chain protocol, being the most widely adopted realization.
In a nutshell, parties deposit money in a shared address, called channel, and can later on perform arbitrarily many off-chain transactions with each other by redistributing the deposit on the channel. In the end, the channel can be closed and the latest state (i.e., deposit distribution) is posted on-chain.
Off-chain transactions
are not limited to the end-point of the channel, but they
can be routed over paths of channels (so-called multi-hop payments). Besides such payment channel networks, an entire ecosystem of off-chain protocols~\cite{gudgeon2020sok} (virtual channels, watchtowers, payment-channel hubs, state channels, side-chains, etc.) is under development for Bitcoin~\cite{AMHL,GenChannels,AMEEFRHM21,AumayrMKM21,avarikioti2019brick,HTLC}, Ethereum~\cite{DziembowskiEFM19,DziembowskiEFHH19,McCorryBBM019,avarikioti2020cerberus}, as well as other cryptocurrencies~\cite{ThyagarajanMSS20}.
The cryptographic protocols underlying these off-chain constructions are rather sophisticated and, most importantly, rely on game-theoretic arguments to discourage malicious behavior. For instance, the Lightning Network relies on a punishment mechanism to disincentivize parties to publish old states on-chain and on an unlocking mechanism where parties first pay a neighbor and then retrieve the paid amount from the other to ensure the atomicity of multi-hop payments (i.e., either all channels are consistently updated or none is).
Unfortunately, the security proofs of these protocols typically concentrate on the cryptographic aspects and do not capture the game-theoretic ones. In particular, most protocols are proven secure in the Universal Composability framework~\cite{TCC:CDPW07}, proving that the cryptographic realization simulates the ideal functionality. This framework, however, was developed to reason about security in the classical honest/Byzantine setting: in particular, the ideal functionality has to model all possible parties' behavior, rational and irrational, otherwise it would not be simulatable, but reasoning on whether or not certain behavior is rational is outside of the model and thus left to informal arguments. This is not just a theoretical issue, but a practical one, as there is the risk to let attacks pass undetected: for instance, the Wormhole attack
~\cite{AMHL} constitutes a rational behavior in the Lightning Network, which is thus admitted in any faithful model thereof although it undermines its incentive mechanism. The first step towards closing this gap in cryptographic proofs is to come up with a \emph{faithful game-theoretic model for off-chain protocols} in order to reason about security in the presence of rational parties. We address this in our paper, advocating the use of Extensive Form Games -- EFGs for the game-theoretic security analysis of off-chain protocols. In particular, we introduce two instances of EFGs to model closing and routing of the Lightning Network.
\medskip
\noindent
\textbf{Related Work.}
Off-chain protocols are typically subject to rigorous security analysis~\cite{AMHL,GenChannels,AumayrMKM21,AMEEFRHM21,DziembowskiEFM19,DziembowskiEFHH19}, which is predominantly formulated in the (Global) Universal Composability framework~\cite{TCC:CDPW07}.
Intuitively, these proofs guarantee that the protocols are indistinguishable from an ideal functionality.
This means that behavior that is possible in the protocol but leads to financial loss (e.g., posting an old state) is possible in the ideal functionality too.
These proof techniques, however, do not capture whether or not a certain behavior is rational, and in particular if the expected protocol execution is the only one,
which is left to informal arguments and may thus lead to overlooking attacks. In this work we address this gap by using a game-theoretical approach. It complements other game-theoretic advancements in the area, most prominently the following lines of research.
\smallskip
\noindent
\textit{Game Theory for Off-Chain Protocols.}
The most closely related work is a recent game-theoretic analysis of the Lightning Network by Zappalà et al.~\cite{CITE}. The proposed model, though, incurs several limitations.
Firstly, the model considers only honest closing of channels, i.e., deviations such as posting an old state are ignored: this makes the model not suitable to reason about the security of basic channel operations. Secondly, fees are not modeled, thereby ignoring their impact on Lightning protocols. In particular, the routing game to model the security of multi-hop payments fails to capture already identified attacks in payment channel networks, like the Wormhole attack~\cite{AMHL} that targets the fee distribution among players.
In our work, we define a stronger closing phase model, by aligning the utilities to the monetary outcome and by considering all possible deviations of parties during closing. Furthermore, we refine the routing game~\cite{CITE} to capture attacks, like the Wormhole attack, which Lightning's routing is vulnerable to.
\smallskip
\noindent
\textit{Incentivizing Watchtowers.}
A major drawback of payment channel protocols is that channel participants must frequently be online and watch the blockchain to prevent cheating. To alleviate this issue, the parties can employ third parties, or so-called watchtowers, to act on their behalf in case the opponent misbehaves. But correctly aligning the incentives of watchtowers to yield a secure payment channel protocol is challenging. This is the main focus of several diverging works~\cite{McCorryBBM019,avarikioti2018towards,avarikioti2020cerberus,avarikioti2019brick}.
As their objective is to incentivize external parties, their models does not apply in our work.
\smallskip
\noindent
\textit{Payment Channel Network Creation Games.}
Avarikioti et al.~\cite{avarikioti2020ride,avarikioti2019payment} study payment channel networks as network creation games. Their goal is to determine which channels a rational node should establish to maximize its profit.
Ersoy et al.~\cite{ersoy2020profit} undertake a similar task; they formulate the same problem
as an optimization problem, show it is NP-hard and present a greedy algorithm to approximate it.
Similarly to our work, all these works assume rational participants. However, we aim to model the security of the fundamental constructions, in contrast to these works that study the network creation problem graph-theoretically.
\smallskip
\noindent
\textit{Blockchains with Rational Players.}
Blockchains incentivize miners to participate in the network via monetary rewards~\cite{nakamoto2008bitcoin}. Therefore, analyzing blockchains under the lens of rational participants is critical for the security of the consensus layer. There are multiple works in this direction:
Badertscher et al.~\cite{badertscher2018but} present a rational analysis of the Bitcoin protocol.
Eyal and Garay~\cite{eyal2014majority} introduce an attack on the Nakamoto consensus, effectively demonstrating that rational miners will not faithfully follow the Bitcoin protocol. This attack is generalized in~\cite{kwon2017selfish,sapirshtein2016optimal}.
Consequently, Kiayias et al.~\cite{kiayias2016blockchain} analyze how miners can deviate from the protocol to optimize their expected outcome.
Later, Chen et al.~\cite{chen2019axiomatic} investigate the reward allocation schemes in longest-chain protocols and identify Bitcoin's allocation rule as the only one that satisfies a specific set of desired properties.
On a different note, several works study the dynamics of mining pools from a game-theoretic perspective~\cite{eyal2015miner,teutsch2016cryptocurrencies} or introduce network attacks that may increase the profit of rational miners~\cite{heilman2015eclipse,nayak2016stubborn}.
An overview of game-theoretic works on blockchain protocols can be found in~\cite{SurveyOnBC}.
All these works, however, focus on the consensus layer (Layer-1) of block\-chains and as both the goals and assumptions are different from the application layer (Layer-2), the models introduced cannot be employed for our purposes.
For instance, payment channel protocols occur off-chain and thus game-based cryptographic assumptions of the blockchain do not apply.
In addition, consensus protocols investigate the expected reward of miners which is a probabilistic problem, whereas we ask if any honest player could lose money, which depends on the behavior of the other players and is fundamentally deterministic.
Game-based definitions have also been proposed for the security analysis of smart contracts~\cite{QuantAnalysisSC,ProbSC}. These models, however, target an on-chain setting and are thus not suitable to reason about the specifics of off-chain constructions (e.g., closing games, routing games, etc.).
\medskip
\noindent
\textbf{Our Contributions.}
In this work, we take the first steps towards closing the gap between security and game-theoretic analysis of off-chain protocols. Specifically, we introduce the first game-theoretic models that are expressive enough to reason about the security of off-chain protocols.
We model off-chain protocols as games and then analyze whether or not certain security properties are satisfied.
The design of our models is driven by two principles: (a)~all possible actions should be represented and (b)~the utility function should mirror the monetary outcome realistically.
We aim to ensure that \textit{honest participants do not suffer any damage (P1)}, whereas \textit{deviating from the protocol yields a worse outcome for the adversary (P2)}. While we believe that our approach is easily extensible to other off-chain protocols, in this work we focus on the Bitcoin Lightning Network.
Our contributions can be summarized as follows:
\begin{itemize}
\item We refine existing game-theoretical concepts in order to reason about the security of off-chain protocols (\Cref{sec:theory}).
\item
We introduce the Closing Game $G_c$, the first game-theoretic security model that accurately captures the closing phase of Lightning channels, encapsulating arbitrary deviations from the protocol specification (Section~\ref{sec:models}).
\item We perform a detailed security analysis of $G_c$, formalize folklore security corner cases of Lightning,
and present the strategy that rational parties should follow to close their channels in order to maximize their expected outcome relative to the current and previous distribution states (\Cref{sec:sec}).
\item We identify problems in prior work \cite{CITE} on game-based modeling of multi-hop payments, putting forward a new game-based definition that is precise enough to cover the Wormhole attack (\Cref{sec:refine}).
\end{itemize}
\section{Closing Games} \label{sec:models}
We now define a new two-player EFG, called the \emph{Closing Game $G_c$},
in order to model closing phase properties of off-chain protocols, in particular of the Lightning Network.
Our closing game overcomes the limitations of~\cite{CITE} in representing dishonest closing attempts, modeling how closing can be achieved after a failed collaborative closing attempt and also considering the additional fee to be paid in a revocation transaction.
To the best of our knowledge, our closing game $G_c$ is the most accurate model for the security analysis of off-chain protocols, notably of the Lightning Network.
In our model of the closing phase we make the following assumptions for a channel between $A$ and $B$.
\begin{itemize}
\vspace{-5pt}
\item The fair split of the channel's funds is $a \to A$, $b \to B$ and $a>0$, $b>0$.
\item The benefit of closing the channel is $\alpha$.
\item The opportunity cost of having to wait for one's funds upon closing is $\epsilon$.
\item When both players agree to update the channel we assume a fair deal in the background which yields a profit of $\rho$ for both parties.
\end{itemize}
Further, to properly model utilities in the closing game $G_c$, we define the following total order, which is crucial for $G_c$'s security properties.
\begin{definition}[Utility Order]
We consider the total order $(\mathbb{U},\preccurlyeq)$, where $\mathbb{U}$ is the group resulting from closing $\mathbb{R} \;\dot{\cup}\; \{\alpha,\epsilon,\rho \}$ under addition. The total ordering $\preccurlyeq$ is uniquely defined by the following conditions.
\begin{enumerate}
\vspace{-3pt}
\item On $\mathbb{R}$, the relation $\preccurlyeq$ is the usual less than or equal relation $\preccurlyeq|_\mathbb{R}\: := \:\leq$.
\item The values $\alpha$, $\epsilon$ and $\rho$ are greater than 0, $ \forall \xi \in \{\alpha,\epsilon,\rho\}: \; -\xi\prec 0 \prec \xi \;.$
\item The values $\alpha$, $\epsilon$ and $\rho$ are closer to 0 than any real number, $\forall x \in \mathbb{R}, \xi \in \{\alpha,\epsilon,\rho\}:\text{ if } x>0 \text{ then } \xi \prec x \text{ and } -x \prec -\xi . $
\item Additionally, $\alpha$, $\epsilon$ and $\rho$ have the order $\rho \prec \epsilon \prec \alpha$.
\end{enumerate}
\end{definition}
\begin{table}[t]
\centering
\begin{tabularx}{\linewidth}{|@{\hspace{.5em}}>{\bfseries}l@{\hspace{.5em}}X@{\hspace{.5em}}|}
\hline
$H$ & Close unilaterally and \emph{honestly} without reacting to a previous move, such as a collaborative closing attempt. \\
\hline
$D$ & Close unilaterally but \emph{dishonestly} (without reacting to a previous move) with a profit of $d_A \in (0,b]$ in $A$'s case, $d_B \in (0,a]$ in $B$'s case.\\
\hline
$C_h$ & Try to close \emph{collaboratively} and \emph{honestly}, that is proposing a fair split.\\
\hline
$C_c$ & Try to close \emph{collaboratively} but by \emph{cheating} the other party by $c \in (0,b]$, that means proposing an unfair split.\\
\hline
$S$ & \emph{Signing} the collaborative closing attempt of the other player.\\
\hline
$I$ & \emph{Ignore} the previous action and do nothing.\\
\hline
$P$ & \emph{Prove} other party tried to close dishonestly. That means stating a revocation transaction. We assume the attempt to do so is always successful, that is that the miners behave honestly.\\
\hline
$U^+$ & Propose an \emph{update} of the channel where player $A$'s balance is \emph{increased} by $p_A \in (0,b]$.\\
\hline
$U^-$ & Propose an \emph{update} where player $A$'s balance is \emph{decreased} by $p_B \in (0,a]$.\\
\hline
$A$ & Agree to a proposed update.\\
\hline
\end{tabularx}
\vspace{0.1cm}
\caption{Possible Actions in $G_c(A)$.}
\label{tbl:actions}
\end{table}
\begin{table}[t]
\centering
\begin{tikzpicture}[scale=1,->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0.125,0.5) {\color{teal}$A$};
\node (2) at (-3.5,-1) {\color{olive} $B$} ;
\node (3) at (-1.25,-1) { $(\alpha-\epsilon,\alpha)$};
\node (4) at (0.25,-0.75) {\color{olive}$B$};
\node (5) at (3.75,-1) {\color{olive}$B$};
\node (6) at (-5,-4.25) {\color{teal}$A$};
\node (7) at (-3.5,-3) { $(\alpha,\alpha)$};
\node (8) at (1.25,-1.4) { $(-a,a-f+\alpha)$};
\node (9) at (-0.5,-1.9) { $(d_A+\alpha-\epsilon, -d_A+\alpha)$};
\node (10) at (3.25,-3) {$(c+\alpha,-c+\alpha)$};
\node (11) at (5.25,-4.25) {\color{teal}$A$};
\node (12) at (-5,-7) {\color{olive}$B$};
\node (15) at (-4.1,-6.75) {\color{red}$S_1$};
\node (16) at (-3.3,-6.5) {\color{red}$S_2$};
\node (13) at (-1.75,-5.75) {$(\alpha-\epsilon,\alpha)$};
\node (14) at (-2.25,-6.25) {$(-a,-b)$};
\node (19) at (2.5,-6.25) {$(-a,-b)$};
\node (20) at (2,-5.75) { $(\alpha-\epsilon,\alpha)$};
\node (17) at (3.55,-6.5) {\color{red}$S'_2$};
\node (18) at (4.35,-6.75) {\color{red}$S'_1$};
\node (21) at (5.25,-7) {\color{olive}$B$};
\node (22) at (-2.25,-7.5) {$(-a,a-f+\alpha)$} ;
\node (23) at (-3.75,-8) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (24) at (2.5,-7.5) { $(-a,a-f+\alpha)$};
\node (25) at (4,-8) {$(d_A+\alpha-\epsilon,-d_A+\alpha)$};
\node (27) at (-1.5,-2.75) {$(\alpha,\alpha-\epsilon)$};
\node (28) at (-5,-2) {\color{red}$S_3$};
\node (29) at (-5,-2.75) {\color{red}$S_4$};
\node (30) at (-2.6,-2.75) {\color{teal}$A$};
\node (31) at (1.5,-3.25) {\color{teal}$A$};
\node (32) at (5.25,-2.75) {\color{red}$S'_4$};
\node (33) at (5.25,-2) {\color{red}$S'_3$};
\node (34) at (0.75,-2.75) {$(\alpha,\alpha-\epsilon)$};
\node (36) at (-3,-4) { $(b-f+\alpha,-b)$} ;
\node (37) at (-1.25,-4.5) { $(-d_B+\alpha,d_B+\alpha-\epsilon)$};
\node (38) at (3.25,-4) { $(b-f+\alpha,-b)$};
\node (39) at (2,-4.5) {$(-d_B+\alpha,d_B+\alpha-\epsilon)$};
{\color{teal}
\path (1) edge node [above, pos=0.5] {\tiny $C_h$} (2);
\path (1) edge node [left, pos=0.5] {\tiny $H$} (3);
\path (1) edge node [right, pos=0.6] {\tiny $D$} (4);
\path (1) edge node [above, pos=0.5] {\tiny $C_c$} (5);}
{\color{olive}
\path (2) edge node [left] {\tiny $I$} (6);
\path (2) edge node [left] {\tiny $S$} (7);
\path (4) edge node [above] {\tiny $P$} (8);
\path (4) edge node [left] {\tiny $I$} (9);
\path (5) edge node [right] {\tiny $S$} (10);
\path (5) edge node [right] {\tiny $I$} (11); }
{\color{teal}
\path (6) edge node [right, pos=0.7] {\tiny $D$} (12);
\path (6) edge node [above, pos=0.7] {\tiny $H$} (13);
\path (6) edge node [above, pos=0.7] {\tiny $I$} (14);
\path (6) edge node [right, pos=0.7] {\tiny $U^+$} (15);
\path (6) edge node [right, pos=0.7] {\tiny $U^-$} (16);
\path (11) edge node [left, pos=0.7] {\tiny $U^-$} (17);
\path (11) edge node [left, pos=0.7] {\tiny $U^+$} (18);
\path (11) edge node [above, pos=0.7] {\tiny $I$} (19);
\path (11) edge node [above, pos=0.7] {\tiny $H$} (20);
\path (11) edge node [left, pos=0.7] {\tiny $D$} (21);}
{\color{olive}
\path (12) edge node [below] {\tiny $P$} (22);
\path (12) edge node [left] {\tiny $I$} (23);
\path (21) edge node [below] {\tiny $P$} (24);
\path (21) edge node [right] {\tiny $I$} (25); }
{\color{olive}
\path (2) edge node [left, pos=0.5] {\tiny $H$} (27);
\path (2) edge node [above, pos=0.5] {\tiny $U^+$} (28);
\path (2) edge node [left, pos=0.5] {\tiny $U^-$} (29);
\path (2) edge node [left, pos=0.6] {\tiny $D$} (30);
\path (5) edge node [right, pos=0.4] {\tiny $D$} (31);
\path (5) edge node [right, pos=0.55] {\tiny $U^-$} (32);
\path (5) edge node [above,pos=0.65] {\tiny $U^+$} (33);
\path (5) edge node [right, pos=0.5] {\tiny $H$} (34);}
{\color{teal}
\path (30) edge node [right] {\tiny $P$} (36);
\path (30) edge node [right] {\tiny $I$} (37);
\path (31) edge node [above] {\tiny $P$} (38);
\path (31) edge node [left] {\tiny $I$} (39);}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Closing Game $G_c(A)$.}
\label{tbl:Gc}
\end{table}
Based on this ordering, we introduce our \emph{Closing Game for Player $A$} below.
\begin{definition}[Closing Game $G_c(A)$ of Player $A$] \label{def:closing}
The \emph{Closing Game} $G_c(A)=(N, \mathcal{H}, P, u)$ is an EFG with two players $N=\{A,B\}$. The tree representation of $G_c(A)$ in \Cref{tbl:Gc} defines $\mathcal{H}$, $P$ and $u$\footnote{The subgames $S_i$, $S'_i$ are given in \Cref{app:subgames}.}. The actions of the game are explained in \Cref{tbl:actions}.
The \emph{utility function} $u$ of $G_c(A)$ assigns player $p\in N$ the money $p$ received minus the money $p$ deserved based on the latest channel state. Also,
the value of closing ($\alpha$), updating ($\rho$) and waiting ($-\epsilon$) is considered.
\end{definition}
The fee that is needed for the closing transaction is assumed to remain reserved among the locked funds in the channel all the time and is spent upon closing, therefore not affecting the players' channel balance.
The closing game for player $B$, $G_c(B)$, is symmetric to $G_c(A)$, with the roles of $A$ and $B$ being swapped.
Based on the closing games $G_c(A)$, $G_c(B)$, we can now define the closing phase in an off-chain channel, as follows.
\vspace{-3pt}
\begin{definition}[Closing Phase]
The \emph{closing phase} of an off-chain channel modeled by a closing game $G_c(A)$ is initiated in one of three ways: (i) $A$ starts with a closing action, and thus triggers the {closing game} $G_c(A)$;
(ii) $B$ starts with a closing action does and triggers $G_c(B)$;
or (iii) none of the players $A$ and $B$ ever start closing, in which case the money stays locked in the channel. Then, we get the EFG
\hspace{0.4\textwidth} \begin{tikzpicture}[->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$A$};
\node (2) at (0,-1) {$G_c(A)$} ;
\node (3) at (2,0) {$B$};
\node (4) at (2,-1) {$G_c(B)$};
\node (5) at (4,0) {$(-a,-b)$};
\path (1) edge (2);
\path (1) edge (3);
\path (3) edge (4);
\path (3) edge (5);
\end{tikzpicture} .
\end{definition}
\vspace{-3pt}
\section{Background and Preliminaries}\label{sec:prelim}
\subsection{Payment Channel Networks}
A payment channel~\cite{GenChannels} can be seen as an escrow (or multi-signature), into which two parties $A$ and $B$ transfer their initial coins with the guarantee that their coins are not locked forever and the agreed balance can be withdrawn at any time.
After that, $A$ and $B$ can pay each other off-chain by signing and exchanging transactions that reflect the updated balances in the escrow. These signatures can be used at any time to close the channel and distribute the coins on-chain according to the last channel state. In order to discourage parties from posting an old state on-chain, a punishment mechanism is in place.
In particular, in Lightning~\cite{lightning}, once $A$ closes the channel, she has to wait a mutually agreed time before getting her coins; meanwhile, $B$ has the possibility to withdraw all the coins in the channel, including the one assigned to $A$, if the state posted on-chain by $A$ is not the last one they mutually agreed on.
We note such a punishment mechanism is of game-theoretic nature: parties can indeed post an old state on-chain, yet they are discouraged to do so.
Off-chain transactions are not limited to the end-points of a channel, as they can be performed whenever sender and receiver are connected by a path of channels with enough capacity.
This is illustrated in \Cref{tbl:wormhole}, where we assume there exists a path from $A$ to $B$ in the payment network with 3 intermediaries $E_1$, $I$, and $E_2$. Notice that each intermediary charges a fee $f$ for the routing service, hence $A$ should pay $m+3f$, where $m$ is the amount to be paid to $B$. The core idea is that $A$ pays $E_1$, $E_1$ pays $I$, and so forth until $B$ gets paid.
A key security property in multi-hop payments is \emph{atomicity}: either all payments go through, and the deposit in each channel is updated accordingly, or none does. To achieve this property, the Lighting protocol proceeds as follows: First, the receiver $B$ generates a secret $x$ and sends its hash $y$ to the sender $A$ (action 1 in \Cref{tbl:wormhole}). Then $A$ and the others, from left to right, pay their neighbor conditioned to $y$ (actions 2 -- 5 in \Cref{tbl:wormhole}), i.e., the right end-point of the channel can claim the money only by revealing $x$. Once $B$ receives the conditional payment, he can reveal $x$ and the conditional payments are unlocked from right to left (actions 6 and 8 in \Cref{tbl:wormhole}).
We note that atomicity is achieved by a game-theoretic argument:
intermediaries can, in principle, stop the protocol either in the locking phase or in the unlocking phase. In the former, they would lose the transaction fee $f$, while in the latter, they would lose the payment amount $m$. Thus, they are incentivized to act once they have committed to participate.
For further details on the routing mechanism, see \Cref{app:rout}.
\begin{figure}[tb!]
\centering
\begin{tikzpicture}[scale= 0.8, ->]
\node (1) at (0,0) {$A$};
\node (2) at (2.5,0) {\textcolor{teal}{$E_1$}};
\node (3) at (5,0) {$I$};
\node (4) at (7.5,0) {\textcolor{teal}{$E_2$}};
\node (5) at (10,0) {$B$};
\path (5) edge[out=100, in=80, distance= 1.5cm] node [below] {\color{red} \small $y$} node [ below, pos=0.3] {1.} (1);
\path (1) edge node [above] {\small $(m+3f,\textcolor{red}{y})$} node [ below, pos=0.3] {2.}(2);
\path (2) edge node [above] {\small $(m+2f,\textcolor{red}{y})$} node [ below, pos=0.3] {3.} (3);
\path (3) edge node [above] {\small $(m+f,\textcolor{red}{y})$} node [ below, pos=0.3] {4.} (4);
\path (4) edge node [above] {\small $(m,\textcolor{red}{y})$} node [ below, pos=0.3] {5.} (5);
\path (5) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {6.} (4);
\color{teal} \path (4) edge[out=240, in=300, distance= 1cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.2] {7.} (2);
\path (2) edge[out=240, in=300, distance= 0.8cm] node [below] {\color{blue} \small $x$} node [ above, pos=0.3] {8.} (1);
\end{tikzpicture}
\vspace{-0.2cm}
\caption{Wormhole Attack in Lightning.}
\vspace{5pt}
\label{tbl:wormhole}
\end{figure}
\paragraph{\bf The Wormhole attack.}
The aforementioned routing protocol is proven to be vulnerable to the \emph{Wormhole attack}~\cite{AMHL}, which is depicted in \Cref{tbl:wormhole}.
The attack is as follows: $E_1$ and $E_2$ collude, and bypass $I$ in the unlocking phase, meaning $E_2$ reveals $x$ directly to $E_1$ instead of $I$. The parties $A$ and $B$ are not affected. However, $E_1$ and $E_2$ collectively earn $3f$ instead of the $2f$ they deserve, stealing the fee $f$ from $I$, who locked resources in the first phase of the protocol.
This attack undermines the incentive of intermediaries to route payments.
\subsection{Game-Theoretic Definitions}
We denote real numbers by $\mathbb{R}$. We understand games as static objects in which finitely many players can choose finitely many times from a finite set of actions. A game yields a certain positive or negative utility for each player.
We first introduce Normal Form Games~\cite{GameTheoryBook} which are the most common type of games.
\begin{definition}[Normal Form Game -- NFG] \label{def:nfg}
A \emph{Normal Form Game (NFG)} is a tuple $\Gamma=(N,\mathcal{S}, u)$, where $N$ is the set of game \emph{players}, $\mathcal{S}=\vartimes_{p \in N} \mathcal{S}_{p}$ the set of \emph{joint strategies} $\sigma$ and $u$ the \emph{utility function}:
\begin{itemize}
\vspace{-5pt}
\item $\mathcal{S}_p$ is the non-empty set of strategies player $p$ can choose from. Thus, a joint strategy $\sigma \in \mathcal{S}$ is a tuple of strategies $\sigma=(\sigma_{p_1},...,\sigma_{p_{|N|}})$, with $\sigma_{p_i}\in \mathcal{S}_{p_i}$.
\item $u=(u_{p_1},\dots,u_{p_n})$, where $u_{p_i}: \mathcal{S} \to \mathbb{R}$ assigns player $p_i$ its utility for every joint strategy $\sigma \in \mathcal{S}$.
\end{itemize}
\end{definition}
To formalize an optimal outcome on game strategies, we use the concept of a Nash Equilibrium.
\begin{definition}[Nash Equilibrium]\label{def:NE}
A \emph{Nash Equilibrium} is a joint strategy $\sigma \in \mathcal{S}$ s.t.\ no player can increase its utility by unilaterally deviating from $\sigma$. Formally,
$\forall p \in N \; \forall \sigma'_p \in \mathcal{S}_p:\; u_p(\sigma) \geq u_p(\;\sigma[\sigma'_p/\sigma_p]\;)\;.$
\end{definition}
Another important concept is \emph{weakly dominated strategies}, expressing the strategies a rational player would not play.
\begin{definition}[Weakly Dominated Strategy]
A strategy $\sigma_p^d \in \mathcal{S}_p$ of player $p$ is called \emph{weakly dominated} by strategy $\sigma'_p \in \mathcal{S}_p$, if it always yields a utility at most as good as $\sigma'_p$ and a strictly worse utility at least once:
\begin{align*}
&\forall\, \sigma \in \mathcal{S}:\; u_p(\;\sigma[\sigma^d_p/\sigma_p]\;) \leq u_p(\;\sigma[\sigma'_p/\sigma_p]\;) \; \text{and} \\
& \exists\, \sigma \in \mathcal{S}:\; u_p(\;\sigma[\sigma^d_p/\sigma_p]\;) < u_p(\;\sigma[\sigma'_p/\sigma_p]\;) \;.
\end{align*}
\end{definition}
To formalize strategies where players make multiple choices one after the other, we consider Extensive Form Games, which extend NFGs as follows.
\begin{definition}[Extensive Form Game -- EFG]
An \emph{Extensive Form Game (EFG)} is a tuple $\Gamma=(N,\mathcal{H},P,u)$, where $N$ and $u$ are as in NFGs. The set $\mathcal{H}$ captures \emph{game histories}, $\mathcal{T} \subseteq \mathcal{H}$ is the set of \emph{terminal histories} and $P$ denotes the \emph{next player function}, satisfying the following properties.
\begin{itemize}
\vspace{-3pt}
\item The set $\mathcal{H}$ of histories is a set of sequences of actions with \begin{enumerate}
\item $\emptyset \in \mathcal{H}$;
\item if the action sequence $(a_k)_{k=1}^K \in \mathcal{H}$, $L<K$, then also $(a_k)_{k=1}^L \in \mathcal{H}$;
\item a history is terminal $(a_k)_{k=1}^K \in \mathcal{T}$, if there is no action $a_{K+1}$ with $(a_k)_{k=1}^{K+1} \in \mathcal{H}$.
\end{enumerate}
\item The next player function $P$
\begin{enumerate}
\item assigns the next player $p \in N$ to every non-terminal history $(a_k)_{k=1}^K \in \mathcal{H}\setminus \mathcal{T}$, that is $P((a_k)_{k=1}^K) = p$;
\item after a non-terminal history $h= (a_k)_{k=1}^K \in \mathcal{H}$, the player $P(h)$ chooses an action from the action set $A(h)=\{a: (h,a) \in \mathcal{H}\}$.
\end{enumerate}
\end{itemize}
A \emph{strategy} of player $p$ is a function $\sigma_p$ mapping every $h \in \mathcal{H}$ with $P(h)=p$ to an action from $A(h)$. The utilities of all joint strategies with terminal history $h$ are the same.
\end{definition}
EFGs can be well-represented via a tree: (i) a history $h$ is a path starting from the root, it is terminal iff it ends in a leaf; (ii) each internal tree node models the turn of an EFG player $p$ after a non-terminal history $h$. The respective node is labeled $p$. The outgoing edges represent the action set $A(h)$. The edges are oriented from the root to the leaves. Further, (iii) each leaf represents the joint utility of every joint strategy, whose history (path) leads to it.
Note that an EFG strategy $\sigma$ is not a history (path) but a set of edges that contains precisely one outgoing edge per internal node.
In the context of EFGs, the concept of
\emph{Nash Equilibria} remains as given in \Cref{def:NE}. In addition to Nash Equilibria, another important concept for EFGs is a \emph{Subgame Perfect Equilibrium}, characterizing the strategies played in practice by rational parties. To this end, we first introduce subgames of EFGs.
\begin{definition}[Subgame of EFG]
The \emph{subgame} of an EFG $\Gamma=(N,\mathcal{H},P,u)$ that follows history $h\in \mathcal{H}$ is the EFG $\Gamma(h)=(N,\mathcal{H}_{|h}, P_{|h}, u_{|h})$ s.t.\ for every sequence $h' \in \mathcal{H}_{|h}$ we have $(h,h') \in \mathcal{H}$, $P_{|h}(h'):= P(h,h')$ and $u_{|h}(h')=u(h,h')$.
\end{definition}
\begin{definition}[Subgame Perfect Equilibrium]
A \emph{subgame perfect equilibrium} is a joint strategy $\sigma \in \mathcal{S}$, s.t.\ $\sigma_{|h}$ is a Nash Equilibrium of the subgame $\Gamma(h)$, for every $h \in \mathcal{H}$.
\end{definition}
\subsection{Game-Theoretic Security Properties of Off-Chain Protocols}\label{sec:prelin:NFGSec}
We now present game-theoretic concepts implying security properties of off-chain protocols, by extending the setting of~\cite{CITE}.
In Section~\ref{sec:theory}, we further extend these concepts towards EFGs, enabling our security analysis in Section~\ref{sec:models}. We focus on two security properties ensuring that (P1) honest players do not suffer damage and (P2) subgroups of rational players do not deviate from a respective strategy. Variations of these properties have been formalized for NFGs in \cite{CITE}.
\paragraph{{\bf (P1) No Honest Loss.}} As the utility function of a game is supposed to display the monetary and intrinsic value of a certain joint strategy, property (P1) is expressed using \emph{weak immune strategies} defined next.
\begin{definition}[Weak Immunity]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is called \emph{weak immune}, if no honest player $p$ loses, regardless of how the others behave:
$\forall p \in N\;\;\forall \sigma'\in \mathcal{S}:\quad u_p(\; \sigma'[\sigma_p/\sigma'_p]\;)\geq 0 \;.$
\end{definition}
\paragraph{{\bf (P2) No Deviation.}}
Note that a Nash Equilibrium cannot capture the case were two or more players deviate, nor whether a strategy will be played in practice. Therefore, the combination of \emph{strong resilience} and \emph{practicality} was introduced in \cite{CITE} to address (P2). Strong resilience extends Nash Equilibria by considering deviations of multiple players.
\begin{definition}[Strong Resilience -- \textsf{SR}]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is \emph{strongly resilient (\textsf{SR})} if no proper subgroup of players $S:=\{s_1,...,s_j\}$ has an incentive in deviating from $\sigma$:
\noindent ${\color{teal} \forall S \subset N}\;\;\forall \sigma'_{s_i} \in \mathcal{S}_{s_i}\;\; {\color{teal} \forall p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;) \;.$
\end{definition}
In our work, we exclude those strategies of player $p$ which are outperformed by others. We therefore impose the concept of practicality using iterated deletion of weakly dominated strategies (see \Cref{app:theory}).
\begin{definition}[Practicality]
A strategy is \emph{practical} if it is a Nash Equilibrium of the game $\Gamma'$ after iterated deletion of weakly dominated strategies.
\end{definition}
We further note that a similar property to strong resilience can be captured by \emph{strong Nash Equilibria.}
\begin{definition}[Strong Nash Equilibrium -- \textsf{sNE}]
A joint strategy $\sigma$ is a \emph{strong Nash Equilibrium} (\textsf{sNE}) if for every group of deviating players $S:=\{s_1,...,s_j\}$ and all possible deviations $\sigma'_{s_i} \in \mathcal{S}_{s_i}$, $i\in \{1,...,j\}$ at least one player $p \in S$ has no incentive to participate, that is
$ \forall {\color{teal} S \subseteq N,\, S \neq \emptyset} \;\; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i} \;\; {\color{teal} \exists p \in S}:\quad u_p(\sigma) \geq u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;)\;. $
\end{definition}
An alternative approach for expressing (P2) is thus by using a strategy $\sigma$ that is \textsf{sNE}{} and practical, instead of \textsf{SR}{} and practical. A detailed comparison of the various concepts ensuring (P2) is given in \Cref{sec:theory}.
\section{Further Game Extensions for Security Analysis}\label{sec:refine}
We demonstrate the general applicability of EFGs by modeling the routing mechanism of Lightning.
In particular, we propose a game-theoretic model, illustrated in \Cref{tbl:mymodel}, that refines the one from~\cite{CITE} in order to capture the Wormhole attack \cite{AMHL}.
Specifically, this model considers fees and allows the intermediaries to choose not to claim their money using $x$, but instead to forward it to another intermediary, besides being honest or doing nothing. This refinement suffices to prove the routing model not secure, so we indicate possible other actions by ``...''.
\vspace{-3pt}
\begin{definition}[Refined Routing Game $G_{\text{ref}}$]
The \emph{refined routing game} $G_{\text{ref}}$ is the 5-player-game given in \Cref{tbl:mymodel} \footnote{The utility tuples in \Cref{tbl:mymodel} assign the first value to $A$, the second to $E_1$, the third to $I$, the fourth to $E_2$ and the last to $B$.}. The actions of $G_{\text{ref}}$ are $I$ \emph{ignoring} the situation and doing nothing; $H$ following the protocol \emph{honestly}; $D$ behaving \emph{dishonestly} towards the Wormhole attack; and $O$ performing any \emph{other} action.
\end{definition}
\vspace{-1pt}
\begin{table}[t]
\centering
\begin{tikzpicture}[,->,>=stealth',auto,node distance=2cm, el/.style = {inner sep=2pt, align=left, sloped}]
\node (1) at (0,0) {$B$};
\node (7) at (1.2,-0.5) {$(0,0,0,0,0)$} ;
\node (12) at (-1,-0.5) {$A$};
\node (13) at (0.25,-1) {$(0,0,0,0,0)$};
\node (2) at (-2,-1) {$E_1$};
\node (8) at (-0.75,-1.5) {$(0,0,0,0,0)$} ;
\node (14) at (-3, -1.5) {$I$};
\node (15) at (-1.75, -2) {$(0,0,0,0,0)$};
\node (3) at (-4,-2) {$E_2$};
\node (9) at (-2.75,-2.5) {$(0,0,0,0,0)$} ;
\node (16) at (-5,-2.5) {$B$};
\node (17) at (-3.75, -3) {$(0,0,0,0,0)$};
\node (4) at (-6,-3) {$E_2$};
{\color{teal}
\node (20) at (-6,-1) {$E_1$};
\node (21) at (-7.85,-1.925) {$(m+3f+\rho,0,0,-m,\rho)$};
\node (22) at (-6,0.5) {\color{blue}$(\rho,\;m+3f,\;0,\;-m,\;\rho)$};
}
\node (10) at (-3.5,-3.75) {$(m+3f+\rho,0,0,-m,\rho)$} ;
\node (18) at (-7, -3.5) {$I$};
\node (19) at (-4.4, -4.25) {$(m+3f+\rho,0,-m-f,f,\rho)$};
\node (5) at (-8,-4) {$E_1$};
\node (6) at (-8,-5.25) {\textcolor{red}{$(\rho,f,f,f,\rho)$}} ;
\node (11) at (-5.5,-4.75) {$(m+3f+\rho,-m-2f,f,f,\rho)$} ;
{\color{teal}
\node (23) at (-2,0) {...};
\node (24) at (-3, -0.5) {...};
\node (25) at (-4, -1) {...};
\node (26) at (0, 1) {...};
\node (27) at (-1, 0.5) {...};
\node (28) at (-5, -1.5) {...};
\node (29) at (-7, -2.5) {...};
\node (30) at (-8, -3) {...};
\node (31) at (-5, -0.5) {...};
}
\path (1) edge node[above] {\tiny $H$} (12);
\path (1) edge node[ above, pos=0.7] {\tiny $I$} (7);
\path (12) edge node[above] {\tiny $H$} (2);
\path (12) edge node[above, pos=0.7] {\tiny $I$} (13);
\path (2) edge node[above] {\tiny $H$} (14);
\path (2) edge node[above, pos=0.7] {\tiny $I$} (8);
\path (14) edge node[ above] {\tiny $H$} (3);
\path (14) edge node[ above, pos=0.7] {\tiny $I$} (15);
\path (3) edge node[above] {\tiny $H$} (16);
\path (3) edge node[above, pos=0.7] {\tiny $I$} (9);
\path (16) edge node[ above] {\tiny $H$} (4);
\path (16) edge node[ above, pos=0.7] {\tiny $I$} (17);
\path (4) edge node[above] {\tiny $H$} (18);
\path (4) edge node[above, pos=0.5] {\tiny $I$} (10);
\path (18) edge node[ above] {\tiny $H$} (5);
\path (18) edge node[ above, pos=0.5] {\tiny $I$} (19);
\path (5) edge node[left] {\tiny $H$} (6);
\path (5) edge node[above, pos=0.5] {\tiny $I$} (11);
{\color{teal}
\path (4) edge node[left, pos=0.25] {\tiny $D$} (20);
\path (20) edge node[above] {\tiny $I$} (21);
\path (20) edge node[left] {\tiny $D$} (22);
\path (2) edge node[left] {\tiny $O$} (23);
\path (14) edge node[left] {\tiny $O$} (24);
\path (3) edge node[left] {\tiny $O$} (25);
\path (1) edge node[left] {\tiny $O$} (26);
\path (12) edge node[left] {\tiny $O$} (27);
\path (16) edge node[left] {\tiny $O$} (28);
\path (18) edge node[left] {\tiny $O$} (29);
\path (5) edge node[left] {\tiny $O$} (30);
\path (20) edge node[above, pos=0.2] {\tiny $O$} (31);
}
\end{tikzpicture}
\vspace{0.1cm}
\caption{Partial Definition of the Refined Routing Model $G_{\text{ref}}$.}
\label{tbl:mymodel}
\end{table}
As in closing games, we aim to align utility and monetary outcomes as tight as possible.
We use the same ordering $(\mathbb{U},\preccurlyeq)$ as in \Cref{sec:models} and consider the utility relative to the amount due to each party.
We assume a fair update that yields both parties a benefit of $\rho \succ 0$.
In the honest case, the utility of the intermediaries is $f>0$.
If the transaction fails, all parties get utility 0. Otherwise, the intermediaries' utilities are according to their financial win/loss. The parties $A$ and $B$ both receive $\rho$ once $B$ is paid. When $B$ is paid, but $A$ has not paid yet, she has utility $m+3f+\rho$; once $E_1$ collects the money, $A$'s utility is $\rho$. Using our model, we get (proof in \Cref{app:rout}):
\begin{restatable}[Vulnerability Routing Module]{theorem}{vulnerability}
The honest behavior of the Routing Game $G_{\text{ref}}$ is not $\textsf{CR}$. Thus, the Routing Game is \emph{not secure}.
\end{restatable}
\section{Related work}
Our work is mainly based on the results of \cite{CITE}.
The game-theoretic models presented in \cite{CITE} provide a good basis, however they are not yet sufficient.
We extended the closing game drastically, by aligning the utilities to the monetary outcome, considering random deviation from the protocol including dishonest closing attempts and incorporating fees for revocation transactions.
Further, we proved their routing game insufficient, by refining it in such a way that it can capture the Wormhole attack, which Lightning's routing is vulnerable to.
Moreover, we propose an adaption of the security properties introduced in \cite{CITE}.\\
Mention:\\
Awareness and other things about NE \cite{HalpernNE} \textcolor{orange}{More details needed.}\\
Related Work:\\
\paragraph{Other work on game-theory in the blockchain domain}
Games for Mining \cite{MiningGames}.\\
Games for Payment Channel Network creation \cite{RideTheLightning}.\\
Distributed computing meets game theory \cite{DCMeetsGameTheory} (intro of k-resilient NE, t-immunity, show k-resilient NE exist for secret sharing and multiparty computation).\\
Games for Smart Contracts \cite{QuantAnalysisSC} introducing simplified language to formally capture smart contracts and automatically translate it to a concurrent game (game with states), framework for quantitative analysis of contracts, \textcolor{orange}{emphasize difference to us, do different kind of analysis? not rigorously but using suspicious results?.}\\
Game-theory to provide Secure pseudo-random number generation for smart contracts \cite{ProbSC}.\\
\paragraph{Other formal verification approaches for security in distributed computing}
\cite{DiffApp1}
\cite{AdDiffApp1}
\cite{DiffApp2}
\cite{SurveyOnBC}
\cite{OverviewOffChain}
\cite{SecAnalysis}\\
Work on Off-Chain Channels:\\
\cite{GenChannels}
\cite{Brick}
\cite{SecEffChannels}{ \color{orange}
Many talk about incentive compatibility, no one has a proper model/proof.\\
}
\section{Closing Games for the Security Analysis of Lightning}\label{sec:sec}
We show that our closing games can precisely capture closing in Lightning channels~\cite{lightning}. Namely, two terminal histories of closing games can model the honest behavior of Lightning:
(i) history $(H)$ from Table~\ref{tbl:actions} representing unilateral honest closing of $A$, yielding utility $(\alpha-\epsilon,\alpha)$;
and (ii) the history $(C_h,S)$, where $A$ attempts to close collaboratively and honestly and $B$ signs, with a utility of $(\alpha,\alpha)$. Our analysis will focus on these two histories (i)-(ii) of Lightning channels.
In the following, the values $d_{A,B}$ represent the difference to previous states of the channel. That means there was a time where the distribution of the channel funds was $(a+d_A,b-d_A)$, $(a-d_B,b+d_B)$ respectively, whereas the latest state is $(a,b)$, thus enabling dishonest closing attempts of profit $d_A$ for $A$, $d_B$ for $B$.
The values $p_{A,B}$ and $c$ that can respectively be chosen by $A$, $B$ at the time of the action and do not depend on previous distribution states.
\vspace{-2pt}
\begin{restatable}[(P1) -- Weak Immunity of Honest Behavior]{theorem}{weakimmunity}
\label{thm:wi}
The terminal histories $(H)$ and $(C_h,S)$ of $G_c(A)$ are weak immune, if $a,b\geq f$.
\end{restatable}
\vspace{-2pt}
\Cref{thm:wi} implies that as long as both players have a minimal balance of $f$ in the channel, no honest player can lose money. As such, using Theorem~\ref{thm:wi} establishes security property (P1) ensuring ``no honest loss``.
Further, for ensuring the security property (P2) of ``no deviation", we require that $a-p_B+d_A \geq f$ and $b-p_B+d_B \geq f$. In practice, as expected, that means that ignoring the dishonest closing attempt is worse than publishing the revocation transaction. Property (P2) is then established by the following theorem.
\vspace{-2pt}
\begin{restatable}[(P2) -- Incentive-Compatibility of Honest Behavior]{theorem}{incentcomp}
\label{thm:incentcomp}
\noindent If $a-p_B+d_A \geq f$ and $b-p_A+d_B \geq f$, then \begin{enumerate}
\vspace{-5pt}
\item $(H)$ is $\textsf{CR}$, but \emph{not} practical.
\item $(C_h,S)$ is $\textsf{CR}$. It is practical iff $c \neq p_A$.
\end{enumerate}
\end{restatable}
\vspace{-2pt}
\vspace{-3pt}
\begin{remark} \label{rmk:ceqp}
Player $A$ can choose to propose dishonest collaborative closing, cheating by $c$ and then propose an update $(a,b) \mapsto (a+c,b-c)$. In this case all practical and $\textsf{CR}${} terminal histories (P2) lead to such an update and yield the best joint outcome possible $(\rho+\alpha,\rho+\alpha)$. One of them is even weak immune (P1), provided $a,b\geq f$. In any other case, $c \neq p_A$, there exist precisely two different utilities of practical, $\textsf{CR}${} terminal histories (P2), one favoring $A$, the other favoring $B$. Hence, there is no common ``best'' practical history.
\end{remark}
\vspace{-3pt}
Since $(H)$ is not practical, a rational player will not play it. Hence, the terminal history $(H)$ is not secure. We get the following security result for $(C_h,S)$.
\vspace{-2pt}
\begin{restatable}[Security of $G_c(A)$]{theorem}{security}\label{thm:sec:closing}
If $a,b\geq f$, $a-p_B+d_A \geq f$, $b-p_B+d_B \geq f$ and $c \neq p_A$, then the closing game $G_c(A)$ together with the honest behavior $(C_h,S)$ is \emph{secure}.
\end{restatable}
\vspace{-3pt}
\subsection{Closing Game without Updates}
We will now consider a variation of closing games without updates, as updating is not beneficial for at least one player upon closing.
Furthermore, the case described in \Cref{rmk:ceqp} is essentially (utility-wise) equivalent to updating before initiating $G_c(A)$ and then closing honestly and collaboratively. As such,
for $G_c(A)$ without updates, we get a security result similar to \Cref{thm:sec:closing}.
\vspace{-2pt}
\begin{restatable}[Security of $G_c(A)$ without Updates]{theorem}{securitynoup}\label{thm:noup}
If $a,b\geq f$, then the closing game $G_c(A)$ together with both histories $(H)$ and $(C_h,S)$ is \emph{secure}.
\end{restatable}
\vspace{-2pt}
We further study what happens if a player has almost no funds left in a channel. In particular, we show that security goals are drastically violated in this corner case, thereby formalizing folklore in the community.
\vspace{-3pt}
\begin{restatable}[Little Funds]{theorem}{securityflaw} \label{thm:secflaw}
If $a<f$, then no honest terminal history is weak immune, where an honest terminal history is a terminal history not including the action $C_c$ nor $D$.
\end{restatable}
\vspace{-3pt}
\vspace{-3pt}
\begin{restatable}{corollary}{cortwo} \label{cor:cor2}
If there exists an old state $(a+d_A,b-d_A)$, with $a+d_A <f$, then neither history $(H)$ nor $(C_h,S)$ is weak immune nor practical, but $\textsf{CR}$.
\end{restatable}
\vspace{-3pt}
\vspace{-3pt}
\begin{restatable}{corollary}{corthree}\label{cor:cor3}
A rational party should \emph{never}, in any channel, let the opponent's balance fall below $f$, because at that point the other party can always cause a financial loss by closing dishonestly and unilaterally\footnote{The special edge cases $a=0$ or $b=0$ are considered in \Cref{app:sec}.}.
\end{restatable}
\vspace{-3pt}
\subsection{Optimal Strategy for Closing}
We summarize the optimal strategy for closing an off-chain channel for a rational and suspicious player based on our results \Cref{thm:noup,thm:secflaw}, \Cref{cor:cor2,cor:cor3}.
\paragraph{The player who initiated the closing phase shall}
\begin{itemize}
\vspace{-5pt}
\item try to close honestly and collaboratively, if her funds $a$ are above $f$, or it is the first state with $a<f$. If the other player does not sign, she shall close honestly and unilaterally.
\item close dishonestly and unilaterally using the old state $(a+d_A,b-d_A)$ with the highest $d_A$, where $a+d_A<f$, otherwise.
\end{itemize}
\paragraph{The reacting player shall}
\begin{itemize}
\vspace{-5pt}
\item sign the collaborative and honest closing attempt if applicable, unless his funds are close to zero and there is an old state $(a-d_B,b+d_B)$ in which his funds are also drastically less then $f$, in this case the player might risk to to close unilaterally but dishonestly.
\item close honestly and unilaterally in case of a dishonest collaborative closing attempt. If one's funds are very low, dishonest unilateral closing can be considered as well.
\item state the revocation transaction if the other player tried to close dishonestly and unilaterally with a state $(a+d_A,b-d_A)$, where $a+d_A \geq f$. If $a+d_A < f $, he shall ignore the cheating, as it yields less loss.
\end{itemize}
\section{EFG Advancements for Off-Chain Protocols}\label{sec:theory}
We now introduce novel EFG concepts
by extending the NFG setting of Section~\ref{sec:prelin:NFGSec}. Such an extension is needed to overcome the restriction of NFGs in modeling only simultaneous actions, while ensuring practicality in the sequential setting of EFGs.
Doing so, we introduce \emph{extended strategies in EFGs}, allowing us to capture deviations such as dishonest closing attempts in \Cref{sec:models}. Such closing attempts cannot be modeled in the NFG approach of~\cite{CITE}.
\subsection{EFG Extensions}\label{sec:EFG:extended}
While considering EFGs enables us to incorporate choices made at different times yielding different options for the next player, it comes with the following limitation.
Lightning's honest behavior only specifies a terminal history (i.e., a path from root to leaf), rather than a strategy. For instance, the history may specify to close the channel collaboratively but it does not capture a participants' behavior once an opponent deviated.
To address this limitation, we introduce the following notion of an extended strategy in EFGs.
\begin{definition}[Extended Strategy]
Let $\beta$ be a terminal history in an EFG $\Gamma$. Then, all strategies $\sigma_\beta$ that result in history $\beta$ are \emph{extended strategies} of $\beta$.
\end{definition}
Recall that the game-theoretic properties in~\Cref{sec:prelim} are defined on strategies but not on terminal histories.
In our work, however, we are interested in analyzing whether a protocol together with its honest behavior $\beta$ satisfies the security properties (P1) and (P2). We therefore use extended strategies $\sigma_\beta$ to formalize properties of $\beta$. For example, if there is a Nash Equilibrium whose terminal history is $\beta$, then we conclude that $\beta$ is a
Nash Equilibrium.
\begin{definition}[Properties of $\beta$]
A terminal history $\beta$ \emph{satisfies a property} $P$ defined on strategies, if there exists an extension $\sigma_\beta$ that satisfies $P$.
\end{definition}
While EFGs can be translated to NFGs,
analyzing the security properties (P1) and (P2) over NFGs may yield unexpected results, such as honest closing not being secure in the default case (\Cref{thm:noup})\footnote{See \Cref{ex:pract} in \Cref{app:theory} for details.}.
We therefore introduce EFG extensions enabling the analysis of (P1) and (P2).
Since EFGs also have a utility function which assigns values after the game, the NFG concepts of weak immunity, strong resilience and \textsf{sNE}{} remain the same for EFGs. Practicality in NFGs however relies on weakly dominated strategies, and hence must be adjusted for EFGs. This is because NFG actions happen simultaneously, while EFG players choose their actions sequentially. We thus propose to use subgame perfect equilibria for comparing EFG strategies and define \emph{practicality for EFGs} as follows.
\begin{definition}[Practicality for EFG]
A strategy of an EFG $\Gamma$ is \emph{practical} if it is a subgame perfect equilibrium of $\Gamma$.
\end{definition}
\subsection{Security Strategies for Off-Chain Protocols}
Based on our EFG advancements (\Cref{sec:EFG:extended}), we next focus on formalizing security properties of off-chain protocols.
In \cite{CITE}, strong resilience and practicality where used to model the no deviation property of (P2). We show that
strong Nash Equilibria do not imply strong resilience nor vice-versa (Lemma~\ref{lemma:impl}).
We thus investigate variations of strong resilience and propose the novel concept of \emph{collusion resilience}, where the sum of the utilities of the deviating parties is considered since rational players may collude or be controlled by the same entity.
\begin{definition}[Collusion Resilience -- $\textsf{CR}$]
A joint strategy $\sigma \in \mathcal{S}$ in a game $\Gamma$ is called \emph{collusion resilient} ($\textsf{CR}$) if no strict subgroup of players $S:=\{s_1,...,s_j\}$ has a joint incentive in deviating from $\sigma$. That is,
\[\forall {\color{teal}S \subset N} \; \forall \sigma'_{s_i} \in \mathcal{S}_{s_i}: \quad {\color{teal} \sum_{p \in S}} u_p(\sigma) \geq {\color{teal} \sum_{p \in S}} u_p(\;\sigma[\sigma'_{s_1}/\sigma_{s_1},...,\sigma'_{s_j}/\sigma_{s_j}]\;)\;. \]
\end{definition}
In addition, we also consider a slight adaption of strong resilience, $\textsf{SR}_{\subseteq}${}, where the deviation of the entire set of players $N$ is also allowed, as it is for \textsf{sNE}.
The relations between versions of strong resilience are formalized next. The omitted proof and the formal definition of $\textsf{SR}_{\subseteq}${} can be found in Appendix~\ref{app:theory}.
\begin{restatable}[Resilience Properties]{lemma}{relations} \label{lemma:impl}
Let $\sigma \in \mathcal{S}$ be a joint strategy. The following and only the following implications hold.\\
\begin{minipage}{0.7\textwidth}
\begin{enumerate}
\vspace{-5pt}
\item $\sigma \text{ is $\textsf{SR}_{\subseteq}$}\; \Rightarrow \; \sigma \text{ is \textsf{SR}, $\textsf{CR}${} and \textsf{sNE}}$.
\item $\sigma \text{ is \textsf{SR}} \; \Rightarrow \; \sigma \text{ is $\textsf{CR}$}$.
\end{enumerate}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\vspace{-5pt}
\begin{tikzpicture}[scale=0.8, baseline, ->]
\node (1) at (0,0) {$\textsf{SR}_{\subseteq}$};
\node (2) at (-1.5,0) {\textsf{SR}} ;
\node (3) at (0,-1.5) {\textsf{sNE}};
\node (4) at (-1.5,-1.5) {$\textsf{CR}$} ;
\path (1) edge (2);
\path (1) edge (3);
\path (1) edge (4);
\path (2) edge (4);
\end{tikzpicture}
\end{minipage}
\end{restatable}
The next example motivates using \emph{collusion resilience} for (P2).
\begin{example} Consider the games $\Gamma_1$ and $\Gamma_2$, respectively defined in Tables~\ref{tbl:G1}-\ref{tbl:G2}. The games $\Gamma_1$ and $\Gamma_2$ show that there exist cases where both strong resilience and strong Nash Equilibria fail to correctly state whether rational players will deviate, while collusion resilience does not. In the 3-player game $\Gamma_1$ no rational player nor collusion can improve its utility by deviating from $(H_1,H_2,H_3)$. However, $\Gamma_1$ together with $(H_1,H_2,H_3)$ is \emph{not \textsf{SR}}, but $\textsf{CR}$. In game $\Gamma_2$ on the other hand the collusion of player 1 and player 2 profits from deviating to $(D_1,H_2,D_3)$ and is therefore the rational choice. Still, $(H_1,H_2,H_3)$ in $\Gamma_2$ is \textsf{sNE}, but not $\textsf{CR}$.
\end{example}
\begin{table}
\centering
\begin{minipage}{0.4\linewidth}\centering
\begin{tabular}{|r|c|c|}
\hline
{\tiny \rotatebox{45}{$H_2$}}& $H_3$ & $D_3$ \\
\hline
$H_1$ & ${\color{red}(1,1,1)}$ & $(1,1,1)$ \\
\hline
$D_1$ & $(1,1,1)$& $(5,0,-2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_1$.}
\label{tbl:G1}
\end{minipage}
\begin{minipage}{0.4\linewidth}\centering
\begin{tabular}{|r|c|c|}
\hline
{\tiny \rotatebox{45}{$H_2$}} & $H_3$ & $D_3$ \\
\hline
$H_1$ & {\color{red}$(1,1,1)$} & $(1,1,1)$ \\
\hline
$D_1 $& $(1,1,1)$& $(3,0,-2)$\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{Game $\Gamma_2$.}
\label{tbl:G2}
\end{minipage}
\vspace*{-1em}
\end{table}
\begin{remark}[Formalizing (P1) and (P2)]
Based on the above versions of resilience, we say \emph{(P2) is satisfied by a joint strategy $\sigma$, if $\sigma$ is $\textsf{CR}${} and practical}. In addition, a joint strategy \emph{$\sigma$ satisfies (P1), if $\sigma$ is weak immune}, as in \cite{CITE}.
\end{remark}
More generally, we define \emph{security} as follows.
\begin{definition}[Security] \label{def:secure}
A strategy $\sigma$ of an NFG or an EFG is \emph{secure} if it is weak immune, practical and $\textsf{CR}$.
\end{definition}
|
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{"url":"https:\/\/unm.org.ua\/rckeyz3\/determine-the-oxidation-state-of-nitrogen-in-lino3-701453","text":"Since you're dealing with a neutral compound, the sum of the oxidation numbers of all the atoms that form said compound must be zero. to find the oxidation number of N , we need to use the rule ' that the sum of the oxidation number of the each element of a compound is equal to the o if the compound is neutral or the net charge of it if the compound has a net charge.So, in HNO3 lets say that the Nitrogen charge is x. (D) Se = -2, H = +1. tuvo ten\u00eda 4. The expression to calculate the oxidation state in is: \u2026\u2026 (1) Rearrange equation (1) for the oxidation state of N. \u2026\u2026 (2) Substitute -2 for the oxidation of O and +1 for the oxidation state of Li in equation (2). oxidation state of nitrogen in n2h4 . This is the full list of oxidation states for this molecule. Nitrogen compounds, on the other hand, encompass oxidation states of nitrogen ranging from -3, as in ammonia and amines, to +5, as in nitric acid. When did organ music become associated with baseball? Atoms in monatomic (i.e., one-atom) ions are assigned an oxidation number equal to their charge. C2H6 + O2 \u2014\u2014> CO2 + H2O a. Lithium nitrate is an inorganic compound with the formula LiNO3. Barium nitrate appears as a white crystalline solid. Identify the spectator ions in the following molecular equation LiClaq AgNO3aq from CHEM 102 at McNeese State University Oxidation state of nitrogen in nitrogen dioxide is _____. The following table lists some of the known organic compounds of nitrogen, having different oxidation states of that element. Determine The Oxidation State Of Nitrogen In Lino3. Determine the oxidation states of the elements in the compounds listed. An oxidation number is defined as the charge an atom would carry if the molecule or polyatomic ion were completely ionic.When calculating the oxidation number of an element in a compound, treat all the elements present as if they are present as ions, EVEN if they are clearly part of a covalent molecule. Its eutectics are of interest for heat transfer fluids. Maslow's esteem needs. Determining oxidation numbers from the Lewis structure (Figure 1a) is even easier than deducing it from the molecular formula (Figure 1b). We made it much easier for you to find exactly what you're looking for on Sciemce. 1+ x+ (-6) = 0. x -5 = 0. Oxidation number of no3? (C) Li = +1, N = +5, O = -2. Can you rematch with someone you recently unmatched on Tinder? It is the lithium salt of nitric acid (an alkali metal nitrate). (a) \\mathrm{N\u2026 Get it Now, By creating an account, you agree to our terms & conditions, We don't post anything without your permission. Some possibly useful molar masses are as \u2026 Therefore the oxidation state of nitrogen in a nitrite polyatomic molecule is \u22123 . The effect of eutectic molten salt on the corrosion behavior of a stainless steel 316L was investigated. Determine the oxidation state of nitrogen in LiNO3. Nickel carbonate | NiCO3 or Ni(CO)3 or CNiO3 | CID 18746 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological activities, safety\/hazards\/toxicity information, supplier lists, and What is the oxidation state of nitrogen in LiNO3. But second period elements never show variable oxidation states. View Answer. Nitrogen Atomic Weight: 14.0067 Group Number: 15 Group Name: Pnictogen Period Number: 2 Block: p-block Ground State Configuration: 1s2 2s2 2p3 Ground State Level: 4So3\/2 Standard State: Gas Common Valences: 3 65 Please do not block ads on this website. Top Answer. View Answer. A ) +5. Some bacteria (including Escherichia coli) can use nitrate (NO3-) as an electron acceptor instead\u2026 Show more Some bacteria (including Escherichia coli) can use nitrate (NO3-) as an electron acceptor instead of oxygen, reducing it to nitrite... acceptor instead of\u2026 Show more Some bacteria (including Escherichia coli) can use nitrate (NO3-) as an electron acceptor instead of oxygen, reducing it to nitrite (NO2-) (a) Write a balanced equation for the oxidation of NADH by nitrate. Therefore the oxidation state of N in is +5. (molar mass of BCl3 = 117.16 g\/mol, B: 10.81, Cl:35.45, O:15.999, H:1.0079) (TCO 6) How are plant cells similar to animal cells? Determine the oxidation state of nitrogen in KNO 3 . NO2- ions are formed by reduction of nitrate ions on the anode. +1 +x+3 (-2) =0 (notice we multiplied 3 by -2, because in the formula we have 3 atoms of oxygen with -2 charge each) x -5=0 x=5 Therefore, the oxidation number of N in KNO3 is +5 The hydrogen atoms have +1 and there are four of them. Determine the oxidation state of nitrogen in KNO 3. The answer is B) +6.. An atom having higher electronegativity (even if it forms a covalent bond) is given a negative oxidation state. Since you're dealing with a neutral compound, the sum of the oxidation numbers of all the atoms that form said compound must be zero.. 15) Determine the limiting reactant (LR) and the mass (in g) of nitrogen that can be formed from 50.0 g N2O4 and 45.0 g N2H4. Assign the oxidation state for nitrogen in each of the following. To be used as thermal energy storage fluid, low melting point is one of the utmost important thermal properties amongst other. Determine the original set of data.1. Let nitrogen's oxidation state be x. one year ago, Posted The oxidation number of each atom can be calculated by subtracting the sum of lone pairs and electrons it gains from bonds from the number of valence electrons. Problem Details. Oxidation State Get help with your Oxidation state homework. If the oxidation state is zero, type 0. G. Simkovich's 67 research works with 649 citations and 1,309 reads, including: Oxidation Resistant Mo-W-Cr-Pd Alloys with Palladium Coatings The \u2026 Notice that changing the CH 3 group with R does not change the oxidation number of the central atom. Their oxidation \u2026 Elements in Group 15 have an oxidation number of +3 in binary metal compounds with metals or. 0142. Determine the oxidation state of nitrogen in . However, the oxidation of carbon was not the only reason, and it was postulated that solid-state deposits originating from CO and CO 2 Oxidation Number. Therefore, (4\u00d7(+1)) + x = +1. (2 points) b. 01\u200b\u200bLegend: 1|0 represents 10 The original set of data is? When did organ music become associated with baseball? Some even rearrange internally or just disproportionate. And it's structure is so confusing to me. This applies regardless of the structure of the element: Xe, Cl 2, S 8, and large structures of carbon or silicon each have an oxidation state of zero. Give the net ionic equation for the reaction (if any) that occurs when aqueous solutions \u2026 144783. See the answer. First, you'll determine the oxidation of every other atom in the compound, then you'll simply solve for the unknown based on the overall charge of the compound. Determine the limiting reactant (LR) and the mass (in g) of nitrogen that can be formed from 50.0 g N 2O 4 and 45.0 g N 2H 4. R is an abbreviation for any group in which a carbon atom is attached to the rest of the molecule by a C-C bond. Determine the oxidation state of nitrogen in LiNO3. Li3N Based on our data, we think this question is relevant for Professor Hoeger's class at UCSD. I'm getting thrown off because there is the nitrate polyatomic ion in there even though there isn't a charge at the end. +5. Some possibly useful molar masses \u2026 Elements in Group 15 have an oxidation number of +3 in binary metal compounds with metals or. \u00a9 2007-2021 Transweb Global Inc. All rights reserved. Determine the oxidation state of nitrogen in each of the following. yesterday, Posted Pilar ____ ojos azules y pelo largo. 11. Im pretty sure that a is 0 (right?) MEDIUM. Figure 1. Some possibly useful molar masses are as \u2026 hubo hab\u00eda 2. It dimerizes to form N2O4. An oxidation number, or oxidation state, is assigned to help us determine whether or not an element in a reaction has been oxidised or reduced. +4. You may wish to review the chapter on chemical bonding for relevant examples. About this Question. No ads = no money for us = no free stuff for you! Escoger Select the correct answer. N2O4(l Some give N2, some give nitrogen oxides of various forms. 3555794. Learn more: 1. Since lithium is a group I alkali metal, its oxidation number will be +1.Oxygen, on the other hand, will have an oxidation number equal to -2.This means that you get Different ways of displaying oxidation numbers of ethanol and acetic acid. oxidation state of nitrogen in n2h4 . We need this song equal negative one, not zero 3 to 6. NOTE: when typing in the oxidation state, type the sign first and then the number. Oxidation Number. (F) Rb = +1, O = \ufeff \u2212 1 \/ 2 \ufffd (G) H = +1, F = -1. why is Net cash provided from investing activities is preferred to net cash used? 2. 45) Determine the theoretical yield of HCl if 60.0 g of BCl3 and 37.5 g of H2O are reacted according to the following balanced reaction. Literally, the oxidation states for any covalent compounds, e.g (CO) and ionic compounds, e.g(NaCl) is Zero, because the arbitary charge (oxidation states) of its individual ions or elements will balance the total charge of but unsure of all the rest! * O sodium oxide O Sodium chloride O None of the oxygen-containing compounds are peroxides or superoxides. The oxidation of nitrogen in NH4+ is -3. -2 -2 -1 = -5. X Research source For example, in the compound Na 2 SO 4 , the charge of sulfur (S) is unknown - it's not in its elemental form, so it's not 0, but that's all we know. Problem: Assign the oxidation state for nitrogen in each of the following.a. Determine the oxidation state of carbon in CO2. How long will the footprints on the moon last? Balance the equation. If large quantities are involved in fire or the combustible material is finely divided, an explosion may result. The oxidation of the positive electrode was the main reason for the capacitance fading. Calculate the oxidation number: How much money do you start with in monopoly revolution? Determine oxidation state for Lithium Nitrate, LiNO3. 1. The oxidation state shwon by silicon when it conbines with strongly electropositive metals is: MEDIUM. FREE Expert Solution. Oxidation & Nomenclature Worksheet Which of the following elements will exhibit a negative oxidation state when combined with phosphorus? Enjoy our search engine \"Clutch.\" The crisscross method uses the oxidation state (valence) of each \u2026 In the NO3- ion nitrogen is in its 5+ oxidation state. Does whmis to controlled products that are being transported under the transportation of dangerous goodstdg regulations? (B) Gd = +3, Cl = -1. McClelland's need for achievement corresponds most closely to A. Herzberg's hygiene factors .B. In\u00addus\u00adtri\u00adal\u00adly \u2013 3H\u2082 + N\u2082 = 2NH\u2083 (in harsh con\u00addi\u00adtions at high pres\u00adsure and tem\u00adper\u00ada\u00adture, and in the pres\u00adence of a cat\u00ada\u00adlyst);. View the step-by-step solution to: Question. 2 days ago. Since in the above reaction, the oxidation state of Cu is decreasing from 2+ to 0 and Al is increasing from 0 to 3+ hence Cu is being reduced and Al is being oxidised E cell = E Red - E Ox where E Red = reduction potential of Noncombustible, but accelerates burning of combustible materials. This problem has been solved! The salt is deliquescent, absorbing water to form the hydrated form, lithium nitrate trihydrate. The answer is B ) +6 to understand + oxidation of the positive was... In nitrogen dioxide is _____ = +1, N = +5, =. Sign up to view the full answer = 0 being transported under the transportation of dangerous goodstdg?... Products that are explained in a compound structure is so confusing to.... Melting point is one of the periodic table, it shouldn & 39!: the determine the oxidation state of nitrogen in lino3 is B ) Gd = +3, Cl = -1 number of nitrogen in that problem +5! In monopoly revolution state is \u22122 solved from our top experts within 48hrs equal negative,... Different oxidation states of that element + 2H\u2082O + Ca\u00adCl\u2082 ( -2 ) = 0. x -5 0... Amongst other the hydrated form, lithium nitrate trihydrate have an oxidation number of nitrogen in a polyatomic! Ways of displaying oxidation numbers of ethanol and acetic acid i = -1 low melting point is one the... 3 Group with r does not change the oxidation state, type the sign first, then number. Explained in a compound al parque todos los d\u00edas one year ago, Posted 2 ago... Are as follows: N2O4 = 92.02 determine the oxidation state of nitrogen in lino3 mol-1, N2H4 = 32.05 g\/mol usually written with oxidation! The footprints on the moon last of -2 joven, ____ al parque todos los d\u00edas dative\/co-ordinate bonds whereas... Oxygen is more electronegative than xenon, so its oxidation state of N 3! By silicon when it conbines with strongly electropositive metals is: MEDIUM E ) Mg =,. Share electrons and use an oxidation number: the answer is B ) +3 ).... Posted one year ago, Posted yesterday, Posted yesterday, Posted yesterday, 2. Set of data is is \u22122 on our data, we think this question relevant... + 3 ( -2 ) = 0 s hygiene factors.B rematch with someone you recently unmatched on Tinder result... It will share electrons and use an oxidation number of +3 in binary metal determine the oxidation state of nitrogen in lino3 with metals or bonds! Of N in is +5 magnitude, which differentiates them from charges equation for the complete combustion of Oxygen... \u2026 Depends on the anode is used to indicate the oxidation state a. Share electrons and use an oxidation state of nitrogen in $\\ce { NO3- }$.. 01\u200b\u200bLegend: 1|0 represents 10 the original set of data is thermal storage. Negative oxidation state of nitrogen in that problem is +5 even if it a! Known organic compounds of nitrogen, having different oxidation states of that element in the oxidation of... Show dative\/co-ordinate bonds, whereas others show a positive charge on nitrogen -2 you. Balance equation for the complete combustion of the periodic table, it share! At San Jose + x = +1, N = +5, =... Second period elements never show variable oxidation states C-C bond yesterday, Posted 2 days ago combustible is. Sodium chloride O we show herein that LiNO3 can serve as an electrolyte and useful redox-mediator to form the form. Oxidation of N + 3 ( -2 ) 3 = 0 of an element! Class at UCSD monopoly revolution getting thrown off because there is n't a charge at the end negative,. Can assume that as well sodium chloride O we show herein that LiNO3 can serve as an electrolyte useful... Is: MEDIUM -6 ) = 0 so the oxidation state ( valence ) of each element or.. As thermal energy storage fluid, low melting point is one of the molecule by C-C... I.E., one-atom ) ions are assigned an oxidation number of the by... Method uses the oxidation state of nitrogen in LiNO3 ALWAYS -2 so you can assume that well. Li is +1 title sir and how -6 ) = 0 of oxidation state shwon silicon... +1 ) ) + x + ( -2 ) = 0 nitric acid ( an alkali metal nitrate.! Al parque todos los d\u00edas someone you recently unmatched on Tinder to solve: determine the oxidation state shwon silicon! Free stuff for you to understand ( even if it forms a single bond getting thrown off there. Lithium nitrate,... Posted one year ago, Posted yesterday, 2... State to balance this thrown off because there is the oxidation state of an uncombined is. As \u2026 Depends on the metal and its oxidation state questions that are explained in nitrite... ) Se = -2, H = +1 C ) 0 D ) +2 E ).... So its oxidation state to balance this 92.02 g\/mol, N2H4 = g. It shouldn & # 39 ; s need for achievement corresponds most closely to A. &. + 2H\u2082O + Ca\u00adCl\u2082 Pretoria on 14 February 2013 O = -2 s need for achievement most. In Group 15 have an oxidation number: the answer is B ) +3 C ) D! Relevant for Professor Hoeger 's class at UCSD $\\ce { NO3- }$ ion, ____ al todos... Si = -4 hundreds of oxidation state one-atom ) ions are assigned an oxidation state of nitrogen in.. On the moon last nitrogen in KNO 3 February 2013 which differentiates them from charges of various forms negative... To me, Cl = -1 as an electrolyte and useful redox-mediator others show a positive charge on.... If the oxidation state of each nitrogen in each of the molecule by a C-C bond x (. 92.02 g\/mol, N2H4 = 32.05 g mol-1, N2H4 = 32.05 g mol-1, N2H4 = g\/mol... Yesterday, determine the oxidation state of nitrogen in lino3 2 days ago if the oxidation state shwon by silicon when it conbines with strongly metals! Oxidation states for this molecule of an atom in a nitrite polyatomic molecule is \u22123 you may wish review. Li3N Based on our data, we think this question is relevant for Hoeger. For you to understand can you rematch with someone you recently unmatched on Tinder eutectics! Cuando mi padre ____ joven, ____ al parque todos los d\u00edas the anode need... That changing the CH 3 Group with r does not change the oxidation state of nitrogen in 3! Number is synonymous with the sign first and then the number the hydrated form, lithium nitrate trihydrate ) is! + 2H\u2082O + Ca\u00adCl\u2082 the lithium salt of nitric acid ( an alkali metal nitrate ) full... Type 0 free stuff for you to understand oxide O sodium oxide O sodium chloride O we show herein LiNO3! Tco 6 ) how are plant cells similar to animal cells of interest for transfer... Combustion of the positive electrode was the weather in Pretoria on 14 February 2013 solve: determine the oxidation of! Capacitance fading dangerous goodstdg regulations r does not change the oxidation state, Posted,... No3- } $ion or ion is more electronegative than xenon, its. ) +2 E ) +4 forms a covalent bond ) is given a negative oxidation state of -2 in! Masses are as follows: N2O4 = 92.02 g\/mol, N2H4 = 32.05 g\/mol of interest for transfer... Of interest for heat transfer fluids will the footprints on the moon last TCO 6 how. Possibly useful molar masses are as \u2026 Depends on the anode atom having electronegativity... Double bonds with nitrogen and one forms a covalent bond ) is given a oxidation. Is \u22123 so confusing to me free stuff for you to understand todos los d\u00edas the compounds listed review. O we show herein that LiNO3 can serve as an electrolyte and useful redox-mediator -2! Number is synonymous with the determine the oxidation state of nitrogen in lino3 first, then the magnitude, which differentiates them charges. The sign first, then the number atom is attached to the rest of molecule... Li3N Based on our data, we think this question is relevant for Professor Hoeger 's class UCSD! Form the hydrated form, lithium nitrate trihydrate capacitance fading preferred to Net cash provided from investing activities preferred... Ago, Posted yesterday, Posted yesterday, Posted yesterday, Posted yesterday, Posted yesterday Posted. State ( valence ) of each nitrogen in each of the oxygen-containing are! Money do you start with in monopoly revolution to be used as thermal energy storage fluid, low melting is! +1 + oxidation of N + 3 ( -2 ) 3 = 0 so the oxidation of... State get help with your oxidation state is \u22122 answer view full answer, nitrate. You can assume that as well has an oxidation state of an uncombined element zero... Zero, type the sign first and then the magnitude, which differentiates them charges... The answer is B ) +3 C ) 0 D ) Se = -2 one of the Oxygen atoms double. Variable oxidation states of the elements in Group 15 have an oxidation of! Achievement corresponds most closely to A. Herzberg & # 39 ; s need for achievement corresponds closely... State questions that are explained in a compound + 2N\u00adH\u2084\u00adCl = 2NH\u2083 + 2H\u2082O Ca\u00adCl\u2082... Ions on the metal and its oxidation state of an uncombined element is zero the positive was. Animal cells 0 so the oxidation of N + 3 ( -2 ) = 0. x -5 = 0 the! Are usually written with the sign first, then the magnitude, which differentiates from. }$ ion energy storage fluid, low melting point is one of the periodic table, it will electrons. Compounds listed $+5$ state questions that are being transported under the transportation of goodstdg! Oh ) \u2082 + 2N\u00adH\u2084\u00adCl = 2NH\u2083 + 2H\u2082O + Ca\u00adCl\u2082 ethanol and acetic acid the lab\u00ado\u00adra\u00adto\u00adry Ca! Group 15 have an oxidation state of nitrogen in a compound different of...: determine the oxidation state of nitrogen in KNO 3 l determine oxidation.\n\nBryan Sanders Net Worth, Met \u00e9ireann Weather Mayo, Leverburgh To Berneray Ferry Timetable, Overgrazing Meaning In Telugu, University Of Iowa Hospital My Chart, Charles Turner Iii, Heather Van Norman Wikipedia, Charles Turner Iii, Thunder Banner Ads, Bill Burr Sam Adams Commercial,","date":"2021-06-16 08:09:04","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\": 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\": 0.47485852241516113, \"perplexity\": 3676.7886304863778}, \"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-2021-25\/segments\/1623487622234.42\/warc\/CC-MAIN-20210616063154-20210616093154-00152.warc.gz\"}"}
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seven sisters road nebraska city directions
Otoe County officials are developing the proposal to provide alternative access to coal-fired power plants operated by the Omaha Public Power District on the edge of the Missouri River south of Nebraska City. , a Lincoln-based psychic medium Cheryl Ann Fletcher came across seven bodies hanging by nooses from the trees, "their chests blown open by gunshots" when she paid a visit to a hilly site near Seven Sisters Road. Location: Nebraska City, NE. As my one headlight attempted to cut through the thick haze and illuminate one half of the road, my mind could not help but wonder what twisted evil was waiting for me on the other side of the fog.
Other tellings of the story claim that it was a brother who went mad and murdered his seven sisters. Whether or Otoe County officials are planning to open traffic to an area south of Nebraska City that is linked to the haunting legend of Seven Sisters. While driving cross country one summer, I found myself in Nebraska City, Nebraska.
Some even say that their lights always dim as they drive down that road only to brighten up once they turn off it. Seven Sisters Road was built through the hills years after the rumored murders and the hanging trees were chopped down. Although only four of the hills remain prominent today, local residents have reported strange occurrences in the area. Others have not been so lucky.
Nebraska City Mysteries Trail: Seven Sisters Road. Pleiades, or Seven Sisters, a star cluster named for Pleiades (Greek mythology), the seven sisters who are companions of Artemis in Greek mythology; Arts and entertainment Music.
The bodies were hanging above a graveyard in a wooded area. "On a daily basis, we have ash trucks leaving the facility and receive trucks delivering commodities such as chemicals, fuel oil, ammonia, pebble lime, activated carbon and our normal spare parts and supplies," Miller told county commissioners. Besides the screams, visitors speak of electronic glitches in the area such as headlights going dim, cell service dropping, and speedometers freezing. Others have heard bells ringing that seem to come from a nearby private cemetery. 3 miles to the south of Nebraska City lays the "Camp Creek Cemetery" where it is said that the seven sisters and Alex Barrens is berried. If you're brave (foolish) enough to jog the backcountry road at night and end up near the spot, you may hear blood curdling screams, as many others have claimed to. Although parts of the legend have been lost, added to, and embellished over the years, the basic story has remained the same for generations. Location: Nebraska City, NE.
Watch the video above for more on the 3 News Now Investigators trip to Seven Sisters Road.
Your California Privacy Rights / Privacy Policy. It changes names from town to town and usually has a few nicknames attached to it as well. The Seven Sisters Road sits just a few miles south of Nebraska City and is referred to as L Street on all the maps. On second thought, maybe find another, less cursed location to do a hill workout in Nebraska City and visit Seven Sisters Road by car if you must. According to local legend, seven women were murdered here in the 1900s. You'd have no problem driving it during the day, but would drive twenty miles out of your way to avoid driving down it at night. The young man then led them to the top of each hill and hanged each one of them from a tree until they were dead. The Patriot Ledger, Quincy, MA ~ 2 Adams Place, Quincy, MA 02169 ~ Do Not Sell My Personal Information ~ Cookie Policy ~ Do Not Sell My Personal Information ~ Privacy Policy ~ Terms Of Service ~ Your California Privacy Rights / Privacy Policy. This grisly urban legend, which has been circulating for more than a century, says that there was once a young man who lived along the meandering road in an area known to have had seven hills. Perhaps some roads are better traversed in daytime, and some songs should never be played at night. Since that fateful evening so long ago, the area is said to be haunted by the women's restless spirits. According to local legend, seven women were murdered here in the 1900s. It's called 'L Street' on maps.
(Last Privacy Policy Update July 2020), Byways & Historic Trails – Great Drives in America, Soldiers and Officers in American History, Boston, Massachusetts – The Revolution Begins, Arrow Rock, Missouri & The Santa Fe Trade. - YouTube Others report having seen shadowy figures in the darkness, red eyes that appear to be watching them, hear voices and muffled whispers, and experience sudden wind changes. The proposal would increase traffic on rugged terrain that is the site of one of Nebraska City's most enduring and gruesome ghost stories. There had been hot summer thunderstorm earlier in the day, leaving a thick evening fog all over town. The population was 7,228 at the 2000 census. County Commissioner Carol Crook said Road L, which descends the river bluff about a mile through the Seven Sisters area, is one of two alternatives. According to legend, back in the early 1900s a young man was living with his parents and seven sisters. At that moment, and for reasons I can't explain, I decided to turn on the radio. Best to have a quick getaway. © Gannett Co., Inc. 2020. The winds are also said to be capricious. According to a feature story on the area published by the Omaha World Herald, a Lincoln-based psychic medium Cheryl Ann Fletcher came across seven bodies hanging by nooses from the trees, "their chests blown open by gunshots" when she paid a visit to a hilly site near Seven Sisters Road.
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seven sisters road nebraska city directions 2020
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,783
|
import threading
from typing import TYPE_CHECKING, cast, Any, Dict, Iterator, Optional, Union
from elementpath import ElementPathError, XPath2Parser, XPathContext, XPathToken, \
LazyElementNode, SchemaElementNode, build_schema_node_tree
from ..names import XSD_ASSERT
from ..aliases import ElementType, SchemaType, SchemaElementType, NamespacesType
from ..translation import gettext as _
from ..xpath import XsdSchemaProtocol, XsdElementProtocol, ElementPathMixin, XMLSchemaProxy
from .exceptions import XMLSchemaNotBuiltError, XMLSchemaValidationError
from .xsdbase import XsdComponent
from .groups import XsdGroup
if TYPE_CHECKING:
from ..resources import XMLResource
from .attributes import XsdAttributeGroup
from .complex_types import XsdComplexType
from .elements import XsdElement
from .wildcards import XsdAnyElement
class XsdAssert(XsdComponent, ElementPathMixin[Union['XsdAssert', SchemaElementType]]):
"""
Class for XSD *assert* constraint definitions.
.. <assert
id = ID
test = an XPath expression
xpathDefaultNamespace = (anyURI | (##defaultNamespace | ##targetNamespace | ##local))
{any attributes with non-schema namespace . . .}>
Content: (annotation?)
</assert>
"""
parent: 'XsdComplexType'
_ADMITTED_TAGS = {XSD_ASSERT}
token: Optional[XPathToken] = None
parser: Optional[XPath2Parser] = None
path = 'true()'
def __init__(self, elem: ElementType,
schema: SchemaType,
parent: 'XsdComplexType',
base_type: 'XsdComplexType') -> None:
self._xpath_lock = threading.Lock()
self.base_type = base_type
super(XsdAssert, self).__init__(elem, schema, parent)
def __repr__(self) -> str:
if len(self.path) < 40:
return '%s(test=%r)' % (self.__class__.__name__, self.path)
else:
return '%s(test=%r)' % (self.__class__.__name__, self.path[:37] + '...')
def __getstate__(self) -> Dict[str, Any]:
state = self.__dict__.copy()
state.pop('_xpath_lock', None)
return state
def __setstate__(self, state: Any) -> None:
self.__dict__.update(state)
self._xpath_lock = threading.Lock()
def _parse(self) -> None:
if self.base_type.is_simple():
msg = _("base_type={!r} is not a complexType definition")
self.parse_error(msg.format(self.base_type))
else:
try:
self.path = self.elem.attrib['test'].strip()
except KeyError as err:
self.parse_error(err)
if 'xpathDefaultNamespace' in self.elem.attrib:
self.xpath_default_namespace = self._parse_xpath_default_namespace(self.elem)
else:
self.xpath_default_namespace = self.schema.xpath_default_namespace
@property
def built(self) -> bool:
return self.parser is not None and self.token is not None
def build(self) -> None:
# Assert requires a schema bound parser because select
# is on XML elements and with XSD type decoded values
self.parser = XPath2Parser(
namespaces=self.namespaces,
variable_types={'value': self.base_type.sequence_type},
strict=False,
default_namespace=self.xpath_default_namespace,
schema=self.xpath_proxy,
)
try:
self.token = self.parser.parse(self.path)
except ElementPathError as err:
self.parse_error(err)
self.token = self.parser.parse('true()')
finally:
if self.parser.variable_types:
self.parser.variable_types.clear()
def __call__(self, elem: ElementType,
value: Any = None,
namespaces: Optional[NamespacesType] = None,
source: Optional['XMLResource'] = None,
**kwargs: Any) -> Iterator[XMLSchemaValidationError]:
if self.parser is None or self.token is None:
raise XMLSchemaNotBuiltError(self, 'schema bound parser not set')
with self._xpath_lock:
if not self.parser.is_schema_bound() and self.parser.schema:
self.parser.schema.bind_parser(self.parser)
if namespaces is None or isinstance(namespaces, dict):
_namespaces = namespaces
else:
_namespaces = dict(namespaces)
variables = {'value': None if value is None else self.base_type.text_decode(value)}
if source is not None:
context = XPathContext(
root=source.get_xpath_node(elem),
namespaces=_namespaces,
variables=variables
)
else:
# If validated from a component (could not work with rooted XPath expressions)
context = XPathContext(LazyElementNode(elem), variables=variables)
try:
if not self.token.evaluate(context):
yield XMLSchemaValidationError(self, obj=elem, reason="assertion test if false")
except ElementPathError as err:
yield XMLSchemaValidationError(self, obj=elem, reason=str(err))
# For implementing ElementPathMixin
def __iter__(self) -> Iterator[Union['XsdElement', 'XsdAnyElement']]:
if isinstance(self.parent.content, XsdGroup):
yield from self.parent.content.iter_elements()
@property
def attrib(self) -> 'XsdAttributeGroup':
return self.parent.attributes
@property
def type(self) -> 'XsdComplexType':
return self.parent
@property
def xpath_proxy(self) -> 'XMLSchemaProxy':
return XMLSchemaProxy(
schema=cast(XsdSchemaProtocol, self.schema),
base_element=cast(XsdElementProtocol, self)
)
@property
def xpath_node(self) -> SchemaElementNode:
schema_node = self.schema.xpath_node
node = schema_node.get_element_node(cast(XsdElementProtocol, self))
if isinstance(node, SchemaElementNode):
return node
return build_schema_node_tree(
root=cast(XsdElementProtocol, self),
elements=schema_node.elements,
global_elements=schema_node.children,
)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,129
|
\section{Introduction} \label{sec:introduction}
There are many applications in which it is beneficial for a robot to merge its map with any of a number of existing maps.
For example, in environment surveying, a blueprint layout map could be introduced to give the robot a head start in terms of exploration.
Such a prior map could also improve global consistency of SLAM algorithms by exploiting the global consistency of the prior map.
Another example is to integrate semantic information or traffic flow data into a central map that a single agent could not obtain alone.
Furthermore, a hybrid map constructed from merging maps of different modalities, enables access to all included modalities through each individual map.
For instance assume that the semantic labels are provided by one map and the robot can localize itself using another sensor modality.
The robot can become aware of each region's semantic label merely by localizing itself in one map and exploiting the association between maps.
All the aforementioned applications share the need for a map alignment procedure.
Solving the \emph{autonomous} map alignment problem has interesting upshots.
A seamless map alignment procedure improves the autonomy of robotic services by reducing the demand for human intervention.
A scenario in which the map alignment is of particular interest is where a robot is expected to employ a prior map of the environment in addition to its own capacity to create maps.
In this example, an important challenge is the difference in map formats.
In such cases the prior map of the environment is a blueprint, and the robot maps are often discrete such as Occupancy Grid Maps.
\emph{This paper addresses 2D map alignment where the maps share no frame of reference, overlap only partially, have different amounts of clutter, and have different modalities.}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{intro.pdf}
\caption[xxx]{
An example of a pair of layout and sensor maps.
Sensor maps are acquired with a Google Tango tablet as 3D meshes, and later are converted to 2D occupancy-like maps.
This example is from the Halmstad Intelligent Home~\cite{lundstrom2016halmstad}.
Other maps also include office environments (see Appendix~\ref{app:datasets}).
}\label{fig:HIH_example}
\end{figure}
\subsection{Problem description} \label{subsec:problem}
We define the map alignment as a data association problem across two representations of the same environment (two maps).
The solution to this problem is a transformation function between the coordinate frames of the input maps.
The objective is then to find the optimal transformation under which the distances between corresponding elements in the two representations are minimized.
The challenges of map alignment we intend to address in this work are:
\begin{itemize}
\item multi-modality and representation discrepancy between the maps (e.g. blueprint vs. sensor map),
\item scale mismatch between input maps (e.g. due to different modalities),
\item repeating patterns and the associated problem of local minima in the alignment objective function (auto-isomorphic graphs in a topological sense, and shape correspondence problem in a geometric sense).
\end{itemize}
By the following set of assumptions we contain the problem description to a more specific application domain:
\begin{itemize}
\item the environments are:
\begin{itemize}
\item well-structured, that is to say their maps could be modeled (abstracted) with a set of 2D geometric curves,
\item composed of meaningful regions that can be segmented (e.g. room, corridor, etc.).
\end{itemize}
\item the maps are:
\begin{itemize}
\item spatial, geometric and 2-dimensional (or could be represented in a 2D plane),
\item globally consistent (not ``broken'') and uniformly scaled.
\end{itemize}
\end{itemize}
The restrictions and limitations caused by these assumptions are further discussed in Section~\ref{sec:conclusion}.
\subsection{Approaches} \label{subsec:approaches}
In this general formulation, the map alignment problem proves to be more challenging than the relaxed versions such as scan matching or image alignment problems.
When the displacement between two frames (maps or scans) are small, optimization based algorithms such as point set registration~\cite{besl1992method}, \cite{Gold19981019}, \cite{Tsin2004}, \cite{NIPS2006_2962}, \cite{5432191} or image registration~\cite{Baker2004} are suitable solutions.
However, these algorithms are vulnerable to local minima, especially in the absence of an initial guess.
This pitfall is exacerbated when the input maps contain repetitive patterns that increase the number of local minima.
Another approach is to employ the Hough transform to structure the search space and decompose the transformation into separate operations of rotation and translation~\cite{carpin2008fast}, \cite{saeedi2012efficient}.
However, these approaches are limited to rigid transformations (i.e. Euclidean transformation that includes translation, rotation and no scaling) and expect homogeneity of the input signal (same modality).
A more common approach to the map alignment problem is to interpret the input maps with an abstract representation that enables a search on the similarity of instances.
For example, graphs capture the canonical points of the open space as vertices, and the connectivity of the open space is represented by the edges between vertices~\cite{huang2005topological}, \cite{schwertfeger2013evaluation}, \cite{kakuma2017alignment}.
Consequently, geometric and/or topological similarities of the vertices and/or edges are used to find a match between two maps.
When maps are of different types, such an interpretation plays an important role.
This interpretation abstracts the input maps into a \emph{shared instantiated representation}, which makes the search for similarities between maps feasible.
A more thorough review of related work is presented in Section~\ref{sec:related}, after which detailing our method and the experimental validation of it are presented in sections~\ref{sec:method} and \ref{sec:results}.
\subsection{Our approach} \label{subsec:our_approach}
In this work, we propose a method based on the aforementioned concept of \emph{shared instantiated representation}.
The underlying representation of our method is a geometric decomposition that outlines the segmentation of regions, namely ``arrangement'' \cite{agarwal2000arrangements} in 2D.
When modeling the occupied regions of the maps (corresponding to the physical elements in the environment), the 2D arrangement explicitly represents both the \emph{boundaries} and the \emph{regions} of the open-spaces.
In addition, it implicitly captures the connectivity of the open-spaces through the regions' adjacency.
Our proposal is to
\begin{inparaenum}[i)]
\item abstract input maps via region decomposition into a shared geometric representation, i.e. 2D arrangement,
\item search for all potential alignments that match the regions of the maps,
\item select the best alignment according to a proposed quality measure.
\end{inparaenum}
The details of the method, the 2D arrangement, and further details are provided in Section~\ref{sec:method}.
The differentiating characteristic of our method is its representation and the consequent search approach.
Most approaches generate hypotheses from some initial cues and follow along the progress of those cues.
In contrast, we exhaust the search space to avoid missing the correct answer.
This is crucial in the case of noisy maps and maps of different modalities.
To our knowledge, no algorithm has been developed for solving the alignment problem in this manner.
The main contributions of this work are as follows:
\begin{itemize}
\item
An algorithm for map alignment that does not rely on rigid transformations, available initial guess or similar representation between maps.
\item
An abstract representation (2D arrangement), which is capable of interpreting maps of different modalities.
We use this interpretation for region segmentation, and for solving the alignment problem.
This interpretation results in a hierarchical representation of maps, where the abstract models on the top level are readily available for other geometric processing and manipulations after alignment.
\item
A publicly available collection of maps, containing forty maps of four different environments.
The source code to our implementation of the proposed method is also made available online.
For the links to these online repositories please see Section~\ref{sec:results}.
\end{itemize}
\section{Related Work} \label{sec:related}
The main underlying problem in map alignment is \emph{data association}, which manifests in a variety of forms according to the application context.
Few examples are image registration (e.g. stereo vision correspondence, optical flow, and visual odometry), laser scan matching and point cloud registration in Simultaneous Localization and Mapping (SLAM), and the correspondence problems in SLAM such as loop closure and partial map merging.
While above-mentioned problems share the underlying challenge of data association, different methods formulate their underlying problem differently depending on the context of their application, data type, and prior assumptions.
While we try to point out some of the seminal works with formulations other than those related to our work, we turn the focus of the literature review to those closest to ours, i.e. map alignment.
\paragraph{Motivations of sensor to prior map alignment}
There are different motivations for fusing prior maps and sensor maps.
For instance, Sanchez and Branaghan argue that abstract maps are easier to learn \cite{sanchez2009interaction}, and accordingly, Georgiou et al.~\cite{georgiou2017constructing} state that a correspondence between an abstract human readable map and robot's sensor map is desired to facilitate collaborative tasks between humans and robots.
Bowen-Biggs et al. claim that sensor maps are not ``natural'' for many high level tasks~\cite{bowen2016sketched}, especially those including semantics or with human in the loop.
In their work, they present a method of fusing two sensor and floor maps, and using the combination for accomplishing elaborate tasks.
However, in their work the map to map correspondence is established manually.
In other examples, the prior map is exploited towards improving the performance of SLAM algorithms, either through exploiting the structure of the prior map, or by aligning local maps to build a global map.
Georgiou et al.~\cite{georgiou2017constructing} formulated the ``structural information from architectural drawings'' as ``informative Bayesian mapping priors'' in order to improve the performance of the SLAM algorithm.
Although, this work does not address the map alignment problem per se.
Instead the SLAM output is structured according to the prior information embedded into the SLAM algorithm.
Vysotska and Stachniss~\cite{vysotska2017improving} proposed an approach to improve SLAM performance by generating constraints from the correspondence between the building information from \emph{OpenSteetMap} and the robot's perception of its surroundings.
They also benefit from the ``localizability'' information available in the OpenSteetMap.
Mielle et al.~\cite{mielle2017slam} proposed a method for applications with extreme conditions (e.g. with dust or smoke) where the information from a ``rough prior'' is incorporated in order to improve the SLAM performance, and enhance the quality of the rough prior map by fusing it with sensor map.
\subsection{Graph matching approaches}
Topological structure of the open spaces is one of the most salient information in the maps, and it is natural that the graph representation of the aforementioned structure draws much attention as a fitting representation.
Two of the sub-problems in graph theory that are most relevant to map alignment are the Maximal Common Sub-graph (MCS) problem, and the error-tolerant sub-graph isomorphism~\cite{huang2005topological}.
Huang and Beevers~\cite{huang2005topological} proposed a method for merging partial maps based on the embedded topological maps.
Their approach is based on a graph matching process inspired by maximal common sub-graph (MCS) and image registration, followed by a second stage in which the geometric consistencies of the match hypotheses are evaluated.
The vertices of the topological map are embedded in a metric space, along with a minimal amount of metric information (e.g., orientation of edges at each vertex and path length for each edge).
Therefore, their method benefits from both the geometric and topological information of the open spaces.
In another work with a similar approach, Wallgr{\"u}n \cite{wallgrun2010voronoi} proposes a map alignment technique with a graph matching method based on the Voronoi graph of the maps.
The objective of his work is localization and mapping, and the underlying data association model of his method is based on an inexact graph matching with graph edit distance, over annotated graphs generated from the Voronoi graphs.
Nodes are annotated with the radii of the maximal inscribed circles used to generate the Voronoi graph, and the edges are annotated with their relative length, the shape of the Voronoi curve beneath the edge, and the edge's traversability.
By assigning such attributes to the elements of the graph, he incorporates geometric constraints into the matching process.
In order to develop an automated process for map quality assessment, Schwertfeger and Birk have developed an interesting method for map alignment~\cite{schwertfeger2013evaluation}.
Their method captures the high-level spatial structures of the maps through Voronoi graphs, and represents with topological graphs that contain the angles between edges and the length of edges.
The map alignment is done by finding similar vertices of the graphs and ``identification of sub-graph isomorphisms through wave-front propagation''~\cite{schwertfeger2013evaluation}.
With experimental results, they show the robustness of their method by detecting brokenness in sensor maps.
In another intriguing work, Mielle et al.~\cite{mielle2016using} proposed a map alignment method based on graph matching, which enables robots to follow navigation orders specified in sketch maps.
Their method converts the Voronoi skeleton to a graph, where vertices are the bifurcation and ending points of the skeleton.
Vertex type (dead-end or junction) and an ordered list of edges are attributed to the graph's vertices in the matching process.
To find the error-tolerant maximal common sub-graph (ETMCS), they developed a modified version of Neuhaus and Bunke's~\cite{neuhaus2004error} graph matching algorithm based on the normalized Levenshtein edit-distance (LED)~\cite{yujian2007normalized}.
By skipping the absolute position values, the interpretation becomes insensitive to noise and inconsistency of the map.
Consequently their method doesn't require global consistency and uniform scaling of the maps.
In order to benefit from semantic information available in floor maps for high level task execution, Kakuma et al. \cite{kakuma2017alignment} proposed a graph matching based method for the alignment of sensor maps to floor plans of the buildings.
Their method constructs a graph from segmented regions of the occupancy map.
Graph matching is carried out with minimizing a matching cost function based on a variation of Graph Edit Distance (GED)~\cite{sanfeliu1983distance} and Hu-Moments~\cite{hu1962visual}.
\subsection{Hough/Radon transform approaches}
Hough (/Radon) transform maps the input signal from the Cartesian to a \emph{parametric} space.
This parametric space has the advantage of capturing the salient, thought maybe latent, structure of the maps.
The core of those methods based on Hough transform is to decompose the alignment problem into rotation and translation estimation.
Such approaches are often deterministic, non-iterative, and fast, thanks to this decomposition.
However, methods in this category are limited to rigid transformation, and work best on maps with same modalities.
For merging partial maps in a multi-robots application, Carpin proposed a method \cite{carpin2008fast} that first finds the rotation alignment via a correlation between the Hough spectra of the two maps.
After the orientation alignment, the translation parameters are estimated from a x-y projection of the maps.
One of the interesting features of this method is that the estimated transformations are weighted and such weights could be treated as uncertainties.
With a conceptually similar approach, Bosse and Zlot~\cite{bosse2008map} tackle the problem of global mapping by merging local maps.
Their method also decouples the rotation and translation estimations, but with some twists in their transformation techniques.
They use an ``orientation histogram of the scan normals'' (yields an output similar to a Hough transform) to determine the orientation alignment.
Then a ``weighted projection histograms created from the orthogonal projections'' (somewhat equivalent to radiography) is used for estimating the translation between the orientationally aligned data.
Saeedi et al.~\cite{saeedi2012efficient}, \cite{saeedi2014map}, \cite{saeedi2014group} proposed a novel technique to represent the topology of the open space with a probabilistic Voronoi graph.
Even though they employ a graph representation, they do not solve the matching problem by graph matching techniques.
First a Radon transform is employed to find the relative orientation between maps, followed by an edge matching technique based on a 2D cross correlation over graphs' edges to find the translation.
One of the very interesting features of their method is the propagation of the uncertainty from input map to the Voronoi graph, and accounting for this uncertainty in the fusion process.
\subsection{Optimization approaches}
One of the most popular categories of techniques for data associations in robot mapping is based on optimization.
A famous example is the Iterative Closest Point (ICP)~\cite{besl1992method} which is a \emph{point set registration} and finds a rigid transformation between two point sets.
Such an approach is inherently susceptible to the problem of local minima.
They are only suitable to problems where a [rough] initial estimate of the displacement between input data is available.
While this is a reliable assumption in incremental mapping, it is not a valid assumption in map alignment.
Furthermore, such methods work on same modality input data.
Other similar approaches in image alignment, such as Lucas-Kanade algorithm~\cite{lucas1981iterative},~\cite{Baker2004} and Enhanced Correlation Coefficient (ECC) Maximization~\cite{evangelidis2008parametric}, also work under similar assumptions and consequently they are prone to similar pitfalls.
One example of optimization based method applied to the map merging problem is presented by Carpin and Birk~\cite{carpin2005stochastic}, \cite{carpin2005map}, \cite{birk2006merging}.
Their approach minimizes a dissimilarity function (overlapping quality index) over the transformation parameters, with a stochastic process (random walk), used for the optimization.
An interesting feature of this method is its ability to robustly handling unstructured environments.
\paragraph{What else?}
It is good to mention some other interesting approaches, even though we did not find them particularly relevant in order to investigate them in detail and experiment with them.
Among those are methods from the multi-robot mapping applications where the alignment of individual maps are determined by localizing each robot in the partial maps of other robots.
Works by Thrun~\cite{thrun2001probabilistic}, Dedeoglu and Sukhatme~\cite{dedeoglu2000landmark}, and Williams et al.~\cite{williams2002towards} are good examples in this category.
These methods are based on the assumption that the input maps are from the same modality.
With a similar application, i.e. multi-robot exploration, some researchers have developed methods to determine the relative transformation between robots' partial maps when the robots can physically meet in the environment.
Examples of the methods based on ``rendezvous'' or ``mutual observation'' are proposed by Howard et al.~\cite{howard2006experiments}, Howard~\cite{howard2004multi}, Fox et al.~\cite{fox2006distributed}, Zhou and Roumeliotis~\cite{zhou2006multi}, and Konolige et al.~\cite{konolige2003map}.
These methods are based on the robots' ability to meet and generate transformation hypotheses from a rendezvous, which is unfeasible for off-line methods.
Erinc et al. proposed a method~\cite{erinc2013heterogeneous} to annotate heterogeneous maps with WIFI signal that provides cues for data association between maps.
This means two essentially different maps are annotated by a shared landmark, which provides a seamless data association cue.
Boniardi et al.~\cite{boniardi2015robot} developed a method for localizing and navigating directly in a sketch map, without the map alignment.
Partial map alignment is an essential component of map merging.
Saeedi et al.~\cite{saeedi2016multiple} provided a thorough review of the multi-robot SLAM field which covers a broad range of such methods.
However, most of these methods, being specifically developed to improve multi-robot mapping applications, are dependent on sources of information that are specific to robot map and not accessible for any arbitrary map (e.g. layout maps).
The work by Bonanni et al.~\cite{bonanni2017map} on merging 3-D maps, and its earlier version on 2D maps targeting the problem of merging ``partially consistent maps''~\cite{bonanni2014merging}, require the pose graph to be available (or computable) for both maps.
Jiang et al.~\cite{jiang2017simultaneous} proposed a method based on ``motion averaging'' for merging multiple local maps.
Their approach is to find transformation between all local map, construct a graph of the inter-map ``motion'' and optimize such motions for optimal alignment.
However, the core of this method is the optimization of alignment between several local maps, and therefore not suitable for aligning a pair of maps.
\section{Method} \label{sec:method}
The essence of our method, as depicted in Figure~\ref{fig:method}, is to abstract the representation of input maps in order to facilitate the search for alignment.
This abstraction consists of modeling the physical entities of the environment with 2D geometric objects (such as lines and circles.)
These models are then used to partition the map into separate regions with a 2D arrangement.
We have shown in our earlier works~\cite{shahbandi2014sensor},~\cite{shahbandi2015semi}, how this representation could be used for semantic annotation and place categorization of occupancy grid maps.
Section~\ref{subsec:interpretation} describes the 2D arrangement representation.
Furthermore, we explain how this representation is adjusted to capture \emph{meaningful} regions, i.e. adjusting the structural decomposition to region segmentation.
Section~\ref{subsec:alignment} describes the alignment procedure, that is the matching of regions in the maps and estimation of alignment transformation for each match, resulting in a pool of plausible hypotheses.
While each hypothesis is estimated from matching only two regions, they are evaluated based on how well they align the two maps as a whole.
We introduce a ``match score'' in Section~\ref{subsec:match_score}, that is used for comparing the quality of alignments, which in turn is used to pick the best hypothesis.
\begin{figure
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=\linewidth]{method_core.pdf}
\caption[xxx]
{The orange blocks represent inputs and output, blue blocks are the intermediate representations, and the green blocks are the alignment processes.}
\label{subfig:method_core}
\end{subfigure}%
\begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{method_detailed.pdf}
\caption{An example of the process on a pair real world maps.}
\label{subfig:method_detailed}
\end{subfigure}%
\caption[xxx]{
The outline of our method in Figure~\ref{subfig:method_core}, and a concrete example demonstrating the process on real maps in Figure~\ref{subfig:method_detailed}.
} \label{fig:method}
\end{figure}
\subsection{Map interpretation} \label{subsec:interpretation
First step towards map alignment is the modeling of maps with an abstract representation, i.e. arrangement \cite{agarwal2000arrangements}.
Algorithm~\ref{alg:interpretation} outlines the process of this abstraction, composed of geometric trait detection, decomposition (arrangement), and pruning of the arrangement from a structural decomposition to a region segmentation.
An arrangement partitions a 2D plane according to a set of geometric objects (such as, but not limited to, lines and circles), referred to as geometric traits and traits for short.
A set of geometric traits $\mathcal{T}$ will result in a unique arrangement $A$ identified by a prime graph $\mathcal{P}$, and a set of faces $F$.
The prime graph $\mathcal{P}$ is the result of intersecting all traits $\mathcal{T}$, and faces are \emph{irreducible} closed-regions (``Jordan Curve'') bounded to edges from the prime graph $V(\mathcal{P})$.
Neighborhood $N(\mathcal{F})$ is an attribute associated with the set of faces $\mathcal{F}$, defined as a set of tuples of faces where each tuple identifies a pair of neighboring faces.
Figure~\ref{fig:arrangment_demo} demonstrates an arrangement and its components on a toy example.
For technical details of the arrangement algorithm, see~\cite{agarwal2000arrangements}.
\begin{figure
\centering
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[width=\linewidth]{arrangement_traits.pdf}
\caption{geometric traits
\label{subfig:arrangement_traits}
\end{subfigure}%
~%
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[width=\linewidth]{arrangement_prime.pdf}
\caption{prime graph
\label{subfig:arrangement_graphs}
\end{subfigure}%
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[width=\linewidth]{arrangement_faces.pdf}
\caption{faces
\label{subfig:arrangement_faces}
\end{subfigure}%
\caption[xxx]{
An arrangement $\mathcal{A}:= (\mathcal{T}, \mathcal{P},\mathcal{F})$.
This example involves straight lines and circles to demonstrates the ability to handle geometric traits beyond straight lines.
However, due to the nature of buildings in our maps, our map interpretation with arrangement only relies on straight lines.
} \label{fig:arrangment_demo}
\end{figure}
\begin{algorithm}[t]
\caption {Map Interpretation} \label{alg:interpretation}
\begin{algorithmic
\Function {Interpret}{Map}
\State $\mathcal{T}$ = \Call{DetectGeometricTraits}{Map}
\State $A(\mathcal{T}, \mathcal{P},\mathcal{F})$ = \Call{Arrangement}{$\mathcal{T}$}
\State /* \emph{Pruning} */
\State // $M_d$: normalized Distance Transform of Map
\State // $N_p(e)$: neighboring cells (pixels) to an edge $e$
\State // $V(e) = \frac{1}{size(N_p(e))} \sum_{p \in N_p(e)} M_d(p) $
\State // $thr_e$: wall/gateway detection threshold ($\sim 0.075$)
\State $E(\mathcal{P}) = E(\mathcal{P}) - \{e \mid e \in E(\mathcal{P}) \land V(e) < thr_e\}$
\State \Call{Update}{$A(\mathcal{T}, \mathcal{P},\mathcal{F})$}
\State \Return $A$
\EndFunction
\\\hrulefill
\Function {Arrangement}{$\mathcal{T}=\{t_i\}$}
\State /* \emph{construct prime graph} $\mathcal{P}$ \emph{from traits} $\mathcal{T}$ */
\State $V(\mathcal{P}): \{ v_i(x,y) \mid (\exists t_j,t_k \in \mathcal{T}) [v_i \in t_j \land v_i \in t_k] \}$
\State $E(\mathcal{P}): \{ e_i(t_k, v^i_s, v^i_e ) \mid (\nexists v_m \in e_i) [v_m \neq v^i_s \land v_m \neq v^i_e] \}$
\State /* \emph{identify irreducible faces} $\mathcal{F}$ \emph{in} $\mathcal{P}$ */
\State $\mathcal{F}: \{ f_i:=\{e_j\} \mid (\forall e_j \in f_i) [\exists! e_k \in f_i \mid v^j_s = v^k_e]\}$
\State $N(\mathcal{F}): \{ (f_i,f_j) \mid (\exists e_k) [e_k\in f_i \land e_k\in f_j] \}$
\State \Return $A\left( \mathcal{T},\mathcal{P},\mathcal{F}\right)$
\EndFunction
\end{algorithmic}
\end{algorithm}
\subsubsection{Geometric traits}
We model the physical elements of the buildings (e.g occupied pixels in occupancy maps) with geometric traits, which represent the boundary between open spaces and occupied (or unexplored) areas.
Accordingly, an arrangement manifests a \emph{dual} characteristic:
\begin{inparaenum}[i)]
\item $\mathcal{F}$ is a geometric representation of the open-space and its boundaries, and
\item $N(\mathcal{F})$ captures the topology of open-space.
\end{inparaenum}
The detection of the geometric traits from 2D maps could be achieved by common algorithms such as \emph{Generalized Hough Transform}~\cite{ballard1981generalizing} and \emph{Radon Transform}~\cite{radon1986determination}.
Given that all maps used in our experiments could be modeled with only straight lines, in this work we use \emph{radiography}, which is a variation of the aforementioned algorithms.
Radiography operates as a Radon Transform that is filtered by the oriented gradient of the image.
That is, the projection of each point is weighted by the magnitude of the image's gradient at that point, multiplied by the difference between the orientation of the image's gradient and the direction of the Radon projection.
We have shown previously~\cite{shahbandi2014sensor} that radiography is more robust in modeling physical elements of the environment (e.g walls) that suffer from a discrepancy in their continuity or too much noise.
Nonetheless, the arrangement representation is neither limited to straight lines, nor dependent on the trait detection technique.
\subsubsection{Abstraction compatibility and arrangement pruning}
Despite its merits in detecting discontinuous traits in noisy maps and capturing the global structure of environments~\cite{shahbandi2014sensor}, radiography detects unbounded traits (e.g. infinite lines instead of line segments).
Consequently the partitioning of the space is not equivalent to a plausible region segmentation, due to over-decomposition of areas that are conceptually a single region (e.g. a kitchen or an office).
Figure~\ref{fig:arrangement_prunning} demonstrates the over-decomposition of a real map.
This inconsistency of region segmentation is non-deterministic, depends on the noise, partiality and inconsistencies of the maps, and could vary from sensor maps to layout maps.
Since the essence of our alignment method is to match corresponding regions, it is crucial for the maps to have representations on the same level of abstraction (regions segmentation), what we call \emph{abstraction compatibility}.
That is to say, if a single face represents a room in one map, the same room must be represented by a similar face in the other map.
Based on empirical observations, the success rate of our method seems to be most sensitive to this compatibility assumption.
However, the sensitivity of the alignment method to abstraction compatibility is not critically obstructive, and not every corresponding region should have compatible abstraction in the maps.
We remedy this challenge by \emph{pruning} the arrangement to a more plausible region segmentation, presented in the function \textproc{Interpret} in Algorithm~\ref{alg:interpretation}.
This pruning is the process of removing all face boundaries (i.e. $e_i \in E(\mathcal{P})$) which do not correspond to a \emph{wall} or \emph{gateway}, followed by merging all adjacent faces whose boundaries are removed.
As described in Algorithm~\ref{alg:interpretation}, an edge is considered a gateway or wall if the average value of the pixels $V(e)$ from the distance map $M_d$ (Figure~\ref{subfig:HIH_dist_map}) within a neighborhood of the that edge $N_p(e)$ is below a certain threshold $thr_e$.
Since the distance map $M_d$ is normalized (scaled to $[0,1]$), the threshold $thr_e$ over the averaged pixel values $V(e)$ is independent of the map scale.
The result of a pruning process applied to the over-decomposed example of Figure~\ref{subfig:HIH_arrangement}, is presented in Figure~\ref{subfig:HIH_arrangement_prune}.
Our pruning of an over-decomposed arrangement to region segmentation is a variation of ``Morphological Region Segmentation'' as presented in the ``Room Segmentation'' survey by Bormann et al.~\cite{bormann2016room}.
In more elaborate scenarios, one could employ other region segmentation methods presented in \cite{bormann2016room}, or more recent works by Fermin-Leon et al.~\cite{leon2017incremental} and Mielle et al.~\cite{mielle2017method}.
Nevertheless, we observed that the arrangement pruning, along with an approximation of faces with \emph{Oriented Minimum Bounding Boxes} (details in Section~\ref{subsec:alignment}), satisfies the abstraction compatibility assumption.
\begin{figure
\centering
\begin{subfigure}{.49\linewidth}
\includegraphics[width=\linewidth]{HIH_original.jpg}
\caption{occupancy map}
\label{subfig:HIH_ogm}
\end{subfigure}%
~%
\begin{subfigure}{.49\linewidth}
\includegraphics[width=\linewidth]{HIH_arrangement.jpg}
\caption{arrangement}
\label{subfig:HIH_arrangement}
\end{subfigure}%
\begin{subfigure}{.49\linewidth}
\includegraphics[width=\linewidth]{HIH_dist_map.jpg}
\caption{distance image}
\label{subfig:HIH_dist_map}
\end{subfigure}%
~%
\begin{subfigure}{.49\linewidth}
\includegraphics[width=\linewidth]{HIH_arrangement_prune.jpg}
\caption{after pruning}
\label{subfig:HIH_arrangement_prune}
\end{subfigure}%
\caption[xxx]{
An Occupancy Map in Figure~\ref{subfig:HIH_ogm},
its original decomposition in Figure~\ref{subfig:HIH_arrangement},
and a cleaned-up version of the arrangement (pruned) in Figure~\ref{subfig:HIH_arrangement_prune},
based on the distance map $M_d$ in Figure~\ref{subfig:HIH_dist_map}.}
\label{fig:arrangement_prunning}
\end{figure}
\subsection{Alignment procedure} \label{subsec:alignment
A hypothesis in the context of this work is a transformation function between the coordinate frames of the two maps.
Hypothesis generation is the process of proposing such plausible transformations.
According to the \emph{uniform scaling assumption} stated in Section~\ref{subsec:our_approach}, the transformations estimation is restricted to ``similarity'' (i.e. translation, rotation and uniform scaling.)
To propose hypotheses, faces of the open space regions with similar \emph{shapes} are associated and a transformation is estimated for each pair of faces with similar shapes.
The \emph{shape descriptor} is an ordered sequence of vertex-edge tuples
\[
\begin{array}{l}
D(f):= ( (v_i,e_l), \cdots, (v_j,e_k))\\
\end{array}
\]
in which, all the entries (vertices and edges) are ordered counter clock-wise from an arbitrary reference point.
Since the choice of reference point in each face is arbitrary, the descriptor also contains all its ``cyclic shifts''.
For example, the descriptor for face $f_1$ of the arrangement from the Figure~\ref{fig:arrangment_demo} with all its $s$-step shifts are
\[
\begin{array}{l l}
D(f^{s=1}_1) & = ( (v_2,e_9) , (v_1,e_1) , (v_3,e_5) )\\
D(f^{s=2}_1) & = ( (v_3,e_5) , (v_2,e_9) , (v_1,e_1) )\\
D(f^{s=3}_1) & = ( (v_1,e_1) , (v_3,e_5) , (v_2,e_9) )\\
\end{array}
\]
In the context of the shape descriptor, vertices denote corners, where corners are defined as vertices with any internal angles other than $\pi$.
The features of the descriptor are the internal angles of vertices (i.e. \emph{corner angle}), and the length of edges normalized to the perimeter of the face (i.e. \emph{edge length ratio}).
Figure~\ref{fig:shape_mismatch} demonstrates the necessity of these features, through examples where the absence of these two features would result in false matches.
Descriptor size equivalency is the first necessary condition for a potential match.
A match is then identified as a $s$-step ``circular shift'' of one descriptor, so that all corresponding entries in the descriptors of the faces are equivalent.
After a correspondence between the two point sets (face corners) is proposed via face matching, a transformation between the two point sets is estimated based on the ``Least-squares estimation'' method proposed by Umeyama~\cite{umeyama1991least}, which uses the singular value decomposition of a covariance matrix of the data points.
Algorithm~\ref{alg:method} presents the map alignment procedure in pseudo-code, and Figure~\ref{subfig:method_detailed} depicts two examples of correct (in green) and wrong (in red) association and their consequent transformation.
\begin{algorithm}[t]
\caption {Map Alignment Procedure} \label{alg:method}
\begin{algorithmic
\State \textbf{Input:} Map$_1$, Map$_2$
\State \textbf{Output:} Alignment (similarity transformation)
\State
\State /* \emph{Map Interpretation} */
\State $A_1(\mathcal{T}_1, \mathcal{P}_1,\mathcal{F}_1)$ = \Call{Interpret}{Map$_1$}
\State $A_2(\mathcal{T}_2, \mathcal{P}_2,\mathcal{F}_2)$ = \Call{Interpret}{Map$_2$}
\State
\State /* \emph{Face Matching} \& \emph{Hypotheses Generation} */
\State // $f.c :$ corners of the face $f$
\State // $D(f) :$ shape descriptor of face $f$
\State // $f^{s} :=$ $s$-step circular shift of corners and $D(f)$
\State $H = \emptyset$
\For {$f_{i} \in \mathcal{F}_1, f_{j} \in \mathcal{F}_2 \mid \Call{Size}{f_i.c}=\Call{Size}{f_j.c}$}
\For {$s$ := 1 to \Call{Size}{$f_i.c$}, step=1}
\If {$D(f^{s}_i) = D(f_j)$}
\State $H = H \cup \{ \Call{EstimateTransform}{f^{s}_i.c, f_j.c} \}$
\EndIf
\EndFor
\EndFor
\State
\State /* \emph{Selecting The Best Hypotheses} */
\State Alignment = $\operatorname*{arg\,max}_{h \in H} $ \Call{MatchScore}{$h, A_1, A_2$}
\end{algorithmic}
\end{algorithm}
\begin{figure
\centering
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=.8\linewidth]{shape_mismatch_a.pdf}
\caption{} \label{subfig:shape_mismatch_a}
\end{subfigure}%
\hfill%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=.8\linewidth]{shape_mismatch_b.pdf}
\caption{} \label{subfig:shape_mismatch_b}
\end{subfigure}%
\hfill%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=.8\linewidth]{shape_mismatch_c.pdf}
\caption{} \label{subfig:shape_mismatch_c}
\end{subfigure}%
\caption[xxx]{
Examples where a missing feature results in wrong matches.
In the absence of \emph{edge length ratio} those faces in Figure~\ref{subfig:shape_mismatch_a} could have three matches instead of one, and the faces in Figure~\ref{subfig:shape_mismatch_b} could have four matches instead of zero.
In the absence of \emph{corner angle} those faces in Fig.~\ref{subfig:shape_mismatch_c} could have five matches instead of none.
} \label{fig:shape_mismatch}
\end{figure}
\paragraph{Simplified Alignment Procedure}
In the presence of too much noise in a map, the pruning of the arrangement might not return clean-cut shapes desired for face matching.
One wrong corner missed in the pruning process will render the shape of that region useless for matching, if the same error does not occur in the other map for the corresponding region.
One example of missed corners is visible at the bottom of Figure~\ref{subfig:HIH_arrangement_prune}.
Alternatively, due to such cases, we simplify shapes with their \emph{Oriented Minimum Bounding Boxes} (OMBB).
This counts as an interpretation of the ``well-structured environments'' assumption stated in Section~\ref{subsec:our_approach}.
This substitution of the shapes with OMBB renders the descriptor size and corner angles redundant, and consequently, the only relevant shape feature would be the aspect-ratio of the OMBBs (i.e. edge length ratio).
We have noticed that it is computationally cheaper to replace this feature (i.e. edge length ratio), with an equivalent process of \emph{False Positive rejection} which is based on the uniform scaling assumption.
Accordingly, the uniform scale assumption is relaxed, where all the transformations are estimated with an affine model, and then any transformation that does not qualify as a similarity transformation is rejected.
Algorithm~\ref{alg:simp_method} is the modified version of the alignment procedure, which reflects the simplification of the shape descriptor and the rejection of non-uniformly scaled transformations.
The result of the False Positive rejection can be seen in Table~\ref{tab:res_layout_sensor}, where $\sim 90\%$ of initial hypotheses are rejected.
We have carefully monitored the hypotheses pools of cases where the method failed to find the correct alignment.
We can report that a correct alignment was never generated.
In other words, we can safely assure that a rejection of potentially correct hypotheses has never been a cause of failure.
\begin{algorithm}[t]
\caption {Map Alignment Procedure\\
with simplified hypotheses generation} \label{alg:simp_method}
\begin{algorithmic
\State /* \emph{\textbf{Input, Output} (see Algorithm~\ref{alg:method})}
\State
\State /* \emph{Map Interpretation (see Algorithm~\ref{alg:method})} */
\State
\State /* \emph{Generate Hypotheses (without face matching)} */
\State // $\hat{f} = \Call{OrientedMinimumBoundingBox}{f}$
\State $H = \emptyset$
\For {$f_{i} \in \mathcal{F}_1, f_{j} \in \mathcal{F}_2$}
\For {$s$ := 1 to 4, step=1}
\State $H = H \cup \{ \Call{EstimateTransform}{\hat{f}^s_i.c, \hat{f}_j.c} \}$
\EndFor
\EndFor
\State
\State /* \emph{Reject False Positives Hypotheses} */
\State // $h.s_x, h.s_y$: scales of transformation in $x/y$ directions
\State // $thr_{s}$: acceptable ratio between scales ($\sim 1.2$)
\For {$h \in H \mid \neg (1/thr_{s} < (h.s_x/h.s_y) < thr_{s}) $}
\State reject $h$
\EndFor
\State
\State /* \emph{Selecting The Best Hypotheses (see Algorithm~\ref{alg:method})} */
\end{algorithmic}
\end{algorithm}
\subsection{Alignment match score} \label{subsec:match_score}
To select the winning hypothesis, each hypothesis is evaluated based on how well the arrangements of the two maps ($A_1$ and $A_2$) are aligned under that transformation.
To this end, an \emph{arrangement match score} ($S_A$) is defined to measure the alignment quality of each hypothesis.
The arrangement match score between two arrangements $A_1$ and $A_2$, under the transformation $^1T_2$, is defined as
\[
S_A(A_1,A_2,^1\!T_2) = \displaystyle\sum_{\substack{
f_i \in \mathcal{F}_1\\
f_j \in \mathcal{F}_2\\
}} min(w(f_i), w(f_j)) \times s_f(f_i,f_j)
\]
where $w(f)$ is a \emph{weight} assigned to individual faces,
and $s_f$ is the \emph{face match score}.
The weight is defined as the relative surface area of faces to the surface area of the whole arrangement they belong to:
\[
w(f_i) = \frac{\textit{area}(f_i)}{\textit{area}(A)}, \quad \textit{area}(A) = \displaystyle\sum_{\substack{f_k \in \mathcal{F}}} \textit{area}(f_k)
\]
The larger a face is, the higher impact it will have in the arrangement match score.
The face match score $s_f$ is defined as:
\[
s_f(f_i,f_j) =
\begin{cases}
\frac{e^{\left(\frac{f_i \cap f_j}{f_i \cup f_j}\right)} - 1}{e-1} & if \quad (f_i,f_j) \in \textit{association}\\
0 & otherwise\\
\end{cases}
\]
where $f_i \cap f_j$ is the surface area of the faces' intersection and $f_i \cup f_j$ is the surface area of the faces' union.
The match score of a face with itself (perfect match) equals one, and the match score of two non-intersecting faces equals zero.
The exponential expression rewards slight improvements close to perfect match more than the slight improvements close to a bad match.
The $\textit{association}$ represents pairs of faces from two arrangements that are associated (not just overlapping) under the transformation.
We define $\textit{association}$ based on three conditions:
\[ \begin{array}{l}
\textit{association}: \{ (f_i,f_j) \mid \forall f_i \in \mathcal{F}_1, f_j \in \mathcal{F}_2, c_1 \land c_2 \land c_3 \} \\
c_1: \quad \textit{center}(f_i) \in f_j \land \textit{center}(f_j) \in f_i\\
c_2: \quad \nexists f_k \in \mathcal{F}_2 \mid \textit{center}(f_k)\in f_i, d(f_i,f_j) > d(f_i,f_k)\\
c_3: \quad \nexists f_k \in \mathcal{F}_1 \mid \textit{center}(f_k)\in f_j, d(f_i,f_j) > d(f_j,f_k)\\
\end{array} \]
where $\textit{center}(f_i)$ is the center of $f_i$ and $d(f_i,f_j)$ is the difference in surface area of $f_i$ and $f_j$.
First, for the two faces $f_i$ and $f_j$ to be associated, they must enclose each other's center.
We define the center of a face as the ``centroid'' (geometric center) of the vertices of the face.
Condition number two assures a one to one association where a face overlaps with multiple faces from the other arrangement.
In such cases, among all faces of $\mathcal{F}_2$ with their centers enclosed by $f_i \in \mathcal{F}_1$, the face ($f_j \in \mathcal{F}_2$) with most similar size (surface area) is associated with $f_i$.
And the third condition is symmetric to the second condition, i.e. vice versa for $f_i$ to $f_j$.
This match score is devised only for the comparison of different hypotheses for a single pair of maps.
That is to say, the alignment of different sensory maps over a layout map could not be compared with this score,
nor is it suitable to detect the layout to which a sensor map belongs (i.e. layout recognition),
and neither is it suitable as a quantified match accuracy measure.
This matter is better observed in Figures~\ref{fig:match_score_matrix} and \ref{fig:match_score_boxplot} from Section~\ref{sec:results}, where it is discussed with experimental observations.
\paragraph{The challenge of face center}
The center points of the faces can be defined differently according to the context of the application, such as ``center of gravity'', ``Chebyshev centers'' and ``polygon centroid''.
However, it proves to be very hard to lay down a definition that guarantees to be enclosed by the region \emph{and} unique.
If a face is non-convex as in Figure~\ref{subfig:centre_challenge_concave}, there is no guarantee that the center of gravity would be enclosed by the face.
Chebyshev centers are defined as the center of either
\begin{inparaenum}[i)]
\item the minimal-radius circle enclosing a region, or
\item the maximal-radius inscribed circle inside the region.
\end{inparaenum}
The example of Figure~\ref{subfig:centre_challenge_concave} shows that the minimal-radius enclosing circle is susceptible to the same problem as the center of gravity, and Figure~\ref{subfig:centre_challenge_multiple} presents an example where the maximal-radius inscribed circle is not necessarily unique.
Another example of tackling this challenge that we have explored, is to extract the Generalized Voronoi Diagram, and picking a point on the skeleton of the influence zone (SKIZ) that has the minimum sum of distance to all other points.
This definition is also prone to degenerate cases, such the example in Figure~\ref{subfig:centre_challenge_concentric} shows.
A variety of definitions could be considered for the center of a region that guarantee to be enclosed by the region.
Alas, they would ultimately depend on the interpretation of ``center point'' with respect to the domain of application, and most are prone to degenerate cases where such a point is not unique.
Ultimately, we have observed that in the setting of our problem such degenerate cases are not so frequent to disturb the performance of the method.
Either of these definition would satisfy the requirements of our method as long as it guarantees uniqueness, and we chose the centeroid of the vertices.
\begin{figure}
\centering
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=.8\linewidth]{center_challenge_concave.pdf}
\caption{} \label{subfig:centre_challenge_concave}
\end{subfigure}%
\hfill%
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=.8\linewidth]{center_challenge_multiple.pdf}
\caption{} \label{subfig:centre_challenge_multiple}
\end{subfigure}%
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=.8\linewidth]{center_challenge_concentric.pdf}
\caption{} \label{subfig:centre_challenge_concentric}
\end{subfigure}%
\hfill%
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=.8\linewidth]{center_challenge_squares.pdf}
\caption{} \label{subfig:centre_challenge_squares}
\end{subfigure}%
\caption[xxx]{
Examples where ``center of gravity'', ``Chebyshev centers'' and ``Voronoi-based'' definitions fail to identify a center point of a region that is both enclosed by the region \emph{and} unique.
Figures~\ref{subfig:centre_challenge_concave}, \ref{subfig:centre_challenge_multiple} and \ref{subfig:centre_challenge_concentric} highlight the failures of each definition under different circumstances.
While those three may not seem probable in a real world scenario, Figure~\ref{subfig:centre_challenge_squares} presents the failure of such definitions in a more realistic example.
} \label{fig:centre_challenge}
\end{figure}
\section{Experimental Results and Verification} \label{sec:results}
In this section we present the results from a series of experiments, on a data-set of forty maps collected specifically for this task.
The experiments are designed to show the method's performance, under different circumstances and in comparison with other methods.
All the experiments are based on an implementation of the method in Python, using many libraries \cite{scipy, NumPy, Matplotlib, SymPy, scikit-image, opencv_library, GPC-python, hagberg2008exploring}.
The source code to our implementation is also available online\footnote{{\scriptsize \url{https://github.com/saeedghsh/Map-Alignment-2D/}}}.
\subsection{Data collection} \label{subsec:data}
To evaluate our method, we collected maps of four different environments in two modalities, of CAD drawings and sensor maps.
All the maps are available online\footnote{{\scriptsize \url{https://github.com/saeedghsh/Halmstad-Robot-Maps/}}}, and presented in Appendix~\ref{app:datasets}.
\paragraph{Modalities}
A series of sensor maps were collected by a \emph{Google Tango tablet}, and the \emph{Tango Constructor application} from Google.
The 3D meshes were sliced horizontally and converted to an occupancy-like bitmap, where all the space is open except for the vertices of the mesh.
From there, we generated a pseudo-occupancy map through an interactive ray-casting process.
Detection of the geometric traits from foregoing maps were done via a variation of the Radon transform, namely radiography~\cite{shahbandi2014sensor}.
As for the other modality, the layout maps were obtained from CAD drawings in Portable Document Format (PDF).
These CAD drawings had to be manually simplified before further processing, due to the presence of furniture and other common appliances.
The process involved removing all elements of the drawings, except for the building's elements (i.e mainly walls).
This simplification can be observed in Figure~\ref{subfig:method_detailed} from Section~\ref{sec:method}.
It should be mentioned that the simplified version of the layout map is not tailored to accurately reflect the real layout and what is captured by the sensor map, with no intention to benefit the alignment method.
For instance, walls are represented with single lines in layout maps (width=$\sim$1-2 pixel), while they are much wider in the sensor maps ($\sim$5-10 pixels).
The drawings were converted to Scalable Vector Graphics (SVG) and the geometric traits were obtained directly by parsing the SVG files~\cite{port2017svgpathtools}.
In order to acquire segmented regions and for the sake of convenience, the SVG files were converted to bitmap format (PNG) and the same process of decomposition and arrangement pruning based on distance transform has been employed.
However, if CAD drawings of the layouts are accessible in a richer format (e.g. DXF or DWG), the process of simplification and parsing could also be automated.
Furthermore, if the regions are accessible in such formats, there would not be a need for conversion to bitmap and distance transform for region segmentation.
While all the sensor maps have the same scale, that is, they could be correctly aligned with each other under a rigid transformation, layout maps have different scales compared to the sensor maps, and a rigid transformation could not correctly align sensor maps to layout maps.
\paragraph{Environment types}
We collected data from four different environments, two of which are homes and the other two are office buildings.
Table~\ref{tab:data_sets} lists the number of available maps for each environment, and all the maps can be found in Appendix~\ref{app:datasets}.
In total there are forty maps, four of which are layout maps and the rest are sensor maps.
Most sensor maps are partial and vary in their coverage of the environment.
\begin{table}
\centering
\begin{tabular}{c | c | c | c}
environment & type & \# sensor maps & \# layout maps\\
\hline
HH\_E5 & office & 14 & 1 \\
HH\_F5 & office & 14 & 1 \\
HIH & home & 4 & 1 \\
KPT & home & 4 & 1 \\
\end{tabular}
\caption[xxx]{A list of all maps of four different environments}
\label{tab:data_sets}
\end{table}
\paragraph{Maps that violate our assumptions}
The map collection contains maps that violate some of the initial assumptions.
For instance, maps HH\_E5\_2, HH\_E5\_3, HH\_E5\_4 and HH\_F5\_2 only cover corridors and halls and do not contain any room, and therefore there are not enough segment-able regions for hypotheses generation.
Other examples include HH\_E5\_12 and HH\_F5\_1 which are bent (deformed) and violate the global consistency assumption.
There exist further minor defects in some other maps.
Consequently, the performance results presented here are not the representative of the method's performance under all the assumptions.
Nevertheless, we include these maps to better observe the dependency of the method on the aforementioned assumptions, and provide a more inclusive performance result under different conditions.
\paragraph{Evaluations are based on success rate}
The performance of each method is provided as success rate, which is a percentage of successful alignments.
We skip a \emph{quantified accuracy measure} for the alignment.
It proved very hard (impossible for our data) to provide a per map \emph{alignment accuracy}, due to:
\begin{inparaenum}[i)]
\item the lack of ground truth for the sensor maps,
\item the inaccuracy of layout maps, and
\item the presence of noise and global inconsistency of the sensor maps.
\end{inparaenum}
Figure~\ref{fig:alignment_correctness} illustrates our quality assessment of the alignments.
\begin{figure
\centering
\begin{subfigure}{.33\linewidth}
\includegraphics[width=\linewidth]{alignment_correct.jpg}
\caption{correct}
\label{subfig:correct_alignment}
\end{subfigure}%
~%
\begin{subfigure}{.33\linewidth}
\includegraphics[width=\linewidth]{alignment_defect.jpg}
\caption{defected}
\label{subfig:defect_alignment}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\includegraphics[width=\linewidth]{alignment_wrong.jpg}
\caption{wrong}
\label{subfig:wrong_alignment}
\end{subfigure}%
\caption[xxx]{
Examples of correct, defected, and wrong alignments.
Both correct and defected are considered as successful alignments.
} \label{fig:alignment_correctness}
\end{figure}
\subsection{Experiments and results} \label{subsec:experiments}
The performance of the method is evaluated under three different experimental setups:
\begin{itemize}
\item \emph{sensor map to layout map alignment}, which is the main objective of the proposed method.
\item \emph{sensor map to sensor map alignment}, where we observe how partial coverage, noise and inconsistency of sensor maps affect the performance.
\item \emph{evaluation of alignment match score}, where the match score is studied for the alignments of intra and inter environment maps.
Accordingly every sensor map is aligned to all other layout maps, whether from the same environment or not.
\end{itemize}
\subsubsection{Sensor map to layout map alignment}
Table~\ref{tab:res_layout_sensor} presents the performance of the method in aligning sensor maps to layout maps (within the same environment).
The column \emph{initial} represents the number of initial estimated transformations,
\emph{after rejection} represents the number of remaining hypotheses after rejecting non-uniformly scaled transformations ($\sim 90\%$ are rejected),
and the last column marks the success of each alignment.
In total, the method has successfully aligned all maps of the home environments, and yielded $~83\%$ in success rate for the office buildings.
According to our investigations, failures are mainly due to the violation of the prior assumptions, such as global inconsistency and not enough segment-able regions in sensor maps.
\begin{table}
\centering
\begin{tabular}{c | c | c | c }
& \multicolumn{2}{c|}{number of hypotheses} & \\
map & initial & after rejection & result\\
\hline
HIH\_01 & 24 & 6 & \checkmark \\
HIH\_02 & 48 & 8 & \checkmark \\
HIH\_03 & 24 & 6 & \checkmark \\
HIH\_04 & 36 & 4 & \checkmark \\
\hline
KPT\_01 & 256 & 14 & \checkmark\\
KPT\_02 & 128 & 10 & \checkmark\\
KPT\_03 & 144 & 10 & \checkmark\\
KPT\_01 & 160 & 10 & \checkmark\\
\hline
HH\_E5\_01 & 9152 & 726 & \checkmark\\
HH\_E5\_02 & 5280 & 458 & $\times$\\
HH\_E5\_03 & 6688 & 704 & $\times$\\
HH\_E5\_04 & 5984 & 458 & $\times$\\
HH\_E5\_05 & 5808 & 368 & \checkmark\\
HH\_E5\_06 & 2992 & 178 & \checkmark\\
HH\_E5\_07 & 3696 & 214 & \checkmark\\
HH\_E5\_08 & 4400 & 352 & $\times$\\
HH\_E5\_09 & 9152 & 616 & \checkmark\\
HH\_E5\_10 & 9152 & 794 & \checkmark\\
HH\_E5\_11 & 5808 & 470 & \checkmark\\
HH\_E5\_12 & 8624 & 644 & \checkmark\\
HH\_E5\_13 & 4928 & 336 & $\times$\\
HH\_E5\_14 & 3344 & 328 & \checkmark\\
\hline
HH\_F5\_01 & 1292 & 128 & \checkmark\\
HH\_F5\_02 & 1088 & 82 & \checkmark\\
HH\_F5\_03 & 816 & 86 & \checkmark\\
HH\_F5\_04 & 680 & 70 & \checkmark\\
HH\_F5\_05 & 544 & 56 & \checkmark\\
HH\_F5\_06 & 408 & 28 & \checkmark\\
HH\_F5\_07 & 476 & 26 & \checkmark\\
HH\_F5\_08 & 3604 & 158 & \checkmark\\
HH\_F5\_09 & 952 & 78 & \checkmark\\
HH\_F5\_10 & 680 & 56 & $\times$\\
HH\_F5\_11 & 680 & 264 & \checkmark\\
HH\_F5\_12 & 680 & 46 & \checkmark\\
HH\_F5\_13 & 1020 & 92 & \checkmark\\
HH\_F5\_14 & 1088 & 158 & \checkmark\\
\end{tabular}
\caption[xxx]{
Performance of the method in aligning sensor maps to layout maps.
The column \emph{initial} represents the number of initial estimated transformations,
\emph{after rejection} represents the number of remaining hypotheses after rejecting non-uniformly scaled transformations ($\sim 90\%$ are rejected),
and the last column marks the success of each alignment.
}
\label{tab:res_layout_sensor}
\end{table}
\subsubsection{Sensor map to sensor map alignment}
Table~\ref{tab:success_rate} compares the success rate of the method in aligning sensor maps to sensor maps, versus aligning sensor maps to layout maps.
It can be observed that the success rate of the method drops in aligning sensor maps to sensor maps.
There are two main reasons for this drop;
\begin{inparaenum}[i)]
\item many sensor maps are partial and consequently they overlap with each other marginally,
\item the violation of initial assumptions.
\end{inparaenum}
In the presence of layout map there is one source of noise and global inconsistency, but in case of aligning two sensor maps the noise and inconsistencies are amplified.
\begin{table}
\centering
\begin{tabular}{c | c | c}
environment & sensor vs layout & sensor vs sensor\\% & sensor map vs sensor map (discarding non overlapping)
\hline
HH\_E5 & $64.28\%$ $(\sfrac{9}{14})$ & $50.54\%$ $(\sfrac{46}{91})$\\
HH\_F5 & $92.85\%$ $(\sfrac{13}{14})$ & $68.13\%$ $(\sfrac{62}{91})$\\
HIH & $100\%$ $(\sfrac{4}{4})$ & $100\%$ $(\sfrac{6}{6})$\\
KPT & $100\%$ $(\sfrac{4}{4})$ & $83.33\%$ $(\sfrac{5}{6})$\\
\end{tabular}
\caption[xxx]{The success rate of the method in aligning sensor maps to sensor maps, versus aligning sensor maps to layout maps.}
\label{tab:success_rate}
\end{table}
\subsubsection{Evaluation of the alignment match score}
Figure~\ref{fig:match_score_matrix} represents the match score of the \emph{winning hypotheses} for all pairs of sensor maps (includes pairing sensor maps of different environment).
Gray-scale encodes the value of the match score ($0\leq S_A \leq 1$).
The cells on diagonal (marked with blue borders) represent the alignment of sensor maps versus the layout maps, and the red lines separate different environments.
Green and red dots mark the success and failure of the alignments respectively.
The squares on diagonal, corresponding to intra environment alignments, are slightly brighter compared to the rest of the matrix which corresponds to inter environment alignments.
However, this is not conclusive enough to employ this measure across different environments and to identify a layout map to which a sensor map belongs.
Under scrutiny it can be seen that maps of a smaller environment (KPT) align with a maps of a bigger environment (HH\_E5) with a high score.
Also, some maps of the same environment have low match score due to the small overlap, even though some are successfully aligned.
Figure~\ref{fig:match_score_boxplot} presents a box plot of the alignment match score for \emph{all hypotheses} in aligning sensor maps to layout maps.
The winning hypotheses are marked red and green, representing the failure and success of each alignment.
There seems to be a cut-off point on the match score value across all maps ($\sim$0.15), which separates successful alignments from failures.
However there is no reliable margin to this cut-off to be used as a threshold between success and failure.
The take away message here is that the value of match score is not a reliable indicator of the alignment success.
In conclusion we can say, even though the \emph{match score} has proven useful in selecting the winning alignment among all hypotheses, yet it is not conclusively reliable to detect to which layout map a sensor map belongs, nor to autonomously detect a successful alignment.
\begin{figure
\centering
\includegraphics[width=\linewidth]{matchscore.pdf}
\caption[xxx]{
The match score of the \emph{winning hypotheses} for all pairs of sensor maps (includes pairing sensor maps of different environment).
Gray-scale encodes the value of the alignment match score ($0\leq S_A \leq 1$).
The cells on diagonal (marked with blue borders) represent the alignment of sensor maps versus the layout maps, and the red lines separate different environments.
Green and red dots mark the success and failure of the alignments respectively.
} \label{fig:match_score_matrix}
\end{figure}
\begin{figure*
\centering
\includegraphics[width=\linewidth]{matchscore_boxplot.pdf}
\caption[xxx]{
The alignment match score for \emph{all hypotheses} in sensor maps to layout maps alignments.
The winning hypotheses are marked red and green, representing the failure and success of each alignment.
} \label{fig:match_score_boxplot}
\end{figure*}
\subsection{Comparison with other methods} \label{sec:comparison}
Our initial investigation and experiments towards map alignment lead to methods which we categorize into two groups.
First the generic approaches in data association, such as image alignment, image registration and point set registration.
And second the map alignment methods, such as Hough transform-based algorithms.
The performance of all methods in terms of success rate is available in Table~\ref{tab:success_rate_comparison}.
And finally, a brief account of computational costs, presented in Table.~\ref{tab:computation_time_comparison}, will follow the performance evaluation.
\subsubsection{Generic data association methods} \label{subsubsec:generic}
Here the performances of three generic data association methods in map alignment are presented:
\begin{inparaenum}[i)]
\item image alignment with Enhanced Correlation Coefficient Maximization (ECC) \cite{evangelidis2008parametric},
\item image registration with Scale-Invariant Feature Transform (SIFT) \cite{lowe1999sift}, and
\item point set registration with Coherent Point Drift (CPD) \cite{NIPS2006_2962}, \cite{5432191}.
\end{inparaenum}
All the performance results are available in Table~\ref{tab:success_rate_comparison}.
We observed that the methods based on ECC and SIFT perform slightly better when they are applied to the distance transform of the maps, instead of the occupancy maps.
Accordingly the presented results are based on distance images of the maps.
ECC maximization performs worse on aligning sensor maps to layouts, due to its higher sensitivity to data-level similarity.
A detailed review of the failed cases in aligning sensor maps to sensor maps reveals that the main causes of failure are the global inconsistencies of the sensor maps and small overlaps between the maps.
Image registration with SIFT~\cite{lowe1999sift} was tested in combination with Fast Approximate Nearest Neighbors \cite{muja2009fast} for feature matching.
This method works best on maps with unique patterns represented by a unique constellation of ``key points'', and consequently has a slightly better performance on bigger maps with more key points.
Although the data-level similarity between sensor maps is in favor of resulting more similar features, however, this method yields better results in aligning sensor maps to layouts of bigger environments thanks to higher overlaps.
For the experiments with CPD, point sets have been generated from the occupied cells of the maps.
CPD is superior to Iterative Closest Point (ICP) as it supports affine transformations.
However, it is computationally expensive and memory demanding, so that the original point sets had to be sub-sampled.
This in turn conceals the structural patterns of the maps, and becomes more sensitive to local minima.
\subsubsection{Map alignment methods}
We have chose the works by Carpin~\cite{carpin2008fast} and Saeedi et al. (PGVD) \cite{saeedi2012efficient} as the representatives of this category.
Implementations of both methods are made publicly available by the authors.
The performances of these methods are presented in Table~\ref{tab:success_rate_comparison}.
One interesting aspect of these methods is their independence from the assumption of maps' ``segment-able regions''.
Therefore they could be considered to have a broader target applications.
For instance, we have observed that such methods perform better on maps that mostly contain corridors, which is a challenge for the region segmentation phase of our method.
Also, thanks to the underlying decoupling of rotation and translation estimation, they could be relatively faster than other methods, specifically the method proposed by Carpin~\cite{carpin2008fast}.
However, these advantages come with a price in performance, while these methods perform better on particular cases, they do have a lower overall success rate over our collection of maps.
By inspecting individual results, we observed that many of the failures were due to a wrong orientation alignment.
And many of those cases which survived the orientation estimation, they still failed at the translation estimation.
Fundamentally, these methods exploit the structural similarities in maps, by finding similarity is Hough spectra and cross correlating the maps after orientation alignment.
We believe the noise, the global inconsistencies, and the repetitive patterns of our maps are the top challenges for such methods.
These methods are limited to rigid transformation, and as a result they could solve only the alignment of sensor to sensor map.
Therefore, we manually adjusted scales of the layout maps, so that these methods could be evaluated over the alignments of sensor to layout maps.
These results (manually adjusted scales) are marked with asterisks in Table~\ref{tab:success_rate_comparison}.
They score very low, which we believe is due to the significant disparity in the representations (i.e. different modalities).
When contrasted with our method, this is an interesting result.
Compared to the alignment of sensor to layout maps, our method scores lower when both maps are sensor maps.
This is mainly due to the amplification of noise, inconsistency and partial coverage when both maps are sensor maps.
On the other hand, these methods (\cite{carpin2008fast}, \cite{saeedi2012efficient}) perform worse when aligning sensor maps to layouts, due to their sensitivity to representation disparity.
On a final note, it is important to note that due to a lack of proper insight to the implementations of these methods, we could not fine-tune them, to maximize their performances in the setting of our experiments.
Therefore we would like to point out, that the success rates of the methods presented in Table~\ref{tab:success_rate_comparison} might not represent their best performances, but rather they provide an insight into advantages and drawbacks of each method.
\begin{table*}
\begin{minipage}{\textwidth}
\centering
\begin{tabular}{l|ccccc|ccccc}
~ &
\multicolumn{5}{c}{sensor to layout} &
\multicolumn{5}{c}{sensor to sensor}\\
method &
HH\_E5 & HH\_F5 & HIH & KPT & total &
HH\_E5 & HH\_F5 & HIH & KPT & total\\
\hline
ECC maximization~\cite{evangelidis2008parametric} &
$0.0$ & $7.14$ & $0.0$ & $0.0$ & $2.77$ &
$37.36$ & $29.67$ & $0.0$ & $16.67$ & $31.9$ \\
SIFT~\cite{lowe1999sift} &
$21.43$ & $50$ & $0.0$ & $0.0$ & $27.7$ &
$17.58$ & $28.57$ & $16.67$ & $0.0$ & $22.1$\\
CPD~\cite{NIPS2006_2962}~\cite{5432191} &
$0.0$ & $0.0$ & $0.0$ & $0.0$ & $0.0$ &
$8.79$ & $3.3$ & $0.0$ & $0.0$ & $5.6$\\
Saeedi et al. (PGVD)~\cite{saeedi2012efficient} &
$0.0^*$ & $0.0^*$ & $25^*$ & $0.0^*$ & $2.77^*$ &
$10.98$ & $12.08$ & $33.33$ & $16.66$ & $12.4$\\
Carpin~\cite{carpin2008fast} &
$0.0^*$ & $7.14^*$ & $0.0^*$ & $0.0^*$ & $2.77^*$ &
$16.48$ & $29.67$ & $100$ & $83.33$ & $27.31$\\
our method &
$64.28$ & $92.85$ & $100$ & $100$ & $83.3$ &
$50.54$ & $68.13$ & $100$ & $83.33$ & $66.5$\\
\end{tabular}
\caption[xxx]{
Success rates (in $\%$) of different methods on map alignment.
Numbers marked with an asterisk (*) refer to the methods that are not able to handle scaling.
In those cases, the experiments were performed on manually scaled maps.}
\label{tab:success_rate_comparison}
\end{minipage}
\end{table*}
\subsubsection{Computation time}
The timings of all methods are provided in Table~\ref{tab:computation_time_comparison}.
All the experiments were carried out on a computer with an Intel$^\circledR$ Core\texttrademark~i5-3340M CPU @ 2.70GHz $\times$4, and 8GiB SODIMM DDR3 Synchronous 1600 MHz of memory, running Ubuntu 14.04.
The timings of experiments are separated into home and office building, which provides a sense of methods' scalability with respect to the size of maps.
The average map size for home environments is $2.2 \cdot 10^5$ pixels, and it is $1.0 \cdot 10^6$ pixels for office buildings (roughly 5 times bigger).
Since CPD is expensive and not scalable, the original point sets were reduced from $1.2 \cdot 10^4$ points on average in small maps and $3.3 \cdot 10^4$ points in bigger maps, to $500$ (close to memory limit of the algorithm on our hardware.)
Therefore a meaningful computation time could not be provided here.
In comparison, our method falls behind some other approaches in terms of computational cost.
Specifically, those methods designed for real-time applications such as Carpin's method~\cite{carpin2008fast} for multi-robot mapping, are extremely fast and hard to beat.
Our method is based on the decomposition of the space and requires an interpretation through abstract models, which is in general computationally more expensive than signal based interpretations such as Hough-spectra.
However, if one intends to exploit the fast speed of the Hough transform-based methods in combination with our method, there is a trade-off between thoroughness of the hypotheses generation and computational time.
In conclusion we speculated that, under certain assumptions (such as orthogonally structured environments), one can create a set of constraints imposed on the hypotheses generation to narrow down the search space.
Although, a better understanding of such potential combination requires further development and more experiments.
At the end, we would like to emphasize that the timings of each method provided here can portray a rough scale, and should not be taken as an accurate computational cost comparison.
This is mainly due to the heterogeneity of the implementations (C++, Python, Matlab).
Furthermore, some of the algorithms are borrowed from other context (e.g. CPD, ECC) and applied to map alignment problem.
Some are intended for offline applications with not much concern for computational time, while others were specifically designed to be fast for real-time applications.
As a result, these computation times are not sufficient to generalize on the performance of each approach.
\begin{table*}
\begin{minipage}{\textwidth}
\centering
\begin{tabular}{l|c|c c}
& & \multicolumn{2}{c}{time in seconds}\\
method & implementation & home & office \\
\hline
ECC maximization~\cite{evangelidis2008parametric} & Python \& C++ & $32.79 (28.24)$ & $73.46 (85.46)$ \\
SIFT~\cite{lowe1999sift} & Python \& C++ & $0.20 (0.05)$ & $0.67 (0.14)$ \\
Saeedi et al. (PGVD)~\cite{saeedi2012efficient} & Matlab & $4.91 (1.42)$ & $50.20 (19.84)$ \\
Carpin~\cite{carpin2008fast} & C++ & $3.07 \cdot 10^{-4}(9.28 \cdot 10^{-5})$ & $2.65 \cdot 10^{-4}(6.72 \cdot 10^{-5})$ \\
our method & Python & $8.86 (2.13)$ & $41.86 (41.92)$
\end{tabular}
\caption[xxx]{Average (and standard deviation) of the computation times (in seconds) of different methods, separated to home and office environments.
}
\label{tab:computation_time_comparison}
\end{minipage}
\end{table*}
\section{Conclusion} \label{sec:conclusion}
In this paper, we present our work and findings on solving the map alignment problem, for 2D spatial maps.
Many interesting approaches have been proposed to address this problem.
However, existing algorithms hinge on assumptions that are not valid in (a number of) interesting use cases, such as aligning partial maps of different modalities.
Most often they are designed to perform map merging where maps are from similar modalities, hence they rely on sensor level similarity of the input maps, and consequently are sensitive to noise and inconsistencies of sensor maps.
In addition, maps of the same modality have similar scale, and as a result, such methods are limited to \emph{rigid transformations}.
Such assumptions do not hold where maps of different modalities, such as sensor maps and layout maps, are to be aligned.
Also, the scaling from one map to the other adds a new dimension to the search space and the desired solution becomes a \emph{similarity transformation} rather than a rigid transformation.
We have shown, with experimental results, the insufficiency of generic data association methods (e.g. SIFT, ECC), and some map alignment methods (designed for aligning maps of same modalities) in solving the problem in our experimental setup.
We have compared the performance of our method with that of other methods both for sensor to sensor map alignment and sensor to layout map alignment.
Except for few examples of similar performance, our method outperforms other methods.
In aligning sensor to sensor maps, we observed that the presence of noise and global inconsistency has been the main challenge for most other approaches.
The representation disparity between maps of different modalities has been even more challenging for those methods in aligning sensor to layout maps.
For the latter experiment, the layout maps were manually scaled to match the sensor maps in size, since other map alignment methods are limited to rigid transformation.
Our method relies on the notion that most human built environments are composed of regions.
Accordingly, our method finds the correct alignment by associating regions and selecting the best hypothesis among all candidates.
By exploiting the notion of regions and founding our method on spatial decomposition, our alignment method operates on a higher level of abstraction.
As a consequence, the method is more robust to dissimilarity and heterogeneity of the sensor-level data.
Furthermore, the approach of aligning regions rather that associating sensor-level data enables our method to handle the scaling factor like any other transformation parameter.
\subsection{Discussion} \label{subsec:disc}
In the result section we tried to provide a thorough performance comparison between our proposed method and other approaches to solve the map alignment problem.
We do not claim, or believe, that our method is superior to other approaches in a generic problem formulation of data association and map alignment.
Rather, we tend to emphasize the particular characteristics and advantages that this method offers over alternatives in specific challenges, namely aligning maps from different modality, severe data level noise, and maps of different scales.
However, there might be some other objectives close to the core of the map alignment problem that our method falls short of.
Examples of such applications are, aligning maps of unstructured environment and real-time applications.
\paragraph{Advantages}
Apart from the higher success rate of our proposed method, we would like to point out some other interesting features of it.
One important aspect, and one of the main motivations behind this work, is the ability to align maps of different modality, and specifically sensor maps to layout maps.
As stated earlier, such a task demands a method that is indifferent to heterogeneity and different scales of input maps.
Our proposed method shows a considerable performance for such cases (success rate $83.3\%$ compared to the best alternative $27.7\%$).
We have developed a region segmentation method based on the arrangement representation and distance transform, but the general framework of our alignment method is not dependent on any specific region segmentation technique.
Our decomposition based algorithm would be able to find the alignment as long as the input maps are effectively interpreted by the arrangement of the 2D plane.
That is to say, as long as the input maps are spatial and could be segmented into meaningful regions, the proposed method in this work could be employed to find the alignment.
We speculate that an improved region segmentation will have a positive effect on the performance of this alignment approach.
It is worth mentioning that the implementation of our proposed method, and the accompanied experiments presented in this paper, convert both maps to occupancy-like bitmaps in advance.
However it is not a requirement of the proposed alignment algorithm, but rather it was a convenient choice.
And finally, the intermediate representation that is constructed for alignment, by itself is a useful representation for different objectives~\cite{shahbandi2015semi},~\cite{shahbandi2014sensor}, and it is not alignment-specific.
\paragraph{Drawbacks and limitations}
The main disadvantage of the proposed method is the computation time.
This means that this method is not suitable for real-time applications.
While exploiting the notion of \emph{meaningful} regions improves the map alignment under difficult circumstances, it also limits the applicability of the method.
Dependency on the region segmentation means it most likely will fail in maps of environments cluttered with furniture, or in a maze-like environment, unless an appropriate region segmentation algorithm is employed.
Partial maps which don't cover multiple regions (e.g. a map of only one room), in applications such as scan matching and incremental mapping, violate one of the initial assumptions and would cause our method to fail.
We speculate that this is a domain where other methods such as the ones proposed by Carpin \cite{carpin2008fast} and Saeedi et al. \cite{saeedi2012efficient} would outperform our method, given the maps are from the same modality.
As stated before, in Section~\ref{sec:results}, not all the maps satisfy our initial assumptions such as global consistency.
We included these maps to better explain the effects of aforementioned assumptions on the method and portray a fair picture of the method's performance under different conditions, even if they violate the assumptions of our method.
Other conditions that make our method unsuitable occurs when the prior assumptions are violated.
Examples are non-uniformly scaled maps like sketch maps, and maze-like environments such as underground tunnels and alike where the notion of meaningful regions might not apply.
\paragraph{Model regression}
Random sample consensus (RANSAC) is a powerful regression technique in estimating a model from noisy data.
However, our empirical observation suggests that RANSAC is not a suitable replacement for the components of our method.
The first possibility is to employ RANSAC for hypothesis generation, i.e. estimating a transformation between faces with known correspondences.
We have found Umeyama's method to be a better fit as a non-iterative deterministic method for this objective.
Alternatively RANSAC could be considered for solving the alignment on the map level with unknown correspondences.
However, the point sets from our representation (vertices of the prime graph) are sparse and do not reflect the skeletal structure.
This challenge is exaggerated with relatively high level of noise and partiality of the maps.
We have experimented with RANSAC in this manner, with a simple setup and a denser sampling of the occupied points, the result of which has not been satisfactory.
This lead to our experimentation with Iterative Closest Points (ICP) and Coherent Point Drift (CPD), a continuation of the attempt in relying on the shape of the distribution of occupied points.
The results of CPD have been included in this manuscript as a representative of this category of approaches.
Despite the inadequacy of RANSAC in estimating the alignment, we speculate that such regression techniques could be beneficial in estimating other models as a part of a more elaborate method.
For instance, RANSAC can be used for the regression of the \emph{geometric coherency of hypotheses}.
That is to say, assuming a correct alignment is represented with multiple hypotheses, the pool of hypotheses is expected to contain clusters of similar transformations.
RANSAC can be used for estimating the geometric coherency of hypotheses and rejecting outliers.
We ran experiments with this idea, although with a clustering algorithm (DBSCAN~\cite{ester1996density}) and not RANSAC.
The challenge is that not always the correct alignment has multiple representatives, specially for small, deformed, and partial maps.
This idea needs further investigation, since treating the pool of hypotheses has to be done carefully with additional considerations.
\subsection{Future work} \label{subsec:future}
In the continuation of this work we intend to address some interesting questions which were raised during the development of this work.
One of those questions is the challenge of autonomous detection of successful alignments.
This problem can be translated to a classification task, where an \emph{alignment match score} could be a multidimensional vector based on other sources of information in addition to arrangement based match score, such as graph matching metrics (e.g. GED), and data level distance between maps.
Towards that objective, we intended to enrich our collection of maps with a wider variety of environments.
Furthermore, we intend to carry out more challenging experiments and with other modalities to inspect the performance of the proposed alignment approach under different circumstances.
The direction of our future work is towards merging maps after alignment.
Specific examples of features to contain in a merging process would be the transferring of semantic labels from layout map to sensor map for high level task planning, and detecting and compensating global inconsistencies in sensor map by relying on the structure of the layout map.
\begin{acknowledgements}
This work was funded by the Swedish Knowledge Foundation (KK-Stiftelsen), and the European Union's Horizon 2020 research and innovation programme under grant agreement No 732737 (ILIAD).
The authors would like to express their gratitude to Dr. Karl Iagnemma and the anonymous reviewers who helped us improve the quality of the manuscript with their valuable and constructive inputs.
\end{acknowledgements}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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**EARLY BIRD BOOKS**
**FRESH EBOOK DEALS, DELIVERED DAILY**
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Uncharted Stars
Andre Norton
FOR PATRICK TERRY
because he has been kind enough
to approve my work
I
It was like any other caravansary at a space port, not providing quarters for a Veep or some off-planet functionary, but not for a belt as sparsely packed with credits as mine was at that moment either. My fingers twitched and I got a cold chill in my middle every time my thoughts strayed to how flat that belt was at present. But there is such a thing as face, or prestige, whatever name you want to give it, and that I must have now or fail completely. And my aching feet, my depressed spirits told me that I was already at the point where one surrendered hope and waited for the inevitable blow to fall. That blow could only fell me in one direction. I would lose what I had played the biggest gamble of my life to win—a ship now sitting on its tail fins in a field I could have sighted from this hotel had I been a Veep and able to afford one of the crown tower rooms with actual windows.
One may be able to buy a ship but thereafter it sits eating up more and more credits in ground fees, field service—more costs than my innocence would have believed possible a planet month earlier. And one cannot lift off world until he has a qualified pilot at the controls, the which I was not, and the which I had not been able to locate.
It had all sounded so easy in the beginning. My thinking had certainly been clouded when I had plunged into this. No—been plunged! Now I centered my gaze on the door which was the entrance to what I could temporarily call "home," and I had very unkind thoughts, approaching the dire, about the partner waiting me behind it.
The past year had certainly not been one to soothe my nerves, or lead me to believe that providence smiled sweetly at me. It had begun as usual. I, Murdoc Jern, had been going about my business in the way any roving gem buyer's apprentice would. Not that our lives, mine and my master Vondar Ustle's, had been without exciting incident. But on Tanth, in the spin of a diabolical "sacred" arrow, everything had broken apart as if a laser ray had been used to sever me not only from Vondar but from any peace of mind or body.
When the sacrifice arrow of the green-robed priests had swung to a stop between Vondar and me, we had not feared; off-worlders were not meat to satisfy their demonic master. Only we had been jumped by the tavern crowd, probably only too glad to see a choice which had not included one of them. Vondar had died from a knife thrust and I had been hunted down the byways of that dark city, to claim sanctuary in the hold of another of their grisly godlings. From there I had, I thought, paid my way for escape on a Free Trader.
But I had only taken a wide stride from a stinking morass into a bush fire—since my rise into space had started me on a series of adventures so wild that, had another recited them to me, I would have thought them the product of fash-smoke breathing, or something he had heard from a story tape.
Suffice it that I was set adrift in space itself, along with a companion whose entrance into my time and space was as weird as his looks. He was born rightly enough, in the proper manner, out of a ship's cat. Only his father was a black stone, or at least several men trained to observe the unusual would state that. Eet and I had been drawn by the zero stone—the zero stone! One might well term that the seed of all disorder!
I had seen it first in my father's hands—dull, lifeless, set in a great ring meant to be worn over the bulk of a space glove. It had been found on the body of an alien on an unknown asteroid. And how long dead its suited owner was might be anyone's guess—up to and including a million years on the average planet. That it had a secret, my father knew, and its fascination held him. In fact, he died to keep it as a threatening heritage for me.
It was the zero stone on my own gloved hand which had drawn me, and Eet, through empty space to a drifting derelict which might or might not have been the very ship its dead owner had once known. And from that a lifeboat had taken us to a world of forest and ruins, where, to keep our secret and our lives, we had fought both the Thieves' Guild (which my father must have defied, though he had once been a respected member of its upper circles) and the Patrol.
Eet had found one cache of the zero stones. By chance we both stumbled on another. And that one was weird enough to make a man remember it for the rest of his days, for it had been carefully laid up in a temporary tomb shared by the bodies of more than one species of alien, as if intended to pay their passage home to distant and unknown planets of origin. And we knew part of their secret. Zero stones had the power to boost any energy they contacted, and they would also home on their fellows, activating such in turn. But that the planet we had landed upon by chance was the source of the stones, Eet denied.
We used the caches for bargaining, not with the Guild, but with the Patrol, and we came out of the deal with, credits for a ship of our own, plus—very sourly given—clean records and our freedom to go as we willed.
Our ship was Eet's suggestion. Eet, a creature I could crush in my two hands (sometimes I thought that solution was an excellent one for me), had an invisible presence which towered higher than any Veep I had ever met. In part, his feline mother had shaped him, though I sometimes speculated as to whether his physical appearance did not continue to change subtly. He was furred, though his tail carried only a ridge of that covering down it. But his feet were bare-skinned and his forepaws were small hands which he could use to purposes which proved them more akin to my palms and fingers than a feline's paws. His ears were small and set close to his head, his body elongated and sinuous.
But it was his mind, not the body he informed me had been "made" for him, which counted. Not only was he telepathic, but the knowledge which abode in his memory, and which he gave me in bits and pieces, must have rivaled the lore of the famed Zacathan libraries, which are crammed with centuries of learning.
Who—or what—Eet was he would never say. But that I would ever be free of him again I greatly doubted. I could resent his calm dictatorship, which steered me on occasion, but there was a fascination (I sometimes speculated as to whether this was deliberately used to entangle me, but if it was a trap it had been very skillfully constructed) which kept me his partner. He had told me many times our companionship was needful, that I provided one part, he the other, to make a greater whole. And I had to admit that it was through him we had come out of our brush with Patrol and Guild as well as we had—with a zero stone still in our possession.
For it was Eet's intention, which I could share at more optimistic times, to search out the source of the stones. Some small things I had noted on the unknown planet of the caches made me sure that Eet knew more about the unknown civilization or confederation which had first used the stones than he had told me. And he was right in that the man who had the secret of their source could name his own price—always providing he could manage to market that secret without winding up knifed, burned, or disintegrated in some messy fashion before he could sell it properly.
We had found a ship in a break-down yard maintained by a Salarik who knew bargaining as even my late master (whom I had heretofore thought unbeatable) did not. I will admit at once that without Eet I would not have lasted ten planet minutes against such skill and would have issued forth owning the most battered junk the alien had sitting lopsidedly on rusting fins. But the Salariki are feline-ancestered, and perhaps Eet's cat mother gave him special insight into the other's mind. The result was we emerged with a useful ship.
It was old, it had been through changes of registry many times, but it was, Eet insisted, sound. And it was small enough for the planet hopping we had in mind. Also, it was, when Eet finished bargaining, within the price we could pay, which in the end included its being serviced for space and moved to the port ready for take-off.
But there it had sat through far too many days, lacking a pilot. Eet might have qualified had he inhabited a body humanoid enough to master the controls. I had never yet come to the end of any branch of knowledge in my companion, who might evade a direct answer to be sure, but whose supreme confidence always led me to believe that he _did_ have the correct one.
It was now a simple problem: We had a ship but no pilot. We were piling up rental on the field and we could not lift. And we were very close to the end of that small sum we had left after we paid for the ship. Such gems as remained in my belt were not enough to do more than pay for a couple more days' reckoning at the caravansary, if I could find a buyer. And that was another worry to tug at my mind.
As Vondar's assistant and apprentice, I had met many of the major gem buyers on scores of planets. But it was to Ustle that they opened their doors and gave confidence. When I dealt on my own I might find the prospect bleak, unless I drifted into what was so often the downfall of the ambitious, the fringes of the black market which dealt in stolen gems or those with dubious pasts. And there I would come face to face with the Guild, a prospect which was enough to warn me off even more than a desire to keep my record clean.
I had not found a pilot. Resolutely now I pushed my worries back into the immediate channel. Deal with one thing at a time, and that, the one facing you. We had to have a pilot to lift, and we _had_ to lift soon, very soon, or lose the ship before making a single venture into space with her.
None of the reputable hiring agencies had available a man who would be willing—at our wages—to ship out on what would seem a desperate venture, the more so when I could not offer any voyage bond. This left the rejects, men black-listed by major lines, written off agency books for some mistake or crime. And to find such a one I must go down into the Off-port, that part of the city where even the Patrol and local police went on sufferance and in couples, where the Guild ruled. To call attention to myself there was asking for a disagreeable future—kidnaping, mind scanning, all the other illegal ways of gaining my knowledge. The Guild had a long and accurate memory.
There was a third course. I could throw up everything—turn on my heel and walk away from the door I was about to activate by thumb pressure on personal seal, take a position in one of the gem shops (if I could find one), forget Eet's wild dream. Even throw the stone in my belt into the nearest disposal to remove the last temptation. In fact, become as ordinary and law-abiding a citizen as I could.
I was greatly tempted. But I was enough of a Jern not to yield. Instead I set thumb to the door and at the same time beamed a thought before me in greeting. As far as I knew, the seals in any caravansary, once set to individual thumbprints, could not be fooled. But there can always be a first time and the Guild is notorious for buying up or otherwise acquiring new methods of achieving results which even the Patrol does not suspect have been discovered. If we had been traced here, then there just might be a reception committee waiting beyond. So I tried mind-touch with Eet for reassurance.
What I got kept me standing where I was, thumb to doorplate, bewildered, then suspicious. Eet was there. I received enough to be sure of that. We had been mind-coupled long enough for even tenuous linkage to be clear to my poorer human senses. But now Eet was withdrawn, concentrating elsewhere. My fumbling attempts to communicate failed.
Only it was not preoccupation with danger, no warn-off. I pressed my thumb down and watched the door roll back into the wall, intent on what lay beyond.
The room was small, not the cubby of a freeze-class traveler, but certainly not the space of a Veep suite. The various fixtures were wall-folded. And now the room was unusually empty, for apparently Eet had sent every chair, as well as the table, desk, and bed back into the walls, leaving the carpeted floor bare, a single bracket light going.
A circle of dazzling radiance was cast by that (I noted at once that it had been set on the highest frequency and a small portion of my mind began calculating how many minutes of that overpower would be added to our bill). Then I saw what was set squarely under it and I was really startled.
As was true of all port caravansaries, this one catered to tourists as well as business travelers. In the lobby was a shop—charging astronomical prices—where one could buy a souvenir or at least a present for one's future host or some member of the family. Most of it was, as always, a parade of eye-catching local handicrafts to prove one had been on Theba, with odds and ends of exotic imports from other planets to attract the attention of the less sophisticated traveler.
There were always in such shops replicas of the native fauna, in miniature for the most part. Some were carved as art, others wrought in furs or fabrics to create a very close likeness of the original, often life-size for smaller beasts, birds, or what-is-its.
What sat now in the full beam of the lamp was a stuffed pookha. It was native to Theba. I had lingered by a pet shop (intrigued in spite of my worries) only that morning to watch three live pookhas. And I could well understand their appeal. They were, even in the stuffed state, luxury items of the first class.
This one was not much larger than Eet when he drew his long thin body together in a hunched position, but it was of a far different shape, being chubby and plump and with the instant appeal to my species that all its kind possess. Its plushy fur was a light green-gray with a faint mottling which gave it the appearance of the watered brocade woven on Astrudia. Its fore-paws were bluntly rounded pads, unclawed, though it was well provided with teeth, which in live pookhas were used for crushing their food—tich leaves. The head was round with no visible ears, but between the points where ears might normally be, from one side of that skull-ball to the other, there stood erect a broad mane of whisker growth fanning out in fine display. The eyes were very large and green, of a shade several tints darker than its fur. It was life-size and very handsome—also very, very expensive. And how it had come here I did not have the slightest idea. I would have moved forward to examine it more closely but a sharp crack of thought from Eet froze me where I stood. It was not a concrete message but a warning not to interfere.
Interfere in what? I looked from the stuffed pookha to my roommate. Though I had been through much with Eet and had thought I had learned not to be surprised at any action of my alien companion, he now succeeded very well in startling me.
He was, as I had seen, hunched on the floor just beyond the circle of intense light cast by the lamp. And he was staring as intently at the toy as if he had been watching the advance of some enemy.
Only Eet was no longer entirely Eet. His slim, almost reptilian body was not only hunched into a contracted position but actually appeared to have become plumper and shorter, aping most grotesquely the outward contours of the pookha. In addition, his dark fur had lightened, held a greenish sheen.
Totally bewildered, yet fascinated by what was occurring before my unbelieving eyes, I watched him turn into a pookha, altering his limbs, head shape, color, and all the rest. Then he shuffled into the light and squatted by the toy to face me. His thought rang loudly in my head.
"Well?"
"You are that one." I pointed a finger, but I could not be sure. To the last raised whisker of crest, the last tuft of soft greenish fur, Eet was twin to the toy he had copied.
"Close your eyes!" His order came so quickly I obeyed without question.
A little irritated, I immediately opened them again, to confront once more two pookhas. I guessed his intent, that I should again choose between them. But to my closest survey there was no difference between the toy and Eet, who had settled without any visible signs of life into the same posture. I put out my hand at last and lifted the nearest, to discover I had the model. And I felt Eet's satisfaction and amusement.
"Why?" I demanded.
"I am unique." Was there a trace of complacency in that remark? "So I would be recognized, remarked upon. It is necessary that I assume another guise."
"But how did you do this?"
He sat back on his haunches. I had gone down on my knees to see him the closer, once more setting the toy beside him and looking from one to the other for some small difference, though I could see none.
"It is a matter of mind." He seemed impatient. "How little you know. Your species is shut into a shell of your own contriving, and I see little signs of your struggling to break out of it." This did not answer my question very well. I still refused to accept the fact that Eet, in spite of all he had been able to do in the past, could _think_ himself into a pookha.
He caught my train of thought easily enough. "Think myself into a hallucination of a pookha," he corrected in that superior manner I found irking.
"Hallucination!" Now _that_ I could believe. I had never seen it done with such skill and exactitude, but there were aliens who dealt in such illusions with great effect and I had heard enough factual tales of such to believe that it could be done, and that one receptive to such influences and patterns could be made to see as they willed. Was it because I had so long companied Eet and at times been under his domination that I was so deceived now? Or would the illusion he had spun hold for others also?
"For whom and as long as I wish," he snapped in reply to my unasked question. "Tactile illusion as well—feel!" He thrust forth a furred forelimb, which I touched. Under my fingers it was little different from the toy, except that it had life and was not just fur laid over stuffing.
"Yes." I sat back on my heels, convinced. Eet was right, as so often he was—often enough to irritate a less logical being such as I. In his own form Eet was strange enough to be noticed, even in a space port, where there is always a coming and going of aliens and unusual pets. He could furnish a clue to our stay here. I had never underrated the Guild or their spy system.
But if they had a reading on Eet, then how much more so they must have me imprinted on their search tapes! I had been their quarry long before I met Eet, ever since after my father's murder, when someone must have guessed that I had taken from his plundered office the zero stone their man had not found. They had set up the trap which had caught Vondar Ustle but not me. And they had laid another trap on the Free Trader, one which Eet had foiled, although I did not know of it until later. On the planet of ruins they had actually held me prisoner until Eet again freed me. So they had had innumerable chances of taping me for their hounds—a fact which was frightening to consider.
"You will think yourself a cover." Eet's calm order cut across my uneasiness.
"I cannot! Remember, I am of a limited species—" I struck back with the baffled anger that realization of my plight aroused in me.
"You have only the limits you yourself set," Eet returned unruffled. "Perceive—"
He waddled on his stumpy pookha legs to the opposite side of the room, and as suddenly flowed back into Eet again, stretching his normal body up against the wall at such a lengthening as I would not have believed even his supple muscles and flesh capable of. With one of his paw-hands he managed to touch a button and the wall provided us with a mirror surface. In that I saw myself.
I am not outstanding in any way. My hair is darkish brown, which is true of billions of males of Terran stock. I have a face which is wide across the eyes, narrowing somewhat to the chin, undistinguished for either good looks or downright ugliness. My eyes are green-brown, and my brows, black, as are my lashes. As a merchant who travels space a great deal, I had had my beard permanently eradicated when it first showed. A beard in a space helmet is unpleasant. And for the same reason I wear my hair cropped short. I am of medium height as my race goes, and I have all the right number of limbs and organs for my own species. I could be anyone—except that the identification patterns the Guild might hold on me could go deeper and be far more searching than a glance at a passing stranger.
Eet flowed back across the room with his usual liquid movement, made one of his effortless springs to my shoulder, and settled down in position behind my neck, his head resting on top of mine, his hand-paws flat on either side of my skull just below my ears.
"Now!" he commanded. "Think of another face—anyone's—"
When so ordered I found that I could not—at first. I looked into the mirror and my reflection was all that was there. I could feel Eet's impatience and that made it even more difficult for me to concentrate. Then that impatience faded and I guessed that he was willing it under control.
"Think of another." He was less demanding, more coaxing. "Close your eyes if you must—"
I did, trying to summon up some sort of picture in my mind—a face which was not my own. Why I settled for Faskel I could not say, but somehow my foster brother's unliked countenance swam out of memory and I concentrated upon it.
It was not clear but I persevered, setting up the long narrow outline—the nose as I had last seen it, jutting out over a straggle of lip-grown hair. Faskel Jern had been my father's true son, while I was but one by adoption. Yet it had always seemed that I was Hywel Jern's son in spirit and Faskel the stranger. I put the purplish scar on Faskel's forehead near his hairline, added the petulant twist of lips which had been his usual expression when facing me in later years, and held to the whole mental picture with determination.
"Look!"
Obediently I opened my eyes to the mirror. And for several startled seconds I looked at someone. He was certainly not me—nor was he Faskel as I remembered him, but an odd, almost distorted combination of us both. It was a sight I did not in the least relish. My head was still gripped in the vise maintained by Eet's hold and I could not turn away. But as I watched, the misty Faskel faded and I was myself again.
"You see—it can be done," was Eet's comment as he released me and flowed down my body to the floor.
" _You_ did it."
"Only in part. There has been, with my help, a breakthrough. Your species use only a small fraction of your brain. You are content to do so. This wastage should shame you forever. Practice will aid you. And with a new face you will not have to fear going where you can find a pilot."
"If we ever can." I push-buttoned a chair out of the wall and sat down with a sigh. My worries were a heavy burden. "We shall have to take a black-listed man if we get any."
" _Ssssss_ —" No sound, only an impression of one in my mind. Eet had flashed to the door of the room, was crouched against it, his whole attitude one of strained listening, as if all his body, not just his ear, served him for that purpose.
I could hear nothing, of course. These rooms were completely screened and soundproofed. And I could use a hall-and-wall detect if I wished to prove it so. Spaceport caravansaries were the few places where one could be truly certain of not being overlooked, overheard, or otherwise checked upon.
But their guards were not proofed against such talents as Eet's, and I guessed from his attitude not only that he was suspicious of what might be arriving outside but that it was to be feared. Then he turned and I caught his thought. I moved to snap over a small luggage compartment and he folded himself into hiding there in an instant. But his thoughts were not hidden.
"Patrol snoop on his way—coming here," he warned, and it was alert enough to prepare me.
II
As yet, the visitor's light had not flashed above the door. I moved, perhaps not with Eet's speed but fast enough, to snap the room's furnishings out and in place so that the compartment would look normal even to the searching study of a trained Patrolman. The Patrol, jealous of its authority after long centuries of supremacy as the greatest law-enforcement body in the galaxy, had neither forgotten nor forgiven the fact that Eet and I had been able to prove them wrong in their too-quick declaration of my outlawry (I had indeed been framed by the Guild). That we had dared, actually dared, to strike a bargain and keep them to it, galled them bitterly. We had rescued their man, saved his skin and his ship for him in the very teeth of the Thieves' Guild. But he had fought bitterly against the idea that we did have the power to bargain and that he had to yield on what were practically our terms. Even now the method of that bargaining made me queasy, for Eet had joined us mind to mind with ruthless dispatch. And such an invasion, mutual as it was, left a kind of unhealed wound.
I have heard it stated that the universe is understood by each species according to the sensory equipment of the creature involved, or rather, the meaning it attaches to the reports of those exploring and testing senses. Therefore, while _our_ universe, as we see it, may be akin to that of an animal, a bird, an alien, it still differs. There are barriers set mercifully in place (and I say mercifully after tasting what can happen when such a barrier goes down) to limit one's conception of the universe to what he is prepared to accept. Shared minds between human and human is not one of the sensations we are fitted to endure. The Patrolman and I had learned enough—too much—of each other to know that a bargain could be made and kept. But I think I would face a laser unarmed before I would undergo that again.
Legally the Patrol had nothing against us, except suspicions perhaps and their own dislike for what we had dared. And I think that they were in a measure pleased that if they had to swear truce, the Guild still held us as a target. And it might well be that once we had lifted from the Patrol base we had been regarded as expendable bait for some future trap in which to catch a Veep of the Guild—a thought which heated me more than a little every time it crossed my mind.
I gave a last hurried glance around the room as the warn light flashed on, and then went to thumb the peephole. What confronted my eye was a wrist, around which was locked, past all counterfeiting, the black and silver of a Patrol badge. I opened the door.
"Yes?" I allowed my real exasperation to creep into my voice as I fronted him.
He was not in uniform, wearing rather the ornate, form-fitting tunic of an inner-world tourist. On him, as the Patrol must keep fit, it looked better than it did on most of the flabby, paunchy specimens I had seen in these halls. But that was not saying much, for its extreme of fashion was too gaudy and fantastic to suit my eyes.
"Gentle Homo Jern—" He did not make a question of my name, and his eyes were more intent on the room behind me than on meeting mine.
"The same. You wish?"
"To speak with you—privately." He moved forward and involuntarily I gave a step before I realized that he had no right to enter. It was the prestige of the badge he wore which won him that first slight advantage and he made the most of it. He was in, with the door rolled into place behind him, before I was prepared to resist.
"We are private. Speak." I did not gesture him to a chair, nor make a single hospitable move.
"You are having difficulty in finding a pilot." He looked at me about half the time now, the rest of his attention still given to the room.
"I am." There was no use in denying a truth which was apparent.
Perhaps he did not believe in wasting time either, for he came directly to the point.
"We can deal—"
That really surprised me. Eet and I had left the Patrol base with the impression that the powers there were gleefully throwing us forth to what they believed certain disaster with the Guild. The only explanation which came to me at the moment was that they had speedily discovered that the information we had given them concerning the zero stones had consisted of the whereabouts of caches only and they suspected the true source was still our secret. In fact, we knew no more than we had told them.
"What deal?" I parried and dared not mind-touch Eet at that moment, much as I wanted his reception to this suggestion. No one knows what secret equipment the Patrol had access to. And it might well be that, knowing Eet was telepathic, they had some ingenious method of monitoring our exchange.
"Sooner or later," he said deliberately, almost as if he savored it, "the Guild is going to close in upon you—"
But I was ready, having thought that out long ago. "So I am bait and you want me for some trap of yours."
He was not in the least disconcerted. "One way of putting it."
"And the right way. What do you want to do, plant one of your men in our ship?"
"As protection for you and, of course, to alert us."
"Very altruistic. But the answer is no." The Patrol's highhanded method of using pawns made me aware that there was something to being their opponent.
"You cannot find a pilot."
"I am beginning to wonder"—and at that moment I was—"how much my present difficulty may be due to the influence of your organization."
He neither affirmed nor denied it. But I believe I was right. Just as a pilot might be black-listed, so had our ship been, before we had even had a chance for a first voyage. No one who wanted to preserve his legal license would sign our log now. So I must turn to the murky outlaw depths if I was to have any luck at all. I would see the ship rust away on its landing fins before I would raise with a Patrol nominee at her controls.
"The Guild can provide you with a man as easily, if you try to hire an off-rolls man, and you will not know it," he remarked, as if he were very sure that I would eventually be forced to accept his offer.
That, too, was true. But not if I took Eet with me on any search. Even if the prospective pilot had been brainwashed and blanked to hide his true affiliation, my companion would be able to read that fact. But that, I hoped, my visitor and those who had sent him did not know. That Eet was telepathic we could not hide—but Eet himself—
"I will make my own mistakes," I allowed myself to snap.
"And die from them," he replied indifferently. He took one last glance at the room and suddenly smiled. "Toys now—I wonder why." With a swoop as quick and sure as that of a harpy hawk he was down and up again, holding the pookha by its whisker mane. "Quite an expensive toy, too, Jern. And you must be running low in funds, unless you have tapped a river running with credits. Now why, I wonder, would you want a stuffed pookha."
I grimaced in return. "Always provide my visitors with a minor mystery. You figure it out. In fact, take it with you—just to make sure it is not a smuggling cover. It might just be, you know. I am a gem buyer—what better way to get some stones off world than in a play pookha's inwards?"
Whether he thought my explanation was as lame as it seemed to me I do not know. But he tossed the toy onto the nearest chair and then, on his way to the door, spoke over his shoulder. "Dial 1-0. Jern, when you have stopped battering your head against a stone wall. And we shall have a man for you, one guaranteed not to sign you over to the Guild."
"No—just to the Patrol," I countered. "When I am ready to be bait, I shall tell you."
He made no formal farewell, just went. I closed the door sharply behind him and was across the room to let Eet out as quickly as I could. My alien companion sat back on his haunches, absent-mindedly smoothing the fur on his stomach.
"They think that they have us." I tried to jolt him—though he must already have picked up everything pertinent from our visitor's mind, unless the latter had worn a shield.
"Which he did," Eet replied to my suspicion. "But not wholly adequate, only what your breed prepares against the mechanical means of detecting thought waves. They are not," he continued complacently, "able to operate against my type of talent. But yes, they believe that they have us sitting on the palm of a hand"—he stretched out his own—"and need only curl their fingers, so—" His clawed digits bent to form a fist. "Such ignorance! However, it will be well, I believe, to move swiftly now that we know the worst."
"Do we?" I asked morosely as I hustled out my flight bag and began to pack. That it was not intelligent to stay where we were with Patrol snoops about, I could well understand. But where we would go next—
"To the Diving Lokworm," Eet replied as if the answer was plain and he was amused that I had not guessed it for myself.
For a moment I was totally adrift. The name he mentioned meant nothing, though it suggested one of those dives which filled the murky shadows of the wrong side of the port, the last place in the world where any sane man would venture with the Guild already sniffing for him.
But at present I was more intent on getting out of this building without being spotted by a Patrol tail. I rolled up my last clean undertunic and counted out three credit disks. In a transit lodging one's daily charges are conspicuous each morning on a small wall plate. And no one can beat the instant force field which locks the room if one does not erase these charges when the scanner below says he is departing. The room might be insured for privacy in other ways, but there are precautions the owners are legally allowed to install.
I dropped the credits into the slot under the charge plate and that winked out. Thus reassured I could get out, I must now figure how. When I turned it was to see that Eet was again a pookha. For a moment I hesitated, not quite sure which of the furry creatures was my companion until he moved out to be picked up.
With Eet in the crook of one arm and my bag in my other hand, I went out into the corridor after a quick look told me it was empty. When I turned toward the down grav shaft Eet spoke:
"Left and back!"
I obeyed. His directions took me where I did not know the territory, bringing me to another grav shaft, that which served the robos who took care of the rooms. There might be scanners here, even though I had paid my bill. This was an exit intended only for machines and one of them rumbled along toward us now.
It was a room-service feeder, a box on wheels, its top studded with call buttons for a choice of meal. I had to squeeze back against the wall to let it by, since this back corridor had never been meant for the human and alien patrons of the caravansary.
"On it!" Eet ordered.
I had no idea what he intended, but I had been brought out of tight corners enough in the past to know that he generally did have some saving plan in mind. So I swung Eet, my bag, and myself to the table top of the feeder, trying to take care that I did not trigger any of the buttons.
My weight apparently was nothing to the machine. It did not pause in its steady roll down the remainder of the corridor. But I was tense and stiff, striving to preserve my balance on this box where there was nothing to grip for safety.
When it moved without pause off the floor and onto the empty air of the grav shaft I could have cried out. But the grav supported its weight and it descended as evenly under me as if it had been a lift platform bringing luggage and passengers out of a liner at the port. A sweeper joined us at the next level, but apparently the machines were equipped with avoid rays, as they did not bump, but kept from scraping against each other. Above and below us, in the dusk of the shaft, I could see other robo-servers descending, as if this was the time when they were through their morning work.
We came down floor by floor, I counting them as we passed, a little more relieved with each one we left behind, knowing that we were that much nearer our goal. But when we reached ground level we faced only blank surface, and my support contined to descend.
The end was some distance below the surface, at least equal, I believed, to three floors above. And the feeder, with us still aboard, rolled out in pitch dark, where the sounds of clanging movement kept me frozen. Nor did Eet suggest any answer to this.
I did gain enough courage to bring out a hand beamer and flash it about us, only to gain disturbing glimpses of machines scuttling hither and thither across a wide expanse of floor. Nor were there any signs of human tenders.
I was now afraid to dismount from my carrier, not knowing whether the avoid rays of the various busy robos would also keep them from running me down. To this hour I had always taken the service department of a caravansary for granted and such an establishment as this I had never imagined.
That the feeder seemed to know just where it was going was apparent, for it rolled purposefully on until we reached a wall with slits in it. The machine locked to one of these and I guessed that the refuse and disposable dishes were being deposited in some sort of refuse system. Not only the feeder was clamped there. Beyond was a sweeper, also dumping its cargo.
A flash of my beamer showed that the wall did not reach the roof, so there might be a passage along its top to take us out of the paths of the roving machines—though such a way might well lead to a dead end.
I stood up cautiously on the feeder, and Eet took the beamer between his stubby pookha paws. The bag was easy to toss to the top of the wall, my furry companion less so, since his new body did not lend itself well to such feats. However, once aloft, he squatted, holding the beamer in his mouth, his teeth gripping more easily than his paws.
With that as my guide I leaped and caught the top of the wall, though I was afraid for a moment my fingers would slip from its slick surface. Then I made an effort which seemed enough to tear my muscles, and drew my whole body up on an unpleasantly narrow surface.
Not only was it narrow but it throbbed and vibrated under me, and I mentally pictured some form of combustion reducing the debris dumped in, or else a conveyer belt running on into a reducer of such refuse.
Above me, near enough to keep me hunched on my hams, was the roof of the place. A careful use of the beamer showed me that the wall on which I crouched ran into a dark opening in another wall met at right angles, as if it were a path leading into a cave.
For want of a better solution I began to edge along, dragging my bag, my destination that hole. Luckily Eet did not need my assistance but balanced on his wide pookha feet behind me.
When I reached that opening I found it large enough to give me standing room in a small cubby. The beam lighted a series of ladder steps bolted to the wall, as though this was an inspection site visited at intervals by a human maintenance man. Blessing my luck, I was ready to try that ladder, for the clanging din of the rushing machines, the whir of their passing rung in my ears, making me dizzy. The sooner I was out of their domain the better.
Eet's paws were not made for climbing, and I wondered if he would loose the disguise for the attempt. I had no desire to carry him; in fact I did not see how I could.
But if he could release the disguise he was not choosing to do so. Thus, in the end, I had to sling the bag on my back by its carrying strap and loosen my tunic to form a sling, with Eet crawling part-way down inside my collar at my shoulders. Both burdens interfered cruelly with my balance as I began to climb. And I had had to put away the beamer, not being conveniently endowed with a third hand.
For the moment all I wanted was to get out of the dark country of the robo-servers, even though I was climbing into the unknown. Perhaps I had come to depend too much on Eet's warnings against approaching dangers. But he had not communicated with me since we had taken transport on the feeder.
"Eet, what is ahead?" I sent that demand urgently as I became aware of just what _might_ lie ahead of us.
"Nothing—yet." But his mind-send was faint, as a voiced whisper might be, or as if most of his mind was occupied with some other pressing problem.
I found, a second or two later, the end of the ladder, as my hand, rising to grope for a new hold, struck painfully instead against a hard surface. I spread my fingers to read what was there. What I traced by touch was a circular depression which must mark a trap door. Having made sure of that, I applied pressure, first gently and then with more force. When there was no reassuring yield I began to be alarmed. If the bolt hole of this door was locked, we would have to recourse but to return to the level of the robos, and I did not want to think of that.
But my final desperate shove must have triggered whatever stiff mechanism held the door and it gave, letting in a weak light. I had wit and control enough left to wait for a very long moment for any warning from Eet.
When he sent nothing I scrambled out into a place where the walls were studded with gauges, levers, and the like, perhaps the nerve center that controlled the robos. Since there was no one there and a very ordinary door in the nearest wall, I breathed a sigh of heart-felt relief and set about making myself more presentable, plucking Eet out of my unsealed tunic and fastening that smoothly. As far as I could tell, examining my clothes with care, I bore no traces of my late venture through the bowels of the caravansary and I should be able to take to the streets without notice. Alway providing that the door opposite me would eventually lead me to freedom.
What it did give on was a very small grav lift. I set the indicator for street level and was wafted up to a short corridor with doors at either end. One gave upon a walled court with an entrance for luggage conveyers. And I hop-skipped with what speed I could along one of those, to drop into an alley where a flitter from the port unloaded heavier transport boxes.
"Now!" Eet had been riding on my shoulder, his pookha body less well adapted to that form of transport than his true form. I felt his paws clamp on either side of my head as he had earlier done when showing me how one's face could be altered. "Wait!"
I did not know his purpose, since he did not demand I "think" a face. And though that waiting period spun out, making me uneasy, he did not alter his position. I was sure he was using his own thought power to provide me with a disguise.
"Best—I—can—do—" The paws fell away from my head and I reached up to catch him as he tumbled from his place. He was shaking as if from extreme fatigue and his eyes were closed, while he breathed in short gasps. Once before I had seen him so drained—even rendered unconscious—when he had forced me to share minds with the Patrolman.
Carrying Eet as I might a child, and shouldering my flight bag, I went down the alley. A back look at the building had given me directions. If I had a tail who had not been confused by our exit, he had no place to hide here.
The side way fed into a packed commercial street where the bulk of the freight from the port must pass. There were six heavy-duty transport belts down its middle, flanked on either side by two light-duty, and there remained room for a single man-way, narrow indeed, which scraped along the sides of the buildings it passed. There was enough travel on it to keep me from being unduly conspicuous, mainly people employed at the port to handle the shipments. I dropped my bag between my feet and stood, letting the way carry me along, not adding speed by walking.
Eet had spoken of the Diving Lokworm, which was still a mystery to me, and I had no intention of visiting the Off-port before nightfall. Daytime visitors, save for tourists herded along on a carefully supervised route, were very noticeable there. Thus I would have to hole up somewhere. Another hotel was the best answer. With what I thought a gift of inspiration I chose one directly across from the Seven Planets, from where I had just made my unusual exit.
This was several steps down from the Seven Planets in class, which suited my reduced means. And I was especially pleased that instead of a human desk clerk, who would have added to the prestige, there was a robo—though I knew that my person was now recorded in the files from its scanners. Whether the confusing tactics on my behalf via Eet's efforts would hold here I did not know.
I accepted the thumb lock plate with its incised number, took the grav to the cheapest second-floor corridor, found my room, inserted the lock, and once inside, relaxed. They could force that door now only with super lasers.
Depositing Eet on the bed, I went to the wall mirror to see what he had done to me. What I did sight was not a new face, but a blurring, and I felt a disinclination to look long at my reflection. To watch with any concentration was upsetting, as if I found my present appearance so distasteful that I could not bear to study it.
I sat down on the chair near the mirror. And as I continued to force myself to look at that reflection I was aware that the odd feeling of disorientation was fading, that in the glass my own features were becoming clearer, sharper, visible and ordinary as they had always been.
That Eet could work such a transformation again when the time came to leave here, I doubted. Such a strain might be too much, especially when it was imperative that his esper talents be fully alert. So I might well walk out straight into the sight of those hunting me. But—could I reproduce Eet's effect by my own powers? My trial with Faskel's features had certainly not been any success. And I had had to call upon Eet's help to achieve even that.
But suppose I did not try for so radical a disguise? Eet had supplied me this time, not with a new face, but with merely an overcast of some weird kind which had made me difficult to look at. Suppose one did not try to change a whole face, but only a portion of it? My mind fastened upon that idea, played with it. Eet did not comment, as I thought he might. I looked to the bed. By all outward appearances he was asleep.
If one did not subtract from a face but added to it—in such a startling fashion that the addition claimed the attention, thus overshadowing features. There had been a time in the immediate past when my skin was piebald, due to Eet's counterfeiting of a plague stigma. I could remember only too well those loathsome purple patches. No return to those! I had no wish to be considered again a plague victim. However, a scar—
My mind wandered to the days when my father had kept the hock-lock shop at the space port on my home planet. Many spacers had sought out his inner room to sell finds into whose origin it was best not to inquire too closely. And more than one of those had been scarred or marked unpleasantly.
A scar—yes. Now where—and what? A healed knife gash, a laser burn, an odd seam set by some unknown wounding? I decided on a laser burn which I had seen and which should fit in well with the Off-port. With it as clear in my mind as I could picture it, I stared into the mirror, striving to pucker and discolor the skin along the left side of my jaw and cheek.
III
It was an exercise against all the logic of my species. Had I not seen it succeed with Eet, seen my partial change under his aid, I would not have believed it possible. Whether I _could_ do it without Eet's help was another question, but one I was eager to prove. My dependence upon the mutant, who tended to dominate our relationship, irked me at times.
There is a saying: If you close doors on all errors, truth also remains outside. Thus I began my struggle with errors aplenty, hoping that a small fraction of the truth would come to my aid. I had not, since I had known Eet, been lax in trying to develop any esper talents I might have. Primarily because, I was sure, it was not in my breed to admit that a creature who looked so much an animal could out-think, out-act a man—though in the galaxy the term "man" is, of course, relative, having to do with a certain level of intelligence rather than a humanoid form. In the beginning, this fact was also difficult for my breed, with their many inborn prejudices, to realize. We learned the hard way until the lesson stuck.
I closed the channels of my mind as best I could, tamping down a mental lid on my worries about our lack of a pilot, a shrinking number of credits, and the fact that I might right now be the quarry in a hunt I could sense but not see or hear. The scar—that must be the most important, the _only_ thing in my mind. I concentrated on my reflection in the mirror, on what I wanted to see there.
Perhaps Eet was right, as he most always was—we of Terran stock do not use the full powers which might be ours. Since I had been Eet's charge, as it were, I must have stretched, pulled, without even being aware of that fact, in a manner totally unknown to my species heretofore. Now something happened which startled me. It was as if, in that part of me which fought to achieve Eet's ability, a ghostly finger set tip to a lever and pressed it firmly. I could almost feel the answering vibration through my body—and following on that, a flood of certainty that this I could do, a heady confidence which yet another part of me observed in alarm and fear.
But the face in the mirror—Yes! I had that disfiguring seam, not raw and new, which would have been a give-away to the observant, but puckered and dark, as though it had not been tended quickly enough by plasta restoration, or else such a repair job had been badly botched—as might be true for a crewman down on his luck, or some survivor of a planetary war raid.
So real! Tentatively I raised my hand, not quite daring to touch that rough, ridged skin. Eet's illusion had been—was—tactile as well as visual. Would mine hold as well? I touched. No, I was not Eet's equal as yet, if I could ever be. My fingers traced no scar, as they seemed to do when I looked into the mirror. But visually the scar was there and that was the best protection I could have.
"A beginning, a promising beginning—"
My head jerked as I was startled out of absorption. Eet was sitting up on the bed, his unblinking pookha eyes watching me in return. Then I feared the break in my concentration and looked back to the mirror. But contrary to my fears, the scar was still there. Not only that, but I had chosen rightly—it drew attention, the face behind it blotted out by that line of seamed and darkened skin—as good as a mask.
"How long will it last?" If I ventured out of this room, went delving into the Off-port as I must, I would not be able to find another hole in a hurry into which I could settle safely for the period of intense concentration I would need to renew my disfigurement.
Eet's round head tilted a little to one side, giving the appearance of critical observation of my thought work.
"It is not a large illusion. You were wise to start small," he commented. "With my aid, I think it will hold for tonight. Which is all we need. Though I shall have to change myself—"
"You? Why?"
"Need you parade your incomprehension of danger?" The whisker mane had already winked out of being. "Take a pookha into the Off-port?"
He was right as ever. Pookhas alive were worth more than their weight in credits. To carry one into the Off-port would be to welcome a stun ray, if lucky, a laser burn if not, with Eet popped into a bag and off to some black-market dealer. I was angry with myself for having made such a display of nonthinking, though it was due to the need for concentration on maintaining the scar.
"You must hold it, yes, but not with your whole mind," Eet said. "You have very much to learn."
I held. Under my eyes Eet changed. The pookha dissolved, vanished as though it were an outer husk of plasta meeting the cold of space and so shattering into bits too tiny for the human eye to see. Now he was Eet again, but as unusual to the observer as the pookha had been.
"Just so," he agreed. "But I shall not be observed. I need not change. It will simply be a matter of not allowing the eye to light on me."
"As you did with my face, coming here?"
"Yes. And the dark will aid. We'll head straight for the Diving Lokworm—"
"Why?"
One of my own species might have given an exaggerated sigh of annoyance. The mental sensation which emanated from my companion was not audible but it had the same meaning.
"The Diving Lokworm is a possible meeting place for the type of pilot we must find. And you need not waste time asking me how I know that. It is the truth."
How much Eet could pick out of nearby minds I did not know; I thought that I did not want to know. But his certainty now convinced me that he had some concrete lead. And I could not argue when I had nothing of my own to offer in return.
He made one of his sudden leaps to my shoulder and there arranged himself in his favorite riding position, curled about my neck as if he were an inanimate roll of fur. I gave a last look into the mirror, to reassure myself that my creation was as solid-seeming as ever, and knew a spark of triumph when I saw that it was, even though I might later have to depend upon Eet to maintain it.
So prepared, we went out and took the main crawl walk toward the port, ready to drop off at the first turn which led to the murk of the Off-port. It was dusk, the clouds spreading like smoke across a dark-green sky in which the first of Theba's moons pricked as a single jewel of light.
But the Off-port was awake as we entered it by the side way. Garish signs, not in any one language (though Basic was the main tongue here), formed the symbols, legible to spacemen of many species and races, which advertised the particular wares or strange delights offered within. Many of them were a medley of colors meant to attract nonhuman races, and so, hurtful to our organs of vision. Thus one was better advised not to look above street level. There was also such a blare of noise as was enough to deafen the passerby, and scents to make one long for the protection of a space suit which could be set to shut out the clamor and provide breathable, filtered air.
To come into this maze was to believe one had been decanted on another world, not only dangerous but inhospitable. How I was to find Eet's Diving Lokworm in this pool of confusion was a problem I saw no way of solving. And to wander, deafened and half asphyxiated, through the streets and lanes was to ask for disaster. I had no belted weapon and I was carrying a flight bag, so perhaps ten or more pairs of eyes had already marked me down as possible prey for a portside rolling.
"Right here—" Eet's thought made as clean a cut as a force blade might make through the muddle of my mind.
Right I turned, out of the stridence of the main street, into a small, very small, lessening of the clamor, with a fraction less light, and perhaps one or two breaths now and then of real air. And Eet seemed to know where we were going, if I did not.
We turned right a second time and then left. The spacemen's rests now about were such holes of crime that I feared to poke a nose into any of them. We were fast approaching the last refuge of the desperate, and the stinking hideups of those who preyed upon them, driven from the fatter profits of the main streets.
The Diving Lokworm had, not its name, but a representation of that unwholesome creature set in glow lines about its door. The designer had chosen to arrange it so that one apparently entered through the open mouth—which was perhaps an apt prophecy of what might really await the unwary within. The stench of the outside was here magnified materially by the fumes of several kinds of drink and drug smoke. Two I recognized as lethal indeed to those who settled down to make their consumption the main business of what little life remained to them.
But it was not dark. The outer Lokworm had here its companions, who writhed about the walls in far too lifelike fashion. And though parts of those gleaming runnels of light had darkened through want of replacement, the whole gave enough radiance so one could actually see the customers' faces after a fashion, if not what might be served in the cups, beakers, tubes, and the like placed before them.
Unlike the drinking and eating places in the more civilized (if that was the proper term) part of the port, the Diving Lokworm had no table dials to finger to produce nourishment, no robo-servers whipping about. The trays were carried by humans or aliens, none of whom had a face to be observed long without acute distaste. Some of them were noticeably female, others—well, it could be a guess. And frankly, had I been drinking the local poison, it would have stopped a second order to have the first slopped down before me by a lizardoid with two pairs of arms. Unless the drink had been more important than what I saw when I looked about me.
The lizardoid was serving three booths along the wall, and doing it most efficiently; four hands were useful. There was a very drunk party of Regillians in the first. In the second something gray, large, and warty squatted. But in the third slumped a Terran, his head supported on one hand, with the elbow of that arm planted firmly on the table top. He had on the remains of a space officer's uniform which had not been cleaned for a long time. One insignia still clung by a few loose threads to his tunic collar, but there was no house or ship badge on the breast, only a dark splotch there to show he had sometime lost that mark of respectability.
To take a man out of this stew was indeed combing the depths. On the other hand, all we really needed to clear the port was a pilot on board. I did not doubt that Eet and I together could get us out by setting automatic for the first jump. And to accept a blacklisted man—always supposing he was not a plant—was our only chance now.
"He is a pilot and a fash-smoker." Eet supplied information, some of which I did not care to hear.
Fash-smoke does not addict, but it does bring about a temporary personality change which is dangerous. And a man who indulges in it is certainly not a pilot to be relied upon. If this derelict was sniffing it now, he was to be my last choice instead of my first. The only bright thought was that fash-smoke is expensive and one who set light to the brazier to inhale it was not likely to patronize the Diving Lokworm.
"Not now," Eet answered. "He is, I believe, drinking veever—"
The cheapest beverage one could buy and enough to make a man as sick as a sudden ripple of color in the tube worm on the wall made this lounger appear. The fact that the light was a sickly green might have had something to do with his queasy expression. But he roused to pull the beaker before him into place and bend his head to catch the suck tube between his lips. And he went on drinking as we came to the side of the booth.
Perhaps he would not have been my first choice. But the stained insignia on his collar was that of a pilot and he was the only one I had sighted here. Also, he was the only humanoid with a face I would halfway trust, and Eet appeared to have singled him out.
He did not look up as I slipped into the bench across from him, but the lizard waiter slithered up and I pointed to the drinker, then raised a finger, ordering a return for my unknown boothmate. The latter glanced at me without dropping the tube from his lip hold. His brows drew together in a scowl and then he spat out his sipper and said in a slurred mumble:
"Blast! Whatever you're offering—I'm not buying."
"You are a pilot," I countered. The lizardoid had made double time to whatever sewer the drinks had been piped from and slammed down another beaker. I flipped a tenth-point credit and one of his second pair of hands clawed it out of the air so fast I never really saw it disappear.
"You're late in your reckoning." He pushed aside his first and now empty beaker, drew the second to him. "I _was_ a pilot."
"System or deep-space ticket?" I asked.
He paused, the sipper only a fraction away from his lips. "Deep space. Do you want to see it all plain and proper?" There was a sneer in his growl. "And what's it to you, anyway?"
There is this about fash-smoking—while it makes a man temporarily belligerent during indulgence, it also alters the flow of emotion so that between bouts, where rage might normally flare, one gets only a flash of weak irritation.
"A lot maybe. Want a job?"
He laughed then, seemingly in real amusement. "Again you're too late. I'm planet-rooted now."
"You offered to show your plate. That hasn't been confiscated?" I persisted.
"No. But that's just because no one cares enough to squawk. I haven't lifted for two planet years, and that's the truth. Quite a spiller tonight, aren't I? Maybe they've cooked some babble stuff into this goop." He stared down into his beaker with dim interest, as if he expected to see something floating on its turgid surface.
Then he mouthed the sipper, but with one hand he pulled at the frayed front seam of his tunic and brought out, in a shaking hand, a badly-worn case, which he dropped on the table top, not pushing it toward me, but rather as if he were indifferent to any interest of mine in its contents. I reached for it just as another ripple of light in the wall pattern gave me sight of the plate within that covering.
It had been issued to one Kano Ryzk, certified pilot for galactic service. The date of issuance was some ten years back, and his age was noted as problematical, since he had been space-born. But what did startle me was the small symbol deeply incised below his name—a symbol which certified him as a Free Trader.
From their beginnings as men who were willing to take risks outside the regular lines, which were the monopolies of the big combines, the Free Traders, loners and explorers by temperament, had become, through several centuries of space travel, more and more a race apart. They tended to look upon their ships as their home worlds, knowing no planet for any length of time, ranging out where only First-in Scouts and such explorers dared to go. In the first years they had lived on the short rations of those who snatch at the remnants of the feast the combines grew fat upon.
Not able to bid at the planet auctions when newly discovered worlds were put up for sale to those wanting their trade, they had to explore, take small gains at high risks, and hope for some trick of fate which would render a big profit. And such happened just often enough to keep them in space.
But seeing their ships as the only worlds to which they owed allegiance, they were a clannish lot, marrying among themselves when they wed at all. They had space-hung ports now, asteroids they had converted, on which they established quasi family life. But they did not contact the planet-born save for business. And to find one such as Ryzk adrift in a port—since the Free Traders cared for their own—was so unusual as to be astounding.
"It is true." He did not raise his eyes from the beaker. He must have encountered the same surprise so many times before that he was weary of it. "I didn't roll some star-stepper to get that plate."
That, too, must be true, since such plates were always carried close to a man's body. If any other besides the rightful owner had kept that plate, the information on it would be totally unreadable by now, since it had a self-erase attuned to personal chemistry.
There was no use in asking what brought a Free Trader shipless into the Diving Lokworm. To inquire might turn him so hostile I would not be able to bargain. But the very fact he was a Free Trader was a point in his favor. A broken combine man would be less likely to take to the kind of spacing we planned.
"I have a ship"—I put it bluntly now—"and I need a pilot."
"Try the Register," he mumbled and held out his hand. I closed the case and laid it on his palm. How much was the exact truth going to serve me?
"I want a man off the lists."
That did make him look at me. His pupils were large and very dark. He might not be on fash-smoke, but he was certainly under some type of mind-dampening cloud.
"You aren't." he said after a moment, "a runner."
"No." I replied. Smuggling was a paying game. However, the Guild had it sewed up so well that only someone with addled brains would try it.
"Then what are you?" His scowl was back.
"Someone who needs a pilot—" I was beginning when Eet's thought pricked me.
"We have stayed here too long. Be ready to guide him."
There was silence. I had not finished my sentence. Ryzk stared at me, but his eyes seemed unfocused, as if he did not really see me at all. Then he grunted and pushed aside the still unfinished second beaker.
"Sleepy," he muttered. "Out of here—"
"Yes," I agreed. "Come to my place." I was on his left, helping him to balance on unsteady feet, my hand slipped under his elbow to guide him. Luckily he was still enough in command of his body to walk. I could not have pulled him along, since though he was several inches shorter than I, his planet days had given him bulk of body which was largely ill-carried lard.
The lizard stepped out as if to bar our way and I felt Eet stir. Whether he planted some warning, as he seemed to have planted the desire to go in Ryzk, I do not know. But the waiter turned abruptly to the next booth, leaving us a free path to the door. And we made it out of the stink of the place without any opposition. Once in the backways of the Off-port, I tried to put on speed, but found that Ryzk, though he did keep on his feet and moving, could not be hurried. And pulling at him seemed to disturb the thought Eet had put in his mind, so I did not dare to put pressure on him. I was haunted by the feeling that we were being followed, or at least watched. Though whether our cover had been detected or we had just been marked down for prey generally by one of the lurking harpies, I did not try to deduce. Either was dangerous.
The floodlights of the port cut out the night, reducing all three moons now progressing at a stately pace over our heads to pallid ghosts of their usual brilliance. To pass the gates and cut across the apron to our ship's berth was the crucial problem. If, as I thought, the Patrol and perhaps the Guild were keeping me under surveillance, there would be a watch on the ship, even if we had lost them in town. And my scar, if I still wore it, would not stand up in the persona scanner at the final check point. Escape might depend on speed, and Ryzk did not have that.
I lingered no longer at the first check point than it took to snap down my own identity plate and Ryzk's. Somehow he had fumbled it out of hiding as we approached, some part of his bemused brain answering Eet's direction. Then I saw a chance to gain more speed. There was a luggage conveyer parked to one side, a luxury item I with my one flight bag had never seen reason to waste half a credit on. But there was need for it now.
Somehow I pushed and pulled Ryzk to it. There was a fine for using it as a passenger vehicle, but such minor points of law did not trouble me at that moment. I got him flat on it, pulled a layer of weather covering over his more obvious outlines, and planted my flight bag squarely on top to suggest that it did carry cargo. Then I punched the berth number for our ship, fed in my credit, and let it go. If Ryzk did not try to disembark en route I could be sure he would eventually arrive at the ramp of our ship.
Meanwhile Eet and I had to reach the same point by the least conspicuous and quickest route. I glanced around for some suggestions as to how to accomplish that. A tourist-class inter-system rocket ship was loading, with a mass of passengers waiting below its ramp and more stragglers headed for it. Many of the travelers were being escorted by family parties or boisterous collections of friends. I joined the tail of one such, matching my pace to keep at the end of the procession. Those I walked with were united in commiserating with a couple of men wearing Guard uniforms and apparently about to lift to an extremely disliked post on Memfors, the next planet out in this sytem, and one which had the reputation of being far from a pleasure spot.
Since most of the crowd were male, and looked like rather hard cases, I did not feel too conspicuous. And it was the best cover I saw. However, I still had to break away when we reached the rocket slot and cross to my own ship. It was during those last few paces I would be clearly seen.
I edged around the fringes of the waiting crowd, putting as many of those between me and the dark as I could, trying to be alert to any attention I might attract But as far as I could see, I might once more be enveloped in Eet's vision-defying blur.
I wanted to run, or to scuttle along under some protective shell like a pictick crab. But both of those safety devices were denied me. Now I dared not even look around as though I feared any pursuit, for wariness alone could betray me.
Ahead I saw the luggage conveyer crawling purposefully on a course which had been more of a straight line than my own. My bag had not shifted from the top, which meant, I trusted, that Ryzk had not moved. It reached the foot of the ramp well before me and stood waiting for the lifting of its burden to release it.
"Watcher—to the right—Patrol—"
Eet came alive with that warning. I did not glance in the direction he indicated.
"Is he moving in?"
"No. He took a video shot of the carrier. He has no orders to prevent take-off—just make sure you do go."
"So they can know the bait is ready and they need only set their trap. Very neat," I commented. But there was no drawing back now, and I did not fear the Patrol at this moment half as much as the Guild. After all, I had some importance to the Patrol—bait has until the moment for sacrificing it comes. Once we were off planet I had the feeling it was not going to be so easy for them to use me as they so arrogantly planned. I still had what they did not suspect I carried—the zero stone.
So I gave no sign that I knew I was under observation as I hauled Ryzk off the luggage carrier, guided him up the ramp, snapped that in, and sealed ship. I stowed my prize, such as he was, in one of the two lower-level cabins, strapped him down, taking his pilot's plate with me, and climbed with Eet to the control cabin.
There I fed Ryzk's plate into the viewer to satisfy the field law and prepared for take-off, Eet guiding me in the setting of the automatics. But I had no trip tape to feed in, which meant that once in space Ryzk would have to play his part or we would find another port only by the slim margin of chance.
IV
Since we lacked a trip tape, we could not go into hyper until Ryzk found us jump co-ordinates. So our initial thrust off world merely set us voyaging within the system itself, an added danger. While a ship in hyper cannot be traced, one system-traveling can readily be picked up. Thus, when I recovered from grav shock, I unstrapped myself and sought out my pilot, Eet making better time, as usual, down the inner stair of the ship.
Our transport, the _Wendwind_ , was not as small as a scout, though not as large as a Free Trader of the D class. She might once have been the private yacht of some Veep. If so, all luxury fittings had long since been torn out, though there were painted-over scars to suggest that my guess was correct. Later she had been on system runs as a general carrier. And her final fate had been confiscation by the Patrol for smuggling, after which she had been bought by the Salarik dealer as a speculation.
She had four cabins besides the regular crew quarters. But three of these had been knocked together for a storage hold. And one feature within attracted me, a persona-pressure sealed strongbox, something a dealer in gems could put to use.
At one time the _Wendwind_ must have mounted strictly illegal G-lasers, judging by the sealed ports and markings on decks and walls. But now she had no such protection.
Ryzk had been left in the last remaining passenger cabin. As I came in he was struggling against the grav straps, looking about him wildly.
"What—where—"
"You are in space, on a ship as pilot." I gave it to him without long explanation. "We are still in system, ready to go into hyper as soon as you can set course—"
He blinked rapidly, and oddly enough, the slack lines of his face appeared to firm, so that under the blurring of planetside indulgence you could see something of the man he had been. He stretched out his hand and laid it palm flat against the wall, as if he needed the reassurance of touch to help him believe that what I said was true.
"What ship?" His voice had lost the slur, just as his face had changed.
"Mine."
"And who are you?" His eyes narrowed as he stared up at me.
"Murdoc Jern. I am a gem buyer."
Eet made one of his sudden leaps from deck to the end of the bunk, where he squatted on his haunches, his handpaws resting on what would have been his knees had he possessed a humanoid body.
Ryzk looked from me to Eet and then back again. "All right, all right! I'll wake up sooner or later."
"Not"—I picked up the thought Eet aimed at Ryzk—"until you set us a course—"
The pilot started, then rubbed his hands across his forehead as if he could so rub away what he had heard, not through his ears, but in his mind.
"A course to where?" he asked, as one humoring some image born out of fash-smoke or veever drink.
"To quadrant 7-10-500." At least I had had plenty of time to lay plans such as these during the past weeks when I feared we would never be space-borne. The sooner we began to earn our way the better. And I had Vondar's experience to suggest a good beginning.
"I haven't set a course in—in—" His voice trailed off. Once more he put his hand to the ship's wall. "This is—this is a ship! I'm not dreaming it!"
"It is a ship. Can you get us into hyper now?" I allowed some of my impatience to show.
He pulled himself out of his bunk, moving unsteadily at first. But perhaps the feel of a ship about him was a tonic, for by the time he reached the core ladder to the control cabin he had picked up speed, and he swung up that with ease. Nor did he wait to be shown the pilot's seat, but crossed to sit there, giving quick, practiced looks to the control board.
"Quadrant 7-10-500—" It was not a question but a repetition, as if it were a key to unlock old knowledge. "Fathfar sector—"
Perhaps I had done far better than I had hoped when I had picked up a planeted Free Trader. A pilot for one of the usual lines would not have known the fringes of the travel lanes which must be my hunting trails now.
Ryzk was pushing buttons, first a little slowly, then picking up speed and sureness, until a series of equations flashed on the small map screen to his left. He studied those, made a correction or two with more buttons, and then spoke the usual warning—"Hyper."
Having seen that he did seem to know what he was doing, I had already retired to the second swing chair in the cabin, Eet curled up tightly against me, ready for that sickening twist which would signal our snap into the hyper space of galactic travel. Though I had been through it before, it had been mostly on passenger flights, where there had been an issue of soothe gas into the cabin to ease one through the wrench.
The ship was silent with a silence that was oppressive as we passed into a dimension which was not ours. Ryzk pushed a little away from the board, flexing his fingers. He looked to me and those firmer underlines of his face were even more in evidence.
"You—I remember you—in the Diving Lokworm." Then his brows drew together in a frown. "You—your face is different."
I had almost forgotten the scar; it must be gone now.
"You on the run?" Ryzk shot at me.
Perhaps he was entitled to more of the truth, since he shared a ship which might prove a target were we unlucky.
"Perhaps—"
But I had no intention of spouting about the past, the secret in my gem belt, and the real reason why we might go questing off into unexplored space, seeking out uncharted stars. However, "perhaps" was certainly not an explanation which would serve me either. I would have to elaborate on it.
"I am bucking the Guild." That gave him the worst, and straight. At least he could not jump ship until we planeted again.
He stared at me. "Like trying to jump the whole nebula, eh? Optimistic, aren't you?" But if he found my admission daunting, it did not appear in any expression or hesitation in his reply. "So we get to the Fathfar sector, and when we set down—on which world by the way?—we may get a warm welcome, crisped right through by lasers!"
"We set down on Lorgal. Do you know it?"
"Lorgal? You picked that heap of sand, rock, and roasting sun for a hide-out? Why? I can give you a nice listing of more attractive places—" It was plain he did know our port. Almost I could suspect he was a plant, except that I had voiced to no one at all my selection for my first essay as a buyer. Lorgal was as grim as his few terse words had said—with hellish windstorms and a few other assorted planetside disasters into the bargain. But its natives could be persuaded to part with zorans. And I knew a place where a selection of zorans, graded as I was competent to do, could give us half a year's supply of credits for cruising expenses.
"I am not hunting a hide-hole. I am after zorans. As I told you, I buy gems."
He shrugged as if he did not believe me but was willing to go along with my story, since it did not matter to him one way or another. But I triggered out the log tape and pushed its recorder to him, setting before him the accompanying pad for his thumbprint to seal the bargain.
Ryzk examined the tape. "A year's contract? And what if I don't sign, if I reserve the right to leave ship at the first port of call? After all, I don't remember any agreement between us before I woke up in this spinner of yours."
"And how long would it take you to find another ship off Lorgal?"
"And how do you know I'll set you down there in the first place? Lorgal is about the worst choice in the Fathfar sector. I can punch out any course I please—"
"Can you?" inquired Eet.
For the second time Ryzk registered startlement. He stared now at the mutant and his gaze was anything but pleasant.
"Telepath!" He spat that out like a curse.
"And more—" I hastened to agree. "Eet has a way of getting things we want done, done."
"You say that you have the Guild after you and you want me to sign on for a year. Your first pick of a landing is a hellhole. And now this—this—"
"Partner of mine," I supplied when he seemed at a loss for the proper term.
"This partner suggests he can make me do as he wishes."
"You had better believe it."
"What do I get out of it? Ship's wages—?"
This was a fair enough protest. I was willing to concede more.
"Take Trade share—"
He stiffened. I saw his hand twitch, his fingers balled into a fist which might have been aimed at me had he not some control over his temper. But I read then his dislike for my knowledge of that fragment of his past. That I had used a Free Trader's term, offering him a Trader deal, was not to his liking at all. But he nodded.
Then he pressed his thumb on the sign pad and recited his license number and name into the recorder, formally accepting duty as pilot for one planet year, to be computed on the scale of the planet from which we had just lifted, which was a matter of four hundred days.
There was little or nothing to do while the ship was in hyper, a matter of concern on the early exploring and trading ships. For idle men caused trouble. It was usually customary for members of a ship's crew to develop hobbies or crafts to keep their minds alert, their hands busy. But if Ryzk had had such in the past, he did not produce them now.
He did, however, make systematic use of the exercise cabin, as I did also, keeping muscles needed planetside from growing flabby in the reduced gravity of space flight. And as time passed he thinned and fined down until he was a far more presentable man than the one we had steered out of the Off-port drinking den.
My own preoccupation was with the mass of records I had managed, with the reluctant assistance of the Patrol, to regain from several storage points used by Vondar Ustle. With some I was familiar, but other tapes, especially those in code, were harder. Vondar had been a rover as well as a gem merchant. He could have made a fortune had he settled down as a designer and retailer on any inner-system planet. But his nature had been attuned to wandering and he had had the restlessness of a First-in Scout.
His designing was an art beyond me, and of his knowledge of stones I had perhaps a tenth—if I was not grossly overestimating what I had been able to assimilate during the years of our master-apprentice relationship. But the tapes, which I could claim under the law as a legally appointed apprentice, were my inheritance and all I had to build a future upon. All that was reasonably certain, that is. For the quest for the source of the zero stone was a gamble on which we could not embark without a backing of credits.
I watched the viewer as I ran the tapes through, concentrating on that which I had not already absorbed in actual tutelage under Vondar. And my own state of ignorance at times depressed me dismally, leaving me to wonder if Eet had somehow moved me into this action as one moves a star against a comet in that most widely spread galactic game of chance, named for its pieces—Stars and Comets.
But I was also sure that if he had, I would never be really sure of that fact, and it was far better for my peace of mind not to delve into such speculation. To keep at my task was the prime need now and I was setting up, with many revisions, deletions, and additions, a possible itinerary for us to follow.
Lorgal had been my first choice, because of the simplicity of its primitive type of exchange barter. In my first solo deal I needed that simplicity. Though I had cut as close as I could in outfitting the _Wendwind_ , I had had to spend some of our very meager store of credits on trade goods. These now occupied less than a third of the improvised storeroom. But the major part of the wares had been selected for dealing on Lorgal.
As wandering people, traveling from one water hole to the next across a land which was for the most part volcanic rock (with some still active cones breathing smoke by day, giving forth a red glow at night), sand, wind to a punishing degree, and pallid vegetation growing in the bottom of sharp-cut gullies, the Lorgalians wanted mainly food for their too often empty bellies, and water, which for far too many days seemed to have vanished from, or rather into, their earth's crust.
I had visited there once with Vondar, and he had achieved instantaneous results with a small solar converter. Into this could be fed the scabrous leaves of the vegetation, the end product emerging as small blocks about a finger in length containing a highly nutritious food which would keep a man going for perhaps five of their dust- and wind-filled days, one of their plodding beasts for three. The machine had been simple, if bulky, and had had no parts so complicated that a nontechnically-inclined people could put it out of running order. The only trouble was that it was so large that it had to be slung between two of their beasts for transport—though that had not deterred the chieftain from welcoming it as he might have a supernatural gift from one of his demon gods.
I had found, in my more recent prowlings through supply warehouses where the residue of scout and exploration ships was turned in for resale, a similar machine which was but half the size of that we had offered before. And while I could raise the price of only two of these, I had hopes that they would more than pay for our voyage.
I knew zorans, and I also knew the market for them. They were one of those special gems whose origin was organic rather than mineral. Lorgal must once have had an extremely wet climate which supported a highly varied vegetable growth. This had vanished, perhaps quite suddenly in a series of volcanic outbreaks. Some gas or other had killed certain of those plants, and their substance was then engulfed in earth fissures which closed to apply great pressure. That, combined with the gas the plants had absorbed, wrought the changes to produce zorans.
In their natural state they were often found still in the form of a mat of crushed leaves or a barked limb, sometimes even with a crystalized insect (if you were very lucky indeed) embedded in them. But once polished and cut, they were a deep purple-blue-green through which ran streaked lines of silver or glittering gold. Or else they were a crystalline yellow (probably depending upon some variation in the plant, or in the gas which had slain it) with flecks of glittering bronze.
The chunks or veins of the stuff were regularly mined by the nomads, who, until the arrival of the first off-world traders, used it mainly to tip their spears. It could be sharpened to a needle point which, upon entering flesh, would break off, to fester and eventually kill, even though the initial wound had not been a deep one.
And during the first cutting a zoran had to be handled with gloves, since any break in the outer layer made it poisonous. Once that had been buffed away, the gems could be shaped easily, even more so by the application of heat than by a cutting tool. Then, plunged into deep freeze, they hardened completely and would not yield again to any treatment. Their cutting was thus a complicated process, but their final beauty made them prized, and even in the rough they brought excellent prices.
So it would be zorans, and from Lorgal we could lift next to Rakipur, where zorans could be sold uncut to the priests of Mankspher and the pearls of lonnex crabs bought. From there perhaps to Rohan for caberon sapphires or—But there was no use planning too far ahead. I had learned long ago that all trading was a gamble and that to concentrate on the immediate future was the best way.
Eet wandered in and out while I studied my tapes. Sometimes he sat on the table to follow with a show of interest some particular one, at other times curling up to sleep. At length Ryzk, probably for lack of something to do, also found his way to where I studied, and his casual interest gave way to genuine attention.
"Rohan," he commented when I ran through Vondar's tape on that world. "Thax Thorman had trading rights on Rohan back in 3949. He made a good thing out of it. Not sapphires, though. He was after mossilk. That was before the thrinx plague wiped out the spinners. They never did find out what started the thrinx, though Thorman had his suspicions."
"Those being?" I asked when he did not continue.
"Well, those were the days when the combines tried to make it hard for the Free Men." He gave their own name to the Free Traders. "And there were a lot of tricks pulled. Thorman bid for Rohan in a syndicate of five Free ships, and he was able to overtop the Bendix Combine for it. The Combine had the auction fixed to go their way and then a Survey referee showed up and their bribed auctioneer couldn't set the computer. So their low bid was knocked out and Thorman got his. It was a chance for him. Bendix had a good idea of what was there, and he was just speculating because he knew they were set on it.
"So—he and the other ships had about four planet years of really skimming the good stuff. Then the thrinx finished that. Wiped out three of the other captains. They had been fool enough to give credit for two years running. But Thorman never trusted Bendix and he kind of expected something might blow up. No way, of course, of proving the B people had a hand in it. Nowadays, since the Free Men have had their own confederation, combines can't pull such tricks. I've seen a couple of those sapphires. Tough to find, aren't they?"
"They wouldn't be if anyone could locate the source. What is discovered are the pieces washed down the north rivers in the spring—loose in the gravel. Been plenty of prospectors who tried to get over the Knife Ridge to hunt the blue earth holes which must be there. Most of them were never heard from again. That's taboo country in there."
"Easier to buy 'em than to hunt them, eh?"
"Sometimes. Other times it is just the opposite. We have our dangers, too." I was somewhat irked by what I thought I detected underlying his comment.
But he was already changing the subject. "We come out of hyper on the yellow signal. Where do you want to set down on Lorgal, western or eastern continent?"
"Eastern. As near the Black River line as you can make it. There is no real port, as perhaps you know."
"Been a lot of time spinning by since I was there. Things could be changed, even a port there. Black River region." He looked over my shoulder at the wall of the cabin as if a map had been video-cast there. "We'll fin down in the Big Pot, unless that has boiled over into rough land again."
The Big Pot was noted on Lorgal, a giant crater with a burned-out heart which was relatively smooth and which had been used as an improvised space port. Though we had not landed there on my one visit to Lorgal, I knew enough from what I had heard then to recognize that Ryzk had chosen the best landing the eastern continent could offer.
Though the Big Pot was off the main nomad route along the series of water holes the Black River had shrunk to, we had a one-man flitter in our tail hold. And that could scout out the nearest camp site, saving a trek over the horribly broken land, which could not be traveled on foot by any off-worlder.
I looked to the recorded time dial. It was solidly blue, which meant that the yellow signal was not too far off. Ryzk arose and stretched.
"After we come out of hyper, it will take us four color spans to get into orbit at Lorgal, then maybe one more to set down, if we are lucky. How long do we stay planetside?"
"I cannot say. Depends upon finding a tribe and setting up a talk fire. Five days, ten, a couple of weeks—"
He grimaced. "On Lorgal that is too long. But you're the owner, it's your ration supply. Only hope you can cut it shorter."
He went out to climb to the control cabin. I packed away the tapes and the viewer. I certainly shared his hope—though I knew that once I entered upon the actual trading, I would find in it the zest which it always held for me. Yet Lorgal was not a world on which one wanted to linger. And now it was for me only a means to an end, the end still lying too far ahead to visualize.
I was not long behind Ryzk in seeking the control cabin and the second seat there. While I could not second his duties, yet I wanted to watch the visa-screen as we came in. This was my first real venture, and success or failure here meant very much. Perhaps Eet was as uncertain as I, for though he curled up in his familiar position against my chest and shoulder, his mind was closed to me.
We snapped out of hyper and it was plain that Ryzk deserved so far the trust I had had to place in him, for the yellow orb was certainly Lorgal. He did not put the ship on automatic, but played with fingers on the controls, setting our course, orbiting us about that golden sphere.
As we cut into atmosphere the contours of the planet cleared. There were the huge scars of old seas, now shrunken into deep pockets in the centers of what had once been their beds, their waters bitterly salt. The continents arose on what were now plateaus, left well above the dried surface of the almost vanished seas. In a short time we could distinguish the broken chains of volcanic mountains, the river valley with lava, country in between.
And then the pockmark of the Big Pot could be seen. But as we rode our deter rockets into that promise of a halfway fair landing, I caught a startling glimpse of something else.
We sat down, waiting that one tense moment to see if it had indeed been a fair three-fin landing. Then, as there came no warning tilt of the cabin, Ryzk triggered the visa-screen, starting its circular sweep of our immediate surroundings. It was only a second before I was able to see that we were indeed not alone in the Big Pot.
There was another ship standing some distance away. It was plainly a trader-for-hire. Which meant dire competition, because Lorgal had only one marketable off-world product—zorans. And the yield in any year from one tribe was not enough to satisfy two gem merchants, not if one had to have a large profit to continue to exist. I could only wonder which one of Vondar's old rivals was now sitting by a talk fire and what he had to offer. The only slim chance which remained to me was the fact that he might not have one of the reduced-in-size converters, and that I could so outbid him.
"Company," Ryzk commented. "Trouble for you?" With that question he disassociated himself from any failure of mine. He was strictly a wage man and would get his pay, from the value of the ship if need be, if I went under.
"We shall see," was the best answer I could make as I unstrapped to go and see the flitter and make a try at finding a nomad camp.
V
My advantage lay in that I had been to Lorgal before, though then the trade responsibility had lain with Vondar, and I had only been an observer. Our success or failure now depended upon how well I remembered what I had observed. The nomads were humanoid, but not of Terran stock, so dealing with them required X-Tee techniques. Even Terrans, or Terran colonist descendants, could not themselves agree over semantics, customs, or moral standards from planet to planet, and dealing with utterly alien mores added just that much more confusion.
The small converter I selected as my best exhibit could be crowded into the flitter's tail storage section. I strapped on the voca-translator and made sure that a water supply and E-rations were to hand. Eet was already curled up inside waiting for me.
"Good luck." Ryzk stood ready to thumb open the hatch. "Be sure to keep contact beam—"
"That is one thing I will not forget!" I promised. Though we had little in common, save that we shared the same ship and some of the duties of keeping it activated, we were two of the same species on an alien world, a situation which tended to make a strong, if temporary, bond between us now.
Ryzk would monitor me all the time the flitter was away from the ship. And I knew that should disaster strike either of us the other would do what he could to aid. It was a ship law, a planet law—one never put onto actual record tape but one which had existed since the first of our breed shot into space.
My memory of my first visit to Lorgal gave me one possible site for a nomad meeting, a deep pool in the river bed which had been excavated time and time again by the wandering tribes until they were always sure of some moisture at its bottom. I set off in that direction, taking my marking from two volcanic cones.
The churned ground passing under the flitter was a nightmare of broken ridges, knife-sharp pinnacles, and pitted holes. I do not believe that even the nomads could have crossed it—not that they ever wandered far from the faint promise of water along the ancient courses of the river.
While most of the rock about the Big Pot had been of a yellow-red-brown shade, here it was gray, showing a shiny, glassy black in patches. We had planeted about midmorning and now the sun caught those gleaming surfaces to make them fountains of glare. There were more and more of these as the flitter dipped over the Black River, where even the sands were of that somber color.
Here the water pits broke the general dark with their side mounds of reddish under-surface sand, which had been laboriously dug out in the past by the few native animals or the nomads. And on the inner sides of those mounds, ringing what small deposits of moisture there might be, grew the stunted plants which were the nomads only attempts at agriculture.
They saved every seed, carrying them where they went, as another race on a more hospitable world might treasure precious stones or metal, planting them one by one in the newly-dug sides of any hole before they left. When they circled back weeks or months later, they found, if they were fortunate, a meager harvest waiting.
Judging by the height of the scrubby brush around the first two pits I dipped to inspect, the Lorgalians had not yet reached them—which meant I must fly farther east to pick up their camp.
I had seen no sign of life about that other ship as I had taken off. Nor had my course taken me close to it. However, I had noted that its flitter hatch was open and guessed that the trader was already out in the field. Time might already have defeated me.
Then the Black River curved and I saw the splotch of tents dotted about. There was movement there, and as I throttled down the flitter to lowest speed and came in for a set-down I knew I was indeed late. For the cloaked and cowled figures of the tribesmen were moving with rhythmic pacing about the circumference of their camp site, each swinging an arm to crack a long-lashed whip at the nothingness beyond, a nothingness which they believed filled with devils who must be drien away by such precautions before any ceremony or serious business could be transacted.
There was another flitter parked here. It had no distinguishing company markings, so I was not about to buck a combine man. Of course I hardly expected to find one here. The pickings, as far as they were concerned, were too small. No, whoever was ready to deal with the camp was a free lance like myself.
I set down a length from the other transport. Now I could hear the high-pitched, almost squealing chant voiced by the devil-routers. With Eet on my shoulders I plunged into dry, stinging air, and the glare of a sun against which my goggles were only part protection. That air rasped against the skin as if it were filled with invisible but very tangible particles of grit. Feeling it, one did not wonder at the long robes, the cowls, the half-masks the natives wore for protection.
As I approached the ring of devil-lashers two of the whips curled out to crack the air on either side, but I did not flinch, knowing that much of nomad custom. Had I shown any surprise or recoil, I would have labeled myself a demon in disguise and a shower of zoran-pointed spears would have followed that exposure of my true nature.
The tribesmen I passed showed no interest in me; they were concentrating on their duty of protection. I cut between two of the closed tents to a clear space where I could see the assembly the whippers were guarding.
There was a huddle of nomads, all males, of course, and so enwrapped in their robes that only the eye slits suggested that they were not just bales of grimy lakis-wool cloth. The lakises themselves, ungainly beasts with bloated bodies to store the food and water for days when there was need, perched on long, thin legs with great wide, flat feet made for desert travel. These were now folded under them, for they lay to serve as windbreaks behind their masters. Their thick necks rested across each other's bodies if they could find a neighbor to so serve them, and their disproportionately small heads had the eyes closed, as if they were all firmly asleep.
Facing this assembly was the suited and helmeted figure of one of my own race. He stood, some packages about his feet, making the Four Gestures of Greeting, which meant, considering his ease, that he had either visited such a camp before, or else had made a careful study of record tapes.
The chieftain, like everyone else in that muffled crowd, could certainly not be recognized by features, but only by his badge of office, the bloated abdomen which was the result of much prideful padding. That layer upon layer of swaddling was not simply a shield against assassination (chieftainship among the Lorgalians was based upon weapon skill, not birthright); to be fat was a sign of wealth and good fortune here. And he who produced a truely noticeable belly was a man of prestige and standing.
I could not even be sure that this was the tribe with whom Vondar had traded. Only luck might help me in that. But surely, even if it was not, they would have heard of the wonder machine he had introduced and would be the more eager to acquire one of their own.
When I had entered the gathering I had come up behind the trader. And the nomads did not stir as they sighted me. Perhaps they thought me one of the stranger's followers. I do not think he was aware of me until I stepped level with him and began my own gestures of greeting, thus signifying that he was _not_ speaking for me, but that I was on my own.
He turned his head and I saw one I knew—Ivor Akki! He had been no match for Vondar Ustle; few were. But he was certainly more than I would have chosen to contend against at the beginning of my independent career. He stared at me intently for a moment and then grinned. And that grin said that in me he saw no threat. We had fronted each other for several hours once at a Salarik bargaining, but there I had been only an onlooker, and he had been easily defeated by Vondar.
He did not pause in his ritual gestures after that one glance to assess his opposition and dismiss it. And I became as unseeing of him. We waved empty hands, pointed north, south, east, and west, to the blazing sun, the cracked, sandy earth under us, outlined symbols of three demons, and that of the lakis, a nomad, and a tent, signifying that by local custom we were devout, honest men, and had come for trade.
By right Akki had the first chance, since he was first on the scene. And I had to wait while he pulled forward several boxes, snapped them open. There was the usual small stuff, mostly plastic—some garish jewelry, some goblets which were fabulous treasure to the eye but all plastic to the touch, and a couple of sun torches. These were all make-gifts—offered to the chief. And seeing their nature I was a little relieved.
For such an array meant this was not a return visit but a first try by Akki. If he were here on spec and had not heard of Vondar's success with the food converter, I could beat him yet. And I had had this much luck, a small flag fluttering by the chieftain's tent told me—this _was_ the tribe Vondar had treated with. And I needed only tell them that I had a more easily transported machine to sweep all the zorans they had to offer out of their bags.
But if I felt triumph for a few seconds it was speedily swept away as Akki opened his last box, setting out a very familiar object and one I had not expected to see.
It was a converter, but still more reduced in size and more portable than those I had chanced upon in the warehouse, undoubtedly a later and yet further improved model. I could only hope that he had just the one and that I might halve or quarter his return by offering two.
He proceeded to demonstrate the converter before that silent, never-moving company. Then he waited.
A hairy hand with long dirty nails flipped out from under the bundle of the chieftain's robes, making a sign. And one of his followers hunched forward to unfold a strip of lakis hide on which were many loops. Each loop held a chunk of zoran and only strict control kept me standing, seemingly indifferent, where I was. Four of those unworked stones were of the crystalline type and each held an insect. It was a better display than I had ever heard of. Vondar had once taken two such stones and the realization of their value off world had seemed fabulous to me. Four—with those I would not have to worry about a year's running of the ship. I would not even have to trade at all. We could be off after the zero stone after a single sale.
Only Akki was the one to whom they were offered, and I knew very well that none of them was ever going to come to me.
He deliberated, of course—that was custom again. Then he made his choice, sweeping up the insect pieces, as well as three of the blue-green-purple stones of size large enough to cut well. What was left after his choices had been fingered seemed refuse.
Then he raised his head to grin at me again as he slipped his hoard into a travel case, clapped his hand twice on the converter, and touched the rest of the goods he had spread out, releasing them all formally.
"Tough luck," he said in Basic. "But you've been having that all along, haven't you, Jern? To expect to fill Ustle's boots—" He shook his head.
"Good fortune," I said, when I would rather have voiced disappointment and frustration. "Good fortune, smooth lifting, with a sale at the end." I gave him a trader's formal farewell.
But he made no move to leave. Instead he added the insulting wave of hand signifying among the Lorgalians a master's introduction of a follower. And that, too, I had to accept for the present, since any dispute between us must be conducted outside the camp. A flare of temper would be swift indication that a devil had entered and all trading would be under ban, lest that unchancy spirit enter into some piece of the trade goods. I was almost tempted to do just that, in order to see Akki's offerings ritually pounded into splinters, the zorans treated the same way. But though such temptation was hot in me for an instant, I withstood it. He had won by the rules, and I would be the smaller were I to defeat him so, to say nothing of destroying all thought of future trade with Lorgal not only for the two of us, but for all other off-worlders.
I could take a chance and try to find another tribe somewhere out in the stark wilderness of the continent. But to withdraw from this camp now without dealing would be a delicate matter and one I did not know quite how to handle. I might offend some local custom past mending. No, like it or not, I would have to take Akki's leavings.
They were waiting and perhaps growing impatient. My hands spun into the sign language, aided by the throaty rasping my translator made as it spoke words in their own sparse tongue.
"This"—I indicated the converter—"I have also—but larger—in the belly of my sky lakis."
Now that I had made that offer there was no turning back. In order to retain the good will of the nomads I would have to trade, or lose face. And inwardly I was aware of my own inaptitude in the whole encounter. I had made my mistake in ever entering the camp after I had seen Akki's flitter already here. The intelligent move would have been then to prospect for another clan. But I had rushed, believing my wares to be unduplicated, and so lost.
Again that hairy hand waved and two of the bundled warriors arose to tail me to the flitter, cracking their whips above us as we crossed the line kept by the lashing guards. I pulled the heavy case from where I had so hopefully wedged it. And with their aid, one protecting us from the devils, the other helping me to carry it, I brought it back to the camp.
We set it before the chieftain. Either by accident or design, it landed next to Akki's, and the difference in bulk was marked. I went through the process of proving it was indeed a food converter and then awaited the chieftain's decision.
He gestured and one of my assistants booted a lakis to its feet, the creature bubbling and complaining bitterly with guttural grunts. It came up with a splayfooted shuffle which, awkward as it looked, would take it at an unvarying pace day after day across this tormented land.
A kick on one foreknee brought it kneeling again and the two converters were set beside it. Then proceeded a demonstration to prove the inferiority of my offering. Akki's machine might be put in a luggage sling on one side of the beast, a load of other equipment on the other—while if it bore the one I had brought, it could carry nothing else.
The chieftain wriggled his fingers and a second roll of lakis hide was produced. I tensed. I had thought I would be offered Akki's leavings, but it would seem I was too pessimistic. My elation lasted, however, only until the roll was opened.
What lay within its loops were zorans right enough. But nothing to compare with those shown to Akki. Nor was I even allowed to choose from his rejects. I had to take what was offered—or else return to the ship empty-handed, with a profitless set-down to my credit, or rather discredit. So I made the best of a very bad bargain and chose. There were, naturally, no insect pieces, and only two of the more attractive yellow ones. The blues had faults and I had to examine each for flaws, taking what I could, though in the end I was certain I had hardly made expenses.
I still had the second converter, and I might just be able to contact another tribe. With that small hope, I concluded the bargain and picked up what still seemed trash compared with Akki's magnificent haul.
He was grinning again as I wrapped the pieces of my choice into a packet and stood to make the farewell gestures. All this time Eet had been as inert as if he were indeed a fur piece about my shoulders. And it was not until I had to walk away from the camp, badly defeated, that I wondered why he had not taken some part in the affair. Or had I come to lean so heavily on him that I was not able to take care of myself? As that thought hit me I was startled and alarmed. Once I had leaned upon my father, feeling secure in his wisdom and experience. Then there had been Vondar, whose knowledge had so far exceeded my own that I had been content to accept his arrangement of both our lives. Soon after disaster had broken that tie, Eet had taken over. And it would seem that I was only half a man, needing the guidance of a stronger will and mind.
I could accept that, become Eet's puppet. Or I could be willing to make my own mistakes, learn by them, hold Eet to a partnership rather than a master-servant relationship. It was up to me, and perhaps Eet wanted me to make such a choice, having deliberately left me to my own bungling today as a test, or even an object lesson as to how helpless I was when I tried to deal on my own.
"Good fortune, smooth lifting—" That was Akki mockingly echoing my farewell of minutes earlier. "Crab pearls next, Jern? Want to wager I will take the best there, too?"
He laughed, not waiting for my answer. It was as if he knew that any defiance on my part would be in the nature of a hollow boast. Instead, he tramped off to his flitter, letting me settle into mine.
I did not take off at once to follow him back to his ship. If he also expected to hunt another camp, I did not want him to follow my path—though he might put a scanner on me.
Triggering the com, I called Ryzk. "Coming in." I would not add to that. The channels of all flitter corns were the same and Akki could pick up anything I now said.
Nor did I try to contact Eet, stubbornly resolved I would leave him in mental retirement as I tried to solve my own problems.
Those problems were not going to become any lighter, I saw as I took off. There was an odd greenish-yellow cast to the sky. And the surface of the ground, wherever there was a deposit of sand, threw up whirling shapes of grit. Seconds later the very sky about us seemed to explode and the flitter was caught in a gust which even her power could not fight.
For a space we were caught in that whirlwind and I knew fear. The flitter was never meant for high altitudes, and skimming the surface beneath the worst of the wind carried with it the danger of being smashed against some escarpment. But I had little choice. And I fought grimly to hold the craft steady.
We were driven south and west, out over the dead sea bottom. And I knew bleakly that even if I did get back to the _Wendwind_ my chances of finding another tribe were finished. Such a storm as this drove them to shelter and I could spend fruitless weeks hunting them. But I was able bit by bit to fight back to the Big Pot. And when I finally entered the hatch I was so weak I slumped forward over the controls and was not really aware of anything more until Ryzk forced a mug of caff into my hands and I knew I was in the mess cabin.
"This pest hole has gone crazy!" He was drumming with his fingers on the edge of the table. "According to our instruments we are sitting over a blowhole now. We up ship, or we are blown out!"
I did not quite realize what he meant and it was not until we had spaced that he explained tersely; the readings of planet stability under the Big Pot had suddenly flared into the danger zone, and he had feared I would not get back before he would be forced to lift. That I had squeezed in by what he considered a very narrow margin he thought luck of a fabulous kind.
But that danger was not real to me, since I had not been aware of it until afterward. The realization of my trade failure was worse. I must lay better plans or lose out as badly as I would have, had we never raised from Theba.
Akki had mentioned crab pearls—which might or might not mean that his itinerary had been planned along the same course as mine. I laid out the poor results of my zoran dealing and considered them fretfully. Akki might have done two things: he might have boastfully warned me off the planet where he was going to trade (his ship had lifted, Ryzk informed me, at once upon his return), or he might just have said that out of malice to make me change my own plans.
I wondered. Eet could tell me. But straightaway I rebelled. I was not going to depend on Eet!
Where was my next-best market? I tried to recall Vondar's listings. There was—Sororis! And it was not from Ustle's notes that memory came, but from my father. Sororis had been an "exit" planet for years, that is, a very far out station in which outlaws could, if they were at the end of their resources and very desperate indeed, find refuge. It had no regular service of either passenger or trade ships, though tramps of very dubious registry would put in there now and then. The refuse of the galaxy's criminal element conjoined around the half-forgotten port and maintained themselves as best they could, or died. They were too useless for even the Guild to recruit.
However, and this was the important fact, there was a native race on Sororis, settled in the north where the off-worlders found the land too inhospitable. And they were supposed to have some formidable weapons of their own to protect themselves against raiders from the port.
The main thing was that they had a well-defined religion and god-gifts were an important part of it. To present their god with an outstanding gift was the only real means of winning status among them. Such presentations gave the donor the freedom of their city for a certain number of days.
My father had been given to telling stories, always supposedly about men he knew during his years as a Guild appraiser. I believed, however, that some concerned his own exploits as a youth. He had told of an adventure on Sororis in detail, and now I could draw upon that for a way to retrieve the Lorgal fiasco.
To the inhabitants of Sororis these chunks of zoran would be rare and strange, since they would not have seen them before. Suppose I presented the largest at the temple, then offered the rest to men who wished to make similar gifts and thus enhance their standing among their fellows? What Sororisan products might be taken in exchange I did not know. But the hero of my father's story had come away with a greenstone unheard of elsewhere. For there was this about the Sororisans—they traded fairly.
It was so wild a chance that no one but a desperate man would think of it. But the combination of my defeat by Akki and the need for asserting my independence of Eet made me consider it. And after I had finished the caff I went to the computer in the control cabin and punched the code for Sororis, wagering with myself that if I received no answer I would accept that as a meaning there was no chance of carrying through such a wild gamble.
Ryzk watched me speculatively as I waited for the computer's answer. And when, in spite of my half-hopes, a series of numbers did appear on the small screen, he read them aloud:
"Sector 5, VI—Norroute 11—Where in the name of Asta-Ivista is that? Or what?"
I was committed now. "That is where we are going." I wondered if he had heard of it. "Sororis."
VI
"Where are your beam lasers and protect screens?" Ryzk asked in the voice, I decided, one used for addressing someone whose mental balance was in doubt. He even glanced at the control board, as if expecting to see such armaments represented there. And so convincing was his question that I found myself echoing that glance—which might not have been so fruitless had the ship still carried what scars proclaimed she once had.
"If you don't have those," he continued, his logic an irritation, "you might just as well blow her tubes and end us all right here without wasting the energy to take us to Sororis—if you do know what awaits any ship crazy enough to planet there. It's a rock prison and those dumped on it will storm any ship for a way to lift off again. To set down at whatever port they do have is simply inviting take-over."
"We are not going in—that is, the ship is not." At least I had planned that far ahead, drawing on my father's very detailed account of how his "friend" had made that single visit to the planet's surface. "There is the LB. It can be fitted with a return mechanism if only one is to use it."
Ryzk looked at me. For a very long moment he did not answer, and when he did, it was obliquely.
"Even a parking orbit there would be risky. They may have a converted flitter able to try a ship raid. And who is going down and why?"
"I am—to Sornuff—" I gave the native city the best pronunciation I could, though its real twist of consonants and vowels was beyond the powers of the human tongue and larynx to produce. The Sororisans were humanoid, but they were not of Terran colony stock, not even mutated colony stock.
"The temple treasures!" His instant realization of what I had in mind told me that his Free Trader's knowledge of the planet's people was more than just surface.
"It has been done," I told him, though I was aware that I was depending perhaps too much on my father's story.
"An orbit park for Sornuff," Ryzk continued, almost as if thinking aloud, "could be polar, and so leave us well away from the entrance route for anything setting down at the real port. As for the LB, yes, there can be lift-off modifications. Only"—he shrugged—"that's a job you don't often tackle in space."
"You can do it?" I demanded. I would admit frankly that I was no mech-tech and such adjustments were beyond either my knowledge or my skill. If Ryzk could not provide the knowledge, then we would have to risk some other and far more dangerous way to gain Sornuff.
"I'll take a look—" He was almost grudging.
But that was all I wanted for now. Free Traders by the very nature of their lives were adept in more fields than the usual spacer. While the fleet men were almost rigorously compartmented as to their skills, the men of the irregular ships had to be able to take over some other's duties when need arose.
The LB must have been periodically overhauled or it would not have had the certification seal on its lock. But it still dated to the original fitting of the ship, and _so_ must have been intended to carry at least five passengers. Thus we were favored in so much room. And Ryzk, dismantling the control board with the ease of one well used to such problems, grunted that it was in better shape for conversion than he had supposed.
It suddenly occurred to me that, as on Lorgal, Eet had made no suggestions or comments. And that started a small nagging worry in my mind, gave me a twinge of foreboding. Had Eet read in my mind my decision for independence? If so, had he some measure of foreknowledge? For never yet had I been able to discover the limit of his esper powers. Whenever I thought I knew, he produced something new, as he had on Theba. So, possessing foreknowledge, was he now preparing to allow me to run into difficulty from which he alone could extricate us, thus proving for once and for all that our association was less a partnership than one of master and servant, with Eet very much in the master's seat? He had closed his mind, offering no comments or suggestions. Nor did he now ever accompany us to the lock where Ryzk and I—I as the unhandy assistant—worked to give us possible entry to a hostile world where I had a thin chance of winning a gamble. I began to suspect he was playing a devious game, which made me more stubborn-set than ever to prove I could plan and carry through a coup which did not depend upon his powers.
On the other hand, I was willing enough to use what I had learned from Eet, even though it now irked me to admit I owed it to him. The hallucinatory disguise was so apt a tool that I systematically worked at the exercise of mind and will which produced the temporary changes. I found that by regular effort I could hold a minor alteration such as the scar I had worked so hard to produce as long as I pleased. But complete change, a totally new face for instance, came less easily. And I must labor doggedly even to produce the slurring of line which would pass me through a crowd unnoticed for a short space. It was Eet's added force which had held that before, and I despaired of ever having enough power to do it myself.
Practice, Eet had said, was the base of any advance I could make, and practice I had time for, in the privacy of my own cabin, with a mirror set up on a shelf to be my guide in success or failure.
At the back of my mind was always the hope that so disguised I might slip through Guild watch at any civilized port. Sororis might be free of their men, but if I won out with a precious cargo, I would have to reach one of the inner planets and there sell my spoil. Stones of unknown value were only offered at auction before the big merchants. Peddled elsewhere, they were suspect and could be confiscated after any informer (who got a percentage of the final sale) turned in a tip. It did not matter if they had been honestly enough acquired on some heretofore unmarked world; auction tax had not been paid on them and that made them contraband.
So I spent our voyage time both acting as an extra pair of inept hands for Ryzk and staring into a mirror trying to reflect there a face which was not that I had seen all my life.
We came out of hyper in the Sororis system with promptitude, which again testified to Ryzk's ability, leading me to wonder what had grounded him in the scum of the Off-port. There were three planets, two, dead worlds, balls of cracked rock with no atmosphere, close enough to the sun to fuse any ship finning down on them like a pot to fry its crew.
On the other hand, Sororis was a frozen world, or largely so, with only a belt of livable land, by the standards of my species, about its middle. It was covered by glaciers north and south of that, save where there were narrow fingers of open land running into that ice cover. In one of these Sornuff was supposed to exist, well away from the outcast settlement about the port.
Ryzk, whom I left at the controls, set up his hold orbit to the north while I packed into the LB what I judged I would need for my visit to the ice-bound city. Co-ordinates would be fed to the director, and that, too, was Ryzk's concern. On such automatic devices would depend my safe arrival not too far from Sornuff and my eventual return to the ship, the latter being even less sure than the former.
If Ryzk's fears were realized and a high-altitude conditioned flitter from the port raised with a pilot skillful or reckless enough to attempt a take-over of the _Wendwind_ , it might be that the ship would be forced out of orbit in some evasive maneuvering during my absence. If so, I had a warning which would keep me planetside until the ship was back on a course the LB was programmed to intercept.
I checked all my gear with double care, as if I had not already checked it at least a dozen times while we were in hyper. I had a small pack containing special rations, if the local food was not to be assimilated, a translator, a mike call Ryzk would pick up if he were safely in orbit, and, of course, the stones from Lorgal. There was no weapon, not even a stunner. I could not have smuggled one on board at Theba. I could only depend upon my knowledge of personal defense until I was able to outfit myself with whatever local weapons were available.
Ryzk's voice rasped over the cabin com to say that all was clear and I picked up the pack. Eet was stretched on the bunk, apparently asleep as he had been every time I had come in recently. Was he sulking, or simply indifferent to my actions now? That small germ of worry his unexpected reaction to my bid for independence had planted in me was fast growing into a full-sized doubt of myself—one I dared not allow if I were to face the tests of my resourcefulness below.
Yet I hesitated just to walk out and leave him. Our growing rift hurt in an obscure way, and I had to hold stubbornly to my purpose to keep from surrender. Now I weakened to the degree that I aimed a thought at him.
"I am going—" That was weakly obvious and I was ashamed I had done it.
Eet opened his eyes calmly. "Good fortune." He stretched out his head as if savoring a comfort he was not in the least desirous of leaving. "Use your hind eyes as well as the fore." He closed his own and snapped our linkage.
"Hind eyes as well as fore" made little sense, but I chewed angrily upon it as I went to the LB, setting the door seals behind me. As I lay down in the hammock I gave the eject signal to Ryzk, and nearly blacked out when the force of my partition from the ship hit.
Since I was set on automatics, using in part the LB's built-in function to seek the nearest planet when disaster struck the ship, I had nothing to do but lie and try to plan for all eventualities. There was an oddly naked feel to traveling without Eet, we had been in company for so long. And I found that my rebellion did not quite blank out that sense of loss.
Still, there was an exultation born of my reckless throwing over of all prudent warnings, trying a wholly new and dangerous venture of my own. This, too, part of me warned against. But I was not to have very long to think about anything. For the cushioning for landing came on and I knew I had made the jump to planet-side and was about to be faced by situations which would demand every bit of my attention.
The LB had set down, I discovered, in the narrow end of one of those claw-shaped valleys which cut into the ice. Perhaps the glacial covering of Sororis was now receding and these were the first signs of thaw. There was water running swiftly and steadily from the very point of the earth claw, forming a good-sized stream by the time it passed the LB. But the air was so chill that its freezing breath was a blow against the few exposed portions of my face. I snapped down the visor of my helmet as I set the LB hatch on persona lock and, taking up my pack, crunched the ice-packed sand under my space boots.
If Ryzk's reckoning had been successful I had only to go down this valley to where it joined a hand-shaped wedge from which other narrow valleys stretched away to the north and I would be in sighting distance of the walls of Sornuff. When I reached that point I must depend upon my father's tale for guidance. And now I realized he had gone into exhaustive detail in describing the country, almost as if he were trying to impress it upon my memory for some reason—though at the time it had not seemed so. But then I had listened eagerly to all his stories, while my foster brother and sister had apparently been bored and restless.
Between me and the city wall was a shrine of the ice spirit Zeeta. While she was not the principal deity of the Sororisans, she had a sizable following, and she had acted for the hero of my father's story as an intermediary with the priests of the major temples in the city. I say "she" for there was a living woman—or priestess—in that icy fane who was deemed to be the earth-bound part of the ice spirit, and was treated as a supernatural being, even differing in body from her followers.
I came to the join of "claw" and "hand" and saw indeed the walls of the city—and not too far away, the shrine of Zeeta.
My landing had been made just a little after dawn, and only now were thin beams of the hardly warm sun reaching to raise glints from the menace of the tall ice wall at my back. There was no sign of any life about the shrine and I wondered, with apprehension, if Zeeta had been, during the years since that other visitor was here, withdrawn, forsaken by those who had petitioned her here.
My worries as to that were quickly over as I came closer to the building of stone, glazed over with glistening ice. It was in the form of a cone, the tip of which had been sliced off, and it was perhaps the size of the _Wendwind_. Outside, a series of tables which were merely slabs of hewn ice as thick as my arm mounted on sturdy pillars of the same frozen substance encircled the whole truncated tower. On each of these were embedded the offerings of Zeeta's worshipers, some of them now so encased in layers of ice that they were only dark shadows, others lying on the surface with but a very thin coat of moisture solidifying over them.
Food, furs, some stalks of vegetable stuff black-blasted by frost lay there. It would seem that Zeeta never took from these supplies, only left them to become part of the growing ice blocks on which they rested.
I walked between two of these chill tables to approach the single break in the rounded wall of the shrine, a door open to the wind and cold. But I was heartened to see further proof of my father's story, a gong suspended by that portal. And I boldly raised my fist to strike it with the back of my gloved hand as lightly as I could—though the booming note which answered my tap seemed to me to reach and echo through the glacier behind.
My translator was fastened to my throat and I had rehearsed what I would say—though the story had not supplied me with any ceremonial greeting and I would have to improvise.
The echoes of the gong continued past the time I thought they would die. And when no one came to answer, I hesitated, uncertain. The fairly fresh offerings spelled occupancy of the shrine, but perhaps that was not so, and Zeeta, or her chosen counterpart, was not in residence.
I had almost made up my mind to go on when there was a flicker of movement within the dark oblong of the door. That movement became a shape which faced me.
It was as muffled as a Lorgalian. But they had appeared to have humanoid bodies covered by ordinary robes. This was as if a creature completely and tightly wound in strips or bandages which reduced it to the likeness of a larva balanced there to confront me.
The coverings, if they were strips of fabric, were crystaled with patterns of ice which had the glory of individual snowflakes and were diamond-bright when the rising sun touched them. But the body beneath was only dimly visible, having at least two lower limbs (were there any arms they were bound fast to the trunk and completely hidden), a torso, and above, a round ball for a head. On the fore of that the crystal encrustrations took the form of two great faceted eyes—at least they were ovals and set where eyes would be had the thing been truly humanoid. There were no other discernible features.
I made what I hoped would be accepted as a gesture of reverence or respect, bowing my head and holding up my hands empty and palm out. And though the thing had no visible ears, I put my plea into speech which emerged from my translator as a rising and falling series of trills, weirdly akin in some strange fashion to the gong note.
"Hail to Zeeta of the clear ice, the ice which holds forever! I seek the favor of Zeeta of the ice lands."
There was a trilling in return, though I could see that the head had no mouth to utter it.
"You are not of the blood, the bones, the flesh of those who seek Zeeta. Why do you trouble me, strange one?"
"I seek Zeeta as one who comes not empty-handed, as one who knows the honor of the Ice Maiden—" I put out my right hand now, laying on the edge of the nearest table the gift I had prepared with some thought—a thin chain of silver on which were threaded rounded lumps of rock crystal. On one of the inner worlds it had no value, but worth is relative to the surroundings and here it flashed bravely in the sunlight as if it were a string of the crystals such as adorned Zeeta's wrappings.
"You are not of the blood, the kind of my people," came her trilling in reply. She made no move to inspect my offering, nor even, as far as I could deduce, to turn her eyes to view it. "But your gift is well given. What ask you of Zeeta? Swift passage across ice and snow? Good thoughts to light your dreams?"
"I ask the word of Zeeta spoken into the ear of mighty Torg, that I may have a daughter's fair will in approaching the father."
"Torg also does not deal with men of your race, stranger. He is the Guardian and Maker of Good for those who are not of your kind."
"But if one brings gifts, is it not meet that the gift-giver be able to approach the Maker of Good to pay him homage?"
"It is our custom, but you are a stranger. Torg may not find it well to swallow what is not of his own people."
"Let Zeeta but give the foreword to those who serve Torg and then let him be the judge of my motives and needs."
"A small thing, and reasonable," was her comment. "So shall it be done."
She did turn her head then so those blazing crystal eyes were looking to the gong. And though she raised nothing to strike its surface, it suddenly trembled and the sound which boomed from it was enough to summon an army to attack.
"It is done, stranger."
Before I could give her any thanks she was gone, as suddenly as if her whole crystal-encrusted body had been a flame and some rise of wind had extinguished it. But though she vanished from my sight, I still lifted my hand in salute and spoke my thanks, lest I be thought lacking in gratitude.
As before, the gong note continued to rumble through the air about me, seemingly not wholly sound but a kind of vibration. So heralded, I began to walk to the city.
The way was not quite so far as it seemed and I came to the gates before I was too tired of trudging over the ice-hardened ground. There were people there and they, too, were strangely enough clad to rivet the attention.
Fur garments are known to many worlds where the temperature is such that the inhabitants must add to their natural covering to survive. Such as these, though, I had not seen. Judging by their appearance, animals as large as a man standing at his full height had been slain to obtain skins of shaggy, golden fur. These had not been cut and remade into conventional garb but had retained their original shape, so that the men of Sornuff displayed humanoid faces looking out of hoods designed from the animal heads and still in one piece with the rest of the hide; the paws, still firm on the limbs, they used as cover for hands and feet. Save for the showing of their faces they might well be beasts lumbering about on their hind legs.
Their faces were many shades darker than the golden fur framing them, and their eyes narrow and slitted, as if after generations of holding them so in protection against the glare of sun on snow and ice this had become a normal characteristic.
They appeared to keep no guard at their gate, but three of them, who must have been summoned by the gong, gestured to me with short crystal rods. Whether these were weapons or badges of office I did not know, but I obediently went with them, down the central street. Sornuff had been built in circular form, and its center hub was another cone temple, much larger than Zeeta's shrine.
The door into it was relatively narrow and oddly fashioned to resemble an open mouth, though above it were no other carvings to indicate the rest of a face. This was Torg's place and the test of my plan now lay before me.
I could sense no change in warmth in the large circular room into which we came. If there was any form of heating in Sornuff it was not used in Torg's temple. But the chill did not in any way seem to bother my guides or the waiting priests. Behind them was the representation of Torg, again a widely open mouth, in the wall facing the door.
"I bring a gift for Torg," I began boldly.
"You are not of the people of Torg." It was not quite a protest, but it carried a faint shadow of warning and it came from one of the priests. Over his fur he wore a collar of red metal from which hung several flat plaques, each set with a different color stone and so masively engraved in an interwined pattern that it could not be followed.
"Yet I bring a gift for the pleasures of Torg, such as perhaps not even his children of the blood have seen." I brought out the best of the zorans, a blue-green roughly oval stone which nearly filled the hollow of my hand when I had unrolled its wrappings and held it forth to the priest.
He bent his head as if he sniffed the stone, and then he shot out a pale tongue, touching its tip to the hard surface. Having to pass it through some strange test, he plucked it out of my hold and turned to face the great mouth in the wall. The zoran he gripped between the thumb and forefinger of each hand, holding it in the air at eye level.
"Behold the food of Torg, and it is good food, a welcome gift," he intoned. I heard a stir and mutter from behind me as if I had been followed into the temple by others.
"It is a welcome gift!" the other priests echoed.
Then he snapped his fingers, or appeared to do so, in an odd way. The zoran spun out and away, falling through the exact center of the waiting mouth, to vanish from sight. The ceremony over, the priest turned once more to face me.
"Stranger you are, but for one sun, one night, two suns, two nights, three suns, three nights, you have the freedom of the city of Torg and may go about such business as is yours within the gates which are under the Guardianship of Torg."
"Thanks be to Torg," I answered and bowed my head. But when I in turn faced around I found that my gift giving had indeed had an audience. There were a dozen at least of the furred people staring intently at me. And though they opened a passage, giving me a free way to the street without, one on the fringe stepped forward and laid a paw-gloved hand on my arm.
"Stranger Who Has Given to Torg." He made a title of address out of that statement. "There is one who would speak with you."
"One is welcome," I replied. "But I am indeed a stranger within your gates and have no house roof under which to speak."
"There is a house roof and it is this way." He trilled that hurriedly, glancing over his shoulder as if he feared interruption. And as it did seem that several others now coming forth from the temple were minded to join us, he kept his grasp on my arm and drew me a step or two away.
Since time was a factor in any trading I would do here, I was willing enough to go with him.
VII
He guided me down one of the side streets to a house which was a miniature copy of shrine and temple, save that the cone tip, though it had been cut away, was mounted with a single lump of stone carved with one of the intricate designs, one which it somehow bothered the eyes to study too closely.
There was no door, not even a curtain, closing the portal, but inside we faced a screen, and had to go between it and the wall for a space to enter the room beyond. Along its walls poles jutted forth to support curtains of fur which divided the outer rim of the single chamber into small nooks of privacy. Most of these were fully drawn. I could hear movement behind them but saw no one. My guide drew me to one, jerked aside the curtain, and motioned me before him into that tent.
From the wall protruded a ledge on which were more furs, as if it might serve as a bed. He waved me to a seat there, then sat, himself, at the other end, leaving a goodly expanse between us as was apparently demanded by courtesy. He came directly to the point.
"To Torg you gave a great gift, stranger."
"That is true," I said when he paused as though expecting some answer. And then I dared my trader's advance. "It is from beyond the skies."
"You come from the place of strangers?"
I thought I could detect suspicion in his voice. And I had no wish to be associated with the derelicts of the off-world settlement.
"No. I had heard of Torg from my father, many sun times ago, and it was told to me beyond the stars. My father had respect for Torg and I came with a gift as my father said must be done."
He plucked absent-mindedly at some wisps of the long fur making a ruff below his shin.
"It is said that there was another stranger who came bringing Torg a gift from the stars. And he was a generous man."
"To Torg?" I prompted when he hesitated for the second time.
"To Torg—and others." He seemed to find it difficult to put into words what he wanted very much to say. "All men want to please Torg with fine gifts. But for some men such fortune never comes."
"You are, perhaps, one of those men?" I dared again to speak plainly, though by such speech I might defeat my own ends. To my mind he wanted encouragement to state the core of the matter and I knew no other way to supply it.
"Perhaps—" he hedged. "The tale of other days is that the stranger who came carried with him not one from-beyond-the-stars wonders but several, and gave these freely to those who asked."
"Now the tale which I heard from my father was not quite akin to that," I replied. "For by my father's words the stranger gave wonders from beyond, yes. But he accepted certain things in return."
The Sororisan blinked. "Oh, aye, there was that. But what he took was token payment only, things which were not worth Torg's noting and of no meaning. Which made him one of generous spirit."
I nodded slowly. "That is surely true. And these things which were of no meaning—of what nature were they?"
"Like unto these." He slipped off the ledge to kneel on the floor, pressing at the front panel of the ledge base immediately below where he had been sitting. That swung open and he brought out a hide bag from which he shook four pieces of rough rock. I forced myself to sit quietly, making no comment. But, though I had never seen greenstone, I had seen recorder tridees enough to know that these were uncut, unpolished gems of that nature. I longed to handle them, to make sure they were unflawed and worth a trade.
"And what are those?" I asked as if I had very little interest in the display.
"Rocks which come from the foot of the great ice wall when it grows the less because the water runs from it. I have them only because—because I, too, had a tale from my father, that once there came a stranger who would give a great treasure for these."
"And no one else in Sornuff has such?"
"Perhaps—but they are of no worth. Why should a man bring them into his house for safekeeping? They have made laughter at me many times when I was a youngling because I believed in old tales and took these."
"May I see these rocks from the old story?"
"Of a surety!" He grabbed up the two largest, pushed them eagerly eagerly, with almost bruising force, into my hands. "Look! Did your tale speak also of such?"
The larger piece had a center flaw, but it could be split, I believed, to gain one medium-sized good stone and maybe two small ones. However, the second was a very good one which would need only a little cutting. And he had two other pieces, both good-sized. With such at auction I had my profit, and a bigger, more certain one than I had planned in my complicated series of tradings beginning with the zorans.
Perhaps I could do even better somewhere else in Sornuff. I remembered those other men who had moved to contact me outside the temple before my present host had hurried me off. On the other hand, if I made this sure trade I would be quicker off world. And somehow I had had an eerie sensation ever since I had left the LB that this was a planet it was better to visit as briefly as possible. There were no indications that the outlaws of the port came this far north, but I could not be sure that they did not. And should I be discovered and the LB found—No, a quick trade and a speedy retreat was as much as I dared now.
I took out my pouch and displayed the two small and inferior zorans I had brought.
"Torg might well look with favor on him who offered these."
The Sororisan lunged forward, his fur-backed hands reaching with the fingers crooked as if to snatch that treasure from me. But that I did not fear. Since I had fed Torg well this morning, I could not be touched for three days or the wrath of Torg would speedily strike down anyone trying such a blasphemous act.
"To gift Torg," the Sororisan said breathlessly. "He who did so—all fortune would be his!"
"We have shared an old tale, you and I, and have believed in it when others made laughter concerning that belief. Is this not so?"
"Stranger, it is so!"
"Then let us prove their laughter naught and bring truth to the tale. Take you these and give me your stones from the cold wall, and it shall be even as the tale said it was in the days of our fathers!"
"Yes—and yes!" He thrust at me the bag with the stones he had not yet given me, seized upon the zorans I had laid down.
"And as was true in the old tale," I added, my uneasiness flooding in now that I had achieved my purpose, "I go again into beyond-the-sky."
He hardly looked up from the stones lying on the fur.
"Yes, let it be so."
When he made no move to see me forth from his house, I stowed the bag of greenstones into the front of my weather suit and went on my own. I could not breathe freely again until I was back in the ship, and the sooner I gained that safety the better.
There was a crowd of Sororisans in the street outside, but oddly enough none of them approached me. Instead they looked to the house from which I had come, almost as if it had been told them what trade had been transacted there. Nor did any of them bar my way or try to prevent my leaving. Since I did not know how far the protection of Torg extended, I kept a wary eye to right and left as I walked (not ran as I wished) to the outer gate.
Across the fields which had been so vacant at my coming a party was advancing. Part of them wore the fur suits of the natives. But among them were two who had on a queer mixture of shabby, patched, off-world weather clothing. And I could only think they must have connection with the port. Yet I could not retreat now; I was sure I had already been sighted. My only hope was to get back to the LB with speed and raise off world.
The suited men halted as they sighted me. They were too far away for me to distinguish features within their helmets, and I was sure they could not see mine. They would only mark my off-world clothing. But that was new, in good condition, which would hint to them that I was not of the port company.
I expected them to break from their traveling companions, to cut me off, and I only hoped they were unarmed. I had been schooled by my father's orders in unarmed combat which combined the lore of more than one planet where man made a science of defending himself using only the weapons with which nature had endowed him. And I thought that if the whole party did not come at me at once I had a thin chance.
But if such an attack was in the mind of the off-worlders, they were not given a chance to put it to the test. For the furred natives closed about them and hustled them on toward the gate of the city. I thought that they might even be prisoners. Judging by the tales I had heard of the port, an inhabitant there might well give reason for retaliation by the natives.
My fast walk had become a trot by the time I passed the shrine of Zeeta and I made the best speed I could back to the LB, panting as I broke the seal and scrambled in. I snapped switches, empowering the boat to rise and latch on to the homing beam to the _Wendwind_ , and threw myself into a hammock for a take-off so ungentle that I blacked out as if a great hand had squeezed half the life out of me.
When I came groggily to my senses again, memory returned and I knew triumph. I had proved my belief in the old story right. Under the breast of my suit was what would make us independent of worry—at least for a while—once we could get it to auction.
I rendezvoused with the ship, thus proving my last worry wrong, and stripped off the weather suit and helmet, to climb to the control cabin. But before I could burst out with my news of success, I saw that Ryzk was frowning.
"They spy-beamed us—"
"What!" From a normal port such a happening might not have been too irregular. After all, a strange ship which did not set down openly but cruised in a tight orbit well away from any entrance lane would have invited a spy beam as a matter of regulation. But by all accounts Sororis had no such equipment. Its port was not defended, needed no defense.
"The port?" I demanded, still unable to believe that.
"On the contrary." For the first time in what seemed to me days, Eet made answer. "It came from the direction of the port, yes, but it was from a ship."
This startled me even more. To my knowledge only a Patroler would mount a spy beam, and that would be a Patroler of the second class, not a roving scout. The Guild, too, of course, had the reputation of having such equipment. But then again, a Guild ship carrying such would be the property of a Veep. And what would any Veep be doing on Sororis? It was a place of exile for the dregs of the criminal world.
"How long?"
"Not long enough to learn anything," Eet returned. "I saw to that. But the very fact that they did not learn will make them question. We had better get into hyper—"
"What course?" Ryzk asked.
"Lylestane."
Not only did the auction there give me a chance to sell the greenstones as quickly as possible, but Lylestane was one of the inner planets, long settled, even over-civilized, if you wish. Of course the Guild would have some connections there; they had with every world on which there was a profit to be made. But it was a well-policed world, one where law had the upper hand. And no Guild ship would dare to follow us boldly into Lylestane skies. So long as we were clear of any taint of illegality, we were, according to our past bargain with the Patrol, free to go as we would.
Ryzk punched a course with flying fingers, and then signaled a hyper entrance, as if he feared that at any moment we might feel the drag of a traction beam holding us fast. His concern was so apparent it banished most of my elation.
But that returned as I brought out the greenstones, examined them for flaws, weighed, measured, set down my minimum bids. Had I had more training, I might have attempted cutting the two smaller. But it was better to take less than to spoil the stones, and I distrusted my skill. I had cut gems, but only inferior stones, suitable for practice.
The largest piece would cut into three, and the next make one flawless one. The other two might provide four stones. Not of the first class. But, because greenstone was so rare, even second- and third-quality stones would find eager bidders.
I had been to auctions on Baltis and Amon with Vondar, though I had never visited the more famous one of Lylestane. Only two planet years ago one of Vondar's friends, whom I knew, had accepted the position of appraiser there, and I did not doubt that he would remember me and be prepared to steer me through the local legalities to offer my stones. He might even suggest a private buyer or two to be warned that such were up for sale. I dreamed my dreams and spun my fantasies, turning the stones around in my fingers and thinking I had redeemed my stupidity on Lorgal.
But when we had set down on Lylestane, being relegated to a far corner of the teeming port, I suddenly realized that coming to such as a spectator, with Vondar responsible for sales and myself merely acting as a combination recording clerk and bodyguard, was far different from this. Alone—For the first time I was almost willing to ask Eet's advice again. Only the need to reassure myself that I could if I wished deal for and by my lone kept me from that plea. But as I put on the best of my limited wardrobe—inner-planet men are apt to dress by station and judge a man by the covering on his back—the mutant sought me out.
"I go with you—" Eet sat on my bunk. But when I turned to face him I saw him become indistinct, hazy, and when the outlines of his person again sharpened I did not see Eet, but rather a pookha. On this world such a pet would indeed be a status symbol.
Nor was I ready to say no. I needed that extra feeling of confidence Eet would supply by just riding on my shoulder. I went out, to meet Ryzk in the corridor.
"Going planetside?" I asked.
He shook his head. "Not here. The Off-port is too rich for anyone less than a combine mate. This air's too thick for me. I'll stay ramp-up. How long will you be?"
"I shall see Kafu, set up the auction entry, if he will do it, then come straight back."
"I'll seal ship. Give me the tone call." I wondered a little at his answer. To seal ship meant expectation of trouble. Yet of all the worlds we might have visited we had the least to fear from violence here.
There were hire flitters in the lanes down-field and I climbed into the nearest, dropping in one of my now very few credit pieces and so engaging it for the rest of the day. At Kafu's name it took off, flying one of the low lanes toward the heart of the city.
Lylestane was so long a settled world that for the most part its four continents were great cities. But for some reason the inhabitants had no liking for building very high in the air. None of the structures stood more than a dozen stories high—though underground each went down level by level deep under the surface.
The robo-flitter set down without a jar on a rooftop and then flipped out an occupied sign and trundled off to a waiting zone. I crossed, to repeat Kafu's name into the disk beside the grav shaft, and received a voiced direction in return:
"Fourth level, second crossing, sixth door."
The grav float was well occupied, mostly by men in the foppish inner-planet dress, wherein even those of lower rank went with laced, puffed, tagged tunics. To my frontier-trained eyes they seemed more ridiculous than in fashion. And my own plain tunic and cropped hair attracted sideways eyeing until I began to wish I had applied some of the hallucinatory arts at least to cloud my appearance.
Fourth level down beneath the ground gave Kafu's standing as one of reasonably high rank. Not that of a Veep, who would have a windowed room or series of rooms above surface, but not down to the two- and three-mile depth of an underling.
I found the second crossing and stopped at the sixth door. There was an announce com screwed in its surface, a pick-up visa-plate above it—a one-way visa-plate which would allow the inhabitant to see me but not reveal himself in return.
I fingered the com to on, saw the visa-plate come to life.
"Murdoc Jern," I said, "assistant to Vondar Ustle."
The wait before any answer came was so long I began to wonder if perhaps Kafu was out. Then there did come a muffled response from the com.
"Leave to enter." The barrier rolled back to let me into a room in vivid contrast to the stone-walled Sororisan house where I had done my last trading.
Though men went in gaudy and colorful wear, this room was in subdued and muted tones. My space boots trod springy summead moss, a living carpet of pale yellow. And along the walls it had raised longer stalks with dangling green berries which had been carefully twined and massed together to form patterns.
There were easirests, the kind which yielded to one's weight and size upon bodily contact, all covered in earth-brown. And the light diffused from the ceiling was that of the gentle sun of spring. Directly ahead of me as I came in, one of the easirests had been set by the wall where the berry stalks had been trained to frame an open space. One might have been looking out of a window, viewing miles upon miles of landscape. And this was not static but flowed after holding for a time into yet another view, and with such changes in vegetation one could well believe that the views were meant to show not just one planet but many.
In the easirest by this "window" sat Kafu. He was a Thothian by birth, below what was considered to be the norm in height for Terran stock. His very brown skin was pulled so tightly over his fragile bones that it would seem he was the victim of starvation, hardly still alive. But from the deep sockets of his prominent skull, his eyes watched me alertly.
Instead of the fripperies of Lylestane he wore the robe of his home world, somewhat primly, and it covered him from throat, a stiffened collar standing up in a frame behind his skull, to ankles, with wide sleeves coming down over his hands to the knucklebones.
Across the easirest a table level had been swung, and set out on that were flashing stones which he was not so much examining as arranging in patterns. They might be counters in some exotic game.
But he swept these together as if he intended to clear the board for business, and they disappeared into a sleeve pocket. He touched his fingers to forehead in the salute of his people.
"I see you, Murdoc Jern."
"And I, you, Kafu." The Thothians accepted no address of honor, making a virtue of an apparent humbleness which was really a very great sense of their own superiority.
"It has been many years—"
"Five." Just as I had been suddenly restless on Sororis, so this room, half alive with its careful tended growth, affected me with a desire to be done with my business and out of it.
Eet shifted weight on my shoulder and I saw, I thought, a flicker of interest in Kafu's eyes.
"You have a new companion. Murdoc Jern."
"A pookha," I returned, tamping down impatience.
"So? Very interesting. But you are thinking now that you did not come to discuss alien life forms or the passage of years. What have you to say to me?"
I was truly startled then. Kafu had thrown aside custom in coming so quickly to the point. Nor had he offered me a seat or refreshment, or gone through any of the forms always used. I did not know whether I faced veiled hostility, or something else. But that I was not received with any desire to please I did know.
And I decided that such an approach might be met by me with its equal in curtness.
"I have gems for auction."
Kafu's hands came up in a gesture which served his race for that repudiation mine signified by a shake of the head.
"You have nothing to sell, Murdoc Jern."
"No? What of these?" I did not advance to spill the greenstones onto his lap table as I might have done had his attitude been welcoming, but held the best on the palm of my hand in the full light of the room. And I saw that that light had special properties—no false, doctored, or flawed stone could reveal aught but its imperfections in that glow. That my greenstones would pass this first test I did not doubt.
"You have nothing to sell. Murdoc Jern. Here or with any of the legally established auctions or merchants."
"Why?" His calmness carried conviction. It was not in such a man as Kafu to use a lie to influence a sale. If he said no sale, that was true and I was going to find every legitimate market closed to me. But the magnitude of such a blow had not yet sunk in, and as yet I only wanted an answer.
"You have been listed as unreliable by the authorities," he told me then.
"The lister?" I clung desperately to that one way of possible clearance. Had my detractor a name, I could legally demand a public hearing, always supposing I could raise the fees to cover it.
"From off world. The name is Vondar Ustle."
"But—he is dead! He was my master and he is dead!"
"Just so," Kafu agreed. "It was done in his name, under his estate seal."
This meant I had no way of fighting it. At least not now, and maybe never, unless I raised the astronomical fees of those legal experts who would be able to fight through perhaps more than one planet's courts.
Listed, I had no hope of dealing with any reputable merchant. And Kafu said I had been listed in the name of a dead man. By whom, and for what purpose? The Patrol, still wishing to use me in some game for the source of the zero stones? Or the Guild? The zero stone—I had not really thought of it for days; I had been too intent on trying my trade again. But perhaps it was like a poison seeping in to disrupt my whole life.
"It is a pity. They look like fine stones—" Kafu continued.
I slapped the gems back in their bag, stowing it inside my tunic. Then I bowed with what outward impassiveness I could summon.
"I beg the Gentle Homo's pardon for troubling him with this matter."
Kafu made another small gesture. "You have some powerful enemy, Murdoc Jern. It would be best for you to walk very softly and look into the shadows."
"If I go walking at all," I muttered and bowed again, somehow getting myself out of that room where all my triumph had been crushed into nothingness.
This was bottom. I would lose the ship now, since I could not pay field fees and it would be attached by the port authorities. I had a small fortune in gems I could not legally sell.
Legally—
"This may be what they wish." Eet followed my thoughts.
"Yes, but when there is only one road left, that is the one you walk," I told him grimly.
VIII
On some worlds I might have moved into the shadowy places with greater ease than I could on Lylestane. I did not know any contacts here. Yet it seemed to me when I had a moment to think that there had been something in Kafu's talk with me—perhaps a small hint—
What had he said? "You have nothing to sell with any of the legally established merchants or auctions—" Had he or had he not stressed that word "legally"? And was he so trying to bait me into an illegal act which would bring him an informer's cut of what I now carried? With a lesser man than Kafu my suspicions might be true. But I believed that the Thothian would not lend his name and reputation to any such murky game. Vondar had considered Kafu one of those he could trust and I knew there had been an old and deep friendship between my late master and the little brown man. Did some small feeling of friendliness born of that lap over to me, so that he had been subtly trying to give me a lead? Or was I now fishing so desperately for anything which might save me that I was letting my imagination rule my common sense?
"Not so—" For the second time Eet interrupted my train of thought. "You are right in supposing he had friendly feelings for you. But there was such in that room that he could not express them—"
"A spy snoop?"
"A pick-up of some sort," Eet returned. "I am not as well attuned to such when they are born of machines rather than the mind. But while this Kafu spoke for more than your ears alone, his thoughts followed different paths, and they were thoughts of regret that he must do this thing. What does the name Tacktile mean to you?"
"Tacktile?" I repeated, speculating now as to why Kafu had been under observation and who had set the spy snoop. My only solution was that the Patrol was not done with me and were bringing pressure to bear so that I would agree to the scheme their man had outlined when he offered me a pilot of their choosing.
"Yes—yes!" Eet was impatient now. "But the past does not matter at this moment—it is the future. Who is Tacktile?"
"I do not know. Why?"
"The name was foremost in this Kafu's mind when he hinted of an illegal sale. And there was a dim picture there also of a building with a sharply pointed roof. But of that I could see little and it was gone in an instant. Kafu has rudimentary esper powers and he felt the mind-touch. Luckily he believed it some refinement of the spy snoop and did not suspect us."
_Us?_ Was Eet trying to flatter me?
"He had a crude shield," the mutant continued.
"Enough of a one to muddle reception when I did not have time to work on him. But this Tacktile, I believe, would be of benefit to you now."
"If he is an IGB—a buyer of illegal gems—he might just be the bait in someone's trap."
"No, I think not. For Kafu saw in him a solution for you but no way to make that clear. And he is on this planet."
"Which is helpful," I returned bitterly, "since I lack the years it could take to run him down on name alone. This is one of the most densely populated worlds in the inner systems."
"True. But if a man such as Kafu saw this Tacktile as your aid, then he would be known to other gem dealers also, would he not? And I would suggest—"
But this time I was ahead of him. "I make the rounds, not accepting Kafu's word that I am listed. While you try to mind-pick those I meet."
It might just work, though I must depend upon Eet's gifts and not my own this time. However, there was also the thin chance that some one of the minor merchants might take a chance at an undercounter sale when they saw the quality of the stones I had to offer. And I decided to begin with these smaller men.
Evening was close when I had finished that round of disappointing refusals. Disappointing, that is, on the surface. For though some of those I had visited looked with greed on what I had to offer, all of them repeated the formula that I was listed and there was no deal. Only Eet had done his picking of minds, and as I sat in the ship's cabin again, very tired, I was not quite so discouraged as I might have been, for we knew now who Tacktile was and that he was right here in the Off-port.
As my father had done, so did Tacktile here—he operated a hock-lock for spacers wherein those who had tasted too deeply of the pleasures of the Off-port parted with small portable treasures in return for enough either to hit the gaming tables unsuccessfully again or to eat until they shipped out.
Being a hock-lock, he undoubtedly had dealings with the Guild, no matter how well policed his establishment might be. But, and this was both strange and significant, he was an alien from Warlock, a male Wyvern, which was queer. Having for some reason fled that matriarchy and reached Lylestane, he kept his own planet's citizenship and had some contact with it still which the Patrol did not challenge. Thus his holding was almost a quasi consulate for the world of his birth. His relationship with the female rulers of Warlock no one understood, but he was able to handle some off-world matters for them and was given a semidiplomatic status here which allowed him the privilege of breaking minor laws.
Tacktile was not his right name, but a human approximation of the sounds of his clacking speech—for audible speech was used by the males of Warlock while the females were telepathic.
"Well"—Ryzk faced me—"what luck?"
There was no reason to keep the worst from him. And I did not think he would jump ship here in a port where he had already decided he could not even afford to visit the spacer's resorts.
"Bad. I am listed. No merchant will buy."
"So? Do we move out now or in the morning?" He leaned back against the wall of the cabin. "I don't have anything to be attached. And I can always try the labor exchange." His tone was dry and what lay behind it was the dull despair of any planet-bound spacer.
"We do nothing—until I make one more visit—to-night." Time, as it had been since the start of our venture, was our enemy. We must raise our port fees in a twenty-four hour period or we would have the ship base-locked and confiscated.
"But not," I continued, "as Murdoc Jern." For I had this one small thread of hope left. If I were listed and suspect, then this ship and its crew of two—for Eet might well be overlooked as a factor in our company—would be watched and known. I would have to go in disguise. And already I was working out how that might be done.
"Dark first, then the port passenger section—" I thought out loud. Ryzk shook his head.
"You'll never make it. Even a Guild runner could be picked up here. That entrance is the focus of every scanner in the place. They screen out all the undesirables when they are funneled through at landing."
"I shall chance it." But I did not tell him how. My attempts at Eet's art were still a secret. And all the advantages of any secret lie in the fact that it is not shared.
We ate and Ryzk went back to his own cabin—I think to consider gloomily what appeared to be a black future. That he had any faith in me was now improbable. And I could not be sure he was not right.
But I set up the mirror in my cabin and sat before it. Nothing as simple as a scar now. I must somehow put on another face. I had already altered my clothing, taking off my good tunic and donning instead the worn coveralls of an undercrew man to a tramp freighter.
Now I concentrated on my reflection. What I had set up as a model was a small tri-dee picture. I could not hope to make my copy perfect, but if I could only create a partial illusion—
It required every bit of my energy, and I was shaking with sheer fatigue when I could see the new face. I had the slightly greenish skin of a Zorastian, plus the large eyes, the show of fanged side teeth under tight-stretched, very thin, and near colorless lips. If I could hold this, no watcher could identify me as Murdoc Jern.
"Not perfect." I was shaken out of my survey of my new self by Eet's comment. "The usual beginner's reach for the outré. But in this case, possible, yes, entirely possible, since this is an inner planet with a big mingling of ship types."
Eet—I had turned to look—was no longer a pookha. Nor was he Eet. Instead there lay on my bunk a serpent shape with a narrow, arrow-shaped head. The kind of a life form it was I could not put name to.
There was no question that Eet was going to accompany me. I could not depend now on my limited human senses alone, and what rested on my visit to Tacktile was more important than my pride.
The reptile wound about my arm, coiled there as a massive and repulsive bracelet, its head a little upraised to view. And we were ready to go, but not openly down the ramp.
Instead I descended through the core of the ship to a hatch above the fins, and in the dark felt for the notches set on one of those supports for the convenience of repair techs. So that we hit ground in the ship's shadow.
I had Ryzk's ident disk, but hoped I would not have to show it. And luckily there was a liberty party from one of the big intersolar ships straggling across the field. As I had done when disembarking from our first port, I tailed this and we tramped in a group through the gate. Any reading on me would be reported as my own and I had the liberty of the port. But the scanners, being robos, would not report that my identity did not match my present outward appearance. Or so I hoped as I continued to tag along behind the spacers, who steered straight for the Off-port.
This was not as garish and strident as that in which I had found Ryzk—at least on the main street. I had a very short distance to go, since the sharply peaked roof of Tacktile's shop could be seen plainly from the gate. He appeared to depend upon the strange shape of his roof rather than a sign for advertisement.
That roof was so sharply slanted that it formed a very narrow angle at the top and the eaves well overhung the sides. There was an entrance door so tall it seemed narrower than it was, but no windows. The door gave easily under my touch.
Hock-locks were no mystery to me. Two counters on either side made a narrow aisle before me. Behind each were shelves along the wall, crowded with hock items, protected by a thin haze of force field. It would seem Tacktile conducted a thriving business, for there were four clerks in attendance, two on either side. One was of Terran blood, and there was a Trystian, his feathered head apparently in molt, as the fronds had a ragged appearance. The gray-skinned, warty-hided clerk nearest me I did not recognize, but beyond him was another whose very presence there was a jarring note.
In the galaxy there is an elder race, of great dignity and learning—the Zacathans, of lizard descent. These are historians, archaeologists, teachers, scholars, and never had I seen one in a mercantile following before. But there was no mistaking the race of the alien, who stood in a negligent pose against the wall, fitting the strip of reader tape in his clawed hands into a recorder.
The gray creature blinked sleepily at me, the Trystian seemed remote in some personal misery, and the Terran grinned ingratiatingly and leaned forward.
"Greetings, Gentle Homo. Your pleasure is our delight." He mouthed the customary welcome of his business. "Credits promptly to hand, no hard bargaining—we please at once!"
I wanted to deal directly with Tacktile and that was going to be a matter of some difficulty—unless the Wyvern had Guild affiliations. If that were so, I could use the knowledge of the correct codes gained from my father to make contact. But I was going to have to walk a very narrow line between discovery and complete disaster. If Tacktile was honest, or wanted to protect a standing with the Patrol, the mere showing of what I carried would lead to denunciation. If he was Guild, the source of my gems would be of interest. Either way I was ripe for betrayal and must make my deal quickly. Yet I knew well the value of what I held and was going to lose no more of the profit than I was forced to.
I gave the Terran what I hoped was a meaningful stare and out of the past I recalled what I hoped would work—unless the code had been changed.
"By the six arms and four stomachs of Saput," I mumbled, "it is pleasing I need now."
The clerk did not show any interest. He was either well schooled or wary.
"You invoke Saput, friend. Are you then late from Jangour?"
"Not so late that I am forgetful enough to wish to return. Her tears make a man remember—too much." I had now given three of the Guild code phrases which in the old days had signified an unusual haul, for the attention of the master of the shop only. They had been well drilled into me when I had stood behind just such a counter in my father's establishment.
"Yes, Saput is none too kind to off-worlders. You will find better treatment here, friend." He had placed one hand palm-down on the counter. With the other he pushed out a dish of candied bic plums, as if I must be wooed as a buyer in one of the Veep shops uptown.
I picked up the top plum, laying the smallest of the greenstones in its place. A quick flicker of eyes told him what I had done. He withdrew the dish, putting it under the counter, where I knew a small vis-com would pick up the sight for Tacktile.
"You have, friend?" he continued smoothly. I laid down one of the lesser zorans from my unhappy Lorgal trade.
"It is flawed." He gave it a quick professional examination. "But as it is the first zoran we have taken in in some time, well, we shall do our best for you. Hock or sale?"
"Sale."
"Ah, we can hock but not buy. For sale you must deal with the master. And sometimes he is not in the mood. You would do better at hock, friend. Three credits—"
I shook my head as might a stupid crewman set for a higher price. "Four credits—outright sale."
"Very well, I shall ask the master. If he says no, it will not even be hock, friend, and you will have lost all." He allowed his finger to hover over the call button set in the counter as if awaiting some change in my mind. I shook my head and with a commiserating shrug he pressed the button.
Why the elaborate byplay I did not know. Except for me there was no one else in the shop, and surely the other clerks were equally well versed in the code. The only answer must be that they feared some type of snoop ray, at least in the public portion of the shop.
A brief spark of light flashed by the button and the clerk motioned me toward the back of the shop. "Don't say you weren't warned, friend. Your stone is not enough to interest the master, and you shall lose all the way."
"I will see." I passed the other clerks, neither of whom looked at me. As I came to the end of the aisle a section of wall swung in and I was in Tacktile's office.
It did not surprise me to see the dish of sticky plums on his desk, the greenstone already laid out conspicuously in a pool of light. He raised his gargoyle head, his deep-set eyes searching me, and I was glad that he lacked that other sense given Wyvern females and could not read my thoughts.
"You have more of these?" He came directly to the point.
"Yes, and better."
"They are listed stones, with a criminal history?"
"No, received in fair trade."
He rapped his blunted talons on the desk top, almost uneasily. "What is the deal?"
"Four thousand credits, on acceptance of value."
"You are one bereft of wits, stranger. These on the open market—"
"At auction they would bring five times that amount." He did not offer me a seat, but I took the stool on the other side of the desk.
"If you want your twenty thousand, let them go at auction," he returned. "If they are indeed clean stones, there is no reason not to."
"There is a reason." I moved two fingers in a sign.
"So that is the way of it." He paused. "Four thousand—well, they can go off world. You want cash?"
I gave an inward sigh of relief. My biggest gamble had paid off—he had accepted me as a Guild runner. Now I shook my head. "Deposit at the port."
"Well, very well." Eet's words were in my mind: "He is too afraid not to be honest with us."
Tacktile pulled a recorder to him. "What name?"
"Eet," I told him. "Port credit, four thousand, to one Eet. To be delivered on a voice order repeating," and I gave him code numerals.
I had come to Lylestane with high hopes. I was getting away with a modest return of port fees and supplies, and the danger of making a contact which could alert my enemies.
Now I produced the greenstones, and the Wyvern rapidly separated them. I could tell by his examination that he had some knowledge of gems. Then he nodded and gave the final signal to the recorder.
I retraced my path through the shop and now none of the clerks noticed me. The word had been passed I was to be invisible. When I reached the outside Eet spoke.
"It might be well to drink to your good fortune at the Purple Star." And so out of the ordinary was that suggestion that I was startled into breaking stride. It would be far wiser and better to get back to the ship, to prepare for take-off and rise off world before we got into any more difficulty. Yet Eet's suggestions were, as I well knew from the past, never to be disregarded.
"Why?" I asked and kept on my way, the port lights directly ahead.
"That Zacathan has been planted in Tacktile's." Eet returned as smoothly as if he were reading it all from a tape. "He is hunting for information. Tacktile has it. The Wyvern is to meet someone at the Purple Star within the hour and it is of vast importance."
"Not to us," I denied. The last thing to do was to become involved in some murky deal, especially one with the Guild—
"Not Guild!" Eet cut into my train of thought. "Tacktile is not of the Guild, though he deals with them. This is something else again. Piracy—or Jack raiding—"
"Not for us!"
"You are listed. If the Patrol has done this, you can perhaps buy your way out with pertinent information."
"As we did before? I do not think we can play that game twice. It would have to be information worth a lot—"
"Tacktile was excited, tempted. He visualized a fortune," Eet continued. "Take me into the Purple Star and I can discover what excites him. If you are listed, what kind of future voyages can you expect? Let us buy our freedom. We are still far from seeking the zero stones."
The source of the zero stones had receded from my mind to a half-remembered dream, smothered by the ever-present need to provide us with a living. All my instincts told me that Eet proposed running us headlong into a meteor storm, but the gamble might go two ways. Supposing he could mind-read a meeting between the Wyvern and some mysterious second party—the affair must be important if the Zacathans had seen fit to plant an agent in the shop. And having a drink in a spacers' bar would add to my disguise as an alien crewman who had made a successful deal at the hock-lock.
"Back four buildings," Eet dictated. And when I turned I saw the purple five-pointed light.
It was one of the better-class drinking places and the door attendant eyed me questioningly as I entered with all the boldness I could muster. I thought he was going to bar me, but if that was so he changed his mind and stepped aside.
"Take the booth to the right under the mask of Iuta," Eet ordered. There was another beyond that but the curtain had been dropped to give its occupants privacy. I settled in and punched the robo-server on the table for the least expensive drink in the house—it was all I could afford and I did not intend to drink it anyway. The lights were dim and the occupants very mixed, but more were of Terran descent than alien. I had no sight of Tacktile. Eet moved on my arm so that his arrow head now pointed to the wall between me and the curtained booth.
"Tacktile has arrived," he announced. "Through a sliding wall panel. And his contact is already there. They are scribo-writing."
I could hear the murmur of voices and guessed that those behind me were discussing some ordinary matter while their fingers were busy with the scribos, which could communicate impervious to any snoop ray. But if their thoughts were intent upon their real business, that dodge would not hide their secrets from Eet.
"It is a Jack operation," my companion reported. "But Tacktile is turning it down. He is too wary—rightly so—the victims are Zacathans."
"Some archaeological find, then—"
"True. One of great value apparently. And this is not the first one to be so Jacked. Tacktile says the risk is too great, but the other one says it has been set up with much care. There is no Patrol ship within light-years, it will be easy. The Wyvern is holding fast, telling the other to try elsewhere. He is going now."
I raised my glass but did not sip the brew it contained.
"Where and when is the raid?"
"Co-ordinates for the where—he thought of them while talking. No when."
"No concrete proof then for the Patrol," I said sourly, and spilled most of my glass's contents on the floor.
"No," Eet agreed with me. "But we do have the coordinates and a warning to the intended victims—"
"Too risky. They might already have been raided and then what? We are caught suspiciously near a Jack raid."
"They are Zacathans," Eet reminded me. "The truth cannot be hid from them, not with one telepath contacting another."
"But you do not know when—it might be now!"
"I do not believe so. They have failed with Tacktile. They must now hunt another buyer, or they may feel they can eventually persuade him. You took a gamble on Sororis. Perhaps this is another for you, with a bigger reward at the end. Get Zacathan backing and your listing will be forgotten."
I got up and went out on the noisy street, the port my goal. In spite of my intentions it would seem that Eet could mold my future, for reason and logic were on his side. Listed, I no longer had a trade. But suppose I did manage to warn some Zacathan expedition of a Jack raid. Not only would it mean that I would gain some very powerful patrons, but the Zacathans dealt only in antiquities and the very great treasure the stranger had used to tempt Tacktile might well be zero stones!
"Just so." There was a smug satisfaction in Eet's thought. "And now I would advise a speedy rise from this far from hospitable planet."
I jogged back to the ship, wondering how Ryzk would accept this latest development. To go up against a Jack raid was no one's idea of an easy life. More often it was quick death. Only, with Zacathans involved, the odds were the least small fraction inclined to our side.
IX
Below us the ball of the planet was a sphere of Sirenean amber, not the honey-amber or the butter-amber of Terra, but ocher very lightly tinged with green. The green areas grew, assumed the markings of seas. There were no very large land masses but rather sprays of islands and archipelagoes, with only two providing possible landing sites.
Ryzk was excited. He had protested the co-ordinates we had brought back from the Purple Star, saying they were in a sector completely off any known map. Now I think all his Free Trader instinct awoke when he realized that we had homed in on an uncharted world.
We orbited with caution, but there was no trace of any city, no sign that this was anything but an empty world. However, we decided at last that the same tactics used at Sororis would be best here—that Eet and I should leave the ship in orbit and make an exploratory trip in the converted LB. And since it seemed logical that the two largest land masses were the most probable sites for any archaeological dig, I made a choice of the northern.
Dawn was the time we descended. Ryzk, having experimented with the LB, had added some refinements to his original adaptations, making it possible to switch from automatics to hand controls. He had run through the drill patiently with me until he thought I could master the craft. Though I did not have the training of a spacer pilot, I had used flitters since I was a child and the techniques of the LB were not too far from that skill.
Eet, once more in his own form, curled up on the second hammock, allowing me to navigate unhindered as we went in. As the landscape became more distinct on the view-plate I saw that its ocher color was due to trees, or rather giant, lacy growths, waving fronds with delicate trunks hardly thicker than my two fists together. They were perhaps twenty or thirty feet tall and swayed and tossed as if they were constantly swept by wind. In color they shaded from a bright rust-brown to a pale green-yellow with brighter tints of reddish tan between. And they seemed to grow uniformly across the ground, with no sign of any clearing where the LB might set down. I had no desire to crash into the growth, which might be far tougher than it looked, and I went on hand controls to cruise above it, searching vainly for some break. So untouched was that willowy expanse that I had about decided my choice of island had been wrong and that we must head south to investigate the other.
Now the fronds gave way from taller to shorter. Then there was a stretch of red sand in which the sunlight awoke points of sharp glitter. This was washed by the green waves of the sea, and such green I had only seen in the flawless surface of a fine Terran emerald.
At this point the beach was wide and in the middle of it was my first signpost, a broad blot of glassified sand blasted by deter rockets, a ship's landing place. I guided the LB past that a little along the fringe of the growth, bringing it down under the overhang of vegetation with a care of which I was rightfully proud. Unless that mark had been left by a scout, I should be able to find traces of the archaeological camp not too far away, or so I hoped.
The atmosphere was breathable without a helmet. But I took with me something Ryzk had put together. We might not be allowed lasers or stunners, but the former Free Trader had patiently created a weapon of his own, a spring gun which shot needle darts. And those darts were tipped with my contribution, made from zorans too flawed to use, cut with a jeweler's tool, and deadly.
I have used a laser and a stunner, but this, at close range, was to my mind an even deadlier weapon, and only the thought that I might have to front a Jack crew prepared me to carry it. Those in space learned long ago that the first instinct of our species, to attack that which is strange as being also dangerous, could not be allowed to influence us. And in consequence, mind blocks were set on the first explorers. Such precautions continued until those who were explorers and colonizers became inhibited against instant hostility. But there were times when we still needed arms, mainly against our own species.
The stunner with its temporary effect on the opponent was the approved weapon. The laser was strictly a war choice and outlawed for most travelers. But as a former Patrol suspect, I could not have my permit to carry either renewed for a year. I was a "pardoned" man, pardoned for an offense I never committed—something they conveniently forgot. And I had no wish to demand a permit and give them some form of control over me again.
Now that I dropped out of the LB, Eet riding on my shoulder, I was very glad Ryzk had found such an arm. Not that this seemed a hostile world. The sun was bright and warm but not burning hot. And the breeze which kept the fronds ever in play was gentle, carrying with it a scent which would have made a Salarik swoon in delight. From ground level I could see that the trunks of those fronds had smaller branches and those bent under the weight of brilliant scarlet flowers rimmed with gold and bronze. Insects buzzed thickly about these.
The soil was a mixture of red sand and a darker brown earth where the beach gave way to forested land. But I kept to the edge between sand and wood, angling along until I was opposite that patch of glass formed by the heat of the rockets at some ship's fin-down.
There I discovered what had not been visible from above, covered by the trees and vegetation—a path back into the interior of the forest. I am no scout, but elementary caution suggested that I not walk that road openly. However, I soon found that forcing a passage along parallel to the route was difficult. The clusters of flowers beat against my head and shoulders, loosing an overpowering scent, which, pleasant as it was, became a cloying, choking fog when close to the nose. That and a shower of floury, rust-yellow pollen which made the skin itch where it settled finally forced me into the path.
Though fronds had been cut down to open that way, yet the press of the thick growth had spread out overhead to again roof in the channel, providing a dusky, cooling shade. On some of the trees the clusters of flowers were gone and pods hung there, pulling the trunks well out of line with their weight.
The path ran straight, and in the ground underfoot were the marks of robo-carriers. But if the camp had been so well established, why had I not been able to sight it from the air as the LB had passed overhead? Certainly they must have cut down enough fronds to make a clearing for their bubble tents.
Suddenly the trail dipped, leaving rising banks on either side. They had not had to cut a path here, for the earth had been scraped away by their carriers to show a pavement, while the fronds growing on the bank spread to cover the cut completely.
I knelt to examine the pavement, sure that it had been set of a purpose a long time ago, that it was no fortuitous rock shelf. Thus the banks on either hand might well be walls long covered by earth.
The passage continued to deepen and narrow, growing darker and more chill as I went. I slowed my advance to a creep, trying to listen, though the constant sighing of the wind through the fronds might cover any sound.
"Eet?" Finally, out of a need for more than my own five senses, I appealed to my companion.
"Nothing—" His head was raised, swaying slowly from side to side. "This is an old place, very old. There have been men here—" Then he stopped short and I could feel his small body tense against mine.
"What is it?"
"Death smell—there is death ahead."
I had my weapon ready. "Danger for us?"
"No, not now. But death here—"
The cut had now led underground, the earth lips closing the slit above, and what lay ahead was totally dark. I had a belt beamer, but to use it might bring on us the very attention which would be danger.
"Is there anyone here?" I demanded of Eet as I halted, unwilling to enter that pocket of utter black.
"Gone," Eet told me. "But not long ago. And—no—there is a trace of life, very faint. I think someone still lives—a little—"
Eet's answer was obscure, and I did not know whether we dared go on.
"No danger to us," he flashed. "I read pain—no thoughts of anger or of waiting our coming—"
I dared then to trigger the beamer, which flashed on stone walls. The blocks had been so set together that only the faintest of lines marked their joining, with no trace of mortar at all, only a sheen on their surface, as if their natural roughness had been either polished away or given a slick coating. They were a dull red in hue, a shade unpleasantly reminiscent of blood.
As we advanced the space widened, the walls almost abruptly expanding on either side to give one the feeling of being on the verge of some vast underground chamber. But my beamer had picked up something else, a tangle of wrecked gear which had been thrown about, burned by lasers. It was as if a battle had been fought in this space.
And there were bodies—
The too-sweet scent of the flowers was gone, lost in the stomach-twisting stench of seared flesh and blood—until I wanted to reel out of that hole into the clean open.
Then I heard it, not so much a moan as a kind of hissing plaint, with that in it which I could not refuse to answer. I detoured around the worst of the shambles to a place near the wall where something had crawled, leaving a ghastly trail of splotches on the floor that glistened evilly in the beam ray.
It was a Zacathan and he had not been burned down in a surprise attack as had the others I had caught glimpses of amid the chaos of the camp. No, this was such treatment as only the most sadistic and barbaric tribe of some backward planet might have dealt a battle slave.
That he still lived was indicative of the strong bodies of his species. That he would continue to live I greatly doubted. But I would do all I could for him.
I summoned up determination enough to search through the welter of the camp until I found their medical supplies. Even these had been smashed about. In fact, the whole mess suggested either a wild hunt for something hidden or else destruction for the mere sake of wanton pillage.
One who roves space must learn a little of first aid and what I knew I applied now to the wounded Zacathan, though I had no idea of how one treated alien ills. But I did my best and left him what small comfort I could before I went to look about the chamber. To take him back to the LB I needed some form of transportation and the camp trail had the marks of robo-carriers. I had not seen any such machines among the wreckage, which might mean they were somewhere in the dark.
I found one at last, its nose smashed against the wall at the far end of that space as if it had been allowed to run on its own until the stone barrier halted it. But beside it was something else, a dark opening where stones had been taken out of the wall, piled carefully to one side.
Curiosity was strong and I pushed in through that slit and flashed the beamer. There was no mistaking the purpose of the crypt. It had been a tomb. Against the wall facing me was a projecting stone outline, still walled up. Instead of being set horizontally as might be expected of a tomb, it was vertical, so that what lay buried there must stand erect.
There were shelves, but all of them were now bare. And I could imagine that what had stood there once had been taken to the camp and was now Jack loot. I had been too late. Perhaps he who had dealt with Tacktile had not known that the raid was already a fact, or had chosen to suppress that knowledge.
I returned to the carrier. In spite of the force with which it had rammed the wall it was still operative, and I put it in low gear, so that it crawled, with a squeal of protesting metal, back to the Zacathan. Since he was both taller and heavier than I, it was an effort to load his inert body on the top of the machine. But fortunately he did not regain consciousness and I thought one of the balms Eet had suggested I employ had acted as an anesthetic.
There was no use searching the wreckage. It was very plain that the raiders had found what they came for. But the wanton smashing was something I did not understand—unless Jacks were a different breed of thief from the calmly efficient Guild.
"Can you run the carrier?" I asked Eet. It obeyed a simple set of buttons, usable, I believed, by his hand-paws. And if he could run it I would be free to act as guard. Though I thought the Jacks had taken off, there was no sense in not being on the alert.
"Easy enough." He leaped to squat behind the controls, starting the machine, though it still complained noisily.
We reached the LB without picking up any sign that the raiders had lingered here or that there were any other survivors of the archaeological party. Getting the Zacathan into the hammock of the craft was an exhausting job. But I did it at last and flipped the automatic return which would take us to the _Wendwind_.
With Ryzk's help I carried the wounded survivor to one of the lower cabins. The pilot surveyed my improvised treatment closely and at last nodded.
"Best we can do for him. These boys are tough. They walk away from crashes that would pulp one of us. What happened down there?"
I described what I had found—the opened tomb, the wreckage of the camp.
"They must have made a real find. Now there's something worth more than all your gem hunting, even if you made a major strike! Forerunner stuff—must have been," Ryzk said eagerly.
The Zacathans are the historians of the galaxy. Being exceptionally long-lived by our accounting of planet years, they have a bent for the keeping of records, the searching out of the source of legends and the archaeological support for such legends. They knew of several star-wide empires which had risen and fallen again before they themselves had come into space. But there were others about whom even the Zacathans knew very little, for the dust of time had buried deep all but the faintest hints.
When we Terrans first came into the star lanes we were young compared to many worlds. We found ruins, degenerate races close to extinction, traces over and over again of those who had proceeded us, risen to heights we had not yet dreamed of seeking, then crashed suddenly or withered slowly away. The Forerunners, the first explorers had called them. But there were many Forerunners, not just of one empire or species, and those Forerunners had Forerunners until the very thought of such lost ages could make a man's head whirl.
But Forerunner artifacts were indeed finds to make a man wealthy beyond everyday reckoning. My father had shown me a few pieces, bracelets of dark metal meant to fit arms which were not of human shape, odds and ends. He had treasured these, speculated about them, until all such interest had centered upon the zero stone. Zero stone—I had seen the ruins with the caches of these stones. Had there been any in this tomb which the Zacathans had explored? Or was this merely another branch of limitless history, having no connection with the Forerunner who had used the stones as sources of fantastic energy?
"The Jacks have it all now anyway," I observed. We had rescued a Zacathan who might well die before we could get him to any outpost of galactic civilization, that was all.
"We did not miss them by too much," Ryzk said. "A ship just took off from the south island—caught it on radar as it cut atmosphere."
So they might have set down there and used a flitter to carry out the raid—which meant they had either scouted the camp carefully or had a straight tip about it. Then what Ryzk had said reached my inner alarms.
"You picked them up—could they have picked us up in return?"
"If they were looking. Maybe they thought we were a supply ship and that's why they cut out so fast. In any case, they will not be coming back if they have what they wanted."
No, they would be too anxious to get their loot into safe hiding. Zacathans, armed with telepathic powers, did not make good enemies, and I thought that the Jacks who had pulled this raid must be very sure of a safe hiding place at some point far from any port or they would not have attempted it at all.
"Makes you think of Waystar," commented Ryzk. "Sort of job those pirates would pull."
A year earlier I would have thought Ryzk subscribing to a legend, one of the tall tales of space. But my own experience, when Eet had informed me that the Free Traders who had taken me off Tanth, apparently to save my life after Vondar's murder, had intended to deliver me at Waystar, had given credibility to the story. At least the crew of that Free Trader had believed in the port to which I had been secretly consigned.
But Ryzk's casual mention of it suddenly awoke my suspicions. I had had that near-fatal brush with one Free Trader crew who had operated on the shady fringe of the Guild. Could I now have taken on board a pilot who was also too knowing of the hidden criminal base? And was Ryzk—had he been planted?
It was Eet who saved me from speculation and suspicion which might have been crippling then.
"No. He is not what you fear. He knows of Waystar through report only."
"He"—I indicated the unconscious Zacathan—"might just as well write off his find then."
My try at re-establishing our credit had failed, unless the Zacathan lived long enough for us to get him to some port. Then perhaps the gratitude of his House might work in my favor. Perhaps a cold-blooded measuring of assistance to a fellow intelligent being. Only I was so ridden by my ever-present burden of worry that it was very much a part of my thinking—though I would not have deserted any living thing found in that plundered camp.
I appealed to Ryzk for the co-ordinates to the nearest port. But, though he searched through the computer for any clue as to where we were, he finally could only suggest return to Lylestane. We were off any chart he knew of and to try an unreckoned jump through hyper was a chance no one took, except a First-in Scout as part of his usual duty.
But we did not decide the matter, for as we were arguing it out Eet broke into our dispute to say that our passenger had regained consciousness.
"Leave it up to him," I said. "The Zacathans must have co-ordinates from some world to reach here. And if he can remember those, we can return him to his home base. Best all around—"
However, I was not at all sure that the alien, as badly wounded as he was, could guide us. Yet a return to Lylestane was for me a retracing of a way which might well lead to more and more trouble. If he died and we turned up with only his body on board, who would believe our story of the Jacked camp? It could be said that we had been responsible for the raid. My thinking was beoming more and more torturous the deeper I went into the muddle. It seemed that nothing had really gone right for me since I had taken the zero stone from its hiding place in my father's room, that each move I made, always hoping for the best, simply pushed me deeper into trouble.
Eet flashed down the ladder at a greater speed than we could make. And we found him settled by the head of the bed we had improvised for the wounded alien. The latter had his bandaged head turned a little, was watching the mutant with his one good eye. That they were conversing telepathically was clear. But their mental wave length was not mine, and when I tried to listen in, the sensation was like that of hearing a muttering of voices at the far side of the room, a low sound which did not split into meaning.
As I came from behind Eet the Zacathan looked up, his eye meeting mine.
"Zilwrich thanks you, Murdoc Jern." His thoughts had a sonorous dignity. "The little one tells me that you have the mind-touch. How is it that you came before the last flutters of my life were done?"
I answered him aloud so Ryzk could also understand, telling in as few words as possible about our overhearing of the Jack plot, and why and how we had come to the amber world.
"It is well for me that you did so, but ill for my comrades that it was not sooner." He, too, spoke Basic now. "You are right that it was a raid for the treasures we found within a tomb. It is a very rich find and a remainder of a civilization not heretofore charted. So it is worth far more than just the value of the pieces—it is worth knowledge!" And he provided that last word with such emphasis as I might accord a flawless gem. "They will sell the treasure to those collectors who value things enough to hide them for just their own delight. And the knowledge will be lost!"
"You know where they take it?" Eet asked.
"To Waystar. So it would seem that that is not a legend after all. They have one there who will buy it from them, as has been done twice lately with such loot. We have tried to find who has betrayed our work to these stit beetles, but as yet we have no knowledge. Where do you take me now?" He changed the subject with an abrupt demand.
"We have no co-ordinates from here except those for return to Lylestane. We can take you there."
"Not so!" His denial was sharp. "To do that would be to lose important time. I am hurt in body, that is true, but the body mends when the will is bent to its aid. I must not lose this trail—"
"They blasted into hyper. We cannot track them." Ryzk shook his head. "And the site of Waystar is the best-guarded secret in the galaxy."
"A mind may be blocked where there is fear of losing such a secret. But a blocked mind is also locked against needful use," returned Zilwrich. "There was one among those eaters of dung who came at the last to look about, see that nothing of value was left. His mind held what we must know—the path to Waystar."
"Oh, no!" I read enough of the thought behind his words to deny what he suggested at once. "Maybe the Fleet could blast their way in there. We cannot."
"We need not blast," corrected Zilwrich. "And the time spent on the way will be used to make our plans."
I stood up. "Give us the co-ordinates of your base world. We will set you down there and you can contact the Patrol. This is an operation for them."
"It is anything but a Patrol operation," he countered. "They would make it a Fleet matter, blast to bits any opposition. And how much would then be left of the treasure? One man, two, three, four"—he could not move his head far but somehow it was as if he had pointed to each of us in turn—"can go with more skill than an army. I shall give you only those co-ordinates."
I had opened my mouth for a firm refusal when Eet's command rang in my head. "Agree! There is an excellent reason."
And, in spite of myself, in spite of knowing that no excellent reason for such stupidity could exist, I found myself agreeing.
X
It was so wild a scheme that I suspected the Zacathan of exerting some mental influence to achieve his ends—though such an act was totally foreign to all I had ever heard of his species. And since we were committed to this folly, we would have to make plans within the framework of it. We dared not go blindly into the unknown.
To my astonishment, Ryzk appeared to accept our destination with equanimity, as if our dash into a dragon's mouth was the most natural thing in the world. But I held a session in which we pooled what we knew of Waystar. Since most was only legend and space tales, it would be of little value, a statement I made gloomily.
But Zilwrich differed. "We Zacathans are sifters of legends, and we have discovered many times that there are rich kernels of truth hidden at their cores. The tale of Waystar has existed for generations of your time, Murdoc Jern, and for two generations of ours—"
"That—that means it antedates our coming into space!" Ryzk interrupted. "But—"
"Why not?" asked the Zacathan. "There have always been those outside the law. Do you think your species alone invented raiding, crime, piracy? Do not congratulate or shame yourselves that this is so. Star empires in plenty have risen and fallen and always they had those who set their own wills and desires, lusts and envies, against the common good. It is perfectly possible that Waystar has long been a hide-out for such, and was rediscovered by some of your kind fleeing the law, who thereafter put it to the same use. Do you know those co-ordinates?" he asked Ryzk.
The pilot shook his head. "They are off any trade lane. In a 'dead' sector."
"And what better place—in a sector where only dead worlds spin about burned-out suns? A place which is avoided, since there is no life to attract it, no trade, no worlds on which living things can move without cumbersome protection which makes life a burden."
"One of those worlds could be Waystar?" I hazarded.
"No. The legend is too plain. Waystar is space-borne. Perhaps it was even once a space station, set up eons ago when the dead worlds lived and bore men who reached for the stars. If so, it has been in existence longer than our records, for those worlds have always been dead to us."
He had given us a conception of time so vast we could not measure it. Ryzk frowned.
"No station could go on functioning, even on atomics—"
"Do not be too sure even of that," Zilwrich told him. "Some of the Forerunners had machines beyond our comprehension. You have certainly heard of the Caverns of Arzor and of that Sargasso planet of Limbo where a device intended for war and left running continued to pull ships to crash on its surface for thousands of years. It is not beyond all reckoning that a space station devised by such aliens would continue to function. But also it could have been converted, by desperate men. And those criminals would thus have a possession of great value, if they could continue to hold it—something worth selling—"
"Safety!" I cut in. Though Waystar was not entirely Guild, yet surely the Guild had some ties there.
"Just so," agreed Eet. "Safety. And if they believe they have utter safety there we may be sure of two things. One, that they do have some defenses which would hold perhaps even against Fleet action, for they cannot think that the situation of their hole would never be discovered. Second, that having been so long in the state of safety, they might relax strict vigilance."
But before Eet had finished, Ryzk shook his head. "We had better believe the former. If anyone not of their kind had gotten in and out again, we would know it. A story like that would sweep the lanes. They have defenses which really work."
I called on imagination. Persona detectors, perhaps locked, not to any one personality, but rather to a state of mind, so that any invader could pass only if he were a criminal or there on business. The Guild was rumored to buy or otherwise acquire inventions which the general public did not know existed. Then they either suppressed them or exploited them with care. No, such a persona detector might be possible.
"But such could be 'jammed,'" was Eet's answer.
Ryzk, who could follow Eet's mental broadcast but not mine (which was good for us both, as I well knew), looked puzzled. I explained. And then he asked Eet:
"How could you jam it? You can't tamper with a persona beam."
"No one ever tried telepathically," returned the mutant. "If disguise can deceive the eye, and careful manipulation of sound waves, the ear, a change in mental channels can do the same for a persona detector of the type Murdoc envisioned."
"That is so," Zilwrich agreed. I must accept the verdict of the two of our company who best knew what was possible with a sixth sense so few of my own species had.
Ryzk leaned back in his seat. "Since we two do not have the right mental equipment, that lets us out. And you, and you"—he nodded to Eet and Zilwrich—"are not able to try it alone."
"Unfortunately your statement is correct," said the alien. "Limited as I now am by my body, I would be a greater hindrance than help—in person—to any such penetration. And if we wait until I am healed"—he could not move enough to shrug—"then we are already lost. For they will have disposed of what they have taken. We were under Patrol watch back there—"
I stiffened. So we had been lucky indeed in our quick descent and exit from the island world. Had we come during a Patrol visit—
"When the expedition's broadcast signal failed they must have been alerted. And since the personnel of our expedition are all listed, they will be aware of my absence. But also they have evidence of the raid. The Jacks must have foreseen this, since they have been acting on a reliable source of information. And so they will be quick to dispose of their loot."
I thought I saw one fallacy in his reasoning. "But if they have taken the loot to Waystar, and they need not fear pursuit there, then they may believe they have plenty of time to wait for a high bid on it and not be so quick to sell."
"They will sell it, probably to some resident buyer. No Jack ship will have the patience to sit on a good haul." Surprisingly Ryzk took up the argument. "They may even have a backer. Some Veep who wants the stuff for a private deal."
"Quite true," said Zilwrich. "But we must get there before the collection is dispersed, or even, Zludda forbid, broken up for the metal and gems! There was that among it—yes, I will tell you so you may know the prime importance of what we seek. There was among the pieces a star map!"
And even I who was sunk in foreboding at that moment knew a thrill at that. A star map—a chart which would give those who could decode it a chance to trace some ancient route, even the boundaries of one of the fabled empires. Such a find had never been made before. It was utterly priceless and yet its worth might not be understood by those who had stolen it.
Not be recognized for what it was—my thoughts clung to that. From it sprang a wilder idea. My father had had fame throughout the Guild for appraising finds, especially antiquities. He had had no ambition to climb to Veep status with always the fear of death from some equally ambitious rival grinning behind his shoulder. He had indeed bought out and presumedly retired when his immediate employer in the system had been eliminated. But he was so widely known that he had become an authority, borrowed at times from his Veep to assist in appraising elsewhere. And he had been noted for dealing with Forerunner treasure.
Who would be the appraiser on Waystar? He would have to be competent, trusted, undoubtedly with Guild affiliations. But supposing that a man of vast reputation turned up at Waystar fleeing the Patrol, which was a very common occupational hazard. He might make his way quietly at first, but then that very reputation would spread to the Veep who had the treasure and he might be asked for an independent report. All a series of if's, and's, but's, but still holding together with a faint logic. The only trouble was that the man who could do this was dead.
I was so intent upon my thoughts that I was only dimly aware that Ryzk had begun to say something and had been silenced by a gesture from Eet. They were all staring at me, the two who were able to follow my thoughts seemingly bemused. My father was dead, and that appeared to put a very definite end to what might have been accomplished had he been alive. It was a useless speculation to follow, yet I continued to think about the advantages my father would have had. Suppose an appraiser in good standing with the Guild when he retired, one with special knowledge of Forerunner artifacts, were to show up at Waystar, settle down without any overt approach to the Veep who had the treasure. It would very logically follow that he would be asked to inspect the loot and then—But at that point my speculation stopped short. I could not foresee action leading to the retaking of the treasure—that could only be planned after the setup on Waystar had been reconnoitered.
Must be planned! I was completely moon-dazed to build on something impossible. Hywel Jern was dead for near to three planet years now. And his death, which had undoubtedly been ordered by the Guild, would be common knowledge. His reputation, in spite of his years of retirement, was too widespread for it to be otherwise. He was _dead!_
"Reports have been wrong before." That suggestion slid easily into my thoughts before I knew Eet had fed it.
"Not in the case of executions carried out by the Guild," I retorted, aroused from my preoccupation with a plan which might have been useful had I only stood in my father's boots.
My father's boots—had that been a sly manipulation of Eet's? No, I was sensitive enough now to his insinuations to be sure that it had been born inside my own mind. When I was a child I had looked forward to being a copy of Hywel Jern. He had filled my life nearly to the exclusion of all else. I did not know until years later that my luke-warm feeling for his wife, son, and daughter must have come from the fact that I was a "duty" child, one of those babies sent from another planet for adoption by a colony family in order to vary what might become too inborn a strain. I had felt myself Jern's son, and I continued to feel that even when my foster mother disclosed the true facts after Jern's death, jealously pointing out that my "brother" Faskel was the rightful heir to Jern's shop and estate.
Hywel Jern had done as well by me as he could. I had been apprenticed to a gem buyer, a man of infinite resources and experience, and I had been given the zero stone, as well as all I could absorb of my father's teachings. He had considered me, I was fully convinced, the son of his spirit, if not of his body.
There might be some record somewhere of my true parentage; I had never cared to pursue the matter. But I thought that the same strain of aloof curiosity and restlessness which had marked Hywel Jern must also have been born into me. Given other circumstances I might well have followed him into the Guild.
So—I had wanted to be like Hywel Jern. Would it be possible for me to _be_ Jern for a period of time? The risk such an imposture would entail would be enormous. But with Eet and his esper powers—"
"I wondered," the mutant thought dryly, "when you would begin to see clearly."
"What's this all about?" Ryzk demanded with some heat. "You"—he looked almost accusingly at me—"you have some plan to get into Waystar?"
But I was answering Eet, though I did so aloud, as if to deny the very help which might be the key to the whole plan. "It is too wild. Jern is dead, they would be sure of that!"
"Who is Jern and what has his death got to do with it?" Ryzk wanted to know.
"Hywel Jern was the top appraiser for one sector Veep of the Guild, and my father." I stated the facts bleakly. "They murdered him—"
"On contract?" asked Ryzk. "If he's dead, how is he of any use to us now? Sure, I can see how an appraiser with Guild rank might get into Waystar. But—" He paused and scowled. "You got some idea of pretending to be your father? But they would know—if there was a contract on him, they'd know."
Only now I was not quite so sure of that. My father had been in retirement. True enough, he had been visited from time to time by Guild men. I had had my proof of that when I had recognized as one of those visitors the captain of the Guild ship who had ordered my questioning on the unknown world of the zero-stone caches. Jern must have been killed by Guild orders for the possession of the zero stone, which his slayers did not find. But supposing they had left a body in which they thought life extinct and my father had revived? There had been a funeral service carried out by his family. But that, too, was an old cover for a man's escape from vengeance. And on the sparsely settled frontier planet he had chosen for his home, they could not have investigated too much for fear of detection.
So, we had Hywel Jern resurrected, smuggled off world perhaps—There were many radical medical techniques—plastic surgery which could alter a man. No, that was wrong. It must be an unmistakable Hywel Jern to enter Waystar. I tried again to dismiss the plan busy fitting itself together piece by piece in my mind—utter folly, logic told me it was. But I could not. I must look like Hywel Jern. And my appearance would be baffling, for who would believe that someone would assume the appearance of a dead man, and one who had been killed by Guild orders? Such a circumstance might give me even quicker access to the Veeps on Waystar. If past rumor spoke true, there was a rivalry between the Veeps of Waystar and the center core of the Guild. The former might well receive a fugitive, one they could use, even if he were now Guild-proscribed. After all, once at their station, he would be largely a prisoner they could control utterly.
Thus—Hywel Jern, running from the Patrol. After all, I _had_ been a quarry of both sides for a while because I had the zero stone. The zero stone. My thoughts circled back to that. I had not put to any use the one I carried next to my body—not experimented to step up the _Wendwind_ 's power as Eet and I had discovered it could do. I had not even looked at it in weeks, merely felt in my belt at intervals to know I still carried it.
To dare even hint that I carried such would make me an instant target for the Guild, break the uneasy truce, if that still held, between the Patrol (who might suspect but could not be sure) and me. No, that I could not use to enter the pirate station. Back to Hywel Jern. He had never been on Waystar. Of that I was reasonably certain. So he would not have to display familiarity with any part of it. And with Eet to pick out of minds what I _should_ know—
But could I be Hywel Jern for the length of time it would—might well—take for the locating of the loot? I had held my scar-faced disguise for only hours, the alien countenance I had devised for the Lylestane venture even less. And I would have to be Hywel Jern perhaps for days, keeping up that façade at all times lest I be snooped or surprised.
"It cannot be done, not by me," I told Eet, since I knew that he, of the three facing me, was the one waiting for my decision, preparing arguments to counter it.
"You could not hold it either," I continued, "not for so long."
"There you speak the truth," he agreed.
"Then it is impossible."
"I have discovered"—Eet assumed that pontifical air which I found most irksome, which acted on me as a spur even when I was determined not to be ridden by him in any direction—"that few things, very few things, are impossible when one has all the facts and examines them carefully. You did well with the scar—for one of your limited ability—your native ability. You did even better with your alien space man. There is no reason why you cannot—"
"I cannot hold it—not for the necessary length of time!" I shot back at him, determined to find, for once and all, an answer which would satisfy my own thoughts as well as the subtle compulsion I sensed coming from both telepaths.
"That, too, can be considered," Eet returned evasively. "But now, rest is needed for our friend."
And I awoke to the fact that the Zacathan had indeed slumped on his bed. His eye was near closed and he appeared to be completely exhausted. Together with Ryzk I worked to make him as comfortable as possible and then I went to my own cabin.
I threw myself on my bunk. But I found that I could not shut off my thoughts, bent as they were, in spite of my desires, on the solving of what seemed to be the first of the insurmountable problems. So I lay staring up at the ceiling of the cabin, trying to break my problem down logically. Hywel Jern might get into Waystar. Possibly I could use Eet's form of disguise to become Hywel Jern. But the exertion of holding that would be a drain which could exhaust both of us and might not leave my mind clear enough to be as alert as I must be to cope with the dangers awaiting us in the heart of the enemies' territory.
If there was only some way to increase my power to hold the illusion without draining myself and Eet. For Eet must have freedom for the mind reading which would be the additional protection we had to have. Increase the power—just as we were able to increase the power of the Patrol scout with the zero stone. The zero stone!
My fingers sought that very small bulge in my belt. I sat up and swung my feet to the cabin floor. For the first time in weeks I unsealed that pocket and brought out the colorless, unattractive lump which was the zero stone in its unawakened phase.
Zero stone—energy, extra energy for machines, for stepping up their power. But when I strove to create the illusions, I used energy of another kind. Still it was energy. But my race had for so long been used to the idea of energy only in connection with machines that this was a new thought. I closed both my hands over the gem, so that its rough edges pressed tightly, painfully, into my flesh.
The zero stone plus a machine already alive with energy meant a heightened flow, an output which had been almost too much for the engine in the scout ship to handle. Zero stones had apparently powered the drifting derelict we had found in space, Eet and I. And it had been their energy broadcast that had activated the stone I then carried, causing it to draw us to the derelict in the first place. Just as on the unnamed planet a similar broadcast had guided us to the long-forsaken ruins where the stones' owners had left their caches.
Energy—But the idea which was in my mind was no wilder than others that had visited me lately. There was a very simple trial. Not on myself, not yet. I was wary of experimentation I might not be able to control. I looked about me hurriedly, seeing Eet curled apparently asleep, on the foot of my bunk. For a moment I hesitated—Eet? There was humor in that, and something else—the desire to see Eet for once startled out of his usual competent control over the situation.
I stared at Eet. I held the zero stone, and I thought—
The cold gem between my hands began to warm, grew hotter. And the lines of Eet's body began to dim. I dared not allow one small spark of triumph to break my concentration. The stone was afire almost past the point where I could continue to hold it. And Eet—Eet was gone! What lay on the foot of my bunk now was what his mother had been, a ship's cat.
I had to drop the stone. The pain was too intense for me to continue to hold it. Eet came to his feet in one of those quick feline movements, stretched his neck to right and left, to look along his body, and then faced me, his cat's ears flattened to his skull, his mouth open in an angry hiss.
"You see!" I was exultant.
But there was no answer to my mind-touch—nothing at all. It was not that I met the barrier which Eet used to cut off communication when he desired to retire into his own thoughts. Rather it seemed that Eet was not!
I sank down on the pull seat to stare back at the angry cat now crouched snarling, as if to spring for my throat. Could it be true that I had done more than create an illusion? It was as if Eet _was_ now a cat and not himself at all! I had indeed stepped up energy and to what disastrous point? Frantically I took the stone tightly into my seared hands, grasped it between my painful palms, and set about undoing what I had done.
No cat, I thought furiously, but Eet—Eet in his mutation from the enraged bundle of fur now facing me with anger enough, had it been larger, to tear out my life. Eet, my thoughts commanded as I fought panic and tried only to concentrate on what I _must_ do—get Eet back again.
Again the stone warmed, burned, but I held it in spite of the torment to my flesh. The furry contours of the cat dimmed, changed. Eet crouched there now, his rage even somehow heightened by the change into his rightful body. But was it truly Eet?
"Fool!" That single word, hurled at me as a laser beam might be aimed, made me relax. This was Eet.
He leaped to the table between us, stalked back and forth, lashing his ridged tail; in his fury, very feline.
"Child playing with fire," he hissed.
I began to laugh then. There had been little to amuse one in the weeks immediately behind us, but the relief of having pulled off this impossibility successfully, plus the pleasure of having at last surprised and bested Eet in his own field, made me continue to laugh helplessly, until I leaned weakly back against the wall of the cabin, unwitting of the pain in my hands.
Eet stopped his angry pacing, sat down in a feline posture (it seemed to me his cat ancestry was more notable than before) with his tail curled about him so that its tip rested on his paws. He had closed his mind tightly, but I was neither alarmed nor abashed by his attitude. I was very sure that Eet's startled reaction to transformation was only momentary and that his alert intelligence would speedily be bent to consider the possibilities of what we had learned.
I stowed the stone carefully in my belt and treated my burned hands with a soothing paste. The mutant continued to sit statue-still and I made no further attempts at mind-touch, waiting for him to make the first move.
That I had made a momentous discovery exhilarated me. At that moment nothing seemed outside my grasp. It was not only machine energy which the zero stone furthered; it could also be mental. As a cat, Eet had been silenced and, I was sure, unable by himself to break the image I had thought on him, even for his own defense. This must mean that any illusion created with the aid of the stone would have no time limit, remaining so until one thought it away.
"Entirely right." Eet came out of his sulk—or perhaps it was a deep study. His rage also seemed to have vanished. "But you were indeed playing with a fire which might have consumed us both!" And I knew that he did not mean the burns on my hands. Even so, I was not going to say that I was sorry the experiment had worked. We needed it. Hywel Jern could indeed go to Waystar and it would require no expenditure of energy to keep the illusion intact as long as he carried the zero stone.
"To take that in," remarked Eet, "is a great hazard." And his reluctance puzzled me.
"You suspect"—I thought I guessed what bothered him—"they might have one, able to pick up emanations from ours?"
"We do not know what the Guild had as their original guide to the stones. And Waystar would be an excellent stronghold for the keeping of such. But I agree that we cannot be choosers. We must take such a chance."
XI
"It must be here." Ryzk had brought us out of hyper in a very old system where the sun was an almost-dead red dwarf, the planets orbiting around it black and burned-out cinders. He indicated a small asteroid. "There is a defense shield up there. And I don't see how you are going to break through that. They must have an entrance code and anything not answering that and getting within range—" He snapped his fingers in a significant gesture of instantaneous extinction.
Zilwrich studied what showed on the small relay visa-screen we had set up in his cabin. He leaned against the back rest we had improvised, his inert head frill crumpled about his neck. But though he appeared very weak, his eye was bright, and I think that the interest in the unusual which motivated his race made him forget his wounds now.
"If I only had my equipment!" He spoke Basic with the hissing intonation of his species. "Somehow I do not believe that is a true asteroid."
"It may be a Forerunner space station. But knowing that is not going to get us in undetected," rasped Ryzk.
"We cannot all go in," I said. "We play the same game over. Eet and I shall take in the LB."
"Blasting through screens?" scoffed Ryzk. "I tell you our detect picked up emanations as strong as any on a defensive Patrol outpost. You'd be lasered out of existence quicker than one could pinch out an angk bug!"
"Suppose one dogged in a ship which did have the pass code," I suggested. "The LB is small enough not to enlarge the warn beep of such a one—"
"And when are you going to pick up a ship to dog in?" Ryzk wanted to know. "We might hang here for days—"
"I think not," Eet cut in. "If this is truly Waystar, then there will be traffic, enough to cut down days of waiting. You are the pilot. Tell us if this could be done—could the LB ride in behind another ship in that way?"
It secretly surprised me that there were some things Eet did _not_ know. Ryzk scowled, his usual prelude to concentrated thought.
"I could rig a distort combined with a weak traction beam. Cut off the power when that connected with another ship. You'd have this in your favor—those defenses may only be set for big stuff. They'd expect the Fleet to burn them out, not a one-man operation. Or they might detect and let you through. Then you'd find a welcome-guard waiting, which would probably be worse than being lasered out at first contact."
He seemed determined to paint the future as black as possible. I had only what I had learned of the zero stone to support me against the very unpleasant possibilities ahead. Yet the confidence my experiment had bred in me wavered only in the slightest degree.
In the end, Ryzk turned his Free Trader's ingenuity to more work on the LB, giving it what defenses he could devise. We could not fight, but we were now provided with distorters which would permit us to approach the blot our ship's radar told us was Waystar, and then wait for the slim chance of making a run into the enemies' most securely guarded fortress.
Meanwhile, the _Wendwind_ set down on the moon of the nearest dead planet, a ball of creviced rock so bleak and black that it should afford a good hiding place. And the co-ordinates of that temporary landing site were fed into the computer of the LB to home us if and when we left the pirate station—though Ryzk was certain we would never be back and said so frankly, demanding at last that I make a ship recording releasing him from contract and responsibility after an agreed-upon length of time. This I did, Zilwrich acting as witness.
All this did not tend to make me set about the next part of our venture with a great belief in success. I kept feeling the lump of the zero stone as a kind of talisman against all that could go wrong, too long a list of possible disasters to count.
Eet made a firm statement as we prepared our disguise.
"I choose my own form!" he said in a manner I dared not question.
We were in my cabin, for I had no wish to share the secret of the zero stone with either Ryzk or the Zacathan—though what they might think of our disguises I could not tell.
But Eet's demand was fair enough. I took the dull, apparently lifeless gem and laid it on the table between us. My own change was already thought out. But in case I needed a reminder of some details, I had something else, a vividly clear tri-dee of my father. He had never willingly allowed such to be taken, but this had belonged to my foster mother and had been the one thing I had taken, besides the zero stone, from my home when his death closed its doors to me. Why I had done so I could not have said—unless there was buried deep inside me a fragment of true esper talent, that of precognition. I had not looked at the tri-dee since the day I had lifted from that planet. Now, studying it carefully, I was very glad I had it. The face I remembered had, as usual, been hazed by time, and I found memory differed from this more exact record.
Warned by the fury of heat in the stone when I had used it on Eet, I touched it now with some care, my attention centering on the tri-dee, concentrating on the face appearing therein. I was only dimly aware that Eet crouched on the table, a clawed hand-paw joining mine in touching the jewel.
I could not be sure of the change in my outward appearance. I felt no different. But after an interval I glanced at the mirror ready for the necessary check, and indeed saw a strange face there. It was my father, yes, but in a subtle way younger than I remembered him last. But then I was using as my guide a picture taken planet years before I knew him, when he had first wed my foster mother.
There could certainly be no mistaking his sharp, almost harsh features by anyone who had ever known him. And I hoped that Eet could help me carry out the rest of the deception by mind reading and supplying me with the memories necessary to make me a passable counterfeit of a man known in Guild circles.
Eet—what had been his choice of disguise? I fully expected something such as the pookha or the reptilian form he had taken on Lylestane. But this I did not foresee. For it was no animal sitting cross-legged on the table, but a humanoid perhaps as large as a human child of five or six years.
The skin was not smooth, but covered with a short plushy fur, much like that of the pookha. On the top of the head this grew longer, into a pointed crest. Only the palms of the hands were bare of the fur, which in color was an inky black, and the skin bared there was red, as were the eyes, large and bulging a little from their sockets, the red broken only by vertical pupils. The nose had a narrow ridge of fur up and down it, giving a greater prominence to that feature. But the mouth showed only very narrow slits of lips and those as black as the fur about them.
To my knowledge I had neither seen nor heard described such a creature, and why Eet had chosen to assume this form first intrigued and then bothered me. Space-rovers were addicted to pets and one met with many oddities accompanying their masters. But this was no pet, unusual as it looked. It had the aura of an intelligent life form, one which could be termed "man."
"Just so." Eet gave his old form of agreement. "But I think you will discover that this pirate hold will have varied life forms aboard. And also this body has possibilities which may be an aid in future difficulties."
"What are you?" curiosity made me ask.
"You have no name for me," Eet returned. "This is a life form which I believe long gone from space."
He ran his red-palmed hands over his furred sides, absent-mindedly scratching his slightly protrudent middle. "You, yourselves, admit you are late-comers to the stars. Let it suffice that this is an adequate body for my present need."
I hoped Eet was right, as there was no use in arguing with him. Now I saw something else. That hand not occupied with methodical hide-scratching hovered near the zero stone—though if Eet was preparing to snatch that treasure I did not see where, in his present unclothed state, he would stow it. However, my fingers closed promptly on the gem and sealed it back in my belt. Eet was apparently not concerned, for his straying hand dropped back on his knee.
We bade good-by to Ryzk and the Zacathan. And I did not miss that Zilwrich watched Eet with an attention which might have been rooted in puzzlement but which grew into a subdued excitement, as if he recognized in that black-furred body something he knew.
Ryzk stared at us. "How long can you keep that on?" It was plain that he thought our appearances the result of some plasta change. But how he could have believed we carried such elaborate equipment with us I did not know.
"As long as necessary," I assured him and we went to board the greatly altered LB.
As we took off, forceably ejected from the parent ship by the original escape method, we aimed in the general direction of the pirate station. But Ryzk's modifications allowed us to hover in space, waiting a guide. And it was Eet in his new form who took over the controls.
How long we would have to patrol was the question. Waiting in any form is far more wearisome than any action. We spent the slowly dragging time in silence. I was trying to recall every small scrap of what my father had said about his days with the Guild. And what lay in Eet's mind I would not have tried to guess. In fact, I was far too occupied with the thought that my father had been remarkably reticent about his Guild activities and that there might be as many pitfalls ahead as those pocking the dead moon, with only hair-thick bridges spanning them.
But our silence was broken at last by a clatter from the control board and I knew our radar had picked up a moving object. The tiny visa-screen gave us a ship heading purposefully for the station. Eet glanced over his shoulder and I thought he was looking at me for orders. The mutant was not accustomed, once a matter had been decided, to wait for permission or agreement. I found myself nodding my head, and his fingers made the necessary adjustments to bring us behind that other ship, a little under its bulk where we might apply that weak traction beam without being sighted, or so we hoped.
The size of the newcomer was in our favor. I had expected something such as a scout ship, or certainly not larger than the smallest Free Trader. But this was a bulk-cargo vessel, of the smallest class, to be sure, but still of a size to be considered only a wallowing second-rate transfer ship.
Our traction beam centered and held, drawing us under the belly of the bigger vessel, which overhung us, if anyone had been out in space to see, as a covering shadow. We waited tensely for some sign that those in the other ship might be alarmed. But as long moments slipped by we breathed more freely, reassured by so much, though it was very little.
However, on the visa-screen what we picked up now was not the ship, but what lay ahead. For additional safety Eet had snapped on the distort beam and through that we could see just a little of the amazing port we neared.
Whatever formed its original core—an asteroid, a moon, an ancient space station—could not be distinguished now. What remained was a mass of ships, derelicts declared so by their broken sides, their general decrepit appearances. They were massed, jammed tightly together into an irregular ovoid except in one place directly before us, where there was a dark gap, into which the ship controlling our path was now headed.
"Looted ships—" I hazarded, ready to believe now in every wild story of Waystar. Pirates had dragged in victim ships to help form their hiding place—though why any such labor was necessary I could not guess. Then I saw—and felt—the faint vibration of a defense screen. The LB shuddered but it did not break linkage with the ship. Then we were through without any attack.
As the wall of those crumpled and broken ships funneled about us, I foresaw a new danger, that we might be scraped or caught by the wreckage, for that space down which we were being towed narrowed the farther we advanced.
Also, though the ships had seemed tightly massed at first sight, this proved not to be so upon closer inspection. There were evidences that they had been intended as an enveloping cover for whatever core lay at the heart. There were girders and patches of skin welded together, anchoring one wreck to another. But it was a loose unity and there were spaces in between, some large enough to hold the LB.
Seeing those, and calculating that we might come to grief ahead were the passage to narrow to the point where only the cargo ship might wedge through, I decided one gamble was better than another.
"Wedge in here"—I made this more a suggestion than an order—"then suit up and go through?"
"Perhaps that is best," Eet answered. However, I suddenly remembered that though I might suit up, there was no protective covering on board which would take Eet's smaller body.
"The disaster bag," Eet reminded me as his hands moved to loose our tie with the bulk of ship overhead.
Of course, the baglike covering intended to serve a seriously injured escapee using the LB, one whose hurt body could not be suited up if the emergency landing had been made on a planet with a hostile atmosphere and it was necessary to leave the boat. I unstrapped, and opened the cupboard where the suit lay at full length. The disaster bag was in tight folds beside its booted feet. Passage in that would leave Eet helpless, wholly dependent on me, but there was hope it would not be for long.
He was busy at the controls, turning the nose of the LB to the left, pointing it into one of those hollows in the mass of wreckage. The impetus left us by the pull of the ship sufficed to give us forward movement, and two girders welded just above the hole we had chosen held the pieces of wreckage forming its walls steady. There was a bump as we scraped in, and another, moments later, as the nose of the LB rammed against some obstacle. We could only hope that the crevice had swallowed us entirely and that our tail was not sticking betrayingly into the ship passage.
I suited up as fast as I could, wanting to make sure of that fact—though what we could do to remedy matters if that had happened I did not have the slightest idea. Then I hauled out the disaster bag and Eet climbed in so that I could make the various sealings tight and inflate its air supply. Since it was made for a man he had ample room, in fact moved about in it in the manner of one swimming in a very limited pool, for there was no gravity in this place and we were in free fall.
Activating the exit port, I crawled out with great care, fearing more than I wanted to admit some raw edge which could piece the protecting fabric of the suit or Eet's bag. But there was space enough to wriggle down the length of the LB, mostly by feel, for I dared not flash a beamer here.
Fortune had served us so far. The tail of the LB was well within the hole. And I had to hitch and pull, the weight of Eet dragging me back, by grasping one piece of wreckage and then the next for several lengths until I was in the main passage.
There was a weak light here, though I could not see its source, enough to take me from one handhold to the next, boring into the unknown. I made that journey with what speed I could, always haunted by the fear that another ship might be coming in or going out and I would be caught and ground against the wreckage.
The band of murdered ships ended suddenly in a clear space, a space which held other ships—three I could see. One was the cargo ship which had brought us in, another was one of those needle-nosed, deadly raiders I had seen used by the Guild, and the third was plainly a yacht. They were in orbit around what was the core of this whole amazing world in space. And it was a station, oval in shape like the protecting mass of wreckage, with landing stages at either end. Its covering was opaque, but with a crystalline look to the outer surface, which was pitted and pocked and had obviously been mended time and time again with substances that did not match the original material.
The cargo ship had opened a hatch and swung out a robo-carrier, heavily laden. I held on to my last anchorage and watched the robo spurt into a landing on a stage. The top half carrying the cargo dropped off and moved into an open hatch of the station while the robo took off for another load. There was no suited overseer to be seen, just robos. And I thought I saw a chance to make use of them to reach the station, just as we had used the robos to leave the caravansary.
Only I was not to have an opportunity to try. Out of nowhere came a beam, the force of which plastered me as tightly to the wreckage at my back as if my suit had indeed been welded in eternal bondage.
There was no breaking that hold. And my captors were very tardy about coming to collect me, finally spurting from the hatch of the yacht on a mini air sled. They lashed me into a tangle cord and used it as a drag to pull me behind them, not back to the ship from which they had issued, but to the landing stage where the robo had set down. Then, dismounting from their narrow craft, they tugged us both through a lock and into the interior of the station, where a weak gravity brought my boots and Eet's relaxed body to the floor.
Those who had taken me prisoner were humanoid, perhaps even of Terran breed, for they had that look. They snapped up their helmets and one did the same for me, letting in breathable air, though it had that peculiar faint odor of reprocessed oxygen. Leaving the tangle about my arms, they loosed me enough to walk, pointing with a laser to enforce my going. One of them took the bag from me and towed Eet, turning now and then to study the mutant narrowly.
So it was as prisoners that we came to the legendary Waystar, and it was an amazing place. The center was open, a diffused light filling it, a greenish light which gave an unpleasant sheen to most of the faces passing. By some unknown means there was a light gravity giving a true up and down to the corridors and balconies opening on that center. I caught sight of what could be labs, passed other doors tightly shut. There was population enough to equal that of a village on an ordinary planet—though, as I guessed, those who used the station as home base were often in space and the permanent dwellers were limited in number.
It was one of the latter I was taken before. He was an Orbsleon, his barrel bulk immersed in a bowl chair with the pink fluid he needed for constant nourishment washing about his wrinkled shoulders, his boneless upper tentacles floating just beneath its surface. His head was very broad in the lower part, dwindling toward a top in which two eyes were set far apart, well to the sides. His far-off ancestor of the squid clan was still recognizable in this descendant. But that alien body housed a very shrewd and keen intelligence. A Veep in Waystar would be a Veep indeed, no matter what form of body held him.
A tentacle tip flashed from the bowl chair to trigger keys on a Basic talker, for the Orbsleon was a tactile communicator.
"You are who?"
"Hywel Jern." I gave him an answer as terse as his question.
Whether that name meant anything to him I had no way of knowing. And I received no aid from Eet. For the first time I doubted that the mutant could carry some of the burden of my impersonation. It might well be that the alien thought process would prove, in some cases, beyond his reading. Then I would be in danger. Was this such a time?
"You came—how?" The tentacle tip played out that question.
"On a one-man ship. I crashed on a moon—took an LB—" I had my story ready. I could only hope it sounded plausible.
"How through?" There was of course no readable expression on the alien's face.
"I saw a cargo ship coming in, hung under it. The LB played out halfway through the passage. Had to suit up and come along—"
"Why come?"
"I am a hunted man. I was Veep Estampha's value expert, I thought to buy out, live in peace. But the Patrol were after me. They sent a man on contract when they could not take me legally. He left me for dead. I have been on the run ever since." So thin a tale it might hold only if I were recognized as Hywel Jern. Now that I was well into this I realized more and more my utter folly.
Suddenly Eet spoke to me. "They have sent for one who knew Jern. Also they did not register 'dead' when you gave your name."
"What do here?" my questioner went on.
"I am an appraiser. There is perhaps need for one here. Also—this is the one place the Patrol is not likely to take me." I kept as bold a front as I could.
A man came in at the slow and rather stately pace the low gravity required. To my knowledge I had not seen him before. He was one of the mutants of Terran stock having the colorless white hair and goggle protected eyes of a Faltharian. Those goggles made his expression hard to read. But Eet was ready.
"He did not know your father well, but had seen him several times in Veep Estampha's quarters. Once he brought him a Forerunner piece, a plaque of irridium set with bes rock. Your father quoted him a price of three hundred credits but he did not want to sell."
"I know you," I said swiftly as Eet's mind read that for me. "You had a piece of Forerunner loot—irridium with bes setting—"
"That is the truth." He spoke Basic with a faint lisp. "I sold it to you."
"Not so! I offered three hundred, you thought you could do better. Did you?"
He did not answer me. Rather his goggled head swung toward the Orbsleon. "He looks like Hywel Jern, he knows what Jern would know."
"Something—you do not like?" queried the tentacles on the keys.
"He is younger—"
I managed what I hoped would register as a superior smile. "A man on the run may not have time or credits enough for a plasta face change, but he can take rejub tablets."
The Faltharian did not reply at once. I wished I could see the whole of his face without those masking goggles. Then, almost reluctantly, he did answer.
"It could be so."
During all those moments the Orbsleon's gaze had held on me. I did not see his small eyes blink; perhaps they did not. Then he played the keys of the talker again.
"You appraiser, maybe use. Stay."
With that, not sure whether I was a prisoner or perhaps now an employee, I was marched out of the room and led to a cubby on a lower level, where Eet and I, having been searched for weapons and had the suit and bag taken from us, were left alone. I tried the door and was not surprised to find it sealed. We were prisoners, but to what degree I could not be sure.
XII
What I needed most at that moment was sleep. Life in space is always lived to an artificial timetable which has little relationship to sun or moon, night or day, in the measured time of planets. In hyper, when there is little to do for the smooth running of the ship, one simply sleeps when tired, eats when hungry, so that regular measurement of time does not apply. I did not know really how long it had been since I had had a meal or slept. But now sleep and hunger warred in me.
The room in which we had been so summarily stowed was a very small one, having little in the way of furnishings. And what there was resembled that planned for the economy of space, such as is found in a ship. There was a pull-down bunk, snapped up into a fold in the wall when not in use, a fresher, into which I would have to pack myself, when needful, with some care, and a food slot. On the off chance that it might be running, I whirled the single dial above it (there seemed to be no choice of menu). And somewhat to my surprise, the warn lights in the panel snapped on and the front flipped open to display a covered ration dish and a sealed container of liquid.
It would appear that the inhabitants of Waystar were on tight rations, or else they believed that uninvited guests were entitled only to the bare minumum of sustenance. For what I uncovered were truly space rations, nutritious and sustaining, to be sure, but practically tasteless—intended to keep a man alive, not in any way to please his taste buds.
Eet and I shared that bounty, as well as the somewhat sickening vita drink in the container. I did have a fleeting suspicion that perhaps some foreign substance had been introduced into either, one of those drugs which will either make a man tell all he knows or eradicate his will, so that for a time thereafter he becomes merely the tool of whoever exerts mastery over him. But that suspicion did not keep me from eating.
As I dumped the empty containers down the disposal unit I knew that just as I had had to eat, so I must now sleep. But it seemed that Eet did not agree, or not as far as he himself was concerned.
"The stone!" He made a command of those two words.
I did not have to ask what stone. My hand was already at the small pocket in my belt.
"Why?"
"Do you expect me to go exploring in the body of a phwat?"
Go exploring? How? I had already tried the cabin door and found it sealed. Nor did I doubt that they had guards outside, perhaps in the very walls about us—scan rays—
"Not here." Eet appeared very sure of that. "As to how—through there." He indicated a narrow duct near the ceiling, an opening which, if the grill over it were removed, might offer a very small exit.
I sat on the bunk and glanced from the hairy man-thing Eet now was to that opening. When we had first tried this kind of change I had believed it all illusion, though tactile as well as visual. But now, had Eet really altered in bulk so that what I saw before me was actually many times the size of my alien companion? If so—how had that been done? And (in me a sharp fear stabbed) if one did not have the stone, would changes remain permanent?
"The stone!" Eet demanded. He did not answer any of my thoughts. It was as if he were suddenly pressed for time and must be off on some important errand from which I detained him.
I knew I was not going to get any answers from Eet until he was ready to give them. But his ability to read minds was perhaps our best key to this venture and if he now saw the necessity for crawling through ventilation ducts, then I must aid him.
I kept my hand cupped about the stone. Though Eet had said there were no snoop rays on us, yet I would not uncover that treasure in Waystar. I stared at Eet where he hunkered on the floor and forced myself to see with the mind's eye, not a furred humanoid, but rather a mutant feline, until just that crouched at my feet.
It was easy to screw out the mesh covering of the duct. And then Eet, using me as a ladder, was up into it with speed. Nor did he leave me with any assurance as to when he would return, or where his journey would lead, though perhaps he did not know himself.
I wanted to keep awake, hoping that Eet might report via mind-touch, but my body needed sleep and I finally collapsed on the bunk into such slumber as might indeed have come from being drugged.
From that I awoke relucantly, opening eyes which seemed glued shut. The first thing I saw was Eet, back in his hairy disguise, rolled in a ball. I sat up dazedly, trying to win out over the stupor of fatigue.
Eet was back, not only in this cell but in his other body. How had he managed the latter? Fear sharpened my senses and sent my hand to my belt again, but I felt with relief the shape of the stone in the pocket.
Even as I watched bleerily, he unwound, sat up blinking, and stretched his arms, as if aroused from a sleep as deep as mine had been.
"Visitors coming." He might give the outward seeming of one only half awake, but his thought was clear.
I shambled to the fresher. Best not let any arrival know I had warning. I used the equipment therein and emerged feeling far more alert. Even as I looked to the food server, the door opened and one of the Orbsleon's followers looked in.
"Veep wants you."
"I have not eaten." I thought it well to show some independence at the suggestion that I was now the Orbsleon's creature.
"All right. Eat now." If he made that concession (and the very fact that he did was a matter of both surprise and returning confidence for me) he was not going to enlarge upon it. For he stood in the doorway watching me dial the unappetizing food and share it with Eet.
"You—" The guard stared at the mutant. "What do you do?"
"No good talking to him," I improvised hurriedly. "You would need a sonic. He is—was—my pilot. Only fourth part intelligence, but good as a tech."
"So. What is he anyway?" Whether he spoke out of idle curiosity or was following an order to learn more, I did not know. But I had made a reasonable start on providing Eet with a background and I enlarged upon it a little with the name he had given himself.
"He is a phwat, from Formalh—" I added to my inventions. With so many planets supporting intelligent or quasi-intelligent life in the galaxy, no one could be expected to know even a thousandth of them.
"He stays here—" As I prepared to leave, the guard stepped in front of Eet.
I shook my head. "He is empathic-oriented. Without me he will will himself to death." Now I referred to something I had always thought a legend—that two species could be so emotionally intertied. But since I had believed, until last year, that the place in which I now stood was also a legend, there might be truth in other strange tales. At least the guard seemed inclined to accept what I said as a fact; he allowed Eet to shamble along behind me.
We did not return to the room in which the Orbsleon had interviewed me, but rather to one which might be a small edition of the hock-locks I well knew. There was a long table with various specto-devices clamped on it. In fact, it was a lab which many an appraiser on a planet might have envied. And on the walls were outlines of "safe" cupboards, each one with the locking thumb hole conspicuous in the center, where only the thumb of one authorized to open it would register to release its contents.
"Snooper ray on us," Eet informed me. But I had already guessed that, knowing why I had been brought here. They were going to prove my claim of being an appraiser, which meant tricky business. I would have to call on all I had learned from the man I seemed to be, all that I had picked up since I had left his tutelage, in order to survive such a test.
The things to be valued were spread on the table, under a protective ull web. I went straight to it, for in that moment my lifetime preoccupation took command.
There were four pieces in all, gemmed and set in metal—their glitter sparking life clear across the room. The first was a necklace—koro stones, those prized gems from out of the Sargolian seas which the Salariki doubly value because of their ability to give forth perfume when warmed by the body heat of the wearer.
I held it up to the light, weighed each of the jewels in my hand, sniffed at each stone. Then I let it slide carelessly from my grasp to the bare surface of the table.
"Synthetic. Probably the work of Ramper of Norstead—or of one of his apprentices—about fifty planet years old. They used marquee scent on it—five, maybe six steepings." I gave my verdict and turned to the next piece, knowing I did not have to impress the guard, or the two other men in the room, but rather those who held the snooper ray on me.
The second piece was set in a very simple mounting. And its dark rich fire held me for a moment or two. Then I put it in the cup below the infrascope and took two readings.
"This purports to be a Terran ruby of the first class. It is unflawed, true enough. But it has been subjected to two forms of treatment. One I can identify, the other is new to me. This has resulted in a color shift. I think it was originally a much lighter shade. It will pass, save for quality lab testing. But any expert gemologist would be uneasy about it."
The third on that table was an arm band of metal which was reddish but carried a golden overcast that shifted across the surface when the ornament was handled. The maker had taken advantage of that overcast in working out the pattern on it, which was of flowers and vine, so that the gold appeared to line some of the leaves at all times. There was no mistaking it and my mind jumped back to the day my father had shown me such work, but then as a small pendant he had sold to a museum.
"This is Forerunner, and it is authentic. The only piece I have previously seen was taken from a Rostandian tomb. That was decided by the archaeologists to be very much older than the tomb even. Perhaps it had been found by the Rostandian buried there. Its origin is unknown as yet."
In contrast to the three other offerings the fourth was dull, leaden-gray, ugly metal set with an ill-formed cluster of badly-cut stones. It was only the center stone, one of perhaps four carats, which seemed to have any real life, and that, too, had been unimaginately treated.
"Kamperel work. The centerpiece is a sol sapphire and would pay recutting. The rest"—I shrugged—"not worth working with. A tourist bauble. If this"—I turned to the two men, who had not spoken—"is the best you have to show me, then indeed, rumor has greatly overrated the take of Waystar."
One of them came around the table to restow the four pieces under the web. I was wondering if I were now to be returned to my cell when the monotonous click of the Veep's voice sounded from some concealed com.
"As you think, this was test. You will see other things. The sol—can you recut?"
Inwardly I sighed with relief. My father had not had that training, I need not be forced to claim it.
"I am an appraiser, not a cutter. It will take skill to make the most out of that stone after it has been mishandled the way it has. I would suggest that it be offered as is"—I thought furiously—"to such a firm as Phatka and Njila." Again I pulled names from my memory, but this time from Vondar's warning about borderline dealers whose inventories of stones were kept in two or three different accountings, those they could sell openly, those to be sold privately. That they had Guild affiliations was suspected but unproved. But my ability to name them would be more proof that I had dealt on the border line of the law.
There was a period of silence. The man who had re-wrapped the treasures in the web now sealed them into one of the wall cubbies. No one commented, nor did the com speak again. I shifted from one foot to the other, wondering what would happen now.
"Bring here—" the com finally clicked.
So I was taken back to the room where the Orbsleon Veep wallowed in his fluid-filled seat. Swung out over the surface of that was a lap table and on it lay a single small piece of metal.
It had no gem and it was an odd size. But the shape I had seen before and knew very well indeed. A ring—meant to fit, not a bare finger, but over the bulky glove of a space suit. Only this had no zero stone, dull and lifeless, in its empty prongs. That it was, or had been, twin to the ring which had caused my father's death, I was sure. Yet the most important part was missing. I knew instantly that this was another test, not of my knowledge as an appraiser, but of how much I might know on another subject. My story must hold enough truth to convince them.
"There is a snooper ray on." Eet had picked up my thought.
"What this?" The Veep wasted no time in coming to the test.
"May I examine it?" I asked.
"Take, look, then say," I was ordered.
I picked up the ring. Without its stone it was even more like a piece of battered junk. How much dared I say? They must know a great deal about my father's "death"—So I would give them all my father had known.
"I have seen one of these before—but that had a stone." I began with the truth. "A dim stone. It had been subjected to some process which rendered it lifeless, of no value at all. The ring was found on the space glove of a dead alien—probably a Forerunner—and brought to me for hock-lock."
"No value," clicked the voice of the Veep. "Yet you bought."
"It was alien, Forerunner. Each bit we learn about such things is knowledge, which makes some men richer. A hint here, a hint there, and one can be led to a find. This in itself has no value, but its age and why it was worn over a space glove—that makes it worth payment."
"Why worn on glove?"
"I do not know. How much do we know of the Forerunners? They were not even all of one civilization, species, or time. The Zacathans list at least four different star empires before they themselves developed a civilization, and claim there are more. Cities can crumble, suns burn out, sometimes artifacts remain—given proper circumstances. Space itself preserves, as you know well. All we can learn of those Forerunners comes in bits and pieces, which makes any bit of value."
"He asks," Eeet told me, "but the questions are now from another."
"Who?"
"One more important than this half-fish." For the first time Eet used a derogatory expression, allowed an aura of contempt to pervade his mind-touch. "That is all I know. The other wears a protective antiesper, antisnoop device."
"This was a ring," I repeated aloud and laid the plundered circlet back on the lap table. "It held a stone now gone, and it resembles the one I held for a time which had been found on a Forerunner."
"You held—now where?"
"Ask that," I returned sharply, "of those who left me for dead when they plundered my shop." False now, but would any snooper detect that? I waited, almost expecting some loud contradiction of my lie. If any had been made perhaps those in the room were not aware of it yet. And if my last statement were accepted as truth, perhaps there might be awkward questions asked inside the ranks of the Guild, the which would do me no harm at all.
"Enough," the voicer clicked. "You go—sales placewatch."
My escort moved for the door. He did not snap to attention as a Patrolman, but he wore a tangler at his belt and I did not dispute his right to see me to where the Veep ordered my attendance.
We passed along one of the balcony corridors which rimmed the open center. It was necessary to shuffle, not lifting the feet much, keeping a handhold on the wall rail, or the low gravity became a hazard. When our way led down on a curled rod with handholds instead of stairsteps, we managed almost as if we were in a grav lift, coming to the third level below that where the Veep had his quarters.
This possessed some of the bustle of a market place. There was a coming and going of many races and species, Terrans, Terran-mutants, humanoids, and nonhumanoid aliens.
Most of them wore ship uniforms, though unmarked by any official badges. And all of them wore stunners, though I saw no lasers. And I thought perhaps there might be some rule against more lethal weapons here.
The booth into which I was ushered lacked the elaborate detection equipment of the lab. Another Orbsleon (plainly of inferior caste, since he still had the crab legs long ago removed from the Veeps) squatted in a bowl with just enough liquid washing in it to keep him on the edge of comfort. It was plain he was in charge and must have expected me. He clicked nothing on his talker, but gestured with one tentacle to a stool back against the wall, where I obediently sat down, Eet hunkering at my feet. There were two others there, and seeing them, I realized, with a shudder I hoped I successfully suppressed, just how far outside the bounds of law I was.
There has always been slavery within the galaxy, sometimes planet-orientated, sometimes spread through a solar system, or systems. But there are kinds of slavery which make men's stomachs turn more readily than the war-captive, farm-labor type most widely known. And these—these— _things_ —were the result of selective breeding in a slavery the Patrol had worked for years to eliminate from any star lane.
The Orbsleon's servants were humanoid—to a point But there had been both surgical and genetic modifications, so that they were not truly "men" as the Lankorox scale defines men in an alien-Terran-mutant society. They were rather living machines, each programmed for a special type of service, knowing nothing else. One sat now with his hands resting limply on the table, his whole puffy body slack, as if even the energy which brought him pseudo life had drained away. The other worked with precise and delicate speed at a piece of jewelry, a gem-studded collar such as is worn at a feast of state by a Warlockian Wyvern. He pried each gem from its setting, sorting them with unerring skill, and at the same time graded them, placing the gems in a row of small boxes before him. The many-lensed orbs in his misshapen, too large, too round head were not turned upon what he did but rather stared straight ahead out of the booth, though they were not focused on anything beyond.
"He is a detect—" Eet told me. "He sees all, reports without defining what he sees. The other is a relay."
"Esper!" I was suddenly afraid, afraid that that loosely sprawling hulk of flesh before us might tune in on Eet, know that we two together were far more than we seemed.
"No, he is on a lower band," Eet returned. "Only if his master wishes—"
He lapsed into silence and I knew he, too, knew the danger.
Why I had been sent here I did not know. Time passed. I watched those go to and from outside. The detect slave continued his work until the collar was entirely denuded of its jewels and then the metal went into a larger box. Now the busy fingers brought out a filigree tiara. Selections were made from the boxes of gems, and with almost the same speed with which they had been pried forth from their first settings they were put into the tiara. Though all the jewels were not used, I could see that the result of the work would be a piece which would easily bring a thousand certified credits in any inner planet shop. But all the time he worked, the slave never looked at what he wrought.
What were to be my duties, if any, I was not told. And while the activities of the detect slave interested me for awhile, it was not enough to hold my attention too long. I found the inactivity wearing and I was restless. But surely anyone in my situation would want employment after awhile and no one would be suspicious if I showed my boredom.
I was shifting on what became an increasingly hard stool the longer I sat on it when a man stepped inside the booth. He wore the tunic of a space captain without any company insignia and he appeared to be familiar with the establishment, as he bypassed the table where the slaves sat and came directly to the Orbsleon.
He had been pressing his left hand against his middle—reminding me of my own frequent check on my gem belt. Now he unsealed his tunic and fumbled under its edge. The alien pushed forward a swing table much like the one his Veep had used to display the ring.
The spacer produced a wad of ull web, picked it apart to show a very familiar spot of color—a zoran. The Orbsleon's tentacle curled about the stone and without warning threw it to me. Only instinct gave me the reflex to catch the flying stone out of the air.
"What!" With a sharp exclamation the captain swung around to eye me, his hand on the butt of his stunner. I was turning the stone around, examining it.
"First grade," I announced. Which it was—about the best I had seen for some time. Also it was not a raw stone but had been carefully cut and mounted in a delicate claw setting, hooked to hang as a pendant.
"Thank you." There was sarcasm in the captain's voice. "And who may you be?" He lost now some of his aggressive suspicion.
"Hywel Jern, appraiser," I answered. "You wish to sell?"
"I wouldn't come here just for you to tell me it's first grade," he retorted. "Since when has Vonu added an appraiser?"
"Since this day." I held the stone between me and the light to look at it again. "A fleck of clouding," I commented.
"Where?" He went across the booth in two strides, snatched the jewel out of my hand. "Any clouding came from your breathing on it. This is a top stone." He swung around to the Orbsleon. "Four trade—"
"Zorans are not four trade," the talker clicked. "Not even top grade."
The captain frowned, half turned, as if to march out of the booth. "Three then."
"One—"
"No! Tardorc will give me more. Three!"
"Go Tadorc. Two only."
"Two and a half—"
I had no idea what they bid, since they did not use the conventional credits. Perhaps Waystar had its own scale of value.
The Orbsleon seemed to have reached a firm decision.
"Two only. Go Tadorc—"
"All right, two." The captain dropped the zoran on the lap table and the alien's other tentacle stretched to a board of small buttons. When that mobile tip punched out a series on it there was no vocal reply. But he used the talker again.
"Two trade—at four wharf—take supplies as needed."
"Two!" The captain made an explosive oath of that word as he left with a force which might have been a stamp in a place of higher gravity.
The alien again threw the zoran, this time to be caught by the detect, who tucked it away in one of his boxes. And it was then that my earlier guide-guard came to the front of the booth.
"You"—he gestured to me—"come along."
Glad for the moment to be released from the boredom of the booth, I went.
XIII
"Top Veep," Eet's warning came, to match my own guess as to where we were being taken. We again climbed through the levels to the higher ways of the station, this time passing the one where the Orbsleon had his quarters. Now the time-roughened walls about us showed dim traces of what had once been ornamentation. Perhaps for whatever creatures had built this station this had been officer territory.
I was motioned through a roll door, my guards remaining outside. They made a half-hearted attempt to stop Eet, but he suddenly developed an agility he had not shown before and pushed past them. I thought it odd they did not follow. Then, a moment later, I discovered why the inhabitant of these particular quarters did not need their attendance, for with another step I struck rather painfully against a force wall.
Also, inside this room the light gravity to which I had partially adjusted not only had become full for my race, but had an added pull, so it was an effort to take a step.
Beyond that invisible barrier the room was furnished as might be one in a luxury caravansary on some inner planet. Yet the furniture did not harmonize but was jammed together, showing even differences in scale size, as if some pieces had been made for bodies smaller or larger than my own. The one thing these had in common was their richness, which in some cases was gaudy and blatantly flamboyant.
Stretched in an easirest was the Veep. He was of Terran descent, but with certain subtle differences, modifications of feature, which suggested mutation. Probably he came from a race which had been among the early colonists. His hair had been cut so that it stood above his partially shaven skull in a stiffened roach, making him resemble one of the mercenaries of old times, and I wondered how he got a space helmet over that crest, if he ever did. His skin was brown, not just space-tanned, and there were two scars, too regular to be anything but inflicted on purpose for a patterning, running from corner of eye to chin on either side of his mouth.
Like the gaudy room, his clothing was a colorful mixture of planetary styles from several worlds. His long legs, stretched out in the rest, which fitted itself to give him greatest comfort, were encased in tight-fitting breech-legging-boots of a pliable, white-furred hide, the fur patterned with a watered rippling. Above his waist he wore the brilliant black-and-silver combination of a Patrol admiral's dress tunic complete with begemmed stars and ribbons of decorations. But the sleeves of that had been cut out, leaving his arms bare to the shoulders. Below his elbows he wore on both arms very wide bracelets or armlets of irridium, one mounted with what could only be Terran rubies of the first _water_ , the other with sol sapphires and lokerals running in alternate rows, the vivid greens and blues in harmonious contrast. Both armlets were barbaric in taste.
In addition, his stiffened top ridge of hair was encircled by a band of mesh metal, green-gold—from which hung, flat against his forehead, a pendant bearing a single koro—about ten carats and very fine. The whole effect was that of what he must be, a pirate chief on display.
Whether he wore that mixture of splendor and bad taste by choice or for the effect such a bold showing of wealth might have on his underlings, I did not know. The Guild men in the upper echelons were usually inclined to be conservative in dress rather than ostentatious. But perhaps as master, or one of the masters, of Waystar, he was not Guild.
He watched me reflectively. Meeting his dark eyes, I had the impression that his clothing was a mask of sorts, meant to bedazzle and mislead those with whom he dealt. He was holding a small plate of white translucent jade in one hand, from time to time raising it to his mouth to touch tongue tip in a small licking movement to the gob of blue paste it held.
"They tell me"—he spoke Basic with no definable accent—"that you know Forerunner material."
"To some extent, Gentle Homo. I have seen, have been able to examine perhaps ten different art forms."
"Over there—" He pointed, not with his empty hand, but with his chin, to my left. "Take a look at what lies there and tell me—is it truly Forerunner?"
A round-topped table of Salodian marble supported what he wanted appraised. There was a long string of interwoven metal threads dotted here and there with tiny brilliant rose-pink gems; it could have been intended as either a necklace or a belt. Next to it was a crown or tiara, save that no human could have worn it in comfort, for it was oval instead of round. There was a bowl or basin, etched with lines and studded here and there with gems, as if they had been scattered by chance or whim rather than in any obvious pattern. And last of all, there was a weapon, still in a sheath or holster—its hilt or butt of several different metals, each of a different color but inlaid and mingled with the others in a way I knew we had no means of duplicating.
But what was more, I knew we had found what we had entered this kolsa's den to seek. This was the larger part of the treasure the Zacathans had found in the tomb; I had been too well briefed by Zilwrich to mistake it. There were four or five other pieces, but the best and most important lay here.
It was the bowl which drew my attention, though I knew if the Veep had not already caught the significance of those seemingly random lines and gems he must not be given a hint, by any action of mine, that it was a star map.
I walked toward the table, coming up against the barrier again before I reached it—a circumstance which gave me a chance to assert myself as I was sure Hywel Jern would have done.
"You cannot expect an appraisal, Gentle Homo, if I cannot inspect closely."
He tapped a stud on the chair arm and I could advance, but I noted that he tapped it again, twice, when I reached the table, and I did not doubt I was now sealed in.
I picked up the woven cord and ran it through my fingers. In the past I had seen many Forerunner artifacts, some in my father's collection, some through the aid of Vondar Ustle. Many others I had studied via tri-dee representation. But this stolen treasure was the richest it had ever been my good fortune to inspect. That the pieces were Forerunner would of course have been apparent even if I had not known their recent past history. But as everyone knew, there were several Forerunner civilizations and this workmanship was new to me. Perhaps the Zacathan expedition had stumbled upon the remains of yet another of those forgotten stellar empires.
"It is Forerunner. But, I believe, a new type," I told the Veep, who still licked at his confection and watched me with an unwavering stare. "As such it is worth much more than its intrinsic value. In fact, I cannot set a price on it. You could offer it to the Vydyke Commission, but you might even go beyond what _they_ could afford—"
"The gems, the metal, if broken up?"
At that moment his question was enough to spark revulsion and then anger in me. To talk of destroying these for the worth of their metal and gems alone was a kind of blasphemy which sickened anyone who knew what they were.
But he had asked me a direct question and I dared not display my reaction. I picked up each piece in turn, longing to linger in my examination of the bowl map, yet not daring to, lest I arouse his suspicion.
"None of the jewels is large," I reported. "Their cutting is not of the modern fashion, which reduces their value, for you would lose even more by attempting to recut. The metal—no. It is the workmanship and history which makes them treasure."
"As I thought." The Veep gave a last lick to his plate and put it aside empty. "Yet a market for such is difficult to find."
"There are collectors, Gentle Homo, who are perhaps not as free-handed as the Vydyke, but who would raise much on all their available resources to have a single piece of what lies here. They would know it for a black deal and so keep what they obtained hidden. Such men are known to the Guild."
He did not answer me at once, but continued to stare, as if he were reading my mind more than concentrating on my words. But I was familiar enough with mindtouch to know he was not trying that. I judged rather that he was considering carefully what I had just said.
But I was now aware of something else which first alarmed and then excited me. There was warmth at my middle, spreading from the pocket which held the zero stone. And that could only mean, since I was not putting it to service, that somewhere near was another of those mysterious gems. I looked to the most obvious setting, that of the crown, but I saw no telltale glow there. Then Eet's thought reached me.
"The bowl!"
I put out my hand, as if to reinspect that piece. And I saw that on the surface nearest me, luckily turned away from the Veep, there was a bright spark of light. One of those seemingly random jewels I had thought were meant to mark stars had come to life!
Picking up the bowl, I turned it idly around, holding my palm to cover the zero stone, and felt both at my middle and from the bowl the heat of life.
"Which do you think of greatest importance?" the Veep asked.
I put down the bowl, the live gem again turned away from him, looking over the whole array as if to make up my mind.
"This perhaps." I touched the strange weapon.
"Why?"
Again I sensed a test, but this time I had failed.
"He knows!" Eet's warning came even as the Veep's hand moved toward the buttons on the chair arm.
I threw the weapon I held. And by some superlative fortune I did not have any right to expect, it crashed against his forehead just beneath that dangling koro stone, as if the force field no longer protected him, or else I was inside it. He did not even cry out, but his eyes closed and he slumped deeper into the hold of the easirest. I whirled to face the door, sure he had alerted his guards. The force field might protect me, but it would also hold me prisoner.
I saw the door open, the guards there. One of them cried out and fired a laser beam. The force field held, deflected that ray enough to send a wave of flame back, and the man farthest into the room staggered, dropped his weapon, and fell against the one behind him.
"There is a way." Eet was by the easirest. He reached up and grabbed at the strange weapon now lying in the Veep's lap. I swept up the other treasures, holding them between my body and arm as I followed Eet to the wall, where he fingered a stud and so opened a hidden door. As that fell into place behind us, he mind-touched again.
"That will not hold them for long, and there are alarms and safeguards all through this wall way. I snouted them out when I explored. They need only throw those into action and we are trapped."
I leaned against the wall, unsealing my tunic and making its front into a bag to hold what I had snatched up. It was so awkward a bundle that I had difficulty in closing the tough fabric over it.
"Did your exploring see a way out?" I asked now. Our escape from that room had been largely a matter of unthinking reflex action. Now I was not sure we had not trapped ourselves.
"These are old repair ways. There are suits in a locker. They still have to patch and repatch the outside. It depends now upon how fast we can reach the suit locker."
The gravity here was practically nonexistent, and we made our way through the dark, which was near absolute, by swimming through the air. Luckily there were handholds at intervals along the outer wall, proving that this method of progression had been used here before. But my mind worried at what lay ahead. Supposing fortune did favor us enough to let us reach the suits, get into them, and out on the outer shell of the station. We still had a long strip of space to cross to the ring of wreckage, and then to find our LB. This time the odds were clearly too high against us. I believed that the whole of Waystar would be alerted to track us down, they to hunt over familiar ground, we lost in their territory.
"Wait—" Eet's warning brought me up with a bump against him. "Trap ahead."
"What do we do—?"
"You do nothing, except not distract me!" he snapped.
I half expected him to make some move forward, for I thought his intention was to disarm what waited us. But he did not. Though no mind-touch was aimed at me, I felt what could only be waves of mental energy striking some distance ahead—and the zero stone in my belt grew uncomfortably warm against my body.
"Well enough," Eet reported. "The energy is now burned out. We have a clear path for a space."
We encountered two more of what Eet declared to be pitfalls, but which I never saw, before we came out of a sliding panel in the wall into a blister compartment on the outer skin of the hull. There we found the suits, just as Eet had foretold. Since I could not stuff myself plus the loot I carried into the one nearest my size, I had to pass the bowl and the tiara on to Eet, who was in the smallest, still much too large for him.
But how we would reach the outer shell of wreckage and the LB, I had not the least idea. The suits were both equipped, it was true, with blast beams, intended to give any worker who was jolted off into open space a chance of returning to the surface of the station. But if we used those, their power might not be enough to take us all the way to the wreckage, and in addition, we would be in plain sight of any watcher or radar screen. However, we did have the treasure and—
"That mistake I made—does the Veep know the importance of the bowl?" I demanded now.
"Part of it. He knows it is a map."
"Which they would not destroy willingly." I hoped that was true.
"You argue from hope, not knowledge," the mutant returned. "But it is all the hope we may have."
I signaled exit from the bubble, and crawled out, the magnetic plates on my boots anchoring me to the surface of the station. Once before Eet and I had so gone into space and I was touched now with the terrible fear which had gripped me then when I had lost my footing on the skin of the Free Trader ship and my contact with security, and floated into empty space.
But here there was a limit to emptiness. The cargo ship which we had followed into this port was gone, but the needle-nosed raider and the yacht were still in orbit, and above, all around, was the mass of wreckage—though I could sight no landmarks there and wondered how we were ever going to discover the narrow inlet in the jagged, tangled mass which hid the LB.
I could see no reason to wait. Either we would coast across to the wreckage or our power would fail. But to wait here any longer was to risk being captured before we had even tried. However, we did take the precaution of linking together by one of the hooked lines meant to anchor a worker to the surface of the station. So united, we took off between the two ships hanging ominously above.
"I cannot reach the controls of my jet—" Eet delivered what might be a final blow, dooming us to capture. Would the power in my own shoulder-borne rocket be enough to take us both over?
I triggered the controls, felt the push thrust which sent me and the suit containing Eet away from the station. My aim was the nearest of wreckage. I might be able to work my way along that in search of the passage if I could get to it. But every moment I expected to be caught by tangle beams, somehow sure that the Veep would not risk an annihilating weapon which would destroy his treasure.
The spurt of thrust behind me continued, in spite of the drag Eet caused as he spun slowly about at the end of the line, and there did not come any pursuit or pressure beam. I did not feel any triumph, only a foreboding which wore on my nerves. It is always worse to wait for an attack. I was certain that we had been sighted and that any moment we would be caught in a net.
The thrust failed while we were still well away from the wreckage. And though I got one more small burst by frantic fingering of the controls, it did little more than set me spinning across a small portion of that gap. Eet had been carried ahead of me by some chance of my own efforts, and now I saw his suit roll from side to side, as if, within it, he fought to reach his controls and so activate his own power.
What he did I could not tell, but suddenly there was a lunge forward of his spinning suit, and he towed me with him. The power of his progress intensified, for he no longer rolled. Now he was as straight as a dart flung at some target, and he dragged me easily behind as he headed for the wreckage. Still I could not guess why we had not been followed.
The splintered and dangerous mass of that wall of derelict ships grew more distinct. I trusted Eet could control his power, so that we would not be hurled straight into it. The merest scrape of some projection could tear suits and kill us in an instant.
Eet was rolling again, fighting against the full force of the power. Though I could do little to control my own passage, I rolled, too, hoping to meet feet first a piece of ship's side which would afford a reasonably smooth landing among the debris.
We whirled on at a faster pace than my own pack had sent us. And I guessed suddenly that Eet was making use of the zero stone on the map to trigger the energy of his rocket.
"Off!" I thought that as an order. "We'll be cut to ribbons if you do not."
Whether he could not control the force now, I did not know, but my feet slammed with bone-shaking impact against the smooth bit I had aimed for. I reached out, trying to grip Eet's suit. He had managed to turn, to coast alongside of the debris, just far enough away not to be entangled in it, yet. The magnetic plates in my boots kept me anchored, but not for long. Though I stopped Eet's advance with a sharp jerk, I was immediately thereafter torn loose by the power which dragged him on.
We nudged along beside the wreckage, twisting and turning as best we could to avoid any contact. Even if we might not be picked up by sight scanners against the camouflaging irregularities of that mass of metal, any heat identification ray could pick us up. And I did not doubt in the least that such equipment was in use at Waystar.
Was it that they dared not attack for fear of losing the treasure? Had they sent ahead of us some command to activate the outer defenses, to keep us bottled up until they could collect us at their leisure? Perhaps when loss of air had rendered us perfectly harmless?
"I think they want you alive." Eet's answer came in response to my last dark speculation. "They guess that you know the value of the map. They want to know why. And perhaps they know that Hywel Jern did not really rise from the dead. I may read minds, but in that nest back there I could not sort out all thoughts."
I was not interested in the motives of the enemy. I was absorbed now in escape, if that was at all possible. Given time, we might work our way completely around the wall of debris to find the entrance. But such time our air supply would not offer.
"Ahead—the ship with the broken hatch," Eet said suddenly. "That I have seen before!"
I could make out the broken hatch. It took the shape of a half-opened mouth. And in me, too, memory stirred. I had set gloved hand to the edge of that very same hatch just before the pressure beam had made us captive. We could not be far now from the entrance, though I could hardly believe in such fortune.
Eet put on an extra burst of speed, drawing out a space from the wreckage, and certainly this energy could not all come from the suit rocket. The spurt was enough to bring us inside the ship passage. And we worked our way back from one handgrip to another, or rather I did so, pulling Eet's suit along. Only the fact that we were both relatively weightless made it possible. And even then, I was weak, shaking with fatigue, not certain I could make the full journey.
Every handhold I won to and from was a struggle. I did not direct my attention to the whole passage yet ahead, but limited it to the next hold only, and then to the next. I even lost my fear of what might lie behind my concentration was so great on just swinging to the next hold—
We gained, I was not quite sure how, the crevice in which we had left the LB and crawled to its hatch. But once I slammed the door shut behind us I lost my last ounce of energy, and slid down, unable to move, watching Eet, in the clumsy suit, lift one arm with visible effort to reach the inner controls, fail, and then with grim patience try again.
Eventually he succeeded. Air hissed in around me and the inner hatch opened. The suit holding Eet squirmed and wriggled, and then the mutant emerged, kicked the suit away in an almost vindictive gesture, and scrambled over to me to fumble with the sealings which held me in the protective covering.
The ship air revived me to the extent that I was able to shed that shell and crawl on into the cabin. Eet had preceded me, and now squatted in the pilot's web, fingering the buttons to ease us out and away.
I dragged myself to the hammock, lay weakly back in it. I did not believe at that moment that we had the least chance of breaking through the outer defenses of Waystar. We and our ship _must_ meet some force field which would hold us, intact, as our captors wanted. But some reckless desire to go down fighting made me take the zero stone out in my shaking hands. I broke the disguise it had given me, or hoped I did. Having no mirror I could not be sure.
Now—there was something I could do which would at least confuse them if they slapped a spy ray on us.
"Such comes now," Eet reported and then closed his mind tightly, intent only on getting us out of the tunnel.
How much time did I have? The stone burned my hands but I held on. I had no mirror to mark the course of my transformation, but I willed it with all the energy and resource I had left. Then I lay back weakly, unable even to put away the precious source of my pain.
I looked blearily down what I could see of my prone body. There were, surely I could not be mistaken, the furred breeches, and above them the brilliance of a space admiral's tunic. I turned my head a fraction from side to side. My arms were bare, below the elbow wearing the gemmed armlets. I was, I hoped, by the power of the zero stone, a complete copy of the Veep. If they now snooped us with a seeing ray, the change might give us a small advantage, a few moments of confusion among our enemies.
Eet did not turn to look at me but his thought rang in my head.
"Very well done. And—here comes their snoop ray!"
Not having his senses, I must take his word for that. I levered myself up in the hammock with what energy I could summon, which was only enough to keep me braced with some small semblance of alertness. Eet suddenly slapped a furred fist on the board and the answering leap of the LB pinned me against the hammock. My head spun, I was sick—then I was swept into darkness.
XIV
When I roused groggily I lay staring at the rounded expanse above me, not able at once to remember where I was, or perhaps even _who_ I was. With what seemed painful and halting slowness, memory of the immediate past returned. At least we were still in existence; we had not been snuffed out by some defense weapon of the pirate stronghold. But were we free? Or held captive by a force beam? I tried to lever myself up and the LB hammock swayed.
But I had had a look at my own body and I was not now wearing the semblance of the Veep—though a furry dwarf still hunched at the controls of the small craft. My hand went to the bulge in my belt. The sooner I was sure I was myself again, the better. I had a strange feeling that I could not think or plan until I was Murdoc Jern outwardly as well as inwardly, as if the outer disguise could change me from myself into a weak copy of the man my father had been. Eet had been a cat, but I had willed that on him without his desire. This I had taken upon myself by my own wish, meant to be outer, not complete. What _did_ make sense any more?
"You are yourself," came Eet's thought.
But there was something else. My hand rested upon a pocket wherein all those days, months, I had carried the zero stone. And there was no reassuring hard lump to be felt. It was flat—empty!
"The stone!" I cried that aloud. I drew myself up, though my body was weak and drained of energy. "The stone—"
Then Eet turned to me. His alien face was a mask as far as I was concerned. I could read no expression there.
"The stone is safe," he thought-flashed.
"But where—?"
"It is safe," he repeated. "And you are Murdoc Jern outwardly again. We are through their defenses. The snooper ray caught you in the Veep's seeming and was deceived long enough for the stone to boost us out of range."
"So that is the way you used it. I will take it now." I held myself upright, though I must still clutch at the hammock to keep that position. Eet had used the zero stone even as we had once used it to boost the power of a Patrol scout ship and so escape capture. I was angry with myself for having overlooked that one weapon in our armament. "I will take it now," I repeated when Eet made no move to show me where it was. Though I had worked on the LB under Ryzk's direction I could not be sure where Eet had put it for the greatest effect in adding to our present drive.
"It is safe," he told me for the third time. Now the evasiveness of that reply made an impression on me.
"It is mine—"
"Ours." He was firm. "Or, rather, it was yours by sufferance."
Now I was thinking clearly again. "The—the time I turned you into a cat.... You are afraid of that—"
"Once warned, I cannot be caught so again. But the stone is danger if used in an irresponsible fashion."
"And you"—I controlled my rising anger with all the strength I had learned—"are going to see that it is not!"
"Just so. The stone is safe. And what is more to the purpose—look here." He pointed with one of his fingers to something which, for the want of other safekeeping, lay in the second hammock.
I loosed one hand to pull that webbing a little toward me. There lay the bowl with the map incised on its outer surface. A moment later I held it close to my eyes.
With the bowl turned over, the bottom was a half sphere on which the small jewels which must be stars winked in the light. And I saw, now that I had the time and chance to view it searchingly, that those varied. My own species rate stars on our charts by color—red, blue, white, yellow, dwarfs and giants. And here it would seem that the unknown maker of this chart had done the same. Save in one place alone, where next to a yellow gem which might denote a sun was a zero stone!
Quickly I spun the bowl around, studying the loose pattern. Yes, there were other planets indicated about those colored suns, but they were done in tiny, amost invisible dots. Only the one was a gem.
"Why, think you?" Eet's question reached me.
"Because it was the source!" I could hardly believe that we might hold the answer to our quest. I think my unbelief was born in the subconscious thought that it would be one of those quests, such as fill the ancient ballads and sagas, wherein the end is never quite in the grasp of mortals.
But it is one thing to hold a star map and another to find on it some already known point. I was no astro-navigator and unless some point of reference marked on this metal matched our known charts, we could spend a lifetime looking, unable even to locate the territory it pictured.
"We know where it was found," Eet suggested.
"Yes, but it may be another case of a relic of an earlier civilization treasured by its finder long after and buried with one who never even knew the life form that fashioned it, let alone the planets it lists."
"The Zacathan may furnish our key, together with Ryzk, who does know these star lanes. The stars this shows may be largely uncharted now. But still, those two together might give us one point from which we can work."
"You will tell them?" That surprised me somewhat, for Eet had never before suggested hinting to anyone that the caches we had disclosed to the Patrol were not the sum total of the stones now in existence. In fact, our quest had been his plan from its inception.
"What is needful. That this is the clue to another treasure. The Zacathan will be drawn by his love of knowledge, Ryzk because it will be a chance for gain."
"But Zilwrich is to be returned with the treasure to the nearest port. Of course—" I began to see that perhaps Eet was not so reckless as he seemed in suggesting that we plunge into the unknown with a map which might be older than my species itself as our only guide. "Of course, we did not say _when_ we would return him."
There was in the back of my mind the thought that the Zacathan might even willingly agree to our plan to go exploring along the bowl route, the thirst for knowledge being as keen as it was among his kind.
But though I held that star map in my hand, my attention returned to the more important point for now.
"The stone, Eet."
"It is safe." He did not enlarge upon that.
There was, of course, this other stone, which, compared to the one we had used, was a mere pin point of substance, now so dull as to be overlooked by anyone not aware of its unusual properties. Did the amount of energy booster depend upon the size of the stone? I remembered how Eet had produced that burst of power which had brought us along the barrier of the wreckage. Had all that come from this dull bit which I could well cover with only a fraction of the tip of my little finger? It must be that we had learned only a small portion of what the stones could do.
I was most eager to get back to the ship, away from Waystar. And as the LB was on course, I began to wonder at the length of our trip. Surely we had not been this far from where we had set down on the dead moon.
"The homer—" I moved to see that dial. Its indicator showed set to bring us back on automatics to the _Wendwind_. Suddenly I doubted its efficiency. Most of the alterations in the controls of the LB had been rigged by Ryzk, were meant to be only temporary, and had been made with difficulty—though it was true that a Free Trader had training in repairs and extempore rigging which the average spacer never learned.
Suppose the linkage with the parent ship was faulty? We could be lost in space. Yet it was true we were holding to a course.
"Certainly," Eet broke into my ominous chain of thought. "But not, I believe, to the moon. And if they go into hyper—"
"You mean—they have taken off? Not waiting for us?" Perhaps that fear, too, had ever lain in the depths of my mind. Our visit to Waystar had been so rash an undertaking that Ryzk and the Zacathan could well have written us off almost as soon as we left for the pirate station. Or Zilwrich might have begun to fail and the pilot, realizing the Zacathan was too far spent to object, and wanting to get him to some aid—There were many reasons I could count for myself for the _Wendwind_ to have taken off. But we were still on course for something—a course which would hold only until the ship went into hyper for a system jump. If that happened, our guide line would snap and we would be adrift—with only a return to Waystar or a landing on one of the dead worlds for our future.
"If they left for out-system they would hyper—"
"If they do not know the system they must reach its outermost planet before they do," Eet reminded me.
"The stone—if we use that to step up energy to join them—"
"Such a journey must be made with great care. To maneuver the LB and the ship together during flight—" But it was apparent that Eet was thinking for himself as well as for my enlightenment. He studied the control board and now he shook his head. "It is a matter of great risk. These are not true controls, only improvised, and so might not serve us at a moment of pressing need."
"A choice between two evils," I pointed out. "We stay here and die, or we take the chance of meeting with the ship. As long as we remain on course we are linked with her. Why doesn't"—I was suddenly struck by a new thought—"Ryzk know we are following? The fact that we are should have registered—"
"The indicator in the ship may have failed. Or perhaps he does not choose to wait."
If the pilot did not want to wait—he had the _Wendwind_ , he had the Zacathan, and he had an excellent excuse for our disappearance. He might return to the nearest port with the rescued archaeologist, the coordinates of Waystar to deliver to the Patrol, a ship he could claim for back wages. All in all, the master stars lay in his hand in this game and we had no comets to cut across the playing board to bring him down—except the zero stone.
"Into the hammock," Eet warned now. "I shall cut in the stone power. And hope that the ship does not hyper before we can catch up."
I lay down again. But Eet remained by the controls. Could the alien body he had wished upon himself stand the strain of not using such protection as the LB afforded? If Eet blacked out, I could not take his place, and we could well strike the _Wendwind_ with projectile speed.
In the past I had been through the strain of take-offs in ships built for speed. But the LB was not such. I could only remember that the original purpose of the craft was to flee a stricken ship, and that it must thus be fit to take the strain of a leap away from danger. To sustain such energy, however, was another matter. Now I lay in the hammock and endured, though I did not quite black out. It seemed as if the very material of the walls about us, protested against the force. And the bowl, which I still held, had a fiery spot of light on its surface where the infinitely smaller stone answered the burst of power from the larger, which Eet had concealed.
I endured and I watched through a haze the furred body of Eet, his arms flung out, his fingers crooked to hold in position at the controls. Then I heard the loud rasp of painful breathing which was not mine alone. And every second I expected a break in the link tying us to the ship, the signal that the _Wendwind_ has gone into hyper, vanished out of the space we knew.
Either my sight was affected by the strain or else Eet was _so_ pinned by our speed that he could not function well, but I saw mistily his one hand creep at a painfully slow rate to thumb a single lever. Then we were free of that punishing pressure. I clawed my way out of the hammock, swung across to elbow Eet aide, and took his place, facing the small battery of winking lights and warnings I did understand and which Ryzk had patiently drilled me to respond to.
We had reached match distance of the _Wendwind_ and must now join her. Automatics had been set up to deal with much of this, but there were certain alarms I must be ready to answer if they were triggered. And if Ryzk had ignored our following signal, he could not, short of winking instantly into hyper, avoid our present homing.
I sweated out those endless seconds at the board, my fingers poised and ready to make any correction, watching the dials whose reading could mean life or death not only to us but to the ship we fought to join. Then we were at our goal. The visa-screen winked on to show the gap of the bay for the LB and we bumped into it. The screen went dark again as the leaves of the bay closed about us. I was weak with relief. But Eet arose from where he had crouched, hanging to one end of the other hammock.
"There is trouble—"
He did not complete that thought. I cannot tell now—there are no words known to my species to describe what happened then—for we were not bedded down, prepared for the transition as was needful. We were not even warned. Seconds only had brought us in before the ship went into hyper.
There was the taste of blood in my mouth. I drooled it forth to flow stickily down my chin. When I opened my eyes I was in the dark, a dark which brought the terror of blindness with it. My whole body was one great ache which, when I tried to move, became sheer agony. But somehow I got my hand to my head, wiped it without knowing across the stickiness of blood. I could not _see!_
"Eet!" I think I screamed that. The sound echoed in my ears, adding to the pain in my head.
There was no answer. The dark continued. I tried to feel about me and my hand struck against solid substance as memory stirred. I was in the LB, we had returned to the ship just an instant before it had gone into hyper.
How badly I was hurt I did not know. As the LBs had originally been fashioned to take care of injured survivors of some space catastrophe, I needed only get back to the hammock and the craft would be activated into treating me.
I felt about me, seeking the touch of webbing. But though my one arm obeyed me, I could not move the other at all. And I touched nothing but wall. I tried to inch my body along, sliding my fingers against that wall, seeking some break, some change in its surface. The quarters of the LB were so confined that surely I could soon find one of the hammocks. I flung my arm up and out, rotating it through the thick darkness. It encountered nothing.
But I _was_ in the LB and it was too small for me not to have found the hammock by now. The thought of the hammock, that it was ready to soothe my pain, to apply restoratives and healing, so filled me that I forced myself to greater efforts to find it. But my agonizing movements, so slow and limited, told me that there was no hammock. And whatever space in which I now lay was not in the LB. My hand fell to the floor and touched a small, inert body. Eet! Not as I had seen him last, my exploring fingers reported. But Eet, the mutant, as he had been from birth.
I drew my fingers down his furred side and thought I detected a very faint fluttering there, as if his heart still beat. Then I tried to discover by touch alone whether he bore any noticeable wounds. The darkness—I would not allow myself to accept the thought that I was _blind_ —took on a heavy, smothering quality. I was gasping as if the lack of light was also a lack of air. Then I feared that it was, and that we had been sealed in somewhere to suffocate.
Eet did not answer my thoughts, which I tried to make coherent. I felt on, beyond him, and sometime later gave up the hope we were in the LB. Instead we lay in a confined space with a door which would not yield to the small force I could exert against it. We must be on board the _Wendwind_ —and I believed we were now imprisoned in one of those stripped lower cabins which had been altered for cargo transport. This could only mean that Ryzk had taken command. What he might have told the Zacathan I did not know. Our actions had been strange enough to give credence to some story that we operated outside the law, and Ryzk could testify truly that we had brought him on board without his knowledge. The Zacathans were esper—telepaths. Ryzk could tell the exact truth and Zilwrich would have to believe him. We could well be on our way now to being delivered to the Patrol as kidnapers and shady dealers with the pirates of Waystar. Yes, as I painfully marshaled the facts as another would see them I realized that Ryzk could make an excellent case, and Zilwrich would back him up.
That we brought back part of the treasure meant nothing. We could have done that and still planned to keep it, and the Zacathan, for ransom. Such deals were far from unknown.
If Ryzk had been black-listed, bringing us in might return him to the rolls. And if we underwent, or I underwent, deep interrogation—the whole affair of the zero stone would be known. It would be clear that we were guilty of what the Patrol might deem double-dealing. Ryzk had only to play a completely honest man at the nearest port and we would have lost our big gamble.
It seemed so hopeless when I thought it all out that I could see no possible counter on our part. Had we one of the zero stones we might—so much had I come to accept the unusual powers of those strange gems-have a fighting chance. Eet—if he were not dead—or dying—might just—
I felt my way back to that small body, gathered it carefully up so that Eet's head rested against me, and put my good arm protectingly around it. I thought now that I no longer felt that small stirring of a heartbeat. There was no answer to my mind-call. So there was good reason to believe that Eet was dead. And in that moment I forgot all my annoyance at his interference in my life, the way he had taken over the ordering of my days. Perhaps I was one who needed such dependence upon a stronger will. There had been my father, then Vondar Ustle, then Eet—
Only I would not accept that this was the end. If Eet was dead, then Ryzk would pay for that death. I had thought of the aid of the stone, and the aid of Eet, and both of them were gone. What remained was myself, and I was not ready to say I was finished.
I had always believed that I was no esper. Certainly no such talent was apparent in me before I met Eet. He had touched my mind for communication and I had learned that use from him. He had at one uncomfortable time given me mental contact with another human in order to prove our innocence to a Patrol officer. Then he had taught me to use the hallucinatory change and I had been the one to discover that the zero stone could bring about an almost total change.
But Eet—he was either dead or very close to it. I had neither Eet nor the stone. I was hurt, how badly I could not tell, and I was a prisoner. There was only one small—very small—spark of hope left—the Zacathan.
He was normally esper, as was Eet. Could I possibly reach him now? Make some appeal?
I stared into a dark which I hoped would not be my portion all the rest of my life, but in my mind I pictured the face of Zilwrich as I had seen it last. And I strove to hold that face in mind, not now for the purpose of making it mine, but rather as a homing point for my thought-seek. And I aimed, not a coherent thought, but a signal for attention, a cry for help.
Then—I touched! It was as if I had put tip of finger to a falder leaf which had instantly coiled away from contact with my flesh. Then—it returned.
But I was racked with disappointment. With Eet mind-touch had been clear, as it had been with the Zacathan when the mutant was present. This was a jumble of a language I did not know, poured at me in a wealth of impressions too fast for me to sort and understand, forming a sickening, chaotic whirl, so that I must retreat, drop touch.
Eet was the connecting link I must have. Otherwise I could only try until that whirl of alien thought drove my brain into mindlessness. I considered the chances. I could stay prisoner here for whatever purpose Ryzk had in mind. Or I could try the Zacathan again. And it was not in me to accept the helplessness of that first choice.
So, warily, as a man might seek a path across a quaking bog ready to swallow him up in a thousand hungry mud mouths, I sent out once more the mind-seek. But this time I thought my message—slowly, impression by impression, and doggedly held to what I had to convey as the stream of the alien mind lapped over it. I did not try to tell Zilwrich anything, as I would have "talked" to Eet. I merely thought out over and over again what I would have him know, letting it lie for him to pick up as he could. Though I feared my slow channel was as unintelligible to him as his frighteningly swift flow was to me.
Once, twice, three times, a fourth, I thought through what I made as my plea. Then I could hold no longer. The pain of my body was as nothing compared to the pain now filling my mind. And I lost contact as well as consciousness, just as I had when we had snapped into hyper.
It was as if I were being pricked over and over again by the sharp point of a needle. I stirred under that torment, which was small and far away at first, and then became so much the greater, more insistent. And I fought to remain in the safety of nothingness. Prick—the summons to what I did not want continued.
"Eet?" But it was not Eet—no—
"Wait—"
Wait for what, who? I did not care. Eet? No, Eet was dead. And I would be dead. Death was not caring, not needing to care, or feel, or think—And I wanted just that—no more stirring of life, which hurt both mind and body. Eet was dead, and I was dead, or would be if the pricking would only stop and leave me in peace.
"Awake—"
Awake? I thought it was "wait." Not that it mattered. Nothing mattered—
"Awake!"
A shouting in my head. I hurt and that hurting came from outside. I turned my head from side to side, as if to shake out the voice in my mind.
"Keep awake!" screamed that order and the pain it caused me aroused me further from my stupor. I was moaning a little, whimpering through the dark a plea to be let alone, left to the death which was rest.
"Keep awake!"
Hammering inside my skull. Now I could hear my own whimpering plaint and was unable to stop. But also with the pain came an awareness which was a barrier against my slipping back into the nothingness.
"Awake—hold—"
Hold what? My rolling head? There was nothing to hold.
Then I sensed, not words echoing through my bruised mind, but something else—a stiffening, a support against which my feeble thoughts could find root and sustenance. And this continued until I stared wide-eyed into the dark, as much another person inwardly as I had been outwardly with the hallucinations born of the zero stone. For only a limited time, somehow I knew, would that support me. And during that time I must make any attempt I could to help myself.
XV
Somehow I got to my feet, still holding Eet against me with my good arm, my other hanging uselessly by my side. I was ready to move, but where, against what—or whom? Realizing I was still helplessly caught in this pocket of dark, I was ready to slump again into a stupor.
"Wait—be ready—" There was a sense of strain in that message, as if he who sent it were making a vast effort.
Well, I was waiting and ready, but for how long? And in this dark time seemed forever and ever, not measured by any standard I had known.
Then came sound, a small grating, and I knew a leap of heart—I was not blind after all! There was a line of light to my right. I lurched in that direction as that line grew from a slit into an opening I could squeeze through—though I was blinking against the discomfort of light.
I brought out and up against the wall of the well which was the core of the ship, too spent for a moment to turn and see who had freed me. But leaning one shoulder against the wall, I was able to face about.
Zilwrich, whom I had last seen lying on the pallet, supported himself with his two arms rigid against the floor, clearly at the end of the flutter of strength which had made him crawl to the door of my cell. He lifted his head with manifest effort.
"You—are—free—To you—the rest—"
Free but weaponless, and as near the end of my resources as the Zacathan, though not yet finished. Somehow I was able to lay Eet on the floor, get my good arm about Zilwrich, and half drag the Zacathan back to the bed he had crawled from. Then I stumbled out, picked up the mutant, and brought him back, nursed against me, though no tending would return life to that small body.
"Tell me." I used the Basic speech, glad to be able to relinquish touch with that bewildering alien mind. "What happened?"
"Ryzk"—Zilwrich spoke slowly as if each word came hard—"would go to Lylestane—return me—the treasure—"
"And turn us in," I ended, "probably as accomplices in Guild plotting."
"He—wishes—reinstatement. I did not know you had returned alive—until your mind-seek. He said—you died—when we went into hyper."
I glanced down at the limp body pressed to mine. "One of us did."
I might be free inside the ship, but that I could do anything to change the course of events I doubted. Ryzk would return us to Lylestane and we—I—would find the balance of justice heavily weighted against me. Not only were circumstances largely in the pilot's favor, but under the scanner they would have out of me all that the zero stone meant. And—the zero stone!
Eet had concealed it somewhere in the LB. As far as I knew Ryzk did not suspect it. If I could get hand on it again—I was not sure how I could use it as a weapon. But that it had possiblities of this sort there was no doubt. The LB-but Eet had hidden the stone and Eet was dead.
The bowl—if I had that I could trace the zero stone by the fire of the one inlaid in it.
"The treasure—where is it?"
"In the lock safe." Zilwrich's eyes were on me with piercing keenness, but he was ready enough with that information.
The lock safe—If Ryzk had sealed that with his own thumb, I had no chance of getting the bowl. The compartment would remain closed until he chose to release it.
"No." It would seem that like Eet the Zacathan could readily read my mind, but that did not matter. "No—it is sealed to me."
"He allowed that?"
"He had to. What is this thing you must have—that the bowl will bring you nearer to—a weapon?"
"I do not know if it can be a weapon. But it is a source of power beyond our reckoning. Eet hid it in the LB; the bowl will find it for me."
"Help me—to the lock safe."
It was a case of the lame leading the crippled. We made a hard journey of a short space. But I was able to steady the alien while he activated the thumb lock and I scooped out the bowl. He held it tightly to him as I guided and supported him back to his bed.
Before he released the bowl to me he turned it around in his hands, examining it closely. Finally one of his finger talons tapped the tiny zero stone.
"This you seek."
"We have long sought it, Eet and I." There was no use in concealing the truth any longer. We might not make the voyage we had planned, going out among the uncharted stars in search of an ancient world which was the source of the stones, but it was the here and now which mattered most—the finding of the one Eet had hidden.
"It is a map, and you hunt the treasure you believe lies at its end?"
"More than such treasure as you found in the tomb." And, as tersely as I could, I told him the story of the zero stones—the one in my father's ring, those of the caches on the unknown planet, that which Eet had secreted, and how we had used it since.
"I see. Take this then." Zilwrich held out the bowl. "Find your hidden stone. It would seem that we were on the edge of a vast discovery when we uncovered this—but one which would unleash perils such as a man thinks twice about loosing."
I held the bowl to me as I had held Eet, using my shoulder against the wall to keep erect, shambling from Zilwrich's cabin to the ladder, down which I fell rather than climbed, to reach the LB's berth. The last steps of that journey were such a drain that I could hardly take them.
Then I was back in the craft which had served us so well. I fought to keep moving, holding the bowl a little away from ne now, watching the zero stone. It glimmered and then broke into vivid life. But it was hard to see how I could use it as a guide, since there seemed no variation in that light. However, I must try.
I moved jerkily, first to the tail, without any change I could detect in the degree of emanation from the bowl stone. But as I came up the right side of the small ship on return the bowl moved in my grasp, fought my hold. I released it. As the zero stone, on its first awakening, had pulled me across space to the derelict ship where others of its kind lay, so did the bowl cross, to hang suspended against a part of the casing. I jerked and tore at the rim of the casing, hoping Eet had not been able to seal in the stone too tightly. As my nails broke and my fingers were lacerated by the sharp edging I began to despair. One-handed there was little I could do to force it.
But I continued to fight, and at last I must have touched what lock was there, for a whole section of panel fell down and I saw the brilliant blaze of the large stone within. The bowl snapped to meet it until stone touched stone, and I did not try to part them. With the bowl I began to retrace my way.
When I subsided beside Zilwrich, the bowl on the floor between us, he looked at the gems but seemed as content as I at that moment to do no more. Not only was I too weak to prod my body to more effort, but my thoughts were dulled, slow. Now that I had found the second stone, I could not see any way to make use of it against Ryzk. It seemed that, having achieved this one small success, I was finished.
Eet lay on the edge of the Zacathan's pallet and one of the alien's scaled hands rested on the mutant's head.
"This one is not dead—"
I was startled out of my lethargy. "But—"
"There is still the spark of life, very low, very dim, but there."
I was no medico, and even if I had been I would have had no knowledge to deduce the mutant's hurts. My own helplessness was an added burden. Eet would die and there was nothing I could do—
Or was there?
For a little beyond Eet's head was the bowl, the stones close-welded together. The zero stone was power. It had the power to turn us into the seeming of others and hold that seeming. And I had been able to turn Eet into a cat because I had sprung that change on him when he did not expect it. Could I will, not change, but will life itself into the mutant's body?
As long as there was a faint spark left, I must try.
I took the left hand on my limp and useless arm with my right, moved the numb palm to rest on the stones, not caring if I would be burned. At least I would not feel it. The right I put on Eet's head. I set my mind to the task, summoning, not some strange disguise for my companion, but rather the sight of him as he was alive. So did I fight my battle—with mind, with a hand which will always bear the scars, with my determination, against death itself, or what Eet's kind knew as the end of existence. And I strove with the power passing through me to find that spark Zilwrich said existed, to fan it into flame.
The stones made a fire to fill one's sight, shutting out the cabin, the Zacathan, even Eet, but I continued to hold the image of the live Eet in my mind. My eyes which had been useless in the dark of the cell were now blinded again, by light. But I held fast in spite of that in me which cringed, and cried, and tried to flee.
Nor was I truly conscious of why I fought that battle, save that it was one which I must face to the end. I was at last done, my seared hand lying palm up on my knee, the bowl and stone hidden from me by a fold of cloth. Eet no longer lay limp, with the semblance of death, but sat on his haunches, his paw-hands folded over his middle, his stance one of alert life, of complete restoration.
I caught communication, or the edge of it, between the Zacathan and my companion. But so difficult was it now for me to hold to any thought that it was more like hearing a murmur or whisper from across a room.
Eet moved with all his old agility, bringing out the aid kit, seeing to my hand, giving me also a shot to counteract the hurt in my arm. But to me this had little or no meaning. I watched the Zacathan agree to something Eet suggested and the mutant carry the bowl out of the room—into hiding again, I supposed. But all I wanted was sleep.
Hunger awoke me. I was still in the Zacathan's cabin. If Ryzk had paid him a visit during the time I slept he had not seen fit to return me to custody. But that I had slept worried me vaguely. There was much to be done and I had failed to do it.
Eet whisked in, almost as if my waking had sent him some signal. He carried in his mouth as he came two of those tubes of E-rations. And seeing them, for a second or two I forgot all else. But when I had squeezed one into my mouth and savored the first few swallows (though normally I would not have considered them appetizing) I had a question:
"Ryzk?"
"We can do nothing while in hyper," Eet reported. "And he has found his own amusement. It seems that this ship was not thoroughly searched when it was taken in as a smuggler. Somehow Ryzk uncovered a supply of vorx and is now having sweet dreams in his cabin."
Vorx was potent enough to give anyone dreams—though whether they were sweet was another question. It was not only an intoxicating drink, but so acted on Terran bodies that it was also hallucinatory. That Ryzk had been searching the ship did not surprise me either. The boredom of space travel would set any man immured within these walls during hyper passage to do such to relieve his tedium. And Ryzk might have known this was a smuggler sold after confiscation.
"He had help—" Eet commented. There was such a bubbling renewal of well-being in him as made me envious, perhaps tired of being on the edge wash of such energy.
"From you?"
"From our distinguished colleague." Eet nodded to the Zacathan.
"It would seem that Ryzk's weakness is drink," Zilwrich agreed. "While it is wrong of anyone to play upon another's weakness, there are times when such a fall from Full Grace is necessary. I deemed that I might take on error-load for once in this way. We need Ryzk's room rather than his company."
"If we come out of hyper in the Lylestane system we shall be in Patrol territory," I replied a little sourly.
"It is possible to come out and go in again before a challenge of boarding can be delivered," Zilwrich returned. "I have a duty to report the raid on our camp, that is true. But I have also a duty to those who sent my party there. This map is such a find as we come upon perhaps once in a thousand years. If we can find a clue to the location of the planet it marks, then a scouting trip thither at this time means more than arousing the law as to what has happened in one raid."
"But Ryzk is pilot. He will not agree to go off known charts. And if he's made up his mind to turn us in—"
"Off the charts," repeated the Zacathan thoughtfully. "Of that we cannot be sure as yet. Look—"
He produced a tri-dee projector which I knew to be part of the equipment of the control cabin. At a push of his finger there flashed on the wall a blowup of a star chart. Being no astro-navigator, I could not read it to any real purpose, save that I could make out the position of stars and sight the coded co-ordinates for hyper jumps under each.
"This is on the edge of the dead strip," Zilwrich informed me. "To your left and third from the corner is the blasted system of Waystar. It must have been scouted three centuries ago, by your time, from the dates on this chart. This is one of the old Blue maps. Now, look upon the bowl, imagine that the dead sun on that system is a red dwarf, turn the bowl two degrees left—"
I held up the bowl and rotated it slowly, looking from it to the tri-dee chart on the wall. Though I was not taught to read such maps I could see he was right! Not only did the blasted system we had just fled appear on the bowl as one about the red-dwarf star—a dying sun—but there was a course to be traced from that to the zero stone.
"No co-ordinates for hyper," I pointed out. "It would be the most reckless kind of guesswork. And even a scout trained for exploring jumps would take chances of two comets to a star of coming out safe."
"Look at the bowl through this." It would seem that Eet must have been gathering aids from all over the ship, for what the Zacathan handed me now was my own jeweler's lens.
As I inspected the constellation engraved on the metal through the magnification of the lens I saw there were minute identations there, though I could not translate any.
" _Their_ hyper code perhaps," the Zacathan continued.
"Still no good to us."
"Of that I am not sure. We have those of the dead system—from that—"
"You can work?" Of course, he was an archaeologist and such puzzles were common to him. I lost something of my mood of depression. Perhaps because my hunger had been satisfied and I could now use my arm and hand to better advantage, I was regaining confidence not only in myself but in the knowledge and ingenuity of my companions.
When I put the bowl on the floor, open side down so that its star-specked dome was revealed, Eet squatted by it. He had taken up the lens, holding it in his paw-hands, his head bent over it as if his nose were smelling out the pictured solar systems.
"It can be done." His thought was not only clear; it was as confident as if there had been no obstructions at all between us and success. "We return to the dead system by reversing Ryzk's tape—"
"And so straight into what may be a vla-wasp nest," I commented. "But continue. Perhaps you have an answer for that also. Then what do we do, unless the Honorable Elder"—I gave Zilwrich the proper title of formal address—"can read these co-ordinates."
Eet did not close his mind as he had upon occasion, but I read a side flash of what might be indecision. I had never read fear in Eet's communications—awareness of danger, but not fear. But this had the aura of just that emotion.
And inspiration hit me in the same instant. " _You_ can read these!" I had not perhaps meant it as an accusation, but it came forth that way.
His head turned on his too-long neck so that he could look at me.
"Old habits, memories, die hard," he answered obliquely, as he sometimes did. He turned the lens about, giving me the impression of uneasiness, of one wanting to escape coming to a decision.
I caught a flicker of alien mind-flow, and for a moment resented that communication I could not share. It was my guess that the alien and the mutant might be in argument about just the knowledge I accused Eet of having.
"Just so." Eet resumed touch with me. "No, I cannot read these. But they are enough like another form of record for me to guess to more purpose than the rest of you." And such was the finality of that answer that I knew better than to try to pry at how he could be familiar with any record approximating that of a Forerunner race living millenniums ago. The old problem of who—or _what_ —Eet was crossed my mind.
Though he made no comment, the impression remained that any guessing he would do would be against his inclination and that he had a personal reason for disliking the situation fortune had forced upon him.
It seemed that now I was to serve as his hands. And back in the control cabin I made ready to follow his instructions to reverse the course Ryzk had set and return us, as soon as we emerged near Lylestane, to the vicinity of Waystar.
Ryzk did not appear. Apparently the smugglers' drink was of great potency. What would have happened when we came out of hyper and he was not at the controls, I do not know. Perhaps we would have aimlessly cruised the Lylestane system as a traffic hazard until some Patrol ship linked beam and dragged us in as a derelict.
I punched out the figures Eet fed me and we were wrenched back on a return course once again from Lylestane. Once more in hyper, we had plenty of time to meditate on the numerous dangers our appearance near Waystar would range against us. Certainly our successful escape with the treasure had alerted all the defenses of the pirate stronghold. They would be expecting a visit from the Patrol on one hand, now that strangers knew the co-ordinates of their hide-out, and trouble from others, perhaps even the Guild, demanding an account of how or why loot could be so summarily removed from what was believed to be an impregnable safe place.
The only answer would be that we dared not linger long enough in the dead system to be detected. Our unarmed ship had no defense against what the Jacks could easily muster. Therefore, we must follow exactly the same procedure we had on emerging near Lylestane: We must have the other course ready to punch in and spend as little time in normal space as we could.
Success in that maneuver would depend entirely on what Zilwrich and Eet could produce in the way of a new course. And since I was no help to them, the ship and Ryzk were my concern.
My most practical answer to Ryzk was to apply a force lock on his cabin. He sobered up when we were back in hyper and his struggle with the door lock led me to state through the intercom that we had taken over. More than that I did not explain, and I turned off the com thereafter, so his demands went unheard. E-rations and water went to him through the regular supply vent and I left him to consider, soberly I hoped, the folly of the immediate past in relationship to the _Wendwind_ and her owners.
For the rest I tinkered in the small repair shop. The crossbows Ryzk had earlier produced I refined, making more zoran heads for their bolts. I had no mind to go exploring on an unknown planet unarmed, as I had once done in the past.
If by some miracle of fortune we did reach the world indicated by the zero stone, we would not know what we might face there. It could be a planet on which those of our kind could not live without suits; it could be inhabited by beings infinitely superior to us in every way, who would be as hostile to strangers as the Veeps of Waystar. Though the civilization the bowl represented must have ended eons past, others could have arisen from the degenerate dregs of that, and we might face such challengers as we could not even imagine. When I got to that point of my speculations, I handled my crossbows with very bleak attention to all their manifest defects.
Our first test would come when we left hyper in the dead system. As that moment approached I was tense and nervy. I saw practically nothing of Eet and Zilwrich except when I supplied them with food and drink. And I was almost tempted to let Ryzk out of his cabin in order to have someone to match fears with.
But when the alarm shattered the too-great silence of the ship, Eet was on hand in the control cabin. He curled into my lap as I settled in the pilot's seat—though he kept his mind closed, as if it were full of some precious knowledge and sharing that too soon might spill what could not be regained.
We came out of hyper and I punched the proper buttons for a reading of our present site. At least fortune had favored us to the point that we had emerged very close to that place where we had entered on our first trip, at the outer edge of the dead system.
But we were given very little time to congratulate ourselves on besting what was perhaps the smallest portion of the ordeal facing us. For there was an alarm ringing wildly through the cabin. We had been caught by a snooper and now we could expect a traction beam. My hands rested on the edge of the control board. I was ready to punch out the course Eet supplied. But would he feed me one, and could I set it quickly enough to avoid the linkage which would hold us for taking by the enemy?
XVI
Eet was ready for me, though the co-ordinates he flashed into my mind had no meaning for me. I was merely the means of putting finger tip to controls to punch them in. Only, it seemed those fingers did not move fast enough. I could feel the force of the locking beam catch at our ship.
We passed into hyper. But once the dizzy spin in my head cleared and I knew we had made the transition, I was aware that we had brought our enemy with us. Instead of snapping the lock beam in our return to hyper, we had, through some balance of force against force, dragged the source of that beam with us! We had danger locked to the ship, ready to attack as soon as we moved into normal space again.
There is no maneuvering in hyper. To do so would be to nullify the co-ordinates. And one would emerge utterly lost in space, if one were lucky, or perhaps in the very heart of a blazing sun. We were both prisoners here until we finished the voyage the Zacathan and Eet had set us. But there was this much: The enemy was as helplesss as we—until we went out. And not being prepared for hyper transfer, they might be badly shaken, though they would have the length of our trip in which to pull themselves together.
"Jern!" Ryzk bawled through the ship's com. "Jern, what are you trying to do?"
It sounded very much as if the pilot not only had recovered from his drinking bout but was genuinely alarmed. Alarmed enough, I speculated, to be willing to work with us? Not that I trusted him now.
I picked up the mike. "We are in hyper—with a companion."
"We're linked!" he roared back.
"I said we had a companion. But he cannot move any better than we. We are both in hyper."
"Going where?"
"You name it!" Our momentary escape was acting on me like a shot of exult. Not that I had ever tried the stuff, but I had heard enough to judge that this must be akin to the heady feeling those addicts gained. When we snapped out of hyper we might be in grave danger, but we had now a respite and time to plan.
But his question echoed in my mind. Going where? To a planet which might or might not still exist. And if it did—what would it be like?
At that moment I felt as if I would more than anything like to be a believer in the gods of the planet-rooted. This was the time when one would prefer to kneel in some fane as did, say, the Alfandi, thrusting a god-call deep into ground already pitted with holes left by other's rods, pulling hard upon the cord which would set its top quivering to give off the faint sound meant to reach the ear—if one might grant a spiritual being an ear—of that High One, and thus alerting the Over-Intelligence to listen to one's plea. I had met with the worshippers of many gods and many demons on many worlds. And complete belief gave a man security which was denied to the onlooker. That there was a purpose behind the Galaxy I would be the first to agree. But I could not bow my head to a planet-based god.
There was one belief I had read in the old tapes, that brain and mind are not the same. That the brain is allied to the body and serves it, while the mind is able to function in more than one dimension—hence esper talents, born of the mind and not the brain.
Now when I came from the control cabin I found Zilwrich seated on his pallet, and it seemed that he tried to prove the truth of this old theory, for he held between his two hands the bowl. His eyes were closed and he was breathing in small, shallow gasps. Eet, who had preceded me at his usual speed, had taken a position which mimicked that of the Zacathan, his small hand-paws resting on the rim of the bowl, his eyes also closed. And there was an aura of esper power which even I could feel.
What they were trying to do I did not know. But I felt that my presence was an intrusion there. I backed away, closing the door behind me. But at the same time my triumph ebbed. And the fact that we had a companion locked to us began to assume the shadow of menace. If Ryzk could only be trusted! Perhaps he could as long as his own skin was in danger. The coordinates which had brought us here—I reclimbed the way to the control cabin. We had used a return of Ryzk's setting to take us back to the dead system. Suppose I now erased those co-ordinates from the tape. Then no move of Ryzk's could return us, only what lay in Eet's and the Zacathan's memories. Loosed in the unknown, the pilot would be no great danger, and we needed badly any knowledge he might have to help us to deal with the enemy once we returned to normal space.
I set the erase on the tape before allowing myself to have second thoughts. Then I went to unseal the pilot's cabin. He lay on his bunk but turned his head to stare at me as I stood in the doorway. I had not brought one of the crossbows. After all, I was trained in a variety of weaponless fighting methods, and I did not think we were less than evenly matched, since he had nothing save similar skills to use against me.
"What are we doing?" He had lost the anger tinged with alarm which had colored his first demand through the com.
"Heading for a point on a Forerunner chart."
"Who's linked with us?"
"Someone out of Waystar is our best guess."
"They followed us!" He was genuinely astonished.
I shook my head. "We came back to the Waystar system. It was the only recognizable point of reference on the chart."
He turned his head away, now looking to the ceiling. "So—what happens when we come out of hyper?"
"With luck we are in a system not on the charts. But—can we break linkage when we come out of hyper?"
He did not answer at once. There was a sharp frown line between his brows. And then he replied to my question with another.
"What are you after, Jern?"
"Perhaps a whole world of Forerunner artifacts. What is that worth?"
"Why ask me? Anyone knows that is not to be reckoned in credits. Is Zilwrich behind this? Or is it your gamble?"
"Both. Zilwrich and Eet together set up the co-ordinates."
He grimaced. "So we sweat out a landing, maybe to be sun-cooked or worse when we come out—"
"And if we are not, but take the others with us?" I brought him back to the matter over which we might have some control.
He sat up. The sickly-sweet smell of the drink was strong. But to my eyes he appeared sober. Now he put his elbows on his knees and bent over to rest his head on his hands. I could no longer see his face. He sighed.
"All right. In hyper we can't switch course. So we can't try to shake them loose. We _can_ set the emerge on high velocity. It will mean blacking out, maybe taking a beating. But it is the only way I know of to break the link. We will have to rig special webbing or we won't survive at all."
"And if we do break the link?"
"If we pulled them in with us, the course is only set on our ship. The break will take us out, not them. They would have to gamble on an emerge. It might land them in the same system, or somewhere else. How do I know? I say it is barely possible. I am not planning on more than one thin chance in ten thousand." And his voice said that was very optimistic odds.
"You can do it?"
"It looks as if we have no choice. Yes, I can rig it, given time enough. What are the odds if we come out still linked?"
"We are unarmed, and they can take us over. They have no use for us, only what we carry."
He sighed again. "About what I thought. You're all fools and I have to go along."
But perhaps he was not wholly convinced until we entered the control cabin and he pushed past me to read the dial above the journey setting.
"Erased!" He whirled to face me, his lips twisted into a snarl.
"No turning back." I braced myself, tensed against attack. Then I saw his eyes change and knew that if he meant me harm in the future, he was willing to wait for such a reckoning. The main interest now must be the ship and our possible manner of escape from our unseen companion.
Just as Eet and Zilwrich in their mysterious occupation with the bowl had given me no explanations, so did Ryzk keep his own counsel about the alterations he made in some wiring. But he did keep me with him as a very ignorant assistant, to hand tools, to hold this or that while he made delicate adjustments.
"This will have to be redone," he said, "before we make a return. It is only temporary. I cannot even swear it will work. We'll need heavy webs—"
We set about providing those, too. The two shock-prepared seats in the control cabin were reinforced with what we could strip off the bunks in our two cubbys. Then we descended to the section where Eet and the Zacathan were in session to provide Zilwrich with such safeguards as we could rig. Eet, I supposed, would share my seat as usual.
I tapped lightly on the door behind which I had left the two enwrapt, with the bowl between them.
"Enter," called Zilwrich.
He lay now, his whole body expressive of a vast exhaustion. I could not see the bowl. Eet, too, lay there, but his head came up and he watched us almost warily.
I explained what we would do.
"This thing _is_ possible?"
Again Ryzk shrugged. "I cannot swear to it on my name, if that is what you mean. It remains theoretical until we prove it one way or another. But if what you say is true, we have little choice."
"Very well," the Zacathan agreed. I waited for some comment, pro or con, from Eet. But such did not come. And that made me uneasy. But I would not press him, lest he confirm my worst doubts. It is better not to be met by pessimism when the situation already looks dark.
But Zilwrich had suggestions as to the rigging we must provide to counteract the strain on his body. And we carried out his instructions with all the skill we could summon. When we fastened the last of the improvised webbing Ryzk arose and stretched.
"I'll take cabin watch," he said as if there was no disputing that. But I did not miss the sudden flicker of eye Zilwrich made in my direction, as though he expected me to protest. However, we did not have Ryzk's experience and training in the pilot's seat. And with the erase on I did not see how he could do any harm.
He could have no reason to wish to surrender to a Waystar force. And they would give him, I was certain, no time to parley if he tried it. He left and I said to Eet via thought-send: "The tape is on erase. He cannot send us back."
"An elementary precaution," Eet returned crushingly. "If he does not kill us all at emerge, and his theory works, we may have a small chance."
"You do not sound too sure of that." My inner uneasiness increased.
"Machines are machines and cannot be made to function too far from their norm, or they will cease to function at all. However, doubtless this is the only answer. And we shall have other matters to consider after the emerge."
"Such as what?" I was not prepared to accept vagueness now. Forewarned is always forearmed.
"We have tried psychometry," the Zacathan broke in. "I am not greatly talented in that direction, but the two of us working so—"
The term he used meant nothing to me and he must have read my ignorance, for he explained, and I was glad that it was he and not the mutant, for he did not condescend.
"One concentrates upon some object and he who has the talent can so gather information concerning its past owners. There is, of course, the belief that any object connected with high emotion in usage, say a sword used in battle, will carry the most vivid impressions to be picked up by the sensitive."
And the bowl?"
"Unfortunately it has been a center point for the emotions of more than one individual, of more than one species even. And some of those owners must have been far removed from the norm we accept today. Thus we received a mass of emotional residue, some violent. Many impressions are overlaid, one upon another. It is as if one took a tattered skin, put over it a second, also rent but in other places, and over that a third such, then tried to see what lay beneath those unmatched rents.
"Our supposition that the bowl might be much older than the tomb in which it was found, belonging to a people different from those with whom it was buried, is right. For we have deduced, though it is very hard to define any one well, at least four overlays left by former possessors."
"And the zero stone?"
"That perhaps is the source of some of the difficulty we encountered. The force which animates it might well govern the unfortunate mixture of impressions. But this we can tell you—the map was of prime importance to those who first wrought it, though the bowl itself meant more to later possessors."
"Suppose we do find the source of the stones," I said. "What then? We cannot hope to control the traffic in them. Any man who has a monopoly on a treasure sets himself up as a target for the rest."
"A logical deduction," Zilwrich agreed. "We are four. And a secret such as this cannot remain a secret long, because of the nature of what we must exploit. Like it or not, you—we—shall have to deal with the authorities, or else live hunted men."
"We can choose the authorities with whom we deal," I replied, an idea forming in my mind.
"Logical and perhaps the best." Eet cut across my thought, picking it up in its half-formed state, following it straight to a decisive conclusion.
"And if those authorities are Zacathan—" I said it aloud.
Zilwrich eyed me. "You pay us much honor."
"By right." It gave me a small quirk of shame to have to answer so, to admit that it was the alien whom I might trust above those of my own species. Yet that was so. And I would hand to any one of their Council the secret of what we found here (if we found anything worth the title of secret) more willingly than I would to any of my own leaders. The Zacathans have never been empire builders, never sought colonies among the stars. They are observers, historians, teachers at times. But they were never swayed by the passions, desires, fanaticism which has from the first made both great heroes and villains among my own kind.
"And if this secret might well be one not to be shared?" Zilwrich asked.
"That, too, I could accept," I said promptly. But I knew that I did not speak for Eet, or for Ryzk, who must now be included as one of our number.
"We shall see," Eet answered, his reservations plain. Not for the first time I wondered whether Eet's dogged insistence that the quest of the stone's source be our main goal did not have some reason he had never shared with me. And then, could I, myself, completely surrender the stones, knowing what I could do with them, knowing that perhaps there was more, much more, we might learn from them? Supposing the Zacathans advised us to hide, destroy, blot out all we know of the gems. Could I agree to that with no regret?
Later I lay in my cabin thinking. Eet, lying beside me, did not touch those thoughts. But at last, to escape a dilemma I could not resolve until we had passed many ifs and buts in the future, I asked the mutant:
"This reading of the past of the bowl, what _did_ you learn of its past?"
"As Zilwrich said, there were several pasts and they were overlaid, mixed with one another until what we gained was so disjointed it was difficult to read any part of it and be sure we were correct. It was not made by those who fashioned the tomb. They came, I believe, long after, finding it themselves as a treasure-trove, leaving it with some ruler to whom they wished to pay funeral honor.
"The source of the stone—" he hesitated and the thought I picked up was one of puzzlement—"was not clear. Save that we do go now, if we have read the co-ordinates right, to that source. And the stone was set in the chart as a guide to those to whom it was very important. But that its native planet was their world of origin—that I do not think is the truth either. However, the reading was enough to set one's mind upside down, and the less I rethink on it the better!" With that he snapped mind-touch and curled into a ball to sleep. A state I followed.
The warning that we were at the end to our journey in hyper came some time later. As the Zacathan had assured us when we rigged his protection that he could manage it by himself, I made speed to the control cabin, Eet with me. Soon I was well wrapped in my webbing, watching Ryzk, in a like cocoon at the controls, trying to relax when the final test of our drastic emerge came.
It was bad, as bad or perhaps a fraction worse than that which had hit when we had joined the ship in the LB before the other jump—Only this time we had all the protection Ryzk's experience had been able to devise, and we came out in better shape.
As soon as I was fully conscious I looked to the radar. There were points registering on it, but they marked planets, not the ship locked to us through hyper.
"We did it!" Ryzk almost shouted. At the same time Eet scrambled along my still nearly immobilized body. I saw then what he held in a forepaw against his upper belly—the zero stone.
It was blazing with a brilliance I had not seen before except when we had put it to action. Yet now it was not adding to any power of ours. The glare grew, hurting the eyes. Eet gave an exclamation of pain and dropped it. He tried to pick it up again, but it was clear he could not use his paw-hand near that spot of fire. Now I could not even look directly at it.
I wondered if it was about to eat its way through the deck by the heat it was engendering.
"Blanket it!" Eet's cry was a warning. "Think dark—black!"
The power of his own thought swept mine along with it. I bent what mental energy I could summon to thinking dark. That we were able to control the surge of energy in the stone by such means astounded me. That awful brilliance faded. However, the stone did not return to its original dull lifelessness; it continued to contain a core of light which set it above any gem I had ever known and it lay in a small hollow which its power had melted out of the substance of the deck.
"Pliers—" I did not know whether they would help, for the heat of the stone might melt any metal touching it. But we could not pick it up in bare fingers and we dared not leave it lie, maybe to eat straight through the fabric of the ship level by level.
Ryzk stared at it, unable to understand just what had happened. But I had pulled out of the cocoon of webbing and managed to reach the box of tools he had used earlier. With pliers in hand I knelt to pick up the gem, fearing I might find it welded to the floor.
But it came away, though I could still feel heat and see that a hole in the deck beneath it was nearly melted through. Once on land, once in space, once on the edge of the wreckage we had used the zero stone as a guide. Could this small gem now bring us to the final goal of its home world?
We did not need it, since the bowl chart had already located the planet for us, fourth out from the sun. And oddly enough, once placed within the bowl, the furious blaze of the loose stone subsided into a fraction of its glow, as if the bowl governed the energy.
Though we kept a watch on the radar, there was no sign that the enemy had followed us into this system. And Ryzk set course for the fourth planet.
I half expected that time would have wrought a change in the sun, that it might have gone nova, imploded into a red dwarf, even burned out. But this was not so. It tested in the same class as was indicated on the ancient chart.
We went into scan orbit, our testers questing to inform us it was truly Arth type, though we were suspicious enough to keep all indicators on alert.
What we picked up on our viewers was amazing. I knew that Terra, from which my species had come into an immeasurably ancient galaxy, had been monstrously overcrowded in the last days before general emigration to the stars began—that cities had soared skyward, tunneled into depths, eaten their way across most of the continental land masses, even swung out into the seas. I knew that, but I had never seen it. Terran by descent I am, but Terra is across the galaxy now and more than half legend. Oh, we see the old tri-dees and listen to archaic tapes which are copied over and over again. But much of what we see is meaningless and there are long arguments as to what really did or did not exist in the days before Terrans roamed the star lanes.
Now I looked upon something like the jostling, crowded—terribly crowded—erections those tri-dees had shown. This was a planet where no empty earth, no sign of vegetation showed. It was covered, on the land masses by buildings, and even across the seas by strings of large platforms which were too regular in outline to be islands. The whole gave one a terrible sensation of claustrophobia, of choking pressure, of erection against erection, or against the earth of its foundations.
We passed from day to night in our orbit. But on the dark side no light showed. If there was life below—
But how could there be? They would be smothered, pushed, wedged out of existence! I could not conceive of life here.
"There is a landing port," Ryzk said suddenly, but he had a keener eye than I, or else we had swung over and past what he had seen. To me there was no break in that infernal mass of structures.
"Can you land?" I asked, knowing that treasure or no treasure, stone or no stone, I must force myself to set foot down there.
"On deters," Ryzk said. "Orbit twice for a bearing. There are no guide beams. Probably deserted." But he looked far from happy, and I thought perhaps he might share some of my feeling about what lay below.
He began to set a course. Then we lay back in our seats, our eyes on the visa-screen, watching the dead city-world reach up—for that was what it seemed to be doing—as if its towers were ready to drag us down to the world they had completely devoured.
XVII
It was a tribute to Ryzk's skill that our landing was three-point, exactly on fins. He rode the ship down her tail rockets as only a master pilot could do. Ad not for the first time I was led to wonder what had exiled him from his kind—drink alone? Then we lay in our webbing watching the visa-screen as our snooper made a complete circuit of what lay about us, reporting it within.
With that report I came to respect Ryzk's skill even more. It was as if we had been threaded into a slit between walls of towers whose assault against the sky was such that one could not immediately adjust one's thoughts to what one's eyes reported. Only now that we were in that forest of man-made giants could we see the hurts time had dealt them.
For the most part they were either gray-brown or a blue-green in color, and there was no sign of seam or join as one might sight with stone blocks or the like. But there were cracks in their once smooth sides, rents in their fabric, which were not windows or doors. We could see no indication of those.
Ryzk turned to check the atmosphere dials. "Arth type, livable," he said. But he made no move to leave his webbing, nor did I.
There was something about those crowding lines of buildings which dwarfed, threatened us, not actively, but by their being. We were as insects, unable to raise ourselves from the dust in which we crawled, confronted by men who were giants with clouds gathering about their barely seen heads. And about it all there hung a feeling that this was a place of old death. Not a decent tomb in which honor had been paid to the one who slept there through the centuries, but rather a place in which decay had reduced to a common anonymity all that had meant aught—men, learning, belief—
Nothing moved out there. No flying thing flitted among the towers. There was no sign of vegetation. It was truly a forest of bones long removed from life. We could see nothing to fear, save that feeling which grew in us, or in me (though Ryzk's actions led me to believe he must share my uneasiness), that life had no place here now.
"Let us move!" That was Eet. There was a tenseness in his small body, a feral eagerness in the way his head darted from side to side, as if he tried to focus more intently on the visa-screen—though as that continued its slow sweep I saw no change in the monotony of the towered vista.
I left the webbing, Ryzk also. The bowl with the zero stone was on the deck, with Eet crouched over it as if he were on guard above its contents. And the stone blazed, though perhaps with not the same intensity as earlier.
We climbed down to join Zilwrich. The Zacathan was on his feet, leaning against the wall. He looked to Eet and I guessed some message passed between them. I lent my shoulders to the Zacathan's support and, together with Ryzk, aided him out of the hatch, down the ramp, to the apron of the space port.
There arose a hollow moaning and the pilot slewed around in a half crouch, looking down one of the narrow passages between the towers. Save for the open pocket of the port, there was gloom unbroken in those ways, such dusk as I had seen in forests of other worlds. The moaning shrilled and then our startlement vanished as we realized it was caused by the wind. Perhaps that acted upon the rents in the building to produce such sounds.
But outside the _Wendwind_ the vast desolation was worse even than it had seemed on the screen. And I had not the slightest desire to go exploring. In fact, I was gripped by the feeling that to venture away from the port was to enter such a maze as one could never issue from again. As to where to search—Seen from the air, this planet-wide city covered all the ground, part of the sea. We might be half, three quarters, or the world away from what we sought, and it would take days, months of searching—
"I think not!" Eet had brought the bowl with him. Now he held it out and we saw the double blaze of the point on its surface and of the jewel within. He turned his head sharply to the right. "That way!"
But whatever lay "that way" might still be leagues from the port. And Zilwrich could certainly not tramp any distance on his unsteady feet, nor would I leave any of our party with the ship this time. We had the flitter—if we could crowd two of us into its cargo space, then we could quest some distance above the surface.
We settled Zilwrich with Eet at the end of the ramp and returned to the ship. What supplies we had room for and the crossbows went into the flitter. Three of us, plus Eet, would make such a heavy load we could not gain much altitude, but it was the best we could do. The LB had been so modified it might take days to alter it again, and we had no time to waste.
Judging by the sun, it was late afternoon when we were ready. I suggested waiting until the morning, but to my surprise the Zacathan and Eet overruled me. They had been in a huddle over the bowl and seemed very sure of what must be done.
As a matter of course Eet took command after we packed ourselves into the small craft, using my hands to his service. We hovered perhaps twice my height from the ground, then headed off sharply to the right, crossing the edge of the port, turning down a dusky channel between the towers.
The dark closed about us more and more as the buildings cut out the sun. Again I wondered how men could have lived here. Away from the port there appeared aerial runways connecting the buildings at different levels, crisscrossing into a net which finally grew so thick as to shut off most of the light from the level at which we traveled. Some of the ways were broken, and the debris of their disintegration weighted those below, or had landed in a heap of remains on the surface of the break below.
We had the beamer on, and I cut the speed to hardly more than a hover lest we crash into one of those piles. Yet Eet seemed entirely sure of our direction, sending me out of one half-filled lower way into another.
Dusk became full night. I had a growing fear we would be utterly lost, forever unable to find our way back to the comparative open of the port. There was a sameness to this level, just here and there the remains of a bridge fallen from the heights, the smooth bases of the buildings totally unbroken by any sign of an entrance.
Then the beamer picked up a flash of movement. It had been so quick that I thought my imagination had betrayed me into thinking I had seen it—until our beam trapped the thing against one of the walls. So cornered, it turned to face us, slavering defiance, or perhaps fear.
I have seen many strange beings on many worlds, so that weird defections from what is the norm to my species were not unknown to me. Yet there was something about this thing in the dark and forgotten ruins which brought an instant reaction of loathing in me. Had I been in the open, a laser in my hand, I think I would have slain it without thought or compassion.
Only for a moment did we see it so, backed against the unyielding buttress, pinned by the light. Then it was gone, with such speed as left me astounded. It had gone on two legs, then dropped to four. And the worst thing was that it looked like a man. Or what might have been a man eons ago, before time had burned out all which makes my kind more than an unthinking creature set upon survival alone.
"So it would seem that the city still has its inhabitants," Zilwrich commented.
"That thing—what was it?" The disgust in Ryzk's voice matched my own emotion. "Where did it go?"
"Turn to the left." Eet appeared unaffected by what we had seen. "In there—"
"There" was the first opening I had seen on the ground level of any building. It was too regular to be another rent. The gap was large enough to accommodate the flitter. But I had a very unpleasant suspicion that it was also where the scuttling creature had disappeared. To search further would mean leaving the craft, and to be trapped by that "thing" or others of its kind—
Yet I obeyed Eet's direction, bringing the flitter to a standing hover within the shell of chamber beyond that doorway. We were in a circular space. If there had been any furnishings, those were long since gone. But the floor was heaped with gritty, flaky stuff which perhaps had once been fittings. This was pathed, beaten solid in some places. And the paths—there were two of them—led directly to another dark opening in the floor, a well.
I moved the flitter cautiously until we nosed the lip of that descent. We could indeed lower into it in the machine. But to do this, unaware of what might lie below, was a peril I was not ready to face. If I had such fears, Eet was not concerned with them. He hung over the bowl in which the gem blazed.
"Down!" he urged, "Now down!"
I would have refused, but the Zacathan spoke.
"It is true. There is a very strong force below us. And if we go with caution—"
I certainly would not descend outside the flitter, but to go in it would give us a small measure of protection. Yet I thought it foolhardy to try at all. I fully expected a protest from Ryzk. Only when I glanced to him I saw he was as bemused by the gem in the bowl as Eet.
Moving out over the well I eased the flitter onto settle-hover, thankful that we were using a craft meant for exploration. And I kept a wary eye on the walls as we began the descent at as slow a speed as I could hold us to.
What had been the original use of this opening we could not know. But that it was also a passage for later users was apparent. Into the once smooth walls had been pounded or wedged a series of projections meant to serve as hand- and footholds, a very crude ladder. And the bits and pieces so used were rough, some of them surely ripped from more complex fittings. The work was very bad, its quality far beneath that of the city constructions, as if it had been done by a race who was at a primitive level.
We were descending by floors, passing dark openings in the walls of the shaft, as if that were a hub of a series of wheels whose spokes were evenly spaced passages. I counted six such levels, yet the circumference of the well did not dwindle in size as I feared it might. And though the crude ladder led to several of the cross-corridor openings, it also continued on down and down, as if it served a vast warren of burrows.
I watched the mouths of any opening the ladder served, but there was no sign of life, and our beamer could not penetrate them very far. Down and down, six levels, ten, a dozen, twenty—the wall grew no smaller. But it was a growing strain to hold the flitter on settle-hover at this slow speed. And always that ladder kept pace with us. Fifty—
"Soon, very soon now!" Eet's thought was excited, more filled with emotion than any I had ever received before. I looked to the dials. We were some miles below the surface. I cut our speed to the lowest and waited. There was a bump, and we had landed. Only a single tunnel mouth faced us now, a little to the right. And it was too small for the flitter. Any further exploration must be on foot, and I had no desire to leave the confines of the small safety offered by that craft.
My prudence was justified. There was movement at the mouth of that tunnel, though I remembered that crude ladder had ended four levels above our present position. Only what came into our beam was a machine, unlike any I had seen before. But there was enough resemblance to things I knew to suggest that the tube rising to aim at us was about to discharge something meaning no good to invaders.
When I put a finger to the rise button, both Eet and the Zacathan spoke, Eet by thought, the alien in Basic.
"Do not!"
Do not? They were crazed. We had to get out of the range of that thing, if we could, before it fired!
"Look—" That was Zilwrich. Eet was still staring at the stone in the bowl.
Look I did, expecting death to come at me from that sinister tube. What I did see was—nothing at all!
"Where—?"
"Esper impressions," Zilwrich answered. "It is known that certain things, trees, water, stones—and perhaps other objects—can hold visual impressions for many years, release them to one in the proper frame of mind for reception. The builders here may have known and used that principle. Or what we have seen may be only a report of its use at some time in the past, action which impelled such heightened emotions in those viewing it that the impression remained to be activated by us."
"We go—there—" Eet brushed aside the need for any explanation. Instead he was pushing the bowl ahead, using it as an indicator that our way led down that dark passage.
In the end he had his way. Otherwise he and the Zacathan would have set off alone. And my pride, such as it was, would not let me hold back. Because we were now a party united against the unseen perils of the unknown, I gave Ryzk once of the crossbows. So armed, we started out, Eet riding on my shoulder, where his weight was something of a problem, Zilwrich and Ryzk on my heels.
I had taken a smaller beamer from our supplies, but we did not need its ray long. Soon the gem in the bowl gave us light. And what it showed ahead for a goodly space was smooth, unbroken walling, as if we were advancing along a great tube.
Distance in the dark underground was relative. I thought we might find lack of air a danger. But apparently whatever system supplied this depths with a breathable atmosphere was still operative.
At last we came to the end of the passage and out. Not into a mine burrowing, as I had come more and more to expect, but into a room crammed with apparatus, equipment, some firmly based on the floor, the rest on tables or long counters. In the middle of this expanse was a blaze of light toward which Eet wanted to go.
A cone-shaped object perhaps as tall as I sat on a table by itself. And in it a transparent porthole allowed one to view an inner rack on which rested a dozen of the zero stones, vibrant with glowing life as we brought the two we carried closer to their container.
Resting beside the cone, on the table, was a second rack to which were clamped a further dozen rough, uncut stones. They were as black as lumps of carbon, yet they did not have the burned-out look of the exhausted zero stones we had found in the derelict space ship on our first trial of the power of the gems.
Eet sprang from my shoulder to the top of the table, put down the bowl, and set about prying at the porthole in the cone, trying to get at the jewels within. But something about that whole array triggered my memory.
There are many ways of cheating known to the experienced gem buyer. Stones may be so treated as to change their color, even hide flaws. Heat will transform amethyst to golden topaz. A combination of heat and chemical skillfully used can make a near undetectable royal rovan of the best crimson hue from a pale-pink one. Heat can do—
I loosened one of the black lumps from the rack and brought out my jeweler's lens. I had no way of testing the thing I held, yet there grew in me the belief that this was the matrix, the true zero stone. They might not be natural gems at all, but manufactured—which could logically give them the power to step up energy.
The thing I held was certainly odd. Its surface was velvety to the eye, but not the touch. If it had been shaped like a seed pod—I drew a deep breath. Memory was playing a strange trick on me. Surely it had to be a trick.
Once before I had found stones, or what appeared to be stones, tumbled in a stream. To the eye, though not to the touch, they had had a velvety, almost furred surface. One of those stones had been appropriated by the ship's cat, who had licked it, swallowed it, to give birth to—Eet! These were hunks of mineral, not rounded, podlike. But their surfaces—
I looked to Eet as I weighed that lump in my hand. He had discovered the secret of the latch on the porthole, jerked it open, and was taking out the rack with the finished gems. Then, to my amazement, as the weight of the tray was lifted from the latches which held it, I saw the cone come to life, a light flash on in its interior. Without thinking (further than wanting) past my desire to prove the truth of my suspicion, I inserted the second rack, saving out only the lump I had taken from it. My fingers were almost trapped as the porthole snapped shut of its own accord. And blazing light, blinding to any direct gaze, gathered behind the view-plate.
I had my answer. "Made stones."
Zilwrich picked up one from the other rack, took from me the black lump to compare.
"Yes, I believe you are correct. And I do not think that this"—he indicated the black lump—"is true ore or matrix either." He turned his bandaged head from right to left to view the room. The light was breaking in fierce waves from the cone, giving us a far radiance. "This was, I am certain, a laboratory."
"Which means," Ryzk commented, "that these are the last stones we may ever see. Unless they left records of how—"
There was sudden horrible shrilling, hurting one's ears, reaching into the brain. I gave one glance at the cone and grabbed for Eet, shouldered Zilwrich back, and cried out a warning. Then fire broke through the top of the oven, fountained up. Somehow I hit the floor, Eet fighting in my hold, the Zacathan's body half under mine.
Then—the light went out!
The following dark was so thick it smothered one. I groped for the beamer at my belt, for the second time unable to be sure whether my eyes or the light itself had failed. But a ray answered my press of button.
I aimed at the table, or where the table had stood. Now there was nothing at all! Nothing but a fan of clear space, as if the power had eaten a path for itself—but away, not toward us. Only one thing still lay there, seemingly unharmed, as if it was armored for all time against destruction—the map bowl. Eet uttered a sound, one of the few he had ever made. He broke from my hold and ran for it. But before he reached it he stopped short and I cried out even louder, moved by emotion in which fear and awe were mingled.
For in the beam of the torch Eet's furred body shimmered. He reared on his hind legs as might an animal caught by a throat collar and tight leash as it reached the end of the slack allowed it.
His hand-paws flailed at the air, and from his jaws came a wail of agony. But no mind-touch. It was as if then he was only animal.
With his back stiff, high-reared on his hind legs, he began to move jerkily, in a kind of weird, manifestly painful dance, round in a circle, the center of which was the bowl. Froth gathered on his muzzle, his eyes rolled wildly, and his body continued to shimmer until he was only a misty column.
That column grew taller, larger. It might be that the atoms which had formed the sustance of Eet's half-feline body were being dispersed, that he was literally being shaken into nothingness. Yet, instead of spreading out then into wisps, the mist began to coalesce again. Still the solidifying column was not as small as Eet, nor was it gathering into the same shape.
I could not move, nor did Zilwrich, nor Ryzk. The beamer had fallen from my hand, but lay so that its ray, if only by chance, held full on Eet, or what had been Eet, and the bowl.
Darker, thicker, and more solid grew the column of that shuddering thing. Eet had been as large as his foster mother, the ship's cat. This was almost as tall as I. At last it stopped growing, and its frenzied circling about the bowl became slower and slower, then finally halted.
I was still held in frozen astonishment.
I had seen Eet take three shapes by hallucinatory disguise: the pookha, the reptilian thing at Lylestane, and the hairy subhuman who had entered Waystar with me. But that he had willed this last change I was certain was not true.
He was humanoid and—
A slender body, yet curved, with long shapely legs, a small waist, and above that—
He—no—SHE—stood very still, staring at her outstretched hands, their skin soft with a pearly sheen to their golden hue. She bent her head as if to view that body, ran her hands up and down it, perhaps to reassure herself that this _was_ what she now saw.
While from Zilwrich broke a single word: "Luar!"
Eet's head turned, she looked at us with large eyes, a deeper and richer golden than her skin, drew her long dark-red hair about her as a cloak. Then she stooped and picked up the bowl. Balancing it on the palm of one hand, she walked to us along the beam of the torch, as if to impress upon us her altered appearance.
"Luar?" Her lips shaped the word. "No—Thalan!"
She hesitated, her eyes not on us for a moment but looking beyond us, as if they saw what we never could. "Luar we knew, yes, and dwelt there for a space, Honorable One, so that we left traces of our passage there. But it was not our home. We are the Searchers, the Born-again ones. Thalan, yes. And before that, others, many others."
She held out the bowl, reversed it so we could see the map. But the wink of the zero stone on it was dead, and that other stone it had held had vanished. "The treasure we sought here—it is now gone. Unless your wise ones, Honorable Elder, can read very forgotten riddles."
"Thanks to you, Jern!"
I staggered as a sudden blow against my arm threw me hard against one of the pieces of equipment based on the floor. I clung to it so as not to go down.
Eet, in one of those lightning movements which had been his—hers—as a feline mutant, snatched up the beamer from the floor. She swung the full light on Ryzk as the pilot was setting another bolt to his crossbow. And from her lips came a clear whistle.
Ryzk twisted as if his body had been caught in the shriveling discharge of a laser. His mouth opened on a scream which remained soundless. And from his now powerless hands dropped his weapon.
"Enough!" Zilwrich, moving with the dignity of his race, picked up the bow. The whistle stopped in mid-note and Ryzk stood, turning his head from side to side, as if he fought against some mind daze and tried thus to shake it away.
Gingerly I investigated my hurt by touch, since what light there was Eet had focused on Ryzk, now weaving back and forth as if his will alone kept him on his feet. I could find no cut, but the flesh was very tender, and I guessed it had been so close a miss that the shaft of the bolt had bruised me sorely.
"Enough!" the Zacathan repeated. He dropped his hand on the pilot's shoulder, steadied him as if they had been comrades-in-arms. "The treasure—the best treasure—still lies about us. Or"—he looked to Eet measuringly—"is now a part of us. You have what you have long wished, One Out of Time. Do not begrudge lesser prizes to others."
She spun the bowl on her hand and her lips curved in a smile. "Of a surety, Honorable Elder, at this hour I wish no hurt to any, having, as you have pointed out, achieved a certain purpose of my own. And knowledge is treasure—"
"No more stones," I said aloud, not really knowing why. "No more trouble. We are luckier without them—"
Ryzk raised his head, blinking in the light. He looked to where I leaned against my support but I think he did not really see me.
"Well enough!" Eet said almost briskly then. "The Honorable Elder is right. We have found a treasure world, which he and his kind are best fitted to exploit. Is this not so?"
"Yes." I had no doubts of that.
Ryzk shook his head once more, but not in denial. It was rather to try and clear his mind.
"The stones—" he said hoarsely.
"Were bait for too many traps," I answered. "Do you want the Guild, those of Waystar, the Patrol, always at your heels?"
He raised his hand, wiped it back and forth across his face. Then he looked to Zilwrich, keeping his eyes carefully from Eet, as if from the Zacathan alone he might expect an answer he could accept as the truth.
"Still treasure?" There was something curiously childlike in that question, as if Eet's strange attack had wiped from the pilot years of suspicion and wariness.
"More than can be reckoned." Zilwrich spoke soothingly.
But treasure no longer interested me. I watched rather Eet. As mutant and trader we had been companions. But what would follow now?
Mind-touch instead of words, amusement in part but delicately so, came swiftly in answer to my chaotic thoughts. "I told you once, Murdoc Jern, we each have in us that which must depend upon the other. I needed your body in the beginning, you needed certain attributes which I possessed in the woefully limited one I acquired. We are not now independent of each other—unless you wish it, just because I have found a body better for my purposes. In fact, one which, as I remember, served my race very well thousands of years ago. But I do not declare our partnership at an end because of that. Do you?"
She came forward then, tossing from her the bowl, the torch, as if both were no longer of service to her. Then her touch was on my body, light, soothing above my bruised hurt.
I had chaffed against Eet's superiority many times, sought to break his—her (I still could not quite accept the change) hold on me, that tie which fate, or Eet, had somehow spun between us since he—she—had been born on my bunk in the Free Trader.
It seemed that her touch now drew away the pain in my arm and side. And I knew that for better or worse, for ill times and good, there was no casting away of what that fate had given me. When I accepted that, all else fell into place.
"Do you—?" Her mind-touch was the faintest of whispers.
"No!" My reply was strong, clear, and I meant it with all of me.
About the Author
For well over a half century, Andre Norton was one of the most popular science fiction and fantasy authors in the world. With series such as Time Traders, Solar Queen, Forerunner, Beast Master, Crosstime, and Janus, as well as many standalone novels, her tales of adventure have drawn countless readers to science fiction. Her fantasy novels, including the bestselling Witch World series, her Magic series, and many other unrelated novels, have been popular with readers for decades. Lauded as a Grand Master by the Science Fiction Writers of America, she is the recipient of a Life Achievement Award from the World Fantasy Convention. An Ohio native, Norton lived for many years in Winter Park, Florida, and died in March 2005 at her home in Murfreesboro, Tennessee.
All rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
This is a work of fiction. Names, characters, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.
Copyright © 1969 by Andre Norton
Cover design by Barbara Brown
ISBN: 978-1-5040-2548-5
This edition published in 2015 by Open Road Integrated Media, Inc.
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{
"redpajama_set_name": "RedPajamaBook"
}
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shopt -s nullglob
#
#Execution anchor
MYCALLPATHNAME=$0
MYCALLNAME=`basename $MYCALLPATHNAME`
MYCALLNAME=${MYCALLNAME%.sh}
MYCALLPATH=`dirname $MYCALLPATHNAME`
MYBOOTSTRAPFILE=$(getPathToBootstrapDir.sh)/bootstrap-03_03_001.sh
. ${MYBOOTSTRAPFILE}
if [ $? -ne 0 ];then
echo "ERROR:Missing bootstrap file:configuration: ${MYBOOTSTRAPFILE}">&2
exit 1
fi
setUTALMbash 1 $*
#
###
#
. $(getPathToLib.sh libutalmfileobjects.sh)
. $(getPathToLib.sh libutalmrefpersistency.sh)
#
r=$(utalm-bash-cli.sh -d 1,f:FILEPATHNAME,title 2>&1)
f="FILEPATHNAME:DATA"
assertWithExit $LINENO $BASH_SOURCE "[[ '$r' == '$f' ]]"
## \endcond
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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For the 24 hours to 23:00 GMT, the AUD rose 0.86% against the USD and closed at 0.7587.
LME Copper prices rose 1.14% or $77.5/MT to $6861.0/MT. Aluminium prices declined 0.81% or $18.5/MT to $2278.5/MT.
In the Asian session, at GMT0300, the pair is trading at 0.7581, with the AUD trading 0.08% lower against the USD from yesterday's close.
The pair is expected to find support at 0.7525, and a fall through could take it to the next support level of 0.7469. The pair is expected to find its first resistance at 0.7615, and a rise through could take it to the next resistance level of 0.7649.
Moving ahead, traders would look forward to Australia's Westpac leading index for April, slated to release overnight.
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{
"redpajama_set_name": "RedPajamaC4"
}
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{"url":"https:\/\/www.groundai.com\/project\/valley-enhanced-fast-relaxation-of-gate-controlled-donor-qubits-in-silicon\/","text":"Valley-enhanced fast relaxation of gate-controlled donor qubits in silicon\n\n# Valley-enhanced fast relaxation of gate-controlled donor qubits in silicon\n\nP\u00e9ter Boross Institute of Physics, E\u00f6tv\u00f6s University, Budapest, Hungary \u2003\u2003 G\u00e1bor Sz\u00e9chenyi Institute of Physics, E\u00f6tv\u00f6s University, Budapest, Hungary \u2003\u2003 Andr\u00e1s P\u00e1lyi Department of Physics and MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, Budapest, Hungary\nSeptember 25, 2019\n###### Abstract\n\nGate control of donor electrons near interfaces is a generic ingredient of donor-based quantum computing. Here, we address the question: how is the phonon-assisted qubit relaxation time affected as the electron is shuttled between the donor and the interface? We focus on the example of the \u2018flip-flop qubit\u2019 [Tosi et al., arXiv:1509.08538v1], defined as a combination of the nuclear and electronic states of a phosphorous donor in silicon, promising fast electrical control and long dephasing times when the electron is halfway between the donor and the interface. We theoretically describe orbital relaxation, flip-flop relaxation, and electron spin relaxation. We estimate that the flip-flop qubit relaxation time can be of the order of , 8 orders of magnitude shorter than the value for an on-donor electron in bulk silicon, and a few orders of magnitude shorter (longer) than the predicted inhomogeneous dephasing time (gate times). All three relaxation processes are boosted by (i) the nontrivial valley structure of the electron-phonon interaction, and (ii) the different valley compositions of the involved electronic states.\n\n## I Introduction\n\nDonor-based spin qubits in siliconKane (1998); Morton\u00a0et\u00a0al. (2011); Zwanenburg\u00a0et\u00a0al. (2013) (Si) are promising building blocks for quantum information processing schemes, mainly due to qubit lifetimes that are prolonged by the weakness of spin-orbit and hyperfine interactions in this materialFeher (1959); Feher\u00a0and\u00a0Gere (1959); Roth (1960); Hasegawa (1960); Tahan\u00a0et\u00a0al. (2002); Tyryshkin\u00a0et\u00a0al. (2003, 2006); Morello\u00a0et\u00a0al. (2010); Tyryshkin\u00a0et\u00a0al. (2012); Tahan\u00a0and\u00a0Joynt (2014). Recent important experimental achievements of the field include initialization, coherent control and readout of electronic and nuclear spins of individual phosphorous (P) donorsMorello\u00a0et\u00a0al. (2010); Pla\u00a0et\u00a0al. (2012, 2013), as well as increasing qubit lifetimes Tyryshkin\u00a0et\u00a0al. (2003, 2006, 2012); Muhonen\u00a0et\u00a0al. (2014); Itoh\u00a0and\u00a0Watanabe (2014) by using isotopically purified samples with strongly increased abundance of the nuclear-spin-free Si-28 isotope.\n\nA ubiquitous ingredient of donor-based quantum-information processing schemes is to use electrical gates to control the wave function of the donor electron (see Fig.\u00a01a). That often means that the electron is shuttled between the donor and a nearby interfaceKane (1998); Vrijen\u00a0et\u00a0al. (2000); Calder\u00f3n\u00a0et\u00a0al. (2006, 2008); Lansbergen\u00a0et\u00a0al. (2008); Rahman\u00a0et\u00a0al. (2009a); Baena\u00a0et\u00a0al. (2012); Laucht\u00a0et\u00a0al. (2015); Tosi\u00a0et\u00a0al. ; Urdampilleta\u00a0et\u00a0al. (2015); Harvey-Collard\u00a0et\u00a0al. . For example, in the Kane proposalKane (1998), gate control is suggested to tune the hyperfine interaction strength and to allow for exchange-based two-qubit operations. Here, we address the following question: how does the phonon-assisted qubit relaxation time depend on the location of the electron, as it is placed in an intermediate position between the donor and the interface? We focus on the example of the recently proposedTosi\u00a0et\u00a0al. flip-flop qubit (see Fig.\u00a01c); it is defined as a combination of the nuclear and electronic states of a phosphorous donor in silicon, and it is expected to allow for fast electrical control and long dephasing times, when the gate-induced electric fields locate the electron halfway between the donor and the interface.\n\n### i.1 Flip-flop qubit\n\nNaturally, most of the coherent-control experiments with donor-based spin qubits are performed using ac magnetic fields in the spirit of paramagnetic resonance. However, for a number of practical reasons, it can be advantageous to substitute the magnetic excitation with electrical driving, which is possible if a sufficiently strong interaction exists between the spin qubit and electric fields. On a single-qubit level, such an interaction allows local control via ac gate-voltage pulsesGolovach\u00a0et\u00a0al. (2006); Flindt\u00a0et\u00a0al. (2006); Nowack\u00a0et\u00a0al. (2007), and dispersive non-demolition readout via probing a nearby electromagnetic resonatorBlais\u00a0et\u00a0al. (2004). It also enables two-qubit operations, either via electric dipole-dipole interactionFlindt\u00a0et\u00a0al. (2006); Trif\u00a0et\u00a0al. (2007); Tosi\u00a0et\u00a0al. ; Salfi\u00a0et\u00a0al. (2015), or via an electromagnetic resonator that mediates interaction between the qubitsBlais\u00a0et\u00a0al. (2004). These two-qubit gates, in contrast to the exchange-based gate, should be robust against donor placement uncertaintiesCullis\u00a0and\u00a0Marko (1970); Kane (1998); Koiller\u00a0et\u00a0al. (2001).\n\nThe flip-flop qubitTosi\u00a0et\u00a0al. is expected to interact strongly with electric fields, and therefore has the potential to realize the desired features outlined above. The qubit is encoded in the composite system of the electronic and nuclear spins of a P donor, such that the qubit basis states are given by the two anti-aligned spin configurations and , where the first (second) arrow represents the electronic (nuclear) spin. Importantly, the flip-flop terms of the hyperfine interaction between the electronic and nuclear spins couple the two qubit basis states. As a consequence, an ac electric field can drive coherent Rabi oscillations of the qubit: the field shakes the electronic wave function, thereby modulates the hyperfine coupling strength, which is in turn felt by the qubit as an ac Hamiltonian matrix element that couples the basis states.\n\nThis interaction between the flip-flop qubit and electric fields can be strongly enhanced in the configuration shown in Fig.\u00a01a. Here, the donor is placed in the vicinity of an interface between silicon and a barrier material (e.g., SiO). If the charge center of the electron is approximately halfway between the donor ion and the interface (ionization point), then the coupling between the qubit and electric fields is maximized. A further advantage of such a setting is the existence of dephasing sweet spots in the space of the control parameters, including second-order clock-transition points where both the first and second derivatives of the qubit\u2019s Larmor frequency with respect to the dc electric field are zero. Tuning the system to such a sweet spot might result in exceptionally strong resilience against electrically-induced dephasing.\n\n### i.2 This work\n\nIn this work, we theoretically describe phonon-mediated relaxation of the flip-flop qubit, and determine the corresponding relaxation time (see Fig.\u00a01c and section IV). Reference Pines\u00a0et\u00a0al., 1957 estimated a very long low-temperature relaxation time of s for a P donor in bulk silicon, set by deformation-induced changes of the effective mass and the dielectric constant. (Phonon-mediated spin relaxation processes involving nuclear-spin ensembles are treated, e.g., in Refs.\u00a0Abragam, 1961; Khaetskii, 2001; Erlingsson\u00a0and\u00a0Nazarov, 2002.) In contrast, here we describe a deformation-potential mechanism that is particularly strong in the proposed working point of the flip-flop qubit, when the electron is at the ionization point, and leads to a characteristic s. This time scale is approximately 8 orders of magnitude shorter than the prediction of Ref.\u00a0Pines\u00a0et\u00a0al., 1957, and a few orders of magnitude shorter (longer) than the predictedTosi\u00a0et\u00a0al. inhomogeneous dephasing time (gate times) of the flip-flop qubit.\n\nThe reason for the relatively fast relaxation is twofold. First, the flip-flop qubit is designed to strongly interact with electric fields at its working point, and that is achieved via hyperfine-induced mixing of the ground-state orbital with a low-lying excited orbitalTosi\u00a0et\u00a0al. ( and , to be introduced below). The same low-lying excited orbital also provides strong interaction between the flip-flop qubit and phonon-induced deformation potentials, leading to relatively fast qubit relaxation. Second, we show that the relaxation process is valley-enhanced, where valley refers to the 6 conduction-band minima of the electronic band structure of silicon. In particular, the relaxation is boosted by the nontrivial valley-related features of the electron-phonon interaction and the involved electronic states.\n\nWe also characterize orbital relaxation, that is, relaxation of the charge qubit ( in Fig.\u00a01b). Since orbital relaxation is conceptually simpler than the flip-flop relaxation, we start with the case of orbital relaxation in section III, and use it to introduce and illustrate the key ingredients of the valley-enhanced mechanism that governs all the three processes we consider (orbital, flip-flop, and electron spin relaxation). Finally, in section V, we describe electron spin relaxationFeher (1959); Feher\u00a0and\u00a0Gere (1959); Roth (1960); Hasegawa (1960); Tahan\u00a0et\u00a0al. (2002); Morello\u00a0et\u00a0al. (2010); Tahan\u00a0and\u00a0Joynt (2014) from the excited state of the flip-flop qubit ( in Fig.\u00a01c): this process is also relevant for the functionality of the flip-flop qubit, as it leads to leakage from the qubit subspace.\n\n## Ii The flip-flop qubit and its model Hamiltonian\n\nHere, based on Ref.\u00a0Tosi\u00a0et\u00a0al., , we discuss the setup in which the flip-flop qubit is envisioned, a simple 8-dimensional model Hamiltonian that captures the essential ingredients of the setup, and the 2-dimensional flip-flop qubit subspace. We note that in Ref.\u00a0Tosi\u00a0et\u00a0al., , this 8-dimensional Hamiltonian was found to reliably reproduce various physical quantities obtained from atomistic tight-binding simulations. This fact promotes this model to a trustable starting point for exploring the relaxation mechanisms of the flip-flop qubit.\n\nIn the absence of gate-induced electric fields, the donor electron is localized at the donor site, occupying the ground-state donor orbital (see Fig.\u00a01a). A voltage applied on the gate electrode induces an electric field along the z axis, and hence can pull the electron to the vicinity of the silicon-barrier interface, where it occupies the orbital state . This two-orbital charge qubit degree of freedom is described by the Pauli matrices , , , where, e.g., . By continuously changing the gate-induced electric field , the electron is continuously moved between the two localized orbitals; the corresponding Hamiltonian reads\n\n Ho=Vt2\u03c3x\u2212e(Ez\u2212E0z)d2\u03c3z, (1)\n\nwhere is the tunnel amplitude between the orbital states and , and is the value of the gate-induced electric field along z where the stationary electron charge is equally distributed among and . The splitting between the energy eigenvalues of is\n\n \u03f5o=\u221aV2t+[e(Ez\u2212E0z)d]2. (2)\n\nAn external homogeneous magnetic field introduces Zeeman splittings for both the electron and the nuclear spin of the donor. For simplicity, for the moment we assume isotropic and location-independent -tensors, yielding the following electronic and nuclear Zeeman Hamiltonians, respectively:\n\n HB,e=h\u03b3e\\boldmath{B}\\boldmath{S}, (3)\n HB,n=h\u03b3n\\boldmath{B}\\boldmath{I}. (4)\n\nIf the electron is located on the donor, then its spin interacts with the nuclear spin of the donor. Hence the hyperfine interaction is described by the following Hamiltonian:\n\n Hhf=A(1\u2212\u03c3z2)% \\boldmath{S}\\boldmath{I}. (5)\n\nHere, both and are represented by times the vector of Pauli matrices. We introduce the secular and non-secular or flip-flop part of the hyperfine Hamiltonian, where the former is defined as\n\n Hhf,sec=A(1\u2212\u03c3z2)(\\boldmath{S}\u22c5\\boldmath{B}B)(\\boldmath{I}\u22c5\\boldmath{B}B). (6)\n\nThat is, incorporates spin components that are parallel to the external magnetic field, whereas incorporates the flip-flop terms.\n\nThe energy eigenstates of the Hamiltonian are direct products of the energy eigenstates and of and electron (, ) and nuclear (, ) spin states pointing along the external magnetic field. These states will be labelled by the above quantum numbers and denoted as, e.g., , and we will call them the unperturbed energy eigenstates.\n\nIf the spectral gaps of are much larger than the energy scale characterizing , then the latter remains a perturbation, and the energy eigenstates of the full Hamiltonian are approximately direct products as above, hence can be labelled with the same quantum numbers, and will be denoted as, e.g., . Using this notation, the basis states of the flip-flop qubit are and . An example parameter set where the above conditions are met, and which was studied extensively in Ref.\u00a0Tosi\u00a0et\u00a0al., , is shown in Table 1. The level diagram consisting of the four energy eigenstates of associated to the ground-state orbital manifold is depicted in Fig.\u00a01c; there, the flip-flop qubit basis states, having an energy separation of , are highlighted as bold black lines.\n\n## Iii Orbital relaxation\n\nFirst, we characterize the phonon-emission-mediated orbital relaxation, that is, relaxation from the excited state of the charge qubit to its ground state , and calculate the corresponding relaxation time , see Fig.\u00a01b. We disregard the spin degrees of freedom for simplicity; the charge qubit is described by the Hamiltonian of Eq.\u00a0(1), and the eigenstates and of are called the charge qubit basis states. The valley-enhanced, deformation-potential-induced relaxation mechanism we describe here, as well as the structure of the calculation itself, is easily translated to treat the flip-flop relaxation and electron spin relaxation processes, which will be discussed in the subsequent sections.\n\n### iii.1 Preliminaries\n\nTo account for the phonons and the electron-phonon interaction, we use a bulk-type description, neglecting any effects arising from inhomogeneities in the nanostructure.\n\nIn the experimentally relevant range of parameters, the charge-qubit energy splitting is resonant with low-energy long-wavelength acoustic phonons. Hence only those are considered here. Their dispersion relations are assumed to be linear and characterized by the sound velocities , where (L,T1,T2) is the polarization index and L (T) refers to longitudinal (transverse).\n\nWe focus on the case of zero temperature and use the corresponding Fermi\u2019s Golden Rule to evaluate the qubit relaxation time:\n\n (7)\n\nHere, bras and kets represent joint states of the composite electron-phonon system, denotes the vacuum of phonons, and () is the wave number (polarization index) of the emitted phonon.\n\nThe mechanism we describe is based on the deformation-potential electron-phonon interaction, which we treat via the silicon-specific Herring-Vogt HamiltonianHerring\u00a0and\u00a0Vogt (1956); Yu\u00a0and\u00a0Cardona (2010):\n\n Heph=\u039eu\u239b\u239c \u239c \u239c \u239c \u239c \u239c \u239c \u239c \u239c\u239d\u03b5xx000000\u03b5xx000000\u03b5yy000000\u03b5yy000000\u03b5zz000000\u03b5zz\u239e\u239f \u239f \u239f \u239f \u239f \u239f \u239f \u239f \u239f\u23a0, (8)\n\nwhere the matrix structure corresponds to valley space, that is, the 6 envelope functions associated to the 6 conduction-band valleys of silicon, denoted and ordered as . In Eq.\u00a0(8), is the uniaxial deformation potential and is the strain tensor. Note that in addition to the right hand side of Eq.\u00a0(8), the Herring-Vogt Hamiltonian incorporates a conventional, valley-independent deformation-potential term, , where is the dilational deformation potential, is the deformation-induced relative volume change, and is the unit matrix; however, we disregard that term here as (i) it does not contribute to the valley-enhanced mechanism to be described here, and (ii) its contributions to the relaxation rates obtained here are much smaller than those of the uniaxial deformation potential term.\n\nThe diagonal elements of the strain tensor, that is, the elements that determine via Eq.\u00a0(8), read\n\n \u03b5jj=i\u221a\u210f2\u03c1V\u2211% \\boldmath{q},\u03bbe\\boldmath{q}\u03bbjqj\u221av\u03bbqei\\boldmath{q}\u22c5\\boldmath{r}(a\\boldmath{q},\u03bb+a\u2020\u2212\\boldmath{q},\u03bb). (9)\n\nHere, , is the mass density of silicon, is the sample volume and is the polarization vector of the phonon mode with wave number and polarization index . For the setup we consider, the wavelength of the phonon emitted by the qubit is much longer than the spatial size of the qubit itself. Therefore, the plane-wave factor in Eq.\u00a0(9) can be approximated as\n\n ei\\boldmath{q}\u22c5\\boldmath{r}\u22481; (10)\n\nthis corresponds to a homogeneous deformation, and as we will show, such a homogeneous deformation is sufficient to induce the described relaxation processes.\n\nTo obtain via Fermi\u2019s Golden Rule (7), we need to provide the envelope-function representation of the localized charge states and . For the purpose of obtaining the order of magnitude and the parameter dependence of the relaxation rates, it is sufficient to use simple \u2018perfectly localized\u2019 envelope functions, dressed by the appropriate valley compositionsKohn\u00a0and\u00a0Luttinger (1955); Baena\u00a0et\u00a0al. (2012); Zwanenburg\u00a0et\u00a0al. (2013). The interface state resembles that of a planar quantum-dot ground state pushed toward the barrier by the gate-induced electric field, hence its wave function resides in the and valleys, evenly distributed. The donor state , on the other hand, is evenly distributed in all the 6 valleys. Using these considerations, we represent the two localized charge states as\n\n = \u221a\u03b4(\\boldmath{r}\u2212\\boldmath{r}i%\u00a0)1\u221a2(0,0,0,0,ei\u03d5z,ei\u03d5\u00afz), (11a) = \u221a\u03b4(\\boldmath{r})1\u221a6(1,1,1,1,1,1), (11b)\n\nwhere is the three-dimensional Dirac delta, the donor position is chosen as the origin of the reference frame, is the center of charge of the orbital , and the phases and are between 0 and , but their actual values turn out to be irrelevant. In Eq.\u00a0(11), the Dirac delta is a strongly simplified representation of the envelope functions associated to the valleys. We emphasize that a more realistic representation, e.g., using Kohn-LuttingerKohn\u00a0and\u00a0Luttinger (1955) envelope functions for the donor orbital , would only lead to minor quantitative corrections of our results.\n\nBefore evaluating the orbital relaxation time, it is instructive to restrict the electron-phonon interaction Hamiltonian to the charge-qubit Hilbert space:\n\n Heph,o=PHephP=\u039eui6\u221a\u210f2\u03c1V\u03a3z\u03c3z, (12)\n\nwhere ,\n\n \u03a3z = \u2211\\boldmath{q},\u03bb(\u2212e% \\boldmath{q}\u03bbxqx\u2212e\\boldmath{q}\u03bbyqy+2e\\boldmath{q}\u03bbzqz)\u221av\u03bbq(a\\boldmath{q},\u03bb+a\u2020\u2212\\boldmath{q},\u03bb). (13)\n\nHere we used Eqs. (8), (9), (10) and (11), and from Eq.\u00a0(12) we omitted an irrelevant term proportional to the unit matrix . Remarkably, is proportional to , which means that there is a deformation-induced potential difference between the interface and donor sites, in spite of the homogeneous nature of the considered deformation component. The appearance of that effective potential difference is due to two factors: the nontrivial valley structure of the Herring-Vogt Hamiltonian, see Eq.\u00a0(8), and the different valley compositions of the two localized orbitals and , see Eq.\u00a0(11).\n\nLet us illustrate that claim, and the corresponding physical mechanism, with a simple example. Take a longitudinal phonon propagating along the x axis. This case corresponds to and a finite . Hence, according to Eq.\u00a0(8), the conduction-band edges in the and valleys are raised by the uniaxial deformation potential , whereas the conduction band edges in the other four valleys are not affected. Then, this effective potential in the and valleys is felt differently by and : the state has no weight in the and valleys [see Eq.\u00a0(11a)], therefore it does not feel the presence of the deformation; the state , however, has a total weight of in the and valleys together [see Eq.\u00a0(11b)], and hence the deformation raises its potential energy by . Therefore we conclude that a homogeneous deformation indeed induces a potential energy difference between the interface orbital and the donor orbital. Furthermore, our argument translates to an effective electron-phonon coupling Hamiltonian with a nontrivial part of , in line with the corresponding term in Eq.\u00a0(12).\n\n### iii.2 Results\n\nTo obtain the orbital relaxation time, Fermi\u2019s Golden Rule (7) is evaluated as\n\n 1T1,o = \u03f5oV2t\u039e2u60\u03c0\u210f4\u03c1\u239b\u239d23v5L+1v5T\u239e\u23a0, (14)\n\nwhere we used\n\n (15)\n\nAt the ionization point, where , and using the working-point parameters specified in Table 1, the orbital relaxation rate is estimated as , corresponding to a relaxation time of s.\n\nUpon detuning from the ionization point, the charge qubit energy splitting increases, and therefore, according to Eq.\u00a0(14), the relaxation speeds up. This is interpreted as the result of a competition between three effects.\n\nFirst, relaxation should slow down upon detuning from the ionization point because the charge qubit basis states and become more localized, which suppresses the relevant matrix element . Second, relaxation should be enhanced upon detuning from the ionization point, as the charge qubit energy splitting increases, and therefore the density of states of the available phonons also increases. These two mechanisms exactly cancel each other.\n\nThe fact that the relaxation speeds up upon detuning from the ionization point is therefore a consequence of a third fact: the vacuum fluctuation of the strain of a phonon mode with energy is proportional to ; that follows from Eqs.\u00a0(9) and (10), and the energy conservation condition embedded in Fermi\u2019s Golden Rule (7). The quadratic form of Fermi\u2019s Golden Rule then implies a dependence due to this factor, which does indeed appear in our result (14).\n\n### iii.3 Valley-enhanced relaxation\n\nWe wish to highlight the fact that the nontrivial features of the setup associated to the valley degree of freedom boost the orbital relaxation process, and will play the same role in the flip-flop relaxation and electron spin relaxation processes to be described below. In that sense, all these can be considered valley-enhanced relaxation processes. Our argument supporting that claim is as follows. The two relevant features are (i) the nontrivial valley structure of the electron-phonon interaction, and (ii) the different valley compositions of the localized charge states and . In the absence of any of these two ingredients, the first, homogeneous-deformation term in the plane-wave expansion of the strain tensor (9) would give a vanishing contribution to the relaxation rate, and therefore the relaxation rate would be suppressed by a factor of . Using the parameter values of Table 1, that factor has the value of [] for longitudinal [transverse] phonons.\n\nIn conclusion, in this section we have described a phonon-emission-mediated orbital relaxation process, characteristic of a charge qubit formed by a gate-tuned electron located between its donor atom and a nearby interface. In particular, we have shown that the relaxation process is enhanced by (i) the nontrivial valley structure of the electron-phonon interaction and (ii) the different valley compositions of the two orbital wave functions forming the charge qubit.\n\n## Iv Flip-flop relaxation\n\nHere, we use the model described in sections II and III to characterize the phonon-emission-mediated relaxation process from the flip-flop qubit excited state to its ground state . This process is labelled in Fig.\u00a01c as . The characteristic time scale of this process for an isolated P donor at low temperature in bulk silicon has been estimatedPines\u00a0et\u00a0al. (1957) as s. Here we show that this time scale can decrease by approximately 8 orders of magnitudes, that is, s is possible, if the flip-flop qubit is tuned to couple strongly to electric fields.\n\nThe flip-flop relaxation mechanism is visualized using the level diagram in Fig.\u00a02a. It can be thought of as a two-step or second-order process, in which matrix elements of the flip-flop part of the hyperfine interaction , depicted as solid arrows in Fig.\u00a02a, and matrix elements of the electron-phonon interaction , denoted as dashed arrows in Fig.\u00a02a, provide relaxation paths via virtual intermediate states.\n\nOur calculation of follows the preliminaries and derivation steps of the calculation of in the previous section. For the flip-flop relaxation rate, Fermi\u2019s Golden Rule reads\n\n (16)\n\nAs long as is a perturbation of , we can use first-order perturbation theory to obtain analytical approximate expressions for the qubit basis states and in terms of the 8 unperturbed energy eigenstates. In fact, the form of guarantees that the flip-flop qubit basis states are linear combinations of the 4 unperturbed energy eigenstates , , , and . The flip-flop relaxation rate is then readily evaluated from Eq.\u00a0(16) as:\n\n 1T1,ff=A2\u039e2uV4t\u03f53B240\u03c0\u210f4\u03c1\u03f52o(\u03f52o\u2212\u03f52B)2\u239b\u239d23v5L+1v5T\u239e\u23a0. (17)\n\nHere, , and Eq.\u00a0(17) shows the leading-order result in the small parameters .\n\nThis result can also be expressed in terms of the orbital relaxation time:\n\n 1T1,ff=14A2V2t\u03f53B\u03f53o(\u03f52o\u2212\u03f52B)21T1,o, (18)\n\ntaking a particularly simple approximate form in the vicinity of the proposed working point, where the electron is placed halfway between the interface and the donor and the energy splittings of the charge qubit and flip-flop qubit are similar ():\n\n 1T1,ff\u2248(A\/4\u03f5% o\u2212\u03f5B)21T1,o. (19)\n\nNote that this result corresponds to the special case when the leftmost virtual transition of Fig.\u00a02a dominates the relaxation process.\n\nWith the parameter values in Table 1, from Eq.\u00a0(17) we obtain , implying a flip-flop relaxation time of . This value is approximately 8 orders of magnitude shorter than the s time scale that was estimated for an on-donor electron in bulk by Ref.\u00a0Pines\u00a0et\u00a0al., 1957. The reason for the fast relaxation at the proposed working point of the flip-flop qubit is is twofold. First, the working point is chosen with the goal of optimizing the speed of electrically driven qubit transitions: , which appears as an energy denominator in the perturbative description of the leftmost virtual process of Fig.\u00a02, is chosen to be relatively small ( MHz), so that the qubit excited state has a relatively large, hyperfine-mediated admixture with the unperturbed energy eigenstate . Second, this flip-flop relaxation process is valley-enhanced, in a similar sense as described in section III.3. That is, due to the nontrivial valley structure of the electron-phonon Hamiltonian and the different valley compositions of the involved electronic orbitals and , even a uniform phonon-induced deformation is capable to induce relaxation.\n\nIf the charge-qubit splitting is much larger than the electronic Zeeman splitting , then Eq.\u00a0(17) implies the power-law relation , see also Fig.\u00a02d. The 3rd power arises as a sum , where the terms, respectively, are associated to the strain vacuum fluctuations and the density of states of three-dimensional acoustic phonons. This is analogous to the low-temperature limiting case of the relaxation mechanism considered in Ref.\u00a0Pines\u00a0et\u00a0al., 1957 for on-donor electrons in bulk: even though the mechanisms considered here and there are different, in both cases a homogeneous deformation is responsible for the relaxation.\n\nIn Fig.\u00a02b, we show the dependence of the qubit relaxation rate on the gate-induced electric field and the magnetic field. To obtain this result, we first numerically computed the eigenvalues and eigenvectors of . Then, we identified the flip-flop qubit ground (excited) state as the energy eigenstate having the largest overlap with (). Finally, we evaluated the relaxation rate according to Eq.\u00a0(16).\n\nThe key features in Fig.\u00a02b are as follows. (i) The qubit relaxation rate is strongly suppressed at low magnetic fields, due to the above-discussed dependence. (ii) The qubit relaxation rate is maximal, taking values around 1 MHz, along the upward-bending hyperbola, which corresponds to , and, therefore, nonperturbative mixing of and . Hence this relaxation rate of 1 MHz reflects the orbital relaxation rate.\n\nComparison of the numerical results of Fig.\u00a02b and the perturbative, analytical expression (17) is shown in Fig.\u00a02c,d. In Fig.\u00a02c, the dashed blue line shows a horizontal cut of Fig.\u00a02b through the working point (white cross), whereas the solid red line is the analytical result. A similar comparison, corresponding to a vertical cut of Fig.\u00a02b through the working point, is shown in Fig.\u00a02d. Note that in Fig.\u00a02d , for magnetic fields slightly higher than the working-point magnetic field, the analytical result deviates from the numerical one and diverges; that behavior is an artefact arising from the breakdown of first-order perturbation theory.\n\nIn conclusion, we have proposed a valley-enhanced relaxation mechanism of the flip-flop qubit, calculated its characteristic relaxation time , and found a relatively short, s time scale in the proposed working point. This is partly due to the presence of a low-lying orbital that is utilized to enhance the coupling of the qubit to the electric field. Another factor boosting the relaxation process is the absence of dipole suppression (see section III.3): thanks to the nontrivial valley structure of the electron-phonon interaction and the involved electronic orbitals and , a homogeneous deformation can induce an effective potential difference between the two orbitals, and hence lead to efficient relaxation.\n\n## V Electron spin relaxation\n\nA further process, leading to leakage from the flip-flop qubit subspace, is electron spin relaxation (henceforth spin relaxation, for short): this is shown in Fig.\u00a01c, labelled as . We first describe a valley-enhanced spin-relaxation mechanism that is enabled by spin-orbit interaction; more precisely, by spin-dependent electron tunnelling between the two localized orbitals and . We also discuss an alternative valley-enhanced relaxation mechanism, which is enabled by the feature that the -tensors characterizing the localized orbitals and are, in general, different and anisotropicRahman\u00a0et\u00a0al. (2009b).\n\n### v.1 Spin relaxation due to spin-dependent tunneling\n\nFirst, we incorporate spin-orbit interaction to our model Hamiltonian described in section II. For simplicity, we assume that the setup is cylindrically symmetric around the z axis. We claim that this symmetry condition, together with the condition that the spin-orbit Hamiltonian must be invariant under time reversal, imply the following simple form for the spin-orbit Hamiltonian:\n\n Hso=Vs\u03c3ySz, (20)\n\nwhere is real. Naturally, this Hamiltonian excludes the nuclear-spin operators. Furthermore, since is an off-diagonal matrix, describes spin-dependent tunneling between the two orbitals and .\n\nThe proof of Eq.\u00a0(20), inspired by a related argument of Ref.\u00a0Danon\u00a0and\u00a0Nazarov, 2009, is as follows. In principle, the spin-orbit Hamiltonian can be expanded in terms of products of charge-qubit Pauli matrices including the unit matrix , and the three spin Pauli matrices: , where represent 12 unknown coefficients. Then, the condition of time reversal invariance renders 9 of the coefficients zero, , for the following reason. Time reversal is represented as with being the complex conjugation, therefore the spin matrices , , and change sign under time reversal, whereas the real matrices , and keep their signs. That implies that the only charge-qubit Pauli matrix allowed in the spin-orbit Hamiltonian is : . However, a finite value of either or would specify a certain direction in the xy plane, which is disallowed by the cylindrical symmetry of the setup around the z axis. With the identification , this concludes the proof of Eq.\u00a0(20). A quantitative characterization of could be obtained from microscopic, e.g., tight-bindingRahman\u00a0et\u00a0al. (2009b), simulations, incorporating the nanostructure geometry and spin-orbit interaction. Here, we treat as a phenomenological parameter.\n\nHaving the spin-orbit Hamiltonian at hand, we now propose the spin relaxation mechanism it enables. The mechanism is analogous to the flip-flop relaxation, and is visualized using the level diagram in Fig.\u00a03a. Here, we parametrize the magnetic-field orientation via its polar angle : , but disregard any orbital effects caused by . Furthermore, recall that the arrows in our state notation (for example, in ) correspond to spin alignments with respect to the external magnetic field, not with respect to z. Then, we conclude that mixes the unperturbed state with . This mixing is depicted as the left solid arrow in Fig.\u00a03a. Since, in turn, is connected to by the electron-phonon interaction (left dashed arrow in Fig.\u00a03a), we conclude that the spin-orbit interaction does indeed enable spin relaxation. A similar two-step process contributing to spin relaxation is depicted by the right solid and dashed arrows.\n\nThe spin relaxation rate arising from these second-order processes can be calculated as\n\n (21)\n\nwhere the states and are perturbed by the spin-orbit interaction, in analogy to Eq.\u00a0(16), where the states are perturbed by the flip-flop terms of the hyperfine interaction. Furthermore, is the energy splitting between the energy eigenstates and .\n\nUsing first-order perturbation theory to account for the spin-orbit-induced mixing of the unperturbed states, we find\n\n 1T1,s=V2ssin2\u03b8\u039e2uV2t\u03f55B15\u03c0\u210f4\u03c1\u03f52%o(\u03f52o\u2212\u03f52B)2\u239b\u239d23v5L+1v5T\u239e\u23a0, (22)\n\nwhich shows the leading-order result in the small parameters .\n\nExpressed with the orbital relaxation rate:\n\n 1T1,s=4V2ssin2\u03b8\u03f55B\u03f53o(\u03f52o\u2212\u03f52B)21T1,o (23)\n\nIn the vicinity of the proposed working point where , this is approximated as\n\n 1T1,s\u2248\u239b\u239d12Vssin\u03b8\u03f5o\u2212\u03f5B\u239e\u23a021T1,o%\u00a0. (24)\n\nAt weak magnetic fields, , the spin-relaxation rate in Eq.\u00a0(22) follows the power-law relation , see Fig.\u00a03d; this is a stronger dependence then the seen in the previous section, and the difference is due to van Vleck cancellationvan Vleck (1940); Khaetskii\u00a0and\u00a0Nazarov (2001).\n\nIn Fig.\u00a03b, we show the dependence of the spin relaxation rate on the gate-induced electric field and the magnetic field, in analogy to Fig.\u00a02, using the numerically computed eigenvalues and eigenvectors of . To produce this plot, the magnetic field is assumed to be aligned with the x axis (), and for not having a calculated or measured value for the spin-dependent tunneling energy , we used an arbitrary value MHz. That implies that even though the parameter dependencies of the spin relaxation rate shown in Fig.\u00a03b,c,d are expected to be accurate, the actual numerical values should not be regarded as predictions. Having a realistic estimate for the spin-dependent tunneling amplitude, the plotted results could be rescaled to provide numerical predictions by multiplying with .\n\nThe key features in Fig.\u00a03b are analogous to those of the flip-flop relaxation. (i) The spin relaxation rate is strongly suppressed at low magnetic fields, due to the above-discussed dependence. (ii) The spin relaxation rate is maximal along an upward-bending hyperbola, corresponding to , and, therefore, nonperturbative mixing of and . (iii) For the working point specified in Table 1 (white cross in Figs. 3b,c,d), the numerical value for the spin relaxation rate evaluated from Eq.\u00a0(22) is , that is, the spin relaxation time is .\n\nA comparison between the exact and perturbative results, for a horizontal (vertical) cut of Fig.\u00a03b across the working point, is shown in Figs.\u00a03 c (d).\n\n### v.2 Spin relaxation due to g-tensor modulation\n\nWe conclude the list of valley-enhanced relaxation mechanisms with spin relaxation due to -tensor modulation. This process is allowed if the -tensors associated to the localized orbitals and are different, and, e.g., that of is anisotropicRahman\u00a0et\u00a0al. (2009b); Tosi\u00a0et\u00a0al. . In that case, a phonon, corresponding to an effective potential difference between the two localized orbitals, redistributes the electron between the two locations and thereby changes the -tensor. In general, that implies that both the length and the direction of the effective Zeeman field felt by the electron changes, leading to spin relaxation.\n\nWe focus on the simple case when the -tensors show cylindrical symmetry along the growth direction z. Importantly, in this case, this relaxation process can be avoided by a perfect in-plane or out-of-plane alignment of the external magnetic field . We further assume that -tensor anisotropy is present only at the interface. That anisotropy is incorporated in our model as the perturbation term\n\n Hgtm=(1+\u03c3z2)h\u03b3e\\boldmath{B}\u239b\u239c \u239c\u239d\u0394\u22a5\u03b3000\u0394\u22a5\u03b3000\u0394\u2225\u03b3\u239e\u239f \u239f\u23a0\\boldmath{S}. (25)\n\nTight-binding nanostructure models predictRahman\u00a0et\u00a0al. (2009b) that the typical absolute value of the relative -tensor anisotropy parameters is in the range .\n\nTo evaluate the corresponding spin relaxation time , we follow the same procedure as in section V.1, but now instead of , we use as the perturbation in the Hamiltonian. (Note that the two mechanisms do interfere in general; we disregard that here and discuss their effects separately for simplicity.) The leading-order perturbative result, expressed using the orbital relaxation time , reads\n\n 1T1,s=116[(\u0394\u22a5\u03b3\u2212\u0394\u2225\u03b3)sin(2\u03b8)]2V2t\u03f55B\u03f53o(\u03f52o\u2212\u03f52B)21T1,o. (26)\n\nIn the vicinity of the proposed working point where , this is approximated as\n\n 1T1,s\u2248\u23a1\u23a3\u03f5B(\u0394\u22a5\u03b3\u2212\u0394\u2225\u03b3)sin(2\u03b8)\/8\u03f5o%\u00a0\u2212\u03f5B\u23a4\u23a621T1,o. (27)\n\nWith the parameters in Table 1, we estimate the maximal spin relaxation rate, corresponding to a B-field polar angle of , as , implying a spin relaxation time of . Recall that under our presumptions, this mechanism can be fully suppressed by aligning the magnetic field in the xy plane or along the z axis; this appears explicitly in the results (26) and (27) via the factor .\n\nIn conclusion, we proposed that spin-orbit-induced spin-dependent tunneling between the localized charge states and can induce a valley-enhanced electron spin relaxation process, which leads to leakage from the flip-flop qubit subspace, and we expressed the parameter dependence of the corresponding relaxation rate. We also discussed how the different -tensors characterizing the two localized charge states and can contribute to electron spin relaxation.\n\n## Vi Discussion\n\n### vi.1 Electrically driven spin resonance\n\nThe spin relaxation process described in section V.1 is allowed by the spin-orbit-induced spin-dependent tunneling matrix element . The same matrix element could also be utilized for electrically driven electron spin resonance: an ac voltage component on the top gate produces an ac electric field along z, which provides the same couplings as the electron-phonon matrix elements depicted as dashed arrows in Fig.\u00a03a, and thereby drives coherent transitions between and . The corresponding Rabi frequency reads\n\n fs,Rabi=\u03f5BVsVtsin\u03b8eEacd2h\u03f5o(\u03f52o\u2212\u03f52B). (28)\n\nThis result is obtained via the following steps: we (i) expressed the energy eigenstates of the model Hamiltonian using first-order perturbation theory in , (ii) projected the driving Hamiltonian onto the two-dimensional subspace spanned by the perturbed energy eigenstates and , and (iii) read off the Rabi frequency as the amplitude of the transverse driving term in the resulting two-dimensional Hamiltonian. In the vicinity of the proposed working point, , and in the presence of an in-plane magnetic field () the result (28) simplifies to\n\n fs,Rabi=VseEacd4h(\u03f5o\u2212\u03f5B). (29)\n\nUsing the parameter values in Table 1, and assuming an in-plane magnetic field (), we find .\n\nFor comparison, we provide the analogous result for the Rabi frequency of the electrically driven transitions of the flip-flop qubit:\n\n fff,Rabi=AV2teEacd4h\u03f5o(\u03f52o\u2212\u03f52B). (30)\n\nIn the vicinity of the proposed working point, , the result (30) simplifies to\n\n fff,Rabi=AeEacd8h(\u03f5o\u2212\u03f5B), (31)\n\nand thereby we recover the corresponding result of Ref.\u00a0Tosi\u00a0et\u00a0al., [2 times the coupling rate in Eq.\u00a0(7) of Ref.\u00a0Tosi\u00a0et\u00a0al., ].\n\nFinally, we highlight a potential use of electrically driven spin resonance in the nuclear-spin-based quantum processor proposed in Ref.\u00a0Tosi\u00a0et\u00a0al., . For that setup, a key ingredient is a magnetic drive of the donor electron spin via an ac magnetic field. Creating such an ac magnetic field requires an extra element, for example, a microwave transmission line, in the setup. Electrically driven spin resonance, allowed by a sufficiently strong spin-dependent tunnel matrix element and driven by an ac gate voltage component, could substitute the ac magnetic field, and hence reduce the complexity of the envisioned architecture. To assess the practical feasibility of electrically driven spin resonance, a quantitative characterization of is required.\n\n### vi.2 Breaking of the approximate cylindrical symmetry affects spin relaxation\n\nIn Ref.\u00a0Tosi\u00a0et\u00a0al., , it is proposed that the interface-donor tunneling amplitude is tuned to the desired value by moving the interface orbital away from the donor orbital along the interface, using an appropriately designed gate stack. Of course, in that case the approximate cylindrical symmetry assumed in our considerations of spin relaxation (section V) is broken, and therefore our symmetry-based results have to be refined accordingly.\n\n### vi.3 Relaxation rates at finite temperature\n\nHere, we evaluated relaxation rates corresponding to zero temperature and spontaneous phonon emission. Induced-emission and absorption rates at finite temperature are obtained by multiplying the corresponding spontaneous-emission rates with the Bose-Einstein factor , where is the energy of the involved phonons and is the Boltzmann constant.\n\n### vi.4 Orbital relaxation: comparison to experiment\n\nA recent experimentUrdampilleta\u00a0et\u00a0al. (2015) reports an orbital relaxation time of a charge qubit, formed in an effective double quantum dot system in a silicon nanowire transistor, where one of the dots is presumably a single P donor, while the other one is gate-defined. The quoted orbital relaxation time, measured at the charge-qubit anticrossing point at a nominal charge-qubit energy splitting of GHz, can be compared to the corresponding prediction of our Eq.\u00a0(14), that is .\n\nNote that even though the two setups, studied in Ref.\u00a0Urdampilleta\u00a0et\u00a0al., 2015 and in this work, share their hybrid dot-donor character, there are also important differences between them: (i) The charge qubit in the experiment is formed by two electrons, in the (1,1)\u2013(0,2) charge configuration, where the first (second) integer is the number of electrons in the quantum dot (on the donor); our result (14) corresponds to the single-electron case. (ii) In the experiment, the quantum dot is formed at the corner of a nanowire, which presumably implies that the valley composition of the occupied electronic state is different from that described by Eq.\u00a0(11a), the latter corresponding to an electronic state at a (001) silicon\/barrier interface. (iii) In the experiment, the inhomogeneous dephasing time of the charge qubit is comparable to its splitting at the (1,1)\u2013(0,2) anticrossing. This indicates the presence of relatively strong electrical noise affecting the charge qubit detuning or tunnel coupling. Therefore, the measured orbital relaxation time should probably be understood as an average over a random ensemble of the charge-qubit parameters.\n\nNote also that the orbital relaxation time was measured for a single setting of the charge qubit. Measuring as a function of the charge-qubit parameters would allow a qualitative comparison with theoretically predicted trends, e.g., Eq.\u00a0(14) of this work, and thereby help identifying the underlying relaxation mechanism.\n\n### vi.5 Prolonging the relaxation times\n\n(1) Controlling the valley composition of the donor orbital . In the valley-enhanced relaxation mechanisms described in this work, a key ingredient is the substantially different valley structure of the electronic wave functions at the interface and donor sites, see Eq.\u00a0(11). Making the valley composition [Eq.\u00a0(11b)] of the donor orbital more similar to that [Eq.\u00a0(11a)] of the interface orbital would prolong the relaxation times. The even valley composition of in Eq.\u00a0(11b) might be altered by a number of mechanisms: for example, by static strain due to a finite germanium concentration in the heterostructureVrijen\u00a0et\u00a0al. (2000); Tahan\u00a0et\u00a0al. (2002); Koiller\u00a0et\u00a0al. (2002), by the close vicinity of an interfaceBaena\u00a0et\u00a0al. (2012); Salfi\u00a0et\u00a0al. (2014), or by an electric fieldFriesen (2005). For example, placing the donor closer to an interface, while keeping all other relevant parameters unchanged, would bring the valley compositions of and closer to each other, and therefore presumably prolong the relaxation times considered here. That speculation is supported by, e.g., the estimate in Ref.\u00a0Salfi\u00a0et\u00a0al., 2014, claiming that the z-valley population of a donor electron at nm below a silicon surface is 40%, in contrast to the bulk value 33%.\n\n(2) Optimizing the working point via weakening the qubit-field interaction. In the vicinity of the working point of Table 1, the estimatedTosi\u00a0et\u00a0al. time required for a cavity-mediated two-qubit gate is . This implies that the number of such operations performed during the flip-flop relaxation time is . In principle, this quality factor can be improved via, e.g., increasing the tunneling amplitude , thereby weakening the hyperfine-induced hybridization of with (see Fig.\u00a02a), and hence weakening the interaction between the flip-flop qubit and the electric fields. For example, approximately a factor of 2 improvement of the above quality factor can be achieved by the following adjustments. (i) The tunnel matrix element is reset to GHz. Essentially, this doubles the energy denominator in the flip-flop relaxation rate (19) as well as in the vacuum Rabi frequency; the latter is obtained from (31) by identifying with the cavity vacuum field. As a result, increases by a factor of 4. (ii) The magnetic field is reset such that the qubit-cavity detuning is halved. As a result of (i) and (ii), increases only with a factor of 2, without a significant change in the gate fidelity; hence the quality factor indeed doubles. In practice, an important consideration that should be added to the above procedure is the change of the inhomogeneous dephasing time of the flip-flop qubit with the adjustments, with the goal of exploiting the expected long times offered by the second-order clock transitionsTosi\u00a0et\u00a0al. . In general, this necessitates a more complex optimization procedure.\n\n## Vii Conclusions\n\nWe described fast, valley-enhanced relaxation mechanisms (orbital, flip-flop and electron spin relaxation) for a gate-controlled P donor electron close to a silicon\/barrier interface. For the flip-flop qubit setup and the proposed qubit working point, we have found that the flip-flop relaxation can be approximately 8 orders of magnitude faster than in bulk. The predicted relaxation time scale is , still longer than the expected single-qubit () and two-qubit () gate timesTosi\u00a0et\u00a0al. . Nevertheless, relaxation might dominate dephasing, if our estimates as well as the inhomogeneous dephasing rate estimateTosi\u00a0et\u00a0al. Hz are reliable. The relevant time scales are listed in Table 2.\n\nWe also discussed analogous, valley-enhanced mechanisms inducing orbital and electron spin relaxation. Since gate control of donor electrons near interfaces is an ubiquitous ingredient of donor-based quantum-computing schemes, the relevance of the mechanisms described here extends beyond the considered specific flip-flop qubit architecture.\n\n###### Acknowledgements.\nWe thank D.\u00a0Culcer, R.\u00a0Joynt, and R.\u00a0Rahman for useful discussions, and M.\u00a0Calder\u00f3n, A.\u00a0Morello, and G.\u00a0Tosi for their helpful and constructive feedback on the manuscript. We acknowledge funding from the EU Marie Curie Career Integration Grant CIG-293834, Hungarian OTKA Grants No.\u00a0PD 100373 and 108676, the Gordon Godfrey Bequest, and the EU ERC Starting Grant 258789. 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(2015) A.\u00a0Laucht, J.\u00a0T.\u00a0Muhonen, F.\u00a0A.\u00a0Mohiyaddin, R.\u00a0Kalra, J.\u00a0P.\u00a0Dehollain, S.\u00a0Freer, F.\u00a0E.\u00a0Hudson, M.\u00a0Veldhorst, R.\u00a0Rahman, G.\u00a0Klimeck, K.\u00a0M.\u00a0Itoh, D.\u00a0N.\u00a0Jamieson, J.\u00a0C.\u00a0McCallum, A.\u00a0S.\u00a0Dzurak, \u00a0and\u00a0A.\u00a0Morello,\u00a0Science Advances\u00a01 (2015),\u00a010.1126\/sciadv.1500022.\n\u2022 (25) G.\u00a0Tosi, F.\u00a0A.\u00a0Mohiyaddin, S.\u00a0B.\u00a0Tenberg, R.\u00a0Rahman, G.\u00a0Klimeck, \u00a0and\u00a0A.\u00a0Morello,\u00a0ArXiv:1509.08538v1.\n\u2022 Urdampilleta\u00a0et\u00a0al. (2015) M.\u00a0Urdampilleta, A.\u00a0Chatterjee, C.\u00a0C.\u00a0Lo, T.\u00a0Kobayashi, J.\u00a0Mansir, S.\u00a0Barraud, A.\u00a0C.\u00a0Betz, S.\u00a0Rogge, M.\u00a0F.\u00a0Gonzalez-Zalba, \u00a0and\u00a0J.\u00a0J.\u00a0L.\u00a0Morton,\u00a0Phys. Rev. X\u00a05,\u00a0031024 (2015).\n\u2022 (27) P.\u00a0Harvey-Collard, N.\u00a0T.\u00a0Jacobson, M.\u00a0Rudolph, J.\u00a0Dominguez, G.\u00a0A.\u00a0T.\u00a0Eyck, J.\u00a0R.\u00a0Wendt, T.\u00a0Pluym, J.\u00a0K.\u00a0Gamble, M.\u00a0P.\u00a0Lilly, M.\u00a0Pioro-Ladri\u00e8re, \u00a0and\u00a0M.\u00a0S.\u00a0Carroll,\u00a0ArXiv:1512.01606v1.\n\u2022 Golovach\u00a0et\u00a0al. (2006) V.\u00a0N.\u00a0Golovach, M.\u00a0Borhani, \u00a0and\u00a0D.\u00a0Loss,\u00a0Phys.\u00a0Rev.\u00a0B\u00a074,\u00a0165319 (2006).\n\u2022 Flindt\u00a0et\u00a0al. (2006) C.\u00a0Flindt, A.\u00a0S.\u00a0S\u00f8rensen, \u00a0and\u00a0K.\u00a0Flensberg,\u00a0Phys. Rev. Lett.\u00a097,\u00a0240501 (2006).\n\u2022 Nowack\u00a0et\u00a0al. (2007) K.\u00a0C.\u00a0Nowack, F.\u00a0H.\u00a0L.\u00a0Koppens, Y.\u00a0V.\u00a0Nazarov, \u00a0and\u00a0L.\u00a0M.\u00a0K.\u00a0Vandersypen,\u00a0Science\u00a0318,\u00a01430 (2007).\n\u2022 Blais\u00a0et\u00a0al. (2004) A.\u00a0Blais, R.-S.\u00a0Huang, A.\u00a0Wallraff, S.\u00a0M.\u00a0Girvin, \u00a0and\u00a0R.\u00a0J.\u00a0Schoelkopf,\u00a0Phys. Rev. A\u00a069,\u00a0062320 (2004).\n\u2022 Trif\u00a0et\u00a0al. (2007) M.\u00a0Trif, V.\u00a0N.\u00a0Golovach, \u00a0and\u00a0D.\u00a0Loss,\u00a0Phys. Rev. B\u00a075,\u00a0085307 (2007).\n\u2022 Salfi\u00a0et\u00a0al. (2015) J.\u00a0Salfi, J.\u00a0A.\u00a0Mol, D.\u00a0Culcer, \u00a0and\u00a0S.\u00a0Rogge,\u00a0arXiv:1508.04259\u00a0 (2015).\n\u2022 Cullis\u00a0and\u00a0Marko (1970) P.\u00a0R.\u00a0Cullis\u00a0and\u00a0J.\u00a0R.\u00a0Marko,\u00a0Phys. Rev. B\u00a01,\u00a0632 (1970).\n\u2022 Koiller\u00a0et\u00a0al. (2001) B.\u00a0Koiller, X.\u00a0Hu, \u00a0and\u00a0S.\u00a0Das Sarma,\u00a0Phys. Rev. Lett.\u00a088,\u00a0027903 (2001).\n\u2022 Pines\u00a0et\u00a0al. (1957) D.\u00a0Pines, J.\u00a0Bardeen, \u00a0and\u00a0C.\u00a0P.\u00a0Slichter,\u00a0Phys. Rev.\u00a0106,\u00a0489 (1957).\n\u2022 Abragam (1961) A.\u00a0Abragam,\u00a0The principles of nuclear magnetism\u00a0(Oxford University Press,\u00a01961).\n\u2022 Khaetskii (2001) A.\u00a0V.\u00a0Khaetskii,\u00a0Physica E\u00a010,\u00a027 (2001).\n\u2022 Erlingsson\u00a0and\u00a0Nazarov (2002) S.\u00a0I.\u00a0Erlingsson\u00a0and\u00a0Y.\u00a0V.\u00a0Nazarov,\u00a0Phys.\u00a0Rev.\u00a0B\u00a066,\u00a0155327 (2002).\n\u2022 Rahman\u00a0et\u00a0al. (2009b) R.\u00a0Rahman, S.\u00a0H.\u00a0Park, T.\u00a0B.\u00a0Boykin, G.\u00a0Klimeck, S.\u00a0Rogge, \u00a0and\u00a0L.\u00a0C.\u00a0L.\u00a0Hollenberg,\u00a0Phys. Rev. B\u00a080,\u00a0155301 (2009b).\n\u2022 Herring\u00a0and\u00a0Vogt (1956) C.\u00a0Herring\u00a0and\u00a0E.\u00a0Vogt,\u00a0Phys. Rev.\u00a0101,\u00a0944 (1956).\n\u2022 Yu\u00a0and\u00a0Cardona (2010) P.\u00a0Y.\u00a0Yu\u00a0and\u00a0M.\u00a0Cardona,\u00a0Fundamentals of Semiconductors\u00a0(Springer,\u00a0Berlin,\u00a02010).\n\u2022 Kohn\u00a0and\u00a0Luttinger (1955) W.\u00a0Kohn\u00a0and\u00a0J.\u00a0M.\u00a0Luttinger,\u00a0Phys. Rev.\u00a098,\u00a0915 (1955).\n\u2022 Danon\u00a0and\u00a0Nazarov (2009) J.\u00a0Danon\u00a0and\u00a0Y.\u00a0V.\u00a0Nazarov,\u00a0Phys. Rev. B\u00a080,\u00a0041301 (2009).\n\u2022 van Vleck (1940) J.\u00a0H.\u00a0van Vleck,\u00a0Phys. Rev.\u00a057,\u00a0426 (1940).\n\u2022 Khaetskii\u00a0and\u00a0Nazarov (2001) A.\u00a0Khaetskii\u00a0and\u00a0Y.\u00a0Nazarov,\u00a0Phys.\u00a0Rev.\u00a0B\u00a064,\u00a0125316 (2001).\n\u2022 Koiller\u00a0et\u00a0al. (2002) B.\u00a0Koiller, X.\u00a0Hu, \u00a0and\u00a0S.\u00a0Das Sarma,\u00a0Phys. Rev. B\u00a066,\u00a0115201 (2002).\n\u2022 Salfi\u00a0et\u00a0al. (2014) J.\u00a0Salfi, J.\u00a0A.\u00a0Mol, R.\u00a0Rahman, G.\u00a0Klimeck, M.\u00a0Y.\u00a0Simmons, L.\u00a0C.\u00a0L.\u00a0Hollenberg, \u00a0and\u00a0S.\u00a0Rogge,\u00a0Nat.\u00a0Mater.\u00a013,\u00a0605 (2014).\n\u2022 Friesen (2005) M.\u00a0Friesen,\u00a0Phys. Rev. Lett.\u00a094,\u00a0186403 (2005).\nYou are adding the first comment!\nHow to quickly get a good reply:\n\u2022 Give credit where it\u2019s due by listing out the positive aspects of a paper before getting into which changes should be made.\n\u2022 Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.\n\u2022 Your comment should inspire ideas to flow and help the author improves the paper.\n\nThe better we are at sharing our knowledge with each other, the faster we move forward.\nThe feedback must be of minimum 40 characters and the title a minimum of 5 characters","date":"2020-07-09 17:25:40","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\": 0.8592460751533508, \"perplexity\": 925.4743693378947}, \"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-2020-29\/segments\/1593655900614.47\/warc\/CC-MAIN-20200709162634-20200709192634-00587.warc.gz\"}"}
| null | null |
(function(){
'use strict';
angular
.module('lolTot.singup')
.controller('HelpController', HelpController);
function HelpController($mdDialog, maxPoints, leagues) {
var vmd = this;
vmd.maxPoints = maxPoints;
vmd.leagues = leagues;
vmd.close = function(){
$mdDialog.hide();
};
}
})();
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,915
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STARTTIME=$(date +%s)
source "$(dirname "${BASH_SOURCE}")/lib/init.sh"
# Update test names
go generate -mod vendor ./test/extended
os::build::setup_env
OUTPUT_PARENT=${OUTPUT_ROOT:-$OS_ROOT}
# If you hit this, please reduce other tests instead of importing more
if [[ "$( cat "${OUTPUT_PARENT}/test/extended/testdata/bindata.go" | wc -c )" -gt 2500000 ]]; then
echo "error: extended bindata is $( cat "${OUTPUT_PARENT}/test/extended/testdata/bindata.go" | wc -c ) bytes, reduce the size of the import" 1>&2
exit 1
fi
ret=$?; ENDTIME=$(date +%s); echo "$0 took $(($ENDTIME - $STARTTIME)) seconds"; exit "$ret"
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,247
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'Star Wars': Han Solo Spin-Off Close To Casting | Film News
16 Mar 2016 — Filed Under: Film, Film News
It's a way ahead in the future, but that doesn't stop the legion of Star Wars fans from being excited about it. Despite having Rogue One: A Star Wars Story and Episode VIII to burn through, they have Han Solo on their minds. With a target release date of May 25, 2018, directors Phil Lord and Chris Miller are now closing in on the man to play their scruffy-looking scoundrel.
Filling the shoes of Harrison Ford will be no easy task no matter who lands the role, but despite this the project has been flooded with hopeful auditions and rehearsal tapes. After much searching, Lord and Miller have narrowed it down to a handful and the most prominent names amoungst the reports are Alden Ehrenreich, Jack Reynor and Taron Egerton.
Taron Egerton has long been in the running to play Solo, being the flavour of the month after good performances in Kingsman: The Secret Service and Testament of Youth. Jack Reynor is best known for appearing in Transformers: Age of Extinction and Macbeth alongside Michael Fassbender. However, he is also filming Transformers 5; will there be room in his schedule for another franchise?
The real surprise is Alden Ehrenreich, who has not been in contention before now. Ehrenreich will be best known for recently appearing in the Coen brothers' Hail, Caesar! and Woody Allen's Blue Jasmine. However, whether any of these names make it to the final post is questionable, as the Han Solo shortlist has proven to be a volatile thing.
Lawrence Kasdan, writer of The Empire Strikes Back, Return of the Jedi and The Force Awakens, has claimed that this spin-off will be set 10 years before the events of A New Hope; before Han met Chewie (great name for a rom-com); before he owned the Millennium Falcon; and before he met Luke and Leia (and before having a child with the latter who proceeded to stab him through the chest).
The question is: can any of these hopefuls pull off one of the greatest action characters of all time?
#Peace.Love.HanSolo
Tom Russell
Film and Television graduate of the University of Lincoln. Aspiring Writer. Golden-Age thinker. Lover of film, history, gaming, theatre and literature. https://tomrussell94.wordpress.com/
Latest posts by Tom Russell (see all)
Fran Kranz Joins Stephen King's 'The Dark Tower' | Film News - April 22, 2016
Superman Prequel Series 'Krypton' Gets A Pilot | TV News - April 22, 2016
Simon Pegg Reveals New Plot Details For 'Star Trek Beyond' | Film News - April 22, 2016
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,528
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Mix the lime & lemon juice in a glass with the limoncello & rum. Add a few ice cubes & then the lemonade. Garnish with a strawberry on the rim of the glass.
OMG! THE BEST Concealer Cocktail for Mature Under Eyes!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,081
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« Zzzzzzz.... | Main | I wasn't going to Bookerthon, but shall I... »
Thoughts on Grayson Perry...
It's possibly ( to my uncertain knowledge) the first time dovegreyreader scribbles has been linked to by a cross dressing website and I feel the moment shouldn't pass by unnoticed.
Port Eliot still whirring around in my mind, as it will do for weeks to come, and in particular the whole Grayson Perry thing which had passed by on the periphery of my consciousness until now, something happening in London to which I had paid little attention.
We can be a bit cut off from reality down here in the West Country and it would indeed stop the traffic and probably cause a few accidents were Grayson to walk down the street, but somehow not at all unusual at Port Eliot, and there's something intriguing about it all.
Of course transvestites not unknown in Devon, indeed, in the days when health visitors visited across the age spectrum of patients registered at the GP surgery they were attached to, I took a referral from a GP and had the pleasure of visiting a particularly glamorous man who had modelled his look on a 1960's film star whose name he had taken too. His make-up always immaculate but living out his lonely last days in chronically poor health, poverty and isolation in a woodland caravan with just a dog for company.
I got to know him/her really well and even found myself popping into the WI clothing store to pick up clothes on his/her behalf in the days when we did helpful things like that too. We'd made a wish list and I'd been careful to check colour preferences etc and caused quite a stir the day I asked for a salmon pink dressing gown, preferably with some lace frills, to fit someone 6ft 5" tall, and could they possibly help me with a pair of ladies, knee-high winter boots (with extra calf room) in a size 14.
It occurred to me how effectively Grayson Perry blasts away stereotypes and demolishes personal prejudices with his style whilst equally I discover on reading up, elicits fierce responses..
"...recently a man in the street called him a paedophile: "It's interesting how that is now the nuclear weapon of insults. The soundbite of the venal hater."
If ever anyone spots a paedophile element it's the health visitor in me and I have to say it was the very last thought that would have come into my head. The whole look is almost like an art form on legs regardless of any personal reasons ceramic artist Grayson Perry may have for adopting his alter ego Claire when he feels so inclined, and the dresses are mind-blowingly clever designs, though interesting to read Grayson's own take on it
"In the past seven years, I have changed from dressing as a woman to being a man in a dress. I am not pretending to be a woman any more. I am a man who is wearing the most ridiculous outfits."
They are utterly gorgeous creations, what better frock to wear to a Flower Show awards ceremony than this one and beautifully made I discovered as I tried to get a good look without invading his personal space. I had to stop myself having a feel of the fabric, but curtain weight I think and it occurred to me that I was probably getting a glimpse of an iconic dress exhibition of the far distant future... when people will queue for hours, much as they have for the Grace Kelly collection from the thirties, to see the wardrobe of a man's frocks from the noughties.
Monday, July 26, 2010 | Permalink
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,138
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Restaurant job growth slowed significantly in July
The restaurant industry continued along its long road to recovery in July, albeit at a much slower pace than the previous two months. Eating and drinking places* added a net 502,000 jobs in July on a seasonally-adjusted basis, according to preliminary data from the Bureau of Labor Statistics (BLS).
In what would typically be a record-breaking payroll expansion in normal times, July's half-million employment gain represented a significant slowdown from the 1.5 million jobs added in both May and June. As a result, employment at eating and drinking places remains well below February's pre-coronavirus level of 12 million.
Looking forward, restaurant job growth will likely be uneven in the coming months, due to the uncertainty associated with spiking COVID-19 case levels and the potential for renewed restrictions in many parts of the country.
[It's important to note that the BLS monthly employment reports count jobs during the payroll period that includes the 12th of each month. Changes in restaurant staffing levels – both negative and positive – have occurred rapidly during the coronavirus pandemic, as restaurants quickly adjust their operating status in response to evolving regulatory and economic conditions. As a result, significant changes likely occurred during the weeks between each measurement period, and the monthly data may not fully capture the total job losses experienced during the coronavirus lockdowns. Still, the figures are a useful indication of the extent to which restaurant employment is recovering.]
Staffing levels are rising across most segments
Although employers across most of the major restaurant segments added jobs in May and June, their distance from pre-coronavirus staffing levels varies significantly. [Note that the segment-level employment figures are lagged by one month, so June is the most current data available.]
The fullservice segment added more than 2 million jobs in May and June, which is more than the other major restaurant segments combined. However, the gain was only a little over half of the jobs that the segment lost in March and April.
Job losses in the limited-service segments were less severe during the lockdowns, as these operations were more likely to retain staff to handle off-premises traffic. The quickservice and fast casual segments added more than 641,000 jobs in May and June, after shedding nearly 1 million jobs during the previous two months.
Snack and nonalcoholic beverage bars – including coffee, donut and ice cream shops – added 284,600 jobs in May and June. This followed a loss of more than 400,000 jobs during March and April.
Among the major restaurant segments, only food service contractors continued to cut jobs in both May and June, according to BLS.
*Eating and drinking places are the primary component of the total restaurant and foodservice industry, which prior the coronavirus outbreak employed 12 million out of the total restaurant and foodservice workforce of 15.6 million.
Read more analysis and commentary from the Association's chief economist Bruce Grindy.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,577
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Turkey adds crypto firms to terror financing rules
in Markets & Economy
Turkey has added cryptocurrency trading platforms to the list of firms covered by anti-money laundering and terrorism financing regulation, it said in a presidential decree published early May 1.
The move came after a ban on using cryptocurrencies for making payments introduced in response to claims that such transactions are too risky took effect in Turkey on April 30.
The presidential decree makes "crypto-asset service providers" responsible for seeing their assets are not used illegally. The decree immediately went into force with its publication in the Official Gazette.
Turkish authorities last month launched fraud investigations into two cryptocurrency exchanges, Thodex and Vebitcoin.
Six suspects linked to the Thodex probe were jailed on April 30, pending trial.
The investigation into Thodex, which handled daily trades of hundreds of millions of dollars, initially led to the detentions of more than 70 people after customers complained of not being able to access their funds.
Interpol issued a detention warrant for the firm's CEO, Faruk Fatih Özer, on Turkey's behalf. Turkey has sent two five-member police teams to Albania, where Özer fled after the Thodex website went offline abruptly and trapped cryptocurrency codes of nearly 400,000 customers on April 19.
Özer met with an Albanian couple in Istanbul nearly a month before running away, daily Milliyet reported yesterday.
The total assets allegedly seized by Özer is around $108 million, legal experts told the Turk-ish Interior Ministry after initial examinations.
Vebitcoin Founder İlker Baş, his wife and two employees were also arrested after it halted operations on April 23. Baş has been accused of making XRP altcoin transactions worth over $24 million to his overseas accounts in a month.
Turks have been increasingly attracted by cryptocurrencies as protection against the decline of the Turkish Lira and double-digit inflation. The inflation rate passed 16 percent in Turkey in March, at the time when seven-week crypto trading volumes hit 218 billion Turkish Liras ($27 billion), up from just over 7 billion liras in the same period a year earlier, according to data from U.S. researcher Chainalysis.
Around 40 cryptocurrency exchange platforms are operating in Turkey. The Central Bank, the Treasury and Finance Ministry and financial watchdogs are expected to finalize a regulation package for the cryptocurrency market in the upcoming days, Central Bank Governor Şahap Kavcıoğlu said on April 23.
Manufacturing PMI down in April
The Pilgrims' Attack on a May Day Celebration
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,580
|
Guys i bought the nokia 6 ten month before and from the beginning i am facing problem with this phone.
1. headphone jack was not working and the service person were replace the motherboard. That time some hardware has been changed. the SIM adaptor contain IMEI no which was replace by a blank adaptor.
2. mobile some time getting automic shutdown and after 2 days it is completely off means dead and it is not getting ON. when we visit to service team , they told me that product is out of warrany. when we showed him bill and date of purchase then he told me that we are not working on behalf of your bill. if it is showing out of warranty means it is. How irresponcible they are?
within 10 month everything is gone. no durability, no performance, not even a good quality of hardware, no support and no responsible. i am totally fade up with such a things. now we will have to move for consumer form and lets see the better solution.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 76
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Маринское () — село в Горностаевском районе Херсонской области Украины.
Население по переписи 2001 года составляло 512 человек. Почтовый индекс — 74611. Телефонный код — 5544. Код КОАТУУ — 6522683301.
Местный совет
74611, Херсонская обл., Горностаевский р-н, с. Маринское, ул. Юбилейная, 36
Ссылки
Маринское на сайте Верховной рады Украины
Примечания
Населённые пункты Горностаевского района
Немецкие колонии в Херсонской области
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,923
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{"url":"http:\/\/icpc.njust.edu.cn\/Problem\/CF\/578A\/","text":"Time Limit: 1 second\n\nMemory Limit: 256 megabytes\n\n## Description\n\nThere is a polyline going through points (0,\u20090)\u2009\u2013\u2009(x,\u2009x)\u2009\u2013\u2009(2x,\u20090)\u2009\u2013\u2009(3x,\u2009x)\u2009\u2013\u2009(4x,\u20090)\u2009\u2013\u2009...\u2009-\u2009(2kx,\u20090)\u2009\u2013\u2009(2kx\u2009+\u2009x,\u2009x)\u2009\u2013\u2009....\n\nWe know that the polyline passes through the point (a,\u2009b). Find minimum positive value x such that it is true or determine that there is no such x.\n\n## Input\n\nOnly one line containing two positive integers a and b (1\u2009\u2264\u2009a,\u2009b\u2009\u2264\u2009109).\n\n## Output\n\nOutput the only line containing the answer. Your answer will be considered correct if its relative or absolute error doesn't exceed 10\u2009-\u20099. If there is no such x then output \u2009-\u20091 as the answer.\n\n## Sample Input\n\nInput3 1Output1.000000000000Input1 3Output-1Input4 1Output1.250000000000\n\n## Sample Output\n\nNone\n\n## Hint\n\nYou can see following graphs for sample 1 and sample 3.\n\nNone","date":"2020-10-27 12:33:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.27864870429039, \"perplexity\": 409.87223906646716}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107894175.55\/warc\/CC-MAIN-20201027111346-20201027141346-00016.warc.gz\"}"}
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\section{Introduction}
Koszul rings and Koszul duality play an important role in commutative algebra, algebraic
topology, (noncommutative) algebraic geometry, and in the theory of quantum groups, representation theory of algebras and Lie theory (see \cite{Pri, Lof, BGS1, Fr, CPS, Sm}, etc.).
Classically, a Koszul ring $A$ means an $\mathbb{N}$-graded ring $A=A_0\oplus A_1\oplus A_2\oplus \cdots$ with $A_0$ artinian semisimple and $A_0 \cong A/A_{\geqslant 1}$ having a linear projective resolution as a graded $A$-module (see \cite[Definition 1.2.1]{BGS1}).
Here are some major results in the classical Koszul theory. Suppose that $A$ is left finite, that is, every $A_n$ is finitely generated as an $A_0$-module, and $A_0$ is artinian semisimple. Then the following are equivalent: $A$ is Koszul; $A^{op}$ is Koszul \cite[Theorem 2.2.1]{BGS1}; the Koszul dual $\gExt^\bullet_A(A_0,A_0)$ (or the Yoneda Ext ring) of $A$ is Koszul \cite[Theorem 2.10.2]{BGS1} (see also \cite[Theorem 6.1]{GM1}); $\gExt^n_A(A_0,A_0)$ is concentrated in degree $-n$ for all $n \geqslant 0$ \cite[Proposition 2.1.3]{BGS1}; $A$ is isomorphic to the Koszul dual of the Koszul dual of $A$ \cite[Theorem 2.4]{GM2} (see also \cite[Theorems 10.1, 10.2]{GM1}, \cite[Theorem 2.10.2]{BGS1}). Moreover, there is a duality between the category of Koszul modules over a Koszul ring $A$ and the category of Koszul modules over the Koszul dual of $A$ \cite[Theorem 5.2]{GM2}.
The Koszul rings in the sense of \cite{BGS1} will be called sometimes classically Koszul later in this paper.
There are several generalized Koszul theories in literature, where $A_0$ is not assumed being artinian semisimple.
To develop a unified approach to Koszul duality and cotilting theory, $T$-Koszul algebras were first defined in \cite{GRS} for left finite graded algebras $A$ with $A_0$ artinian, where $T$ is a Wakamatsu cotilting $A_0$-module. For locally finite graded algebra $A$ such that $A_0$ has finite global dimension, an equivalent definition of $T$-Koszul algebras was given in \cite[Definition 4.1.1]{Ma}.
Similar to the classical Koszul theory, the following statements were proved: $A^{op}$ is $T^*$-Koszul if $A$ is $T$-Koszul \cite[Theorem 4.1.2]{Ma}; each $T$-Koszul algebra has a $T^*$-Koszul dual algebra which is the Yoneda Ext algebra of $T$ \cite[Theorem 4.2.1(a)]{Ma}; the $T^*$-Koszul dual of the $T$-Koszul dual algebra is isomorphic to the original algebra \cite[Theorem 4.2.1(b)]{Ma}; and there is a duality between categories of $T$-Koszul modules over a $T$-Koszul algebra and its $T$-Koszul dual algebra \cite[Theorem 4.3.1]{Ma}.
If $A_0$ is artinian semisimple and $T=A_0$, then $T$-Koszul algebras are classically Koszul. $T$-Koszul theory specializes to classical Koszul
theory and to Wakamatsu tilting theory (see \cite{Ma} and the references therein).
An earlier generalization of a graded ring
$A$ being Koszul was given by Woodcock in \cite{Wo}, where $A$ is
both a left and a right projective $A_0$-module, but with no restrictions on $A_0$.
This is not necessarily the case in the setting of \cite{GRS}.
The point of view in \cite{Wo} is considering the Koszul complex
as the defining property of the so called Koszul modules.
However, assuming that $A$ is a
left finite graded ring generated in degree $1$ with $A_0$ artinian, and choosing $T=(A_0)^*$, the vector space dual of $A$, one can see that $A$ is $T$-Koszul in the sense of \cite{GRS}.
To apply Koszul theory to study the Ext groups of representations of finite EI categories, Li developed a generalized Koszul theory for locally finite $\mathbb{N}$-graded $k$-algebra $A$ with $A_0$ self-injective instead of semisimple \cite{Li1}. The results were proved to be true more generally for $A_0$ with finitistic dimension $0$ \cite{Li2}. In fact, both finite dimensional local algebras and self-injective algebras have finitistic dimension $0$. In \cite{Li1, Li2}, generalized Koszul modules and algebras were defined via linear projective resolutions as in the classical case.
The following results were proved in \cite[Section 1]{Li2}: if further $A$ is projective as an $A_0$-module then a graded $A$-module $M$ is generalized Koszul if and only if it is a projective $A_0$-module and $\gExt_A^1(A_0,A_0)\cdot \gExt_A^{n-1}(M,A_0)=\gExt^{n}_A(M,A_0)$ for all $n > 0$; if $A$ is generalized Koszul then there is a duality between the generalized Koszul modules over $A$ and over $\gExt_A^{\bullet}(A_0,A_0)$ via $M \mapsto \gExt_A^{\bullet}(M,A_0)$; $A$ is generalized Koszul if and only if $A$ is $A_0$-projective and $A/\mathfrak{R}$ is classically Koszul where $\mathfrak{R}=AJ(A_0)A$; if $A$ is $A_0$-projective, then a graded $A$-module $M$ is generalized Koszul if and only if it is $A_0$-projective and $M/{\mathfrak{R}M}$ is a classically Koszul $A/\mathfrak{R}$-module.
The concept of Koszul algebras has been generalized to higher Koszul algebras \cite{Ber, GMMZ, HY, L3, LHL} and to categories \cite{Man, BGS2, MOS}, which are not the topics in this paper.
A linear projective resolution of a graded module $M$ can be characterized by the property of (the radical filtration of) syzygies of $M$ \cite[Proposition 3.1 and Lemma 5.1]{GM1} (see Proposition \ref{qk and classical koszul}). Using this characterization, Green and Martin{\'e}z-Villa defined (strongly) quasi-Koszul modules and rings, and developed a Koszul theory for noetherian semiperfect rings \cite{GM1,GM2}.
In this paper, we develop a generalized Koszul theory for $\mathbb{N}$-graded rings $A$ with the degree $0$ part $A_0$ noetherian semiperfect (not necessarily semisimple), which specializes to the classical graded Koszul theory if $A_0$ is artinian semisimple and the ungraded Koszul theory for noetherian semiperfect ring if $A$ is concentrated in degree $0$.
More explicitly, let $A$ be an $\mathbb{N}$-graded ring with $A_0$ noetherian semiperfect, $J=J(A_0)\oplus A_{\geqslant 1}$ be the graded Jacobson radical of $A$ and $S=A/J$.
Let $M$ be a left finite and bounded below graded $A$-module. It follows from Proposition \ref{minimal projective resolution} that $M$ has a minimal graded projective resolution.
Following the idea in \cite{GM1}, a graded $A$-module $M$ with a minimal graded projective resolution $P_\bullet \to M \to 0$ is called Koszul (resp. quasi-Koszul) if $J^k\Ker d_n = \Ker d_n\cap J^{k+1}P_n$ for all $n,k\geqslant 0$ (resp. $J\Ker d_n = \Ker d_n\cap J^2P_n$ for all $n$) (see Definition \ref{our-definition}).
Then, a left finite graded ring $A$ generated in degree $1$ with $A_0$ noetherian semiperfect is called a (quasi-)Koszul ring if $S \cong A_0/{J(A_0)}$ is a (quasi-)Koszul $A$-module. Here, we rename ``strongly quasi-Koszul" in \cite{GM1} by ``Koszul" but remain the name ``quasi-Koszul".
Let $E(A)=\gExt_A^{\bullet}(A/J,A/J)$ be the Yoneda Ext ring of $A$.
The first result says that $A$ is a quasi-Koszul ring if and only if $E(A)$ is generated in degree $1$; and if $A$ is quasi-Koszul then $M$ is quasi-Koszul if and only if $\gExt_A^{\bullet}(M,A/J)$ is generated
in degree $0$ as an $E(A)$-module (see Theorem \ref{qK iff E(M) generated in degree 0}). Sometimes, $E(A)$ is called the Koszul dual of $A$.
\begin{theorem} \label{theorem 1
\begin{itemize}
\item[(1)] $M$ is quasi-Koszul if and only if, for any $n>0$, $$\gExt_A^1(A/J,A/J)\cdot \gExt_A^{n-1}(M,A/J)=\gExt^n_A(M, A/J).$$
\item[(2)] $A$ is a quasi-Koszul ring if and only if $\gExt_A^{\bullet}(A/J,A/J)$ is generated by $\gExt^1_A(A/J,A/J)$ over $\gHom_A(A/J,A/J)$.
\end{itemize}
\end{theorem}
We describe in Example \ref{qk-but-not-sqk} a quasi-Koszul ring which is not a Koszul ring.
Koszul rings and modules are characterized in the following by using Theorem \ref{theorem 1} (see Theorem \ref{A sqk implies E(A) Koszul}).
\begin{theorem}\label{theorem 2
\begin{itemize}
\item[(1)] $A$ is a Koszul ring if and only if $\gExt_A^\bullet(A/J,A/J)$ is a classical Koszul ring.
\item[(2)] Suppose that $A$ is Koszul. Then $M$ is a Koszul $A$-module if and only if $\gExt_A^{\bullet}(M,A/J)$ is a classical Koszul $E(A)$-module.
\end{itemize}
\end{theorem}
Let $\Grj A=\oplus (J^i/J^{i+1})$, and $\Grj M=\oplus (J^iM/J^{i+1}M)$ for any graded $A$-module $M$. Then $\Grj M$ is a graded $\Grj A$-module. If $A_0$ is semisimple then $\Grj A\cong A$ and $\Grj M\cong M$.
We emphasize that $\Grj A$ is viewed as a graded ring via the graded degree induced by $J$-adic filtration, and $E(A)$ is viewed as a graded ring via the homological degree.
Let $\mathcal{K}_A$, $\mathcal{K}_{E(A)}$ and $\mathcal{K}_{\Grj A}$ be the full subcategories of finitely generated Koszul $A$-modules, Koszul ${E(A)}$-modules and Koszul ${\Grj A}$-modules respectively in the corresponding categories.
Let
$$\mathcal{E}=\gExt_A^\bullet(-,A/J),
\mathcal{F}=\gExt_{E(A)}^\bullet(-,E(A)/J_{E(A)})
\mathcal{G}= \gExt^\bullet_{\Grj A}(-,A/J).$$
The following is a generalized version of Koszul algebra duality and Koszul module duality, which generalizes \cite[Theorem 2.10.2]{BGS1}, \cite[Theorem 6.3 (4)]{Sm} and \cite[Theorems 10.1, 10.4]{GM1}.
\begin{theorem}\label{theorem 3}
Let $A$ be a Koszul ring. Then
\begin{itemize}
\item[(1)] $E(E(A))\cong \Grj A$ as graded rings.
\item[(2)]
The functors $\mathcal{E}, \mathcal{F}$ and $\mathcal{G}$ restrict to
$$\xymatrix{
\mathcal{K}_A\ar[r]^{\mathcal{E}} &\mathcal{K}_{E(A)}\ar@<1mm>[r]^{\mathcal{F}}& \mathcal{K}_{\Grj A}\ar@<1mm>[l]^{\mathcal{G}}.
}$$
For any $M\in \mathcal{K}_{A}$, $\mathcal{F}\mathcal{E}(M)\cong \Grj M$ as graded $\Grj A$-modules.
\item[(3)] The functors $\mathcal{F}$ and $\mathcal{G}$ give a duality between $\mathcal{K}_{E(A)}$ and $\mathcal{K}_{\Grj A}$.
\end{itemize}
\end{theorem}
The following characterization for Koszul rings is proved under the assumption that $A_0$ is artinian. Similar result holds for Koszul modules (see Theorem \ref{main-resut}).
\begin{theorem} \label{theorem 4}
Suppose that $A$ is a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ artinian. Then the following are equivalent.
\begin{itemize}
\item[(1)] $A$ is a Koszul ring.
\item[(2)] $E(A)$ is a classical Koszul ring.
\item[(3)] $\Grj A$ is a classical Koszul ring.
\item[(4)] $E(A)$ is generated in degree $1$, and
$E(E(A))\cong \Grj A$ as graded rings.
\end{itemize}
Moreover if $A$ is right finite then the above statements are also equivalent to
\begin{itemize}
\item[(5)] $A^{op}$ is a Koszul ring.
\end{itemize}
\end{theorem}
In the representation theory of quasi-hereditary algebras $A$, people are interested in the Koszul property of $\Grj A$ because of its connection with Kazhdan-Lusztig theory. Theorem \ref{theorem 4} gives a characterization of $A$ when $\Grj A$ is classically Koszul. In some sense Theorem \ref{theorem 4} answers the Questions in \cite{CPS} which concern the Koszulity of $A$ and $\Grj A$ (see subsection \ref{questions-in-CPS}).
For locally finite $\mathbb{N}$-graded algebras (with degree zero part not necessarily semisimple), generalized Artin-Schelter (for short, AS) regular property was studied in \cite{MV2, MS, MM, RR1} (see Definition \ref{definition of GAS regluar} and Theorem \ref{equivalent definitions of AS regular}). Generalized AS-regular algebras are closely related to twisted Calabi-Yau algebras (see \cite[Thoeorem 1.5]{RR1}). It is well known that a connected Koszul algebra of finite global dimension is AS regular if and only if its Yoneda Ext algebra is Frobenius \cite[Proposition 5.10]{Sm}. The final result in this paper
is to prove that this fact holds for basic locally finite Koszul algebras of finite global dimension. The proof is reduced to the classically Koszul case via $\Grj A$, which was proved essentially in \cite[Theorem 5.1]{MV1} (see Theorems \ref{A is generalized AS regular iff GrA is} and \ref{Char-generalized Koszul AS-regular algebra}). Recall that an $\mathbb{N}$-graded $k$-algebra $A$ is called basic if the degree $0$ part of $A/J$ is a finite direct sum of $k$.
\begin{theorem}\label{theorem 5}
Suppose that $A$ is a basic locally finite Koszul algebra of finite global dimension. Let $E(A)=\gExt_A^\bullet(A/J,A/J)$ be the Yoneda Ext algebra of $A$. The following are equivalent.
\begin{itemize}
\item[(1)] $A$ is generalized AS regular.
\item[(2)] $\Gr_J A$ is generalized AS regular.
\item[(3)] $E(A)$ is a self-injective algebra.
\end{itemize}
\end{theorem}
The paper is organized as follows. In section 2, we introduce Yoneda products and minimal graded projective resolutions, whose existence is given for bounded below modules over some $\mathbb{N}$-graded rings. In section 3, we define (quasi-)Koszul modules and rings following the ideas in \cite{GM1}, and prove Theorem \ref{theorem 1}, which is about the generating property of the Yoneda Ext rings (resp. modules) of quasi-Koszul rings (resp. modules). In section 4, we prove Theorem \ref{theorem 2} and Theorem \ref{theorem 3}. In section 5, we consider the associated graded rings (resp. modules) of Koszul rings (resp. modules) with respect to the $J$-adic filtration and prove Theorem \ref{theorem 4}. In the last section, we focus on the locally finite algebras, and prove that if $A$ is Koszul then $A$ is generalized AS regular if and only if so is $\Grj A$ and if and only if $E(A)$ is a self-injective.
\section{Preliminaries}
Yoneda products play a key role in this paper and are repeatedly used. We recall the definition of Yoneda products first.
Let $R$ be a ring and $R^{op}$ be its opposite ring. When we say a module, it always means a left module. An $R^{op}$-module is exactly a right $R$-module.
\subsection{Yoneda products}\label{Yoneda Products}
Let $X$, $Y$ and $Z$ be $R$-modules. The map $\Ext_R^j(Y,Z) \times \Ext_R^i(X,Y)\to \Ext_R^{i+j}(X,Z)$ defined in the following is called the {\it Yoneda product} of the Ext groups.
Let $P_\bullet \to X \to 0$ and $Q_\bullet\to Y \to 0$ be projective resolutions. For any $\bar{\alpha}\in \Ext_R^i(X,Y)$ and $\bar{\beta}\in \Ext_R^j(Y,Z)$ represented by $\alpha\in \Hom_R(P_i, Y)$ and $\beta\in \Hom_R(Q_j, Z)$ respectively, there is a commutative diagram
\begin{center}
\begin{tikzcd}
P_{i+j} \arrow[r] \arrow[d, "\alpha^j"] & \cdots \arrow[r] & P_i \arrow[r] \arrow[d, "\alpha^0"] \arrow[rd, "\alpha"] & \cdots \arrow[r] & X \arrow[r] & 0 \\
Q_j \arrow[r] \arrow[d, "\beta"] & \cdots \arrow[r] & Q_0 \arrow[r] & Y \arrow[r] & 0 & \\
Z & & & & &
\end{tikzcd}
\end{center}
where $\alpha^0,\cdots, \alpha^j$ are the lifting of $\alpha$. Then
$$\bar{\beta}\cdot \bar{\alpha}: =\overline{\beta\circ \alpha^j}\in \Ext_R^{i+j}(X, Z)$$
is well-defined, and it is called the Yoneda product of $\bar{\alpha}$ and $\bar{\beta}$ (see, for example, \cite[Chapter 3]{ML}).
Let $\Ext_R^\bullet(X,Y)=\mathop{\oplus}\limits_{i\geqslant 0}\Ext^i_R(X,Y)$. Then, with the Yoneda product, $\Ext_R^\bullet(Y,Y)$ is a graded ring, and $\Ext_R^\bullet(X,Y)$ is a graded $\Ext_R^\bullet(Y,Y)$-module.
The graded ring $\Ext_R^\bullet(S,S)$ is called the {\it Yoneda Ext ring} of $R$ where $S=R/J_R$ with $J_R$ the Jacobson radical of $R$.
Yoneda products can be defined similarly in graded module categories.
\subsection{Minimal (graded) projective resolution}
Recall that a ring $R$ is semiperfect (resp. left perfect) if $R/J(R)$ is artinian semisimple and any idempotent of $R/J(R)$ can be lifted to $R$ (resp. $J(R)$ is left $T$-nilpotent) where $J(R)$ is the Jacobson radical of $R$. A projective cover of an $R$-module $M$ is a surjective morphism $\pi:P\to M$ where $P$ is an $R$-projective module and its kernel is a superfluous submodule.
Projective cover of a module is unique up to isomorphism if it exists.
Perfect rings and semiperfect rings are characterized by the existence of projective covers (see, for instance, \cite[Theorem 24.16]{L}).
\begin{proposition}
For any ring $R$, the following are equivalent.
\begin{itemize}
\item[(1)] $R$ is semiperfect (resp. left perfect).
\item[(2)] Every finitely generated left $R$-module (resp. Every left $R$-module) has a projective cover.
\end{itemize}
\end{proposition}
We are working on graded rings and modules.
For basic definitions and facts concerning graded rings and filtered rings refer to \cite{LO} or \cite{NO} for references.
Let $A$ be an $\mathbb{N}$-graded ring, $J=J(A_0)\oplus A_{\geqslant 1}$ be the graded Jacobson radical of $A$ and $S=A/J$. Let $M$ be a graded $A$-module. For any integer $l$, the $l$-shift $M(l)$ of $M$ is a graded module with $M(l)=M$ as ungraded modules but with the grading $M(l)_i=M_{i+l}$.
A graded projective resolution of a graded $A$-module $M$
\begin{equation}\label{proj-reso}
\cdots \to P_{i+1}\xrightarrow{d_{i+1}} P_i\xrightarrow{d_i}\cdots \to P_0\xrightarrow{d_0} M\to 0
\end{equation}
is called {\it minimal} if $\im d_{i+1}\subseteq JP_i$ for all $i \geqslant 0$. If each $P_i$ in the projective resolution \eqref{proj-reso} is generated in degree $i$, then we say $M$ has a {\it linear projective resolution} or \eqref{proj-reso} is a linear projective resolution of $M$.
Any linear projective resolution is minimal.
For any nonzero bounded below graded $A$-module $M$, let $l(M)$ to be the minimal integer $i$ such that $M_{i}\neq 0$.
The following is a more general version of Nakayama's lemma.
\begin{lemma}\label{Nak lemma}
Let $A$ be an $\mathbb{N}$-graded ring.
\begin{itemize}
\item[(1)] If $M$ is a nonzero bounded below graded $A$-module with $M_{l(M)}$ finitely generated as an $A_0$-module, then $JM \lneq M$.
\item[(2)] $J(A_0)$ is left $T$-nilpotent if and only if $JM \lneq M$ for any nonzero bounded below graded $A$-module $M$.
\end{itemize}
\end{lemma}
\begin{proof} (1) If
$M=JM=J(A_0)M+A_{\geqslant 1}M$, then $M_i\subseteq J(A_0)M_i$ when $i=l(M)$, and so $M_i = J(A_0)M_i$, which implies that $M_i=0$.
(2) By \cite[Lemma 28.3]{AF}, $J(A_0)$ is left $T$-nilpotent if and only if $JM_0 \lneq M_0$ for any nonzero $A_0$-module $M_0$.
\end{proof}
Whenever we say that Nakayama's lemma holds for $M$ we mean $JM \lneq M$.
If $P_\bullet\to M \to 0$ is a minimal graded projective resolution of $M$, and Nakayama's lemma holds for the quotients of all $P_i$, then $P_i$ is a graded projective cover of $\im d_{i}$, that is, $\ker d_i$ is a graded superfluous submodule of $P_i$. In particular, $P_0$ is a graded projective cover of $M$.
An $\mathbb{N}$-graded ring $A$ is called {\it left (resp. right) finite} if $A_i$ is a finitely generated left (resp. right) $A_0$-module for all $i$. A graded left (resp. right) $A$-module $M$ is called {\it left (resp. right) finite} if $M_i$ is a finitely generated left (resp. right) $A_0$-module for all $i$.
In this paper, a noetherian or artinian ring means a left noetherian or left artinian ring.
\begin{proposition}\label{minimal projective resolution}
Let $A$ be an $\mathbb{N}$-graded ring, $M$ be a bounded below graded $A$-module. \begin{itemize}
\item[(1)] If $A$ is left finite with $A_0$ noetherian semiperfect, and $M$ is left finite, then $M$ has a minimal graded projective resolution.
\item[(2)] If $A_0$ is left perfect, then $M$ has a minimal graded projective resolution.
\end{itemize}
\end{proposition}
\begin{proof} (1)
Since $A_0$ is semiperfect, it has a finite complete set of orthogonal primitive idempotents $\{e_i\}_{i=1}^n$ which is also a finite complete set of orthogonal primitive idempotents of $A$. Each indecomposable $A_0$-projective module ${A_0e_i}$ can be viewed as a graded quotient $Ae_i/{A_{\geqslant 1} e_i}$ of the indecomposable projective $A$-module $Ae_i$.
Let $\overline{M}=M/JM=\oplus \overline{M}_i$. Then, for any $i$, $\overline{M}_i$ is a finitely generated $A_0$-module. Let $P_i'$ be aprojective cover of $A_0$-module $\overline{M}_i$. By lifting $P_i'$ to a projective $A$-module $P_i$ via the $(Ae_j)'s$, we have a natural graded $A$-module morphism $\pi_i$ which is the composition of $P_i\to P_i'\to \overline{M}_i$. By a degree shifting, we may assume that $P_i$ is a graded projective cover of $A$-module $\overline{M}_i$ with $l(P_i) \geqslant i$ by Nakayama's lemma.
Since $P_i$ is finitely generated, it is left finite. It follows from $l(P_i) \geqslant i$ that $P=\oplus_i P_i$ is left finite. Then $P$ is a graded projective cover of $\overline{M}$ with the surjective morphism $\pi=\oplus \pi_i$. Since $P$ is graded projective, there is a graded morphism $\pi': P \to M$ so that the following diagram is commutative, that is, $\epsilon\circ\pi'=\pi$.
\begin{center}
\begin{tikzcd}
& P \arrow[ld, "\pi'"', dashed] \arrow[d, "\pi"] & \\
M \arrow[r, "\epsilon"] & \overline{M} \arrow[r] & 0
\end{tikzcd}
\end{center}
Then $\pi'(P)+JM=M$. By Nakayama's lemma, $\pi'$ is surjective. Hence $P$ is a graded projective cover of $M$ because $\Ker \pi'\subseteq \Ker \pi\subseteq JP$.
Since $A_0$ is noetherian, $\Ker \pi'$ is also left finite. Repeating the process by replacing $M$ with $\Ker \pi'$, we may construct a minimal graded projective resolution of $M$ inductively.
(2) By a similar proof of (1) and Lemma \ref{Nak lemma} (2).
\end{proof}
It follows from the proof of Proposition \ref{minimal projective resolution} that any left finite bounded below projective $A$-module is isomorphic to a direct sum of the shifts of the indecomposable projective $A$-modules $Ae_i$ if $A_0$ is semiperfect.
\section{Koszul rings and modules}
In this section we first recall the definitions of classical graded Koszul rings, Koszul modules \cite{BGS1}, and their characterizations by the property of syzygies given in \cite{GM1}.
By using the characterizations in graded case, Green and Martin{\'e}z-Villa defined (strongly) quasi-Koszul modules and rings in \cite{GM1} for ungraded noetherian semiperfect rings.
Following the idea in \cite{GM1} we define (quasi-)Koszul rings and modules in a more general setting. Theorem \ref{qK iff E(M) generated in degree 0} is the main result in this section, which characterizes the quasi-Koszul property via the generating property of the Yoneda Ext ring and module, and hence generalizes the results in both classical graded and ungraded cases.
\subsection{Definition of (quasi-)Koszul modules and rings}
\begin{definition}\cite[Definitions 1.2.1, 2.14.1]{BGS1}
Let $A$ be an $\mathbb{N}$-graded ring with $A_0$ artinian semisimple. Suppose $M$ is a graded $A$-module generated in degree $0$. If $M$ has a linear projective resolution, then $M$ is called {\it classically Koszul}.
If $A_0$ considered as a graded $A$-module is a classically Koszul module, then $A$ is called a {\it classically Koszul ring}.
\end{definition}
Recall that an $\mathbb{N}$-graded ring $A=\oplus A_{i \geqslant 0}$ is called {\it generated in degree $1$} if $A_iA_j=A_{i+j}$ for all $i, j \geqslant 0$ \cite[Lemma 2.1]{GM1}. In this case, $A$ is also said to be generated by $A_1$ over $A_0$ \cite[Definition 1.2.2]{BGS1}.
An graded $k$-algebra $A$ is called {\it locally finite} if $A_i$ is finite dimensional as a $k$-vector space for all $i\geqslant 0$.
An $\mathbb{N}$-graded locally finite $k$-algebra generated in degree $1$ with $A_0$ being a direct sum of $k$ is called a {\it graded quiver algebra}, as it can be viewed as a quotient of a path algebra $kQ$ for a finite quiver $Q$ \cite[Proposition 1.1.1]{MV1}. In fact, most results proved in \cite{GM1, GM2} for graded quiver algebras hold for left finite $\mathbb{N}$-graded ring with $A_0$ artinian semisimple.
Here is a characterization of classical Koszul modules proved in \cite[Proposition 3.1 and Lemma 5.1]{GM1}.
\begin{proposition}\label{qk and classical koszul}
Let $A$ be an
$\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ artinian semisimple.
Suppose that $M$ is a graded $A$-module generated in degree $0$, and
$\cdots \to P_n\xrightarrow{d_n} P_{n-1}\to\cdots\to P_0\xrightarrow{d_0} M\to 0$
is a minimal graded projective resolution of $M$. Then the following are equivalent:
\begin{itemize}
\item[(1)] $M$ is a classical Koszul module.
\item[(2)] For any $n\geqslant 0$, $J\Ker d_n = \Ker d_n\cap J^2P_n$.
\item[(3)] For any $n,k\geqslant 0$, $J^k\Ker d_n = \Ker d_n\cap J^{k+1}P_n$.
\end{itemize}
\end{proposition}
The property (3) can be interpreted as follows: the $1$-shifting of the $J$-adic filtration on the syzygies is exactly the submodule filtration induced from the $J$-adic filtration of the projective modules in the minimal projective resolution.
Quasi-Koszul rings and modules were defined in \cite{GM1} first by the generating property of Koszul dual and were characterized by the condition (2) in Proposition \ref{qk and classical koszul} (see Theorem \ref{Theorem 4.4 in {GM1}}).
\begin{definition}\cite[page 263]{GM1}\label{ungraded qK ring}
Let $R$ be a noetherian semiperfect ring, $J$ be its Jacobson radical and $S=R/J$.
\begin{itemize}
\item[(1)]
A finitely generated $R$-module $M$ is called {\it quasi-Koszul} if, for any $n>0$,
$\Ext_R^1(S,S)\cdot \Ext_R^{n-1}(M,S)=\Ext^n_R(M,S)$.
\item[(2)] The ring $R$ is called {\it quasi-Koszul} if $S$ is a quasi-Koszul module, that is, $\Ext_R^{\bullet}(S,S)$ is generated by $\Ext^1_A(S,S)$ over $\Hom_A(S,S)$.
\end{itemize}
\end{definition}
This means that $R$ is quasi-Koszul if and only if the Yoneda Ext ring $\Ext_R^{\bullet}(S,S)$ of $R$ is generated in degree $1$, and if $R$ is quasi-Koszul, then $M$ is quasi-Koszul if and only if $\Ext_R^{\bullet}(M,S)$ is generated in degree $0$ as a graded $\Ext_R^{\bullet}(S,S)$-module.
\begin{theorem} \cite[Theorem 4.4]{GM1} \label{Theorem 4.4 in {GM1}} Let $R$ be a noetherian semiperfect ring, $J$ be its Jacobson radical and $S=R/J$.
\begin{itemize}
\item[(1)] A finitely generated $R$-module $M$ is quasi-Koszul if and only if $M$ has a minimal projective resolution $P_\bullet \to M \to 0$ such that $J\Ker d_n = \Ker d_n\cap J^2P_n$ for all $n$.
\item[(2)] The ring $R$ is quasi-Koszul if and only if ${}_RS$ has a minimal projective resolution $P_\bullet \to S \to 0$ such that $J\Ker d_n = \Ker d_n\cap J^2P_n$ for all $n$.
\end{itemize}
\end{theorem}
Strongly Quasi-Koszul rings and modules were defined in \cite{GM1} by the condition (3) in Proposition \ref{qk and classical koszul}.
\begin{definition}\cite{GM1}
Let $R$ be a noetherian semiperfect ring and $J$ be its Jacobson radical.
\begin{itemize}
\item[(1)]
A finitely generated
$R$-module $M$ is called {\it strongly quasi-Koszul} if $J^k\Ker d_n = \Ker d_n\cap J^{k+1}P_n$ for any $n,k\geqslant 0$, where $P_\bullet \to M \to 0$ is
a minimal projective resolution of $M$.
\item[(2)] $R$ is called a {\it strongly quasi-Koszul ring} if $R/J$ is a strongly quasi-Koszul $R$-module.
\end{itemize}
\end{definition}
The Auslander algebra of a finite-dimensional algebra of finite type over an algebraically closed field is a quasi-Koszul algebra \cite[Theorem 9.6]{GM1}. If $R$ is further assumed being hereditary, then any quasi-Koszul $R$-module is strongly quasi-Koszul \cite[Lemma 5.1(b)]{GM1}.
A ring which is quasi-Koszul but not strongly quasi-Koszul is given in Example \ref{qk-but-not-sqk}.
To unify the notion of the classical graded Koszulity and the ungraded (strongly) quasi-Koszulity, we propose a definition of (quasi) Koszulity as follows.
\begin{definition} \label{our-definition}
Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect, $J$ be its graded Jacobson radical.
Let $M$ be a graded $A$-module with a minimal graded projective resolution
$$\cdots\to P_n\xrightarrow{d_n} P_{n-1}\to \cdots P_0 \xrightarrow{d_0} M\to 0.$$
\begin{itemize}
\item[(1)] If $J\Ker d_n = \Ker d_n\cap J^2P_n$ for all $n$, then $M$ is called {\it quasi-Koszul}.
\item[(2)] If $J^k\Ker d_n = \Ker d_n\cap J^{k+1}P_n$ for any $n,k\geqslant 0$, then $M$ is called {\it Koszul}.
\end{itemize}
If $S=A/J$ is (quasi-)Koszul as a graded $A$-module, then $A$ is called a {\it (quasi-)Koszul ring}.
\end{definition}
It follows from the definition that any finite direct sum and any direct summand of (quasi-)Koszul modules are (quasi-)Koszul. The shift of (quasi-)Koszul modules is also (quasi-)Koszul, unlike the classical graded Koszulity defined by linear projective resolutions.
\begin{remark}
Suppose that $M$ is a left finite bounded below graded $A$-module. If $M$ is (quasi-)Koszul then $M_{l(M)}$ is (quasi-)Koszul as an $A_0$-module. In particular, if $A$ is (quasi-)Koszul then $A_0$ is (quasi-)Koszul. In fact,
without loss of generality, we may assume that $l(M)=0$. Suppose
$$\cdots\to P_n\xrightarrow{d_n} P_{n-1}\to \cdots \to P_0 \xrightarrow{d_0} M\to 0,$$
is a minimal graded projective resolution of $M$. Then it is clear that
$$\cdots\to (P_n)_0\xrightarrow{} (P_{n-1})_0\to \cdots \to (P_0)_0\xrightarrow{} M_0\to 0$$
is a minimal projective resolution of $M_0$ as an $A_0$-module.
\end{remark}
\begin{remark}
Note that if $A$ is a classical graded Koszul ring, then $A$ is generated in degree $1$ by \cite[Proposition 1.2.3]{BGS1}. So for a (quasi-) Koszul ring $A$ and a graded $A$-module $M$, if $A_0$ is artinian semisimple and $M$ is generated in degree $0$, our definition of (quasi-)Koszul modules coincides with the definition of the classical graded Koszul module by Proposition \ref{qk and classical koszul}; if $A$ is concentrated in degree $0$ and $M$ is finitely generated concentrated in degree $0$, then our definition of (quasi-)Koszul modules coincides with the definition in the ungraded case (see Definition \ref{ungraded qK ring} and Theorem \ref{Theorem 4.4 in {GM1}}). Therefore Definition \ref{our-definition} unifies and generalizes the concepts of classical graded Koszul rings and ungraded (strongly) quasi-Koszul rings.
Although the finiteness condition is not necessary in the definition of classical Koszul rings and modules, it is needed to have the duality, as remarked before \cite[Definition 1.2.4]{BGS1} (see also Theorem \ref{qK iff E(M) generated in degree 0}). In \cite{GM1,GM2}, the algebras considered are the graded quiver algebras, so the finiteness condition is satisfied automatically. In our case, the finiteness condition imposed is to guarantee the existence of minimal graded projective resolutions (see Proposition \ref{minimal projective resolution}).
\end{remark}
In the rest of this section we always assume that $A$ is a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect, $J$ is the graded Jacobson radical of $A$ and $S=A/J$.
\begin{example}
If $A$ has global dimension one, then $A$ is Koszul, as $0\to J\to A\to S\to 0$ is a minimal graded projective resolution of ${}_AS$.
If $A=kQ$, where $Q$ is a quiver of the following form, with at least one arrow of positive weight,
\begin{center}
\begin{tikzcd}
\bullet \arrow[r] & \bullet \arrow[r] & \cdots \arrow[r] & \bullet \arrow[lll, bend right]
\end{tikzcd}
\end{center}
it follows from \cite[Proposition 6.6]{RR2} that $A$ is a generalized Artin-Schelter regular algebra (see Definition \ref{definition of GAS regluar}) of dimension one. In particular, if there is at least one arrow of weight zero and at least one arrow of weight one, then $A_0$ is a non-semisimple finite-dimensional algebra and $A$ is a locally finite graded algebra generated by $A_1$ over $A_0$. This is a simple example of Koszul rings.
\end{example}
Next, for later use, we show that any finitely generated quasi-Koszul module has a finitely generated minimal projective resolution.
\begin{lemma}\label{JM is fg}
If $M$ is a finitely generated graded $A$-module, then so is $JM$.
\end{lemma}
\begin{proof} Since $A_0$ is noetherian, $J(A_0)=A_0x_1+\cdots + A_0x_r$ for some $x_i\in A_0$.
Since $A$ is left finite, we may assume that $A_1=A_0y_1+\cdots+A_0y_s$ for some $y_j \in A_1$. Then $A_{\geqslant 1}=Ay_1+\cdots+Ay_s$ as $A$ is generated in degree $1$. Therefore $J=Ax_1+\cdots + Ax_r+Ay_1 +\cdots +Ay_s$ is a finitely generated $A$-module.
Suppose that $M=Am_1+\cdots +Am_t$. Then $JM=Jm_1+\cdots + Jm_t=\Sigma Ax_im_j+\Sigma Ay_{i}m_j$ is a finitely generated $A$-module.
\end{proof}
\begin{proposition}\label{minimal is fg}
If ${}_AM$ is a finitely generated quasi-Koszul module, then any module $P_n$ in the minimal graded projective resolution $P_\bullet \to M \to 0$ of $M$ is finitely generated.
\end{proposition}
\begin{proof}
Let $\cdots\to P_n\xrightarrow{d_n} P_{n-1}\to \cdots P_0 \xrightarrow{d_0} M\to 0$ be a minimal graded projective resolution of $M$. Since $M$ is finitely generated, $P_0$ is finitely generated.
By Lemma \ref{JM is fg}, $JP_0$ is finitely generated.
Since $J\Ker d_0=J^2P_0\cap \Ker d_0$, the natural map $\Ker d_0/J\Ker d_0\to JP_0/J^2P_0$ is injective. Hence $\Ker d_0/J\Ker d_0$ is a finitely generated $A/J$-module, and so it is a finitely generated $A$-module. By Nakayama's lemma, $\Ker d_0$ is finitely generated. The proof is completed by induction.
\end{proof}
\subsection{Generating property of quasi-Koszul modules and rings}
The following result characterizes quasi-Koszul modules (quasi-Koszul rings) in terms of Yoneda Ext-groups. The proof is based on the ideas in \cite[Sections 3 and 4]{GM1}.
Let $\Omega^i$ be the $i$-th syzygy functor.
\begin{theorem}\label{qK iff E(M) generated in degree 0}
Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect.
\begin{itemize}
\item[(1)] Suppose that $M$ is a left finite bounded below graded $A$-module. Then $M$ is quasi-Koszul if and only if, for any $n>0$, $$\gExt_A^1(S,S)\cdot \gExt_A^{n-1}(M,S)=\gExt^n_A(M,S).$$
\item[(2)] $A$ is a quasi-Koszul ring if and only if $\gExt_A^{\bullet}(S,S)$ is generated by $\gExt^1_A(S,S)$ over $\gHom_A(S,S)$, that is, generated in degree $1$.
\end{itemize}
\end{theorem}
\begin{proof}
It suffices to prove the first statement. Suppose $M$ is quasi-Koszul. Let $\cdots\to P_n\xrightarrow{d_n} P_{n-1}\to \cdots \to P_0 \xrightarrow{d_0} M\to 0$ be a minimal graded projective resolution of $M$.
By Proposition \ref{minimal projective resolution}, each $P_n$ is left finite.
It follows from the minimalism of the projective resolution that
$$\gExt_A^n(M,S)=\gHom_A(P_n,S)\cong \gHom_A(\Omega^n(M),S).$$
Let $\Omega^n=\Omega^n(M)=\im d_n$. Suppose $g\in \gHom_A(\Omega^n,S)$ is a homogeneous element, say, of degree $t$. Since $g(J\Omega^n)=0$, $g$ factors through $\Omega^n/J\Omega^n$, that is, $g=\bar{g}p$ where $p: \Omega^n\to \Omega^n/J\Omega^n$ is the projection and $\bar{g}:\Omega^n/J\Omega^n\to S$ is induced by $g$.
\begin{center}
\begin{tikzcd}
\Omega^n \arrow[dd, "p"] \arrow[rr, "i", hook] \arrow[rd, "g"] & & JP_{n-1} \arrow[dd] \arrow[ld, "g'"', dashed] \arrow[lldd, "p'", dashed, bend left=20] \\
& S & \\
\Omega^n/J\Omega^n \arrow[rr, "\bar{i}"', tail] \arrow[ru, "\bar{g}", dashed] & & JP_{n-1}/J^2P_{n-1}
\end{tikzcd}
\end{center}
Since $J\Omega^n=\Omega^n \cap J^2P_{n-1}$ by assumption, the induced morphism $\bar{i}$ is injective. Since both $\Omega^n/J\Omega^n$ and $JP_{n-1}/J^2P_{n-1}$ are semisimple $S$-modules, $\bar{i}$ splits and there is a morphism $p':JP_{n-1}\to \Omega^n/J\Omega^n$ such that $p=p'i$ .
Let $g'=\bar{g}p'$. Then $g=g'i$.
Let $\pi:P_{n-1}\to P_{n-1}/JP_{n-1}$ be the natural projection. Then $\pi d_n=0$ and $\overline{\pi}\in \gExt_A^{n-1}(M,P_{n-1}/JP_{n-1})$.
We may assume that $P_{n-1}/JP_{n-1} = \mathop{\oplus}\limits_{j}S_j$ where $S_j$ are some graded simple $A$-modules. Since $P_{n-1}$ is left finite, there are only finitely many $S_j$ of degree $t$. Suppose $Q_j\to P'_j\to S_j\to 0$ is the starting part of a minimal graded projective resolution of $S_j$ such that $P_{n-1} = \mathop{\oplus}\limits_{j} P'_j$ and $\mathop{\oplus}\limits_{j}Q_j \stackrel{\pi_2}{\to} JP_{n-1} \to 0$ is a projective cover of $JP_{n-1}$. Let $Q =\mathop{\oplus}\limits_{j}Q_j$.
Then we have the following commutative diagram
\begin{center}
\begin{tikzcd}
P_n \arrow[r, "\pi_1", two heads] \arrow[dd, "g\pi_1"', bend right] \arrow[d, "\exists h", dashed] & \Omega^n \arrow[d, "i", hook] \arrow[r, hook] & P_{n-1} \arrow[rd, "\pi"] \arrow[d, "="] \arrow[r] & \cdots \arrow[r] & M \arrow[r] & 0 \\
Q \arrow[r, "\pi_2", two heads] \arrow[d, "g'\pi_2"] & JP_{n-1} \arrow[ld, "g'"] \arrow[r, hook] & P_{n-1} \arrow[r, "\pi"] & \frac{P_{n-1}}{JP_{n-1}} \arrow[r] & 0 & \\
S & & & & &
\end{tikzcd}
\end{center}
where $h$ is a lifting of $\pi$, and $d_n: P_n \to P_{n-1}$ factors through $\Omega^n$ via $\pi_1$.
Now
$g'\pi_2 \in \gHom_A(Q,S) = \gExt_A^1(\dfrac{P_{n-1}}{JP_{n-1}}, S)$, which is of degree $t$. Since $Q =\mathop{\oplus}\limits_{j}Q_j$ is left finite, there is a finite subset $\{Q_1, Q_2, \cdots, Q_m\}$ of $\{Q_j\}$ such that $$g'\pi_2 \in \gHom_A(\mathop{\oplus}\limits_{j=1}^mQ_j, S) \cong \mathop{\oplus}\limits_{j=1}^m\gHom_A(Q_j,S) = \mathop{\oplus}\limits_{j=1}^m\gExt_A^1(S_j,S).$$
Let $g_j = g'\pi_2l_j: Q_j \to S$ and $h_j = p_jh: P_n \to Q_j$ for $1 \leqslant j \leqslant m$, where $l_j: Q_j \to Q$ and $p_j: Q \to Q_j$ are the canonical injections and projections respectively. It follows from $g\pi_1 = g'\pi_2 h$ that $g\pi_1=\sum\limits_{j=1}^m g'\pi_2 l_j p_jh=\sum\limits_{j=1}^m g_jh_j$, and by the definition of Yoneda products,
$$\gExt^n_A(M,S) \ni g\pi_1=\sum\limits_{j=1}^m g_j h_j\in \gExt_A^1(S_j,S)\cdot \gExt_A^{n-1}(M,S_j).$$
By a degree shifting, we may assume that $S_j$ is concentrated in degree $0$, and so $\gExt_A^{n-1}(M,S_j)$ and $\gExt_A^1(S_j,S)$ can be viewed as direct summands of the groups $\gExt_A^{n-1}(M,S)$ and $\gExt_A^1(S,S)$ respectively. Therefore $$g\pi_1\in \gExt_A^1(S,S)\cdot \gExt_A^{n-1}(M,S).$$
It follows that
$\gExt_A^1(S,S)\cdot \gExt_A^{n-1}(M,S)=\gExt^n_A(M,S).$
Conversely, suppose $\gExt_A^1(S,S)\cdot \gExt_A^{n-1}(M,S)=\gExt^n_A(M,S)$ for all $n>0$.
Let $\Omega=\Omega(M)=\Ker d_0$.
First, we claim that any $g \in \gHom_A(\Omega,S)$ factors through $JP_0$.
Since
$g \in \gHom_A(\Omega,S) \cong \gExt_A^1(M,S)=\gExt_A^1(S,S)\cdot \gHom_A(M,S),$
$g\pi_1=\sum y_j\cdot x_j$ for some $x_j\in \gHom_A(M,S)$ and $y_j\in\gExt^1_A(S,S)$.
Let $Q \to J \to 0$ be a graded projective cover of $J$. Then
$$y_j\in\gExt^1_A(S,S) =\gHom_A(Q,S) \cong \gHom_A(J,S).$$
Now, for any $j$, we have the following commutative diagram
\begin{center}
\begin{tikzcd}
P_1 \arrow[d, "x_j^1"] \arrow[r, "\pi_1", two heads] & \Omega \arrow[r, hook] \arrow[d, "x_j'"] & P_0 \arrow[d, "x_j^0"] \arrow[r] & M \arrow[r] \arrow[d, "x_j"] & 0 \\
Q \arrow[d, "y_j"] \arrow[r, two heads] & J \arrow[ld, "y'_j"] \arrow[r, hook] & A \arrow[r] & S \arrow[r] & 0 \\
S & & & &
\end{tikzcd}
\end{center}
where $x_j^0,x_j^1,x_j'$ are the lifting of $x_j$. Then $g=\sum y_j'x_j'$.
Consider the following commutative diagram
\begin{center}
\begin{tikzcd}
0 \arrow[r] & \Omega \arrow[d, "x_j'"] \arrow[r] & P_0 \arrow[r] \arrow[d, "x_j^0"] & M \arrow[r] \arrow[d, "x_j"] & 0 \\
0 \arrow[r] & J \arrow[d, "y_j'"] \arrow[hook, r] & A \arrow[d, "f_j"] \arrow[r] & S \arrow[r] \arrow[d, "="] & 0 \\
0 \arrow[r] & S \arrow[r, "\alpha"] & E \arrow[r, "\beta"] & S \arrow[r] & 0
\end{tikzcd}
\end{center}
where $E$ is the push-out of $J\hookrightarrow A$ and $J\xrightarrow{y_j'} S$.
Since $f_jx_j^0(JP_0)\subseteq \Ker \beta=\alpha(S)$, $y_j'x_j'$ factors through $JP_0$. It follows that $g$ factors through $JP_0$.
Hence $g(x)=0$ for any $x\in \Omega \cap J^2P_0$. Therefore
$$\Omega \cap J^2P_0\subseteq \cap \{\Ker g \mid g\in \gHom_A(\Omega,S)\}=J\Omega \subseteq \Omega \cap J^2P_0.$$
Hence $J\Ker d_0 = \Ker d_0 \cap J^2P_0$.
Note that
\begin{align*}
\gExt_A^1(\Ker d_0,S)&\cong \gHom_A(\Ker d_1,S)\\
&\cong \gExt_A^2(M,S)\\
&=\gExt^1_A(S,S)\cdot\gExt_A^1(M,S)\\
&=\gExt^1_A(S,S)\cdot\gHom_A(\Ker d_0,S).
\end{align*}
The proof is completed by replacing $M$ with $\Ker d_0$ and by induction.
\end{proof}
\section{Yoneda Ext rings and Koszul Duality}
In this section we study the Yoneda Ext ring $\gExt_A^\bullet(S,S)$ of $A$, where $A$ is a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect and $S=A/J$. We show that $A$ is a Koszul ring if and only if the Yoneda Ext ring $\gExt_A^\bullet(S,S)$ is a left finite classical Koszul ring
in the classical sense \cite[Definitions 1.2.1 and 1.2.4]{BGS1}. A Koszul duality theory is also given in this section.
\subsection{Some preparatory results}
The following lemmas and propositions were proved essentially in \cite[Section 5]{GM1}.
They are still true in our more general framework.
\begin{lemma}\label{fact 1}
Let $0\to X\to Y\to Z\to 0$ be an exact sequence of left finite bounded below graded $A$-modules such that $JX=X \cap JY$. If $P'$ and $P''$ are graded projective covers of $X$ and $Z$ respectively, then $P=P'\oplus P''$ is a graded projective cover of $Y$.
\end{lemma}
\begin{proof}
Obviously, there is an exact commutative diagram
\begin{center}
\begin{tikzcd}
0 \arrow[r] & P' \arrow[r] \arrow[d] & P \arrow[r] \arrow[d] & P'' \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & X \arrow[d] \arrow[r] & Y \arrow[d] \arrow[r] & Z \arrow[d] \arrow[r] & 0 \\
& 0 & 0 & 0
\end{tikzcd}
\end{center}
Since $JX=X \cap JY$, we get the following exact commutative diagram by tensoring the above diagram with $A/J\otimes_A -$.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & P'/JP' \arrow[r] \arrow[d] & P/JP \arrow[r] \arrow[d] & P''/JP'' \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & X/JX \arrow[r] & Y/JY \arrow[r] & Z/JZ \arrow[r] & 0
\end{tikzcd}
\end{center}
It follows from $P'/JP'\cong X/JX$ and $P''/JP''\cong Z/JZ$ that $P/JP \cong Y/JY$. By Proposition \ref{minimal projective resolution} (1), both $P'$ and $P''$ are bounded below and left finite, so is $P$. Hence $P$ is a graded projective cover of $Y$.
\end{proof}
By Lemma \ref{fact 1}, there is an exact commutative diagram, which is used frequently later in this paper.
\begin{equation}\label{comm-diagram-2}
\begin{tikzcd}
& 0 \arrow[d] & 0 \arrow[d] & 0 \arrow[d] & \\
0 \arrow[r] & \Omega(X) \arrow[r] \arrow[d] & \Omega(Y) \arrow[r] \arrow[d] & \Omega(Z) \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & JP' \arrow[r] \arrow[d] & JP \arrow[r] \arrow[d] & JP'' \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & JX \arrow[d] \arrow[r] & JY \arrow[d] \arrow[r] & JZ \arrow[d] \arrow[r] & 0 \\
& 0 & 0 & 0 &
\end{tikzcd}
\end{equation}
\begin{lemma}\label{exact sequence of syzygy}
Let $0\to X\to Y\to Z\to 0$ be an exact sequence of left finite bounded below graded $A$-modules with $JX = X \cap JY$. Suppose that $X$ is quasi-Koszul (resp. Koszul). Then for any $n > 0$,
$$0\to \Omega^n(X)\to \Omega^n(Y)\to \Omega^n(Z)\to 0$$
is exact, and $J\Omega^n(X)=\Omega^n(X) \cap J\Omega^n(Y)$ (resp. $J^k\Omega^n(X)=\Omega^n(X) \cap J^k\Omega^n(Y)$ for all $k > 0$).
\end{lemma}
\begin{proof}
Suppose $X$ is quasi-Koszul.
By applying the functor $A/J\otimes_A -$ to diagram \eqref{comm-diagram-2}, it follows from the quasi-Koszulity of $X$ that the following is an exact commutative diagram.
\begin{tikzcd}[column sep=small]
& 0 \arrow[d] & & & \\
& A/J\otimes_A\Omega(X) \arrow[r] \arrow[d] & A/J\otimes_A\Omega(Y) \arrow[r] \arrow[d] & A/J\otimes_A\Omega(Z) \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & A/J\otimes_AJP' \arrow[r] \arrow[d] & A/J\otimes_AJP \arrow[r] \arrow[d] & A/J\otimes_AJP'' \arrow[r] \arrow[d] & 0 \\
& A/J\otimes_AJX \arrow[d] \arrow[r] & A/J\otimes_AJY \arrow[d] \arrow[r] & A/J\otimes_AJZ \arrow[d] \arrow[r] & 0 \\
& 0 & 0 & 0 &
\end{tikzcd}
Hence $A/J\otimes_A \Omega(X)\to A/J\otimes_A \Omega(Y)$ is injective, that is, $J\Omega(X)= \Omega(X) \cap J\Omega(Y)$.
Therefore, the exact sequence $0\to \Omega(X)\to \Omega(Y)\to \Omega(Z)\to 0$ satisfies the same hypothesis as $0\to X\to Y\to Z\to 0$ do.
Hence the proof can be completed by induction.
For the Koszul case, it is left to prove that $J^k\Omega^n(X)=\Omega^n(X) \cap J^k\Omega^n(Y)$ for all $k > 0$ which can be shown similarly by applying the functor $A/J^k\otimes_A -$ instead of $A/J\otimes_A -$.
\end{proof}
\begin{corollary}\label{strongly quasi Koszul of exact sequence}
Let $0\to X\to Y\to Z\to 0$ be an exact sequence of left finite bounded below graded $A$-modules with $JX = X \cap JY$. If $X$ is Koszul (resp. quasi-Koszul), then $Z$ is Koszul (resp. quasi-Koszul) if and only if $Y$ is Koszul and $J^kX = X \cap J^kY$ for all $k$ (resp. quasi-Koszul and $J^2\Omega^n(X) = \Omega^n(X) \cap J^2\Omega^n(Y)$ for all $n \geqslant 0$).
\end{corollary}
\begin{proof} As in the proof of Lemma \ref{exact sequence of syzygy}, for the Koszul case, we get a new commutative diagram by tensoring diagram \eqref{comm-diagram-2} with the functor $A/J^k\otimes_A -$.
Then, by Snake lemma, $A/J^k\otimes_A \Omega(Z)\to A/J^k\otimes_A JP''$ is injective if and only if $A/J^k\otimes_A \Omega(Y)\to A/J^k\otimes_A JP$ is injective and $J^kX = X \cap J^kY$ for all $k$. Therefore, $J^k\Omega(Z)=\Omega(Z) \cap J^{k+1}P''$ if and only if $J^k\Omega(Y)=\Omega(Y) \cap J^{k+1}P$ and $J^kX = X \cap J^kY$ for all $k$.
Hence, the proof is completed inductively by Lemma \ref{exact sequence of syzygy} and by replacing $X,Y$ and $Z$ with $\Omega(X),\Omega(Y)$ and $\Omega(Z)$ respectively.
The proof of the quasi-Koszul case is similar.
\end{proof}
\begin{proposition}\label{JM is qK}
Let $A$ be a quasi-Koszul (resp. Koszul) ring. If $M$ is a left finite bounded below Koszul module, then $JM$ is quasi-Koszul (resp. Koszul).
\end{proposition}
\begin{proof}
Since ${}_A(A/J)$ is quasi-Koszul (resp. Koszul),
$M/JM$ is quasi-Koszul (resp. Koszul). Let $P$ be a graded projective cover of $M$. Since $M$ is Koszul and $M/JM$ is quasi-Koszul (resp. Koszul), $\Omega(M)$ is Koszul and $JP=\Omega(M/JM)$ is quasi-Koszul (resp. Koszul). Note that $0\to \Omega(M)\to JP\to JM\to 0$ is exact and $J^k\Omega(M)=\Omega(M) \cap J^{k+1}P$ for all $k \geqslant 0$. By Lemma \ref{exact sequence of syzygy}, $J^2\Omega^{n+1}(M)=\Omega^{n+1}(M) \cap J^2 \Omega^n(JP)$.
It follows from Corollary \ref{strongly quasi Koszul of exact sequence} that $JM$ is quasi-Koszul (resp. Koszul).
\end{proof}
\begin{proposition}\label{exact sequence of ext group}
If $M$ is a left finite bounded below quasi-Koszul $A$-module, then for any $n>0$
\begin{itemize}
\item[(1)] $0\to \Omega^n(M)\to \Omega^n(M/JM)\to \Omega^{n-1}(JM)\to 0$ is exact and $J \,\Omega^n(M)=\Omega^n(M) \cap J\, \Omega^n(M/JM)$.
\item[(2)] $0\to \gExt^{n-1}_A(JM,S)\to \gExt_A^n(M/JM,S)\to \gExt^n_A(M,S)\to 0$ is exact.
\end{itemize}
If moreover $A$ is a Koszul ring, $M$ is a left finite and bounded below Koszul
$A$-module, then
\begin{itemize}
\item[$(1)'$] $0\to \Omega^n(J^{k-1}M)\to \Omega^n(J^{k-1}M/J^kM)\to \Omega^{n-1}(J^kM)\to 0$
is exact and $J^l\Omega^n(J^{k-1}M)=J^l\Omega^n(J^{k-1}M/J^kM)\cap \Omega^n(J^{n-1}M)$ for any $k, l >0$.
\item[$(2)'$]
\begin{small}
$0 \to \gExt^{n-1}_A(J^kM,S) \to \gExt_A^n(\dfrac{J^{k-1}M}{J^kM},S) \to \gExt^n_A(J^{k-1}M,S) \to 0$
\end{small}
is exact for any $k>0$.
\end{itemize}
\end{proposition}
\begin{proof}
(1)
As noted before, $0\to \Omega(M)\to \Omega(M/JM)\to JM\to 0$ is exact.
Since $M$ is quasi-Koszul and $JP=\Omega(M/JM)$, $$J\,\Omega(M)=\Omega(M) \cap J^2P =\Omega(M) \cap J\,\Omega(M/JM).$$
Therefore, the conclusion follows from Lemma \ref{exact sequence of syzygy}.
(2) It follows from (1) that
$$0\to \dfrac{\Omega^n(M)}{J\Omega^n(M)}\to \dfrac{\Omega^n(M/JM)}{J\Omega^n(M/JM)}\to \dfrac{\Omega^{n-1}(JM)}{J\Omega^{n-1}(JM)}\to 0$$
is exact.
By acting on
$0\to \Omega^n(M)\to \Omega^n(M/JM)\to \Omega^{n-1}(JM)\to 0$
with the functor
$$\gHom_A(-,S)\cong \gHom_A(-, \gHom_S(A/J, S))\cong \gHom_S(A/J \otimes_A -,S),$$
it implies that
\begin{small}
$$ 0\!\to\! \gHom_A(\Omega^{n-1}(JM),S)\!\to\! \gHom_A(\Omega^n(M/JM),S)\!\to\! \gHom_A(\Omega^n(M),S)\!\to\! 0$$
\end{small}
is exact, as $\gHom_S(-,S)$ is an exact functor.
Note that $\gExt_A^n(X, S)\cong \gHom_A(\Omega^n(X),S)$ for any graded $A$-module $X$ with a minimal graded projective resolution. Therefore,
$$0\to \gExt^{n-1}_A(JM,S)\to \gExt_A^n(M/JM,S)\to \gExt^n_A(M,S)\to 0$$
is exact for any $n>0$.
Note this means that the long exact Ext-group sequence induced by applying the functor $\gHom_A(-, S)$ to the short exact sequence $ 0 \to JM \to M \to M/JM \to 0$ is divided into short exact sequences.
If $A$ and $M$ are Koszul, then $J^kM$ is Koszul by Proposition \ref{JM is qK}. The proof of $(1)'$ and $(2)'$ are completed by replacing $M$ with $J^{k-1}M$.
\end{proof}
\subsection{Characterizations of Koszulity via Koszulity of Yoneda Ext rings}
Let $\mathcal{E}=\gExt_A^\bullet(-,S)$ be the functor from the category of graded $A$-modules to the category of graded $E(A)$-modules. Note that $E(A)_0=\gExt_A^\bullet(S,S)_0=\gHom_A(S,S)\cong S^{op}$ is artinian semisimple.
The following is a generalized version of \cite[Theorems 6.1 and 9.1]{GM1}.
\begin{theorem}\label{A sqk implies E(A) Koszul}
Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect.
\begin{itemize}
\item[(1)] $A$ is a Koszul ring if and only if $E(A)$ is a
classical Koszul ring (in the sense of \cite[Defintion 1.2.1]{BGS1}).
\item[(2)] Suppose $A$ is Koszul. Then, for any left finite bounded below graded $A$-module $M$, $M$ is a Koszul $A$-module if and only if $\mathcal{E}(M)$ is a classical
Koszul $E(A)$-module.
\end{itemize}
\end{theorem}
\begin{proof}
{\bf ``only if" part of (1)}. We show that $E(A)$ is a classical Koszul ring by constructing a linear projective resolution of $E(A)_0$ as an $E(A)$-module.
Let $\cdots\to P_n\xrightarrow{d_n} P_{n-1}\to \cdots \to P_0 \xrightarrow{d_0} S\to 0$ be a minimal graded projective resolution of $S$ as an $A$-module. Let $P'_0 \xrightarrow{d'_0} E(A)_0\to 0$ be the projective cover of $E(A)_0$ as a graded $E(A)$-module, where $P'_0 = E(A)$ and $\ker d_0'=J'=E(A)_{\geqslant 1}$ is the graded Jacobson radical of $E(A)$.
Then
$$ \ker d_0' = \mathop{\oplus}\limits_{i\geqslant 1}\gExt^i_A(S,S)=\mathop{\oplus}\limits_{i\geqslant 1}\gExt_A^{i-1}(JP_0,S)=\gExt_A^\bullet(JP_0,S)\,(-1)
$$ where $(-)$ denotes the degree shifting of the homological grading.
Since $JP_0/J^2P_0$ is a finite direct sum of graded simple modules,
$\gExt_A^\bullet(JP_0/J^2P_0,S)$ is graded $E(A)$-projective.
By Proposition \ref{exact sequence of ext group},
$$0 \to \gExt_A^\bullet(J^2P_0,S)(-2) \to \gExt_A^\bullet(JP_0/J^2P_0,S)\,(-1) \to
\ker d_0' \to 0$$
is an exact sequence of graded $E(A)$-modules.
Let $P'_1:= \gExt_A^\bullet(JP_0/J^2P_0,S)\,(-1)$. It follows from Theorem \ref{qK iff E(M) generated in degree 0} that $P'_1$ is generated in degree $1$ as an $E(A)$-module, and
$\gExt_A^\bullet(J^2P_0,S)(-2) \subseteq J'P_1'$.
Hence $P'_1 \to J'P'_0 \to 0$ is a graded projective cover of $J'P'_0$.
By using Proposition \ref{exact sequence of ext group} $(2)'$ repeatedly, one deduces that for any $k \geqslant 1$,
\begin{equation}\label{reso-of-E(A)_0}
P'_k
\gExt_A^\bullet(J^kP_0/J^{k+1}P_0,S)\,(-k)
\end{equation}
is generated in degree $k$. Thus $E(A)_0$ has a linear projective resolution
\begin{equation}\label{reso-of-E(A)_0-long}
\cdots\to P'_k \to P'_{k-1}\to \cdots \to P'_0 \to E(A)_0\to 0
\end{equation}
as a graded $E(A)$-module, and so $E(A)$ is a classical Koszul ring.
{\bf ``only if" part of (2)}. We first claim that $\gExt_A^\bullet(M/JM,S)$ is graded $E(A)$-projective if $M$ is bounded below and left finite.
As $M/JM=\oplus_{j\geqslant l} \bar{M_j}$ where $l=l(M)$ and $\bar{M_j}$ is a finite direct sum of graded simple modules concentrated in degree $j$, it suffices to prove
$\gExt_A^\bullet(\oplus_{j\geqslant l} S(-j), S)$ is graded $E(A)$-projective. Let $P_\bullet \to S \to 0$ be a minimal graded
projective resolution of ${}_AS$. Then $\oplus_{j\geqslant l} P_\bullet(-j) \to \oplus_{j\geqslant l} S(-j) \to 0$ is a minimal graded projective resolution.
Note that $\Hom_{Gr}(\oplus_{j\geqslant l} P_i(-j), S)=\oplus_{j\geqslant l} \Hom_{Gr}( P_i(-j), S)$ for each $i$ as $P_i$ is bounded below.
Then
\begin{align*}\gExt_A^\bullet(\oplus_{j\geqslant l} S(-j), S) = &\oplus_i \gHom_A(\oplus_{j\geqslant l} P_i(-j), S)\\
=&\oplus_i (\oplus_n \Hom_{Gr}(\oplus_{j\geqslant l} P_i(-j)(-n), S))\\
=&\oplus_i (\oplus_n \Hom_{Gr}(\oplus_{j\geqslant l} P_i(-j), S)(n))\\
=&\oplus_i (\oplus_n (\oplus_{j\geqslant l} \Hom_{Gr}( P_i(-j), S))(n))\\
=&\oplus_i (\oplus_{j\geqslant l} (\oplus_n \Hom_{Gr}( P_i(-j), S))(n))\\
=&\oplus_i (\oplus_{j\geqslant l} \gHom_A( P_i(-j), S))\\
=&\oplus_{j\geqslant l} (\oplus_i \gHom_A( P_{i}(-j), S))\\
=&\oplus_{j\geqslant l} (\oplus_i \gExt^i_A( S(-j), S))\\
=&\oplus_{j\geqslant l} \gExt^\bullet_A( S(-j), S).
\end{align*}
This finishes the proof of the claim.
So, $\gExt_A^\bullet(J^{k-1}M/J^kM,S)$ is graded $E(A)$-projective for all $k \geqslant 1$.
Now for any left finite Koszul module $M$, it follows from Proposition \ref{JM is qK} that $J^{k-1}M$ is left finite Koszul.
By Proposition \ref{exact sequence of ext group} $(2)'$,
\small{$$0\to \gExt^\bullet_A(J^kM,S)(-1)\to \gExt_A^\bullet(J^{k-1}M/J^kM,S)\to \gExt^\bullet_A(J^{k-1}M,S)\to 0$$}
is exact. So we can construct a linear $E(A)$-projective resolution for $\mathcal{E}(M)$. Therefore, $\mathcal{E}(M)$ is a classical Koszul $E(A)$-module.
We postpone the ``if part" proof of (1) and (2) of the theorem, for which a modified version of Lemma \ref{exact sequence of syzygy} is needed.
\end{proof}
A graded $A$-module $M$ with a minimal graded projective resolution is called {\it $l$-quasi-Koszul}, where $l>0$ is an integer, if its minimal projective resolution $P_\bullet$ satisfies that $J^k\Omega^n(M)=\Omega^n(M)\cap J^{k+1}P_{n-1}$ for all $0\leqslant k\leqslant l$ and $n>0$. Quasi-Koszul modules are exactly $1$-quasi-Koszul modules.
\begin{lemma}\label{fact 3}
Let $0\to X\to Y\to Z\to 0$ be an exact sequence of left finite bounded below graded $A$-modules with $JX=X\cap JY$. If $X$, $Y$ and $Z$ are all $l$-quasi-Koszul modules, then, for any $n \geqslant 0$,
$0\to \Omega^n(X)\to \Omega^n(Y)\to \Omega^n(Z)\to 0$ is exact, and
for any $1\leqslant k\leqslant l+1$, $$J^k\Omega^n(X)=\Omega^n(X)\cap J^k\Omega^n(Y).$$
\end{lemma}
\begin{proof} Since $X$ is 1-quasi-Koszul, by Lemma \ref{exact sequence of syzygy}, for any $n > 0$,
$$0\to \Omega^n(X)\to \Omega^n(Y)\to \Omega^n(Z)\to 0$$ is exact, and $J\Omega^n(X)=\Omega^n(X)\cap J\Omega^n(Y)$.
By applying the functor $A/J^i\otimes_A -$ to diagram \eqref{comm-diagram-2}, it follows from the $l$-quasi-Koszulity of $X$, $Y$ and $Z$ that $J^i\Omega(X)=\Omega(X)\cap J^i\Omega(Y)$, and
$J^{i+1}X = JX \cap J^{i+1} Y = X \cap JY \cap J^{i+1} Y = X \cap J^{i+1} Y$ for $0\leqslant i\leqslant l$.
The proof is finished by replacing $0\to X\to Y\to Z\to 0$ with the exact sequence
$0\to \Omega^n(X)\to \Omega^n(Y)\to \Omega^n(Z)\to 0.$
\end{proof}
Now we continue the proof of ``if part" of Theorem \ref{A sqk implies E(A) Koszul}.
\begin{proof}{\bf ``if part" of (1)}.
Suppose $E(A)$ is a
classical Koszul ring. Note $E(A)_0$ is semisimple. By \cite[Proposition 1.2.3]{BGS1}, $E(A)$ is generated in degree $1$. Thus by Theorem \ref{qK iff E(M) generated in degree 0}, $A$ is a quasi-Koszul ring.
To prove that $A$ is Koszul, it suffices to prove that all the graded simple $A$-modules are Koszul. This is done by proving the following claim inductively.
Claim: all the graded simple $A$-modules $N$ and all $J^2Q$, where $Q$ are left finite bounded below graded projective $A$-modules, are $l$-quasi-Koszul for any integer $l$.
Since $A$ is quasi-Koszul, $N$ is quasi-Koszul, that is, $1$-quasi-Koszul.
For any left finite, bounded below graded projective $A$-module $Q$, $\mathcal{E}(Q)$ is a classical Koszul $E(A)$-module as $E(A)$ is a classical Koszul ring. It follows from the quasi-Koszulity of $Q$ and Proposition \ref{exact sequence of ext group} that
$$0\to \mathcal{E}(JQ)(-1)\to \mathcal{E}(Q/JQ)\to \mathcal{E}(Q)\to 0$$
is exact. Since $\mathcal{E}(Q)$ is a classical Koszul module, $\mathcal{E}(JQ)(-1)=\Omega(\mathcal{E}(Q))$ is generated in degree $1$. By
Theorem \ref{qK iff E(M) generated in degree 0} (1),
$JQ$ is a quasi-Koszul $A$-module. Suppose $J^{k-1}Q$ is quasi-Koszul. Then
$$0\to \mathcal{E}(J^{k}Q)(-1)\to \mathcal{E}(J^{k-1}Q/J^k Q)\to \mathcal{E}(J^{k-1}Q)\to 0$$
is exact, and $\mathcal{E}(J^k Q)(-k)=\Omega(\mathcal{E}(J^{k-1}Q)(-k+1))=\cdots=\Omega^k(\mathcal{E}(Q))$ which is generated in degree $k$. By Theorem \ref{qK iff E(M) generated in degree 0} (1) again, $J^k Q$ is quasi-Koszul for any $k\geqslant 0$ by induction.
So, our claim is true for $l=1$.
Now, assume all graded simple $A$-modules $N$ and all $J^2Q$ are $l$-quasi-Koszul for $l \geqslant 1$, where $Q$ are left finite, bounded below graded projective $A$-module $A$-modules.
Let $P_\bullet$ be a minimal graded projective resolution of $N$. Then $P_0$ is an indecomposable graded projective $A$-module.
Note $P_1 \to JP_0 \to 0$ is a projective cover, $\Omega(JP_0)=\Omega^2(N)$ and $JP_1 = \Omega(JP_0/J^2P_0)$. Then
\begin{equation} \label{important-exact=seq}
0\to \Omega(JP_0) \to \Omega(JP_0/J^2P_0)\to J^2P_0\to 0
\end{equation}
is an exact sequence, and all the modules in above exact sequence are $l$-quasi-Koszul by hypothesis.
Since $N$ is quasi-Koszul, $$J\Omega(JP_0)= J \Omega^2(N)= \Omega^2(N) \cap J^2P_1 = \Omega(JP_0) \cap J \Omega(JP_0/J^2P_0).$$
Then Lemma \ref{fact 3} applies to the exact sequence \eqref{important-exact=seq}, and we have, for all $n \geqslant 1$ and $k\leqslant l+1$,
\begin{equation}\label{equation 1}J^k\Omega^n(JP_0)=\Omega^n(JP_0)\cap J^k\Omega^n(JP_0/J^2P_0).
\end{equation}
Next, we show that $N$ is $l+1$-quasi-Koszul, that is, for any $n$ and $ k\leqslant l+1$,
\begin{equation}\label{key equation}
J^k\Omega^n(N)=\Omega^n(N) \cap J^{k+1}P_{n-1}.
\end{equation}
Since $\Omega(N)=JP_0$,
$J^k\Omega(N)=J^{k+1}P_0= \Omega(N) \cap J^{k+1}P_0$ for all $k \geqslant 0$.
So, \eqref{key equation} holds for $n=1$.
Since $\Omega(JP_0/J^2P_0)=JP_1$, by \eqref{equation 1}, for all $0\leqslant k\leqslant l+1$,
\begin{small}
\begin{equation}\label{equation 2}
J^k\Omega^2(N)=J^k\Omega(JP_0)=\Omega(JP_0) \cap J^k\Omega(JP_0/J^2P_0) =\Omega^2(N) \cap J^{k+1}P_1.
\end{equation}
\end{small}
So, \eqref{key equation} holds for $n=2$.
By Lemma \ref{fact 1}, we have the following exact commutative diagram, where $P_2' \to J^2P_0 \to 0$ is a projective cover of $J^2P_0$ and ${P_2}''=P_2\oplus {P'}_2$.
\begin{center}
\begin{tikzcd}
& 0 \arrow[d] & 0 \arrow[d] & 0 \arrow[d] & \\
0 \arrow[r] & \Omega^2(JP_0) \arrow[r] \arrow[d] & \Omega^2(JP_0/J^2P_0) \arrow[r] \arrow[d] & \Omega(J^2P_0) \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & P_2 \arrow[r] \arrow[d] & {P_2}'' \arrow[r] \arrow[d] & {P'}_2 \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & \Omega(JP_0) \arrow[r] \arrow[d] & \Omega(JP_0/J^2P_0) \arrow[r] \arrow[d] & J^2P_0 \arrow[r] \arrow[d] & 0 \\
& 0 & 0 & 0 &
\end{tikzcd}
\end{center}
Note that \eqref{equation 2} holds for all graded simple module $N$. So, it holds for $N=JP_0/J^2P_0$. Then, by \eqref{equation 1} and \eqref{equation 2}, for all $0\leqslant k\leqslant l+1$,
\begin{align*}
J^k\Omega^3(N)&=J^k\Omega^2(JP_0)=\Omega^2(JP_0) \cap J^k\Omega^2(JP_0/J^2P_0) \\
&=\Omega^2(JP_0) \cap \Omega^2(JP_0/J^2P_0)\cap J^{k+1}{P_2}''\\
&=\Omega^2(JP_0) \cap J^{k+1}{P_2}''\\
&=\Omega^2(JP_0)\cap P_2\cap J^{k+1}{P_2}''\\
&=\Omega^2(JP_0) \cap J^{k+1}P_2\\
&=\Omega^3(N) \cap J^{k+1}P_2.
\end{align*}
Hence, \eqref{key equation} holds for $n=3$.
Inductively, we have, for any $n$ and $0\leqslant k\leqslant l+1$,
$$J^k\Omega^n(N)=\Omega^n(N) \cap J^{k+1}P_{n-1}.$$
Thus $N$ is an $(l+1)$-quasi-Koszul module.
Hence $\Omega(JP_0)= \Omega^2(N)$ and $\Omega(JP_0/J^2P_0)$ are $(l+1)$-quasi-Koszul. Similarly to the proof of Corollary \ref{strongly quasi Koszul of exact sequence}, we can show that $J^2P_0$ is $(l+1)$-quasi-Koszul.
Since $N$ is an arbitrary graded simple $A$-module and $P_0$ is its graded projective cover, the induction is completed.
Thus, every graded simple $A$-module is $l$-quasi-Koszul for any $l>0$, so it is Koszul.
{\bf ``if part" of (2)}. By (1), $E(A)$ is a classical Koszul algebra, so it is generated in degree $1$. If $\mathcal{E}(M)$ is a
classical Koszul module, it follows from Theorem \ref{qK iff E(M) generated in degree 0} (1) that $M$ is a quasi-Koszul module.
Since $A$ is Koszul, it follows from Proposition \ref{JM is qK} that $JQ$ is Koszul for all left finite bounded below graded projective $A$-module $Q$.
By Proposition \ref{exact sequence of ext group},
$0\to \Omega^n(M)\to \Omega^n(M/JM)\to \Omega^{n-1}(JM)\to 0$ is exact and $J \,\Omega^n(M)=\Omega^n(M) \cap J\, \Omega^n(M/JM)$ for any $n > 0$.
Now suppose that $M$ is $l$-quasi-Koszul for $l \geqslant 1$.
A similar proof to Proposition \ref{JM is qK} shows that $JM$ is $l$-quasi-Koszul. It follows from Lemma \ref{fact 3} that for any $n$ and $1 \leqslant k \leqslant l+1$,
\begin{equation}
J^k\Omega^n(M)=\Omega^n(M)\cap J^k\Omega^n(M/JM).
\end{equation}
Let $Q_\bullet \to M \to 0$ and $Q''_\bullet \to JQ_0 \to 0$ be minimal graded projective resolutions of $M$ and $JQ_0$ respectively. Then, it follows from the exact sequence $0 \to \Omega(M) \to JQ_0 \to JM \to 0$ and the Koszulity of $JQ_0$ that, for any $1 \leqslant k \leqslant l+1$,
\begin{align*}
J^k\Omega^n(M)=&\Omega^n(M)\cap J^k\Omega^n(M/JM)\\
=&\Omega^n(M)\cap J^k\Omega^{n-1}(JQ_0)\\
=&\Omega^n(M)\cap \Omega^{n-1}(JQ_0) \cap J^{k+1} Q''_{n-2}\\
=&\Omega^n(M)\cap Q_{n-1} \cap J^{k+1} Q''_{n-2}\\
=&\Omega^n(M)\cap J^{k+1} Q_{n-1}.
\end{align*}
Hence, $M$ is $(l+1)$-quasi-Koszul. This finishes the proof.
\end{proof}
\begin{remark} \label{E(A)-left-finite}
It should be noted that if a quasi-Koszul module $M$ is finitely generated then $\mathcal{E}(M)$ is a left finite $E(A)$-module.
Let $Q_\bullet \to M \to 0$ be a minimal graded projective resolution of $M$. It follows from Proposition \ref{minimal is fg} that every $Q_n$ is finitely generated.
Therefore
$$\gExt_A^n(M,S)=\gHom_A(Q_n, S) \cong \gHom_S(Q/JQ, S)$$ is a finitely generated $S^{op}$-module. It is easy to see that the $E(A)_0$-module structure of $\gExt_A^n(M,S)$ given by the Yoneda product corresponds to the canonical $S^{op}$-module structure of $\gHom_A(J,S)$. Hence $\gExt_A^n(M,S)$ is finitely generated as an $E(A)_0$-module. In particular, $E(A)$ is a left finite classical Koszul algebra if $A$ is a Koszul algebra.
But, in general, $\mathcal{E}(M)$ will not be left finite as an $E(A)$-module, even when $M$ is Koszul (for example, $M=\oplus_{i\geqslant 0} A(-i)$).
\end{remark}
\subsection{Koszul Duality for rings and modules}
In the classical Koszul theory, if $A$ is left finite classically Koszul then $E(A)$ is left finite classically Koszul and the Yoneda Ext ring of $E(A)$ is isomorphic to $A$ \cite[Theorem 1.2.5]{BGS1}. In our setting, the Koszul dual of the Koszul dual of $A$ will not be $A$ in general but $\Gr_J A$ (see Theorem \ref{FE(S) cong Grj A}). If $A_0$ is semisimple, it recovers the classical results.
As defined in the Introduction, let
\begin{align*}
&\mathcal{E}=\gExt_A^\bullet(-,A/J),\\
&\mathcal{F}=\gExt_{E(A)}^\bullet(-,E(A)/J_{E(A)}),\\
&\mathcal{G}=\gExt^\bullet_{\Grj A}(-,A/J).
\end{align*}
Note that when we consider the Koszulity, $E(A)$ and $\Grj A$ are viewed as a graded ring via the homological degree and the graded degree induced by the $J$-adic filtration respectively, although both $E(A)$ and $\Grj A$ are bigraded rings.
Let $\mathcal{K}_A$, $\mathcal{K}_{E(A)}$ and $\mathcal{K}_{\Grj A}$ be the full subcategories of finitely generated Koszul $A$-modules, classical Koszul ${E(A)}$-modules and classical Koszul ${\Grj A}$-modules respectively in the corresponding categories.
Now we are ready to prove Theorem \ref{theorem 3} which is a generalized version of Koszul algebra duality and Koszul module duality.
\begin{theorem}\label{FE(S) cong Grj A}
Let $A$ be a Koszul ring. Then
\begin{itemize}
\item[(1)] $E(E(A))\cong \Grj A$ as graded rings (in fact, as bigraded rings).
\item[(2)] The functors $\mathcal{E}$ and $\mathcal{F}$ restrict to
$\xymatrix{
\mathcal{K}_A\ar[r]^{\mathcal{E}} &\mathcal{K}_{E(A)}\ar[r]^{\mathcal{F}}& \mathcal{K}_{\Grj A},
}$
such that, for any $M\in \mathcal{K}_{A}$, $\mathcal{F}\mathcal{E}(M)\cong \Grj M$ as graded $\Grj A$-modules.
\item[(3)] The functors $\mathcal{F}$ and $\mathcal{G}$ restrict to
$\xymatrix{
\mathcal{K}_{E(A)}\ar@<1mm>[r]^{\mathcal{F}}& \mathcal{K}_{\Grj A}\ar@<1mm>[l]^{\mathcal{G}}
}$, which gives a duality of categories.
\end{itemize}
\end{theorem}
\begin{proof} (1) It follows from Lemma \ref{JM is fg} that $J^i/J^{i+1}$ is finitely generated as an $S$-module for all $i \geqslant 0$.
So, there is a canonical $S$-module isomorphism given by the evaluation map
$$J^i/J^{i+1} \to \Hom_{S^{op}}(\Hom_S(J^i/J^{i+1}, S), S), \, \bar{x} \mapsto \big(\bar{x}^{**}: \varphi \to \varphi(\bar{x})\big).$$
By Theorem \ref{A sqk implies E(A) Koszul}, $E(A)$ is a classical Koszul ring. Let $P_\bullet' \to E(A)_0 \to 0$ be the minimal graded projective resolution of $E(A)_0$ given in the proof of Theorem \ref{A sqk implies E(A) Koszul}. Then $P_i'$
is generated in degree $i$ for all $i$. By using the equality \eqref{reso-of-E(A)_0} in the proof of Theorem \ref{A sqk implies E(A) Koszul} and the fact $E(A)_0 \cong S^{op}$,
\begin{align*}
&\gExt^i_{E(A)}(E(A)_0,E(A)_0)\\
=&\gHom_{E(A)}(P_i',E(A)_0)\\
\cong & \Hom_{E(A)_0}((P_i')_i, E(A)_0)\\
= &\Hom_{S^{op}}(\gHom_S(J^i/J^{i+1},S), S)\\
\cong & J^i/J^{i+1}
\end{align*}
as $S$-modules.
It induces a graded $S$-module isomorphism $$\theta: E(E(A))\to \mathop{\oplus}\limits_{i\geqslant 0}J^i/J^{i+1},\, g \mapsto \overline{y}$$
for any $g\in \gHom_{E(A)}(P_i',E(A)_0)$, where $\overline{y} \in J^i/J^{i+1}$ satisfies that $\overline{y}^{**}$ is equal to the action of
$g$ restricting to the degree $0$ part.
In fact, $\theta$ is a graded ring isomorphism as we show next.
For any
$$f\in \gExt_{E(A)}^j(E(A)_0,E(A)_0) \cong \Hom_{S^{op}}(\gHom_S(J^j/J^{j+1},S),S)$$
and
$$g\in \gExt^i_{E(A)}(E(A)_0,E(A)_0)\cong \Hom_{S^{op}}(\gHom_S(J^i/J^{i+1},S),S)$$
with $\theta(f)= \overline{x}\in J^j/J^{j+1}$ and $\theta(g)= \overline{y}\in J^i/J^{i+1}$,
we have to show $\theta(f\cdot g)= \theta(f)\, \theta(g)$, that is, to show that $(f\cdot g)(\varphi)=\varphi(\overline{xy})$ for any $\varphi\in \gHom_S(J^{j+i}/J^{j+i+1},S)$.
Consider first the case that $j=0$.
For any
$\varphi\in
\gHom_S(J^i/J^{i+1},S), $
$g(\varphi)=\varphi(\overline{y})\in S,$
and for any $\psi \in
\gHom_S(A/J,S), $
$f(\psi)=\psi(\overline{x})\in S$. It follows from the definition of Yoneda products that
$$(f\cdot g)(\varphi)=f(\varphi(\overline{y}))=\overline{x}\varphi(\overline{y})=\varphi(\overline{xy}).$$
Therefore, $\theta(f\cdot g)= \overline{xy} =\theta(f)\, \theta(g)\in J^i/J^{i+1}$.
Consider next the case that $j=1$. For any
$$\varphi\in \gHom_S(J^{i+1}/J^{i+2},S) \cong \gHom_A(J^{i+1},S),$$
$\varphi$ corresponds to an element of $P_i'$ via the injective map
\begin{equation}\label{description-of-phi}
0 \to \gHom_A(J^{i+1},S) \to \gExt_A^1(J^i/J^{i+1},S) = (P'_i)_{i+1}
\end{equation}
given by Proposition \ref{exact sequence of ext group} $(2)'$. Let us describe this exact sequence.
Let $\pi: Q\to J^i \to 0$ be a graded projective cover of $J^i$. Then $\Omega(J^i/J^{i+1})=JQ$, and there is an exact commutative diagram
\begin{center}
\begin{tikzcd}
& 0 \arrow[r] & \Omega(J^i) \arrow[d] \arrow[r] & \Omega(J^i/J^{i+1}) \arrow[d] \arrow[r, "\pi"] & J^{i+1} \arrow[r] & 0 \\
& & Q \arrow[d, "\pi"] \arrow[r] & Q \arrow[d] & & \\
0 \arrow[r] & J^{i+1} \arrow[r] & J^i \arrow[r] & J^i/J^{i+1} \arrow[r] & 0. &
\end{tikzcd}
\end{center}
It follows from the construction of the long exact sequence of Ext-groups that the exact sequence \eqref{description-of-phi} is
$$0 \to \gHom_A(J^{i+1},S) \xrightarrow{{\pi}^*} \gExt_A^1(J^i/J^{i+1},S) = (P'_i)_{i+1}.$$
Hence $\varphi\in \gHom_S(J^{i+1}/J^{i+2},S)$ corresponds to the element $\alpha:={\pi}^*(\varphi)=\varphi\circ \pi \in (P'_i)_{i+1}= \gHom_A(\Omega(J^i/J^{i+1}),S)$ of $P_i'$.
Suppose $J^i/J^{i+1}= \mathop{\oplus}\limits_{k=1}^t S_k$ where $S_k$ are some graded simple $A$-modules, and $Q = \mathop{\oplus}\limits_{k=1}^t Q_k$ with $Q_k$ being a graded projective cover of $S_k$.
Let $p_k:J^i/J^{i+1} \to S_k$ and $p'_k:JQ \to JQ_k$ be the projections induced by the canonical projection $Q \to Q_k$.
Let $\alpha_k \in\gHom_A(JQ_k,S)$ be the composition $JQ_k \subseteq JQ \stackrel{\alpha}{\rightarrow}S$.
By the definition of Yoneda products and the following commutative diagram
\begin{center}
\begin{tikzcd}[column sep=small]
0 \arrow[r] & \Omega(J^i/J^{i+1}) \arrow[r] \arrow[d, "="] \arrow[d] & Q \arrow[r] \arrow[d, "="] & J^i/J^{i+1} \arrow[r] \arrow[d, "="] & 0 \\
0 \arrow[r] & JQ=\Omega(Q/JQ) \arrow[r] \arrow[d, "p'_k"] & Q \arrow[r] \arrow[d] & Q/JQ\cong J^i/J^{i+1} \arrow[r] \arrow[d, "p_k"] & 0 \\
0 \arrow[r] & J(Q_k)= \Omega(S_k) \arrow[r] \arrow[d, "\alpha_k"] & Q= Q_k \arrow[r] & S_k \arrow[r] & 0 \\
& S & & &
\end{tikzcd}
\end{center}
we have $\alpha =\sum_k \alpha_k p'_k= \sum_k \alpha_k \cdot p_k \in \gExt^1_A(S_k,S) \cdot \gHom_A(J^i/J^{i+1},S_k)$.
By taking a suitable grading shift, we may view
$$\alpha = \sum_k \alpha_k \cdot p_k \in \gExt^1_A(S,S) \cdot \gHom_A(J^i/J^{i+1},S).$$
Suppose $\tilde{y}\in Q$ such that $\pi(\tilde{y}) = y \in J^i$.
Then $\pi(x\tilde{y})=xy \in J^{i+1}$. Let
$\tilde{y}=\sum\limits_{k=1}^t \tilde{y}_k$ such that $\tilde{y}_k\in Q_k$ and $\overline{\pi(\tilde{y}_k)}= p_k(\overline{y})\in S_k$.
Hence
\begin{equation}\label{action-of-phi}
\varphi(\overline{xy})=\varphi(\overline{\pi(x\tilde{y})})=\alpha(x\tilde{y})=\sum\alpha_k(x\tilde{y}_k).
\end{equation}
To finish the proof of the $j=1$ case, it is left to show
\begin{equation} \label{action-of-phi-2}
(f\cdot g)(\varphi)=\sum\alpha_k(x\tilde{y}_k).
\end{equation}
Now, consider the following commutative diagram
\begin{center}
\begin{tikzcd}
P_{i+1}' \arrow[r, two heads] \arrow[d, "g_1"] & \Omega^{i+1}(E(A)_0) \arrow[d, "g_0'"] \arrow[r, hook] & P'_i \arrow[d, "g_0"] \arrow[r, "d'_i"] & \Omega^i(E(A)_0) \arrow[d, "g"] \arrow[r] & 0 \\
P'_1 \arrow[r, phantom] \arrow[r, two heads] & \Omega^1(E(A)_0) \arrow[r, hook] \arrow[d, "f"] & P'_0 \arrow[r] & E(A)_0 \arrow[r] & 0 \\
& E(A)_0 & & &
\end{tikzcd}
\end{center}
where $g_0,g_0'$ and $g_1$ are the lifting of $g$. Since $P'_i$ is generated in degree $i$, we may assume that when restricting to degree $i$ part the action of $g_0$ is the same as that of $g$. Note that $g_0'$ is the restriction of $g_0$ on $\Omega^{i+1}(E(A)_0)$, $\Omega^{i+1}(E(A)_0)= \gExt^\bullet_A(J^{i+1}, S)[-i-1]$ and $$\Omega^{i+1}(E(A)_0)_{i+1}= \big(\gExt^\bullet_A(J^{i+1}, S)[-i-1]\big)_{i+1}=\gHom_A(J^{i+1}, S).$$
By the definition of Yoneda products,
$$(f\cdot g)(\varphi)=(f\circ g_0')(\varphi)=f(g_0(\alpha))=f(g_0(\sum \alpha_k\cdot p_k)).$$
Since $g_0$ is $E(A)$-linear,
$$f(g_0(\sum \alpha_k\cdot p_k))=f(\sum\alpha_k\cdot g_0(p_k))=f(\sum\alpha_k\cdot p_k(\overline{y})_r)$$
where $p_k(\overline{y})_r \in \gHom_S(A/J, S_k)$ is the right multiplication of $p_k(\overline{y}) \in S_k$.
To see the Yoneda product of $\alpha_k$ with $p_k(\overline{y})_r$, we need the following commutative diagram
\begin{center}
\begin{tikzcd}
0 \arrow[r] & JQ_k \arrow[d, "(\tilde{y}_k)_r"] \arrow[r] & Q_k \arrow[d, "(\tilde{y}_k)_r"] \arrow[r] & S_k \arrow[d, "(p_k(\overline{y}))_r"] \arrow[r] & 0 \\
0 \arrow[r] & JQ_k \arrow[d, "\alpha_k"] \arrow[r] & Q_k \arrow[r] & S_k \arrow[r] & 0 \\
& S & & &
\end{tikzcd}
\end{center}
where $(*)_r$ is the right multiplication given by $*$. Then $$\alpha_k\cdot p_k(\overline{y})=\alpha_k\circ (\tilde{y}_k)_r$$
and $f(\sum\alpha_k\cdot p_k(\overline{y}))=f(\sum\alpha_k\circ (\tilde{y}_k)_r)
=(\sum\alpha_k\circ (\tilde{y}_k)_r)(x)=\sum\alpha_k(x\tilde{y}_k)$.
Hence
$ (f\cdot g)(\varphi)=\sum\alpha_k(x\tilde{y}_k)$, that is, \eqref{action-of-phi-2} holds.
Therefore $(f\cdot g)(\varphi)=\varphi(\overline{xy})$ and so $\theta(f\cdot g)=\overline{xy}=\theta(f) \theta(g)$. The proof of the $j=1$ case is finished.
Finally, consider the general case that $j > 1$. By Theorem \ref{qK iff E(M) generated in degree 0},
$$\gExt^{j}_{E(A)}(E(A)_0,E(A)_0) \ni f=\sum\limits_{p=1}^s h_p\cdot f_p$$
for some $h_p\in\gExt^1_{E(A)}(E(A)_0, E(A)_0)$ and $f_p\in \gExt^{j-1}_{E(A)}(E(A)_0,E(A)_0)$.
Suppose
$\theta(h_p)=\overline{x}_p\in J/J^2$ and $\theta(f_p)=\overline{x}'_p\in J^{j-1}/J^{j}.$
Then, by the $j=1$ case already proved and by induction hypothesis,
$$\theta(f)=\sum \overline{x_px'_p}\in J^{j}/J^{j+1} \textrm{ and } \theta(f_p\cdot g)=\overline{x'_py}\in J^{i-1+j}/J^{i+j}.$$
Thus by the $j=1$ case again,
$$\theta(f\cdot g)=\theta(\sum h_p\cdot (f_p\cdot g))=\sum\overline{x_p}\cdot \overline{x'_py}=\sum\overline{x_px'_py}=\theta(f)\theta(g).$$
It follows that $\theta$ is an isomorphism of graded rings.
(2) By definition, $\mathcal{F}$ is a functor from the category of graded $E(A)$-modules to the category of graded $E(E(A))$-modules. Since $E(E(A)) \cong \Gr_J A$ as graded rings by (1), $\mathcal{F}$ can be naturally viewed as a functor from the category of graded $E(A)$-modules to the category of graded $\Gr_J A$-modules. It follows from the linear $E(A)$-projective resolution
$$ \cdots \to \mathcal{E}(J^iM/J^{i+1}M)[-i]\to \cdots \to \mathcal{E}(M/JM) \to \mathcal{E}(M) \to 0$$
(see the proof of Theorem \ref{A sqk implies E(A) Koszul} (2)) that
\begin{align*}
\mathcal{F}(\mathcal{E}(M))_i & = \gHom_{E(A)}(\mathcal{E}(J^iM/J^{i+1}M)[-i], E(A)_0)\\
& \cong \Hom_{E(A)_0}(\gHom_S(J^iM/J^{i+1}M, S), E(A)_0)\\
& = \Hom_{S^{op}}(\gHom_S(J^iM/J^{i+1}M, S),S) \\
& \cong J^iM/J^{i+1}M,
\end{align*}
where the $\cong$ holds because $J^iM/J^{i+1}M$ is finitely generated.
A similar argument to (1) shows that $\mathcal{F}(\mathcal{E}(M))\cong \Grj M$ as graded $\Grj A$-modules.
(3) Since both $E(A)_0$ and $(\Grj A)_0$ are artinian semisimple, by replacing $A$ with $E(A)$ and $\Grj A$ respectively and repeating the proof above, it follows that $\mathcal{F},\mathcal{G}$ give a duality between $\mathcal{K}_{E(A)}$ and $\mathcal{K}_{\Grj A}$.
Or by Theorem \ref{A sqk implies E(A) Koszul} and Remark \ref{E(A)-left-finite}, $E(A)$ is left finite classical Koszul with $E(A)_0$ artinian semisimple, then the conclusion follows from the classical Koszul theory.
\end{proof}
Under the assumptions in Theorem \ref{FE(S) cong Grj A}, if furthermore $A_0$ is semisimple, then $A\cong \Grj A$ as graded rings. Therefore, Theorem \ref{FE(S) cong Grj A} reduces to the classical Koszul duality (\cite[Theorem 5.2]{GM2}, \cite[Theorem 2.10.2]{BGS1}).
The following result gives the converse statements of Theorem \ref{FE(S) cong Grj A} (1) and (2) in some sense.
\begin{theorem}\label{FE(M) cong Grj M imply M sqk}
Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect.
\begin{itemize}
\item[(1)] If $E(A)$ is generated in degree $1$ and $E(E(A))\cong \Grj A$ as graded rings, then $A$ is Koszul.
\item[(2)] Suppose that $A$ is Koszul and $M$ is a finitely generated graded $A$-module. If $\mathcal{E}(M)$ is generated in degree $0$ and $\mathcal{F}(\mathcal{E}(M))\cong \Grj M$ as graded $\Grj A$-modules, then $M$ is Koszul.
\end{itemize}
\end{theorem}
\begin{proof} (1) Since $E(A)$ is generated in degree $1$, it follows from Theorem \ref{qK iff E(M) generated in degree 0} that $A$ is quasi-Koszul. Then $E(A)$ is left finite as we see in Remark \ref{E(A)-left-finite}.
If $E(E(A))\cong \Grj A$, then it is generated in degree $1$. Thus by Theorem \ref{qK iff E(M) generated in degree 0} again, $E(A)$ is quasi-Koszul. By Proposition \ref{qk and classical koszul}, $E(A)$ is a classical Koszul ring. Therefore $A$ is Koszul by Theorem \ref{A sqk implies E(A) Koszul}.
(2) The proof is similar to that of (1)
\end{proof}
\section{More characterizations of Koszul property}
In this section
we first prove that if $M$ is a Koszul $A$-module, then $\Grj M$ is a classical Koszul $\Grj A$-module. The converse statement is true under an additional condition that $J(A_0)$ is nilpotent. As a corollary, it is proved that $A$ is a Koszul ring if and only if so is $A^{op}$. More characterizations of the Koszulity are given under the condition that $A_0$ is artinian.
\subsection{(Quasi-)Koszulity of A versus Koszulity of GrA }
The following result is trivial if $A_0$ is semisimple.
\begin{theorem}\label{A sqk implies Gr A Koszul}
Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect.
\begin{enumerate}
\item If $M$ is a Koszul $A$-module, then $\Grj M$ is a classical Koszul $\Grj A$-module.
\item If $A$ is a Koszul ring, then $\Grj A$ is a classical Koszul ring.
\end{enumerate}
\end{theorem}
\begin{proof} It suffices to prove (1).
Let $P_\bullet\to M\to 0$ be a minimal graded projective resolution of ${}_AM$. Since $M$ is Koszul, it follows that, with the $J$-adic filtration,
$$0\to \Ker d_0[-1] \xrightarrow{i_0} P_0\xrightarrow{d_0} M\to 0$$
is strict exact, where $\Ker d_0[-1]$ is the shift of the $J$-adic filtration.
Therefore, there is an exact sequence of $\Grj A$-modules
$$0\to \Grj \Ker d_0[-1]\to \Grj P_0\to \Grj M\to 0.$$
It is easy to see that $\Grj P_0$ is a graded projective cover of the $\Grj A$-module $\Grj M$. By replacing $M$ with $\Ker d_0[-1]$, and doing this repeatedly, we can construct a minimal graded projective resolution of the $\Grj A$-module $\Grj M$:
$$\cdots\to \Grj P_i[-i]\to\cdots\to \Grj P_0\to \Grj M\to 0$$
which is a linear projective resolution. Thus $\Grj M$ is a classical Koszul $\Grj A$-module.
\end{proof}
A graded left ideal $I$ of an $\mathbb{N}$-graded ring $A$ is called degree-wise nilpotent, if for any positive integer $t$, there is an integer $N$ such that for all $n\geqslant N$, $(I^n)_{\leqslant t}=0$.
\begin{lemma}\label{locally nilpotent}
Let $A$ be an $\mathbb{N}$-graded ring and $I$ be a graded left ideal of $A$.
Then $I$ is degree-wise nilpotent if and only if $I_0$ is nilpotent.
\end{lemma}
\begin{proof}
For any integer $l \geqslant 0$, the degree $l$ part of $I^n$ has the form
$$(I^n)_l=\sum\limits_{i_1+\cdots+i_{s+1}=n-s, j_1 + \cdots + j_s =l}I_0^{i_1}I_{j_1}I_0^{i_2}I_{j_2}\cdots I_0^{j_s}I_{j_s}I_0^{i_{s+1}}$$
where $j_1, \cdots, j_s \geq 1$. Note that $i_1+\cdots+i_{s+1}=n-s \geqslant n - l$.
If $I_0$ is nilpotent, then, for any fixed positive integer $t$, $(I^n)_{\leqslant t}=0$ for sufficiently large $n$.
The other direction is trivial.
\end{proof}
Hence, the graded Jacobson radical $J$ of $A$ is degree-wise nilpotent if and only if $J(A_0)$ is nilpotent, under the hypothesis of Theorem \ref{A sqk implies Gr A Koszul}. Recall that a ring is noetherian semiperfect with nilpotent Jacobson radical if and only if it is a noetherian perfect ring if and only if it is an artinian ring.
\begin{theorem}\label{GrM is Koszul implies M is qK}
Suppose that $A$ is a left finite $\mathbb{N}$-graded ring generated in degree $1$ such that $A_0$ is artinian.
\begin{enumerate}
\item If $M$ is a left finite bounded below graded $A$-module such that $\Grj M$ is a classical Koszul $\Grj A$-module, then $M$ is a Koszul $A$-module.
\item If $\Grj A$ is a classical Koszul ring, then $A$ is a Koszul ring.
\end{enumerate}
\end{theorem}
\begin{proof} It suffices to prove (1).
Let $P_\bullet\to M\to 0$ be a minimal graded projective resolution of ${}_AM$.
Then $\Grj P_0 \to \Grj M\to 0$ is a graded projective cover of $\Grj M$.
Consider the exact sequence
$0\to \Ker d_0\xrightarrow{i_0} P_0\xrightarrow{d_0} M\to 0$
which is strict exact if we endow $P_0$, $M$ with the $J$-adic filtration and $\Ker d_0$ with the induced submodule filtration, that is, $F_n\Ker d_0=\Ker d_0 \cap J^nP_0$. Then
$$0\to \Gr \Ker d_0\to \Grj P_0\to \Grj M\to 0$$
is exact, where $\Gr \Ker d_0=\oplus (F_n\Ker d_0/F_{n+1}\Ker d_0)$.
Since $\Grj M$ is classically Koszul, $\Gr \Ker d_0$ is generated in degree $1$. Therefore
\begin{small}
$$
\dfrac{\Ker d_0 \cap J^nP_0}{\Ker d_0 \cap J^{n+1}P_0}=\dfrac{J^{n-1}}{J^n} \dfrac{\Ker d_0}{\Ker d_0 \cap J^2 P_0}=\dfrac{J^{n-1}\Ker d_0+\Ker d_0 \cap J^{n+1}P_0}{\Ker d_0 \cap J^{n+1}P_0}.
$$
\end{small}
Hence
\begin{align*}
\Ker d_0 \cap J^nP_0 &=J^{n-1}\Ker d_0+ \Ker d_0 \cap J^{n+1}P_0 \\
&=J^{n-1}\Ker d_0+J^{n}\Ker d_0+\Ker d_0 \cap J^{n+2}P_0 \\
&=\cdots\\
&=J^{n-1}\Ker d_0+\Ker d_0 \cap J^{n+m}P_0
\end{align*}
for any positive integers $n$ and $m$.
Since $A_0$ is artinian, it follows from Lemma \ref{locally nilpotent} that $J$ is degree-wise nilpotent. Hence, for any fixed $n$ and any fixed $t$, there is a large enough integer $m$ such that $(J^{n+m}P_0)_{\leqslant t}=0$. Therefore
\begin{align*}
(\Ker d_0 \cap J^nP_0 )_{\leqslant t}&=(J^{n-1}\Ker d_0)_{\leqslant t}+ (\Ker d_0 \cap J^{n+m}P_0)_{\leqslant t}\\
&=(J^{n-1}\Ker d_0)_{\leqslant t}.
\end{align*}
By the arbitrariness of $t$,
$J^{n-1}\Ker d_0 = \Ker d_0 \cap J^nP_0$ for all $n \geqslant 1$.
Replacing $M$ by $\Ker d_0$ and by induction, it follows that
$$J^{n-1}\Ker d_i=\Ker d_i \cap J^nP_i$$ for all $n \geqslant 1$ and $i \geqslant 0$. Hence $M$ is Koszul.
\end{proof}
\begin{corollary}\label{minimal projecitve resolution of filtration module}
Keep the same assumptions for $A$ as in Theorem \ref{GrM is Koszul implies M is qK}. Suppose that $M$ is a left finite bounded below Koszul $A$-module.
\begin{enumerate}
\item If $\cdots\to P_i\to \cdots \to P_0\to M\to 0$ is a minimal graded projective resolution of ${}_AM$, then $$\cdots\to \Grj P_i[-i]\to\cdots\to \Grj P_0\to \Grj M\to 0$$ is a minimal graded projective resolution of $\Grj\! A$-module $\Grj\! M$.
\item $\pdim {}_AM=\pdim {}_{Gr_J A}\Grj M$, where $\pdim$ means the projective dimension.
\end{enumerate}
\end{corollary}
\begin{corollary}\label{A is sqk iff Aop is}
Suppose that $A$ is a left and right finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ artinian. Then $A$ is a Koszul ring if and only if so is $A^{op}$.
\end{corollary}
\begin{proof}
By \cite[Proposition 2.2.1]{BGS1}, $\Grj A$ is a classical Koszul ring if and only if so is $(\Grj A)^{op}$. The conclusion follows from $\Grj A^{op}\cong (\Grj A)^{op}$, Theorem \ref{A sqk implies Gr A Koszul} and Theorem \ref{GrM is Koszul implies M is qK}.
\end{proof}
Corollary \ref{A is sqk iff Aop is} is a generalization of \cite[Corollary 4.3]{GM2} and \cite[Proposition 2.2.1]{BGS1}.
Combining the results of this section and the previous section, we have the following theorem, which is one of the main results in this paper.
\begin{theorem} \label{main-resut}
Suppose that $A$ is a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ artinian. Then the following are equivalent.
\begin{itemize}
\item[(1)] $A$ is a Koszul ring.
\item[(2)] $E(A)$ is a classical Koszul ring.
\item[(3)] $\Grj A$ is a classical Koszul ring.
\item[(4)] $E(A)$ is generated in degree $1$, and as graded rings
$E(E(A)\cong \Grj A.$
\end{itemize}
Moreover if $A$ is right finite then the above statements are also equivalent to
\begin{itemize}
\item[(5)] $A^{op}$ is a Koszul ring.
\end{itemize}
If $A$ is a Koszul ring, then for any left finite bounded below $A$-module $M$, the following are equivalent.
\begin{itemize}
\item[(1)] $M$ is a Koszul $A$-module.
\item[(2)] $E(M)$ is a classical Koszul $E(A)$-module.
\item[(3)] $\Grj M$ is a classical Koszul $\Grj A$-module.
\end{itemize}
If $M$ is finitely generated, then they are also equivalent to
\begin{itemize}
\item[(4)] $\mathcal{E}(M)$ is generated in degree $0$ and $\mathcal{F}(\mathcal{E}(M))\cong \Grj M$ as graded $\Grj A$-modules.
\end{itemize}
\end{theorem}
This theorem is a generalization of \cite[Theorems 2.5, 4.3]{MZ}.
Now we can give an example of quasi-Koszul ring but not Koszul.
\begin{example} \label{qk-but-not-sqk}
Let $R=k[x, y]/(x^2 + y^3, xy)$, which is a noetherian local ring with
$J=J(R)=(\bar{x}, \bar{y})$. A computation
due to Sj?din \cite[Theorem 5]{Sj} shows
$\Ext_R^\bullet(R/J, R/J) = T\langle u, v \rangle/(v^2, u^2v-vu^2)$, where $T\langle u, v \rangle$ is the free algebra in variables $u, v$ over $R/J$, and $|u| = 1 = |v|$.
In particular $\Ext_R^\bullet(R/J, R/J)$ is generated
by $\Ext_R^1(R/J, R/J)$. Hence $R$ is quasi-Koszul. However, $R$ is not a Koszul ring by Theorem \ref{main-resut} since the associated graded algebra is
$k[x, y]/(x^2, xy, y^4)$, which is not classically Koszul.
\end{example}
\subsection{Questions in [CPS]}\label{questions-in-CPS}
In the representation theory of finite dimensional algebras, $\Grj A$, the associated graded algebra of $A$ with respect to the $J$-adic filtration, plays an important role. It reflects the homological property of $E(A)=\Ext^\bullet(A/J, A/J)$, which is often called the homological dual of $A$. For quasi-hereditary algebras $A$ appeared in Kazhdan-Lusztig theories, we often have $\Grj A \cong E(E(A))$ \cite[Theorem 2.2.1]{CPS}.
In a series of papers, Cline, Parshall and Scott have been searching for Koszul structures in the quasi-hereditary algebras of interest in modular representation theory of algebraic groups. The
general question they considered are what good properties of a finite dimensional algebra $A$
imply that the graded algebra $\Grj A$ is classically Koszul.
The following two questions were proposed in \cite[Section 3]{CPS}.
\begin{itemize}
\item[(1)] Can it be determined if $A$ is classically Koszul entirely from knowledge of $E(A)$?
\item[(2)] Does the classical Koszul property for $\Grj A$ imply the same property for $A$?
\end{itemize}
A counterexample was given to show both of them have negative answers. Here is their example \cite[Example 3.2]{CPS}.
Let $B$ be the basic algebra with quiver
\begin{center}
\begin{tikzcd}
& & b \arrow[ld, "\beta", shift left] & \\
c \arrow[r, "\zeta", shift left] & a' \arrow[ru, "\delta", shift left] \arrow[l, "\xi", shift left] & & a \arrow[ld, "\gamma"] \arrow[lu, "\epsilon"'] \\
& & b' \arrow[lu, "\alpha"] &
\end{tikzcd}
\end{center}
and defining relations
$$\xi\alpha=\xi\zeta=\delta\zeta=\alpha\gamma-\zeta\xi\beta\epsilon=\beta\delta-\zeta\xi=0.$$
Let $J_B$ be the ideal generated by all arrows. By setting the suitable weights of arrows, $B$ can be viewed as a locally finite graded algebra generated by $B_1$ over $B_0$. For example, let $\alpha$ and $\epsilon$ be of weight one and all others be of weight zero. Then, $J_B$ is the graded Jacobson radical of $B$ in any case.
It was shown in \cite{CPS} that
$$\Grj B\cong
E(E(B)),E(\Grj B) = \gExt_{\Grj B}^\bullet(B/J_B,B/J_B)$$
and they are classically Koszul, but $B \ncong \Grj B$. Thus $B$ is not a classical Koszul algebra.
In fact, it follows from Theorem \ref{main-resut} that $B$ is Koszul in our sense.
In general, for any finite dimensional algebra $A$, it follows from Theorem \ref{main-resut} that
\begin{itemize}
\item[(1)] if $\Grj A$ is classically Koszul, then $A$ is Koszul viewed as a graded ring concentrated on degree $0$ in our sense.
\item[(2)] $A$ is Koszul in our sense if and only if that $E(A)$ is generated in degree $1$ and $ E(E(A)) \cong \Grj A$ as graded rings.
\end{itemize}
Thus, the two questions in \cite{CPS} are answered in some sense.
\section{Generalized AS regular Koszul algebras}
In this section we assume that $A$ is a locally finite $\mathbb{N}$-graded $k$-algebra generated in degree $1$, where $k$ is a field.
If we assume further that $A$ is a Koszul algebra, then it is proved that $A$ is generalized AS regular if and only if so is $\Grj A$ (Theorem \ref{A is generalized AS regular iff GrA is}).
It is well known that, for any connected graded classical Koszul algebra $A$ with finite global dimension, $A$ is an AS regular algebra if and only if $E(A)=\gExt^\bullet_A(k,k)$ is a Frobenius algebra (see \cite[Proposition 5.10]{Sm}). This result was generalized to graded quiver algebra in \cite{MV1} (see also \cite{MV2}). The proof in \cite{MV1} works for classical Koszul algebras as stated in Proposition \ref{Classical case about the relation of Yoneda and generalized AS regular}.
We prove that this result holds for locally finite $\mathbb{N}$-graded Koszul algebras of finite global dimension by combining Theorem \ref{A is generalized AS regular iff GrA is} and Proposition \ref{Classical case about the relation of Yoneda and generalized AS regular}.
\subsection{Generalized AS regular algebras}
Here is the definition of generalized AS regular algebras \cite[Definition 1.4 and Theoren 5.4]{RR1} (see also \cite[Definition 3.15]{MM} and \cite{MV2, MS}).
\begin{definition}\label{definition of GAS regluar}
A locally finite $\mathbb{N}$-graded algebra $A$ is called {\it generalized Artin-Schelter regular} (for short, AS regular) of dimension $d$ if the following conditions hold.
\begin{itemize}
\item[(1)] $A$ has global dimension $d < \infty$.
\item[(2)] For every graded simple $A$-module $M$, $\gExt^i_A(M,A)=0$ if $i\neq d$.
\item[(3)] $\gExt^d_A(-,A)$ induces a bijection between the isomorphism classes of graded simple $A$-modules and graded simple $A^{op}$-modules.
\end{itemize}
\end{definition}
In fact, by \cite[Theorem 1.5]{RR1}, $A$ is twisted Calabi-Yau (or equivalently, $A$ has graded Van den Bergh duality) if and only if $A$ is generalized AS-regular and $S=A/J$ is a separated $k$-algebra.
For any graded ring $A$, let $A$-$\gr$ be the category of finitely generated graded $A$-modules.
\begin{theorem}\cite[Theorem 5.2]{RR1}\label{equivalent definitions of AS regular}
Suppose that $A$ is a locally finite $\mathbb{N}$-graded algebra of finite global dimension $d$. If, for every graded simple $A$-module $M$, $\gExt^i_A(M,A)=0$ for $i\neq d$, then the following are equivalent:
\begin{itemize}
\item[(1)] $\gExt^d_A(-,A)$ gives a contravariant equivalence from $A_0$-$\gr$ to $A_0^{op}$-$\gr$.
\item[(2)] $\gExt^d_A(-,A)$ gives a contravariant equivalence from $S$-$\gr$ to $S^{op}$-$\gr$.
\item[(3)] $\gExt^d_A(S,A)\cong V$ as right $S$-modules, for some graded invertible $S$-bimodule $V$.
\item[(4)] $\gExt^d_A(S,A)\cong V$ as $S$-bimodules, for some graded invertible $S$-bimodule $V$.
\item[(5)] $\gExt^d_A(A_0,A)\cong \gHom_k(A_0, k)\otimes_{A_0} W$ as right $A_0$-modules, for some graded invertible $A_0$-bimodule $W$.
\item[(6)] $\gExt^d_A(A_0,A)\cong \gHom_k(A_0, k)\otimes_{A_0} W$ as $A_0$-bimodules, for some graded invertible $A_0$-bimodule $W$.
\item[(7)] $A$ is a generalized AS regular algebra.
\end{itemize}
\end{theorem}
\subsection{Characterization of generalized AS regular Koszul algebras}
Suppose that $M$ and $N$ are two graded $A$-modules, endowed with the $J$-adic filtration. Then $\gHom_A(M,N)$ is a filtered abelian group endowed with the induced filtration on Hom. In fact, in this case,
$$F_n\gHom_A(M,N) = \{f \in \gHom_A(M,N) \mid f(M) \in J^nN\}.$$
This filtration is always exhaustive. It follows from the degree-wise nilpotency of $J$ that the filtration is separated if $N$ is bounded below.
There is a natural map
$$\varphi:\Gr \gHom_A(M,N)\to \gHom_{\Grj A}(\Grj M,\Grj N)$$
given by $\varphi(\bar{f})(\bar{x})=f(x)+J^{n+i+1}N$, where $f\in F_n\,\gHom_A(M,N)$ and $x\in J^iM$.
If $f\in \gHom_A(M,N),g\in \gHom_A(N,K)$, then $\varphi(\overline{gf})=\varphi(\bar{g})\varphi(\bar{f})$.
\begin{lemma}\label{varphi is isomorphism}
In general, $\varphi$ is injective. If $M$ is a graded projective $A$-module, then $\varphi$ is an isomorphism.
\end{lemma}
\begin{proof}
It follows from the graded projetivity of ${}_AM$ that $M$ is $J$-adic filtered projective. The conclusion follows from \cite[Lemma I.6.9]{LO}.
\end{proof}
\begin{theorem}\label{A is generalized AS regular iff GrA is}
Suppose that $A$ is a locally finite Koszul algebra. Then $A$ is a generalized AS regular algebra of dimension $d$ if and only if $\Grj A$ is a generalized AS regular algebra of dimension $d$.
\end{theorem}
\begin{proof}
By Theorem \ref{A sqk implies Gr A Koszul}, $\Grj A$ is a classical Koszul algebra. It follows from Corollary \ref{minimal projecitve resolution of filtration module} that $\gldim A=\pdim {}_AS=\pdim {}_{\Grj A}\Grj S=\gldim \Grj A$.
Let $P_\bullet\to S\to 0$ be a minimal graded projective resolution of ${}_AS$.
By Proposition \ref{minimal is fg}, each $P_i$ is a finitely generated $A$-module.
Now suppose that $A$ is generalized AS regular of dimension $d$.
Let $(-)^*$ be the functor $\gHom_A(-,A)$. Then
$$0\to P_0^*\xrightarrow{d_1^*} P_1^*\to \cdots \xrightarrow{d_d^*} P_d^*\to \gExt_A^d(S,A)\to 0$$
is a minimal graded projective resolution of $\gExt^d_A(S,A)$ as an $A^{op}$-module.
By definition $F_nP_i^*=\{f \in P_i^* \mid f(P_i)\subseteq J^n\}$. In fact, it is the same as the $J$-adic filtration on the right $A$-module $P_i^*$, that is, $F_nP_i^*=P_i^*\, J^n$.
If $f \in P_i^*J^n$, then $f=\sum g_jx_j$ for some $g_j\in P_i^*$ and $x_j\in J^n$, and $f(P_i)=(\sum g_jx_j)(P_i)=\sum g_j(P_i)x_j\subseteq J^n$.
On the other hand, suppose $f \in P_i^*$ such that $f(P_i)\subseteq J^n$. Then $f=\sum_{\alpha} x_{\alpha}^* f(x_{\alpha}) \in P_i^*\, J^n$, where $\{x_{\alpha}, x_{\alpha}^*\}$ is a dual basis of the finitely generated graded projective module $P_i$.
Since $A$ is a locally finite Koszul algebra, it follows from Corollary \ref{A is sqk iff Aop is} that $A^{op}$ is a Koszul algebra. Thus $\gExt_A^d(S,A)$ is a Koszul $A^{op}$-module. By Corollary \ref{minimal projecitve resolution of filtration module},
$$0\to \Grj P_0^*[-d]\to \cdots \to\Grj P_d^*\to \Grj\gExt_A^d(S,A)\to 0$$
is a minimal graded projective resolution of $\Grj\gExt_A^d(S,A)$ as a $(\Gr_J A)^{op}$-module.
It follows from Lemma \ref{varphi is isomorphism} that
\begin{align*}
\Grj P_i^*&=\oplus (P_i^*J^n/P_i^*J^{n+1}) =\oplus (F_nP_i^*/F_{n+1}P_i^*)\\
&= \Gr_J (\gHom_A(P_i, A)) \cong \gHom_{\Grj A}(\Grj P_i,\Grj A),
\end{align*}
and the following is an isomorphism of complexes,
\begin{center}
\begin{tikzcd}[column sep=tiny]
0 \arrow[r] & \Grj P_0^*[-d] \arrow[r] \arrow[d] & \Grj P_1^*[-d+1] \arrow[r] \arrow[d] & \cdots \arrow[r] & \Grj P_d^* \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & (\Grj P_0)^*[-d] \arrow[r] & (\Grj P_1)^*[-d+1] \arrow[r] & \cdots \arrow[r] & (\Grj P_d)^* \arrow[r] & 0 {}
\end{tikzcd}
\end{center}
where $(\Grj P_i)^*=\gHom_{\Grj A}(\Grj P_i,\Grj A)$.
Again by Corollary \ref{minimal projecitve resolution of filtration module} and $S=\Grj S$,
$$\gExt_{\Grj A}^i(S,\Grj A)\left\{
\begin{aligned}
&=0,&i\neq d\\
&\cong \Grj\gExt_A^d(S,A)[d], &i=d
\end{aligned}
\right.
$$
as right $\Grj A$-modules.
By Theorem \ref{equivalent definitions of AS regular}, $J\gExt_A^d(S,A)=\gExt_A^d(S,A)J$ and $\Grj\gExt_A^d(S,A)\cong \gExt_A^d(S,A)$ is invertible as $S$-bimodule. By Theorem \ref{equivalent definitions of AS regular} again, $\Grj A$ is a generalized AS regular algebra.
Conversely, suppose $\Grj A$ is a generalized AS regular algebra of dimension $d$.
Since $\Grj A$ is a classical Koszul algebra, $S$ as a $\Grj A$-module has a linear projective resolution:
$$0\to Q_d\to Q_{d-1}\to \cdots \to Q_0\to S\to 0.$$
Then
$$0\to Q_0^*\to \cdots \to Q_d^*\to \gExt_{\Grj A}^d(S,\Grj A)\to 0$$
is a projective resolution of $\gExt_{\Grj A}^d(S,\Grj A)$. Since $Q_d^*$ is generated in degree $-d$ and $\gExt_{\Grj A}^d(S,\Grj A)$ is semisimple, $\gExt_{\Grj A}^d(S,\Grj A)$ is concentrated in degree $-d$. Therefore,
$\gExt_{\Grj A}^d(S,\Grj A)_i=0,\forall i\neq -d$ and $\gExt_{\Grj A}^d(S,\Grj A)_{-d}$ is an invertible $S$-bimodule.
For convenience, let us denote the complex $\gHom_A(P_{\bullet}, A)$ by $C^\bullet$:
$$0\to \gHom_A(P_0,A)\xrightarrow{d_1^*} \gHom_A(P_1,A)\xrightarrow{d_2^*}\cdots \xrightarrow{d_d^*} \gHom_A(P_d,A)\to 0.$$
Consider the $i$-th shift of the $J$-adic filtration on $P_i^*$. For convenience, we still use $F_n$ to denote the new filtration, that is,
$$F_n\gHom_A(P_i,A):=\{f \mid f(P_i)\subseteq J^{n+i}\}.$$
For any $f\in F_n\gHom_A(P_i,A)$,
$$d_{i+1}^*(f)(P_{i+1})=f(d_{i+1}(P_{i+1}))=f(\Ker d_i)\subseteq f(JP_i)\subseteq J^{n+i+1},$$
which means $d_{i+1}^*(f)\in F_n\gHom_A(P_{i+1},A)$. Therefore, $C^\bullet$ is a filtered cochain complex.
Now we look into the spectral sequence of the filtered complex $FC^\bullet$. We will use the same terminology and notation as \cite[Section 5.4, 5.5]{W}, but the cohomological version.
The objects in $0$-page are
\begin{align*}
E_0^{pq}&=\dfrac{F_p\gHom_A(P_{p+q},A)}{F_{p+1}\gHom_A(P_{p+q},A)}\\
&\cong \big(\Gr \gHom_A(P_{p+q},A)[p+q]\big)_p\\
&\cong \big(\gHom_{\Grj A}(\Grj P_{p+q},\Grj A)[p+q]\big)_p\\
&=\gHom_{\Grj A}((\Grj P_{p+q})[-p-q],\Grj A)_p.
\end{align*}
The objects in $1$-page are
$$E_1^{pq}=\gExt_{\Grj A}^{p+q}(S,\Grj A)_p\left\{
\begin{aligned}
&=0,&(p,q)\neq (-d,2d)\\
&\cong \gExt_{\Grj A}^d(S,\Grj A)_{-d}, &(p,q)= (-d,2d)
\end{aligned}
\right.
$$ as right $\Grj A$-modules.
So, the spectral sequence $\{E_n^{pq}\}$ is bounded, and
$$E_\infty^{pq}\left\{
\begin{aligned}
&=0,&(p,q)\neq (-d,2d)\\
&\cong \gExt_{\Grj A}^d(S,\Grj A)_{-d}, &(p,q)= (-d,2d)
\end{aligned}
\right.
$$ as right $\Grj A$-modules.
Although the filtration is not bounded below, and the classical convergence theorem \cite[Theorem 5.5.1]{W} does not apply directly, this spectral sequence is still convergent to the cohomological groups of $C^\bullet$, which we check in the following.
Let us recall the notations first. Let $\eta_p:F_pC^{p+q}\to F_pC^{p+q}/F_{p+1}C^{p+q}$ be the natural projection and $\partial$ be the differentials of $C^\bullet$.
\begin{align*}
&A_r^{pq}=\{f\in F_pC^{p+q}\mid \partial(f)\in F_{p+r}C^{p+q+1}\}\\
&Z_r^{pq}=\eta_p(A_r^{pq})=\dfrac{A_r^{pq}+F_{p+1}C^{p+q}}{F_{p+1}C^{p+q}}\\
&B_r^{pq}=\eta_p(\partial(A_{r-1}^{p-r+1,q+r-2}))=\dfrac{\partial(A_{r-1}^{p-r+1,q+r-2})+F_{p+1}C^{p+q}}{F_{p+1}C^{p+q}}\\
&Z_\infty^{pq}=\cap_rZ_r^{pq}=\dfrac{\cap_r(A_r^{pq}+F_{p+1}C^{p+q})}{F_{p+1}C^{p+q}}\\
&B_\infty^{pq}=\cup_r B_r^{pq}=\dfrac{\partial(C^{p+q-1})\cap F_pC^{p+q}+F_{p+1}C^{p+q}}{F_{p+1}C^{p+q}} \\%(\text{exhuastive})
& E_\infty^{pq}=Z_\infty^{pq}/B_\infty^{pq}
\end{align*}
Let $z_\infty^{pq}=\dfrac{\Ker \partial\cap F_pC^{p+q}+F_{p+1}C^{p+q}}{F_{p+1}C^{p+q} }\subseteq Z_\infty^{pq}$ and $e_\infty^{pq}=z_\infty^{pq}/B_\infty^{pq}$.
In general $e_\infty^{pq} \subseteq E_\infty^{pq}$.
The cohomological groups $\{H^\bullet\}$ of $C^\bullet$ have the standard filtration induced by $FC^\bullet$ given by
$$F_pH^{p+q}:=\dfrac{\Ker\partial\cap F_pC^{p+q}+\partial(C^{p+q-1})}{\partial(C^{p+q-1})}.$$
Then
\begin{align*}
\dfrac{F_pH^{p+q}}{F_{p+1}H^{p+q}}&\cong\dfrac{\Ker\partial\cap F_pC^{p+q}+\partial(C^{p+q-1})}{\Ker\partial\cap F_{p+1}C^{p+q}+\partial(C^{p+q-1})}\\
&\cong \dfrac{\Ker\partial\cap F_pC^{p+q}}{\Ker\partial\cap F_{p+1}C^{p+q}+\partial(C^{p+q-1})\cap F_pC^{p+q}}.
\end{align*}
Hence
\begin{align*}
e_\infty^{pq}&\cong \dfrac{\Ker \partial\cap F_pC^{p+q}+F_{p+1}C^{p+q}}{\partial(C^{p+q-1})\cap F_pC^{p+q}+F_{p+1}C^{p+q}}\\
&\cong \dfrac{\Ker\partial\cap F_pC^{p+q}}{\Ker\partial\cap F_{p+1}C^{p+q}+\partial(C^{p+q-1})\cap F_pC^{p+q}}\\
&\cong \dfrac{F_pH^{p+q}}{F_{p+1}H^{p+q}}.
\end{align*}
Clearly, $\cap_rA_r^{pq}+F_{p+1}C^{p+q}\subseteq\cap_r(A_r^{pq}+F_{p+1}C^{p+q})$.
Now, in our case, $F_pC^n=C^nJ^{p+n}$ is a bounded below graded $A^{op}$-module. So, $A_r^{pq}=\partial^{-1}(F_{p+r}C^{p+q+1})\cap F_pC^{p+q}$ is also a bounded below graded $A^{op}$-module.
For a fixed degree $i$, by Lemma \ref{locally nilpotent}, there is an integer $r_i$ such that for all $r\geqslant r_i$, $(F_{p+r}C^{p+q+1})_i=0$. Thus $(A_r^{pq})_i=(\Ker \partial)_i\cap (F_pC^{p+q})_i$ for all $r\geqslant r_i$. Therefore
\begin{align*}
&\mathop{\cap}\limits_{r}(A_r^{pq}+F_{p+1}C^{p+q})_i=\mathop{\cap}\limits_{r}((A_r^{pq})_i+(F_{p+1}C^{p+q})_i)\\
=&\mathop{\cap}\limits_{r=0}^{r_i}((A_r^{pq})_i+(F_{p+1}C^{p+q})_i=(A_{r_i}^{pq})_i+(F_{p+1}C^{p+q})_i \\
=&(\mathop{\cap}\limits_{r} A_r^{pq})_i + (F_{p+1}C^{p+q})_i=(\mathop{\cap}\limits_{r}A_r^{pq}+F_{p+1}C^{p+q})_i.
\end{align*}
Then $\mathop{\cap}\limits_{r}(A_r^{pq}+F_{p+1}C^{p+q})=\mathop{\cap}\limits_{r}A_r^{pq}+F_{p+1}C^{p+q}$, as $i$ is arbitrary. So
\begin{displaymath}
Z_\infty^{pq}=\dfrac{\cap_rA_r^{pq}+F_{p+1}C^{p+q}}{F_{p+1}C^{p+q}}.
\end{displaymath}
If $f\in \cap_rA_r^{pq}$, then for any $r$,
$$\partial(f)=f\circ d_{p+q+1}\in F_{p+r}\gHom_A(P_{p+q+1},A)$$ and
$$f\circ d_{p+q+1}(P_{p+q+1})\subseteq J^{p+r+p+q+1}.$$
By the arbitrariness of $r$ and Lemma \ref{locally nilpotent}, $f\circ d_{p+q+1}(P_{p+q+1})\subseteq\cap_i J^i = 0$. Thus $f\in\Ker \partial$ and $\cap_rA_r^{pq}=\Ker \partial\cap F_pC^{p+q}$.
It follows that $Z_\infty^{pq}=\dfrac{\Ker \partial\cap F_pC^{p+q} + F_{p+1}C^{p+q}}{F_{p+1}C^{p+q}}$ and $e_\infty^{pq}=E_\infty^{pq}$.
By definition, $F_{-n}C^n=C^n$, and so $F_{-n}H^n=\dfrac{\Ker d_{n+1}^*}{\im d_{n}^*}=\gExt^{n}_A(S,A)$ for any $n$,
If $n\neq d$,
then $F_iH^n/{F_{i+1}H^n} \cong E^{i, n-i}_{\infty}=0$.
Hence $$F_{-n}H^{n}=F_{-n+1}H^{n}=\cdots=F_iH^{n}=\dfrac{\Ker d_{n+1}^*\cap F_iC^{n}+\im d_{n}^*}{\im d_{n}^*}=\cdots$$
for all $i > -n$. It follows that, for any $i>-n$,
$$\Ker d_{n+1}^*=\Ker d_{n+1}^*\cap F_iC^{n}+\im d_{n}^*.$$
By degree-wise nilpotency again,
$\Ker d_{n+1}^*=\im d_{n}^*.$
Therefore, $H^n=0$, that is, $\gExt_A^n(S,A)=0$ for all $n\neq d$.
If $n=d$, then $F_{-d+1}H^d=F_iH^d$ for all $i>-d$ as only $E_\infty^{-d,2d}\neq 0$. It follows that, for all $i>-d$,
$$\Ker d^*_{d+1}\cap F_{-d+1}C^d+\im d_d^*=\Ker d^*_{d+1}\cap F_i C^d+\im d_d^*.$$
Similarly, by the degree-wise nilpotency,
$$\Ker d^*_{d+1}\cap F_{-d+1}\gHom_A(P_d,A)+\im d_d^*=\im d_d^*.$$
Therefore $F_{-d+1}H^d=F_{-d+2}H^d=\cdots=0.$
Hence
$$F_{-d}H^d/F_{-d+1}H^d=H^d=\gExt^d_A(S,A)\cong E^{-d,2d}_\infty\cong\gExt_{\Grj A}^d(S,\Grj A)_{-d}$$
as right $S$-modules. Therefore, as right $A$-modules,
$$\gExt_A^d(S,A)\cong \gExt_{\Grj A}^d(S,\Grj A)_{-d}.$$
Since $\gExt_{\Grj A}^d(S,\Grj A)_{-d}$ is an invertible $S$-bimodule,
it follows from Theorem \ref{equivalent definitions of AS regular} that $A$ is a generalized AS regular algebra.
\end{proof}
The following proposition studies the relations between the generalized AS regular property of a graded classical Koszul quiver algebra and its Yoneda Ext algebra, see \cite[Theorem 5.1]{MV1} or \cite[Proposition 4.1]{MV2}.
\begin{proposition}\label{Classical case about the relation of Yoneda and generalized AS regular}
Suppose that $A$ is a graded classical Koszul quiver algebra. Let $E(A)=\gExt_A^\bullet(S,S)$ be the Yoneda Ext algebra of $A$. Then for any graded simple $A$-module $N$ with $\pdim N=d$, the following are equivalent.
\begin{itemize}
\item[(1)] $N$ satisfies
$$\gExt_A^i(N,A)=\left\{
\begin{aligned}
&0,&i\neq d\\
&N', &i=d
\end{aligned}
\right.
$$
where $N'$ is a graded simple right $A$-module.
\item[(2)] $\gExt_A^\bullet(N,S)$ is an injective $E(A)$-module.
\end{itemize}
\end{proposition}
Now we are ready to give the main result in this section. Recall that an $\mathbb{N}$-graded algebra $A$ is called basic if the degree $0$ part of $A/J$ is a finite direct sum of $k$.
\begin{theorem}\label{Char-generalized Koszul AS-regular algebra}
Suppose that $A$ is a basic locally finite Koszul algebra of global dimension $d$. Let $E(A)=\gExt_A^\bullet(S,S)$ be the Yoneda Ext algebra of $A$. Then the following are equivalent.
\begin{itemize}
\item[(1)] $A$ is a generalized AS regular algebra of dimension $d$.
\item[(2)] $E(A)$ is a self-injective algebra.
\end{itemize}
\end{theorem}
\begin{proof}
It follows from Theorem \ref{A is generalized AS regular iff GrA is}, Proposition \ref{Classical case about the relation of Yoneda and generalized AS regular} and Theorem \ref{A sqk implies Gr A Koszul}.
\end{proof}
\section*{Acknowledgements} This research is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0713200) and the National Science Foundation of China (Grant No. 11771085).
\thebibliography{plain}
\bibitem[AF]{AF} F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, GTM 13, New York Springer-Verlag, 1973
\bibitem[Ber]{Ber} R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2001), 705-734.
\bibitem[BGS1]{BGS1}A. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473--527.
\bibitem[BGS2]{BGS2} A. Beilinson, V. Ginzburg, and V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1998), 317--350.
\bibitem[CPS]{CPS} E. Cline, B. Parshall, and L. Scott, Graded and ungraded Kazhdan-Lusztig theories. Algebraic groups and Lie groups, 105--125, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997.
\bibitem[Fr{\"o}]{Fr} R. Fr{\"o}berg, Koszul algebras, in: Advances in Commutative Ring Theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999, 337--350.
\bibitem[GM1]{GM1} E. L. Green, R. Martin{\'e}z-Villa, Koszul and Yoneda algebras, Canad. Math. Soc. Conf. Proc. 18 (1996), 247--298.
\bibitem[GM2]{GM2} E. L. Green, R. Martin{\'e}z-Villa, Koszul and Yoneda algebras \uppercase\expandafter{\romannumeral2}, Canad. Math. Soc. Conf. Proc. 24 (1998), 227--244.
\bibitem[GMMZ]{GMMZ} E. L. Green, E.N. Marcos, R. Martin{\'e}z-Villa, P. Zhang, D-Koszul algebras, J. Pure Appl. Algebra 193 (2004), 141--162.
\bibitem[GRS]{GRS} E. L. Green, Idun Reiten, and \O. Solberg, Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc. 159 (2002), no. 754.
\bibitem[HY]{HY} J.-W. He, Y. Ye, On the Yoneda-Ext algebras of semiperfect algebras, Alg. Colloq. 15 (2008), 207--222.
\bibitem[Lam]{L} T. Y. Lam, A first course in noncommutative rings, Second edition, Springer, 2001.
\bibitem[Li1]{Li1} L.-P. Li, A generalized Koszul theory and its application, Trans. Amer. Math. Soc. 366 (2014), 931--977.
\bibitem[Li2]{Li2} L.-P. Li, A generalized Koszul theory and its relation to the classical theory, J. Algebra 420 (2014), 217--241.
\bibitem[L{\"u}]{L3} J.-F. L{\"u}, On modules with d-Koszul-type submodules, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1015--1030.
\bibitem[LHL]{LHL} J.-F. L{\"u}, J.-W. He, D.-M. Lu, Piecewise-Koszul algebras, Sci. China Ser. A 50 (2007), 1795--1804.
\bibitem[LO]{LO}H.-S. Li, F. van Oystaeyen, Zariskian Filtrations, Kluwer Academic Publishers, K-Monographs in Mathematics, V. 2, Springer Science \& Business Media, B.V., Berlin, 1996.
\bibitem[Lof]{Lof} C. L$\ddot{o}$fwall, On the subalgebra generated by the one-dimensional elements in the Yoneda ext-algebra, Lecture Notes in Mathematics, V. 1183, Springer, 1986, pp. 291--
338.
\bibitem[Mac]{ML} S. MacLane, Homology, Grundlehren der mathematischen Wissenschaften, vol. 114, Springer Verlag, 1963.
\bibitem[Ma]{Ma} D. Madsen, On a common generalization of Koszul duality and tilting equivalence, Adv. Math. 227 (2011), 2327--2348.
\bibitem[Man]{Man} Y. I. Manin, Some remarks on Koszul algebras and quantum groups,
Ann. Inst. Fourier 37 (1987), 191--205.
\bibitem[MM]{MM} H. Minamoto, I. Mori, The structure of AS-Gorenstein algebras, Adv. Math. 226 (2011), 4061--4095.
\bibitem[MOS]{MOS} V. Mazorchuk, S. Ovsienko, C. Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), 1129--1172.
\bibitem[MS]{MS} R. Martin{\'e}z-Villa, \O. Solberg, Artin-Schelter regular algebras and categories, J. Pure Appl. Algebra 215 (2011), 546--565.
\bibitem[MV1]{MV1} R. Martin{\'e}z-Villa, Graded, Selfinjective, and Koszul algebras, J. Algebra 215 (1999), 34--72.
\bibitem[MV2]{MV2} R. Martin{\'e}z-Villa, Koszul algebras and the Gorenstein condition, Representations of algebras (S$\tilde{a}$o Paulo, 1999), 135--156, Lecture Notes in Pure and Appl. Math., 224, Dekker, New York, 2002.
\bibitem[MV3]{MV3} R. Martin{\'e}z-Villa, Introduction to Koszul algebras, Rev. Un. Mat. Argentina 48 (2007), 67--95.
\bibitem[MZ]{MZ} R. Martin{\'e}z-Villa, D. Zacharia, Approximations with modules having linear resolutions, J. Algebra 266 (2003), 671--697.
\bibitem[NO]{NO} C. N$\breve{\mathrm{a}}$st$\breve{\mathrm{a}}$sescu, F. van Oystaeyen, Graded ring theory, North-Holland Math. Library, 28. North-Holland Publishing Co., Amsterdam-New York, 1982.
\bibitem[Pri]{Pri} S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39--60.
\bibitem[RR1]{RR1} M. L. Reyes, D. Rogalski, Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular, Nagoya Math. J. 245 (2022), 100--153.
\bibitem[RR2]{RR2} M. L. Reyes, D. Rogalski, Growth of graded twisted Calabi-Yau algebras, J. Algebra 539 (2019), 201--259.
\bibitem[Sj]{Sj} G. Sj{\"o}din, A set of generators for $\Ext_R^\bullet(k, k)$, Math. Scand. 38 (1976), 1--12.
\bibitem[Sm]{Sm} S. P. Smith, Some finite-dimensional algebras related to elliptic curves, Representation theory of algebras and related topics (Mexico City, 1994), 315--348, CMS Conf. Proc., 19, Amer. Math. Soc., Providence, RI, 1996.
\bibitem[Wei]{W} C. A. Weibel, An introduction to homological algebra, Cambridge University Press, 1994.
\bibitem[Wo]{Wo} D. Woodcock, Cohen-Macaulay complexes and Koszul rings, J. London Math. Soc. (2) 57 (1998), 398--410.
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,969
|
package com.kolich.havalo.exceptions.authentication;
public final class BadCredentialsException extends AuthenticationException {
private static final long serialVersionUID = 8271081711610449505L;
public BadCredentialsException(String message, Exception cause) {
super(message, cause);
}
public BadCredentialsException(String message) {
super(message);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,246
|
If you are an existing customer, thank you! We hope you're pleased with Genevieve's fundraisers and products. Please take a moment to share your feedback and suggestions. This will help us serve you better in the future.
If you are not yet a Genevieve's customer, please let us know how we can help. Perhaps we offer a fundraising program that interests you. Or, maybe you have questions about implementing a fundraiser. Either way, we're here to assist you. Sign up below for information or to speak to a Genevieve's representative who can answer all of your questions. You'll be glad you did.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,518
|
"""Tests for lift_to_graph."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.python.eager import def_function
from tensorflow.python.eager import lift_to_graph
from tensorflow.python.eager import test
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import func_graph
from tensorflow.python.framework import ops as framework_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import resource_variable_ops
from tensorflow.python.util import compat
class LiftToGraphTest(test.TestCase):
def testCaptureOrdering(self):
v1 = resource_variable_ops.ResourceVariable(1.0)
v2 = resource_variable_ops.ResourceVariable(2.0)
v3 = resource_variable_ops.ResourceVariable(3.0)
@def_function.function
def fn():
return v1 + v2 + v3
concrete_fn = fn.get_concrete_function()
original_captures = concrete_fn.graph.internal_captures
outputs = concrete_fn.graph.outputs
for _ in range(100):
g = func_graph.FuncGraph('lifted')
lift_to_graph.lift_to_graph(
outputs, g, add_sources=True, handle_captures=True)
lifted_captures = g.internal_captures
self.assertLen(lifted_captures, 3)
for original, lifted in zip(original_captures, lifted_captures):
self.assertEqual(original.name, lifted.name)
def testClassAttrsRemoved(self):
"""Tests that _class attrs (from colocate_with()) are removed."""
@def_function.function
def fn():
two = constant_op.constant(2.0, name='two')
ten = constant_op.constant(10.0, name='ten')
twenty = math_ops.multiply(two, ten, name='twenty')
three = constant_op.constant(3.0, name='three')
with framework_ops.colocate_with(twenty):
thirty = math_ops.multiply(three, ten, name='thirty')
return ten, twenty, thirty
concrete_fn = fn.get_concrete_function()
self.assertItemsEqual( # Before lifting, 'fn' has colocation attrs.
concrete_fn.graph.get_operation_by_name('thirty').colocation_groups(),
[compat.as_bytes('loc:@twenty')])
thirty_out = concrete_fn.graph.outputs[2]
g = func_graph.FuncGraph('lifted')
lift_to_graph.lift_to_graph([thirty_out], g)
# After lifting, colocation attrs are gone.
ops = g.get_operations()
self.assertItemsEqual([op.name for op in ops],
['three', 'ten', 'thirty', # Lifted from `fn` body.
thirty_out.op.name]) # Wrapper for output.
for op in ops:
with self.assertRaises(ValueError):
class_attr = op.get_attr('_class') # Expected not to exist.
print('Unexpected class_attr', class_attr, 'on', op.name)
self.assertItemsEqual(op.colocation_groups(), # Expect default self-ref.
[compat.as_bytes('loc:@%s' % op.name)])
if __name__ == '__main__':
test.main()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 50
|
using System;
using System.Data;
using System.Configuration;
using System.Web;
using System.Web.Security;
using System.Web.UI;
using System.Web.UI.WebControls;
using System.Web.UI.WebControls.WebParts;
using System.Web.UI.HtmlControls;
using WebsitePanel.Providers.Mail;
namespace WebsitePanel.Portal
{
/// <summary>
/// Summary description for IMailEditForwardingControl
/// </summary>
public interface IMailEditForwardingControl
{
void BindItem(MailAlias item);
void SaveItem(MailAlias item);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,768
|
{"url":"https:\/\/scicomp.stackexchange.com\/questions\/30009\/level-scheduling-of-triangular-sparse-matrices","text":"# Level scheduling of triangular sparse matrices\n\nAssume one has a triangular sparse matrix and want to solve $Lx=b$ where $b$ and $L$ are known. This can be done easily by using forward substitution when $L$ is a lower triangular matrix. Forward substitution is highly sequential and hard to implement in parallel. I have read some articles about level scheduling where $L$ is reordered so that different levels appear which can be solved in parallel. A good reference seems to be the book of Saad where also the forward substitution with level scheduling is explained.\n\nBut I do not really get how the level scheduling is performed and the vector $q$ is filled. Could someone please provide an example or something?\n\n\u2022 Even if parallelized, the problem as stated is intrinsically BLAS2 and your opportunities for speedup might be limited (you're probably already memory-bound, adding more processors won't change this). That said, solving by multiple right hand sides is BLAS3, and does require some thought\/strategy to parallelize, so I do think the question is worth asking and answering. \u2013\u00a0rchilton1980 Aug 8 '18 at 16:06\n\u2022 Memory bound code still needs to be parallel to realize full system bandwidth. \u2013\u00a0Reid.Atcheson Aug 8 '18 at 16:12\n\u2022 On many contemporary x86_64 desktop systems, one or two (of 8 or even 16) cores can easily saturate the memory bandwidth. There's not a lot of benefit to parallelizing operations like matrix-vector multiplication on such systems. \u2013\u00a0Brian Borchers Aug 10 '18 at 1:29\n\u2022 \"the book of Saad\" is a particularly bad reference\/citation... \u2013\u00a0Jakub Klinkovsk\u00fd Sep 25 '18 at 19:28","date":"2020-02-19 11:37:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5758044719696045, \"perplexity\": 816.6278827125201}, \"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-2020-10\/segments\/1581875144111.17\/warc\/CC-MAIN-20200219092153-20200219122153-00387.warc.gz\"}"}
| null | null |
{-# LANGUAGE TupleSections, DeriveDataTypeable, DeriveFoldable, ViewPatterns, DeriveFunctor, DeriveTraversable, TemplateHaskell, GeneralizedNewtypeDeriving #-}
module Music.Score.Export2.StandardNotation where
import Control.Applicative
import Control.Lens (over, preview, set, to,
under, view, _head, at)
import Control.Lens.Operators
import Control.Lens.TH (makeLenses)
import Control.Monad.Except
import Control.Monad.Plus
import Control.Monad.Writer
import Data.AffineSpace hiding (Sum)
import Data.Colour (Colour)
import Data.Colour.Names as Color
import Data.Foldable (Foldable)
import Data.Functor.Identity (Identity)
import Data.Map (Map)
import qualified Data.Char
import qualified Data.List
import qualified Data.Map
import qualified Data.List.Split
import qualified Data.Maybe
import qualified Data.Music.Lilypond as Lilypond
import qualified Data.Music.MusicXml.Simple as MusicXml
import Data.Semigroup
import Data.Traversable (Traversable)
import qualified Data.Traversable
import Data.VectorSpace hiding (Sum)
import qualified Music.Articulation
import qualified Music.Dynamics
import Music.Dynamics.Literal (DynamicsL(..), fromDynamics)
import Music.Parts (Group (..))
import qualified Music.Parts
import qualified Music.Pitch
import qualified Music.Pitch
import qualified Music.Pitch.Literal
import Music.Score (MVoice)
import qualified Music.Score
import Music.Score.Articulation (ArticulationT)
import Music.Score.Dynamics (DynamicT)
import qualified Music.Score.Export.ArticulationNotation
import qualified Music.Score.Export.ArticulationNotation as AN
import qualified Music.Score.Export.DynamicNotation
import qualified Music.Score.Export.DynamicNotation as DN
import qualified Music.Score.Internal.Export
import Music.Score.Internal.Quantize (Rhythm (..), dotMod,
quantize, rewrite)
import qualified Music.Score.Internal.Util
import qualified Music.Score.Internal.Instances ()
import Music.Score.Internal.Data (getData)
import qualified Music.Score.Meta
import qualified Music.Score.Meta.Attribution
import qualified Music.Score.Meta.Title
import qualified Music.Score.Meta.Key
import qualified Music.Score.Meta.RehearsalMark
import qualified Music.Score.Meta.Tempo
import qualified Music.Score.Meta.Time
import Music.Score.Part (PartT)
import Music.Score.Pitch ()
import Music.Score.Ties (TieT (..))
import Music.Score.Tremolo (TremoloT, runTremoloT)
import Music.Time
import Music.Time.Meta (meta)
import qualified Text.Pretty as Pretty
import qualified System.Process --DEBUG
{-
type StandardNote =
PartT
Part
(ColorT
(TextT
(TremoloT
(HarmonicT
(SlideT
(ArticulationT Articulation (DynamicT Dynamics [TieT Pitch])))))))
-}
-- TODO
instance Show Music.Score.Meta.Key.KeySignature where show = const "TEMPkeysig"
-- Annotated tree
data LabelTree b a = Branch b [LabelTree b a] | Leaf a
deriving (Functor, Foldable, Traversable, Eq, Ord, Show)
foldLabelTree :: (a -> c) -> (b -> [c] -> c) -> LabelTree b a -> c
foldLabelTree f g (Leaf x) = f x
foldLabelTree f g (Branch b xs) = g b (fmap (foldLabelTree f g) xs)
-- data AltList a b = Nil | Cons a (AltList b a)
-- deriving (Eq, Ord, Show)
-- fromList :: [a] -> AltList () a
-- fromList [] = Nil
-- fromList (x:xs) = Cons () (Cons x (fromList xs))
--
-- toList Nil = []
-- toList (Cons x Nil) = x : toList xs
-- toList (Cons () ) = x : toList xs
type BarNumber = Int
type TimeSignature = Music.Score.Meta.Time.TimeSignature
type KeySignature = Music.Score.Meta.Key.KeySignature
type RehearsalMark = Music.Score.Meta.RehearsalMark.RehearsalMark
type TempoMark = Music.Score.Meta.Tempo.Tempo
-- TODO w/wo connecting barlines
data BracketType = NoBracket | Bracket | Brace | Subbracket deriving (Eq, Ord, Show)
type SpecialBarline = () -- TODO
-- type BarLines = (Maybe SpecialBarline, Maybe SpecialBarline) -- (prev,next) biased to next
-- TODO lyrics
data SystemBar = SystemBar {
_barNumbers::Maybe BarNumber,
_timeSignature::Maybe TimeSignature,
_keySignature::Maybe KeySignature,
_rehearsalMark::Maybe RehearsalMark,
_tempoMark::Maybe TempoMark
-- ,_barLines::BarLines -- Tricky because of ambiguity. Use balanced pair or an alt-list in SystemStaff.
} deriving (Eq,Ord,Show)
instance Monoid SystemBar where
mempty = SystemBar Nothing Nothing Nothing Nothing Nothing
type SystemStaff = [SystemBar]
type InstrumentShortName = String
type InstrumentFullName = String
type Transposition = Music.Pitch.Interval
type SibeliusFriendlyName = String
type SmallOrLarge = Any -- def False
type ScoreOrder = Sum Double -- def 0
data StaffInfo = StaffInfo {
-- TODO instrument part no. (I, II.1 etc)
_instrumentShortName::InstrumentShortName,
_instrumentFullName::InstrumentFullName,
_sibeliusFriendlyName::SibeliusFriendlyName,
-- TODO allow for clef/instrument changes within staff
_instrumentDefaultClef::Music.Pitch.Clef,
_transposition::Transposition,
_smallOrLarge::SmallOrLarge,
_scoreOrder::ScoreOrder
}
deriving (Eq,Ord,Show)
instance Monoid StaffInfo where
mempty = StaffInfo mempty mempty mempty Music.Pitch.trebleClef mempty mempty mempty
type Pitch = Music.Pitch.Pitch
data ArpeggioNotation = Arpeggio | UpArpeggio | DownArpeggio
deriving (Eq,Ord,Show)
-- As written, i.e. 1/16-notes twice, can be represented as 1/8 note with 1 beams
-- TODO No way to represent 2-pitch tremolo
data TremoloNotation = BeamedTremolo Int | UnmeasuredTremolo
deriving (Eq,Ord,Show)
-- type UpDown = Up | Down
-- data CrossStaff = NoCrossStaff | NextNoteCrossStaff UpDown | PreviousNoteCrossStaff UpDown
-- data BreathNotation = Fermata | PauseAfter | CaesuraAfter
data BreathNotation = Comma | Caesura | CaesuraWithFermata
deriving (Eq,Ord,Show)
type ArticulationNotation = Music.Score.Export.ArticulationNotation.ArticulationNotation
type DynamicNotation = Music.Score.Export.DynamicNotation.DynamicNotation
type HarmonicNotation = (Any, Sum Int) -- (artificial?, partial number)
type SlideNotation = ((Any,Any),(Any,Any)) -- (endGliss?,endSlide?),(beginGliss?,beginSlide?)
type Ties = (Any,Any) -- (endTie?,beginTie?)
-- TODO appogiatura/acciatura
-- TODO beaming
-- Rests, single-notes and chords (most attributes are not shown for rests)
data Chord = Chord {
_pitches::[Pitch],
_arpeggioNotation::Maybe ArpeggioNotation,
_tremoloNotation::Maybe TremoloNotation,
_breathNotation::Maybe BreathNotation,
_articulationNotation::Maybe ArticulationNotation, -- I'd like to put this in a separate layer, but neither Lily nor MusicXML thinks this way
_dynamicNotation::Maybe DynamicNotation,
_chordColor::Maybe (Colour Double),
_chordText::[String],
_harmonicNotation::HarmonicNotation,
_slideNotation::SlideNotation,
_ties::Ties
}
deriving (Eq, Show)
instance Monoid Chord where
mempty = Chord [] Nothing Nothing Nothing Nothing Nothing mempty mempty mempty mempty mempty
type PitchLayer = Rhythm Chord
-- type DynamicLayer = Rhythm (Maybe DynamicNotation)
data Bar = Bar {_pitchLayers::[PitchLayer] {-, _dynamicLayer::DynamicLayer-}}
deriving (Eq, Show)
data Staff = Staff {_staffInfo::StaffInfo,_bars::[Bar]}
deriving (Eq, Show)
type Title = String
type Annotations = [(Span, String)]
type Attribution = Map String String -- composer, lyricist etc
data MovementInfo = MovementInfo {
_movementTitle::Title,
_movementAnnotations::Annotations,
_movementAttribution::Attribution
}
deriving (Eq, Show)
instance Monoid MovementInfo where
mempty = MovementInfo mempty mempty mempty
data Movement = Movement {
_movementInfo::MovementInfo,
_systemStaff::SystemStaff,
_staves::LabelTree BracketType Staff -- Don't allow names for staff groups, only staves
}
deriving (Eq, Show)
data WorkInfo = WorkInfo { _title::Title, _annotations::Annotations, _attribution::Attribution}
deriving (Eq, Show)
instance Monoid WorkInfo where
mempty = WorkInfo mempty mempty mempty
data Work = Work { _workInfo::WorkInfo, _movements::[Movement] }
deriving (Show)
makeLenses ''SystemBar
makeLenses ''StaffInfo
makeLenses ''Chord
makeLenses ''Bar
makeLenses ''Staff
makeLenses ''MovementInfo
makeLenses ''Movement
makeLenses ''WorkInfo
makeLenses ''Work
----------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------
-- Log and failure monad
newtype E a = E { runE :: WriterT String (ExceptT String Identity) a }
deriving (Functor, Applicative, Monad, Alternative, MonadPlus, MonadError String, MonadWriter String)
runENoLog :: E b -> Either String b
runENoLog = fmap fst . runExcept . runWriterT . runE
toLy :: Work -> E (String, Lilypond.Music)
toLy w = do
-- TODO assumes one movement
r <- case w^?movements._head of
Nothing -> throwError "StandardNotation: Expected a one-movement piece"
Just x -> return x
let headerTempl = Data.Map.fromList [
("title", (r^.movementInfo.movementTitle)),
("composer", Data.Maybe.fromMaybe "" $ r^.movementInfo.movementAttribution.at "composer")
]
let header = getData "ly_big_score.ily" `expandTemplate` headerTempl
m <- toLyMusic $ r
return (header, m)
type Template = String
-- |
-- One-function templating system.
--
-- >>> expand "me : $(name)" (Map.fromList [("name","Hans")])
-- "me : Hans"
--
expandTemplate :: Template -> Map String String -> String
expandTemplate t vs = (composed $ fmap (expander vs) $ Data.Map.keys $ vs) t
where
expander vs k = replace ("$(" ++ k ++ ")") (Data.Maybe.fromJust $ Data.Map.lookup k vs)
composed = foldr (.) id
replace old new = Data.List.intercalate new . Data.List.Split.splitOn old
toCamel (x:xs) = Data.Char.toUpper x : xs
toLyMusic :: Movement -> E Lilypond.Music
toLyMusic m = do
-- We will copy system-staff info to each bar (time sigs, key sigs and so on, which seems to be what Lilypond expects),
-- so the system staff is included in the rendering of each staff
renderedStaves <- Data.Traversable.mapM (toLyStaff $ m^.systemStaff) (m^.staves)
-- Now we still have (LabelTree BracketType), which is converted to a parallel music expression, using \StaffGroup etc
toLyStaffGroup renderedStaves
toLyStaff :: SystemStaff -> Staff -> E Lilypond.Music
toLyStaff sysBars staff = id
<$> Lilypond.New "Staff" Nothing
<$> Lilypond.Sequential
<$> addPartName (staff^.staffInfo.instrumentFullName)
<$> addClef (toLyClef $ staff^.staffInfo.instrumentDefaultClef)
-- TODO Currently score is always in C with no oct-transp.
-- To get a transposing score, add \transpose <written> <sounding>
<$> (sequence $ zipWith toLyBar sysBars (staff^.bars))
toLyClef c
| c == Music.Pitch.trebleClef = Lilypond.Treble
| c == Music.Pitch.altoClef = Lilypond.Alto
| c == Music.Pitch.tenorClef = Lilypond.Tenor
| c == Music.Pitch.bassClef = Lilypond.Bass
| otherwise = Lilypond.Treble
addClef c xs = Lilypond.Clef c : xs
addPartName partName xs = longName : shortName : xs
where
longName = Lilypond.Set "Staff.instrumentName" (Lilypond.toValue partName)
shortName = Lilypond.Set "Staff.shortInstrumentName" (Lilypond.toValue partName)
toLyBar :: SystemBar -> Bar -> E Lilypond.Music
toLyBar sysBar bar = do
let layers = bar^.pitchLayers
-- TODO emit \new Voice for eachlayer
sim <$> sysStuff <$> mapM (toLyLayer) layers
where
-- System information need not be replicated in all layers
-- TODO other system stuff (reh marks, special barlines etc)
sysStuff [] = []
sysStuff (x:xs) = (addTimeSignature (sysBar^.timeSignature) x:xs)
sim [x] = x
sim xs = Lilypond.Simultaneous False xs
addTimeSignature :: Maybe Music.Score.Meta.Time.TimeSignature -> Lilypond.Music -> Lilypond.Music
addTimeSignature timeSignature x = (setTimeSignature `ifJust` timeSignature) x
where
ifJust = maybe id
setTimeSignature (Music.Score.getTimeSignature -> (ms, n)) x = Lilypond.Sequential [Lilypond.Time (sum ms) n, x]
toLyLayer :: Rhythm Chord -> E Lilypond.Music
toLyLayer (Beat d x) = toLyChord d x
toLyLayer (Dotted n (Beat d x)) = toLyChord (dotMod n * d) x
toLyLayer (Dotted n _) = error "FIXME"
toLyLayer (Group rs) = Lilypond.Sequential <$> mapM toLyLayer rs
toLyLayer (Tuplet m r) = Lilypond.Times (realToFrac m) <$> (toLyLayer r)
where
(a,b) = bimap fromIntegral fromIntegral $ unRatio $ realToFrac m
unRatio = Music.Score.Internal.Util.unRatio
bimap = Music.Score.bimap
{-
TODO _arpeggioNotation::Maybe ArpeggioNotation,
TODO _tremoloNotation::Maybe TremoloNotation,
TODO _breathNotation::Maybe BreathNotation,
-}
toLyChord :: Duration -> Chord -> E Lilypond.Music
toLyChord d chord = id
<$> notateTies (chord^.ties)
<$> notateGliss (chord^.slideNotation)
<$> notateHarmonic (chord^.harmonicNotation)
<$> notateText (chord^.chordText)
<$> notateColor (chord^.chordColor)
<$> maybe id notateDynamic (chord^.dynamicNotation)
<$> maybe id notateArticulation (chord^.articulationNotation)
<$> notatePitches d (chord^.pitches)
where
notatePitches :: Duration -> [Pitch] -> E Lilypond.Music
notatePitches d pitches = case pitches of
[] -> return $ Lilypond.Rest (Just (realToFrac d)) []
[x] -> return $ Lilypond.Note (toLyNote x) (Just (realToFrac d)) []
xs -> return $ Lilypond.Chord (fmap ((,[]) . toLyNote) xs) (Just (realToFrac d)) []
toLyNote :: Pitch -> Lilypond.Note
toLyNote p = (`Lilypond.NotePitch` Nothing) $ Lilypond.Pitch (
toEnum (fromEnum $ Music.Pitch.name p),
-- FIXME catch if (abs accidental)>2 (or simply normalize)
fromIntegral (Music.Pitch.accidental p),
-- Lilypond expects SPN, so middle c is octave 4
fromIntegral $ Music.Pitch.octaves (p.-.Music.Score.octavesDown (4+1) Music.Pitch.Literal.c)
)
-- notateDynamic :: Maybe DynamicNotation -> Lilypond.Music -> Lilypond.Music
-- notateArticulation :: Maybe ArticulationNotation -> Lilypond.Music -> Lilypond.Music
-- notateColor :: Maybe (Colour Double) -> Lilypond.Music -> Lilypond.Music
-- notateTremolo :: Maybe Int -> Duration -> (Lilypond.Music -> Lilypond.Music, Duration)
-- notateText :: [String] -> Lilypond.Music -> Lilypond.Music
-- notateHarmonic :: (Any, Sum Int) -> Lilypond.Music -> Lilypond.Music
-- notateGliss :: ((Any, Any), (Any, Any)) -> Lilypond.Music -> Lilypond.Music
-- (endGliss,endSlide),(beginGliss,beginSlide)
-- notateTies :: (Any, Any) -> Lilypond.Music -> Lilypond.Music
-- (endTie,beginTie)
notateDynamic :: DynamicNotation -> Lilypond.Music -> Lilypond.Music
notateDynamic (DN.DynamicNotation (crescDims, level))
= rcomposed (fmap notateCrescDim crescDims)
. notateLevel level
notateCrescDim crescDims = case crescDims of
DN.NoCrescDim -> id
DN.BeginCresc -> Lilypond.beginCresc
DN.EndCresc -> Lilypond.endCresc
DN.BeginDim -> Lilypond.beginDim
DN.EndDim -> Lilypond.endDim
-- TODO these literals are not so nice...
notateLevel showLevel = case showLevel of
Nothing -> id
Just lvl -> Lilypond.addDynamics (fromDynamics (DynamicsL (Just (fixLevel . realToFrac $ lvl), Nothing)))
fixLevel :: Double -> Double
fixLevel x = fromIntegral (round (x - 0.5)) + 0.5
notateArticulation :: ArticulationNotation -> Lilypond.Music -> Lilypond.Music
notateArticulation (AN.ArticulationNotation (slurs, marks))
= rcomposed (fmap notateMark marks)
. rcomposed (fmap notateSlur slurs)
notateMark mark = case mark of
AN.NoMark -> id
AN.Staccato -> Lilypond.addStaccato
AN.MoltoStaccato -> Lilypond.addStaccatissimo
AN.Marcato -> Lilypond.addMarcato
AN.Accent -> Lilypond.addAccent
AN.Tenuto -> Lilypond.addTenuto
notateSlur slurs = case slurs of
AN.NoSlur -> id
AN.BeginSlur -> Lilypond.beginSlur
AN.EndSlur -> Lilypond.endSlur
-- TODO This syntax might change in future Lilypond versions
-- TODO handle any color
notateColor :: Maybe (Colour Double) -> Lilypond.Music -> Lilypond.Music
notateColor Nothing = id
notateColor (Just color) = \x -> Lilypond.Sequential [
Lilypond.Override "NoteHead#' color"
(Lilypond.toLiteralValue $ "#" ++ colorName color),
x,
Lilypond.Revert "NoteHead#' color"
]
colorName c
| c == Color.black = "black"
| c == Color.red = "red"
| c == Color.blue = "blue"
| otherwise = error "Lilypond backend: Unkown color"
-- Note: must use returned duration
notateTremolo :: Maybe Int -> Duration -> (Lilypond.Music -> Lilypond.Music, Duration)
notateTremolo Nothing d = (id, d)
notateTremolo (Just 0) d = (id, d)
notateTremolo (Just n) d = let
scale = 2^n
newDur = (d `min` (1/4)) / scale
repeats = d / newDur
in (Lilypond.Tremolo (round repeats), newDur)
notateText :: [String] -> Lilypond.Music -> Lilypond.Music
notateText texts = composed (fmap Lilypond.addText texts)
notateHarmonic :: (Any, Sum Int) -> Lilypond.Music -> Lilypond.Music
notateHarmonic (Any isNat, Sum n) = case (isNat, n) of
(_, 0) -> id
(True, n) -> notateNatural n
(False, n) -> notateArtificial n
where
notateNatural n = Lilypond.addFlageolet -- addOpen?
notateArtificial n = id -- TODO
notateGliss :: ((Any, Any), (Any, Any)) -> Lilypond.Music -> Lilypond.Music
notateGliss ((Any eg, Any es),(Any bg, Any bs))
| bg = Lilypond.beginGlissando
| bs = Lilypond.beginGlissando
| otherwise = id
notateTies :: (Any, Any) -> Lilypond.Music -> Lilypond.Music
notateTies (Any ta, Any tb)
| ta && tb = Lilypond.beginTie
| tb = Lilypond.beginTie
| ta = id
| otherwise = id
-- Use rcomposed as notateDynamic returns "mark" order, not application order
composed = Music.Score.Internal.Util.composed
rcomposed = Music.Score.Internal.Util.composed . reverse
toLyStaffGroup :: LabelTree BracketType (Lilypond.Music) -> E Lilypond.Music
toLyStaffGroup = return . foldLabelTree id g
where
-- Note: PianoStaff is handled in toLyStaffGroup
-- Note: Nothing for name (we dump everything inside staves, so no need to identify them)
g NoBracket ms = k ms
g Bracket ms = Lilypond.New "StaffGroup" Nothing $ k ms
g Subbracket ms = Lilypond.New "GrandStaff" Nothing $ k ms
g Brace ms = Lilypond.New "GrandStaff" Nothing $ k ms
-- Why False? No separation mark is necessary as the wrapped music is all in separate staves
k = Lilypond.Simultaneous False
----------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------
toXml :: Work -> MusicXml.Score
toXml = undefined
----------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------
type Asp1 = (PartT Music.Parts.Part
(ArticulationT Music.Articulation.Articulation
(DynamicT Music.Dynamics.Dynamics Pitch)))
type Asp1B = (PartT Music.Parts.Part
(ArticulationT AN.ArticulationNotation
(DynamicT DN.DynamicNotation Pitch)))
-- We require all notes in a chords to have the same kind of ties
type Asp2 = TieT (PartT Music.Parts.Part
(ArticulationT Music.Articulation.Articulation
(DynamicT Music.Dynamics.Dynamics
[Pitch])))
type Asp3 = TieT (PartT Music.Parts.Part
(ArticulationT AN.ArticulationNotation
(DynamicT DN.DynamicNotation
[Pitch])))
type Asp = Score Asp1
asp1ToAsp2 :: Asp1 -> Asp2
asp1ToAsp2 = pureTieT . (fmap.fmap.fmap) (:[])
{-
Note:
Both addDynCon and addArtCon should *not* be used on scores for the time being, due to the faulty
(HasPhrases Score) instance. See comment in Music.Score.Phrases.
We use the MVoice instance here, so this is safe.
-}
asp2ToAsp3 :: Voice (Maybe Asp2) -> Voice (Maybe Asp3)
asp2ToAsp3 = id
. (DN.removeCloseDynMarks . over Music.Score.dynamics DN.notateDynamic . Music.Score.addDynCon)
. (over Music.Score.articulations AN.notateArticulation . Music.Score.addArtCon)
-- . fmap2 (over Music.Score.articulation (const ()))
aspectsToChord :: Maybe Asp3 -> Chord
aspectsToChord Nothing = mempty
aspectsToChord (Just asp) = id
$ ties .~ (Any endTie,Any beginTie)
$ dynamicNotation .~ (Just $ asp^.(Music.Score.dynamic))
$ articulationNotation .~ (Just $ asp^.(Music.Score.articulation))
$ pitches .~ (asp^..(Music.Score.pitches)) $ mempty
where
(endTie,beginTie) = Music.Score.isTieEndBeginning asp
aspectsToBar :: Rhythm (Maybe Asp3) -> Bar
aspectsToBar rh = Bar [layer1] -- TODO more layers (see below)
where
layer1 = fmap aspectsToChord rh
aspectsToStaff :: (Music.Parts.Part, [Rhythm (Maybe Asp3)]) -> Staff
aspectsToStaff (part,bars) = Staff info (fmap aspectsToBar bars)
where
info = id
$ transposition .~ (part^.(Music.Parts._instrument).(to Music.Parts.transposition))
$ instrumentDefaultClef .~ Data.Maybe.fromMaybe (error "FIXME") (part^.(Music.Parts._instrument).(to Music.Parts.standardClef))
$ instrumentShortName .~ Data.Maybe.fromMaybe "" (part^.(Music.Parts._instrument).(to Music.Parts.shortName))
$ instrumentFullName .~ (Data.List.intercalate " " $ Data.Maybe.catMaybes [soloStr, nameStr, subpartStr])
$ mempty
where
soloStr = if (part^.(Music.Parts._solo)) == Music.Parts.Solo then Just "Solo" else Nothing
nameStr = (part^.(Music.Parts._instrument).(to Music.Parts.fullName))
subpartStr = Just $ show (part^.(Music.Parts._subpart))
toLayer :: Music.Parts.Part -> Score a -> E (MVoice a)
toLayer p = maybe (throwError $ "Overlapping events in part: " ++ show p) return . preview Music.Score.singleMVoice
fromAspects :: Asp -> E Work
fromAspects sc = do
-- Part extraction
let postPartExtract = Music.Score.extractPartsWithInfo normScore
-- postPartExtract :: [(Music.Parts.Part,Score Asp1)]
-- Change aspect type as we need Semigroup to compose all simultanous notes
-- Merge simultanous notes into chords, to simplify voice-separation
let postChordMerge = fmap2 (simultaneous . fmap asp1ToAsp2) postPartExtract
-- postChordMerge :: [(Music.Parts.Part,Score Asp2)]
-- Separate voices (called "layers" to avoid confusion)
-- This is currently a trivial algorithm that assumes overlapping notes are in different parts
postVoiceSeparation <- Data.Traversable.mapM (\a@(p,_) -> Data.Traversable.mapM (toLayer p) a) $ postChordMerge
-- Rewrite dynamics and articulation to be context-sensitive
-- This changes the aspect type again
postContextSensitiveNotationRewrite <- return $ fmap2 asp2ToAsp3 $ postVoiceSeparation
-- postContextSensitiveNotationRewrite :: [(Music.Parts.Part,Voice (Maybe Asp3))]
-- Split each part into bars, splitting notes and adding ties when necessary
-- Resulting list is list of bars, there is no layering (yet)
let postTieSplit = fmap2 (Music.Score.splitTiesAt barDurations) $ postContextSensitiveNotationRewrite
-- postTieSplit :: [(Music.Parts.Part,[Voice (Maybe Asp3)])]
-- For each bar, quantize all layers. This is where tuplets/note values are generated.
postQuantize <- Data.Traversable.mapM (Data.Traversable.mapM (Data.Traversable.mapM quantizeBar)) postTieSplit
-- postQuantize :: [(Music.Parts.Part,[Rhythm (Maybe Asp3)])]
-- TODO all steps above that start with fmap or mapM can be factored out (functor law)
-- Group staves, generating brackets and braces
let postStaffGrouping = generateStaffGrouping postQuantize
-- postStaffGrouping :: LabelTree (BracketType) (Music.Parts.Part, [Rhythm (Maybe Asp3)])
return $ Work mempty [Movement info systemStaff (fmap aspectsToStaff postStaffGrouping)]
where
info = id
$ movementTitle .~ (
Data.Maybe.fromMaybe "" $ flip Music.Score.Meta.Title.getTitleAt 0 $ Music.Score.Meta.metaAtStart sc
)
$ (movementAttribution.at "composer") .~ (
flip Music.Score.Meta.Attribution.getAttribution "composer" $ Music.Score.Meta.metaAtStart sc
) $ mempty
systemStaff :: SystemStaff
systemStaff = fmap (\ts -> timeSignature .~ ts $ mempty) timeSignatureMarks
(timeSignatureMarks, barDurations) = extractTimeSignatures normScore
normScore = normalizeScore sc -- TODO not necessarliy set to 0...
-- TODO log rewriting etc
quantizeBar :: Music.Score.Tiable a => Voice (Maybe a) -> E (Rhythm (Maybe a))
quantizeBar = fmap rewrite . quantize' . view Music.Score.pairs
where
quantize' x = case quantize x of
Left e -> throwError $ "Quantization failed: " ++ e
Right x -> return x
pureTieT :: a -> TieT a
pureTieT = pure
extractTimeSignatures
:: Score a -> ([Maybe Music.Score.Meta.Time.TimeSignature], [Duration])
extractTimeSignatures = Music.Score.Internal.Export.extractTimeSignatures
generateStaffGrouping :: [(Music.Parts.Part, a)] -> LabelTree (BracketType) (Music.Parts.Part, a)
generateStaffGrouping = groupToLabelTree . partDefault
partDefault :: [(Music.Parts.Part, a)] -> Music.Parts.Group (Music.Parts.Part, a)
partDefault xs = Music.Parts.groupDefault $ fmap (\(p,x) -> (p^.(Music.Parts._instrument),(p,x))) xs
groupToLabelTree :: Group a -> LabelTree (BracketType) a
groupToLabelTree (Single (_,a)) = Leaf a
groupToLabelTree (Many gt _ xs) = (Branch (k gt) (fmap groupToLabelTree xs))
where
k Music.Parts.Bracket = Bracket
k Music.Parts.Invisible = NoBracket
-- k Music.Parts.Subbracket = Just SubBracket
k Music.Parts.PianoStaff = Brace
k Music.Parts.GrandStaff = Brace
-- Util
fmap2 = fmap.fmap
-- pcatL :: [Lilypond.Music] -> Lilypond.Music
-- pcatL = pcatL' False
--
-- pcatL' :: Bool -> [Lilypond.Music] -> Lilypond.Music
-- pcatL' p = foldr Lilypond.simultaneous (Lilypond.Simultaneous p [])
--
-- scatL :: [Lilypond.Music] -> Lilypond.Music
-- scatL = foldr Lilypond.sequential (Lilypond.Sequential [])
--
-- spellL :: Integer -> Lilypond.Note
-- spellL a = Lilypond.NotePitch (spellL' a) Nothing
--
-- spellL' :: Integer -> Lilypond.Pitch
-- spellL' p = Lilypond.Pitch (
-- toEnum $ fromIntegral pc,
-- fromIntegral alt,
-- fromIntegral oct
-- )
-- where (pc,alt,oct) = Music.Score.Internal.Export.spellPitch (p + 72)
-- Test
test = runENoLog $ toLy $
Work mempty [Movement mempty [mempty] (
Branch Bracket [
Leaf (Staff mempty [Bar [Beat 1 mempty]]),
Leaf (Staff mempty [Bar [Beat 1 mempty]])
])]
test2 x = runENoLog $ toLy =<< fromAspects x
test3 x = do
let r = test2 x
case r of
Left e -> fail ("test3: "++e)
Right (h,ly) -> do
let ly2 = h ++ show (Pretty.pretty ly)
-- putStrLn ly2
writeFile "t.ly" $ ly2
void $ System.Process.system "lilypond t.ly"
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,148
|
Q: Docker isolated network receives packets from outside I set up a docker bridge network (on Linux) for the purpose of testing how network traffic of individual applications (containers) looks like. Therefore, a key requirement for the network is that it is completely isolated from traffic that originates from other applications or devices.
A simple example I created with compose is a ping-container that sends ICMP-packets to another one, with a third container running tcpdump to collect the traffic:
version: '3'
services:
ping:
image: 'detlearsom/ping'
environment:
- HOSTNAME=blank
- TIMEOUT=2
sysctls:
- net.ipv6.conf.all.disable_ipv6=1
networks:
- capture
blank:
image: 'alpine'
command: sleep 300
sysctls:
- net.ipv6.conf.all.disable_ipv6=1
networks:
- capture
tcpdump:
image: 'detlearsom/tcpdump'
volumes:
- '$PWD/data:/data'
sysctls:
- net.ipv6.conf.all.disable_ipv6=1
network_mode: 'service:ping'
command: -v -w "/data/dump-011-ping2-${CAPTURETIME}.pcap"
networks:
capture:
driver: "bridge"
internal: true
Note that I have set the network to internal, and I have also disabled IPV6. However, when I run it and collect the traffic, additional to the expected ICMP packets I get IPV6 packets:
10:42:40.863619 IP6 fe80::42:2aff:fe42:e303 > ip6-allrouters: ICMP6, router solicitation, length 16
10:42:43.135167 IP6 fe80::e437:76ff:fe9e:36b4.mdns > ff02::fb.mdns: 0 [2q] PTR (QM)? _ipps._tcp.local. PTR (QM)? _ipp._tcp.local.
10:42:37.875646 IP6 fe80::e437:76ff:fe9e:36b4.mdns > ff02::fb.mdns: 0*- [0q] 2/0/0 (Cache flush) PTR he...F.local., (Cache flush) AAAA fe80::e437:76ff:fe9e:36b4 (161)
What is even stranger is that I receive UDP packets from port 57621:
10:42:51.868199 IP 172.25.0.1.57621 > 172.25.255.255.57621: UDP, length 44
This port corresponds to spotify traffic and most likely originates from my spotify application that is running on the host machine.
My question: Why do I see this traffic in my network that is supposed to be isolated?
For anyone interested, here is the network configuration:
[
{
"Name": "capture-011-ping2_capture",
"Id": "35512f852332351a9f677f75b522982aa6bd288e813a31a3c36477baa005c0fd",
"Created": "2018-08-07T10:42:31.610178964+01:00",
"Scope": "local",
"Driver": "bridge",
"EnableIPv6": false,
"IPAM": {
"Driver": "default",
"Options": null,
"Config": [
{
"Subnet": "172.25.0.0/16",
"Gateway": "172.25.0.1"
}
]
},
"Internal": true,
"Attachable": true,
"Ingress": false,
"ConfigFrom": {
"Network": ""
},
"ConfigOnly": false,
"Containers": {
"dac25cb8810b2c786735a76c9b8387d1cfb4d6006dbb7549f5c7c3f381d884c2": {
"Name": "capture-011-ping2_tcpdump_1",
"EndpointID": "2463a46cf00a35c8c77ff9f224ff052aea7f061684b7a24b41dab150496f5c3d",
"MacAddress": "02:42:ac:19:00:02",
"IPv4Address": "172.25.0.2/16",
"IPv6Address": ""
}
},
"Options": {},
"Labels": {
"com.docker.compose.network": "capture",
"com.docker.compose.project": "capture-011-ping2",
"com.docker.compose.version": "1.22.0"
}
}
]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,670
|
import os
import sys
import time
import inspect
import traceback
saved_path = sys.path[:]
sys.path.append(os.path.dirname(os.path.abspath(inspect.getsourcefile(lambda:0))))
from internal.memcached_connection import MemcachedTextConnection
port = int(iproto.uri.split(':')[1])
mc_client = MemcachedTextConnection('localhost', port)
blobs_list = [ "mooo\0", "mumble\0\0\0\0\r\rblarg", "\0", "\r" ]
for i in range(len(blobs_list)):
key = "foo_%d" % i
blob = blobs_list[i]
blob_len = len(blob)
mc_client("set %s 0 0 %d\r\n%s\r\n" % (key, blob_len, blob))
mc_client("get %s\r\n" % key)
mc_client("flush_all\r\n")
sys.path = saved_path
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,598
|
Q: When is using the C# ref keyword ever a good idea? The more I see ref used in production code, the more misuse I encounter and the more pain it causes me. I have come to hate this keyword, because from a framework-building standpoint, it seems silly. When would it be a good idea to communicate to users of your code the notion of maybe changing an object reference/value out from beneath them?
By contrast, I love out keywords and I love even more when no keywords are used at all, in both cases because of the guarantees you're given when using them. Ref on the other hand makes no guarantees, except that you'll be forced to initialize the parameter before you pass it in, even though nothing may be changed about it.
I'm no sage developer though; I'm sure it's got practically applicable uses. I'd just like to know what they are.
A: Any time you want to change the value of a value type - this happens a lot in cases where you want to efficiently update a pair of related values (i.e. rather than returning a struct containing two ints, you pass (ref int x, ref int y))
A: The Framework Design Guidelines (a book by Krzysztof Cwalina and Brad Abrams) recommend to avoid both ref and out parameters.
AVOID using out or ref parameters.
Using out or ref parameters requires experience with pointers, understanding how value types and reference types differ, and handling methods with multiple return values. Also, the difference between out and ref parameters is not widely understood. Framework architects designing for a general audience should not expect users to master working with out or ref parameters.
The Framework Design Guidelines cite the canonical Swap method as a valid exception:
void Swap<T>(ref T obj1, ref T obj2)
{
T temp = obj1;
obj1 = obj2;
obj2 = temp;
}
but at the same time a comment remarks
Swap always comes up in these discussions, but I have not written code that actually needed a swap method since college. Unless you've got a very good reason, avoid out and ref altogether.
A: Maybe when you have a struct (which is a value type):
struct Foo
{
int i;
public void Test()
{
i++;
}
}
static void update(ref Foo foo)
{
foo.Test();
}
and
Foo b = new Foo();
update(ref b);
Here you would to use two-parameters with out like:
static void update(Foo foo, out Foo outFoo) //Yes I know you could return one foo instead of a out but look below
{
foo.Test();
outFoo = foo;
}
imaging the method having more than one Foo then you would get twice the parameters with out versus ref. An alternative is to return a N-tuple. I don't have a real-world example on when to use this stuff.
Add on: Different .TryParse methods could also have avoided out if they returned Nullable<T> instead which essentially is a tuple of boolean * T.
A: Most of the Interlocked methods use ref parameters for (I'm sure you agree) good reason.
A: I try to avoid it on public APIs, but it definitely has uses. Mutable value-types is an important one, especially on things like CF (where mutable structs are more common, due to platform requirements). However, perhaps the most common time I use it is when refactoring parts of a complex algorithm out into a few methods, where a state object is overkill and I need to pass multiple values around:
i.e.
var x = .....
var y = .....
// some local code...
var z = DoSomethingSpecific(ref x, ref y); // needs and updates x/y
// more local code...
etc. Where DoSomethingSpecific is a private method, just moved out to keep method responsibility manageable.
A: How about if one wishes to pass an array to a function which might or might not change its size and do something else to it. Often, one would wrap the array in another object, but if one wishes to handle the array directly passing by reference would seem the most natural approach.
A: It's useful when you need efficient in-place algorithms on bignums.
A: Hypothetically, I'd guess that you might use a lot of ref/out arguments if you intended to mimic the architecture of older procedural software, for example old game engines and so on. I've scanned the source code of one, I think it was Duke Nukem 3D, and it's procedural with lots of subroutines modifying variables in place, and almost no functions. Obviously, you'd be unlikely to program like this for a real production application unless you had some specific aim in mind.
A: I'm using ref quite often. Just think about functions with multiple return values.
It doesn't make sense to create a return object (helper object) or even using hashtables for this purpose.
Example:
getTreeNodeValues(ref selectedValue, ref selectedText);
Edit:
It's better to use out here - as commented.
getTreeNodeValues(out selectedValue, out selectedText);
I'm using it for processing objects:
MyCar car = new MyCar { Name="TestCar"; Wieght=1000; }
UpdateWeight(ref car, 2000);
A: Another useful example in addition to swap<> is this:
Prompter.getString("Name ? ", ref firstName);
Prompter.getString("Lastname ? ", ref lastName);
Prompter.getString("Birthday ? ", ref firstName);
Prompter.getInt("Id ? ", ref id);
Prompter.getChar("Id type: <n = national id, p = passport, d = driver licence, m = medicare> \n? ", ref c);
public static class Prompter
{
public static void getKey(string msg, ref string key)
{
Console.Write(msg);
ConsoleKeyInfo cki = Console.ReadKey();
string k = cki.Key.ToString();
if (k.Length == 1)
key = k;
}
public static void getChar(string msg, ref char key)
{
Console.Write(msg);
key = Console.ReadKey().KeyChar;
Console.WriteLine();
}
public static void getString(string msg, ref string s)
{
Console.Write(msg);
string input = Console.ReadLine();
if (input.Length != 0)
s = input;
}
public static void getInt(string msg, ref int i)
{
int result;
string s;
Console.Write(msg);
s = Console.ReadLine();
int.TryParse(s, out result);
if (result != 0)
i = result;
}
// not implemented yet
public static string getDate(string msg)
{
// I should use DateTime.ParseExact(dateString, format, provider);
throw new NotImplementedException();
}
}
Use out here it's not an option
|
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{"url":"https:\/\/cmup.fc.up.pt\/main\/all-publications?f%5Bauthor%5D=120","text":"# Publications\n\nFound 34 results\nAuthor Title Type [ Year]\nFilters: Author is Manuel Delgado\u00a0\u00a0[Clear All Filters]\nIn Press\nOn a question of Eliahou and a conjecture of Wilf. Mathematische Zeitschrift. In Press.Edit\n2016\nNumerical semigroups with a given set of pseudo-Frobenius numbers. LMS Journal of Computation and Mathematics. 2016;19(1):186-205.Edit\nnumericalsgps, a GAP package for numerical semigroups. ACM Communications in Computer Algebra. 2016;50(1):12-24.Edit\n$\\ssfnumericalsgps$, a $\\ssfGAP$ package for numerical semigroups. ACM Commun. Comput. Algebra. 2016;50:12-24.Edit\n2011\nOn iterated Mal'cev products with a pseudovariety of groups. Internat. J. Algebra Comput.. 2011;21:1285-1304.Edit\nPreface [Proceedings of the International Conference on Semigroups and Related Topics]. Internat. J. Algebra Comput.. 2011;21:v\u2013vi.Edit\n2010\nOn the relative solvability of certain inverse monoids. Semigroup Forum. 2010;81:531-547.\n2009\n2006\nComputing relative abelian kernels of finite monoids. J. Algebra. 2006;303:642-654.\nOn the Frobenius number of a proportionally modular Diophantine inequality. Port. Math. (N.S.). 2006;63:415-425.Edit\nOn the GAP package \\it numericalsgps. In: Fifth Conference on Discrete Mathematics and Computer Science (Spanish). Vol 23. Univ. Valladolid, Secr. Publ. Intercamb. Ed., Valladolid; 2006. 2. p. 271-278p. (Ciencias (Valladolid); vol 23).Edit\nRelative abelian kernels of some classes of transformation monoids. Bull. Austral. Math. Soc.. 2006;73:375-404.Edit","date":"2020-09-26 17:35:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6946243047714233, \"perplexity\": 12414.407827592586}, \"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-2020-40\/segments\/1600400244353.70\/warc\/CC-MAIN-20200926165308-20200926195308-00201.warc.gz\"}"}
| null | null |
Q: quasicompact scheme are finite union of affine scheme This is a problem from Liu`s book. Show that a scheme $X$ is quasi-compact if and only if it is a finite union of affine schemes. If the scheme is quasicompact then it is obviously a finite union of affine schemes. How to show the converse? Do we need that the affine schemes should be open in $X$?
A: I think Liu meant affine open subschemes, but in the end it doesn't matter (at least here). If a topological space is a finite union of quasi-compact subspaces, then it is again quasi-compact. In particular, a scheme which is a finite union (of course one means the underlying topological space ...) of affine schemes, then it has to be quasi-compact. Conversely, any quasi-compact scheme is a finite union of affine open subschemes, since we may find a finite subcovering of any given affine open covering.
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Everyone Literally Dies: The Chronicles of Narnia Reread Project Part 7
And so it begins and ends. The Last Battle.
The Last Gif Cap
Our stories begins somewhere different, Narnia itself.
We meet two people, Puzzle the Donkey, and Shift his abusive 'friend'.
They are muddling about when they see the shine of gold through a waterfall. Shift, being the lazy albeit it clever, ass he is, emotionally manipulates Puzzle into going into the waterfall to get the shiny thing.
Puzzle has a rough time treading through water. Encounters a whirlpool, is pulled under, fun times.
He finally gets it back to Shift. It turns out to be an old lion's skin that some hunter threw into the river. Shift immediately sees the opportunity to run a confidence game using Narnia's belief in Aslan, and now wants Puzzle to act as Aslan!
Puzzle is not the messiah.
Shift forces Puzzle to wear the skin, saying how much the real Aslan will appreciate it. A loud thunderstroke lands near them, but Shift reassures Puzzle that its a sign that they should try to act this out for Aslan.
The story then shifts to the novel's unrequited bromance of Jewel and King Tirian. They are both so as giddy as a schoolboy at Aslan being spotted.
Sadly Roonwit the Centaur shows up and damped everyone's spirits with reality that it can't be Aslan.
He goes off to get the Narnian army and Jewel and Tirian set out to investigate this Aslan everyone's talking about.
They come just in time to see enslaved Narnians, Calormen, and the early effects of modernization.
In rage they kill two innocents? Calormen and run away. Entrapped in guilt, they take themselves to see the false Aslan in judgment.
However they instead are captured. Jewel is taken into a stable, the same stable Aslan is said to be in, and Tirian is tied up to a tree.
Tirian then appearers in front of a bunch of strangers, and then wakes up to find none other than Jill and Eustace.
They were riding a train…before seeing a bright flash and ending up in Narnia.
They quickly free Tirian and they escape to an outpost.
The trio prepares to rescue Jewel and set out disguised as Calormen, with armor and brown facing.
I'm perturbed by this too Percy.
With the stable in sight, they infiltrate and discover that Aslan is actually a donkey.
Everyone then escapes and they then come across a group of captive dwarfs, and free them.
Turns out they're atheists now and live only for themselves. Dwarfs for dwarfs!
Dawkins would be proud.
So they're no longer helpful and worse still, Farsight the eagle comes with bad news, The Narnian capital been taken by Calormen!
As such, the Narnian army won't come. No Vale army for our team.
All our team can do now is try to take down the ape at Stablehill. They prepare for the battle. Knowing the terrible odds, how they won't win against so many Calormen.
The team heads back to Stablehill to see the ape turn the animals love of Aslan into fear and distrust.
So the final battle begins. It isn't glorious, it isn't fun, it isn't epic. It is tedious, sad, slow, and incredibly well written.
It's the filter of good writing Jackie, the filter of good writing.
Tirian is forced into the stable. There he finds some peace amongst Peter, Edmund, Lucy, Digory, Polly, Eustace, Jill, everyone sans Susan.
The atheist dwarfs are in the stable too, and are completely and literally blind to the beautiful world they're now in.
Aslan then shows up, and ends the world.
It's really intense, and the process slowly dismantles everything of Narnia till there is nothing left but blackness.
Everyone wakes up in a new land. Here they meet old friends, feel no fear, and discover they all have lived in a world that's only a shadow of something truer.
"I knew it", is what Plato would've probably said. We can't know since he's dead, but I'm sure he would've.
They then meet their parents and even Aslan, who tells them they all died from a train crash.
Damnit Gordon
He then reveals his true form to them, and the story continues in that world, far from where we are now.
Thoughts on it Now
Damn. Lewis actually did it. He completely destroyed a fantasy world he created. I mean, this is the only series I've ever read or even heard of where a writer has the gall to not only write a story about a world's creation, but its complete destruction. I'm really impressed.
This is the best Narnia book. A compliant I've sometimes had is how Lewis has issues balancing his PoV characters. So I'm happily surprised that he is able to balance everyone out. Another plus is that we finally have a battle from a PoV of someone in the battle. These battles have the captivation of a car wreck. It's brutal and depressing, but so well written you want to read it, regardless the terror going on.
The series continuity is excellent. I think people and places from all of the other books showed up. Lewis cared a lot about this books, and it's awesome the series continuity has been so strong.
Puzzle and Shift's relationship is a scary, accurate depiction of abuse, so called 'freindships', and the people who get into these relationships. Shift abuses and manipulates Puzzle by making him feel inferior. He berates Puzzle for being so stupid, and places himself as a superior. This gives Puzzle a terrible inferiority complex. He so meek and unsure of himself, but also much wiser than Puzzle. He has the intelligence to recognize Shift's plan to use the lion skin as evil. But because of his complex, he lets himself be shut up and used by Shift.
Shift's plan however falls apart as he loses Puzzle. Further he's gone all Cersei and starts drinking, dulling his senses. He ends up throw into the maw of Tash. For being so clever, the ape wasn't very smart, he ends of being outsmarted by a Littlefinger the tabby cat (He also goes by Ginger) and the Calormen.
Oh the Calormen. The evil brown people of the story, well except for the one. Initially I really hated how these people were portrayed, but I have the nagging feeling that it isn't as bad as it seams. In fact, parts of Calormen religion is even validated as a way to serve Aslan, instead of being inferior to Narnian beliefs. Instead of being just a Narnian god, he is god for everyone. Likewise Tash steals away both a Narnian and a Calormen, and fails at getting Tabby. This implication makes it so good Narnians and Calormen are reworlded regardless of race while bad Narnians and Calormen are punished. This makes Emeth an important and fleshed out character. The same applies to Rishda, who is complex in the evil way. He doesn't even believe in either god, and is then ironically taken by Tash. Both of these characters have opposite arcs about beliefs.
The Calormen invasion of Narnia also serves as anti-colonialist narrative. As the Calormen civilize the country and force its natives to work for them. They take over the government and try to rule over its people without merit or agreement. It seams that destroying a culture is detrimental to those of that culture, regardless of it.
Could the Calormen have been handled better? Perhaps, but I can't think of anything without messing with the rest of the story. Maybe I would cut out the bit about the brown-face, but even then, its use in the story doesn't feel so much racist as practical disguising as an enemy.
The ending of the last battle is one the best endings to anything I've ever read. I cried thinking over the entire book series. I was happy to see everyone going through the true Narnia, without any fear, and sad that they weren't dead. I wanted to be with them in this glorious place. It's funny that a book that kills everyone is so hopeful. I never expected to be happy at my favorite characters being killed. But are they truly dead? The world is described as fresher than anything here on earth. Life after death is heavenly. I'm happy because death doesn't stop Peter, Jill, or anyone else from living.
This of course ties into the most heavy theme of The Last Battle, religion. The series relationship with religion has been very subtle up to this point. The Last Battle is when things get much more explicit. Aslan is revealed to be god not only of Narnia, but of everything. It's said in all but name. Other than Aslan, we have have Shift becoming a religious leader, who manipulates his fellow Narnians from fear of Aslan's punishment. This forces the Narnian's to become slaves in all but name to the Calormen and Shift. It's like Animal Farm, for kids!
This narrative shows the organized religion created by Shift is detrimental to both the environment and the people. In name of Aslan, forests and cut and forest spirits murdered. People live in fear. And it provides a means for people like Tabby and Rishda. The name of Aslan itself is perverted in Tashlan, an admittedly nice sounding portmanteau. Organization and taming of something unorganized and untamable has disastrous effects that destroy the entire world.
This is one of the best books I've ever read, and of all of the novels, this is the one I'd suggest people listen to at least once. If anything, do it for Patrick Stewart. He was the best choice for the final novel. His Tash and panicked Tabby voice are moments I'll treasure till my death. Thank Aslan he did this. There are frankly so many small things about this book I get into. This is ultimately the best Narnia. It's the grimmest and bloodiest. Everyone dies, and some people are even cast into hell. But it's also the happiest, and has one the sweetest, but not stupid or overbearing. It felt so real.
Illustration Courtesy of Pauline Baynes
Cameron, the writer formerly known as Nick.
In this article:Chronicles of Narnia, Reread, The Last Battle
|
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Q: AzDo to pull into 2 different remote/target branches at same time We have 2 branches:
*
*Master
*Development
Development is always equated to Master and deployed until the Non-Prd environment and features are based on the Development branch.
So, when there is a bug and needed to fix in PRD, then we create a new branch out of the Master and call it as Hotfix. and once it's tested, we merge it into Master and rolled out to PRD as any other process, and at the same time, we also merge into Development, so that the feature branches has this feature. But quite recently we noticed that sometimes after the hotfix and merge to Master, we forget to merge into Development, which later stage causing big issues in conflicts and so not. So, my question is there anyway, in AzDo pipeline that with a click of one button it will merge into both Master and Development? not a button but in an Automated way so, AzDo will trigger auto merging in both branches and this manual error can be fixed.
Added my thoughts:(it's kind of like Start a pull request for multiple target branches)
Something like this, whenever final pull requests will be done, then if the branch is Hotfix* then pull will always show like this:
A: It somewhat appears like you are using the GitFlow branch strategy. You could consider using some custom scripts or extensions to manage this. If you wanted to leverage Git-Flow extension, when you finish a hotfix it would do those two merges for you.
git flow hotfix finish VERSION
If you are using Visual Studio, you could also use the GitFlow for Visual Studio 2019 which when "finishing" the hotfix should do both merges.
A: *
*If you want to merge branches in Azure Pipeline, you can try to use
git merge task. You can find this task in the marketplace. Please
note that you need to use a self-hosted agent.
*Since you mentioned that you want a button to decide whether to merge
to master and development branches, I suggest you can add parameters
in your yaml file.
Here is my sample:
pool: Default
parameters:
- name: MergeToMaster
type: boolean
default: false
- name: MergeToDevelopment
type: boolean
default: false
jobs:
- job: MergeToMaster
condition: eq(${{parameters.MergeToMaster}}, 'true')
steps:
- task: gitMerge@0
inputs:
mergeType: 'merge'
targetBranch: 'master'
repoUrl: 'https://{Organization} @dev.azure.com/{Organization}/{Project} /_git/{repo name}'
remoteName: 'origin'
pat: '{PAT}'
- job: MergeToDevelopment
condition: eq(${{parameters.MergeToDevelopment}}, 'true')
steps:
- task: gitMerge@0
inputs:
mergeType: 'merge'
targetBranch: 'Development'
repoUrl: 'https://{Organization} @dev.azure.com/{Organization}/{Project} /_git/{repo name}'
remoteName: 'origin'
pat: '{PAT}'
You can see two options when you run pipelines. Selecting the option will merge to the corresponding branches.
Update:
You can use REST API to create pull request to master and development branches. Please use powershell task to replace git merger task. Here is my sample:
jobs:
- job: CreatePRToMaster
condition: eq(${{parameters.MergeToMaster}}, 'true')
steps:
- task: PowerShell@2
inputs:
targetType: 'inline'
script: |
$url = "https://dev.azure.com/{Organization name}/{Project name}/_apis/git/repositories/{Repo ID}/pullrequests?api-version=6.1-preview.1"
$contentType = "application/json"
$user="user"
$token="{PAT}"
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $user,$token)))
$body= @'
{
"sourceRefName": "refs/heads/hotfix",
"targetRefName": "refs/heads/master",
"title": "test",
"description": "test"
}
'@
Invoke-RestMethod -Uri $url -Method Post -ContentType $contentType -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} -Body $body
- job: CreatePRToDevelopment
condition: eq(${{parameters.MergeToDevelopment}}, 'true')
steps:
- task: PowerShell@2
inputs:
targetType: 'inline'
script: |
$url = "https://dev.azure.com/{Organization name}/{Project name}/_apis/git/repositories/{Repo ID}/pullrequests?api-version=6.1-preview.1"
$contentType = "application/json"
$user="user"
$token="{PAT}"
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $user,$token)))
$body= @'
{
"sourceRefName": "refs/heads/hotfix",
"targetRefName": "refs/heads/Development",
"title": "test",
"description": "test"
}
'@
Invoke-RestMethod -Uri $url -Method Post -ContentType $contentType -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} -Body $body
Selecting the options when you run pipeline will create pull requests to the corresponding branches.
Result:
|
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| 6,887
|
Blonder Tango ist ein auf dem 1982 erschienenen gleichnamigen Roman von Omar Saavedra Santis basierender DEFA-Spielfilm. Regie führte Lothar Warneke. Die Inszenierung wurde am 10. April 1986 uraufgeführt.
Handlung
Der Chilene Rogelio musste nach dem Putsch in Chile 1973 seine Heimat verlassen. Er hat in der DDR Asyl gefunden und arbeitet als Beleuchter eines kleinen Theaters. Er leidet unter sozialer Isolation. Um seinen in Chile gebliebenen Angehörigen eine Freude zu machen, erfindet er eine Braut und schließlich sogar ein gemeinsames Kind und schickt Fotos fremder Kinder nach Hause. Aus den Briefen seiner Mutter erfährt er, dass alle Angehörigen sich über sein Wohlbefinden und den Familienzuwachs freuen. Doch mit der Zeit wird es immer schwieriger, das Lügengebäude aufrechtzuerhalten. Am Ende erfährt Rogelio, dass seine Mutter schon vor Jahren gestorben ist und die Briefe unter ihrem Namen von den Verwandten geschrieben wurden, die ihn mit dieser schrecklichen Tatsache nicht belasten wollten.
Kritik
Auszeichnungen
1986: Bester Film auf dem 4. Nationalen Spielfilmfestival der DDR
1986: Findlingspreis für Lothar Warneke
1986: Bester Hauptdarsteller: Alejandro Quintana Contreras
1986: Beste Nebendarstellerin: Johanna Schall
1987: Bester DEFA-Film (Kritikerpreis des Verbandes der Film- und Fernsehschaffenden der DDR)
1987: Bester Hauptdarsteller: Alejandro Quintana Contreras (Kritikerpreis des Verbandes der Film- und Fernsehschaffenden der DDR)
Überlieferung
Eine DVD-Edition liegt bis heute (2018) nicht vor.
Siehe auch
Isabel auf der Treppe
Weblinks
Blonder Tango bei der DEFA-Stiftung
Einzelnachweise
Filmtitel 1986
DDR-Film
Literaturverfilmung
Filmdrama
Flüchtlingsthematik im Film
Pinochet-Diktatur
|
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The title of Nikita Koloff's autobiography -- NIKITA: A Tale of the Ring and Redemption -- is a bit misleading, mainly because the "redemption" aspect never really shows up, at least in the way of typical wrestling redemptions over drugs or alcohol. It indirectly refers to his personal redemption when he became a devout Christian as was the focus of Koloff's previous book, Breaking the Chains.
While the book does focus on Koloff's wrestling career, it should not be considered an autobiography. Just to be clear, that is not a negative critique. Rather, Koloff's book should be considered "Pro Wrestling History 101." Koloff definitely comes from the George Santanaya philosophy that "Those who do not learn from history are doomed to repeat it." For anybody with the slightest interest in professional wrestling from the territorial days up to the beginning of the Monday Night Wars, this is a definite read.
Teaming with author William Murdock, who penned Jack Brisco's book as well, Koloff goes into considerable detail discussing the history of professional wrestling, as well as the history of the NWA title belts, with explanations of how and why certain belts were defended. Given that NIKITA is published by Crowbar Press, which has published other wrestling autobiographies such as ASSASSIN: The Man Behind the Mask, and ATLAS: Too Much, Too Soon, the historical aspect no doubt gets a greater emphasis than at a traditional publishing house.
Ever since George Orwell's 1959 book, the year 1984 was considered a landmark year in Western history: the Cold War was fortunately coming to a close and a new professional wrestling era commonly known as "Hulkamania" began. That same year, Nelson Scott Simpson gave up his dream of playing professional football and became Nikita Koloff. He didn't do it partway, either -- he legally changed his name to Nikita Koloff, learned to speak Russian and had his birthplace listed as Lithuania on his child's birth certificate.
Koloff's gimmick was simple: he was the nephew of former WWE (then known as the WWWF) Champion "The Russian Bear" Ivan Koloff, who had come to America for the purpose of taking Ric Flair's NWA title back to Russia. Thus, "The Russian Nightmare" was born.
For a man who never considered a career in professional wrestling, he ended up with a career most wrestlers would be envious of.
For me, the most fascinating part about Koloff's life was everything he did before coming Nikita Koloff. Simpson went to high school and played sports with legendary wrestlers including Demolition's Smash (Barry Darsow), "Mr. Perfect" Curt Hennig and "Ravishing" Rick Rude. In college, Simpson was recruited alongside and played football with Road Warrior Animal, the man who ended up getting him into the wrestling business.
Koloff also does a great job of translating the wrestling vernacular for those who do not live pro wrestling. There is a great balance of explaining wrestling concepts to newcomers without making knowledgeable people feel like this is a dumbed-down book. Even seasoned pros in wrestling history can learn a few things from this book. I had never heard of a match going "Broadway" before reading this and picked up some interesting tidbits of knowledge about the Lou Thesz era of pro wrestling.
Like so many pro wrestlers before him, Koloff's career was cut short due to an in-ring injury. Rather than risk permanent damage, after eight years in the business, Nikita quietly retired from pro wrestling to focus on his religion and fitness training. Koloff joined forces with Lex Luger, a man Koloff personally spoon-fed while Luger was paralyzed, to create the Power Hour and Total Package Fitness, both religious and health seminars. Koloff also runs the Koloff for Christ Ministries.
NIKITA: A Tale of the Ring and Redemption is a fascinating read, just not exactly what I expected when I picked it up.
Matthew Asher hopes one day to have Nikita Koloff also consider him a prominent pro wrestling historian, like SLAM! Wrestling Producer Greg Oliver, who gets a couple of shoutouts in Nikita's book.
|
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This is a guest article by David Bohl.
If you know someone who doesn't experience stress, they're probably not human. We stress over our finances, relationships, health, and careers or businesses. We get stressed when things don't go our way and we are not getting the results we want. We experience stress when we feel rushed and we run out of time to do all that we want to do. What causes you to feel stressed?
Of course, there's good stress and bad stress. Good stress is when you're stretching beyond your comfort zone and taking risks. You feel discomfort, while also feeling good about doing new things and taking new paths. Bad stress causes anguish, negative emotions, and oftentimes physical problems like pain, headaches, ulcers, and worse.
How you react to stress is a choice you make. You either react or act. See what's causing you stress and look for the pluses rather than minuses. That person who cut you off on the freeway and caused you to be five minutes late to work, could have prevented you from getting into the traffic accident that happened five minutes earlier. Choose the attitude and thoughts that give you the most power–they're usually the positive, productive ones.
Stress takes a toll on our bodies, so learning to master your energy can help keep you alert to stressful situations. The minute you feel yourself reacting, stop what you're doing and take five, slow, deep breaths. This will calm your mind and body and put you in a position to make a choice about how you want to respond to the stressful situation.
If your body is well nourished and well rested, you are less likely to react to stressful situations. Often our reactive mode is fueled by being tired, hungry, on a sugar or caffeine high, or just being in poor health. Get regular health check ups, drink water throughout the day, and eat a healthy diet of fresh foods and free of overly processed foods. Get enough sleep and exercise regularly to maintain optimum health.
During the typical work day, you need to take mental breaks so you don't burn out. A mind that is on 24/7 is a prime candidate for stressful reactions. If you're at work and notice you start to go blank or are attempting to do 10 things at once, give yourself a time out. You'll come back with a clearer mind and your work efforts will be more productive.
If you can pull yourself out of the stressful situation and see the bigger picture, you may be able to avoid falling into the reactive trap. Maybe Jack got promoted instead of you, but you really had your eye on a different career path, so let go and move on. Don't get so attached to outcomes, or you'll continually be stressed when life doesn't meet your expectations.
6. Balance work and play.
When you take time out to play, you come back to work renewed. The body and brain need variety and stimulation. With the appropriate balance, you'll be much less likely to get stressed out when you hit roadblocks. Do activities you enjoy like hobbies, travel, laughing, movies, being with friends and family.
If you're not doing work you love, chances are your stress level may be pretty high. Some people may feel stuck in their careers and don't see how they can pursue their dream job or business. It may not be feasible now, but sit down with a coach and plan your exit strategy. You deserve to spend your 40+ hours/week doing work you love and serving others with your talents.
8. Know what calms you.
It's important to have some built-in stress reduction activities so if you find yourself over the top with stress, you'll know how to calm down. Some people enjoy deep breathing or meditation, while others soak in a warm aromatherapy-scented bath or enjoy a deep massage.
9. Have support and love in your life.
We all need to be surrounded by positive, supportive, loving people or we will wither up. Make sure plenty of these types of people are around you, and reduce the number of negative types. If you're experiencing great stress, you need someone to talk to.
10. Be on a growth path.
Anything that is not growing is stagnant or dying. Continue to work with a coach, journal your feelings, read inspiring books, take growth-oriented seminars, and be aware of your stress responses. Most important of all, practice self love. You are the most important person in your life, so take good care of yourself.
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{"url":"http:\/\/www.mscroggs.co.uk\/blog\/tags\/approximation","text":"mscroggs.co.uk\nmscroggs.co.uk\n\nsubscribe\n\n# Blog\n\n2018-09-13\nThis is a post I wrote for round 2 of The Aperiodical's Big Internet Math-Off 2018. As I went out in round 1 of the Big Math-Off, you got to read about the real projective plane instead of this.\nPolynomials are very nice functions: they're easy to integrate and differentiate, it's quick to calculate their value at points, and they're generally friendly to deal with. Because of this, it can often be useful to find a polynomial that closely approximates a more complicated function.\nImagine a function defined for $$x$$ between -1 and 1. Pick $$n-1$$ points that lie on the function. There is a unique degree $$n$$ polynomial (a polynomial whose highest power of $$x$$ is $$x^n$$) that passes through these points. This polynomial is called an interpolating polynomial, and it sounds like it ought to be a pretty good approximation of the function.\nSo let's try taking points on a function at equally spaced values of $$x$$, and try to approximate the function:\n$$f(x)=\\frac1{1+25x^2}$$\nPolynomial interpolations of $$\\displaystyle f(x)=\\frac1{1+25x^2}$$ using equally spaced points\nI'm sure you'll agree that these approximations are pretty terrible, and they get worse as more points are added. The high error towards 1 and -1 is called Runge's phenomenon, and was discovered in 1901 by Carl David Tolm\u00e9 Runge.\nAll hope of finding a good polynomial approximation is not lost, however: by choosing the points more carefully, it's possible to avoid Runge's phenomenon. Chebyshev points (named after Pafnuty Chebyshev) are defined by taking the $$x$$ co-ordinate of equally spaced points on a circle.\nEight Chebyshev points\nThe following GIF shows interpolating polynomials of the same function as before using Chebyshev points.\nNice, we've found a polynomial that closely approximates the function... But I guess you're now wondering how well the Chebyshev interpolation will approximate other functions. To find out, let's try it out on the votes over time of my first round Big Internet Math-Off match.\nScroggs vs Parker, 6-8 July 2018\nThe graphs below show the results of the match over time interpolated using 16 uniform points (left) and 16 Chebyshev points (right). You can see that the uniform interpolation is all over the place, but the Chebyshev interpolation is very close the the actual results.\nScroggs vs Parker, 6-8 July 2018, approximated using uniform points (left) and Chebyshev points (right)\nBut maybe you still want to see how good Chebyshev interpolation is for a function of your choice... To help you find out, I've written @RungeBot, a Twitter bot that can compare interpolations with equispaced and Chebyshev points. Just tweet it a function, and it'll show you how bad Runge's phenomenon is for that function, and how much better Chebysheb points are.\nA list of constants and functions that RungeBot understands can be found here.\n\n### Similar posts\n\n Big Internet Math-Off stickers 2019 Mathsteroids realhats A non-converging LaTeX document\n\nComments in green were written by me. Comments in blue were not written by me.","date":"2019-11-15 15:12:14","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\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.608686089515686, \"perplexity\": 765.7267297338482}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496668682.16\/warc\/CC-MAIN-20191115144109-20191115172109-00410.warc.gz\"}"}
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Today I bring you a new delivery of chairs with style. I have prepared for you a small selection with 4 chairs that seem interesting. The Chair Arachnid imitates the structure of a spider. For your time Pages and chairs the Chair project designer Shira Paldi are quite innovative. Although the most important characteristic is perhaps the collection of chairs Martz. The your elegant design with modern touches will charm you. Let's see all of them in more detail.
Arachnid Bank of Topher Gent. People who suffer from Arachnophobia, it is best to not acquire this Bank because the effect the designer Topher Gent got is magnificent. The legs of this stool are reminiscent of those of a long-legged spider. The your design is at least disturbing.
The eight legs of these unconventional style bench are made of steel while your seat is of cherry wood. I have to admit that despite the spiders make me an impression, this Arachnid Bank appears to be very attractive.
Flexible back Chair Shira Paldi. This Chair gives us a fairly revolutionary concept since it presents us with a surprising solution to keep the Chair: this hides under the table. Until now we had seen chairs that folded to occupy less space to store them, but this has a flexible back to fold and put under the table.
The flexibility of the back of the Chair is perfectly studied, being flexible enough to be able to bend but strong enough for the Chair to be comfortable. The backs are made of steel while the rest of the Chair is made of Beechwood.
Collection of chairs Marts Edition. The industrial design Jean-Pierre Martz proposes these amazing chairs of simple lines and futuristic aspect. In relation to the embodiment, the author proposes a wide range of options. The structure can be acquired in curved wood, steel or aluminum. The seat can be found in several types of fabrics, vinyls and even leather. And all this with a lot of colors.
But even in this article have spoken only of chairs. The collection Martz Edition not only consists of chairs. You can also find banks, chairs and even a Chair for children. And each one of the pieces of this collection presents a fantastic design.
You can purchase any of these products through the page of Jean-Pierre Martz getting in touch directly with him.
Pages 6474 Studio Chairs. These amazing chairs are clearly inspired by the universe of the books. The "pages" that make up the seat allow the user to adjust the height of both the seat and the back. And as these are padded, the convenience is ensured.
This design so imaginative ferments user interaction in addition to give a unique aesthetic to the project. This innovative and colorful seat stimulate the creativity of the person who is sitting on it.
If you want to see previous deliveries of chairs with style can make it through these addresses: r-style Chairs, Chairs with style II and III-style Chairs.
|
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\section{Introduction}
\bigskip We are interested in the regularity of weak solutions to the
viscous incompressible magnetohydrodynamics (MHD) equations in $\mathbb{R}%
^{3}$
\begin{equation}
\left\{
\begin{array}{c}
\partial _{t}u+(u\cdot \nabla )u-\left( b\cdot \nabla \right) b-\Delta
u+\nabla \pi =0, \\
\partial _{t}b+(u\cdot \nabla )b-(b\cdot \nabla )u-\Delta b=0, \\
\nabla \cdot u=\nabla \cdot b=0, \\
u(x,0)=u_{0}(x),\text{ \ }b(x,0)=b_{0}(x),%
\end{array}%
\right. \label{eq1.1}
\end{equation}%
where $u=(u_{1},u_{2},u_{3})$ is the velocity field, $b=(b_{1},b_{2},b_{3})$
is the magnetic field, and $\pi $ is the scalar pressure, while $u_{0}$ and $%
b_{0}$ are the corresponding initial data satisfying $\nabla \cdot
u_{0}=\nabla \cdot b_{0}=0$ in the sense of distribution.
Local existence and uniqueness theories of solutions to the MHD equations
have been studied by many mathematicians and physicists (see, e.g., \cite%
{CW, DL, ST}). But due to the presence of Navier-Stokes equations in the
system (\ref{eq1.1}) whether this unique local solution can exist globally
is an outstanding challenge problem. For this reason, there are many
regularity criteria of weak solutions for the MHD equations has been
investigated by many authors over past years (see e.g., \cite{DJZ, D, FJNZ,
G1, GR1, GR2, GRZ, LD, NGZ, Z1, Z2} and references therein). Note that the
literatures listed here are far from being complete, we refer the readers to
see for example \cite{GR20, JZ1, JZ2, JZ3, JZ4} for expositions and more
references.
More recently, Beir\~{a}o and Yang \cite{BY} proved the following regularity
criterion for the mixed pressure-velocity in Lorentz spaces for Leray-Hopf
weak solutions to 3D Navier-Stokes equations
\begin{equation}
{\frac{\pi }{\left( e^{-\left\vert x\right\vert ^{2}}+\left\vert
u\right\vert \right) ^{\theta }}\in L}^{p}(0,T;L^{q,\infty }(\mathbb{R}%
^{3})),\text{ \ where \ }0\leq \theta \leq 1\text{ and }\frac{2}{p}+\frac{3}{%
q}=2-\theta , \label{eq7}
\end{equation}%
where $L^{q,\infty }(\mathbb{R}^{3})$ denotes the Lorentz space (c.f. \cite%
{Tri}).
Motivated by the recent work of \cite{BY}, the purpose of this note is to
establish the regularity for the MHD equations (\ref{eq1.1}) with the mixed
pressure-velocity-magnetic in Lorentz spaces. Our main result can be stated
as follows:
\begin{thm}
\label{th1}Suppose that $(u_{0},b_{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(%
\mathbb{R}^{3})$ with $\nabla \cdot u_{0}=\nabla \cdot b_{0}=0$ in the sense
of distribution.\ Let $\left( u,b\right) $ be a weak solution to the MHD
equations on some interval $\left[ 0,T\right] $ with $0<T\leq \infty $.\
Assume that $0\leq \theta \leq 1$ and that
\begin{equation}
{\frac{\pi }{\left( e^{-\left\vert x\right\vert ^{2}}+\left\vert
u\right\vert +\left\vert b\right\vert \right) ^{\theta }}\in L}%
^{p}(0,T;L^{q,\infty }(\mathbb{R}^{3})),\text{ \ where }\frac{2}{p}+\frac{3}{%
q}=2-\theta \label{eq15}
\end{equation}%
then the weak $\left( u,b\right) $ is regular on $(0,T].$
\end{thm}
\begin{re}
A special consequence of Theorem \ref{th1} and its proof is the regularity
criterion of the 3D Navier-Stokes equations with the mixed pressure-velocity
in Lorentz spaces. This generalizes those of \cite{BY}.
\end{re}
In order to derive the regularity criterion of weak solutions to the MHD
equations (\ref{eq1.1}), we introduce the definition of weak solution.
Next, let us writing
\begin{equation*}
w^{\pm }=u\pm b,\ \ \text{\ }w_{0}^{\pm }=u_{0}\pm b_{0}.
\end{equation*}%
We reformulate equation (\ref{eq1.1}) as follows. Formally, if the first
equation of MHD equations (\ref{eq1.1}) plus and minus the second one,
respectively, then MHD equations (\ref{eq1.1}) can be re-written as:
\begin{equation}
\left\{
\begin{array}{l}
\partial _{t}w^{+}-\Delta w^{+}+(w^{-}\cdot \nabla )w^{+}+\nabla \pi =0, \\%
[3mm]
\partial _{t}w^{-}-\Delta w^{-}+(w^{+}\cdot \nabla )w^{-}+\nabla \pi =0, \\%
[2mm]
\mathrm{div}~w^{+}=0,~~~~\mathrm{div}~w^{-}=0, \\
w^{+}(x,0)=w_{0}^{+}(x),\text{ \ \ }w^{-}(x,0)=w_{0}^{-}(x).%
\end{array}%
\right. \label{eq1.3}
\end{equation}%
The advantage is that the equations becomes symmetric.
\section{Proof of Theorem \protect\ref{th1}}
This section is devoted to the proof of Theorem \ref{th1}. In order to do
it, we first recall the following estimates for the pressure in terms of $u$
and $b$ (see e.g., \cite{GR20}) :
\begin{equation}
\left\Vert \pi \right\Vert _{L^{q}}\leq C\left( \left\Vert u\right\Vert
_{L^{2q}}^{2}+\left\Vert b\right\Vert _{L^{2q}}^{2}\right) ,\text{ \ \textrm{%
with} \ }1<q<\infty . \label{eq120}
\end{equation}
We are now in position to prove our main result.
\begin{pf}
Multiplying the first and the second equations of (\ref{eq1.3}) by $%
\left\vert w^{+}\right\vert ^{2}w^{+}$ and $\left\vert w^{-}\right\vert
^{2}w^{-}$ , respectively, integrating by parts and summing up, we have
\begin{eqnarray*}
&&\frac{1}{4}\frac{d}{dt}(\left\Vert w^{+}\right\Vert
_{L^{4}}^{4}+\left\Vert w^{-}\right\Vert _{L^{4}}^{4})+\int_{\mathbb{R}%
^{3}}(\left\vert \nabla w^{+}\right\vert ^{2}\left\vert w^{+}\right\vert
^{2}+\left\vert \nabla w^{-}\right\vert ^{2}\left\vert w^{-}\right\vert
^{2})dx+\frac{1}{2}\int_{\mathbb{R}^{3}}(\left\vert \nabla \left\vert
w^{+}\right\vert ^{2}\right\vert ^{2}+\left\vert \nabla \left\vert
w^{-}\right\vert ^{2}\right\vert ^{2})dx \\
&=&-\int_{\mathbb{R}^{3}}\nabla \pi \cdot (w^{+}\left\vert w^{+}\right\vert
^{2}+w^{-}\left\vert w^{-}\right\vert ^{2})dx \\
&=&\int_{\mathbb{R}^{3}}\pi \cdot \mathrm{div}(w^{+}\left\vert
w^{+}\right\vert ^{2}+w^{-}\left\vert w^{-}\right\vert ^{2})dx \\
&\leq &\int_{\mathbb{R}^{3}}\left\vert \pi \right\vert (\left\vert
w^{+}\right\vert +\left\vert w^{-}\right\vert )(\nabla \left\vert
w^{+}\right\vert ^{2}+\nabla \left\vert w^{-}\right\vert ^{2})dx \\
&\leq &C\int_{\mathbb{R}^{3}}\left\vert \pi \right\vert ^{2}(\left\vert
w^{+}\right\vert +\left\vert w^{-}\right\vert )^{2}dx+\frac{1}{4}\int_{%
\mathbb{R}^{3}}(\left\vert \nabla \left\vert w^{+}\right\vert
^{2}\right\vert ^{2}+\left\vert \nabla \left\vert w^{-}\right\vert
^{2}\right\vert ^{2})dx.
\end{eqnarray*}%
Notice that $u=\frac{1}{2}(w^{+}+w^{-})$ and $b=\frac{1}{2}(w^{+}-w^{-})$,
then the above inequality means that
\begin{eqnarray}
&&\frac{d}{dt}(\left\Vert u\right\Vert _{L^{4}}^{4}+\left\Vert b\right\Vert
_{L^{4}}^{4})+2\left\Vert \nabla \left\vert u\right\vert ^{2}\right\Vert
_{L^{2}}^{2}+2\left\Vert \nabla \left\vert b\right\vert ^{2}\right\Vert
_{L^{2}}^{2} \notag \\
&&+2\left\Vert \left\vert u\right\vert \left\vert \nabla u\right\vert
\right\Vert _{L^{2}}^{2}+2\left\Vert \left\vert b\right\vert \left\vert
\nabla b\right\vert \right\Vert _{L^{2}}^{2}+2\left\Vert \left\vert
u\right\vert \left\vert \nabla b\right\vert \right\Vert
_{L^{2}}^{2}+2\left\Vert \left\vert b\right\vert \left\vert \nabla
u\right\vert \right\Vert _{L^{2}}^{2} \notag \\
&\leq &C\int_{\mathbb{R}^{3}}\left\vert \pi \right\vert ^{2}(\left\vert
u\right\vert +\left\vert b\right\vert )^{2}dx=K, \label{eq21}
\end{eqnarray}%
where we have used
\begin{equation*}
\left\vert w^{+}\right\vert +\left\vert w^{-}\right\vert \leq \left\vert
w^{+}+w^{-}\right\vert +\left\vert w^{+}-w^{-}\right\vert .
\end{equation*}%
For $K$, borrowing the arguments in \cite{BY}, we set%
\begin{equation*}
V=e^{-\left\vert x\right\vert ^{2}}+\left\vert u\right\vert +\left\vert
b\right\vert \text{ \ \ and \ \ }\widetilde{\pi }={\frac{\pi }{\left(
e^{-\left\vert x\right\vert ^{2}}+\left\vert u\right\vert +\left\vert
b\right\vert \right) ^{\theta }}.}
\end{equation*}%
By the H\"{o}lder inequality and the following interpolation in Lorentz
space (see \cite{Tri})
\begin{equation*}
\left\Vert f^{\alpha }\right\Vert _{L^{p,q}(\mathbb{R}^{3})}\leq C\left\Vert
f\right\Vert _{L^{\alpha p,\alpha q}(\mathbb{R}^{3})}^{\alpha }\text{ \ \
for \ }\alpha >0,\text{ }p>0,\text{ }q>0,
\end{equation*}
we have%
\begin{eqnarray*}
K &=&\int_{\mathbb{R}^{3}}\left\vert \pi \right\vert ^{\lambda }V^{-\lambda
\theta }\left\vert \pi \right\vert ^{2-\lambda }V^{\lambda \theta
}(\left\vert u\right\vert +\left\vert b\right\vert )^{2}dx \\
&\leq &\int_{\mathbb{R}^{3}}\left\vert \widetilde{\pi }\right\vert ^{\lambda
}\left\vert \pi \right\vert ^{2-\lambda }V^{2+\lambda \theta }dx \\
&\leq &\left\Vert \left\vert \widetilde{\pi }\right\vert ^{\lambda
}\right\Vert _{L^{\frac{q}{\lambda },\infty }}\left\Vert \left\vert \pi
\right\vert ^{2-\lambda }\right\Vert _{L^{s,\frac{2}{2-\lambda }}}\left\Vert
V^{2\lambda }\right\Vert _{L^{r,\frac{2}{\lambda }}} \\
&=&\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\lambda
}\left\Vert \pi \right\Vert _{L^{s(2-\lambda ),2}}^{2-\lambda }\left\Vert
V^{2}\right\Vert _{L^{\lambda r,2}}^{\lambda },
\end{eqnarray*}%
where
\begin{equation*}
\frac{\lambda }{q}+\frac{1}{s}+\frac{1}{r}=1\text{ \ and \ }\lambda =\frac{2%
}{2-\theta }.
\end{equation*}%
By (\ref{eq120}), we have%
\begin{eqnarray*}
K &\leq &\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\lambda
}\left( \left\Vert \left\vert u\right\vert ^{2}\right\Vert _{L^{s(2-\lambda
),2}}+\left\Vert \left\vert b\right\vert ^{2}\right\Vert _{L^{s(2-\lambda
),2}}\right) ^{2-\lambda }\left\Vert V^{2}\right\Vert _{L^{\lambda
r,2}}^{\lambda } \\
&\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\lambda
}\left\Vert V^{2}\right\Vert _{L^{s(2-\lambda ),2}}^{2-\lambda }\left\Vert
V^{2}\right\Vert _{L^{\lambda r,2}}^{\lambda }.
\end{eqnarray*}%
By the interpolation and Sobolev inequalities in Lorentz spaces, it follows
that%
\begin{equation}
\left\{
\begin{array}{c}
\left\Vert V^{2}\right\Vert _{L^{s(2-\lambda ),2}}\leq C\left\Vert
V^{2}\right\Vert _{L^{2,2}}^{1-\delta _{1}}\left\Vert V^{2}\right\Vert
_{L^{6,2}}^{\delta _{1}}\leq C\left\Vert V^{2}\right\Vert _{L^{2}}^{1-\delta
_{1}}\left\Vert \nabla V^{2}\right\Vert _{L^{2}}^{\delta _{1}}, \\
\left\Vert V^{2}\right\Vert _{L^{\lambda r,2}}\leq C\left\Vert
V^{2}\right\Vert _{L^{2,2}}^{1-\delta _{2}}\left\Vert V^{2}\right\Vert
_{L^{6,2}}^{\delta _{2}}\leq C\left\Vert V^{2}\right\Vert _{L^{2}}^{1-\delta
_{2}}\left\Vert \nabla V^{2}\right\Vert _{L^{2}}^{\delta _{2}},%
\end{array}%
\right. \label{eq6.6}
\end{equation}%
where $0<\delta _{1},\delta _{2}<1$ and
\begin{equation*}
\frac{1}{s(2-\lambda )}=\frac{1-\delta _{1}}{2}+\frac{\delta _{1}}{6},\text{
\ }\frac{1}{\lambda r}=\frac{1-\delta _{2}}{2}+\frac{\delta _{2}}{6}.
\end{equation*}%
Hence from (\ref{eq6.6}) and Young inequality, it follows that%
\begin{eqnarray*}
K &\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\lambda
}\left\Vert V^{2}\right\Vert _{L^{2}}^{(2-\lambda )(1-\delta _{1})+\lambda
(1-\delta _{2})}\left\Vert \nabla V^{2}\right\Vert _{L^{2}}^{(2-\lambda
)\delta _{1}+\lambda \delta _{2}} \\
&\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\frac{%
2\lambda }{2-(2-\lambda )\delta _{1}-\lambda \delta _{2}}}\left\Vert
V^{2}\right\Vert _{L^{2}}^{2}+\frac{1}{2}\left\Vert \nabla V^{2}\right\Vert
_{L^{2}}^{2}.
\end{eqnarray*}%
Due to the definition of $V$, we see that
\begin{equation*}
\left\Vert V^{2}\right\Vert _{L^{2}}^{2}\leq C(1+\left\Vert \left\vert
u\right\vert +\left\vert b\right\vert \right\Vert _{L^{2}}^{2}+\left\Vert
\left\vert u\right\vert ^{2}+\left\vert b\right\vert ^{2}\right\Vert
_{L^{2}}^{2}),
\end{equation*}%
and%
\begin{equation*}
\left\Vert \nabla V^{2}\right\Vert _{L^{2}}^{2}\leq C(1+\left\Vert
\left\vert u\right\vert +\left\vert b\right\vert \right\Vert
_{L^{2}}^{2}+\left\Vert \nabla (\left\vert u\right\vert +\left\vert
b\right\vert )\right\Vert _{L^{2}}^{2}+\left\Vert \nabla (\left\vert
u\right\vert ^{2}+\left\vert b\right\vert ^{2})\right\Vert _{L^{2}}^{2}).
\end{equation*}%
Consequently, we get%
\begin{eqnarray*}
K &\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\frac{%
2\lambda }{2-(2-\lambda )\delta _{1}-\lambda \delta _{2}}}(1+\left\Vert
\left\vert u\right\vert +\left\vert b\right\vert \right\Vert
_{L^{2}}^{2}+\left\Vert \left\vert u\right\vert ^{2}+\left\vert b\right\vert
^{2}\right\Vert _{L^{2}}^{2}) \\
&&+C(1+\left\Vert \left\vert u\right\vert +\left\vert b\right\vert
\right\Vert _{L^{2}}^{2}+\left\Vert \nabla (\left\vert u\right\vert
+\left\vert b\right\vert )\right\Vert _{L^{2}}^{2})+\frac{1}{2}\left\Vert
\nabla (\left\vert u\right\vert ^{2}+\left\vert b\right\vert
^{2})\right\Vert _{L^{2}}^{2} \\
&\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\frac{%
2\lambda }{2-(2-\lambda )\delta _{1}-\lambda \delta _{2}}}(1+\left\Vert
u\right\Vert _{L^{2}}^{2}+\left\Vert b\right\Vert _{L^{2}}^{2}+\left\Vert
u\right\Vert _{L^{4}}^{4}+\left\Vert b\right\Vert _{L^{4}}^{4}) \\
&&+C(1+\left\Vert u\right\Vert _{L^{2}}^{2}+\left\Vert b\right\Vert
_{L^{2}}^{2}+\left\Vert \nabla u\right\Vert _{L^{2}}^{2}+\left\Vert \nabla
b\right\Vert _{L^{2}}^{2})+\frac{1}{2}\left\Vert \nabla \left\vert
u\right\vert ^{2}\right\Vert _{L^{2}}^{2}+\frac{1}{2}\left\Vert \nabla
\left\vert b\right\vert ^{2}\right\Vert _{L^{2}}^{2}.
\end{eqnarray*}%
Since $(u,b)$ is a weak solution to (\ref{eq1.1}), then $(u,b)$ satisfies%
\begin{equation*}
(u,b)\in L^{\infty }(0,T;L^{2}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{R%
}^{3})).
\end{equation*}%
Inserting the above estimates into (\ref{eq21}), we obtain
\begin{eqnarray*}
&&\frac{d}{dt}(\left\Vert u\right\Vert _{L^{4}}^{4}+\left\Vert b\right\Vert
_{L^{4}}^{4})+\left\Vert \nabla \left\vert u\right\vert ^{2}\right\Vert
_{L^{2}}^{2}+\left\Vert \nabla \left\vert b\right\vert ^{2}\right\Vert
_{L^{2}}^{2} \\
&&+2\left\Vert \left\vert u\right\vert \left\vert \nabla u\right\vert
\right\Vert _{L^{2}}^{2}+2\left\Vert \left\vert b\right\vert \left\vert
\nabla b\right\vert \right\Vert _{L^{2}}^{2}+2\left\Vert \left\vert
u\right\vert \left\vert \nabla b\right\vert \right\Vert
_{L^{2}}^{2}+2\left\Vert \left\vert b\right\vert \left\vert \nabla
u\right\vert \right\Vert _{L^{2}}^{2} \\
&\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\frac{%
2\lambda }{2-(2-\lambda )\delta _{1}-\lambda \delta _{2}}}(1+\left\Vert
u\right\Vert _{L^{2}}^{2}+\left\Vert b\right\Vert _{L^{2}}^{2}+\left\Vert
u\right\Vert _{L^{4}}^{4}+\left\Vert b\right\Vert _{L^{4}}^{4}) \\
&&+C(1+\left\Vert u\right\Vert _{L^{2}}^{2}+\left\Vert b\right\Vert
_{L^{2}}^{2}+\left\Vert \nabla u\right\Vert _{L^{2}}^{2}+\left\Vert \nabla
b\right\Vert _{L^{2}}^{2}) \\
&\leq &C\left\Vert \widetilde{\pi }\right\Vert _{L^{q,\infty }}^{\frac{%
2\lambda }{2-(2-\lambda )\delta _{1}-\lambda \delta _{2}}}(1+\left\Vert
u\right\Vert _{L^{4}}^{4}+\left\Vert b\right\Vert
_{L^{4}}^{4})+C(1+\left\Vert \nabla u\right\Vert _{L^{2}}^{2}+\left\Vert
\nabla b\right\Vert _{L^{2}}^{2}),
\end{eqnarray*}%
Using Gronwall's inequality with the assumption (\ref{eq15}), we deduce that%
\begin{equation*}
(u,b)\in L^{\infty }(0,T;L^{4}(\mathbb{R}^{3}))\subset L^{8}(0,T;L^{4}(%
\mathbb{R}^{3})).
\end{equation*}%
We complete the proof of Theorem \ref{th1}.
\end{pf}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 388
|
Khenaisser v. Jewell
MAZEN KHENAISSER, Plaintiff,
SALLY JEWELL, et al., Defendants.
CAROLYN K. DELANEY UNITED STATES MAGISTRATE JUDGE.
Defendants' motion to dismiss came on regularly for hearing on June 15, 2016. Plaintiff Mazen Khenaisser appeared in propria persona. Chi Soo Kim appeared for defendants. Upon review of the documents in support and opposition, upon hearing the arguments of plaintiff and counsel, and good cause appearing therefor, THE COURT FINDS AS FOLLOWS:
In this action, plaintiff alleges claims of discrimination arising out of his employment as a civil engineer in the Bureau of Reclamations Design and Construction Division, Mid-Pacific Region. This matter was previously heard on defendants' motion to dismiss, which was granted with leave to amend as to certain claims. ECF No. 29. Plaintiff filed a first amended complaint on March 11, 2016. ECF No. 30.[1] Defendants move to dismiss the amended complaint with prejudice, contending that this court lacks subject matter jurisdiction over certain claims and that the remaining claims are subject to dismissal under Federal Rule of Civil Procedure 12(b)(6). ECF No. 32.
In the first amended complaint, plaintiff alleges a claim for defamation, arising in part from allegations made by agency employees that plaintiff put his fist in a manager's face. ECF No. 30 at p. 20, ¶ 10(d). Plaintiff also alleges defamation of character and that he was slandered by an employee "putting words in my mouth." ECF No. 30, ¶¶ 3(b), 4(c), 5(c), 6(a), 10(b), 10(g), 10(h), 12(f). Defendant moves to dismiss for lack of subject matter jurisdiction plaintiff's claim for defamation. Defendant contends this court lacks jurisdiction because the United States has not waived sovereign immunity over such a claim. This contention is correct.
Federal Rule of Civil Procedure 12(b)(1) allows a defendant to raise the defense, by motion, that the court lacks jurisdiction over the subject matter of an entire action or of specific claims alleged in the action. "A motion to dismiss for lack of subject matter jurisdiction may either attack the allegations of the complaint or may be made as a 'speaking motion' attacking the existence of subject matter jurisdiction in fact." Thornhill Publ'g Co. v. Gen. Tel. & Elecs. Corp., 594 F.2d 730, 733 (9th Cir. 1979).
When a Rule 12(b)(1) motion attacks the existence of subject matter jurisdiction in fact, no presumption of truthfulness attaches to the plaintiff's allegations. Thornhill Publ'g Co., 594 F.2d at 733. "[T]he district court is not restricted to the face of the pleadings, but may review any evidence, such as affidavits and testimony, to resolve factual disputes concerning the existence of jurisdiction." McCarthy v. United States, 850 F.2d 558, 560 (9th Cir. 1988). When a Rule 12(b)(1) motion attacks the existence of subject matter jurisdiction in fact, plaintiff has the burden of proving that jurisdiction does in fact exist. Thornhill Publ'g Co., 594 F.2d at 733.
Absent a waiver, sovereign immunity shields the United States and its agencies from suit. See Loeffler v. Frank, 486 U.S. 549, 554 (1988). Sovereign immunity is jurisdictional in nature. See United States v. Mitchell, 463 U.S. 206, 212 (1983) ("It is axiomatic that the United States may not be sued without its consent and that the existence of consent is a prerequisite for jurisdiction"). This court has jurisdiction over plaintiff's claims against the United States for defamation only where there is an express waiver of sovereign immunity. See United States v. Nordic Village, Inc., 503 U.S. 30, 33-34 (1992).
Under the Federal Torts Claims Act ("FTCA"), 28 U.S.C. § 2680(h), any claims for libel, slander, misrepresentation or deceit are expressly excluded from the general waiver of sovereign immunity for tort claims. See 28 U.S.C. § 2674(b) ("The United States [is] liable ... in the same manner and to the same extent as a private individual under like circumstances."); see also 28 U.S.C. § 1346(b)(1) (conferring original jurisdiction on district court over tort claims). Accordingly, plaintiff's defamation claims are barred in this court and must be dismissed without leave to amend. See Thomas-Lazear v. F.B.I., 851 F.2d 1202, 1206-1207 (9th Cir. 1988) (slander and libel claims barred under 28 U.S.C. § 2680(h) dismissed without leave to amend).
Defendant also moves to dismiss the amended complaint for failure to state a claim. In considering a motion to dismiss under Federal Rule of Civil Procedure 12(b)(6) for failure to state a claim upon which relief can be granted, the court must accept as true the allegations of the complaint in question, Erickson v. Pardus, 127 S.Ct. 2197, 2200 (2007), and construe the pleading in the light most favorable to the plaintiff, see Scheuer v. Rhodes, 416 U.S. 232, 236 (1974).
In order to avoid dismissal for failure to state a claim a complaint must contain more than "naked assertions, " "labels and conclusions" or "a formulaic recitation of the elements of a cause of action." Bell Atlantic Corp. v. Twombly, 550 U.S. 544, 555-557 (2007). In other words, "[t]hreadbare recitals of the elements of a cause of action, supported by mere conclusory statements do not suffice." Ashcroft v. Iqbal, 556 U.S. 662, 678 (2009). Furthermore, a claim upon which the court can grant relief has facial plausibility. Twombly, 550 U.S. at 570. "A claim has facial plausibility when the plaintiff pleads factual content that allows the court to draw the reasonable inference that the defendant is liable for the misconduct alleged." Iqbal, 556 U.S. at 678.
In the first amended complaint (ECF No. 30), plaintiff names as a defendant Benjamin Wagner, United States Attorney. No allegations are made against this defendant in the body of the complaint. As noted in the prior findings and recommendations (ECF No.24), which were adopted by the District Court (ECF No. 29), the only properly named defendant is defendant Sally Jewell, the Secretary of the Interior, in her official capacity. See 42 U.S.C. § 2000e-16(c) (Title VII); 29 U.S.C. § 794a(a)(1); Vinieratos v. United States, 939 F.2d 762, 772 (9th Cir. 1991). As such, defendant Benjamin Wagner should be dismissed with prejudice.
In the amended complaint, plaintiff has restated his claims for disability discrimination. However, the first amended complaint does not cure the deficiencies this court found in the original complaint. Plaintiff's disability claim is premised on "back discomfort" allegedly caused by improper seating arrangements. Such a claim falls far short of the requirement under section 501 of the Rehabilitation Act[2] that plaintiff have a physical or mental impairment that substantially limits one or more of the major life activities in order to state a claim under that Act. See Walton v. U.S. Marshals Serv., 492 F.3d 998, 1005 (9th Cir. 2007); 42 U.S.C. §§ 12112(a), 12102; 29 U.S.C. § 705(9)(B); 29 C.F.R. § 1630.2(g).
The amended complaint also reprises plaintiff's discrimination claims on the basis of religion, national origin and race. Again, as with the original complaint, plaintiff fails to establish a prima facie case under Title VII. See Leong v. Potter, 347 F.3d 1117, 1124 (9th Cir. 2003) (to state a prima facie discrimination claim, plaintiff must show he belongs to protected class, was qualified for position, subjected to adverse employment action, and similarly situated individuals outside protected class were treated more favorably). The amended complaint alleges discrimination on the basis of religion (non-Jewish) but does not allege any discrimination based on religion other than alleging that the Branch Chief collaborated with another Jew because they went to the same synagogue together. This conclusory allegation does not set forth a sufficient basis for religious discrimination. Similarly, the amended complaint sets forth no allegations related to national origin. Also deficient are the allegations relating to racial discrimination in which plaintiff alleges that the Branch Chief is a racist because the Branch Chief responded to plaintiff's Union grievance by informing plaintiff that discrimination was an EEO matter and was excluded from contract. In sum, all of plaintiff's discrimination claims contain only conclusory allegations which are insufficient to support a claim for discrimination on the basis of religion, national origin and race.
Plaintiff's claim for retaliation is similarly deficient. Plaintiff asserts that the Branch Chief retaliated against plaintiff by subjecting him to "Direct Orders" which provided instruction of weekly substantive work tasks and allocated plaintiff's time between these tasks. To establish a prima facie case of retaliation, plaintiff must show that he engaged in statutorily protected activity, that an adverse employment action was thereafter taken against him, and a causal link between the two events. See Villiarimo v. Aloha ...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,077
|
\section{Introduction}\label{intro}
The Hall coefficient, $R_{\rm H}$, in the high-$T_{\rm c}$ superconducting (SC) cuprates has attracted considerable attention due to its peculiar behavior since the early stage of the high-$T_{\rm c}$ research.
The $R_{\rm H}$ in the normal state is strongly dependent on temperature, which is unusual in conventional metallic superconductors.
The temperature dependence of $R_{\rm H}$ at high temperatures has been explained based upon the Fermi-liquid model~\cite{kontani} or the two-carrier model~\cite{ono} taking into account spin fluctuations or charge fluctuations, respectively, but the behavior at low temperatures has not yet been understood.
The experimental results of $R_{\rm H}$ at low temperatures in the La-214 cuprates such as La$_{2-x}$Ba$_x$CuO$_4$ (LBCO)~\cite{sera,ada-prb,ada-jpcs} and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (LNSCO)~\cite{noda} around $x=1/8$ have revealed that $R_{\rm H}$ markedly decreases with decreasing temperature below the structural-phase-transition temperature between the tetragonal low-temperature (TLT) phase (space group: $P4_2/ncm$) and the orthorhombic mid-temperature (OMT) phase ({\it Bmab}), $T_{\rm d2}$.
This has been explained as being due to the disappearance of the Hall voltage in the one-dimensional (1D) charge stripe-ordered state~\cite{tranquada} stabilized through the structural phase transition~\cite{noda} or due to the cancellation of the Hall voltage by equal numbers of holes and electrons in the charge domain in the stripe-ordered state,~\cite{prelovsek} although it is still controversial.
A gradual decrease in $R_{\rm H}$ in the normal state with decreasing temperature at low temperatures has been observed in YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) with $\delta = 0.15 - 0.40$ (Ref. \cite{segawa}) and slightly Zn-substituted La$_{2-x}$Sr$_x$Cu$_{1-y}$Zn$_y$O$_4$ with $x=0.115$ and 0.15,~\cite{ada-jltp} which has been discussed in relation to the formation of the charge stripe order.
Recently, measurements of $R_{\rm H}$ in high magnetic fields for underdoped YBCO with the hole concentration per Cu in the CuO$_2$ plane, $p$, $=0.10-0.14$ by LeBoeuf {\it et al}.~\cite{leboeuf} have revealed a sign change of $R_{\rm H}$ at low temperatures.
They have proposed that the negative $R_{\rm H}$ originates from the formation of an electron pocket through the reconstruction of the Fermi surface caused by the possible formation of the charge stripe order.
In YBCO, however, effects of the possible charge stripe order on various physical properties are relatively weak compared with those in the La-214 cuprates, preventing one from understanding explicitly the relation between the electron pocket and the charge stripe order.
In this Letter, $R_{\rm H}$ in the charge stripe-ordered state is investigated in La-214 cuprates with various values of $p$.
It has been found for the first time, to our knowledge, in LBCO with $x=0.08-0.12$ and LNSCO with $x=0.12$ that the behavior of $R_{\rm H}$ including the sign at low temperatures exhibits a significant dependence on $p$.
That is, $R_{\rm H}$ for $x=0.10-0.12$ undergoing the phase transition to the TLT phase markedly decreases with decreasing temperature below $T_{\rm d2}$.
Moreover, a sign change of $R_{\rm H}$ is observed at low temperatures for $x=0.10$ and the absolute value of the negative $R_{\rm H}$ decreases with increasing $x$, followed by almost zero for $x=0.12$ where the charge stripe order is completely stabilized.
These results indicate that $R_{\rm H}$ is zero in the ground state of the charge-spin stripe order at $x \sim 1/8$ and that $R_{\rm H}$ becomes negative in the less-stabilized state of the charge stripe.
It appears that there exists a close correlation between the stability of the charge stripe order, the sign of $R_{\rm H}$ and the topology of the Fermi surface discussed later.
Single crystals of LBCO with $x=0.08$, 0.10, 0.11, 0.12 and LNSCO with $x=0.12$ were grown by the traveling-solvent floating-zone method.
The detailed procedures are described elsewhere.~\cite{ada-prb,ada-prbmag}
The composition of each crystal was analyzed by the inductively-coupled-plasma analysis.
For LBCO and LNSCO with $x=0.10-0.12$, the charge stripe order is formed at low temperatures below $T_{\rm d2}$,~\cite{tranquada,fujita,fujita-physc} while it is not for LBCO with $x=0.08$.~\cite{fujita-physc}
Both $R_{\rm H}$ and the ab-plane electrical resistivity, $\rho_{\rm ab}$, were measured by the standard ac six-probe method in magnetic fields parallel to the c-axis up to 9 T, using a commercial apparatus (Quantum Design, PPMS).
Both temperature and magnetic-field dependences of the Hall voltage were measured, resulting in good agreement of values of $R_{\rm H}$ each other.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\linewidth]{nfig1.eps}
\end{center}
\caption{(color online) Temperature dependence of the ab-plane electrical resistivity, $\rho_{\rm ab}$, in zero field and a magnetic field of 9 T parallel to the c-axis normalized by its value at 80 K, $\rho_{\rm ab}^{\rm 80K}$, for La$_{2-x}$Ba$_x$CuO$_4$ with $x=0.08 - 0.12$ and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (LNSCO) with $x=0.12$. Note for La$_{2-x}$Ba$_x$CuO$_4$ with $x=0.12$ that the descriptions 'A' and 'B' correspond to different batches of the crystal. Arrows indicate the structural-phase-transition temperature between the TLT and OMT phases, $T_{\rm d2}$.}
\label{p-d}
\end{figure}
Figure 1 shows the temperature dependence of $\rho_{\rm ab}$ in zero field and a magnetic field of 9 T parallel to the c-axis normalized by its value at 80 K for LBCO and LNSCO with $x=0.08-0.12$.
A jump in $\rho_{\rm ab}$ due to the structural phase transition to the TLT phase is observed for $x=0.10-0.12$, as shown by arrows.
For LBCO, $T_{\rm d2}$ systematically increases with an increase in $x$, the values of which are almost in agreement with those formerly reported.~\cite{suzuki}
In zero field, $\rho_{\rm ab}$ exhibits a metallic behavior below $T_{\rm d2}$ for $x=0.10$ and 0.11, whereas it is less metallic or semiconducting for $x=0.12$.
The behavior of $\rho_{\rm ab}$ for $x=0.12$ is typical of the sample with $p \sim 1/8$ where the charge stripe order is stabilized.
For $x=0.12$ in LBCO, $\rho_{\rm ab}$ gradually decreases with decreasing temperature roughly below 30 K and goes to zero, which may be due to tiny inclusion of regions of $x<0.12$ having higher SC transition temperatures, $T_{\rm c}$'s, in the sample.
It is also probable that the different behavior of $\rho_{\rm ab}$ between the sample 'A' and 'B' of $x=0.12$ is due to the different amount of tiny inclusion of $x<0.12$.
From the magnetic susceptibility measurements, in fact, $T_{\rm c}$, defined as the cross point between the extrapolated line of the steepest part of the shielding diamagnetism and zero susceptibility, is estimated to be as low as 4.2 K, indicating that most regions in the sample are made of $x=0.12$.
It is noted that the present behavior of $\rho_{\rm ab}$ for $x=0.12$ in LBCO is different from that reported by Li {\it et al.},~\cite{li} because their reported $\rho_{\rm ab}$ in zero field is metallic even below $T_{\rm d2}$, suddenly drops around 40 K and decreases gradually toward zero with decreasing temperature.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.85\linewidth]{nfig2.eps}
\end{center}
\caption{(color online) Temperature dependences of the Hall coefficient, $R_{\rm H}$, (left axis) and the ab-plane electrical resistivity, $\rho_{\rm ab}$, (right axis) in various magnetic fields parallel to the c-axis up to 9 T for La$_{2-x}$Ba$_x$CuO$_4$ with $x=0.10$. The arrow indicates the structural-phase-transition temperature between the TLT and OMT phases, $T_{\rm d2}$.}
\label{p-d}
\end{figure}
Temperature dependences of $R_{\rm H}$ and $\rho_{\rm ab}$ in various magnetic fields up to 9 T are displayed in Fig. 2 for LBCO with $x=0.10$.
It is noted that similar temperature dependences of $R_{\rm H}$ and $\rho_{\rm ab}$ are observed for LBCO with $x=0.11$ (Ref. \cite{ada-jpcs}).
With increasing field, the SC transition curve in $\rho_{\rm ab}$ vs. $T$ exhibits broadening characteristic of the underdoped high-$T_{\rm c}$ cuprates.~\cite{ada-prbmag}
The $R_{\rm H}$ is almost independent of temperature above $T_{\rm d2}$, whereas it suddenly decreases below $T_{\rm d2}$.
Moreover, a sign change of $R_{\rm H}$ is observed below about 26 K and eventually $R_{\rm H}$ goes to zero at low temperatures.
Since temperatures at which $R_{\rm H}$ and $\rho_{\rm ab}$ become zero are in good agreement with each other, the behavior of $R_{\rm H}$ going to zero at low temperatures is due to the SC transition.
Accordingly, it is concluded that the sign of $R_{\rm H}$ in the ground state of LBCO with $x=0.10$ is negative.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.95\linewidth]{nfig3.eps}
\end{center}
\caption{(color online) Temperature dependence of the Hall coefficient, $R_{\rm H}$, in 9 T parallel to the c-axis for La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) with $x=0.08-0.12$ and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (LNSCO) with $x=0.12$. Note for LBCO with $x=0.12$ that the descriptions 'A' and 'B' correspond to different batches of the crystal. Solid arrows indicate the structural-phase-transition temperature between the TLT and OMT phases, $T_{\rm d2}$. Open arrows indicate the onset temperature of the SC transition, $T_{\rm c}^{\rm onset}$, in 9 T estimated from $\rho_{\rm ab}$ shown in Fig. 1. Open diamonds show data of $R_{\rm H}$ obtained from the measurements of the magnetic-field dependence of the Hall voltage for LBCO with $x=0.11$. The inset is a schematic drawing of one possible reconstruction of the Fermi surface through the formation of a commensurate antiferromagnetic order or a $d$-density-wave order~\cite{chakravarty} for LBCO with $x=0.10$ and 0.12.}
\label{p-d}
\end{figure}
Figure 3 shows the $p$-dependent behavior of $R_{\rm H}$ in 9 T parallel to the c-axis.
For LBCO with $x=0.11$, data of $R_{\rm H}$ obtained from the measurements of the magnetic-field dependence of the Hall voltage at 70 K (above $T_{\rm d2}$), 45 K (a little below $T_{\rm d2}$) and 30 K (at a temperature with negative $R_{\rm H}$) are also plotted, exhibiting a good agreement with data of $R_{\rm H}$ obtained from the measurements of the temperature dependence of the Hall voltage in 9 T.
At high temperatures around 80 K, $R_{\rm H}$ is found to be positive and decrease with increasing $x$, as in the case of La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) single crystals.~\cite{ono}
For $x=0.08$, $R_{\rm H}$ above the onset temperature of the SC transition, $T_{\rm c}^{\rm onset}$, estimated from $\rho_{\rm ab}$ in 9 T shown in Fig. 1, is almost independent of temperature, whereas it gradually decreases with decreasing temperature below $T_{\rm c}^{\rm onset}$ due to the SC transition.
For $x=0.11$ and 0.12, on the other hand, $R_{\rm H}$ markedly decreases below $T_{\rm d2}$ as well as for $x=0.10$ shown in Fig. 2.
Since $T_{\rm c}^{\rm onset}$ is far below $T_{\rm d2}$, the marked decrease in $R_{\rm H}$ is irrespective of the SC transition.
Comparing the data of $x=0.10$ with those of $x=0.11$, the extrapolation of the data above $T_{\rm c}^{\rm onset}$ to zero temperature suggests that the sign change of $R_{\rm H}$ is more marked for $x=0.10$ than for 0.11, as shown by broken lines in Fig. 3.
For LBCO and LNSCO with $x=0.12$, on the other hand, $R_{\rm H}$ becomes almost zero at low temperatures.
These results are summarized as follows.
For the samples in the TLT phase and therefore in the charge stripe-ordered state, $R_{\rm H}$ markedly decreases with decreasing temperature below $T_{\rm d2}$.
Moreover, the sign change of $R_{\rm H}$ at low temperatures exhibits a significant dependence on $p$.
First, the marked decrease in $R_{\rm H}$ below $T_{\rm d2}$ is discussed.
The sudden decrease in $R_{\rm H}$ accompanied by the formation of the charge stripe order has formerly been observed in LBCO~\cite{sera,ada-prb,ada-jpcs} and LNSCO.~\cite{noda}
These former results of $R_{\rm H}$ are compatible with the present ones.
A possible origin of the decrease in $R_{\rm H}$ together with the formation of the charge stripe order has been proposed by Noda {\it et al}.~\cite{noda} to be the disappearance of the Hall voltage due to the formation of a 1D charge domain in the stripe-ordered state.
From direct numerical calculations including a stripe potential within the $t$-$J$ model, on the other hand, Prelov\v{s}ek {\it et al.}~\cite{prelovsek} have suggested that equal numbers of holes and electrons due to the half occupancy of holes in the charge domain leads to the disappearance of the Hall voltage.
For $x=0.10$ and 0.11 where the half occupancy in the 1D charge domain is guessed to be maintained,~\cite{fujita-physc} however, $R_{\rm H}$ exhibits negative finite values at low temperatures, as shown in Fig. 3.
Therefore, these two ways of thinking are not enough to explain the low-temperature behavior of $R_{\rm H}$ in the stripe-ordered state.
The sign change of $R_{\rm H}$ is often observed in the SC fluctuation regime, which is understood to be due to the vortex motion.
In this case, it has been suggested that the application of magnetic field tends to suppress the sign change.~\cite{matsuda}
As shown in Fig. 2, however, the sign change is not suppressed but enhanced by the application of magnetic field, and moreover, $R_{\rm H}$ is negative even in the normal state above $T_{\rm c}^{\rm onset}$ for $x=0.10$ and 0.11.
Therefore, the vortex motion is not the origin of the sign change of $R_{\rm H}$ in LBCO with $x=0.10$ and 0.11.
Simply thinking, the sign of $R_{\rm H}$ is sensitive to the subtle curvature of the Fermi surface.
As mentioned before, recently, LeBoeuf {\it et al.}~\cite{leboeuf} have found a pronounced sign change of $R_{\rm H}$ at low temperatures in strong magnetic fields in underdoped YBCO with $p=0.10-0.14$.
Since quantum oscillations have been observed in YBCO with $p=0.10$,~\cite{leyraud} they have suggested that the negative $R_{\rm H}$ is a product of the formation of an electron pocket through the Fermi-surface reconstruction caused by the possible formation of the charge stripe order.
A similar behavior of $R_{\rm H}$ has formerly been found in 2H-TaSe$_2$ where both a strong decrease and a sign change of $R_{\rm H}$ are observed accompanied by the transition to the charge-density-wave (CDW) state.~\cite{lee}
These have theoretically been explained in terms of the Fermi-surface reconstruction due to the opening of the CDW gap.~\cite{evtushinsky}
Accordingly, the sign change of $R_{\rm H}$ at low temperatures in LBCO with $x=0.10$ and 0.11 is possibly explained as being due to the creation of an electron pocket on the Fermi surface caused by the formation of the charge stripe order.
Then, why does the behavior of $R_{\rm H}$ at low temperatures depend on $p$ significantly?
Supposed that the Fermi surface in the underdoped cuprates is reconstructed through the formation of a commensurate antiferromagnetic (AF) order or a $d$-density-wave ($d$DW) order,~\cite{chakravarty} both hole pockets and an electron pocket are created on the Fermi surface, located around ($\pm \pi/2$, $\pm \pi/2$) and ($\pm \pi$, 0), (0, $\pm \pi$) in the reciprocal lattice space, respectively, which is schematically shown in the inset of Fig. 3.
Actually, recent angle-resolved photoemission experiments have revealed a pocket around ($\pi/2$, $\pi/2$).~\cite{chang,chinese}
Considering the $p$-dependent change of the Fermi-surface topology,~\cite{ino} the possible electron pocket around ($\pm \pi$, 0), (0, $\pm \pi$) tends to shrink with increasing $p$, as shown in the inset of Fig. 3.
This is naturally suggestive of the weakening of the electron aspect and the development of the hole aspect on $R_{\rm H}$ with increasing $p$.
This picture is consistent with the present results that the sign change of $R_{\rm H}$ gradually weakens and $R_{\rm H}$ at low temperatures becomes zero with increasing $x$ from $x=0.10$ to 0.12.
It is noted that $R_{\rm H}$ is positive at low temperatures for LNSCO with $x=0.15$,~\cite{noda} which is also consistent with this picture.
Accordingly, the $p$-dependent change of $R_{\rm H}$ in the ground state is able to be understood by the delicate balance of the contributions of hole and electron pockets.
Millis {\it et al.}~\cite{millis} and Lin {\it et al.}~\cite{lin} have theoretically investigated the Fermi-surface reconstruction within the tight-binding model including the charge and spin stripe potential.
According to their calculations, rather complicated arrangement of electron pockets and hole pockets is reconstructed and the development of the spin stripe correlation makes $R_{\rm H}$ negative, while the development of the charge stripe correlation makes $R_{\rm H}$ positive.
That is, in the charge-spin stripe-ordered state, the sign of $R_{\rm H}$ depends on the delicate balance of the developments of the charge and spin stripe order.
Based upon their calculations, it follows that $R_{\rm H}$ becomes zero due to complete developments of both charge and spin stripe order at $x=1/8$ and that $R_{\rm H}$ becomes negative due to the incomplete development of the charge stripe order for $x<1/8$.~\cite{fujita-physc}
Actually, the sign change of $R_{\rm H}$ has been observed in YBCO with $p=0.10-0.14$ (Ref.~\cite{leboeuf}) and LSCO with $x=0.12$ (Ref.~\cite{suzuki-lysco}) where the spin stripe correlation is developed but the charge stripe order is not.
Accordingly, it may be the case that $R_{\rm H}$ becomes zero due to the stabilization of the charge-spin stripe order at $x=1/8$, while it becomes negative due to the instability of the charge stripe order for $x<1/8$.
Finally, we comment on the value of $R_{\rm H}$ in the ground state of LBCO.
For $x=0.10$, the extrapolated value of $R_{\rm H}$ to zero temperature is estimated from Fig. 3 to be $-0.00375$ cm$^3$/C.
Thus, based on the simple one-carrier model, the carrier density $n_{\rm Hall}$ is calculated to be $n_{\rm Hall} = - V_{\rm cell} / eR_{\rm H} / 2 = 0.0795$ per Cu in the CuO$_2$ plane, where $V_{\rm cell}$ is the volume of the unit cell in the notation of the tetragonal at high temperature ($I4/mmm$).
For YBCO with $p=0.10$,~\cite{leboeuf} on the other hand, it has been reported that $n_{\rm Hall} = 0.0145$ per Cu in the CuO$_2$ plane.
This value is comparable to the deduced value of 0.019 per Cu in the CuO$_2$ plane from the quantum-oscillation experiments.~\cite{leyraud}
Assuming the negative value of $R_{\rm H}$ is originated from a possible pocket on the Fermi surface, a simple comparison of $n_{\rm Hall}$ between LBCO and YBCO results in a larger pocket in LBCO than in YBCO.
This appears to be inconsistent with that expected from the experimentally observed Fermi-surface topology,~\cite{ino,nakayama} because the reconstructed Fermi surface due to the commensurate AF order or the $d$DW order is expected to produce a larger electron pocket around $(\pm \pi,0)$, $(0,\pm \pi)$ in YBCO than in LBCO.
Accordingly, it is hard to make quantitative discussion on the negative value of $R_{\rm H}$ in the ground state using the simple one-carrier model.
In conclusion, $R_{\rm H}$ in the stripe-ordered LBCO and LNSCO in the TLT phase markedly decreases due to the formation of the charge-spin stripe order.
In the ground state, on the other hand, $R_{\rm H}$ is zero in the completely ordered charge-spin stripe state at $x=1/8$, while it is negative in the less-stabilized state of the charge stripe for $x<1/8$.
The $p$-dependent behavior of $R_{\rm H}$ including its sign is interpreted as being due to the delicate balance of the contributions of the hole-like Fermi surface and the possible electron pocket arising from the formation of the charge-spin stripe order.
Fruitful discussions with T. Tohyama are gratefully acknowledged.
This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, Culture and Technology, Japan.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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|
Q: Parsing XML webservice in black berry I need to parse below xml file in black berry
<Vehicles>
<Car>
<ID>000001</ID>
<Make>Audi</Make>
</Car>
<Car>
<ID>000002</ID>
<Make>Buick</Make>
</Car>
<Car>
<ID>000015</ID>
<Make>Chevrolet</Make>
</Car>
<Car>
<ID>000003</ID>
<Make>Chrysler</Make>
</Car>
</Vehicles>
Can any one pleae tell me how to parse it.
A: Please look into the following posts in stackoverflow.com
Parse XML file on BlackBerry
BlackBerry/J2ME - SAX parse collection of objects with attributes
Note: Please search in stackoverflow.com before post an issue.
A: As a starting point check the "xmldemo" sample app. It is a part of the BB SDK placed on your PC. XMLDemoScreen has an example of a DOM-parser implementaion.
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{
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|
consumer revolution - a large and rapid increase in the consumption of consumer goods such as tableware, curtains, pictures, and cutlery, a lust for objects - preceded the Industrial Revolution, both in England and elsewhere in northern Europe.
www.busseycapital.com 251.968.0001 . Without a process, without a discipline on managing money, there is no plan and that is a plan for failure.
|
{
"redpajama_set_name": "RedPajamaC4"
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Petr Pokorný (* 28. prosince 1975, Hradec Králové) je český fotbalista, obránce. Jeho otec je Ladislav Pokorný, který získal s Hradcem jediný mistrovský titul.
Fotbalová kariéra
Hrál za SK Chrudim 1887, SK Hradec Králové, FK Teplice, SC Xaverov Horní Počernice, FK Mladá Boleslav, Zagłębie Lubin, Śląsk Wrocław a Górnik Polkowice. V české lize nastoupil ve 131 utkáních a dal 4 góly.
Ligová bilance
Externí odkazy
Worldfootball.net
90minut
Player History
Čeští fotbalisté
Fotbalisté FC Hradec Králové
Fotbalisté FK Teplice
Fotbalisté FK Mladá Boleslav
Fotbalisté Zagłębie Lubin
Fotbalisté Śląsku Wrocław
Narození v roce 1975
Muži
Fotbalisté SC Xaverov Horní Počernice
|
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<iframe scrolling="no" frameborder="0" width="100%" height="130px" style="border: 1px solid #ECECEC" class="ndfp" data-src="https://products.gobankingrates.com/pub/d825c752-95cf-486e-ac70-78708c937813?targeting%5Bcategory%5D=money&targeting%5Bsubcategory%5D=economy&targeting%5Bkeyword%5D=&targeting%5Btags%5D=&targeting%5Binternal_id%5D=1357476&targeting%5Bcurrent_internal_id%5D=1170800&targeting%5Bpagetype%5D=article&targeting%5Bpage_layout%5D=user-first&targeting%5Benv%5D=&targeting%5Btheme%5D=theme-news-events&resize=1"></iframe> <!-- /wp:html --> <!-- wp:html --> <style> .desktop-mobile-adverts-templates--render { margin-left: 10px; margin-right: 10px; } </style><!-- /wp:html --><!-- wp:paragraph --><p></p> <!-- /wp:paragraph -->
Money / Economy
How the G7 Nations' Ban on Russian Gold Will Impact Metal Prices
By Yaёl Bizouati-Kennedy
Bet_Noire / Getty Images/iStockphoto
The G7 — which is gathering in Germany during a three-day summit — has announced a ban on Russian gold on June 28, following promises of as much tweeted by several of the group nation's leaders. The economic sanctions were reported as being "largely symbolic," according to Bloomberg.
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The U.S. Department of the Treasury's Office of Foreign Assets Control (OFAC) announced that it was implementing the G7 commitments by prohibiting gold imports of Russian Federation origin, with immediate effect, according to a press release.
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"As announced at the G7 Summit, the United States is joined in taking action against Russian gold, the country's biggest non-energy export, by the United Kingdom, Canada, and Japan," according to the release.
"As a result, the importation into the United States of gold of Russian Federation origin is prohibited, except to the extent provided by law, or unless licensed or otherwise authorized by OFAC. This determination excludes gold of Russian Federation origin that was located outside of the Russian Federation prior to today."
Joe Biden, Boris Johnson Hinted At Russian Gold Sanctions
"The United States has imposed unprecedented costs on Putin to deny him the revenue he needs to fund his war against Ukraine. Together, the G7 will announce that we will ban the import of Russian gold, a major export that rakes in tens of billions of dollars for Russia," President Joe Biden tweeted on June 26.
The official Twitter account for the U.K. Prime Minister's office tweeted: "Sanctions update: New Russian gold will no longer be allowed to be exported to the UK, US, Japan and Canada — targeting Putin's war machine. UK import restrictions now apply to over £13.5 billion of Russian exports."
In a call with reporters, a senior White House official said that gold, after energy, is the second-largest export for Russia and a source of significant revenue for Putin and Russia, according to a transcript of the call.
"The United States Treasury will issue a determination to prohibit the import of new gold into the United States on Tuesday, which will further isolate Russia from the global economy by preventing its participation in the gold market," the official said.
Related: 6 Alternative Investments to Consider for 2022
As for the U.K. announcement, it explains that new exports of Russian gold will no longer be allowed to enter the U.K., Canada, U.S. and Japan thanks to tough new measures to be announced at the G7 Summit "designed to ratchet up the pressure on Putin's war machine."
"Gold is a major Russian export, worth £12.6 billion ($15.4 billion) to the Russian economy in 2021. Its value to the Russian elite has also increased in recent months with oligarchs rushing to buy gold bullion in an attempt to avoid the financial impact of western sanctions. London is a major global gold trading hub and UK sanctions, which will be the first of their kind to be implemented against Russia anywhere in the world, will have a huge impact on Putin's ability to raise funds," the U.K. statement reads, in part.
According to Edward Moya — senior market analys for the Americas, OANDA — while much attention was given to the news that the G7 would announce a ban on new Russian gold, Western countries have already been limiting their transactions with Russia, so this ban would merely confirm a general trend. He added that, "gold is acting like a surfer and currently paddling through a wave of rising Treasury yields. Gold will 'pop up' once Wall Street is convinced they nailed down how high the Fed will take rates and then it can rally on global recession fears," according to a note sent to GOBankingRates.
Indeed, as Bloomberg reports, while Britain's government said over the weekend that the measure will have global reach, "analysts played down the potential impacts as the London Bullion Market Association, which sets standards for that market, removed Russian gold refiners from its accredited list in March."
Bloomberg added that while bullion rose 0.8% on Monday, it then erased much of the gain.
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"Prices may have received a little initial support from the news, but most analysts said the sanctions are unlikely to have a longer-term impact," Bloomberg writers Ranjeetha Pakiam and Eddie Spence added.
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Yaёl Bizouati-Kennedy
Yaël Bizouati-Kennedy is a full-time financial journalist and has written for several publications, including Dow Jones, The Financial Times Group, Bloomberg and Business Insider. She also worked as a vice president/senior content writer for major NYC-based financial companies, including New York Life and MSCI. Yaël is now freelancing and most recently, she co-authored the book "Blockchain for Medical Research: Accelerating Trust in Healthcare," with Dr. Sean Manion. (CRC Press, April 2020) She holds two master's degrees, including one in Journalism from New York University and one in Russian Studies from Université Toulouse-Jean Jaurès, France.
Debt Ceiling: 6 Million Jobs, 7% Unemployment Rate Are on the Line If Government Defaults
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\section{Introduction}
Many important questions in biology are fundamentally statistical. For
instance, deciphering the neural code requires knowledge of the
probability of observing patterns of activity in response
to stimuli \cite{Rieke97}; determining which features of a protein are important for
correct folding requires knowledge of the probability that a
particular sequence of amino acids folds naturally \cite{Russ05,Socolich05}; and determining
the patterns of foraging of animals and their social and individual
behavior requires knowledge of the distribution of food and species
over both space and time \cite{Oates87,Wrangham87,Eisenberg72}.
Building statistical descriptions of biological systems is, however,
hard. There are several reasons for this: i) biological systems are composed of large numbers of elements, and so
can exhibit a huge number of configurations, in fact, an exponentially
large number, ii) the elements typically interact with
each other, making it impossible to view the system as a collection of
independent entities, and iii) because of technological
considerations, the descriptions of biological systems have to be
built from very little data. For example, with current technology in
neuroscience, we can record simultaneously from only about 100 neurons
out of approximately 100 billion in the human brain. So, not only are
we faced with the problem of estimating probability distributions in
high dimensional spaces, we must make the estimates based on very
little information.
Despite these apparent difficulties, recent work has suggested that
the situation may be less bleak than it seems. There is evidence that
accurate statistical description of systems can be achieved without having
to examine all possible configurations
\cite{Schneidman05,Shlens06,Tang08,Bethge07,Yu08,Russ05,Socolich05}.
One merely has to measure the probability distribution over pairs of
elements and use those to build the full distribution.
These ``pairwise models'' potentially offer a fundamental
simplification, since the number of pairs is quadratic in the number
of elements, not exponential. However, support for the efficacy of
pairwise models has, necessarily, come from relatively small
subsystems -- small enough that the true probability
distribution could be measured experimentally, allowing direct
comparison of the pairwise distribution to the true one
\cite{Schneidman05,Shlens06,Yu08,Tang08}.
While these studies have provided a
key first step, a critical question remains: will the results from the
analysis of these small subsystems extrapolate to large ones? That is,
if a pairwise model predicts the probability distribution for a subset
of the elements in a system, will it also predict the probability
distribution for the whole system? Here we find that, for a
biologically relevant class of systems, this question can be answered
quantitatively and, importantly, generically -- independent of many of
the details of the biological system under consideration. And the
answer is, generally, ``no.'' In this paper, we explain, both
analytically and with simulations, why this is the case.
\section{Results}
\subsection{The extrapolation problem}
\label{problem:extrapolation}
To gain intuition into the extrapolation problem, let us consider a
specific example: neuronal spike trains. Figure \ref{Fig1}A shows a
typical spike train for a small population of neurons. Although the
raw spike times provide a complete description, they are not a useful
representation, as they are too high-dimensional. Therefore, we divide time into bins and re-represent the spike train as 0s and 1s: 0 if there is no
spike in a bin; 1 otherwise (Fig.~\ref{Fig1}B)
\cite{Schneidman05,Shlens06,Yu08,Tang08}. For now we assume that the
bins are independent (an assumption whose validity we discuss below,
and in more detail in Sec.~\ref{small-time-bin-wrong}). The problem, then, is
to find $p_{true}(\mathbf{r}) \equiv p_{true}(r_1, r_2, ..., r_N)$ where
$r_i$ is a binary variable indicating no spike ($r_i=0$) or one or
more spikes ($r_i=1$) on neuron $i$. Since this, too, is a high
dimensional problem (though less so than the original spike time
representation), suppose that we instead construct a pairwise
approximation to $p_{true}$, which we denote $p_{pair}$, for a
population of size $N$. (The pairwise model derives its name from the
fact that it has the same mean and pairwise correlations as the true
model.) Our question, then, is: if $p_{pair}$ is close to
$p_{true}$ for small $N$, what can we say about how close the two
distributions are for large $N$?
\begin{SCfigure}
\centering
\includegraphics[height=11cm,width=7cm]
{figs/bins.eps}
\caption{Transforming spike trains to spike count.
{\bf A}. Spike rasters. Tick marks indicate spike times; different
rows correspond to different neurons. The horizontal dashed lines are the
bin boundaries.
{\bf B}. Spike count in each bin. In this example the bins are
small enough that there is at most one spike per bin, but this is not necessary -- one
could use bigger bins and have larger spike counts.
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
}
\label{Fig1}
\end{SCfigure}
To answer this question quantitatively, we need a measure of distance.
The measure we use, denoted $\Delta_N$, is defined in Eq.\
(\ref{Delta_N}) below, but all we need to know about it for now is
that if $\Delta_N=0$ then $p_{pair} = p_{true}$, and if $\Delta_N$ is
near one then $p_{pair}$ is far from $p_{true}$. In terms of
$\Delta_N$, our main results are as follows: first, for small $N$, in
what we call the perturbative regime, $\Delta_N$ is proportional to
$N-2$. In other words, as the population size increases, the pairwise
model becomes a worse and worse approximation to the true
distribution. Second, this behavior is entirely generic: for small
$N$, $\Delta_N$ increases linearly, no matter what the true
distribution is. This is illustrated schematically in
Fig.~\ref{cartoon_DN}, which shows the generic behavior of $\Delta_N$.
The solid red part of the curve is the perturbative regime, where
$\Delta_N$ is a linearly increasing function of $N$; the dashed curves
show possible behavior beyond the perturbative regime.
These results have an important corollary: if one does an
experiment and finds that $\Delta_N$ is increasing linearly with $N$, then
one has no information at all about the true distribution. The flip
side of this is more encouraging: if one can measure the true
distribution for sufficiently large $N$ that $\Delta_N$ saturates, as
in the dashed blue line in Fig.~\ref{cartoon_DN}, then one can have
some confidence that extrapolation to large $N$ {\em is} meaningful.
The implications for the interpretation of experiments is, then, that
extrapolation to large $N$ is valid only if one can analyze data past
the perturbative regime.
\begin{SCfigure}
\centering
\includegraphics[height=8cm,width=8cm]
{figs/cartoon_Delta_N.eps}
\caption{Cartoon illustrating the dependence of $\Delta_N$ on
$N$. For small $N$ there is always a
perturbative regime in which $\Delta_N$ increases
linearly with $N$ (solid red line). When $N$ becomes large, on the
other hand, $\Delta_N$
may continue increasing with $N$ (red and black dashed lines) or it may
plateau (cyan dashed line), depending on $p_{true}$. The
observation that $\Delta_N$ increases linearly with $N$ does not,
therefore, provide much, if any
information about the large $N$ behavior.
}
\label{cartoon_DN}
\end{SCfigure}
Under what conditions is a subsystem in the perturbative regime? The
answer turns out to be simple: the size of the system, $N$, times the
average probability of observing a spike in a bin, must be small
compared to $1$. For example, if the average probability is $1/100$,
then a system will be in the perturbative regime if the number of
neurons is small compared to $100$. This last observation would seem
to be good news for studies in which spikes are binned across time and
temporal correlations are ignored. For such binned spike trains, the
probability of a spike can be made arbitrarily small by simply
shrinking the time bins, and so the size of the population for which
the pairwise model appears good can be made arbitrarily large. The
problem with this, though, is that temporal correlations can be
ignored only when time bins are large compared to the autocorrelation
time. This leads to a kind of catch-22: pairwise models are guaranteed
to work well (in the sense that they describe spike trains in which
temporal correlations are ignored) if one uses small time bins, but
small time bins is the one regime where ignoring temporal
correlations is not a valid approximation.
In the next several sections we quantify the qualitative picture
presented above: we write down an
explicit expression for $\Delta_N$, explain why it increases linearly
with $N$, when $N$ is small, and provide additional tests, besides
assessing the linearity of $\Delta_N$, to determine whether or not one
is in the perturbative regime.
\subsection{A measure of goodness of fit}
A natural measure of the distance between $p_{pair}$ and $p_{true}$ is
the Kullback-Leibler (KL) divergence \cite{Kullback51}, denoted
$D_{KL}(p_{true}||p_{pair})$ and defined as
\begin{equation}
D_{KL}(p_{true}||p_{pair})=\sum_{\mathbf{r}} p_{true} (\mathbf{r}) \log_2
{p_{true}(\mathbf{r}) \over p_{pair}(\mathbf{r})}
\, .
\label{KL-def}
\end{equation}
\noindent
The KL divergence is zero if the two distributions are equal; otherwise
it is nonzero.
Although the KL divergence is a very natural measure, it is not easy
to interpret (except, of course, when it is exactly zero). That is
because a nonzero KL divergence tells us is that $p_{pair} \ne
p_{true}$, but it does not give us any real handle on how much we
benefit by including the pairwise correlations in our approximation. To
make sense of the KL divergence, then, we need something to compare
it to. A reasonable reference quantity, used by a number of authors
\cite{Schneidman05,Shlens06,Tang08}, is the KL divergence between the
true distribution and the independent one, the latter denoted
$p_{ind}$. The independent distribution, as its name suggests, is a
distribution in which the variables are taken to be independent,
\begin{equation} p_{ind}(r_1,\dots,r_N)= \prod_i p_i(r_i) \, ,
\label{ind} \end{equation}
\noindent
where $p_i(r_i)$ is the distribution of the response of the $i^{\rm
th}$ neuron, $r_i$. With this choice for a comparison, we define our
measure of goodness of fit as
\begin{equation}
\Delta_N= \frac{D_{KL}(p_{true}||p_{pair})}{D_{KL}(p_{true}||p_{ind})}.
\label{DeltaN-def}
\end{equation}
\noindent
Note that the denominator in this expression,
$D_{KL}(p_{true}||p_{ind})$, is usually referred to as
the multi-information \cite{Friedman01,Schneidman05,Slonim06}.
The quantity $\Delta_N$, which we introduced in the previous section,
lies between 0 and 1, and measures how well a pairwise model does
relative to an independent model. If it is 0, the pairwise model is
equal to the true model ($p_{pair}(\mathbf{r})=p_{true}(\mathbf{r})$); if it is near
1, the pairwise model offers little improvement over the independent
model; and if it is exactly 1, the pairwise model
is equal to the independent model
($p_{pair}(\mathbf{r})=p_{ind}(\mathbf{r})$), and so offers no improvement.
(Our assertion that $\Delta_N$ cannot
exceed 1 assumes that the pairwise model cannot be worse than the
independent one, something that is reasonable in practice but not
guaranteed in general.)
How do we attach physical meaning to the two divergences
$D_{KL}(p_{true}||p_{pair})$ and $D_{KL}(p_{true}||p_{ind})$?
For the latter, we use the fact that, as is easy to show,
\begin{equation}
D_{KL}(p_{true}||p_{ind})=S_{ind}-S_{true},
\label{KL-Ent-ind}
\end{equation}
\noindent
where $S_{ind}$ and $S_{true}$ are the entropies
\cite{Shannon49,Cover91} of $p_{ind}$ and
$p_{true}$, respectively, defined, as usual,
to be $S[p] = -\sum_{\mathbf{r}} p(\mathbf{r})
\log_2 p(\mathbf{r})$.
For the former, we use the definition of the KL divergence to write
\begin{equation}
D_{KL}(p_{true}||p_{pair})
=
- \sum_{\mathbf{r}} p_{true} (\mathbf{r}) \log_2 (p_{pair}(\mathbf{r}))
- S_{true}
\equiv
\tilde{S}_{pair}-S_{true}
\, .
\label{KL-Ent-maxent}
\end{equation}
\noindent
The quantity $\tilde{S}_{pair}$ has the flavor of an entropy, although
it is a true entropy only
when $p_{pair}$ is maximum entropy as well as pairwise (the
maximum entropy pairwise model, or maximum entropy model for short;
see Eq.~\eqref{maxent}). For other pairwise distributions,
all we need to know is that $\tilde{S}_{pair}$ lies between $S_{true}$
and $S_{ind}$, something that is guaranteed by our assumption that the
pairwise model is no worse than the independent model.
What Eqs.~\eqref{KL-Ent-ind}
and \eqref{KL-Ent-maxent} tell us is that
$\Delta_N$ (Eq.~\eqref{DeltaN-def}) is the ratio of the amount
of ``entropy'' {\em not} explained by the pairwise
model to the amount of entropy {\em not} explained by the
independent model. A plot illustrating the relationship
between $\Delta_N$, the two entropies $S_{ind}$ and $S_{true}$, and
the entropy-like quantity $\tilde{S}_{pair}$,
is shown in Fig.~\ref{entropies}.
\begin{SCfigure}[][t!!!]
\epsfig{file=figs/entropies.eps,height=8cm,width=8cm}
\caption{Schematic plot of
$S_{ind}$ (black line), $\tilde{S}_{pair}$ (cyan line) and $S_{true}$
(red line). The better the pairwise model, the closer
$\tilde{S}_{pair}$ is to $S_{true}$. This is reflected in the cost
function $\Delta_N$, which is the distance between the red
and cyan lines divided by the distance between the red and black
lines.
\bigskip
\bigskip
}
\label{entropies}
\end{SCfigure}
\subsection{$\Delta_N$ in the perturbative regime}
\label{N-dependence}
The extrapolation problem discussed above is the problem of
determining $\Delta_N$ in the large $N$ limit. This is hard to do
in general, but, as we show in
Methods, Sec.~\ref{perturbative_expansion},
there is a perturbative regime in which it is possible.
The small parameter that defines this regime is the average number of
spikes per bin; this is written quantitatively as $N \overline{\nu} \delta t$
where $\delta t$ is the bin size and
$\overline{\nu}$ the average firing rate. Letting
and $\nu_i$ the firing rate of neuron $i$, the latter quantity is
given by
\begin{equation}
\label{nubar}
\overline{\nu} \equiv {1 \over N} \sum_i \nu_i
\, .
\end{equation}
\noindent
Note that our small parameter depends on $N$, which means that
for $\overline{\nu}$ and $\delta t$ fixed, there is a maximum population
size we can access perturbatively. We return to this point below, as
it has major implications for experimental studies.
The first step in the perturbation expansion is to compute the
two quantities that make up $\Delta_N$: $D_{KL}(p_{true}||p_{ind})$
and $D_{KL}(p_{true}||p_{pair})$. As we show in
Methods, Sec.~\ref{perturbative_expansion}, these are given by
\begin{subequations}
\begin{align}
D_{KL}(p_{true}||p_{ind})&=g_{ind} N(N-1)\ (\overline{\nu} \delta t)^2 +
{\cal O} \left((N \overline{\nu} \delta t)^3 \right) \label{KLs_Na}
\\
D_{KL}(p_{true}||p_{pair})&=g_{pair} N(N-1)(N-2)\ (\overline{\nu} \delta t)^3 +
{\cal O} \left((N \overline{\nu} \delta t)^4 \right) \label{KLs_Nb}
\, .
\end{align}
\label{KLs_N}
\end{subequations}
\noindent
The exact forms of the prefactors $g_{ind}$ and $g_{pair}$ are given in
Eqs.\ (\ref{gind}) and (\ref{gpair}). The details, however, are not so
important; the important
things to know about them is that they are
independent of $N$ and $\overline{\nu} \delta t$, and they depend on the low
order statistics of the spike trains: $g_{ind}$ depends on the second
order normalized correlations function, and $g_{pair}$ depends on
the second and third order normalized correlations function, as
defined below in Eq.~\eqref{rho_norm}.
The $N$-dependence in the first term on
the right hand side of Eq.~\eqref{KLs_Na} has
been noted previously \cite{Schneidman05}, although the authors did
not compute the prefactor, $g_{ind}$.
Inserting Eq.~\eqref{KLs_N} into Eq.~\eqref{DeltaN-def} (into the
definition of $\Delta_N$), we arrive at our main result,
\begin{equation}
\Delta_N=\frac{g_{pair}}{g_{ind}} (N-2)\ \overline{\nu} \delta t +
{\cal O} \left((N \overline{\nu} \delta t)^2 \right)
\, .
\label{Delta_N}
\end{equation}
\noindent
This expression tells us how $\Delta_N$ scales with $N$ in the
perturbative regime --
the regime in which $N \overline{\nu} \delta t \ll 1$.
The key observation about this scaling is that it is
independent of the details of the true distribution, $p_{true}$. This
has a very important consequence, one that has major implications for
experimental data: if one does an experiment and finds that that
$\Delta_N$ is proportional to $N-2$, then the system is in the
perturbative regime, and one does not know whether
$p_{pair}$ will remain close to $p_{true}$ as $N$ increases.
What this means in practical terms is that if one wants to know
whether a particular pairwise
model is a good one for large systems,
it is necessary to consider values of $N$ that are significantly
greater than $N_c$, where
\begin{equation}
N_{c} \equiv \frac{1}{\overline{\nu} \delta t}
\, .
\label{N-c}
\end{equation}
We interpret $N_c$ as the value at which there is a crossover
in the behavior of the pairwise model.
Specifically, if $N \ll N_c$, the system is in the perturbative
regime and the pairwise model is not informative
about the large $N$ behavior, whereas if $N \gg N_c$, the
system is in a regime in which it may be
possible to make inferences about the behavior of the full system.
\subsection{The dangers of extrapolation}
\label{dangers:extrapolation}
Although the behavior of $\Delta_N$ in the perturbative regime does
not tell us much about its behavior at large $N$, it
is possible that other quantities that can be calculated in
the perturbative regime, $g_{ind}$, $g_{pair}$,
and $S_{ind}$ (the last one exactly), are informative,
as others have suggested \cite{Schneidman05}. Here we
show that they are also uninformative.
The easiest way to relate the perturbative regime to the large $N$
regime is to extrapolate Eqs.~(\ref{KLs_N}a) and (\ref{KLs_N}b), and
ask what their large $N$ behavior tells us. Generic versions of
these extrapolations, plotted on a log-log scale, are shown in
Fig.~\ref{extrapolation}A, along with a plot of the independent
entropy, $S_{ind}$ (which is necessarily linear in $N$; see
Sec.~\ref{s_asymptotic}). The first thing we notice about the
extrapolations is that they do not, technically, have a large $N$
behavior: one terminates at the point labeled $N_{ind}$, which is
where $D_{KL}(p_{true}||p_{ind}) = S_{ind}$ (and thus $S_{true} = 0$;
continuing the extrapolation implies negative true entropy);
the other at the point labeled $N_{pair}$, which is where
$D_{KL}(p_{true}||p_{pair})=S_{ind}$ (and thus $S_{true} \le 0$, since
$\tilde{S}_{pair} \le S_{ind}$).
\begin{figure}
\epsfig{file=figs/extrap.eps,height=6.5cm,width=16cm}
\caption{Cartoon showing the extrapolation of entropy and KL
divergences, and illustrating why the two natural quantities
derived from it, $N_{ind}$ and $N_{pair}$, occur beyond the point at which the
extrapolation is meaningful.
{\bf A.} Extrapolations on a log-log scale.
Black: $S_{ind}$ versus $N$; green: extrapolation of
$D_{KL}(p_{true}||p_{ind})$;
cyan: extrapolation of $D_{KL}(p_{true}||p_{maxent})$.
The red points are the data.
The points $N_{ind}$ and $N_{pair}$ label the
intersections of the two extrapolations with the independent entropy,
$S_{ind}$.
{\bf B.} Extrapolation of the entropies rather than the KL
divergences, plotted on a linear-linear scale.
The data, again shown in red, is barely visible in the lower left hand
corner. Black: $S_{ind}$ versus $N$; solid maroon: extrapolation of
$\tilde{S}_{pair}$;
solid orange: extrapolation of $S_{true}$.
The dashed maroon and orange lines are the extrapolations
of the true pairwise ``entropy'' and true entropy, respectively.
\bigskip
\bigskip
}
\label{extrapolation}
\end{figure}
Despite the fact that the extrapolations end abruptly, they still
might provide information about the large $N$ regime. For example,
$N_{pair}$ and/or $N_{ind}$ might be values of $N$ at which something
interesting happens. To see if this is the case, in
Fig.~\ref{extrapolation}B we plot the naive extrapolations of
$\tilde{S}_{pair}$ and $S_{true}$, as given by Eq.~\eqref{KLs_N}, on a
linear-linear plot, along with $S_{ind}$ (solid lines). This plot
contains no new information compared to Fig.~\ref{extrapolation}A, but
it does elucidate the meaning of the extrapolations. Perhaps its most
striking feature is that the naive extrapolation of $S_{true}$ has a
decreasing portion. As is easy to show mathematically, this cannot
happen (intuitively, that is because observing one additional neuron
cannot decrease the entropy of previously observed neurons). Thus,
$N_{ind}$, which occurs well beyond the point where the naive
extrapolation of $S_{true}$ is decreasing, has essentially no meaning,
something that has been pointed out previously
by Bethge et al. \cite{Bethge07}. The other potentially important value of $N$ is $N_{pair}$.
This, though, suffers from similar problems: either
$N_{pair}>N_{ind}$, in which case the entropy is
negative, or it crosses the green curve in Fig.~\ref{extrapolation}A
from below, meaning $\Delta_N > 1$. Either way, it also doesn't have much
meaning.
How do the naively extrapolated entropies -- the solid lines in
Fig.~\ref{extrapolation}B -- compare to the true entropies? To answer this, in
Fig.~\ref{extrapolation}B we show the true behavior of $S_{true}$ and
$\tilde{S}_{pair}$ versus $N$ (dashed lines). Note that $S_{true}$ is
asymptotically linear in $N$, even though the neurons are correlated,
a fact that forces $\tilde{S}_{pair}$ to be linear in $N$, as it is
sandwiched between $S_{true}$ and $S_{ind}$. (The asymptotically
linear behavior of $S_{true}$ is typical, even in highly correlated systems.
Although this is not always appreciated, it is easy to show; see
Sec.~\ref{s_asymptotic}.) Comparing the dashed and solid lines, we see
that the naively extrapolated and true entropies, and thus the naively
extrapolated and true values of $\Delta_N$, have extremely different
behavior. This further suggests that there is very little connection
between the perturbative and large $N$ regimes.
These observations can be summarized by noting that $g_{ind}$ and
$g_{pair}$ depend only on correlation coefficients up to third order
(see Eqs.~\eqref{gind} and \eqref{gpair}),
whereas the large $N$ behavior depends on
correlations at all orders. Thus, since $g_{ind}$ and $g_{pair}$ tell us very
little, if anything, about higher order correlations, it is not
surprising that they tell us very little about the behavior of
$\Delta_N$ in the large $N$ limit.
\subsection{Numerical simulations}
\label{numerical_simulations}
To check that our perturbation expansions, Eqs.~\eqref{KLs_N}
and \eqref{Delta_N}, are correct, and to
investigate the regime in which they are valid, we performed numerical
simulations. We generated, from synthetic data, a set of true
distributions, computed $D_{KL}(p_{true}||p_{ind})$,
$D_{KL}(p_{true}||p_{pair})$, and $\Delta_N$ numerically for each of
them, and compared to the values predicted by Eqs.~\eqref{KLs_N} and
\eqref{Delta_N}. The results are shown in Fig.~\ref{KL_Delta}. Before
discussing that figure, though, we explain our procedure for
constructing true distributions.
The set of true distributions we used were generated from a
third order model (so named because it includes up to third
order interactions). This model has the form
\begin{equation}
p_{true}(r_1, \dots, r_{N^*}) =
\frac{1}{Z_{true}}
\exp\left[
\sum_i h^{true}_i r_i +
\sum_{i < j} J^{true}_{ij} r_i r_j +
\sum_{i < j < k} K^{true}_{ijk} r_i r_j r_k
\right]
\label{ptrue}
\end{equation}
\noindent where $Z_{true}$ is a normalization constant, chosen to ensure the the
probability distribution sums to 1, and the sums over $i$, $j$ and $k$
run from 1 to $N^*$. The parameters $h^{true}_i, J^{true}_{ij}$ and
$K^{true}_{ijk}$ were chosen by sampling from distributions (see
Methods, Sec.~\ref{Generating_syn_data}), which allowed us
to generate many different true distributions.
For a particular simulation (corresponding to a column in
Fig.~\ref{KL_Delta}), we generated a true distribution with $N^*=15$,
randomly chose 5 neurons, and marginalized over them. This gave us a
10-neuron true distribution. True distributions with $N < 10$ were
constructed by marginalizing over additional neurons within our
10-neuron population. To achieve a
representative sample, we considered all possible marginalizations
(of which there are $10$ choose $N$, or $10!/[N!(10-N)!]$).
The results in Fig.~\ref{KL_Delta} are averages over these
marginalizations.
For neural data, the most commonly
used pairwise model is the maximum entropy model. Therefore, we use that
one here. To emphasize the maximum entropy nature of this model,
we replace the label ``{\em pair}'' that we
have been using so far with ``{\em maxent}.''
The maximum entropy distribution has the form
\begin{equation}
p_{maxent}(\mathbf{r}) =
\frac{1}{Z}
\exp\left[
\sum_i h_i r_i + \sum_{i < j} J_{ij} r_i r_j
\right]
\, .
\label{maxent}
\end{equation}
Fitting this distribution requires that we choose the $h_i$
and $J_{ij}$ so that the first and second moments match those of the
true distribution. Quantitatively, these conditions are
\begin{subequations}
\begin{align}
\langle r_i \rangle_{maxent} & = \langle r_i \rangle_{true}
\\
\langle r_i r_j \rangle_{maxent} & = \langle r_i r_j \rangle_{true}
\end{align}
\label{pair-fit}
\end{subequations}
\noindent
where the angle brackets, $\langle \dots \rangle_{maxent}$ and
$\langle \dots \rangle_{true}$, represent averages with respect to
$p_{maxent}$ and $p_{true}$, respectively.
Once we have $h_i$ and $J_{ij}$ that satisfy Eq.~\eqref{pair-fit}, we calculate the KL
divergences, Eqs. (\ref{KLs_N}), and use those to compute $\Delta_N$.
\begin{figure}
\centerline{
\epsfig{file=figs/theory_vs_exp1,height=12cm,width=12cm}
}
\caption{The $N$ dependence of the KL divergences and the goodness of
fit, $\Delta_N$. Data was generated from a third order model, as
explained in Sec.~\ref{Generating_syn_data} (Methods), and
fit to maximum entropy pairwise model and independent
models. All data points correspond to averages over
marginalizations of the true distribution
(see text for details).
The red points were computed directly from the model fits,
Eqs.~\eqref{KL-def}, \eqref{DeltaN-def} and \eqref{KL-Ent-ind};
the blue points are predictions of the perturbative
expansions,
Eqs.~\eqref{KLs_N} and \eqref{Delta_N}.
The three columns correspond to $\overline{\nu} \delta t =$ 0.024, 0.029, and
0.037, from left to right.
{\bf A}, {\bf B}, {\bf C} ($\overline{\nu} \delta t = 0.024$).
Predictions from the perturbative expansion are in good agreement with
the measurements up to $N=10$,
indicating that the data is in the perturbative regime.
{\bf D}, {\bf E}, {\bf F} ($\overline{\nu} \delta t = 0.029$).
Predictions from the perturbative expansion are in good agreement with
the measurements up to $N=7$, indicating that the data is only
partially in the perturbative regime.
{\bf G}, {\bf H}, {\bf I} ($\overline{\nu} \delta t = 0.037$).
Predictions from the perturbative expansion are not in good agreement with
the measurements, even for small $N$, indicating that the data is
outside the perturbative regime.
}
\label{KL_Delta}
\end{figure}
The results are shown in Fig.~\ref{KL_Delta}. The rows correspond to our three
quantities of interest: $D_{KL}(p_{true}||p_{ind})$,
$D_{KL}(p_{true}||p_{pair})$, and $\Delta_N$ (top to bottom). The
columns correspond to different values of $\overline{\nu} \delta t$, with the
smallest $\overline{\nu} \delta t$ on the left and the largest on the right.
Red circles are the actual values of these quantities; blue ones are
the predictions from Eqs.~\eqref{KLs_N} and \eqref{Delta_N}.
As suggested by our perturbation analysis, the smaller the value of
$\overline{\nu} \delta t$, the better the agreement between computed and
predicted values. Our simulations corroborate this: the left column of
Fig.~\ref{KL_Delta}
has $\overline{\nu} \delta t = 0.024$, and agreement is almost perfect out
to $N=10$; the middle column has $\overline{\nu} \delta t = 0.029$, and
agreement is almost perfect out to $N=7$; and the right column
has $\overline{\nu} \delta t = 0.037$, and agreement is not good for any
value of $N$.
These results validate the perturbation expansions in
Eqs.~\eqref{KLs_N} and \eqref{Delta_N}, and show that those expansions
provide sensible predictions -- at least for some parameters. They
also suggest a natural way to assess the significance of one's data:
plot $D_{KL}(p_{true}||p_{ind})$, $D_{KL}(p_{true}||p_{pair})$, and
$\Delta_N$ versus $N$, and look for agreement with the predictions of
the perturbation expansion. If agreement is good, as in the left
column of Fig.~\ref{KL_Delta}, then one is in the perturbative regime, and one
knows very little about the true distribution. If, on the other hand,
agreement is bad, as in the right column, then one is out of the
perturbative regime, and there is hope of extracting meaningful
information about the relationship between the true and pairwise
models.
That said, the qualifier ``at least for some parameters'' is an
important one. This is because the perturbation
expansion is essentially an expansion that depends on
the normalized correlation coefficients, and there is an
underlying assumption that they don't
exhibit pathological behavior. The $k^{\rm th}$ normalized
correlation coefficient for the distribution $p$,
denoted $\rho^p_{i_1 i_2 \dots i_k}$, is written
\begin{equation}
\rho^p_{i_1 i_2 \dots i_k} \equiv
\frac{
\langle
(\langle r_{i_1} - \langle r_{i_1} \rangle_p)
(\langle r_{i_2} - \langle r_{i_2} \rangle_p)
\dots
(\langle r_{i_k} - \langle r_{i_k} \rangle_p)
\rangle_p
}
{
\langle r_{i_1} \rangle_p
\langle r_{i_2} \rangle_p
\dots
\langle r_{i_k} \rangle_p
}
\, .
\label{rho_norm}
\end{equation}
\noindent
A potentially problematic feature
of the correlation coefficients is that the denominator is
a product over mean activities. If the mean activities are
small, the
denominator can become very small, leading to very large correlation
coefficients. Although our perturbation expansion is always
valid for sufficiently small time bins (because the correlation
coefficients eventually becomes independent of bin size; see Methods,
Sec.~\ref{Bin-size}), ``sufficiently small'' can depend in detail on the
parameters. For instance, at the maximum population size tested
($N=10$) and for the true distributions that had
$\overline{\nu} \delta t < 0.03$, the absolute
error of the prediction had a median of approximately $16\%$. However,
about 11\% of the runs had errors larger than $60\%$.
Thus, the exact size of the small parameter at which the
perturbative expansion breaks down can depend on the details of the
true distribution.
\subsection{Local fields and pairwise couplings have a simple
dependence on firing rates and correlation coefficients
in the perturbative regime}
\label{fields-Js}
Estimation of the KL divergences and $\Delta_N$ from real data can be
hard, in the sense that it takes a large amount of data for them
to converge to their true values. We therefore provide a
second set of relationships that can be used to determine whether or
not a particular data set is in the perturbative regime. These
relationships are between the parameters of the maximum entropy model,
the $h_i$ and $J_{ij}$, and the mean activity and normalized second
order correlation coefficient (the latter defined in
Eq.~\eqref{rho_norm}).
Since the quantity $\overline{\nu} \delta t$ plays a central role in our
analysis, we replace it with a single parameter, which we denote
$\delta$,
\begin{equation}
\delta \equiv \overline{\nu} \delta t
\, .
\label{delta}
\end{equation}
\noindent
In terms of this parameter,
we find, (using the same perturbative approach that led us to Eqs.\
(\ref{KLs_N}) and (\ref{Delta_N}); see Sec.~\ref{local-fields}), that
\begin{subequations}
\begin{align}
h_i &=
\log \left[ \langle r_i \rangle^{-1} -1\right]+ {\cal O}(N\delta)
\\
J_{ij} &=
\log \left[ 1+\rho_{ij} \right]+ {\cal O}(N \delta)
\end{align}
\label{hJ}
\end{subequations}
\noindent
where $\rho_{ij}$, the normalized second order
correlation coefficient, is defined in Eq.~\eqref{rho_norm} with
$k=2$; it is given explicitly by
\begin{equation}
\rho_{ij}=\frac{\langle r_i r_j \rangle-\langle r_i \rangle_ \langle
r_j \rangle}{\langle r_i \rangle_ \langle r_j \rangle}
\, .
\label{norm-corr}
\end{equation}
\noindent
(We don't need a superscript on $\rho$ or a subscript on the
angle brackets because the first and second moments are
the same under the true and pairwise distributions.)
Equation \eqref{hJ} tells us that the $N$-dependence of the
$h_i$ and $J_{ij}$, the local fields and pairwise couplings, are very
weak. In Fig.~\ref{h_J_N} we confirm these theoretical predictions
through numerical simulations.
\begin{figure}
\epsfig{file=figs/theory_vs_exp3.eps,height=12.5cm,width=15.5cm}
\caption{Comparison of the true local fields ($h_i$, top row) and pairwise
interactions ($J_{ij}$, bottom row), to
the predictions from the perturbation expansion, Eq.~\eqref{hJ}.
Values of $N$ ranging from 5 to 10 are shown, with different
colors corresponding to different $N$s.
For each value of $N$, fits are shown for 45 realization of the true
distribution. Insets show the $N$-dependence of the mean
local fields (top) and mean pairwise interactions (bottom).
The three columns correspond exactly to the columns in Fig.~\ref{KL_Delta}.
{\bf A}, {\bf B} ($\overline{\nu} \delta t = 0.024$).
There is a very good match between the theoretical and fit values
of both local fields and pairwise interactions.
{\bf C}, {\bf D} ($\overline{\nu} \delta t = 0.029$).
Even though $\overline{\nu} \delta t$ has increased, the match is still good.
{\bf E}, {\bf F} ($\overline{\nu} \delta t = 0.037$).
The predicted and fit local fields and pairwise interactions do not match
as well as the cases shown in A,B, C and D. There is also now a
strong $N$-dependence in the mean local
fields, and a somewhat weaker dependence in the pairwise interactions.
This indicates that the perturbative expansion is breaking down.
}
\label{h_J_N}
\end{figure}
As an aside, we should point out
that the $N$-dependence is a function of the
variables used to represent the firing patterns. Here we use 0 for silence and 1
for firing, but another, possibly more common,
representation, derived from the Ising model
and used in a number of studies \cite{Schneidman05,Tang08,Yu08}, is to use $-1$ for silence and $+1$ for firing. This
amounts to making the change of variables $s_i = 2r_i-1$. In terms of
$s_i$, the maximum entropy model has the form
$p(\mathbf{r}) \sim \exp \left[ \sum_i h^{ising}_i s_i + \sum_{i < j}
J^{ising}_{ij} s_i s_j \right]$ where
$h_i^{ising}$ and $J_{ij}^{ising}$ are given by
\begin{subequations}
\begin{align}
h_i^{ising} &= \frac{h_i}{2} +
\sum_{j \ne i} \frac{J_{ij}}{4}
\\
J_{ij}^{ising} &= \frac{J_{ij}}{4}
\, .
\end{align}
\label{ising}
\end{subequations}
\noindent
The second term on the right hand side of Eq.~(\ref{ising}a) is
proportional to $N-1$, which means the local fields in the Ising
representation acquire a linear $N$-dependence that was not present in
our 0/1 representation. The two
studies that reported the $N$-dependence of the local fields
\cite{Schneidman05,Tang08} used this
representation, and, as predicted, their local fields
had a component that was linear in $N$.
Equation (\ref{hJ}b) does more than just predict a lack of
$N$-dependence; it also provides a functional relationship between the
pairwise couplings and the normalized pairwise correlations function,
$\rho_{ij}$.
In Figs.~\ref{rho_J}A-C we plot the pairwise couplings, $J_{ij}$,
versus the normalized pairwise correlation coefficient,
$\rho_{ij}$ (blue dots),
along with the prediction from Eq.\ (\ref{hJ}b) (black line).
Consistent with with our
predictions, the data in Figs.~\ref{rho_J}A-C essentially
follows a line -- the one predicted by Eq.~(\ref{hJ}b).
\begin{figure}
\centering
\epsfig{file=figs/theory_vs_exp2.eps,height=10cm,width=10cm}
\caption{Pairwise couplings versus pairwise correlations,
showing that there is a simple relation between
$J_{ij}$ and $\rho_{ij}$ but not between $J_{ij}$ and $c_{ij}$.
Top row: $J_{ij}$ versus the normalized coefficients,
$\rho_{ij}$ (blue points), along with predicted relationship, via
Eq.~(\ref{hJ}b) (black line).
Bottom row: $J_{ij}$ versus the Pearson correlation
coefficients, $c_{ij}$, Eq.~\eqref{c-ij} (blue points).
The three columns correspond exactly to the columns in
Fig.~\ref{KL_Delta}, for which $\overline{\nu} \delta t$ = 0.024, 0.029, and
0.037, from left to right. The prediction in the top row (black line)
matches the data well, even in the rightmost column.}
\label{rho_J}
\end{figure}
A relationship between the pairwise couplings and the correlations
coefficients has been sought previously, but
for the more standard Pearson correlation
coefficient \cite{Schneidman05,Yu08,Tang08}.
Our analysis explains why it was not found. The Pearson correlation
coefficient, denoted $c_{ij}$, is given by
\begin{equation}
c_{ij}\equiv \frac{\langle r_i r_j \rangle-\langle r_i \rangle \langle r_j \rangle }
{\big[(\langle r_i^2 \rangle-\langle r_i \rangle^2)(\langle r_j^2
\rangle-\langle r_j \rangle^2)\big]^{1/2}}
\, .
\label{c-ij}
\end{equation}
\noindent
In the small $\langle r_i \rangle$ limit -- the limit of interest --
the right hand side of Eq.~\eqref{c-ij} is approximately equal to
$[\langle r_i \rangle\langle r_j \rangle]^{1/2} \, \rho_{ij}$. Because
$[\langle r_i \rangle\langle r_j \rangle]^{1/2}$ depends on the local
fields, $h_i$ and $h_j$ (see Eq.\ (\ref{hJ}a)) {\em and} there is a
one-to-one relationship between $\rho_{ij}$ and $J_{ij}$
(Eq.~(\ref{hJ}b)), there can't be a one-to-one relationship between
$c_{ij}$ and $J_{ij}$. We verify the lack of a relationship in
Figs.~\ref{rho_J}D-E, where we again plot $J_{ij}$, but this time
versus the standard correlation coefficient, $c_{ij}$. As predicted,
the data in Figs.~\ref{rho_J}D-E is scattered over two
dimensions. This suggests that $\rho_{ij}$, not $c_{ij}$, is the
natural measure of the correlation between two neurons when the have a
binary representation, something that has also been suggested by Amari
based on information-geometric arguments \cite{Amari08}.
Note that the lack of a simple relationship between the pairwise
couplings and the standard correlation coefficient has been a major
motivation in building maximum entropy models
\cite{Schneidman05,Yu08}. This is for good reason: if there is a
simple relationship, knowing the $J_{ij}$'s adds essentially
nothing. Thus, plotting $J_{ij}$ versus $\rho_{ij}$ (but not $c_{ij}$)
is an important test of one's data, and if the two quantities fall on
the curve predicted by Eq.~(\ref{hJ}b), the maximum entropy model is
adding very little information, if any.
\section{Is there anything wrong with using small time bins?}
\label{small-time-bin-wrong}
An outcome of our perturbative approach is that the goodness of fit
measure, $\Delta_N$, decreases linearly with bin size (see
Eq.~\eqref{DeltaN-def}). This suggests that one could make the
pairwise model look better and better simply by making the bin size
smaller and smaller. Is there anything wrong with this? The answer is
yes, for reasons we discussed in Sec.~\ref{problem:extrapolation};
here we emphasize and expand on this issue, as it is an important one
for making sense of experimental results.
The problem arises because what we have been calling the ``true''
distribution is not really the true distribution of spike trains. It
is the distribution assuming independent time bins, an assumption that
becomes worse and worse as we make the bins smaller and smaller.
(We use this potentially confusing nomenclature primarily
because all studies of
neuronal data carried out so far have assumed temporal independence,
and compared the pairwise distribution to the temporally independent
-- but still correlated across neurons -- distribution
\cite{Schneidman05,Shlens06,Yu08,Tang08}. In addition, the correct
name ``true under the assumption of temporal
independence,'' is unwieldy.)
Here we quantify
how much worse. In particular, we show that if one uses time bins that are
small compared to the characteristic correlation time in the spike
trains, the pairwise model will not provide a good description of the
data. Essentially, we show that, when the time bins are too small, the
error one makes in ignoring temporal correlations is larger than the
error one makes in ignoring correlations across neurons.
As usual, we divide time into bins of size $\delta t$. However,
because we are dropping the independence assumption, we
use $\mathbf{r}^t$, rather than $\mathbf{r}$, to denote the response in bin $t$.
The full probability distribution over all time bins is denoted
${\cal P}(\mathbf{r}^1, \dots,\mathbf{r}^M)$. Here $M$ is the number of bins; it is
equal to $T/\delta t$ for spike trains of length $T$.
If time bins are approximately independent, we can write
\begin{equation}
{\cal P}(\mathbf{r}^1, \dots,\mathbf{r}^M)\approx \prod_t p_{true}(\mathbf{r}^t)
\, ,
\label{temp-ind}
\end{equation}
\noindent
and if the pairwise model is a good one, we have
\begin{equation}
p_{true}(\mathbf{r}^t)\approx p_{pair}(\mathbf{r}^t).
\label{pair-approx}
\end{equation}
\noindent
Combining Eqs.\ (\ref{temp-ind})
and Eq.\ (\ref{pair-approx}) then gives us an especially simple
expression for the full probability distribution:
{${\cal P}(\mathbf{r}^1, \dots,\mathbf{r}^M)\approx \prod_t p_{pair}(\mathbf{r}^t)$.
The problem with small time bins lies in Eq.~(\ref{temp-ind}):
the right hand side is a good approximation to the true
distribution} when the time bins are large compared to the spike train
correlation time, but it is a bad approximation when
the time bins are small (because adjacent time bins become
highly correlated). To quantify how bad, we compare the error one makes
assuming independence across time to the error one makes assuming
independence across neurons. The ratio of those two errors, denoted
$\gamma$, is given by
\begin{equation}
\gamma=\frac{D_{KL}\left({\cal P}(\mathbf{r}^1, \dots,\mathbf{r}^M) \big| \big|
\prod_t p_{pair}(\mathbf{r}^t) \right)}{M D_{KL}(p(\mathbf{r})||p_{ind}(\mathbf{r}))}
\, .
\label{gamma-def}
\end{equation}
\noindent
It is relatively easy to compute
$\gamma$ in the limit of small time bins (see Sec. \ref{ind-assumption}),
and we find that
\begin{equation}
\gamma = \Delta_N + (M-1) +
{\log_2 M \over g_{pair} \delta}
\, . \label{gamma}
\end{equation}
As expected, this reduces to our old result, $\Delta_N$, when there is
only one time bin ($M=1$). When $M$ is larger than 1, however (which
is, of course, the case of interest), the second term is always at
least one, and for small bin size, the third term is much larger than
one. Consequently, if we use bins that are small compared to the
temporal correlation time of the spike train, the pairwise model will
do a very bad job describing the full, temporally correlated
spike trains.
\section{Discussion}
Probability distributions over the configurations of biological
systems are extremely important quantities. However, because of the
large number of interacting elements comprising such systems, these
distributions can almost never be determined directly from
experimental data. Using parametric models to approximate the true
distribution is the only existing alternative. While such models are
promising, they are typically applied only to small subsystems, not
the full system. This raises the question: are they good models of the
full system?.
We answered this question for a class of parametric models known as
pairwise models. We focused on a particular application, neuronal
spike trains, and our main result is as follows: if one were to record
spikes from multiple neurons, use sufficiently small time bins and a
sufficiently small number of
cells, and assume temporal independence, then a
pairwise model would almost always succeed in
matching the true (but temporally independent)
distribution -- whether or not it would match the
true (but still temporally independent)
distribution for large time bins or a large number of cells.
In other words, pairwise
models in the ``sufficiently small'' regime, what we refer to as the
perturbative regime, have almost no predictive value for what will
happen with large populations. This makes extrapolation from small to
large systems dangerous.
This observation is important because pairwise models, and in
particular maximum entropy pairwise models, have recently attracted a
great deal of attention: they have been applied to salamander and
guinea pig retinas \cite{Schneidman05}, primate retina
\cite{Shlens06}, primate cortex \cite{Tang08}, cultured cortical
networks \cite{Schneidman05}, and cat visual cortex \cite{Yu08}.
These studies have mainly operated close
to the perturbative regime.
For example, Schneidman et al.\ \cite{Schneidman05} had $N \overline{\nu}
\delta t \approx 0.35$, Tang et al.\ \cite{Tang08} had $N
\overline{\nu} \delta t \approx 0.06$ to $0.4$
(depending on the preparation), and Yu et al.\ had $N \overline{\nu}
\delta t \approx 0.2$.
For these studies, then, it would be
hard to justify extrapolating to large populations.
The study by Shlens et al.\ \cite{Shlens06}, on the other hand, might
be more amenable to extrapolation. This is because spatially localized
stimuli were used to stimulate retinal ganglion cells, for which a
nearest neighbor maximum entropy models provided a good fit to their
data. (Nearest neighbor means $J_{ij}$ is zero unless cell $i$ and
cell $j$ are adjacent.) As is not hard to show, for nearest neighbor
models the small parameter in the perturbative expansion is $K \overline{\nu}
\delta t$ where $K$ is the number of nearest neighbors. Since $K$ is
fixed, independent of the population size, the small parameter will
not change as the population size increases. Thus, Shlens et al.\ may
have tapped into the large population behavior even though they
considered only a few cells at a time in their analysis.
\subsection{Time bins and population size}
That the pairwise model is always good if $N \overline{\nu} \delta t$ is
sufficiently small has strong implications: if we want to build a good
model for a particular $N$, we can simply choose a bin size that is
small compared to $1/N \overline{\nu}$. However, one of the assumptions in all
pairwise models used on neural data is that bins at different times
are independent. This produces a tension between small time bins and
temporal independence: small time bins essentially ensure that a
pairwise model will provide a close approximation to
a model with independent bins, but
they make adjacent bins highly correlated. Large time bins
come with no such assurance, but they make adjacent bins
independent. Unfortunately, this tension is often unresolvable in
large populations, in the sense that pairwise models are assured
to work only up to populations of size $1/(\overline{\nu} \tau_{\rm corr})$
where $\tau_{\rm corr}$ is the typical correlation time. Given that
$\overline{\nu}$ is at least several Hz, for experimental paradigms in which
the correlation time is more than a few hundred ms, $1/(\overline{\nu}
\tau_{\rm corr})$ is about one, and this assurance does not apply to
even moderately sized populations of neurons.
These observations are especially relevant for studies that use small
time bins to model spike trains driven by natural stimuli. This is
because the long correlation times inherent in natural stimuli are
passed on to the spike trains, so the assumption of independence
across time (which is required for the independence assumption to be
valid) breaks badly. Knowing that these models are successful in
describing spike trains under the independence assumption, then, does
not tell us whether they
will be successful in describing full, temporally correlated,
spike trains. Note that for studies that
use stimuli with short correlation times (e.g., non-natural stimuli
such as white noise), the temporal correlations in the spike trains
are likely to be short, and using small time bins may be perfectly
valid.
The only study that has investigated the issue of temporal
correlations in maximum entropy models does indeed support the above
picture \cite{Tang08}: for the parameters used in that study
($N \overline{\nu} \delta t = 0.06$ to $0.4$), the maximum entropy pairwise
model provided a good fit to the data ($\Delta_N$ was typically
smaller than 0.1), but it did not do a good
job modeling the temporal structure of the spike trains.
\subsection{
Other biological problems that have been approached with pairwise
models, e.g, protein folding
}
As mentioned in the Introduction, in addition to the studies on
neuronal data, studies on protein folding have also emphasized the
role of pairwise interactions \cite{Socolich05,Russ05}. Briefly,
proteins consist of strings of amino acids, and a major question in
structural biology is: what is the probability distribution of amino
acid strings in naturally folding proteins? One way to answer this is
to approximate the full probability distribution of naturally folding
proteins from knowledge of single-site and pairwise distributions. One
can show that there is a perturbative regime for proteins as well.
This can be readily seen using the celebrated HP protein model
\cite{Dill85}), where a protein is composed of only two types of amino
acids: if, at each site, one amino acid type is preferred and occurs
with high probability, say $1-\delta$ ($\delta \ll 1$), then a protein of
length shorter than $1/\delta$ will be in the perturbative regime,
and, therefore, a good match between the true distribution and the
pairwise distribution for such a protein is virtually guaranteed.
Fortunately, though, the properties of real proteins generally prevent
this from happening: at the majority of sites in a protein, the
distribution of amino acids is {\em not} sharply peaked around one
amino acid. Even for those sites that are sharply peaked (the
evolutionarily-conserved sites), the probability of the most likely
amino acid rarely exceeds $90\%$ \cite{Lockless99,Vargas-Madraz94}.
This puts proteins consisting of only a few amino acids out of the
perturbative regime, and puts longer proteins -- the ones usually
studied using pairwise models -- well out of it.
This difference is fundamental: because many of the studies that have
been carried out on neural data were in the perturbative regime, the
conclusions of those studies -- specifically, the conclusion that
pairwise models provide accurate descriptions of large populations of
neurons -- is not yet supported. This is not the case for the protein
studies, because they are not in the perturbative regime. Thus, the
evidence that pairwise models provide accurate descriptions of protein
folding remain strong and exceedingly promising.
\subsection{Outlook}
We have developed a framework for assessing the validity of pairwise
models applied to small systems. Essentially, we developed a set
of tests to determine whether one's
data is in the perturbative regime, a regime in which extrapolation to
large populations is not warranted. This should serve as a useful
guide, not just for analyzing experiments, but also for designing
them.
Although our framework is general, we focused primarily on its
application to neural data. One of our main results is that
the bin size carries an important tradeoff: if the bin size is small,
then pairwise models work well, but at the price of ignoring temporal
correlations; if the bin size is large enough so that adjacent bins
are weakly correlated, then there is no guarantee that pairwise models
will work at all. Pairwise models with small time bins,
therefore, might be rescued by a
small modification: take into account correlations across time as well
as neurons. This would increase the complexity of the models,
but the amount of data one needs to fit them would still not be so
large, as
only pairwise correlations and single neuron firing rates need
to be estimated.
Whether this modification would produce good models is
not clear, but if it did it would bring us much closer to a
fundamental understanding of neural systems.
\section{Methods}
\subsection{The behavior of the true entropy in the large $N$ limit}
\label{s_asymptotic}
To understand how the true entropy behaves in the large $N$ limit,
we need only
express the difference of the entropies as a mutual information.
Using $S_N$ to denote the true entropy of $N$ neurons and $I(1;N)$ to
denote the mutual information between one neuron and the other $N$
neurons in a population of size $N+1$, we have
\begin{equation}
(S_N + S_1) - S_{N+1} = I(1;N) \ \ \ \Rightarrow \ \ \ S_{N+1} - S_N
= S_1 - I(1;N)
\, .
\label{deltasdn}
\end{equation}
\noindent
If knowing the activity of $N$ neurons does not constrain the firing of
neuron $N+1$, then the single neuron entropy, $S_1$, will exceed the
mutual information, $I(1;N)$, and the entropy will be an increasing
function of $N$. For the entropy to be linear in $N$, all we need is
that the mutual information saturates with $N$. Because of synaptic
failures, this is a reasonable assumption for
networks of neurons: even if we observed all the neurons, there is
still residual noise associated with uncertainty about which vesicles
release neurotransmitter. Thus, using $I(1;\infty)$ to denote the asymptotic
value of the mutual information and $\langle S_1 \rangle$ to denote
the average single-neuron entropy, we have
\begin{equation}
S_N = N[\langle S_1 \rangle - I(1;\infty)] + {\rm corrections}
\, ,
\label{dsdn}
\end{equation}
\noindent
where the corrections are sublinear in $N$.
\subsection{Perturbative Expansion}
\label{perturbative_expansion}
Our main quantitative result, given in Eq.~\eqref{KLs_N}, is that the
KL divergence between the true distribution and both the independent
and pairwise distributions can be computed perturbatively as an
expansion in powers of $N \delta$. Here we carry out this
expansion, and derive explicit expressions for the quantities $g_{ind}$
and $g_{pair}$.
To simplify our notation, here we use $p(\mathbf{r})$ for the true
distribution. The critical step in computing the KL divergences
perturbatively is to use the Sarmanov-Lancaster
expansion \cite{Sarmonov62,Sarmonov63,Lancaster58,Lancaster58,Lancaster63,Bahadur61} for
$p(\mathbf{r})$,
\begin{equation}
p(\mathbf{r}) = p_{ind}(\mathbf{r}) \, (1+\xi_p(\mathbf{r}) )
\label{p}
\end{equation}
\noindent
where
\begin{subequations}
\begin{align}
p_{ind}(\mathbf{r}) & = \frac{\exp \sum_i {{\cal H}}^p_i r_i }
{\prod_i \left[ 1 + \exp ({\cal H}^p_i r_i) \right]}
\\
\xi_p(\mathbf{r}) &
\equiv \sum_{i<j} {\cal J}^p_{ij} \delta r_i \delta r_j + \sum_{i<j<k}
{\cal K}^p_{ijk} \delta r_i \delta r_j \delta r_k + \dots
\\
\delta r_i & \equiv r_i- \bar{r}_i
\\
\bar{r}_i & \equiv (1+\exp(-{\cal H}_i))^{-1}
\, .
\end{align}
\label{p_expansion}
\end{subequations}
This expansion has a number of important, but not immediately obvious,
properties. First, as can be shown by a direct calculation,
\begin{equation}
\langle r_i \rangle_p =
\langle r_i \rangle_{ind} =
\bar{r}_i
\label{ribar}
\end{equation}
\noindent
where the subscripts $p$ and $ind$ indicate an average with respect to
$p(\mathbf{r})$ and $p_{ind}(\mathbf{r})$, respectively. This has an immediate
corollary,
\begin{equation}
\langle \delta r_i \rangle_{ind} = 0
\, .
\end{equation}
\noindent
This last relationship is important, because it tells us that
if a product of $\delta r$'s contains any terms linear in one of the
$\delta r_i$, the whole product averages to zero under the independent
distribution. This can be used to show that
\begin{equation}
\langle \xi_p(\mathbf{r}) \rangle_{ind} = 0
\label{xibar}
\end{equation}
\noindent
from which it follows that
\begin{equation}
\sum_\mathbf{r} p(\mathbf{r}) = \langle (1 + \xi_p(\mathbf{r}) \rangle_{ind} = 1
\, ,
\end{equation}
\noindent
which tells us that $p(\mathbf{r})$ is properly normalized.
\noindent
Finally, a slightly more involved calculations provides us with a
relationship between the parameters of the model and the moments:
for $i \ne j \ne k$,
\begin{subequations}
\begin{align}
\langle \delta r_i \delta r_j \rangle_p & =
\bar{r}_i (1-\bar{r}_i )\bar{r}_j (1-\bar{r}_j) {\cal J}^p_{ij}
\\
\langle \delta r_i \delta r_j \delta r_k \rangle_p &=
\bar{r}_i (1-\bar{r}_i )\bar{r}_j (1-\bar{r}_j)\bar{r}_k (1-\bar{r}_k)
{\cal K}^p_{ijk}
\, .
\end{align}
\label{moments}
\end{subequations}
\noindent
Virtually identical expressions hold for higher order moments.
It is this last set of relationships that make the
Sarmanov-Lancaster expansion
so useful.
Note that Eqs.~(\ref{moments}a) and (\ref{moments}b), along with the
expression for the normalized correlation coefficients given in
Eq.~\eqref{rho_norm}, imply that
\begin{subequations}
\begin{align}
(1-\bar{r}_i ) (1-\bar{r}_j) {\cal J}^p_{ij} &= \rho^p_{ij}
\\
(1-\bar{r}_i ) (1-\bar{r}_j) (1-\bar{r}_k) {\cal K}^p_{ijk} &= \rho^p_{ijk}
\, .
\end{align}
\label{jk_moments}
\end{subequations}
\noindent
These identities will be extremely useful for simplifying expressions
later on.
Because the moments are so closely related to the parameters of
the distribution, moment matching is especially convenient: to
construct a distribution whose moments match those of $p(\mathbf{r})$ up to some
order, one simply needs to ensure that the parameters of that
distribution, ${\cal H}_i$, ${\cal J}_{ij}$, ${\cal K}_{ijk}$, etc., are identical to
those of the true distributions up to the order of interest.
In particular, let us write down a new distribution, $q(\mathbf{r})$,
\begin{subequations}
\begin{align}
q(\mathbf{r}) & = p_{ind}(\mathbf{r})
\, (1+\xi_q(\mathbf{r}) )
\\
\xi_q(\mathbf{r}) &
=\sum_{i<j} {\cal J}^q_{ij} \delta r_i \delta r_j + \sum_{i<j<k}
{\cal K}^q_{ijk} \delta r_i \delta r_j \delta r_k + \dots
\, .
\end{align}
\label{q_expansion}
\end{subequations}
\noindent
We can recover the independent distribution by
letting $\xi_q(\mathbf{r})=0$, and we can recover the pairwise distribution by
letting ${\cal J}^q_{ij} = {\cal J}^p_{ij}$. This allows us
to compute $D_{KL}(p||q)$ in the general case, and then either set
$\xi_q$ to zero or set ${\cal J}^q_{ij}$ to ${\cal J}^p_{ij}$.
Note that expressions analogous to those in
Eq.~(\ref{xibar}-\ref{jk_moments})
exist for averages with respect to $q(\mathbf{r})$; the only difference is
that $p$ is replaced by $q$.
\subsubsection{The KL divergence in the Sarmanov-Lancaster representation}
\label{section:KL}
Using Eqs.~\eqref{p} and (\ref{q_expansion}a) and a small amount of
algebra, the KL
divergence between $p(\mathbf{r})$ and $q(\mathbf{r})$ may be written
\begin{equation}
D_{KL}(p||q)=
\frac{1}{\ln 2} \left\langle f(\xi_p(\mathbf{r}), \xi_q(\mathbf{r}))\right\rangle_{ind}
\label{fxybar}
\end{equation}
\noindent
where
\begin{equation}
f(x, y) \equiv (1+x)[ \ln(1+x) - \ln(1+y)] - (x-y)
\, .
\label{fxy}
\end{equation}
\noindent
To derive Eq.~\eqref{fxybar}, we used the fact that
$\langle \xi_p \rangle_{ind}
=\langle \xi_q \rangle_{ind} = 0$ (see Eq.~\eqref{xibar}).
The extra term $(x-y)$ was included to ensure that
$f(x,y)$ and its first derivatives vanish at $x=y=0$, something that
greatly simplifies our analysis later on.
Our approach is to Taylor expand the right hand side of
Eq.~\eqref{fxybar} around $\xi_p =
\xi_q = 0$, compute each term, and then sum the {\em whole} series. Using
$a_{nm}$ to denote the coefficients of the Taylor series, we have
\begin{equation}
D_{KL}(p||q)= \frac{1}{\ln 2}
\sum_{mn} a_{mn} \left\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n
\right\rangle_{ind}
\, .
\label{taylor}
\end{equation}
\noindent
Note that because $f(x,y)$ and its first derivatives vanish
at $x=y=0$, all terms in this sum have $m+n \ge 2$.
Because both $\xi_p$ and $\xi_q$ are themselves sums, an exact
calculation of the terms in Eq.~\eqref{taylor} would be difficult.
However, as we show in Sec.~\ref{section:averages} (in particular
Eqs.~\eqref{xipq2_sum} and \eqref{xipq3_sum}), they can be
computed to lowest order in $N\delta$, and the result is
\begin{eqnarray}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle_{ind} & = &
\frac{1}{\ln 2}
\sum_{i < j}
\bar{r}_i \bar{r}_j (\rho^p_{ij})^m (\rho^q_{ij})^n
+
\left[ \bar{r}_j (-\bar{r}_i \rho^p_{ij})^m (-\bar{r}_i \rho^q_{ij})^n
+
\bar{r}_i \leftrightarrow \bar{r}_j
\right]
\ \ \
\nonumber
\\
& + &
\frac{1}{\ln 2}
\sum_{i < j < k}
\bar{r}_i \bar{r}_j \bar{r}_k
({\tilde{\rho}}^p_{ijk})^m
({\tilde{\rho}}^q_{ijk})^n
+ {\cal O} \left((N\delta)^4\right)
\label{xipq_final}
\end{eqnarray}
\noindent
where ${\tilde{\rho}}^p_{ijk}$ and ${\tilde{\rho}}^q_{ijk}$ are given by
\begin{equation}
{\tilde{\rho}}^x_{ijk} \equiv \rho^x_{ijk}
+ \rho^x_{ij} + \rho^x_{ik} + \rho^x_{jk}
=
\frac{\langle r_i r_j r_k \rangle_x - \bar{r}_i \bar{r}_j \bar{r}_k}
{\bar{r}_i \bar{r}_j \bar{r}_k}
\, ,
\label{rhothree}
\end{equation}
\noindent
$x=p,q$. The last equality in Eq.~\eqref{rhothree}
follows from a small amount of
algebra and the definition of the correlation coefficients given in
Eq.~\eqref{rho_norm}. Equation \eqref{xipq_final} is valid only
when $m+n \ge 2$, which is the case of interest to us (since the
Taylor expansion of $f(x,y)$ has only terms with $m+n \ge 2$).
The important point about Eq.~\eqref{xipq_final} is that the $m$ and $n$
dependence follows that of the original Taylor expansion. Thus, when
we insert this equation back into Eq.~\eqref{taylor}, we recover our original
function -- all we have to do is interchange the sums. For example,
consider inserting the first term in Eq.~\eqref{xipq_final} into
Eq.~\eqref{taylor},
\begin{equation}
\sum_{m,n} a_{mn} \sum_{i<j} \bar{r}_i \bar{r}_j (\rho^p_{ij})^m (\rho^q_{ij})^n
= \sum_{i<j} \bar{r}_i \bar{r}_j \sum_{m,n} a_{mn} (\rho^p_{ij})^m (\rho^q_{ij})^n
= \sum_{i<j} \bar{r}_i \bar{r}_j f(\rho^p_{ij}, \rho^q_{ij})
\, .
\nonumber
\end{equation}
\noindent
Performing the same set of manipulations on all of
Eq.~\eqref{xipq_final} leads
to
\begin{eqnarray}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle_{ind} & = &
\frac{1}{\ln 2}
\sum_{i < j}
\bar{r}_i \bar{r}_j f(\rho^p_{ij}, \rho^q_{ij})
+
\bar{r}_j f(-\bar{r}_i \rho^p_{ij}, -\bar{r}_i \rho^q_{ij})
+
\bar{r}_i f(-\bar{r}_j \rho^p_{ij}, -\bar{r}_j \rho^q_{ij})
\ \ \
\nonumber
\\
& + &
\frac{1}{\ln 2}
\sum_{i < j < k}
\bar{r}_i \bar{r}_j \bar{r}_k
f({\tilde{\rho}}^p_{ijk}, {\tilde{\rho}}^q_{ijk})
+ {\cal O} \left(N\delta)^4\right)
\, .
\label{xipq}
\end{eqnarray}
This expression is true in general (except for some technical
considerations; see Sec.~\ref{section:averages}); to restrict it to the KL
divergences of interest we set $p(\mathbf{r})$ to $p_{true}(\mathbf{r})$ and
$q(\mathbf{r})$ to either $p_{ind}(\mathbf{r})$ or $p_{pair}(\mathbf{r})$. In the first
case, $q(\mathbf{r}) = p_{ind}(\mathbf{r})$, $\xi_q(\mathbf{r}) = 0$, which in term implies
that ${\cal J}^q_{ij}=0$, and thus $\rho^q_{ij} = 0$.
Using Eq.~\eqref{xipq}, we have (to lowest nonvanishing order in
$N\delta$),
\begin{equation}
D_{KL}(p_{true}||p_{ind})=
\frac{1}{\ln 2}
\sum_{i < j} \bar{r}_i \bar{r}_j f(\rho^p_{ij}, 0)
+ {\cal O}\left((N \delta)^3\right)
\, .
\label{dkl_ind}
\end{equation}
\noindent
Then, defining
\begin{equation}
g_{ind} \equiv \frac{1}{N(N-1)\ln(2)}\sum_{i<j}
\frac{\bar{r}_i}{\delta}\frac{\bar{r}_j}{\delta} f(\rho^p_{ij}, 0)
\label{gind}
\, ,
\end{equation}
\noindent
and recalling that $\delta = \overline{\nu} \delta t$, we see that
Eq.~\eqref{dkl_ind} is equivalent to Eq.~\eqref{KLs_Na}.
In the second case, $q(\mathbf{r}) = p_{pair}(\mathbf{r})$, the first and second
moments of $p_{pair}(\mathbf{r})$ and $p_{true}(\mathbf{r})$ are equal. This in turn
implies, using Eq.~\eqref{moments}, that ${\cal J}^q_{ij}={\cal J}^p_{ij}$,
and thus $\rho^p_{ij}=\rho^q_{ij}$. Because $f(x,x)=0$ (see
Eq.~\eqref{fxy}), we see that the first three terms on the right hand
side of Eq.~\eqref{xipq} -- those involving $i$ and $j$ but not
$k$ -- vanish. The next order term does not vanish, and yields
\begin{equation}
D_{KL}(p_{true}||p_{pair}) =
\frac{1}{\ln 2}
\sum_{i < j < k} \bar{r}_i \bar{r}_j \bar{r}_k
f({\tilde{\rho}}^p_{ijk}, {\tilde{\rho}}^q_{ijk})
+ {\cal O} \left( (N\delta)^4 \right)
\, .
\label{dkl2_contracted}
\end{equation}
\noindent
Defining
\begin{equation}
g_{pair} \equiv
\frac{1}{N(N-1)(N-2)\ln(2)}\sum_{i<j<k} \frac{r_i}{\delta}
\frac{r_j}{\delta} \frac{r_k}{\delta}
f({\tilde{\rho}}^p_{ijk},{\tilde{\rho}}^q_{ijk})
\, ,
\label{gpair}
\end{equation}
\noindent
we see that Eq.~\eqref{dkl2_contracted} reduces to Eq.~\eqref{KLs_Nb}.
\subsubsection{Local fields, pairwise couplings and moments}
\label{local-fields}
In this section we derive, to leading order in $N\delta$, expressions
relating the local fields and pairwise couplings of the maximum
entropy model, $h_i$ and $J_{ij}$, to the first and second moments.
These are the expressions reported in Eq.~\eqref{hJ}.
To do this, we simply compute the first and second moment under the
assumption that $N\delta$ is small. This calculation proceeds along
the same lines as in the previous section, with one extra
consideration: the
quadratic term in the maximum entropy distribution,
Eq.~\eqref{maxent}, is proportional to $r_i r_j$, not $\delta r_i
r_j$. However, to lowest order in $N\delta$, this doesn't matter.
That's because
\begin{equation}
\sum_{i < j} J_{ij} r_i r_j = \sum_{i < j} J_{ij} \delta r_i \delta r_j
+ r_i \sum_{j \ne i} J_{ij} \bar{r}_j
+ \hbox{constants}
\, .
\nonumber
\end{equation}
\noindent
where $\bar{r}_i$ is defined as in Eq.~(\ref{p_expansion}d) except
with ${\cal H}^p_i$ replaced by $h_i$, and we used the fact that
$J_{ij} = J_{ji}$. The second term
introduces a correction to the local fields, $h_i$. However, the
correction is ${\cal O}(N\delta)$, so we drop it. We should keep in
mind, though, that our final expression for $h_i$ will have
corrections of ${\cal O}(N\delta)$.
Using Eq.~\eqref{maxent}, but with $r_i$ replaced by $\delta r_i$
where it appears with $J_{ij}$, we may write (after a small amount of
algebra)
\begin{equation}
p_{maxent}(\mathbf{r}) = p_{ind}(\mathbf{r})
\frac{1 + \xi_x(\mathbf{r}) + \psi(\xi_x(\mathbf{r}))}
{1 + \langle \xi_x(\mathbf{r}) + \psi(\xi_x(\mathbf{r})) \rangle_{ind}}
\label{p-me}
\end{equation}
\noindent
where $p_{ind}(\mathbf{r})$ is the same as the function $p_{ind}(\mathbf{r})$
defined in Eq.~(\ref{p_expansion}a), except that ${\cal H}^p_{ij}$ is
replaced by $h$, the subscript ``{\em ind}'' indicates, as usual, an
average with respect to $p_{ind}(\mathbf{r})$, and the two functions
$\xi_x(\mathbf{r})$ and $\psi(x)$ are given by
\begin{equation}
\xi_x(\mathbf{r}) \equiv
\sum_{i < j} J_{ij} \delta r_i \delta r_j
\label{xi_x}
\end{equation}
\noindent
and
\begin{equation}
\psi(x) \equiv e^x - 1 - x
\, .
\label{psi}
\end{equation}
Given this setup, we can use
Eqs.~\eqref{phi_g1} and \eqref{phi_g2} to compute the moments under
the maximum entropy model. That's because
both $\psi(x)$ and its first derivative vanish at $x=0$, which
are the two conditions required for
these equations to be valid. Using also the
fact that $\langle \delta r_i \rangle_{ind} = 0$,
Eqs.~\eqref{phi_g1} and \eqref{phi_g2} imply that
\begin{subequations}
\begin{align}
\langle \xi_x(\mathbf{r}) + \psi(\xi_x(\mathbf{r})) \rangle_{ind} & =
\sum_{i < j} \bar{r}_i \bar{r}_j \psi(J_{ij})
+ {\cal O} \left( (N\delta)^3 \right)
\\
\langle r_i \rangle_{maxent} & = \left(1 + \exp(-h_i) \right)^{-1}
+ {\cal O} \left( N\delta^2 \right)
\\
\langle \delta r_i \delta r_j \rangle_{maxent} &=
\bar{r}_i \bar{r}_j [ \psi(J_{ij}) + J_{ij}]
+ {\cal O} \left( N\delta^3 \right)
\, .
\end{align}
\label{moments-me}
\end{subequations}
\noindent
The term ``$+J_{ij}$'' in Eq.~(\ref{moments-me}c) came from
$\langle \delta r_i \delta r_j \xi_x(\mathbf{r}) \rangle_{ind}$;
see numerator in Eq.~\eqref{p-me}.
Note that for the second two equations, we used the fact that, to
lowest order in $N\delta$, the
denominator in Eq.~\eqref{p-me} is equal to 1.
Finally, using Eq.~\eqref{norm-corr} for the normalized correlation
coefficient, dropping the subscript ``{\em maxent}'' (since the
first and second moments are the same under the maxent and true
distributions), and inverting Eqs.~(\ref{moments-me}b) and
(\ref{moments-me}c) to express the local fields and
coupling coefficients in terms of the first and second moments, we
arrive at Eq~\eqref{hJ}.
\subsubsection{Averages of powers of $\xi_p$ and $\xi_q$}
\label{section:averages}
Here we compute $\langle \xi_p^m \xi_q^n \rangle_{ind}$, which, as can
be seen in Eq.~\eqref{taylor}, is the the key quantity in our
perturbation expansion. Our starting point is to (formally) expand the
sums that make up $\xi_p$ and $\xi_q$ (see
Eqs.~(\ref{p_expansion}b) and ~(\ref{q_expansion}b)), which yields
\begin{equation}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle_{ind} =
\sum_{l=2}^\infty
\sum_{\{m_1, \dots, m_l\}}
\psi_{m_1, \dots, m_l}^{(l)}
\sum_{i_1 < \dots < i_l}
\langle \delta r_{i_1}^{m_1} \dots, \delta r_{i_l}^{m_l}
\rangle_{ind}
\, .
\label{xipq_sum}
\end{equation}
\noindent
The sum over ${\{m_1, \dots, m_l\}}$ is a sum
over all possible configurations of the $m_i$. The
coefficient $\psi^{(l)}_{m_1, \dots, m_l}$ are complicated functions
of the ${\cal J}^p_{ij}, {\cal J}^q_{ij}, {\cal K}^p{ijk}, {\cal K}^q_{ijk}$, etc.
Computing these functions is straightforward, although somewhat
tedious, especially when $l$ is large;
below we compute them only for $l=2$ and 3. The reason $l$
starts at 2 is that $m+n \ge 2$; see Eq.~\eqref{taylor}.
What we will show is that all terms with superscript $(l)$ are
${\cal O}(\delta^l)$. To do this, we first note that, because
the right hand
side of Eq.~\eqref{xipq_sum} is an average with respect to the independent
distribution, the average of the product is the product of the
averages,
\begin{equation}
\langle \delta r_{i_1}^{m_1} \delta r_{i_2}^{m_2} \dots, \delta r_{i_l}^{m_l}
\rangle_{ind}
=
\langle \delta r_{i_1}^{m_1} \rangle_{ind} \langle \delta
r_{i_2}^{m_2} \rangle_{ind} \dots, \langle \delta r_{i_l}^{m_l} \rangle_{ind}
\, .
\label{drmm}
\end{equation}
\noindent
Then, using the fact that $\delta r_i=(1-\bar{r}_i)$ with probability $\bar{r}_i$
and $\delta r_i=(1-\bar{r}_i)$ with probability $(1-\bar{r}_i)$ (see Eq.~(\ref{p_expansion}c)), we have
\begin{equation}
\langle \delta r_i^m \rangle_{ind} = \bar{r}_i
(1-\bar{r}_i)^m+(1-\bar{r}_i)(-\bar{r}_i)^m
= \bar{r}_i (1 - \bar{r}_i)^m \left[ 1 -
\left(\frac{-\bar{r}_i}{1-\bar{r}_i} \right)^{m-1}
\right]
\label{drm}
\, .
\end{equation}
\noindent
The significance of this expression is that, for $m > 1$,
$\langle \delta r_i^m \rangle_{ind} \sim {\cal O}(r_i) \sim {\cal O}(\delta)$,
independent
of $m$. Consequently, if all the $m_i$ in Eq.~\eqref{drmm} are greater
than 1, then the right hand side is ${\cal O}(\delta^l)$. This shows
that, as promised above,
the superscript $(l)$ labels the
order of the terms.
As we saw in Sec.~\ref{section:KL}, we need to go to third order in
$\delta$, which means we need to compute the terms on the right hand
side of Eq.~\eqref{xipq_sum} with $l=2$ and 3. Let us start with $l=2$,
which picks out only on those terms with two unique
indices. Examining the expressions for $\xi_p$ and $\xi_q$ given in
Eqs.~(\ref{p_expansion}b) and (\ref{q_expansion}b), we see
that we must keep only terms involving
${\cal J}_{ij}$, since ${\cal K}_{ijk}$ has
three indices, and higher order terms have more. Thus, the $l=2$
contribution to the average in Eq.~\eqref{xipq_sum}, which we denote
$\langle \xi_p(\mathbf{r}) \xi_q(\mathbf{r}) \rangle_{ind}^{(2)}$, is given by
\begin{equation}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle^{(2)}_{ind} =
\sum_{i < j}
\left\langle
\left({\cal J}^p_{ij} \delta r_i \delta r_j \right)^m
\left({\cal J}^q_{ij} \delta r_i \delta r_j \right)^n
\right\rangle_{ind}
\, .
\nonumber
\end{equation}
\noindent
Pulling ${\cal J}^p_{ij}$ and ${\cal J}^q_{ij}$ out of the averages,
using Eq.~(\ref{jk_moments}a)
to eliminate ${\cal J}^p_{ij}$ and ${\cal J}^q_{ij}$ in favor of
$\rho^p_{ij}$ and $\rho^q_{ij}$, and applying
Eq.~\eqref{drm} (while throwing away some of the terms in that
equation that are
higher than second order in $\delta$), the above expression may be
written
\begin{equation}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle^{(2)}_{ind} =
\sum_{i < j}
\bar{r}_i \bar{r}_j
(\rho^p_{ij})^m
(\rho^q_{ij})^n
\left[1-(-\bar{r}_i)^{m+n-1}-(-\bar{r}_i)^{m+n-1}\right]
\, .
\label{xipq2_sum}
\end{equation}
\noindent
Note that we were not quite consistent in our
ordering with respect to $\delta$, in the sense that we kept some
higher order terms and not others. We did this so that we could
use $\rho_{ij}$ rather than ${\cal J}_{ij}$, as the former
are directly observable.
For $l=3$ the calculation is more involved, but not substantially so.
Including terms with exactly three unique indices in the sum on the
right hand side of Eq.~\eqref{xipq_sum} gives us
\begin{eqnarray}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle^{(3)}_{ind} & = &
\sum_{i < j < k}
\left\langle
\left({\cal K}^p_{ijk} \delta r_i \delta r_j \delta r_k
+ {\cal J}^p_{ij} \delta r_i \delta r_j
+ {\cal J}^p_{ik} \delta r_i \delta r_k
+ {\cal J}^p_{jk} \delta r_j \delta r_k
\right)^m
\right.
\label{xipq3_sum}
\\
\nonumber
& &
\ \ \ \ \ \ \ \ \
\left.
\left({\cal K}^q_{ijk} \delta r_i \delta r_j \delta r_k
+ {\cal J}^q_{ij} \delta r_i \delta r_j
+ {\cal J}^q_{ik} \delta r_i \delta r_k
+ {\cal J}^q_{jk} \delta r_j \delta r_k
\right)^n
\right\rangle_{ind}
\, .
\ \ \ \ \ \
\end{eqnarray}
This expression is not quite correct, since some of its terms contain
only two unique indices -- these are the terms proportional to
$({\cal J}^p_{ij})^m({\cal J}^p_{ij})^n$ -- whereas it should contain only
terms with exactly three unique indices. Fortunately, this turns out
not to matter, for reasons we discuss at the end of the section.
To perform the averages in Eq.~\eqref{xipq3_sum}, we would need to
use multinomial expansions, and then
average over the resulting powers of $\delta r$'s.
For the latter, we can work to
lowest order in the $\delta r_i$, which means we only take the first
term in Eq.~\eqref{drm}. This
amounts to replacing every $\delta r_i$ with $1-\bar{r}_i$ (and
similarly for $j$ and $k$),
and in addition multiplying the whole expression by an overall
factor of $\bar{r}_i \bar{r}_j \bar{r}_k$. For example, if $m=1$ and $n=2$,
one of the terms in the multinomial expansion is
${\cal K}^p_{ijk} {\cal J}^q_{ij} {\cal J}^q_{ik}
\langle \delta r_i^3 \delta r_j^2 \delta r_k^2 \rangle_{ind}$.
This average would yield, using Eq.~\eqref{drm} and considering only
the lowest order term,
$\bar{r}_i \bar{r}_j \bar{r}_k (1-\bar{r}_i)^3 (1-\bar{r}_j)^2 (1-\bar{r}_k)^2$.
This procedure also is not quite correct, since
terms with only one factor of $\delta r_i$, which average to zero,
are replaced with $1-\bar{r}_i$. This also turns out not to matter;
again, we discuss why at the end of the section.
We can, then, go ahead and use the above ``replace blindly''
algorithm. Note that the factors of $1-\bar{r}_i$, $1-\bar{r}_j$ and
$1-\bar{r}_k$ turn ${\cal J}_{ij}$ and ${\cal K}_{ijk}$
into normalized correlation coefficients (see
Eq.~\eqref{jk_moments}), which considerably simplifies our equations.
Using also Eq.~\eqref{rhothree} for $\tilde{\rho}_{ijk}$,
Eq.~\eqref{xipq3_sum} becomes
\begin{equation}
\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle^{(3)}_{ind} =
\sum_{i < j < k}
\bar{r}_i \bar{r}_j \bar{r}_k
({\tilde{\rho}}^p_{ijk})^m
({\tilde{\rho}}^q_{ijk})^n
\, .
\label{xipq3_loc}
\end{equation}
\noindent
We can now combine Eqs.~\eqref{xipq2_sum} and \eqref{xipq3_loc}, and
insert them into Eq.~\eqref{xipq_sum}. This gives us the first
two terms in the perturbative expansion of
$\langle \xi_p(\mathbf{r})^m \xi_q(\mathbf{r})^n \rangle_{ind}$; the result is
written down in Eq.~\eqref{xipq_final} above.
Why can we ignore the overcounting associated with terms in which an
index appears exactly zero or one times? We clearly can't do this in
general, because for such terms, replacing $\delta r_i$ with
$1-\bar{r}_i$ fails -- either because the terms didn't exist in the
first place (when one of the indices never appeared) or because they
averaged to zero (when an index appeared exactly once). In our case,
however, such terms do not appear in the Taylor expansion. To
see why, note first of all that, because of the
form of
$f(x,y)$, its Taylor expansion
can be written $(x-y)^2 \tilde{f}(x,y)$ where
$\tilde{f}(x,y)$ is finite at $x=y$ (see Eq.~\eqref{fxy}).
Consequently, the expression inside the sum over $i, j$ and $k$
in Eq.~\eqref{xipq3_sum} should really contain a multiplicative factor
that arises from $(\xi_p-\xi_q)^2$, and thus has the form
\begin{equation}
\left(({\cal K}^p_{ijk} - {\cal K}^q_{ijk}) \delta r_i \delta r_j \delta r_k
+ ({\cal J}^p_{ij} - {\cal J}^q_{ij}) \delta r_i \delta r_j
+ ({\cal J}^p_{ik} - {\cal J}^q_{ik}) \delta r_i \delta r_k
+ ({\cal J}^p_{jk} - {\cal J}^q_{jk}) \delta r_j \delta r_k
\right)^2
\, .
\nonumber
\end{equation}
As we saw in the previous section, we are interested in the
third order term only to compute $D_{KL}(p_{true}||p_{pair})$, for
which ${\cal J}^p_{ij} = {\cal J}^q_{ij}$. Therefore, the above
multiplicative factor reduces to
$({\cal K}^p_{ijk} - {\cal K}^q_{ijk})^2 (\delta r_i \delta r_j \delta r_k)^2$.
It is that last factor of $(\delta r_i \delta r_j \delta r_k)^2$ that
is important, since it guarantees that every term in the Taylor
expansion will have all indices appearing at least twice.
Therefore, although Eq.~\eqref{xipq3_sum} is not true in general, it
is valid for our analysis.
We end this section by pointing out that there is a very simple
procedure for computing averages to second order in $\delta$. Consider
a function $\phi(\xi_p, \xi_q)$ such that
$\phi(\xi_p, \xi_q)$ and its first derivatives vanish at
$\xi_p=\xi_q = 0$. Then, based on the above analysis, we have
\begin{equation}
\langle \phi(\xi_p, \xi_q) \rangle_{ind}
= \sum_{i < j} \bar{r}_i \bar{r}_j \phi({\cal J}^p_{ij}, {\cal J}^q_{ij})
+ {\cal O} \left((N\delta)^3 \right)
\, .
\label{phi_g1}
\end{equation}
Two easy corollaries of this are: for $k$ and $l$ positive integers,
\begin{subequations}
\begin{align}
\langle \delta r_i^k \phi(\xi_p, \xi_q) \rangle_{ind}
& =
\sum_{j \ne i} \bar{r}_i \bar{r}_j \phi({\cal J}^p_{ij}, {\cal J}^q_{ij})
+ {\cal O} \left(N^2\delta^3 \right)
\\
\langle \delta r_i^k \delta r_j^l \phi(\xi_p, \xi_q) \rangle_{ind}
& =
\bar{r}_i \bar{r}_j \phi({\cal J}^p_{ij}, {\cal J}^q_{ij})
+ {\cal O} \left(N\delta^3 \right)
\end{align}
\label{phi_g2}
\end{subequations}
\noindent
where the sum in Eq.~(\ref{phi_g2}a) run over $j$ only, and we used
the fact that both ${\cal J}^p_{ij}$ and ${\cal J}^q_{ij}$ are symmetric
with respect to the interchange of $i$ and $j$.
\subsection{Generating synthetic data}
\label{Generating_syn_data}
As can be seen in Eq.~\eqref{ptrue}, synthetic data depends
on three sets of parameters:
$h^{true}_i, J^{true}_{ij}$, and $K^{true}_{ijk}$. Here we describe how
they were generated.
Our first step was to generate the $h^{true}_i$. To do that, we chose a
vector $\mathbf{r}^*=[r^*_1 \dots r^*_{N^*}]$ (where, recall, $N^* = 15$ is
the number of neurons in our base distribution), from an exponential
distribution with mean 0.02. From this we chose the local field
according to Eq.~(\ref{hJ}a),
\begin{equation}
h^{true}_i=-\log\left(\frac{1}{r^*_i}-1\right).
\nonumber
\end{equation}
\noindent
In the perturbative regime, a distribution generated with these values
of the local fields will have firing rates approximately equal to the
$r^*_i$; see Eq.~(\ref{hJ}a) and Fig.~\ref{h_J_N}.
We then draw $J^{true}_{ij}$ and $K^{true}_{ijk}$ from Gaussian
distributions with means equal to $0.05$ and $0.02$ and standard
deviations of $0.8$ and $0.5$, respectively. Using non-zero values for
${\cal K}_{ijk}$, means that the distribution is not pairwise.
\subsection{Bin size and the correlation coefficients}
\label{Bin-size}
One of our main claims is that $\Delta_N$ is linear in bin size,
$\delta t$. This is true, however, only if $g_{ind}$ and $g_{pair}$ are
independent of $\delta t$, as can be seen from Eq.~\eqref{DeltaN-def}.
In this section we show that independence is satisfied if $\delta t$
is smaller than the typical correlation time of the responses. For
$\delta t$ larger than such correlation times, $g_{ind}$ and $g_{pair}$ do
depend on $\delta t$, and $\Delta_N$ is no longer linear in $\delta
t$. Note, though, that the correlation time is always
finite, so there will
always be a bin size below which the linear relationship, $\Delta \sim
\delta t$, is guaranteed.
Examining Eqs.~\eqref{gind} and \eqref{gpair}, we see that
$g_{ind}$ and $g_{pair}$ depend on the normalized correlation
coefficients, $\rho_{ij}$, ${\tilde{\rho}}_{ijk}$ (we drop superscripts,
since our discussion will be generic).
Thus,
to understand how $g_{ind}$ and $g_{pair}$ depend on bin size, we need to
understand how the normalized correlation coefficients depend on bin
size.
We start with the second order correlation
coefficient, since it is simplest. The corresponding
cross-correlogram, which we denote $C_{ij}(\tau)$, is given by
\begin{equation}
C_{ij}(\tau) = {1 \over \nu_i \nu_j}
\lim_{T \rightarrow \infty} {1 \over T}
\sum_{kl} \delta(t_i^k - t_j^l - \tau)
\label{C}
\end{equation}
\noindent
where $t_i^k$ is the time of the $k^{\rm th}$ spike on neuron $i$ (and
similarly for $t_j^l$) and $\delta(\cdot)$ is the Dirac
$\delta$-function. The normalization in Eq.~\eqref{C} is slightly
non-standard -- more typical is to divide by something with units of
firing rate ($\nu_i$, $\nu_j$ or $(\nu_i \nu_j)^{1/2}$), to give units
of spikes/s. The normalization we use is convenient, however, because
in the limit of large $\tau$, $C_{ij}(\tau)$ approaches one.
It is slightly tedious, but otherwise straightforward, to show that
when $\delta t$ is sufficiently small that only one spike can occur in
a time bin, $\rho_{ij}$ is related to $C_{ij}(\tau)$ via
\begin{equation}
\rho_{ij} = {1 \over \delta t}
\int_{-\delta t}^{\delta t} d \tau \, (1 - |\tau|/\delta t) \,
(C_{ij}(\tau) - 1)
\, .\label{rhoC}
\end{equation}
\noindent
The (unimportant) factor $(1 - |\tau|/\delta t)$ comes from
the fact that the spikes occur at random locations within a bin.
Equation (\ref{rhoC}) has a simple interpretation: $\rho_{ij}$ is the
average height of the central peak of the cross-correlogram relative
to baseline. How strongly $\rho_{ij}$ depends on $\delta t$ is thus
determined by the shape of the cross-correlogram. If it is smooth,
then $\rho_{ij}$ approaches a constant as $\delta t$ becomes small.
If, on the other hand, there is a sharp peak at $\tau=0$, then
$\rho_{ij} \sim 1/\overline{\nu} \delta t = 1/\delta$
for small $\delta t$, so
long as $\delta t$ is larger than the width of the peak. (The factor
of $\overline{\nu}$ included in the scaling is approximate; it is a
placeholder for an effective firing rate that depends on the indices
$i$ and $j$. It is, however, sufficiently accurate for our purposes.)
A similar relationship exists between the third order correlogram and
the correlation coefficient. Thus, ${\tilde{\rho}}_{ijk}$
is also independent of $\delta t$ in the small $\delta
t$ limit, whereas if the central peak is sharp it scales as
$1/\delta^2$.
The upshot of this analysis is that the shape of the cross-correlogram
has a profound effect on the correlation coefficients and, therefore, on
$\Delta_N$. What is the shape in real
networks? The answer typically depends
on the physical distance between cells. If two neurons
are close, they are likely to receive common input and thus
exhibit a narrow central peak in their
cross-correlogram. If, on the other hand, the neurons are far
apart, they are less likely to receive common input. In this case, the
correlations come from external stimuli, so the central peak tends to
have a characteristic width given by the temporal
correlation time of the stimulus, typically 100s of milliseconds.
Although clearly both kinds of cross-correlograms exist in any single
population of neurons, it is convenient to analyze them separately. We
have already considered networks in which the cross-correlograms were
broad and perfectly flat, so that the correlation coefficients
were strictly independent of bin size. Based on our analysis in
Sec.~\ref{perturbative_expansion}, we
can also consider the opposite
extreme: networks in which the
the cross-correlograms (both second and higher order) among
nearby neurons exhibit sharp peaks
while those among distant neurons are uniformly equal to 1.
In this regime, the correlation coefficients
depend on $\delta t$: as discussed above, the second order ones scale as
$1/\delta$ and the third as
$1/\delta^2$.
This means that the arguments of
$f(\rho_{ij},0)$ and $f({\tilde{\rho}}^{true}_{ijk}, {\tilde{\rho}}^{pair}_{ijk})$ are
large. From the definition of $f(x,y)$ in Eq.~\eqref{fxy},
in this regime both are approximately linear in
their arguments (ignoring log corrections).
Consequently,
$f(\rho_{ij}, 0) \sim 1/\delta$ and
$f({\tilde{\rho}}^{true}_{ijk}, {\tilde{\rho}}^{pair}_{ijk}) \sim 1/\delta^2$.
This implies that $g_{ind}$ and $g_{pair}$ scale as $N \delta$ and $N^2
\delta$, respectively, and so $\Delta_N \sim N$, independent of
$\delta$. Thus, if the bin size is large compared the the correlation
time, $\Delta_N$ will be approximately independent of bin size.
\subsection{Assessing goodness of fit for independence across time assumption}
\label{ind-assumption}
In this section we derive the expression for $\gamma$ given in
Eq.~\eqref{gamma}. Our starting point is its definition,
Eq.~\eqref{gamma-def}. It is convenient to define $\mathbf{R}$ to be a
concatenation of the responses in $M$ time bins,
\begin{equation}
\mathbf{R} \equiv (\mathbf{r}^1, \mathbf{r}^2, ..., \mathbf{r}^M)
\end{equation}
\noindent
where, as in Sec.~\ref{small-time-bin-wrong}, the superscript labels
time, so ${\cal P}(\mathbf{R})$ is the full, temporally correlated, distribution.
With this definition, we may write the numerator in
Eq.~\eqref{gamma-def} as
\begin{equation}
D_{KL}\Big({\cal P}(\mathbf{R}) || \prod_t p_{pair}(\mathbf{r}^t) \Big)
=
- S^M_{true} -
\sum_t \sum_{\mathbf{r}} p^t_{true}(\mathbf{r}) \log_2 p_{pair}(\mathbf{r})
\label{DKL_P}
\end{equation}
\noindent
where $S_{full}^M$ is the entropy of the full distribution,
${\cal P}(\mathbf{R})$, the last sum follows from a marginalization
over all but one element of ${\cal P}(\mathbf{R})$, and $p^t_{true}(\mathbf{r})$ is
the true distribution at time $\mathbf{r}$. Note that
$p_{pair}(\mathbf{r})$ is independent of time, since it is computed from a
distribution averaged over all bins. That distribution, which we have
called $p_{true}(\mathbf{r})$, is given in terms of $p^t_{true}(\mathbf{r})$ as
\begin{equation}
p_{true}(\mathbf{r}) = \frac{1}{M} \sum_t p^t_{true}(\mathbf{r})
\, .
\nonumber
\end{equation}
Inserting this definition into Eq.~\eqref{DKL_P}
eliminates the sum over $t$, and replaces it with $M p_{true}(\mathbf{r})$.
For simplicity we consider the
maximum entropy pairwise model. In this case,
because $p_{pair}(\mathbf{r})$ is in the exponential family, and the first
and second moments are the same under the true and maximum entropy
distributions, we can replace $p_{true}(\mathbf{r})$ with $p_{maxent}(\mathbf{r})$.
Consequently, Eq.~\eqref{DKL_P} becomes
\begin{equation}
D_{KL}\Big({\cal P}(\mathbf{R}) || \prod_t p_{pair}(\mathbf{r}^t) \Big) =
M S_{maxent} - S^M_{true}
\nonumber
\, .
\end{equation}
\noindent
This gives us the numerator in the expression for $\gamma$
(Eq.~\eqref{gamma-def}). The denominator,
$D_{KL}(p_{true}||p_{ind})$,
is equal to $S_{ind} - S_{true}$ (see Eq.~\eqref{KL-Ent-ind}). This
leads to
\begin{equation}
\gamma =
{M(S_{maxent} - S_{true})
\over M (S_{ind} - S_{true})}
+
{MS_{true} - S^M_{true}
\over M (S_{ind} - S_{true})}
\, .
\label{gamma-1}
\end{equation}
\noindent
where we added and subtracted $MS_{true}$ to the numerator.
The first term on the right hand side of Eq.~\eqref{gamma-1}
we recognize, from Eqs.~\eqref{DeltaN-def},
\eqref{KL-Ent-ind} and \eqref{KL-Ent-maxent}, as $\Delta_N$.
To cast the second into a reasonable form, we
define $S_{ind}^M$ to be the entropy of the
distribution that retains the temporal correlations within each neuron
but is independent across neurons. Then, adding and subtracting this
quantity to the numerator in Eq.~\eqref{gamma-1}, and also adding and
subtracting $MS_{ind}$, we have
\begin{equation}
\gamma = \Delta_N +
{
(S^M_{ind} - S^M_{true}) -
M(S_{ind}-S_{true}) +
(M S_{ind} - S^M_{ind})
\over
M(S_{ind} - S_{true})}
\, .
\label{gamma-2}
\end{equation}
\noindent
The key observation is that if $M \delta \ll 1$, then
\begin{equation}
S^M_{ind} - S^M_{true} = g_{ind} N(N-1) (M\delta)^2
\, .
\nonumber
\end{equation}
\noindent
Comparing this with Eq.~\eqref{KLs_Na}, we see that
$S^M_{ind} - S^M_{true}$ is a factor of $M^2$ times larger than
$S_{ind} - S_{true}$. We thus have
\begin{equation}
\gamma = \Delta_N + (M-1) +
{
M S_{ind} - S^M_{ind}
\over
M(S_{ind} - S_{true})}
\label{gamma-final}
\, .
\end{equation}
\noindent
Again assuming $M \delta \ll 1$, and defining
$h(x) \equiv -x \log_2 x - (1-x) \log_2(1-x)$, the last term in this
expression may be written
\begin{equation}
M S_{ind} - S^M_{ind}
= M \sum_i h(r_i) - \sum_i h(Mr_i) \approx M \sum_i r_i \log M
= N\delta \, M \log_2 M
\, .
\end{equation}
\noindent
Inserting this into Eq.~\eqref{gamma-final} and using Eq.~\eqref{KLs_Na}
yields Eq.~\eqref{gamma}.
We have assumed here that $M \delta \ll 1$; what happens when $M
\delta \sim 1$, or larger? To answer this, we rewrite
Eq.~\eqref{gamma-1} as
\begin{equation}
\gamma =
\frac{S_{maxent} - S^M_{true}/M}
{\Delta_N}
\, .
\label{gamma-full}
\end{equation}
\noindent
We argue that in general, as $M$ increases, $S^M_{true}/M$
becomes increasingly different from $S_{maxent}$, since the former was
derived under the assumption that the responses at different time bins
were independent. Thus, Eq.~\eqref{gamma} should be considered a lower
bound on $\gamma$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,330
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Q: $A\bar{B}$ is normal if and only if there is a unitary $U \in M_n$ such that $A = U \Sigma U^T$, $B = U \Lambda U^T$. Let $A,B \in M_n$ be symmetric. Show that $A\bar{B}$ is normal if and only if there is a unitary $U \in M_n$ such that $A = U \Sigma U^T$, $B = U \Lambda U^T$, $\Sigma, \Lambda \in M_n$ are diagonal and the diagonal entries of $\Sigma$ are nonnegative.
I am able to prove that $A\bar{B}$ is normal given a unitary $U \in M_n$ such that $A = U \Sigma U^T$, $B = U \Lambda U^T$, $\Sigma, \Lambda \in M_n$ are diagonal and the diagonal entries of $\Sigma$ are nonnegative.
I am facing difficulty in proving the other direction.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,870
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\section{Introduction}
Since the unanticipated discovery of high-temperature superconductivity in the cuprates, the single-band Hubbard model \cite{Hubbard_PRSLA_1993} has been the focus of an unparalleled level of theoretical scrutiny and associated algorithmic development.\cite{LeBlanc_Gull_PRX_2015,Lieb_Wu_RPL_1968,Scalapino_Brooks_NY_2007,Gukelberger_Werner_PRB_2015} Nevertheless, most materials exhibiting strong correlation, including most transition metal oxides\cite{KuneA_Pickett_NM_2008,Laad_Hartmann_PRB_2006,Maeno_Ikeda_MSE_1999} as well as the pnictides,\cite{Si_Abrahams_NRM_2016,Yin_Kotliar_NM_2011,Haule_Kotliar_NP_2009} fullerides,\cite{Nomura_JPCM_2016,Nomura_Science_2015} and chalcogenides\cite{Sun_Zhao_Nature_2012,Si_Abrahams_NRM_2016,Yin_Kotliar_NM_2011} possess multiple bands that cross their Fermi levels and are therefore fundamentally multi-band in nature.\cite{Georges_AnnRev_2013} In recent years, it has become increasingly evident that some of the most significant effects in such multi-band materials stem from Hund's coupling.\cite{Yin_Kotliar_NM_2011,Haule_Kotliar_NP_2009,Johannes_Mazin_PRB_2009} According to Hund's rules, electrons favor maximizing their total spin by first occupying different, degenerate bands in the same shell with parallel spins; only after they fill all available bands do they then doubly occupy the same bands.\cite{Georges_AnnRev_2013} As such, the effective Coulomb repulsion among electrons in a half-filled shell is increased due to Hund's rules, while that at any other filling is decreased. Hund's effects therefore drive half-filled $d$- and $f$-electron materials closer to a Mott transition for a given Coulomb repulsion, yet drive non-half-filled materials away from a Mott transition while also increasing the correlation within their metallic phases. The consequences of these effects are perhaps best illustrated in 4$d$ transition metal oxides that have more than a single electron or hole in their 4$d$ shells. \cite{Koster_Beasley_RMP_2012,Frandkin_Mackenzie_ARCMP_2010,Dang_Millis_PRB_2015, Han_Millis_PRB_2016} Unlike their rhodate counterparts, which possess a single hole in their shells, many ruthenates and molybdenates exhibit substantial mass enhancements,\cite{Ikeda_JPSJ_2000} unexpected Mott Insulator transitions,\cite{Liebsch_PRL_2007,Gorelov_PRL_2010,Sutter_Chang_NC_2017} novel quantum phase transitions,\cite{Grigera_Science_2001} and even superconducting phases\cite{Mackenzie_RMP_2003} -- all of which may be attributed to Hund's physics.
Despite both the prevalence and importance of Hund's effects, they remain a challenge to describe. Most analytical and numerical treatments revolve around solving a multi-band Hubbard model, most often the Hubbard-Kanamori (HK) model,\cite{Kanamori_PTP_1963} containing a mixture of kinetic, Coulomb $U$, and Hund's $J$ terms. Although analytical studies have been performed,\cite{Roth_PR_1966,LyonCaen_Cyrot_JPC_1975,Khomskii_SolidStateComm_1973,Kugel_Khomskii_Sov_1982,Ishihara_PRB_1997} just as in the case of the single-band Hubbard model containing a repulsive $U$ term,\cite{LeBlanc_Gull_PRX_2015,Zheng_Science_2017} accurate treatments of these models necessitate methods capable of treating strong correlation non-perturbatively. However, because these models possess significantly larger state spaces and involve additional pair-hopping and Hund's exchange terms, they are often even more difficult to treat than the Hubbard model.
Due to the complicated interactions involved, there is no general analytical solution for these problems. Thus, numerical treatments are in high demand. To date, most numerical studies of multi-band models have employed Dynamical Mean Field Theory (DMFT)\cite{Georges_RevModPhys_1996,Metzner_Vollhardt_PRL_1989} either on its own or in combination with Density Functional Theory (DFT)\cite{Anisimov_JPhysCondMat_1997} because of DMFT's ability to treat band and atomic effects on equal footing by self-consistently solving an impurity problem within a larger bath. DMFT has been very successful at mapping out multi-band phase diagrams at finite temperatures.\cite{Gorelov_PRL_2010,Han_Millis_PRL_2018,Han_Millis_PRB_2016,Dang_Millis_PRB_2015,Dang_PRL_2015,Werner_Andrew_PRL_2008,Inaba_PRB_2005} Nevertheless, DMFT is fundamentally limited by the accuracy and scaling of its impurity model solver. Some DMFT studies rely upon exact diagonalization (ED) to solve their impurity models, yet the computational cost of ED grows exponentially with the number of bands involved, thus thwarting its application to many-band models. Some DMFT algorithms employ continuous-time quantum Monte Carlo (CTQMC)\cite{Rubtsov_PRB_2005} to solve their impurity models. CTQMC can solve larger impurity models than ED, but is still hampered by the sign problem, an exponential decrease in the signal to noise ratio observed in stochastic simulations,\cite{Loh_PRB_1990} in certain parameter regimes and low temperature calculations remain difficult.\cite{Gull_RMP_2011} A method that can accurately simulate larger system sizes at lower temperatures is thus in need.
One suite of techniques particularly well-suited for studying the large state spaces inherent to multi-band models are quantum Monte Carlo (QMC) techniques.\cite{Foulkes_RMP_2001,Motta_Wiley_2018} Both finite temperature QMC methods, including CTQMC\cite{Gull_RMP_2011} and Hirsch-Fye QMC\cite{Hirsch_PRL_1986} algorithms that have been employed as impurity solvers within DMFT, and ground state\cite{Motome_Imada_JPSJ_1997,Motome_JPSJ_1998} QMC algorithms have been developed and applied to the multi-band Hubbard model. Nonetheless, the Hund's terms of the HK Hamiltonian have posed challenges for all of these methods. This is because Hund's terms are not readily expressed as products of density operators and are therefore not readily amenable to standard QMC transformations. Straightforward decoupling of the exchange and pair hopping terms leads to a severe sign problem.\cite{Held_Vollhardt_EPJB_1998} Attempts have therefore been made to simplify the Hund's contribution to the Hamiltonian to make it more palatable to QMC methods by constraining its direction to the z-axis,\cite{Held_Vollhardt_EPJB_1998,Han_PRB_1998} but such treatments sometimes fail to properly capture the model's expected physics. Several Hund's-specific transformations have been proposed, including a discrete transformation by Aoki \cite{Sakai_Aoki_PRB_2004,Sakai_Aoki_PRB_2006} and a continuous transformation by Imada. \cite{Motome_Imada_JPSJ_1997,Motome_JPSJ_1998} Nevertheless, these transformations ultimately do not eliminate the sign problem and are limited to parameter regimes with only high signal to noise ratios. These parameter constraints obscure our fundamental understanding of multi-band physics.
In this paper, we present an Auxiliary Field Quantum Monte Carlo (AFQMC) framework especially suited for the study of ground state multi-band Hubbard models and demonstrate its accuracy over a range of realistic parameters using different signal-preserving approximations and trial wave functions. Key to our approach is the strategic use of two forms of both the continuous and discrete Hubbard-Stratonovich (HS) Transformations to decouple the Hund's term: a charge decomposition for negative values of the Hund's coupling parameter, and a spin decomposition for positive values of the Hund's coupling parameter. We also employ an unconventional form of importance sampling in which we shift propagators instead of auxiliary fields so as to enable importance sampling of discrete transformed propagators. Unlike previous works, we furthermore utilize flexible Generalized Hartree-Fock (GHF) trial wave functions combined with the constrained path and phaseless approximations to tame the sign and phase problems, respectively. Altogether, we find that these improvements yield promising results for a variety of HK model benchmarks. Although the algorithm presented is designed for the ground state, it can easily be adapted for use in finite temperature methods.\cite{Liu_JCTC_2018,Zhang_PRL_1999} Our algorithm therefore paves the way to the high accuracy modeling of the low temperature physics of a wide range of multi-band models and materials over a dramatically larger portion of the phase diagram.
The remainder of the paper is organized as follows. In Section \ref{method}, we outline the HK model, summarize the key features of the AFQMC method, and describe how the conventional AFQMC technique may be modified to best accommodate the HK Hamiltonian. In Section \ref{results}, we then present benchmarks of our method's performance within different parameter regimes, using different trial wave functions, and employing different approximations on two- and three-band HK models for which ED results may be obtained. Towards the end of this section, we also demonstrate the accuracy with which our techniques can predict the charge gaps and magnetic ordering of two-dimensional lattice models far beyond the reach of most other techniques. We conclude with a discussion of the broader implications of this work and future directions in Section \ref{conclusions}.
\section{Methods}
\label{method}
\subsection{Hubbard Kanamori Model Hamiltonian}
The HK model is a multi-band version of the Hubbard model designed to account for the competition between the spin and band degrees of freedom observed in the physics of $d$- and $f$- electron material.\cite{Kanamori_PTP_1963,Georges_AnnRev_2013} In order to accomplish this, the model includes not only standard Hubbard on-site density-density interactions, but also inter-band density, exchange, and pair hopping terms. The full HK Hamiltonian, written as general as possible, reads
\begin{equation}
\hat{H} \equiv \hat{H}_{1} + \hat{H}_{2} \equiv \hat{H}_{1} + \hat{H}_{U} + \hat{H}_{J},
\label{Hamiltonian}
\end{equation}
where
\begin{equation}
\hat{H}_{1} = \sum_{i m \sigma}\sum_{j m^\prime \sigma^\prime} t_{im,jm^\prime}^{\sigma\sigma^\prime}\hat{c}_{im\sigma}^{\dagger}\hat{c}_{jm^\prime \sigma^\prime},
\label{HOne_Term}
\end{equation}
\begin{eqnarray}
\hat{H}_{U} &=&
\sum_{i,m} U_{im} \hat{n}_{im\uparrow}\hat{n}_{im\downarrow} \nonumber + \sum_{i, m \neq m^\prime} U^{\prime}_{imm'} \hat{n}_{im \uparrow} \hat{n}_{im^\prime \downarrow } \nonumber \\
&+& \sum_{i,m<m^\prime, \sigma } (U^{\prime}_{imm'} - J_{imm'}) \hat{n}_{im \sigma}\hat{n}_{im^\prime \sigma},
\label{HU_Term}
\end{eqnarray}
and
\begin{equation}
\begin{split}
\hat{H}_{J}
= \sum_{i, m \neq m^\prime} J_{imm^\prime} &(\hat{c}_{im\uparrow}^{\dagger} \hat{c}_{im^\prime\downarrow}^{\dagger} \hat{c}_{im\downarrow} \hat{c}_{im^\prime\uparrow} \\
&+\hat{c}_{im\uparrow}^{\dagger}\hat{c}_{im\downarrow}^{\dagger}\hat{c}_{im^\prime\downarrow}\hat{c}_{im^\prime\uparrow} + H.c.).
\end{split}
\label{HJ_Term}
\end{equation}
In the above, $\hat{c}_{im\sigma}^{\dagger}$($\hat{c}_{im\sigma}$) creates (annihilates) an electron with spin $\sigma$ in band $m$ at site $i$. $\hat{n}$ denotes the number operator and $\hat{n}_{im\uparrow}$, for example, represents the number of spin-up electrons at site $i$ in band $m$. $\hat{H}_{1}$ contains all one-body contributions to the Hamiltonian, including terms parameterized by the constants $t_{im,jm^\prime}^{\sigma \sigma'}$ that describe spin-orbit coupling and the hopping of electrons in different bands between sites $i$ and $j$. $\hat{H}_{2}$ denotes the collection of all two-body operators. $\hat{H}_{U}$ contains all density-density interactions, including the intraband ($U$) and interband ($U^\prime$) Coulomb interactions, and the $z$- (or Ising) component of the Hund's coupling. In contrast, $\hat{H}_{J}$ contains all of the terms that cannot be written as density-density interactions, which consist of the $x$- and $y$- components (spin-exchange) of the Hund's coupling, ($\hat{c}_{im\uparrow}^{\dagger} \hat{c}_{im^\prime\downarrow}^{\dagger} \hat{c}_{im\downarrow} \hat{c}_{im^\prime\uparrow} + H.c. $),
as well as the pair-hopping interaction ($\hat{c}_{im\uparrow}^{\dagger}\hat{c}_{im\downarrow}^{\dagger}\hat{c}_{im^\prime\downarrow}\hat{c}_{im^\prime\uparrow} +H.c.$), in which two electrons in a given band transfer as a pair to other bands. $J$ denotes the Hund's coupling constant. Note that our formalism is general and allows for band- and site-dependent $U$, $U$', and $J$ constants.
\subsection{Modified Hubbard Kanamori Model Hamiltonian}
In order to facilitate programming and the generalization of this HK Hamiltonian into a form in which all coupling constants are independent, we map the Hamiltonian given by Equations \eqref{Hamiltonian}-\eqref{HJ_Term} into a one-band model whose terms only depend upon their band indices. If we now let $i$ and $j$ denote superindices that combine both lattice site and band information, then
\begin{equation}
\begin{split}
\hat{H}
&= \hat{H}_{1} + \hat{H}_{2} \\
&= \sum_{ij,\sigma\sigma^\prime} t_{ij}^{\sigma\sigma^\prime} \hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma^\prime}\\
&+ \sum_{i} U^{i} \hat{n}_{i\uparrow} \hat{n}_{i\downarrow}\\
&+ \sum_{i<j} U_{1}^{ij} (\hat{n}_{i\uparrow} \hat{n}_{j\downarrow}+\hat{n}_{i\downarrow}\hat{n}_{j\uparrow})\\
&+ \sum_{i<j} U_{2}^{ij} (\hat{n}_{i\uparrow}\hat{n}_{j\uparrow}+\hat{n}_{i\downarrow}\hat{n}_{j\downarrow})\\
&+ \sum_{i<j} J^{ij} (\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow}
+\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow} \\
&+\hat{c}_{j\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}\hat{c}_{j\downarrow}\hat{c}_{i\uparrow}
+\hat{c}_{j\uparrow}^{\dagger}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{i\downarrow}\hat{c}_{i\uparrow}).
\end{split}
\label{Reformed_Hamiltonian}
\end{equation}
$t_{ij}^{\sigma\sigma^\prime}$ describes the hopping and spin-orbit coupling between different sites and bands. In keeping with the $\sum_{i, m<m'}$ and $\sum_{i, m\neq m'}$ summations in Equations \eqref{HU_Term} and \eqref{HJ_Term}, $\sum_{i<j}$ only sums over index combinations that reference different bands on the same site. In this modified HK Model, the $U$ term describes density-density interactions only between electrons with opposite spins in the same band, the $U_{1}$ term describes interactions between electrons with opposite spins in different bands on the same site, the $U_{2}$ term describes interactions between electrons with parallel spins in different bands on the same site, and the $J$ term describes spin-exchange and pair hopping interactions on the same site. Thus, in going from Equations \eqref{HU_Term} and \eqref{HJ_Term} to Equation \eqref{Reformed_Hamiltonian}, the original $U'$ term has become the $U_{1}$ term, the original $(U'-J)$ term has become the $U_{2}$ term, and the $J$ term has been re-expressed. Using Equation \eqref{Reformed_Hamiltonian}, we map a multi-band model into a single-band model in which the number of lattice sites has been enlarged into the number of bands. Since there is no explicit index in our model, we can deal with any number of bands as long as the mapping is done correctly.
\subsection{Overview of AFQMC}
In the remainder of this work, AFQMC will be employed to obtain accurate numerical solutions to the HK Model. AFQMC is a quantum many-body method that solves the ground state Schrodinger Equation by randomly sampling an overcomplete space of non-orthogonal Slater determinants\cite{Zhang_Gubernatis_PRB_1997,Zhang_Book_2003,Zhang_Book_2013} and has consistently been demonstrated to be among the most accurate of modern many-body methods for modeling the Hubbard model over a wide range of parameter regimes. \cite{LeBlanc_Gull_PRX_2015,Zheng_Science_2017,Chang_PRL_2010,Chang_PRB_2008,Shi_PRB_2013} At its heart, AFQMC is an imaginary-time projection quantum Monte Carlo technique that applies a projection operator, $e^{-\beta \hat{H}}$, onto an initial wave function, $|\Psi_{I} \rangle$,
\begin{equation}
|\Psi_{0} \rangle \propto \lim_{\beta \rightarrow \infty} \left(e^{-\beta \hat{H}}\right) | \Psi_{I} \rangle.
\end{equation}
In the limit of infinite imaginary projection time $(\beta \rightarrow \infty)$, it converges to the ground state wave function, $|\Psi_{0}\rangle$, as long as the initial wave function is not orthogonal to the ground state wave function. Because the projection operator cannot be evaluated for large values of $\beta$, it is discretized into $n = \beta/\Delta\tau$ smaller time slices for which it can be evaluated
\begin{equation}
|\Psi_{0} \rangle \propto \lim_{n \rightarrow \infty} \left(e^{-\Delta \tau \hat{H}}\right) ^{n} | \Psi_{I} \rangle,
\end{equation}
and the projection is carried out iteratively as follows
\begin{equation}
|\Psi^{(n+1)} \rangle = e^{-\Delta \tau \hat{H}} | \Psi^{(n)} \rangle.
\label{iteration}
\end{equation}
For sufficiently small $\Delta \tau$, the projection operator may be factored into one- and two-body pieces via Suzuki-Trotter Factorization\cite{Suzuki_ProgTheorPhys_1976,Trotter_PAMS_1959}
\begin{equation}
e^{-\Delta\tau \hat{H}} \approx e^{-\Delta\tau \hat{H}_{1}/2} e^{-\Delta\tau \hat{H}_{2}} e^{-\Delta\tau \hat{H}_{1}/2}.
\label{Projection_Equation}
\end{equation}
The two-body propagator may be further decomposed into the four terms given in Equation \eqref{Reformed_Hamiltonian}
\begin{equation}
\begin{split}
e^{-\Delta\tau \hat{H}_{2}}
&\approx e^{-\Delta\tau \hat{H}_{U}}
e^{-\Delta\tau \hat{H}_{U_1}}e^{-\Delta\tau \hat{H}_{U_2}}e^{-\Delta\tau \hat{H}_{J}}\\
&=e^{-\Delta\tau \sum\limits_{i} U_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow}}
e^{-\Delta\tau \sum\limits_{i<j} U_1^{ij} (\hat{n}_{i\uparrow} \hat{n}_{j\downarrow}+\hat{n}_{i\downarrow} \hat{n}_{j\uparrow})}\\
&e^{-\Delta\tau \sum\limits_{i<j} U_2^{ij} (\hat{n}_{i\uparrow} \hat{n}_{j\uparrow}+\hat{n}_{i\downarrow} \hat{n}_{j\downarrow})}\\
&e^{-\Delta\tau \sum\limits_{i<j}J^{ij}(\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow}
+\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow}
+H.c.)}.
\end{split}
\label{two-body propagator}
\end{equation}
A time step extrapolation is needed to make sure the Trotter error is negligible in the Monte Carlo simulation.
\subsection{Hubbard-Stratonovich Transformation of the Modified Hubbard Kanamori Hamiltonian \label{HSTransform}}
According to Thouless's Theorem, \cite{Thouless_NP_1960} acting the exponential of a one-body operator on a determinant results in another determinant, reducing the process of projecting a one-body operator onto the wave function into standard matrix multiplication. Nevertheless, no such theorem applies to exponentials of two-body operators, which necessitates re-expressing these operators into integrals over one-body operators using the so-called Hubbard-Stratonovich transformation. \cite{Hubbard_PRL_1959}
In order to transform the two-body propagator given by Equation \eqref{two-body propagator}, both discrete\cite{Hirsch_PRB_1983,Gubernatis_Werner} and continuous\cite{Buendia_PRB_1986} HS transformations need to be performed. The $U$, $U_{1}$, and $U_{2}$ terms are products of density operators, much like the conventional Hubbard $U$ term, and may therefore be decomposed using discrete transformations. For $\alpha < 0$, where $\alpha$ may be denote $U$, $U^{1}$, or $U^{2}$, it is usually better to use the discrete charge decomposition
\begin{equation}
e^{-\Delta\tau\alpha \hat{n}_{1}\hat{n}_{2}}=e^{-\Delta\tau\alpha (\hat{n}_{1}+\hat{n}_{2}-1)/2}\sum_{x=\pm{1}}\frac{1}{2}e^{\gamma x (\hat{n}_{1}+ \hat{n}_{2}-1)},
\label{density_charge}
\end{equation}
where
$\cosh(\gamma)=e^{-\Delta\tau\alpha/2}$, while for $\alpha > 0 $, it is usually better to use the spin decomposition
\begin{equation}
e^{-\Delta\tau\alpha \hat{n}_{1}\hat{n}_{2}}=e^{-\Delta\tau\alpha (\hat{n}_{1}+\hat{n}_{2})/2}\sum_{x=\pm{1}}\frac{1}{2}e^{\gamma x (\hat{n}_{1}-\hat{n}_{2})},
\label{density_spin}
\end{equation}
where $\cosh(\gamma)=e^{\Delta\tau\alpha/2}$. In both Equations \eqref{density_charge} and \eqref{density_spin}, $x$ represents the namesake auxiliary field that may assume the discrete values of $+1$ or $-1$. For the subsequent discussion, note that the charge decomposition is so named because it produces a one-body propagator involving the sum of $\hat{n}_{1}+\hat{n}_{2}$, which would be equivalent to the charge on a site if $1$ represented an up and $2$ a down spin on that site. Along similar lines, the spin decomposition is so named because it involves the difference between $\hat{n}_{1}$ and $\hat{n}_{2}$, which would represent the spin on a site under the same assumptions.
Because $\hat{H}_{J}$ contains terms that are not simple products of density operators, decomposing it is a much more challenging task. Past attempts have either neglected or simplified $\hat{H}_{J}$.\cite{Held_Vollhardt_EPJB_1998,Han_PRB_1998} Several techniques have employed exact decompositions,\cite{Motome_Imada_JPSJ_1997,Sakai_Aoki_PRB_2006,Sakai_Aoki_PRB_2004} but all such decompositions are accompanied by a sign problem that thwarts explorations of wide swaths of the phase diagram. Unlike these past attempts, in the following, we define a unique decomposition that can be employed in both continuous and discrete transformations, and accompany it by importance sampling that first mitigates and the constrained path and phaseless approximations that eliminate the sign and phase problems. As part of our decomposition of $e^{-\Delta \tau \hat{H}_{J}}$, we first re-expressed $\hat{H}_{J}$ in terms of squares of one-body operators. Let \begin{equation}
\hat{\rho}_{ij} \equiv \sum_{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + \hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma}).
\end{equation}
Then,
\begin{equation}
\hat{\rho}_{ij}^{2} = \sum_{\sigma\sigma^\prime}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + \hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma}) (\hat{c}_{i\sigma^\prime}^{\dagger}\hat{c}_{j\sigma^\prime} + \hat{c}_{j\sigma^\prime}^{\dagger}\hat{c}_{i\sigma^\prime}),
\end{equation}
and $\hat{H}_{J}$ may be re-expressed as (see the Supplemental Materials for more details)
\begin{equation}
\begin{split}
\hat{H}_{J}
&=\sum_{i<j} J^{ij}(\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow}+\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow}+ H.c.) \\
&=\sum_{i<j} \frac{J^{ij}}{2} [\hat{\rho}_{ij}^{2} - \sum_{\sigma} (\hat{n}_{i\sigma}+\hat{n}_{j\sigma} - \hat{n}_{i\sigma}\hat{n}_{j\sigma}-\hat{n}_{j\sigma}\hat{n}_{i\sigma})] \\
&=\sum_{i<j} \frac{J^{ij}}{2}\hat{\rho}_{ij}^{2} - \sum_{i<j, \sigma} \frac{J^{ij}}{2} (\hat{n}_{i\sigma}+\hat{n}_{j\sigma}) +\sum_{i<j, \sigma} J^{ij} \hat{n}_{i\sigma}\hat{n}_{j\sigma}.
\end{split}
\label{transformation_1}
\end{equation}
The second term of Equation \eqref{transformation_1} consists of one-body operators and can be combined with the other one-body operators into $\hat{H}_{1}$. The third term consists of a product of density operators and can therefore be transformed according to either Equations \eqref{density_charge} or \eqref{density_spin}. The first term, however, consists of a square that cannot be resolved into products of density operators. In order to decouple this two-body term, a continuous HS transformation must be employed. In general, the continuous HS transformation may be written as
\begin{equation}
e^{-\Delta \tau \hat{A}^{2}/2} = \int dx \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2} e^{x\sqrt{-\Delta \tau}\hat{A}}
\label{continuous_HS_transformation},
\end{equation}
where $\hat{A}$ represents any one-body operator and $x$ denotes an auxiliary field, as before. Letting $\hat{A} \equiv \hat{\rho}_{ij}$, it follows that the most obvious way to transform the exponential formed from the first term of Equation \eqref{transformation_1} is using the charge decomposition
\begin{equation}
\begin{split}
& e^{-\Delta\tau \sum\limits_{i<j}
\frac{J^{ij} }{2}[\sum\limits_{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}+\hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma})]^{2}} \\
&=
\prod_{i<j}\int dx_{ij} \frac{1}{\sqrt{2\pi}} e^{- x_{ij}^{2}/2} e^{x_{ij}\sqrt{-\Delta\tau J^{ij}} [\sum\limits_{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}+\hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma})]}.
\end{split}
\label{transformation_charge}
\end{equation}
As long as $J^{ij}<0$ for all $i,j$, all of the propagators produced by this transformation will be real, as is desirable within AFQMC simulations. However, if any of the $J^{ij}$ are greater than 0, $\sqrt{-\Delta\tau J^{ij}}$ will be complex resulting in a complex propagator that immediately introduces a complex phase into simulations. To prevent complexity from being introduced into the operators, in certain cases, we take a cue from the discrete case and define a continuous spin decomposition that involves the difference between spin up and down operators. Let
\begin{equation}
\hat{\rho}_{ij} = \sum_{\sigma} \delta _{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + \hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma}),
\label{Rho_Equation}
\end{equation}
where $\delta _{\uparrow}=1$ and $\delta _{\downarrow}=-1$, then (see the Supplemental Materials for further details)
\begin{equation}
\hat{\rho}_{ij}^{2} = \sum_{\sigma\sigma^\prime}
\delta _{\sigma}\delta _{\sigma^\prime}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + \hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma}) (\hat{c}_{i\sigma^\prime}^{\dagger}\hat{c}_{j\sigma^\prime} + \hat{c}_{j\sigma^\prime}^{\dagger}\hat{c}_{i\sigma^\prime}).
\label{Rho_Equation_Squared}
\end{equation}
Using this to re-express $\hat{H}_{J}$, we have
\begin{equation}
\begin{split}
\hat{H}_{J}
&=\sum_{i<j} J^{ij}(\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow}+\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow}+ H .c.) \\
&=\sum_{i<j} - \frac{J^{ij}}{2} [\hat{\rho}_{ij}^{2} - \sum_{\sigma} (\hat{n}_{i\sigma}+\hat{n}_{j\sigma} - \hat{n}_{i\sigma}\hat{n}_{j\sigma}-\hat{n}_{j\sigma}\hat{n}_{i\sigma})] \\
&=\sum_{i<j} -\frac{J^{ij}}{2}\hat{\rho}_{ij}^{2} +\sum_{i<j, \sigma} \frac{J^{ij}}{2} (\hat{n}_{i\sigma}+\hat{n}_{j\sigma}) -\sum_{i<j, \sigma} J^{ij} \hat{n}_{i\sigma}\hat{n}_{j\sigma}.
\end{split}
\label{transformation_3}
\end{equation}
Employing this form for the decomposition, the exponential that stems from the first term of Equation \eqref{transformation_3} may now be transformed to yield
\begin{equation}
\begin{split}
& e^{\Delta\tau \sum\limits_{i<j}
\frac{J^{ij} }{2}[\sum\limits_{\sigma}\delta _{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}+\hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma})]^{2}} \\
&=
\prod_{i<j}\int dx_{ij} \frac{1}{\sqrt{2\pi}} e^{- x_{ij}^{2}/2} e^{x_{ij}\sqrt{\Delta\tau J^{ij}} [\sum\limits_{\sigma}\delta _{\sigma}(\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}+\hat{c}_{j\sigma}^{\dagger}\hat{c}_{i\sigma})]},
\end{split}
\label{transformation_spin}
\end{equation}
which is real for $J^{ij} > 0$. Using the charge decomposition (Equation \eqref{transformation_charge}) when $J^{ij}<0$ and the spin decomposition (Equation \eqref{transformation_spin}) when $J^{ij}>0$ thus completely eliminates complex propagators, easing simulation. In Section \ref{two-band}, we compare the merits of using this mixed decomposition approach to exclusively relying upon the complex charge decomposition on the accuracy of our overall results.
Inserting the HS transformations defined by Equations \eqref{density_charge}, \eqref{density_spin}, \eqref{transformation_charge}, and \eqref{transformation_spin} into Equations \eqref{Projection_Equation} and \eqref{two-body propagator} and combining terms, one arrives at the final AFQMC expression for the projection operator
\begin{equation}
e^{-\Delta \tau \hat{H}} = \int d{\bf x} p({\bf x}) \hat{B}({\bf x}),
\label{effective_propagation}
\end{equation}
where ${\bf x} = \{x_1, x_2, . . . , x_{N_{F}}
\}$ denotes the set of $N_{F}$ total normally distributed auxiliary
fields sampled at a given time slice, $\hat{B}(x)$ represents the amalgamation of all one-body operators, and $p(x)$ is a combination of all scalar functions of the fields. Example expressions for $\hat{B}(x)$ and $p(x)$ are given in the Supplemental Information. As is clear from Equation \eqref{effective_propagation}, the series of HS Transformations described ultimately maps the original two-body propagator into a weighted integral over one-body propagators that are functions of external auxiliary fields.
\subsection{Sampling in AFQMC}
\subsubsection{The Sampling Process}
One of the most computationally efficient ways of evaluating many dimensional integrals such as that given by Equation \eqref{effective_propagation} is to use Monte Carlo sampling techniques. As described in more detail in previous publications,\cite{Hirsch_PRB_1985, Zhang_Gubernatis_PRB_1997,Zhang_Book_2003,Zhang_Book_2013} if $|\Psi_{I}\rangle$ is represented by a single Slater determinant, after each application of the projection operator, a new Slater determinant will be produced. Thus, if $k$ instances (so-called ``walkers'') are initialized to $|\Psi_{I}\rangle$ and the projection operation given by Equation \eqref{effective_propagation} is applied to each of them by independently sampling sets of fields, then a random walk through the space of non-orthogonal determinants is realized in which the overall wave function at time slice $n$, $|\Psi^{(n)}\rangle$, is represented by an ensemble of $k$ wave functions $|\psi_{k}^{(n)}\rangle$ with weights $w_{k}^{(n)}$
\begin{equation}
| \Psi^{(n)}\rangle = \sum_{k} w_k^{(n)} | \psi_k^{(n)} \rangle .
\label{wave function}
\end{equation}
Here, the $w_{k}^{(n)}$ consist of the products of numbers accumulated over all time slices by walker $k$, which can be a complex number.
Ground state observables at each time slice, such as the energies reported below, may then be computed by evaluating the mixed estimator\cite{Foulkes_RMP_2001} over the ensemble
\begin{eqnarray}
\langle \hat{A} \rangle_{mix} &=& \frac{\langle \Psi_{T} | \hat{A} | \Psi^{(n)} \rangle}{\langle \Psi_{T} | \Psi^{(n)} \rangle} \nonumber \\
&=& \frac{ \sum_{k} w_{k}^{(n)} \langle \Psi_{T} | \hat{A} | \psi_{k}^{(n)} \rangle}{ \sum_{k} w_{k}^{(n)} \langle \Psi_{T} | \psi_{k}^{(n)} \rangle} ,
\label{Mixed}
\end{eqnarray}
where $|\Psi_{T} \rangle$ denotes a trial wave function that approximates the true ground state wave function. To facilitate the evaluation of the mixed estimator, it is common to introduce the local energy
\begin{equation}
E_{L}[\Psi_{T}, \Phi] \equiv \frac{\langle \Psi_{T}| \hat{H} | \Phi \rangle}{\langle \Psi_{T} | \Phi \rangle},
\end{equation}
such that Equation \eqref{Mixed} may be simplified to
\begin{equation}
\langle \hat{A} \rangle_{mix} = \frac{ \sum_{k} w_{k}^{(n)} \langle \Psi_{T} | \psi_{k}^{(n)} \rangle E_{L}[\Psi_{T}, \psi_{k}^{(n)}]}{ \sum_{k} w_{k}^{(n)} \langle \Psi_{T} | \psi_{k}^{(n)} \rangle}.
\label{Mixed_Again}
\end{equation}
After a sufficiently large number of time slices such that $|\Psi^{(n)}\rangle$ approaches the ground state, final estimates of $\langle \hat{A}\rangle$ may be obtained by averaging over each time slice expectation value.
A population control procedure \cite{Calandra_Sorella_PRB_1998} is needed during the random walk. During this procedure, walkers with larger weights are replicated and those with smaller weights are eliminated probabilistically. The weight used in population control is
\begin{equation}
W^{(n)}_{k} = w_{k}^{(n)} \langle \Psi_{T} | \psi_{k}^{(n)} \rangle .
\label{weight}
\end{equation}
When there is a sign or phase problem, $ W^{(n)}_{k} $ may become negative or complex. As described in Section (\ref{cpa}) and Section (\ref{pha}), $ W^{(n)}_{k} $ is always positive or zero if the constrained path or phaseless approximations are employed.
\subsubsection{The Sign and Phase Problems}
Unfortunately, the ``free'' projection process just described is typically beset by either the sign\cite{Loh_PRB_1990,Ceperley1991} or phase problems.\cite{Zhang_Krakauer_PRL_2003} These problems fundamentally stem from the fact that observables computed using a single Slater determinant, $|\Psi\rangle$, remain invariant to arbitrary rotations, $e^{i\theta} |\Psi \rangle$, of that determinant, where $\theta$ is a phase angle. Consequently, during the course of an AFQMC simulation involving complex propagators, walkers may accumulate infinitely many possible phases (as there are infinitely many possible phase angles, $\theta \in [0, 2\pi)$), resulting in infinitely many possible determinants. Since these phases are directly multiplied into the walker weights of Equations \eqref{Mixed} and \eqref{Mixed_Again}, after many iterations, the walker weights end up populating the entire complex plane and many of the terms summed to compute weighted averages of observables cancel one another out. This cancellation leads to an exponential decline in observable signal to noise ratios that manifests as infinite variances\cite{Shi_PRB_2013} called the phase problem. If transformations that preclude propagators from becoming complex are employed as described above, positive and negative versions of each determinant may still be generated, resulting in a somewhat less pernicious cancellation of positive and negative weights termed the sign problem. If left unchecked, the sign and phase problems render obtaining meaningful observable averages nearly impossible, thwarting AFQMC simulations. We therefore mitigate these problems using a combination of background subtraction, importance sampling, and either the constrained path (for the sign problem) or phaseless (for the phase problem) approximations.
\subsubsection{Background Subtraction}
One of the simplest ways of reducing variances within AFQMC is via background subtraction.\cite{Purwanto_PRA_2005} As part of background subtraction, the two-body portion of a Hamiltonian is rewritten so that a mean field average is subtracted from each one-body operator. Thus, if the original two-body operator may be written as a square such that $\hat{V} = -\frac{1}{2} \sum_{i} \hat{v}_{i}^{2}$ to make it amenable to a HS Transformation, as part of background subtraction, it would be re-expressed as
\begin{equation}
\hat{V} = -\frac{1}{2} \sum_{i} \left(\hat{v}_{i} - \langle \hat{v}_{i} \rangle \right)^{2} - \sum_{i} \hat{v}_{i} \langle \hat{v}_{i} \rangle + \frac{1}{2} \sum_{i} \langle \hat{v}_{i} \rangle^{2},
\end{equation}
where $\langle \hat{v}_{i} \rangle$ denotes the mean field average of the operator $\hat{v}_{i}$ (see the Supplemental Materials for more details on how this mean field average is obtained). Because the modified $\hat{v}_{i} - \langle \hat{v}_{i} \rangle$ operator will be smaller in magnitude than the bare $\hat{v}_{i}$ operator, background subtraction reduces the variance involved in AFQMC simulations. In this work, we perform background subtraction on the only term in the Hamiltonian that is not a product of on-site densities, the $\frac{J^{ij}}{2}\hat{\rho}_{ij}^{2}$ term of Equation \eqref{transformation_1} or the $-\frac{J^{ij}}{2}\hat{\rho}_{ij}^{2}$ term of Equation \eqref{transformation_3}, yielding
\begin{eqnarray}
\sum_{i<j} \frac{J^{ij}}{2} \hat{\rho}_{ij}^{2}
&=& \sum_{i<j} \frac{J^{ij}}{2} (\hat{\rho}_{ij}-\langle \hat{\rho}_{ij} \rangle)^{2}
-\sum_{i<j} \frac{J^{ij}}{2} \langle \hat{\rho}_{ij} \rangle^{2} \nonumber \\
&+& \sum_{i<j}J^{ij}\langle \hat{\rho}_{ij} \rangle \hat{\rho}_{ij}
\end{eqnarray}
and
\begin{eqnarray}
\sum_{i<j} -\frac{J^{ij}}{2} \hat{\rho}_{ij}^{2}
&=&\sum_{i<j} -\frac{J^{ij}}{2} (\hat{\rho}_{ij}-\langle \hat{\rho}_{ij} \rangle)^{2} +\sum_{i<j} \frac{J^{ij}}{2} \langle \hat{\rho}_{ij} \rangle^{2} \nonumber \\
&-& \sum_{i<j}J^{ij} \langle \hat{\rho}_{ij} \rangle \hat{\rho}_{ij},
\end{eqnarray}
respectively.
\subsubsection{Importance Sampling}
In order to further reduce the variance of walker weights and to make our simulations more amenable to the constrained path and phaseless approximations, we additionally perform importance sampling, which aims to shift the center of the distribution from which we sample our auxiliary fields so that the most important fields are sampled more frequently. The conventional way of performing importance sampling in AFQMC simulations is by introducing a force bias that shifts each sampled field by an amount dependent upon the operator being transformed and the current walker wave function.\cite{Purwanto_PRE_2004,Zhang_Krakauer_PRL_2003,Rom_Neuhauser_CPL_1997,Shi_Zhang_PRA_2015} Because we utilize a mixture of discrete and continuous transformations and force bias importance sampling is only applicable to continuous transformations, in this work, we employ a formally equivalent strategy in which we shift \emph{the propagators instead of the auxiliary fields}.
For continuous HS Transformations, this may be accomplished by shifting the operator $\hat{A}$ by $\langle \hat{A} \rangle$ in Equation \eqref{continuous_HS_transformation}
\begin{equation}
\begin{split}
e^{-\Delta \tau \hat{A}^{2}/2}
&= \int dx \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2} e^{x\sqrt{-\Delta \tau}\hat{A}} \\
&= \int dx \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2} e^{x\sqrt{-\Delta \tau}\langle \hat{A} \rangle} e^{x\sqrt{-\Delta \tau}(\hat{A}-\langle \hat{A} \rangle)},
\label{importance_sampling_continuous}
\end{split}
\end{equation}
where $\langle \hat{A} \rangle$ is the mixed estimator of $\hat{A}$
\begin{equation}
\langle \hat{A} \rangle
\equiv \frac{\langle \Psi_T | \hat{A} | \psi_k^{(n)} \rangle }{\langle \Psi_T | \psi_k^{(n)} \rangle }.
\end{equation}
If we define the dynamic force as $F \equiv \sqrt{-\Delta \tau}\langle \hat{A} \rangle$, then Equation \eqref{importance_sampling_continuous} may be re-expressed as
\begin{equation}
\begin{split}
e^{-\Delta \tau \hat{A}^{2}/2}
&= \int dx \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2} e^{xF} e^{x\sqrt{-\Delta \tau}\hat{A}-xF} \\
&= \int dx \frac{1}{\sqrt{2\pi}} e^{-(x-F)^{2}/2} e^{\frac{1}{2}F^2}e^{x\sqrt{-\Delta \tau}\hat{A}-xF} \\
&= \int dx \frac{1}{\sqrt{2\pi}} e^{-(x-F)^{2}/2} e^{\frac{1}{2}F^2-xF}e^{x\sqrt{-\Delta \tau}\hat{A}}
\label{importance_sampling_continuous-2}.
\end{split}
\end{equation}
In order to realize this transformation, fields are sampled from the shifted Gaussian probability density function, $\frac{1}{\sqrt{2\pi}} e^{-(x-F)^{2}/2}$, and the propagator $e^{x\sqrt{-\Delta \tau} \hat{A}}$ is applied with weight $e^{\frac{1}{2}F^{2}-xF}$. The field distributions are now centered around the dynamic force, which can be shown to minimize the variance. If the dynamic force $F$ is a complex number, our auxiliary fields will have the same imaginary part to ensure the $x-F$ is real. Then the probability function $\frac{1}{\sqrt{2\pi}} e^{-(x-F)^{2}/2}$ stays in the real axis, which can be sampled by Monte Carlo.
Shifting the propagator within a discrete transformation proceeds in exactly the same fashion. Comparing Equations \eqref{continuous_HS_transformation} and \eqref{density_spin}, the dynamic force needed to shift the propagator in Equation \eqref{density_spin}, for example, would be $F\equiv \gamma(\langle \hat{n}_{1} \rangle - \langle \hat{n}_{2} \rangle)$, resulting in the transformation
\begin{equation}
\begin{split}
e^{-\Delta\tau\alpha \hat{n}_{1}\hat{n}_{2}}
&=e^{-\Delta\tau\alpha (\hat{n}_{1}+\hat{n}_{2})/2}\sum_{x=\pm{1}}\frac{1}{2}e^{\gamma x (\hat{n}_{1}-\hat{n}_{2})} \\
&= e^{-\Delta\tau\alpha (\hat{n}_{1}+\hat{n}_{2})/2} \sum_{x=\pm{1}}\frac{1}{2} \left( \frac{e^{xF}}{W} \right) W e^{\gamma x (\hat{n}_{1}-\hat{n}_{2})-xF}. \\
\label{importance_sampling-discrete}
\end{split}
\end{equation}
As in the continuous case, in order to realize this transformation, fields are now sampled from a shifted probability density function, $e^{xF}/W$, where $W$ is the normalization factor, $W=e^{xF} + e^{-xF}$, and the propagator $e^{(-\Delta\tau\alpha/2 + \gamma x)\hat{n}_1}e^{(-\Delta\tau\alpha/2 - \gamma x)\hat{n}_2}$ is applied with weight $\frac{1}{2}We^{-xF}$. A shifted transformation may similarly be constructed for the discrete charge decomposition given by Equation \eqref{density_charge}. Propagators that include background subtraction may be shifted by simply replacing $\hat{A}$ with $\hat{A}-\langle \hat{A} \rangle$ in Equations \eqref{importance_sampling_continuous} and \eqref{importance_sampling_continuous-2} above (see the Supplemental Materials).
It can readily be proven that shifting auxiliary fields is equivalent to shifting propagators.\cite{Rom_Neuhauser_CPL_1997,Rom_JCP_1998,Shi_Zhang_PRA_2015} Shifting propagators therefore entails a convenient way of introducing importance sampling when discrete transformations are involved. Overall, the importance sampled propagation produces the same observable averages as free propagation, but favors the sampling of determinants with larger overlaps with the trial wave function and suppressing the sampling of determinants with no overlap.
\subsubsection{Constrained Path Approximation}
\label{cpa}
In order to address the sign problem that may emerge when our propagators, $\hat{B}(\vec{x})$, are real, we employ the constrained path approximation. \cite{Zhang_Gubernatis_PRB_1997} Here, we impose this approximation by requiring that all walkers maintain a positive overlap with the trial wave function after each propagation step
\begin{equation}
w_k^{(n)}\langle \Psi_T | \psi_k^{(n)} \rangle >0.
\label{overlap}
\end{equation}
As in typical constrained path implementations, walkers with negative overlaps with the trial wave function will be killed (have their weights set equal to zero), preventing them from being propagated further. This condition will select for only walkers with positive determinants, eliminating the sign problem. It can be shown that if the trial wave function is the exact ground state wave function, this condition will be exact;\cite{Carlson_PRB_1999} however, since the trial wave function is typically unknown, constraining the propagation path in this way results in a small, but consequential approximation.\cite{Chang_PRB_2016,Shi_PRB_2013}
\subsubsection{Phaseless Approximation}
\label{pha}
In cases in which our propagators are complex, instead of employing the constrained path approximation, we employ the more general phaseless approximation.\cite{Zhang_Krakauer_PRL_2003,Purwanto_PRA_2005} The phaseless approximation controls the phase problem by projecting complex walker weights onto the positive real axis according to the equation
\begin{equation}
\label{cosProjection}
W^{(n)}_{k} = |W^{(n)}_{k}| \times \max (0, \cos(\Delta \theta)),
\end{equation}
where $W^{(n)}_{k}$ is defined in Equation \eqref{weight} and $\Delta \theta$, the phase angle, is defined as
\begin{equation}
\begin{split}
&\Delta \theta =Arg \left[ \frac{\langle \Psi_T | \hat{B}(x) | \psi_k^{(n)} \rangle }{\langle \Psi_T | \psi_k^{(n)} \rangle } \right] \approx O(Im(xF)).
\end{split}
\end{equation}
The use of the cosine function to project also ensures that the density of the walkers will vanish at the origin. Because this cosine projection does not affect walkers with real weights, in practical implementations, we apply Equation \ref{cosProjection} to realize both the constrained path and phaseless Approximations.
\subsection{Trial and Initial Wave Functions}
Although AFQMC can readily accommodate multi-determinant trial wave functions, we restrict ourselves to employing single determinant trial wave functions that satisfy certain symmetries\cite{Shi_PRB_2013} such as the free electron (FE), restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF), and generalized Hartree-Fock (GHF) wave functions. RHF wave functions preserve spin symmetry. While RHF and UHF wave functions separately conserve the number spin up and down electrons, GHF only fix the total number of electrons. Details about how these wave functions are generated may be found in the Supplemental Materials.
As illustrated in what follows, because GHF wave functions do not impose any spin symmetries and are therefore the most flexible of these wave function ansatzes, they enable the fastest AFQMC wave function relaxation to the global energy minimum. Nevertheless, when the number of up and down electrons must be fixed, UHF/RHF wave functions were employed instead. Even though our formalism permits our initial wave functions to differ from our trial wave functions, we take our initial and trial wave functions to be the same, except where otherwise noted.
\section{Results and Discussion}
\label{results}
\subsection{Two-Band Hubbard Kanamori Model Benchmarks}
\label{two-band}
In order to test the accuracy of our theoretical framework, we began by benchmarking our method against ED results for the one-dimensional, two-band HK Model on $5 \times 1$ and $6 \times 1$ lattices with periodic boundary conditions small enough to diagonalize. For these benchmarks, we simplify the Hamiltonian given by Equations \eqref{Hamiltonian} and \eqref{HOne_Term} so that hopping can only occur between adjacent sites within the same bands and may be described by a single site- and spin-invariant constant $t$, such that
\begin{equation}
\hat{H}_{1}^{'} = -t \sum_{\langle ij \rangle,\sigma }\sum_{m=1}^{2} \hat{c}_{im\sigma}^{\dagger}\hat{c}_{jm\sigma}.
\end{equation}
We moreover assume that the parameters are site-invariant, such that U$^{i}$=U, U$_1^{ij}$ = U$_1$, U$_2^{ij}$ = U$_2$, and J$^{ij}$ = J.
\textbf{\begin{table}[htbp]
\footnotesize
\caption{The ground state energy of the two-band, 6$\times$1 HK model with $N_{\uparrow}=N_{\downarrow}=6$ over a range of parameters using ED and AFQMC. All energies and parameters are reported in units of $t$.}
\label{table2}
\begin{ruledtabular}
\begin{tabular}{cccccc}
U & U$_{1}$ & U$_{2}$ & J & ED & AFQMC \\
\hline
2.0 & 1.5 & 1.0 & 0.5 & -3.773268 & -3.774(3) \\
2.0 & 1.5 & 1.0 & 1.0 & -4.234037 & -4.230(6) \\
2.0 & 1.5 & 3.0 & 0.5 & 0.758540 & 0.755(4) \\
3.0 & 5.0 & 1.0 & 0.5 & 2.460374 & 2.466(5) \\
6.0 & 1.5 & 1.0 & 0.5 & 1.496509 & 1.503(6) \\
\end{tabular}
\end{ruledtabular}
\end{table}}
Table \ref{table2} presents our results for a $6\times 1$ HK model over a representative sampling of parameters at half filling. All of the calculations presented were initialized using 560 walkers and employed FE trial and initial wave functions, except for the $U$=3.0,$U_{1}$=5.0,$U_{2}$=1.0,$J$=0.5 case. In this case, it was found that an RHF trial wave function yielded a lower trial energy and manifested a different spin order (antiferromagnetic (AFM) order between two bands) than the FE solution. Thus, an RHF trial wave function was employed instead. This demonstrates that trial wave functions should first be analyzed to determine whether their global minima exhibit the correct order before using them to guide propagation within AFQMC. Unless otherwise noted, all of the results presented in this section were obtained using a charge decomposition for $J$ and the phaseless approximation to tame the related phase problem that emerges.
As is clear from the table, AFQMC results, are within $0.01t$ or less of exact results, with the smallest discrepancy occurring for the $U=2.0$, $U_{1}=1.5$ case and the largest occurring for the $U=6.0$ case. In all of these cases, exact results are within two standard derivations of the Monte Carlo results, despite the use of the phaseless approximation.
To pinpoint AFQMC systematic bias, as well as to better understand which regions of the phase diagram are the most challenging for AFQMC, we independently scanned through each of the $U$, $U_{1}$, $U_{2}$, and $J$ parameters holding the others fixed for a $5 \times 1$ HK model. In Figures \ref{f1} and \ref{f2}, we present our scans over $U$ and $J$; figures of our $U_{1}$ and $U_{2}$ scans are presented in the Supplemental Materials.
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=10.5cm]{two_band_U.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{AFQMC ground state energy vs. the density-density parameter $U$ for the two-band, 5$\times$1 HK model using the charge decomposition and FE trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1$, $U_1$=0, $U_2$=0, and $J=0$ with $N_{\uparrow}=N_{\downarrow}=6$. Relative errors, $\Delta E$, taken with respect to ED result are plotted in the inset for clarity.}
\label{f1}
\end{figure}
\end{center}
As shown in Figure \ref{f1}, although the magnitude of the error bars grows with $U$, the relative error remains within 0.1\% to 1\% throughout this range. Similar trends are observed for $U_{1}$ and $U_{2}$. This gives us reason to believe that our method can readily accommodate some of the even larger $U$ values used in studies of strongly correlated materials. Nevertheless, much larger relative errors are observed as $J$ is varied, as depicted in Figure \ref{f2}. This is consistent with previous work, which also implicates the $J$ terms as being most conducive to QMC errors.\cite{Held_Vollhardt_EPJB_1998} Fortunately, for most real materials, $J$ is usually a small fraction of $U$. For small $J$ values, the relative errors are observed to remain less than 1\% and are therefore controllable.
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=8.5cm]{two_band_J.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{AFQMC ground state energy vs. the Hund's coupling parameter $J$ for the two-band, 5$\times$1 HK model using the charge decomposition and FE/RHF trial wave functions (WF). Here, all of the other Hamiltonian parameters are held fixed at $t=1$, $U=0$, $U_1$=0, and $U_2$=0 with $N_{\uparrow}=N_{\downarrow}=6$. Relative errors, $\Delta E$, taken with respect to ED results are plotted in the inset for clarity.}
\label{f2}
\end{figure}
\end{center}
What may also be gleaned from Figure \ref{f2} is that the quality of the $J>1.5$ energies depends upon the type of trial wave function employed. While free propagation calculations yield results that are independent of the trial wave function, the quality of the constrained path and phaseless approximations fundamentally depend on the accuracy of the trial wave function employed. As depicted in Figure \ref{f2}, the relative errors in the energies produced by FE trial wave functions surpass 10\% and increase with increasing $J$; in contrast, the relative errors produced by RHF trial wave functions not only remain less than 10\%, but plateau as a function of $J$. As $J$ increases, the RHF electron density becomes non-uniform, yielding a lower variational energy than the FE wave function. Figure \ref{f2} thus demonstrates that AFQMC becomes more accurate as trial wave functions better describe the ground state. Note that we also tested UHF and GHF wave functions, which all converged to the same states as RHF wave functions.
\begin{figure}[htp]
\includegraphics[width=9cm]{two_band_decomp_J.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{Comparison of phaseless and constrained path AFQMC energy errors as a function of $J$ for a two-band, 5$\times$1 HK model. Open circles denote parameters at which the constrained path approximation was employed, while closed circles denote parameters at which the phaseless approximation was employed. Here, we set N$_{\uparrow}$= N$_{\downarrow}$=6, $t=1$, $U=0$, $U_{1}=0$, and $U_{2}=0$. FE trial wave functions were used for both the initial and trial wave functions, and 560 walkers were employed in each calculation.}
\label{f3}
\end{figure}
The accuracy of AFQMC predictions are also influenced by the constrained path and phaseless approximations employed. In Figure \ref{f3}, we compare the errors produced by these approximations. As discussed in Section \ref{HSTransform}, for $J>0$, the spin decomposition will yield real propagators that we constrain using the constrained path approximation, while for $J<0$, the spin decomposition will yield complex propagators that we constrain using the phaseless approximation. The charge decomposition behaves in the opposite fashion with respect to $J$. As shown in Figure \ref{f3}, the constrained path approximation behaves significantly better than the phaseless approximation, which appreciably differs from the exact results for $|J|>1.5$. Indeed, the constrained path approximation nearly reproduces the exact results for $J<0$, only manifesting a slight deviation for larger positive values of $J$. These results attest to the fact that using the transformations we describe to prevent the phase problem from emerging is key to maintaining AFQMC accuracy. They also underscore that our method is capable of simulating -$J$ values, which have been unattainable in previous QMC simulations. We expect these trends in accuracy to generalize to models with more bands and higher dimensionality.
\subsection{Application to Three-Band Hubbard Kanamori Models}
\label{three_band}
In order to understand how our techniques generalize to models that approximate more realistic materials and their magnetic phase transitions, we constructed a three-band model with an adjustable band gap. As illustrated in Figure \ref{three_band_structure}, in this model, three bands are located at each site, one band of which is lower in energy by a `band gap' parameter, $\Delta$, than the other two degenerate bands. When $\Delta = 0$, all three bands are completely degenerate. Similar to the two-band model, the hopping occurs between adjacent sites within the same bands, with hopping constant $t_{ij}=1$. While the band gap would be fixed in any given material, creating a separate $\Delta$ parameter enables us to sample a range of band gaps and, by extension, to drive magnetic ordering transitions. We moreover assume that $U^{i}=U$ and $J^{ij}=J=0.15 U$ with $U_1^{ij}=U_1=U-2J$ and $U_2^{ij}=U_2=U-3J$, which are appropriate for the description of transition-metal oxides with a partially occupied $t_{2g}$ shell.\cite{sugano_tanabe_kamimura_1970} In the following discussion, we fix our filling such that an average of four electrons occupy the three bands at each lattice site.
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=7.5cm]{three_band_structure.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{Schematic of our three-band model on a 4x4 lattice. At each site, there is one atom with three bands, one of which is lower in energy by $\Delta$ than the other two degenerate bands. The top right box illustrates a situation in which AFM order is present between adjacent lattice sites.}
\label{three_band_structure}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=10.5cm]{three_band_mu.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{AFQMC ground state energy as a function of the band gap magnitude, $\Delta$, for the three-band, 2$\times$2 HK model using the charge decomposition and GHF trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1$, $U=6$, $U_1=U-2J$, $U_2=U-3J$, and $J=0.15U$ with $N_{\uparrow}=N_{\downarrow}=8$. Relative errors, $\Delta E$, taken with respect to ED results are plotted in the inset for clarity.}
\label{three_band_mu}
\end{figure}
\end{center}
As an initial step, we benchmarked our AFQMC method against ED results. Diverging from our previous two-band analysis, as part of our three-band benchmarks, we studied our model on two-dimensional lattices with periodic boundary conditions, only varying $\Delta$ and $U$ while keeping the other parameter relationships fixed in order to preserve realism. Our simulations were initialized with 560 walkers and GHF initial and trial wave functions for all of the benchmarks described below. The charge decomposition with the phaseless approximation was employed throughout this section.
In Figure \ref{three_band_mu}, we illustrate how the energy and relative errors change as $\Delta$ is varied from 0 to 1 with $U=6$ on a 2$\times$2 lattice. At fixed $U$, the relative error remains fairly stable and less than 0.1\% throughout this range. This may be anticipated since the band gap only modifies the magnitude of the one-body terms and does not change the phase of the model, which do not directly contribute to our method's stochastic errors.
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=10.5cm]{three_band_U.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{AFQMC ground state energy vs. $U$ for the three-band, 2$\times$2 HK model using the charge decomposition and GHF trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1$, $\Delta =0.8$, $U_1=U-2J$, $U_2=U-3J$, and $J=0.15U$ with $N_{\uparrow}=N_{\downarrow}=8$. Relative errors, $\Delta E$, are taken with respect to ED results are plotted in the inset for clarity.}
\label{three_band_U}
\end{figure}
\end{center}
In Figure \ref{three_band_U}, instead of scanning $\Delta$, we scan $U$ with $\Delta = 0.8$. As shown in Figure \ref{three_band_U}, the relative errors are larger in this case, but still range from 0.1\% for $U<6$ to 1\% for $U>6$. Errors would be expected to grow in this manner as the system becomes more correlated. Overall, the magnitudes of these relative errors suggest that AFQMC's performance is promising.
The rationale for introducing the band gap $\Delta$ parameter is to enable tuning of the magnetic order of the model system. Intuitively, when the band gap is small, the three bands are nearly degenerate and the four electrons have the largest freedom to move among the bands. Such a situation would favor ferromagnetic (FM) order. However, when the band gap becomes sufficiently large, two electrons will populate the lower band, forcing the other two electrons to reside on the higher energy bands. Such a situation would favor AFM order.
This intuition was confirmed by comparing the AFQMC energies attained using trial wave functions with FM and AFM order, respectively (see Figure~\ref{three_band_phase}). Typically, GHF calculations converge to the lowest state with the same magnetic order as the initial state. Thus, in order to construct wave functions with FM order, a randomly initialized density matrix was supplied to the GHF self-consistent equations; to construct wave functions with AFM order, an AFM-ordered initial density matrix was supplied. Several independent GHF calculations were conducted for each system studied to guarantee that the final GHF wave functions produced attained their global minima. At large $\Delta$ ($\Delta \gtrsim 1.1$) values at which ferromagnetic order is disfavored, GHF calculations initialized with random density matrices often developed order. In these situations, FM wave functions produced at smaller values of $\Delta$ were used as trial wave functions in ``FM'' AFQMC calculations performed at larger $\Delta$ values. Figure \ref{three_band_phase} depicts the energies of AFQMC simulations performed with AFM and FM trial wave functions, respectively, as a function of band gap. All of the AFQMC energies presented here are the lowest energies we can obtain at each $\Delta$. At smaller $\Delta$s, trial wave functions with FM order led to the lowest AFQMC energies, while at larger $\Delta$s, AFM trial wave functions did so. This confirms that our model undergoes a ferromagnetic to antiferromagnetic transition at roughly $\Delta = 1.1$. In contrast, Hartree-Fock theory predicts the transition at $\Delta = 0.5$, which is reasonable since Hartree-Fock theory tends to fall into AFM order sector. An illustration of the AFM order exhibited by our model is depicted in Figure~\ref{three_band_structure}.
\begin{center}
\begin{figure}[htbp]
\includegraphics[width=10.5cm]{three_band_phase_trans.pdf}
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{AFQMC ground state energy vs. band gap magnitude, $\Delta$, for the three-band, 4$\times$4 HK model using the charge decomposition. GHF trial wave functions with both FM order and AFM order are used. A QMC predicted phase transition occurred at around $\Delta=1.1$. Hartree-Fock predicted transition point is at $\Delta = 0.5$ illustrated in green dotted line. Here, all of the other Hamiltonian parameters are held fixed at $t=1$, $U=6$, $U_1=U-2J$, $U_2=U-3J$, and $J=0.15$ $U$ with $N_{\uparrow}=N_{\downarrow}=32$.}
\label{three_band_phase}
\end{figure}
\end{center}
To further corroborate the phase transition we observe, we extrapolated the magnitude of the charge gap at $\Delta=0.2$ and $\Delta = 1.5$. To do so, we computed the ground state AFQMC energies of 4$\times$4, 6$\times$6, and 8$\times$8-site systems, with three bands filled with four electrons situated at each site. The charge gap may be determined by computing $E_{N-1}+E_{N+1}-2E_{N}$, where $N$ denotes the total number of electrons in the system. To determine the charge gap in the thermodynamic limit, we fit a $1/L$ form, where $L$ denotes the total number of lattice sites, to the energies and extrapolated to the infinite $L$ limit (see the Supplemental Materials for more details). The energies produced using FM initial trial wave functions were used to ascertain the $\Delta = 0.2$ charge gap, while those produced using AFM wave functions were employed to ascertain the $\Delta = 1.5$ charge gap. The charge gaps obtained are presented in Table \ref{charge_gap}. After extrapolations, the $\Delta=0.2$ charge gap converged to -0.006(47) and the $\Delta=1.5$ charge gap converged to 1.201(41). As one would expect antiferromagnetic, not ferromagnetic, order to be accompanied by a charge gap, these extrapolations support our previous conclusions.
\textbf{\begin{table}[htbp]
\footnotesize
\captionsetup{font=footnotesize, justification=raggedright, singlelinecheck=false}
\caption{The charge gaps of the three-band model at $\Delta = 0.2$ and $\Delta=1.5$ for different system sizes calculated using AFQMC. GHF trial wave functions with FM order and AFM order are used at $\Delta = 0.2$ and $\Delta = 1.5$, respectively. All of the other Hamiltonian parameters are held fixed at $t=1$, $U=6$, $U_1=U-2J$, $U_2=U-3J$, and $J=0.15U$. The electron density per band is $4/3$.}
\label{charge_gap}
\begin{ruledtabular}
\begin{tabular}{ccc}
\# of bands & Charge Gap ($\Delta = 0.2 $) & Charge Gap ($\Delta = 1.5 $) \\
\hline
4x4x3 & 0.222(29) & 1.311(32) \\
6x6x3 & 0.103(27) & 1.268(35) \\
8x8x3 & 0.015(72) & 1.225(36) \\
$\infty$ & -0.006(47) & 1.201(41) \\
\end{tabular}
\end{ruledtabular}
\end{table}}
The successful determination of the magnetic order and charge gaps in this model system illustrate our method's promise for accurately modeling realistic materials.
\section{Conclusions}
\label{conclusions}
In summary, we have presented a ground state AFQMC framework suited for the study of the HK model, a multi-band model designed to capture the Hund's physics of many $d$- and $f$-electron materials. Diverging with past QMC studies of the HK model, we employ a novel set of HS transformations to decouple the Hund's coupling term while preserving the term's essential physics. We find that by carefully combining these transformations with a form of importance sampling that shifts our propagators, well-optimized GHF wave functions, and the constrained path and phaseless approximations, we can accurately predict the energetics of benchmark lattice models and the magnetic order of much larger models that approximate realistic materials. Overall, we find that the phaseless version of our method produces nearly exact energies for small models for $-3<J<3$, a range of $J$ values which contains those commonly observed in experiment. This bodes well for the generalization of our method to other systems.
Our method may readily be extended to include spin-orbit coupling effects and negative $J$ values, which opens the doors to the highly accurate study of exotic, -$J$ fulleride physics.\cite{Nomura_JPCM_2016,Nomura_Science_2015} In order to describe superconducting physics, our method can be adapted to use superconducting trial wave function forms, including Bardeen-Cooper-Schrieffer\cite{Shi_Zhang_PRA_2015,Carlson_Zhang_PRA_2011} and Hartree-Fock-Bogoliubov\cite{Shi_Zhang_PRB_2017} wave functions. We foresee our method having the most immediate impact as a way to delineate low-temperature phase diagrams currently beyond the reach of DMFT methods.\cite{} As the same transformations and importance sampling techniques may readily be adapted into finite temperature AFQMC formalisms,\cite{Hirsch_PRL_1986,Liu_JCTC_2018,Zhang_PRL_1999} the same methods may be used to develop lower scaling, sign and phase problem free impurity solvers. We look forward to employing our methods to more accurately elucidate the complex many body physics of 4$d$ transition metal oxides such as the ruthenates, rhodates, and molybedenates in the near future.
\begin{acknowledgments}
H.H., B.R., and H.S. thank Andrew Millis, Antoine Georges, and Shiwei Zhang for stimulating discussions, and Qiang Han for providing data and insight. H.H. and B.R. acknowledge support from NSF grant DMR-1726213 and DOE grant DE-SC0019441. H.S. thanks the Flatiron Institute for research support. The Flatiron Institute is a division of the Simons Foundation. This work was conducted using computational
resources and services at the Brown University Center for Computation and Visualization, the Flatiron Institute, and the Extreme Science and Engineering Discovery Environment (XSEDE).
\end{acknowledgments}
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{"url":"https:\/\/zbmath.org\/?q=an:1283.05249","text":"# zbMATH \u2014 the first resource for mathematics\n\nGeometric preferential attachment in non-uniform metric spaces. (English) Zbl\u00a01283.05249\nThis paper deals with models of random graphs which have both an element of being \u2018geometric\u2019, in the sense that, as the graph evolves, the probability that a new vertex is adjacent to an existing vertex depends on their proximity to each other (they are put down randomly in some suitable metric space), but also a \\` preferential attachment element\\' in that the probability of adjacency depends on the degree of the existing vertex. Models of this sort were initially developed around 2006 by Flaxman, Frieze and Weiss in two papers. The author [Adv. Appl. Probab. 42, No. 2, 319\u2013330 (2010; Zbl 1210.05158)] showed, under non-trivial assumptions both on the probability measure determining where the vertices are put down in the metric space and the strength of the effect of distance on connection probabilities, that the limiting proportion of vertices with degree $$d$$ is the same as in the (rigorous formulation of) the Barab\u00e1si-Albert model in [B. Bollob\u00e1s et al., Random Struct. Algorithms 18, No. 3, 279\u2013290 (2001; Zbl 0985.05047)].\nThe paper under review extends the author\u2019s earlier results in two directions: firstly by weakening the assumption that the measure of the open ball of radius $$r$$ round a point $$x$$ of the metric space be independent of $$x$$: this can lead to degree distribution being similar to that found in the model of preferential attachment with multiplicative fitness considered in [C. Borgs and J. Chayes, in: Proceedings of the 39th annual ACM symposium on theory of computing, San Diego, CA, USA, STOC 07. New York, NY: Association for Computing Machinery (ACM) 135\u2013144 (2007; Zbl 1232.68018)].\nThe other extension is to make the attractiveness of a vertex at location $$x$$ to one at vertex $$y$$ be not necessarily the same as the attractiveness of one at vertex $$y$$ to one at vertex $$x$$: in other words, to introduce asymmetry to the notion of attractiveness.\nWe omit detailed statements of these main results as they are technical. The proofs work by starting with the underlying metric space being finite, and using stochastic approximation techniques to show convergence of measures, and then using a coupling argument to extend to the infinite case.\n\n##### MSC:\n 05C80 Random graphs (graph-theoretic aspects) 05C82 Small world graphs, complex networks (graph-theoretic aspects) 60D05 Geometric probability and stochastic geometry\n##### Keywords:\ngeometric random graphs; preferential attachment\nFull Text:","date":"2021-09-21 15:00:20","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\": 0.664344072341919, \"perplexity\": 459.5492903406724}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057225.38\/warc\/CC-MAIN-20210921131252-20210921161252-00475.warc.gz\"}"}
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Q: How can I more rigorously check for errors at design or compile-time when using rollup.js in a Svelte app? (Please excuse any naivety about this question, I'm not sure if it's significant whether or not I'm using Svelte)
I'm using rollup.js as the compiler for a Svelte app which uses ES2015. I'm not seeing anything that will warn me if it's possible for a value to be undefined (or warn me about any other potential deficiencies of the code), other than an ordinary compile failure. Is there a way to catch defects earlier, so I don't need to discover them at run-time? For example is there anything that gives me Stylecop-like capability? Or can I somehow configure a less forgiving compilation?
A: You can use Typescript. Here's an article about how to incorporate it with Svelte.
Once you've got it working, you can set strict compilation rules for Typescript so that it will warn you about errors and type issues before letting you compile.
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Wilhelm Kimmich (20 May 1897 in Lauterbach, Baden-Württemberg – 18 September 1986 in Lauterbach), was a German painter and is considered one of the most important Black Forest painters of the 20th century.
Biography
The early years
From 1904 to 1911 Wilhelm Kimmich attended primary school in Lauterbach and made first attempts in drawing as early as 1909. He had a business training and was a soldier in World War I from 1916 to 1918 and returned from a POW camp in 1920. From 1926 till his retirement in 1960 he worked for the Lauterbach Volksbank, since 1929 as a member of the executive board.
He took also part in World War II since 1943 and was released from French captivity in 1946.
Since 1916 Kimmich had been active as a draughtsman and painter and he took drawing lessons with Hans Lembke in Freiburg in the 1920s and with Hermann Gehri in the 1930s, although he later referred to himself as a "self-taught" painter.
Since 1934 Kimmich took part in group exhibitions and had his first individual exhibition in 1937.
The years of artistic maturity
Since 1956 Wilhelm Kimmich, together with his painter friend Professor Hermann Anselment
(1905-1981), had travelled to Ticino and to Italy.
Although, as a result of these trips, he created landscapes depicting those southern areas Kimmich soon returned to his proper motifs, i. e. the Black Forest and its people. Since the middle of the 1950s he "completely refurbished" Black Forest painting from a stylistic point of view. This accomplishment determines Kimmich's standing in the history of art in the southwest of Germany.
As for his stylistic influences, Kimmich stuck to traditional values. He admitted to influences of Impressionism as well as Expressionism and in his works the influence of Claude Monet and Paul Cézanne are visible as well as that of Henri Matisse and Emil Nolde.
In 1970 Kimmich moved to his new house on Lauterbach's Fohrenbühl, where he had his own studio. He was awarded an honorary citizenship of his birthplace Lauterbach in 1977
His wife Hildegard Lutz from Munderkingen, whom he had married in 1949, died in 1980. For the last four years of his life Wilhelm Kimmich lived together with Elisabeth Sandfort.
The last years
Kimmich's friend and biographer Egon Rieble comments on Kimmich's last period of artistic creativity: "In his late period of artistic production Kimmich gives up the art of beautiful appearance, to render individual impression to his imagination. In Kimmich's last paintings the fragility of the traditional Black Forest myth is felt in a menacing way."(see Kimmich's last painting)
Until his death on 18 September 1986 the painter had created about 2,000 paintings, in addition to numerous drawings and sketches.
What works were still his own were bequeathed to the municipality of Lauterbach along with the condition to present them to the public in a foundation.
Kimmich's standing as a German painter
The emphasis on Kimmich's role as a painter of the Black Forest has often led to neglecting other aspects of his work. There are a number of paintings and drawings of other areas in the South West of Germany and − as mentioned before − of the Ticino and Northern Italy. He was impressed by the people in his neighbourhood and portrayed them very accurately, e. g. his neighbour Oberbauer David in 1972.
Kimmich's Legacy
Two years after Kimmich's death, Lauterbach established a gallery in the old town hall, where many of Kimmich's paintings, but also works of other, in particular regional, artists are shown. Initiated by Manfred Schlayer (1934 − 2005), the longstanding mayor of Lauterbach, the Kunstverein Wilhelm Kimmich (see external links) was founded in 1997.
This voluntary association is instrumental in preserving and presenting Kimmich's work in the gallery, and in publishing a complete register of his legacy.
The first part of volume I was published in 1999 and comprises the paintings in private possession.
The second part of volume I, comprising the paintings in private possession, was published in 2007.
Further reading
Egon Rieble: Wilhelm Kimmich - der Maler des Schwarzwalds, Stuttgart 1982
Kunstverein Wilhelm Kimmich et al.(edd.):Wilhelm Kimmich: Werkverzeichnis, Band 1, Das malerische Werk, Teil 1, Gemälde in öffentlichem Besitz, Lauterbach-Rottweil 1999
Kunstverein Wilhelm Kimmich u.a.(Hrsg.): Wilhelm Kimmich: Werkverzeichnis, Band 1, Das malerische Werk, Teil 2, Gemälde in Privatbesitz, Lauterbach–Rottweil 2007,
Notes
External links
Website of Kunstvereins Wilhelm Kimmich in Lauterbach
1897 births
1986 deaths
20th-century German painters
20th-century German male artists
Black Forest
German male painters
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Tigress translocated to Panna Tiger Reserve
4 Mar 2009, 1330 hrs IST, PTI
BHOPAL: A tigress has been translocated from the Bandhavgarh Tiger Reserve to the Panna Tiger Reserve in eastern Madhya Pradesh, with an aim to increase the population of the feline in Panna.
The tigress reached the reserve, spread in an area of 542 sq km in Panna and Chhatarpur districts, by road in the wee hours, forest officials said today, adding one more feline will be translocated to Panna from the same reserve.
"The tigress was tranquilized in the presence of the Wildlife Institute of India (WII) officials and captured last evening. The feline left our park around 1600 hours yesterday for Panna," Bandhavgarh Reserve Director Aseem Shrivastava said.
Initially, the plan was to airlift the tigress by a helicopter but as the Indian Air Force (IAF) chopper was not available till March 8, the forest officials decided to translocate the big cat by road, Shrivastava said.
All necessary precautions were taken to take the tigress from Bandhavgarh to Panna by road, covering a distance of around 250 km.
Forest officials from MP took up the exercise of shifting the feline from Bandhavgarh tiger reserve following reports of dwindling tiger population in the Panna reserve, forest department sources said.
http://timesofindia.indiatimes.com/Earth/Tigress-shifted-to-Panna/articleshow/4222724.cms
Bandhavgarh tigress reaches Panna
Abandoned tiger cub cared for by vets
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Mike Bliss (né le à Milwaukie, Oregon) est un pilote américain de NASCAR dans la Nextel Cup. Il pilote la voiture .
Références
Naissance à Milwaukie
Naissance en avril 1965
Pilote de NASCAR
Pilote automobile américain
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\section{Introduction}
\label{introduction}
Numerical Relativity emerged as a way to provide answers to problems in General Relativity that require solving Einstein equations in physical situations where we do not have neither exact solutions nor good approximation schemes, or at least not at the level of precision that is required. In particular, for systems characterized either by extreme gravitational fields or by relativistic speeds, or both. After more than five decades of developments, Numerical Relativity has become a mature field that has produced revolutionary discoveries for very different types of problems (for some accounts of Numerical Relativity can found in~\cite{Lehner:2001wq,Grandclement:2007sb,Alcubierre:1138167,Bona:2009bo,Baumgarte:2010bs,Lehner:2014asa,Cardoso:2014uka}: From gravitational collapse to cosmological physics, including the description of the non-linear dynamics of binary systems and their gravitational wave emission. Clearly, the problems that Numerical Relativity has addressed are very demanding in terms of the complexity of the algorithms (including formulating a well-posed initial-value problem) and also in terms of computational cost. This has lead to the development of more and more sophisticated algorithms that use state-of-the-art techniques from the fields of numerical analysis and computer science. Nevertheless, there is a question that has not been sufficiently discussed in Numerical Relativity, the question of numerical accuracy in relation to the numerical precision offered by current digital computers. The modelling of physical phenomena requires choosing the appropriate type of numerical algorithm that ensures a satisfactory answer in terms of accuracy, reliability, and computational cost. For many problems, the common $64$-bit (double precision) floating-point arithmetic, i.e. fifteen or sixteen significant digits, is enough to obtain accurate results. In some cases, even $32$-bit (single precision) floating-point arithmetic, i.e. seven or eight significant figures, can be sufficient. However, there are problems that demand a very high degree of precision~\cite{math3020337}. For instance, in cases where solutions at late times strongly depend on the initial conditions or in cases where the physical properties are very sensitive to the value of certain parameters, the ability to increase the precision of the numerics can be an essential ingredient to reach satisfactory results (see, e.g.~\cite{Khanna:2013}).
Different types of numerical algorithms have been used in numerical relativity. Finite Differences provide simple and easy ways to design algorithms while Finite Element methods are in general more robust and modular, which is the reason why they are very frequently used in engineering problems. Nevertheless, if the priority is to achieve high accuracy, then spectral and pseudospectral methods are a convenient choice due to their great convergence properties: They converge exponentially for smooth problems. It has also been shown that they provide highly precise solutions in a variety of problems, from fluid dynamics to astrophysics (see, e.g.~\cite{canuto2007spectral,Bourke1988,ehrendorfer2012spectral}). In Numerical Relativity, they have successfully been applied to the simulation of the collision of orbiting binary black holes and their emission of gravitational waves~\cite{Szilagyi:2009qz,Haas:2016cop} (see also e.g.~\cite{Grandclement:2007sb,Canizares:2009ay,Canizares:2010yx,Macedo:2014bfa,Oltean:2018szc}).
The main goal of this paper is to show that the powerful convergence properties of PseudoSpectral Collocation (PSC) methods make them an ideal option to go beyond the typical $64$-bit floating-point arithmetic for problems in the context of Numerical Relativity.
For the common {\em double} precision, maximum accuracy is usually reached with a relatively quite low number of collocation (discretization) points. Therefore, going beyond this precision does not constitute a high increase in computational cost. In addition, PSC methods provide a high-compression of the information describing the solution of our problem. Indeed, the number of collocation points (or modes in the spectral picture) needed to reach very high accuracy is much smaller than the number of points in other methods (e.g., Finite Differences) so that the memory demands get reduced drastically. Moreover, the structure of the algorithms, and in consequence of the numerical codes that implement them, is independent of the number of collocation points, which is also another advantage of the PSC method since we do not have to touch the algorithms in order to increase the precision, just to change the number of collocation points. This is contrast with Finite Differences methods, where by increasing the number of discretization points, within a wide range, we reach machine precision and then a new Finite Differences algorithm with better truncation error would have to be implemented (i.e. with a different stencil that would require, in general, more discretization points).
In order to show in practice all these claims, we have developed a new numerical library, ANETO (Arbitrary precisioN solvEr with pseudo-specTral MethOds)~\cite{anetolib}, that we use to perform the numerical simulations described in this paper. The ANETO library provides complete freedom in the numerical precision in the sense that we have the possibility of adjusting the bit precision of our algorithms to fulfil the exact accuracy requirements of a given problem. It also contains a tool to translate numerical codes from the standard double precision to arbitrary precision. The work presented in this paper originates from the experience of the authors in the study of gravitational collapse in several scenarios that require the use of General Relativity (see~\cite{Olivan:2015fmy,SantosOlivan:2016djn}), and where the accuracy requirements on the numerical solutions are very high. In the sense, we have developed numerical codes that use the ANETO library to study two different scenarios of gravitational collapse:
(i) The collapse of a spherically-symmetric massless scalar field in asymptotically-flat spacetimes, the scenario where Choptuik found critical behaviour~\cite{Choptuik:1992jv}. We have developed a new characteristic pseduospectral code using the ANETO library and show that we can essentially have arbitrary precision in the estimation of the location where an apparent horizon is formed.
(ii) The collapse of a spherically-symmetric massless scalar field in asymptotically-Anti de Sitter (AdS) spacetimes, which have recently attracted a lot fo attention in the context of string theory and the gauge/gravity duality. The fact that AdS, the maximally symmetric solution of Einstein's equations with a negative cosmological constant (see more in~\cite{Hawking:1973uf}), has the remarkable property that light rays can reach the AdS boundary in a finite time (in contrast to massive particles that need an infinite time as in flat spacetime) has important consequences for the study of gravitational collapse. At the same time, these numerical studies demand a very high degree of precision. We adapt the code used in~\cite{Olivan:2015fmy,SantosOlivan:2016djn} to be compatible with the ANETO library to study the long-term dynamics in AdS spacetimes, showing that we can keep the spacetime mass constant to very high degrees of precision.
The plan of this paper is as follows:
In Sec.~\ref{sec_basics_PSC} we introduce the main techniques that we have developed to combine the PSC method with arbitrary precision arithmetic for the study of the dynamics of the general relativistic gravitational collapse. This includes the main operators, a multidomain scheme, a discussion on computational time and parallelization.
In Sec.~\ref{grav-collapse-Minkowski}, we report on simulations of gravitational collapse in asymptotically-flat spacetimes using the ANETO library and we do the same in Sec.~\ref{grav-collapse-anti-de-Sitter} for the case of gravitational collapse in asymptotically-anti de Sitter spacetimes.
We finish with some conclusions and a discussion of future prospects in Sec.~\ref{conclusions-perspectives}.
\section{The Numerical Method: PseudoSpectral Collocation Methods with Arbitrary Precision Arithmetic}\label{sec_basics_PSC}
There are problems in the context of NR that require high precision numerical computations, in which the usual {\em double precision} used in most codes is not enough. To deal with this type of demanding problems we have to consider the possibility of using other representations of real numbers with a larger number of significant digits, taking always into account the limited memory of a digital computer. The usual approach is based on the so-called {\em floating-point} representation (see~\cite{ieee745-2019}), similar to scientific notation, where a real number $x$ can be expressed/approximated by:
\begin{equation}
fl(x) = \left( 1 + \sum_{i=1}^{b^{}_m-1} b^{}_i\;\times 2^{-i} \right) \;\times\; 2^E\,,
\label{formula_fpa}
\end{equation}
where $i$ is the position of the bit $b^{}_{i}$ of the mantissa from the left, $b_m$ is the number of bits of the mantissa (the precision), and $E$ is the number of bits of the exponent. Then, $E$ establishes the maximum range of our variables and $b_m$ determines the machine precision (or machine roundoff error), which is what we are interested in. A measure of the level of roundoff error in the floating-point number system is:
\begin{equation}
\epsilon_{\rm mp}= \max_{x \neq 0} \frac{|x-fl(x)|}{|x|} \,,
\end{equation}
When rounding is made by chopping, we have that $\epsilon_{\rm mp}=2^{1-b_{m}}$. In this work, the standard numerical precisions that we consider for reference are: {\em single precision}, where $b_{m} = 24$ (eight significant digits); {\em double precision}, with $b_{m} = 53$, corresponding to $15-16$ significant digits; and {\em quadruple precision}, where $b_{m} = 113$, corresponding to approximately $34$ significant digits. Apart from that, we also use arbitrary precision so that we manually select the bit precision $b_{m}\,$.
On the other hand, given the additional computational cost of high-order arithmetic, we have to think carefully what type of discretization algorithms we use to solve the partial differential equations involved in our physical problem. There two important factors, one is the question of the computational cost of the operations and the size of the storage required to store the information associated with the variables involved in the computation. The other one is the question of whether we need to adapt the discretization algorithms (and their programming language implementation) in terms of the particular numerical precision chosen. In this paper we advocate for the use of the PSC method~\cite{Boyd,Fornberg:1996psc,Canutoetal:2006sm1,Grandclement:2007sb} for spatial discretization of our variables. The reason is that due to spectral convergence of the PSC method for smooth problems we just need in general much less grid points as compared with other discretization methods, which reduces the storage needs. On the other hand, we do not need to change the algorithm code in the PSC method to increase the precision, just the number of collocation points. All the PSC algorithms that use arbitrary precision arithmetic in this work have been developed using the ANETO library. The library has been developed using C++ templates, which allows for the use of any kind of data type. In several of the computations done for this paper, we have used standard types like {\em float} and {\em double} for single and double precision respectively. For the {\em quadruple precision} type we have used the {\em float128} type implemented in the Boost Multiprecision library~\cite{boostmultiprecisionlib}. For general bit precision, we have used the GNU's Multiple Precision Floating-Point Reliable (MPFR) Library~\cite{Fousse:2007}, a C library for multiple-precision floating-point computations with correct rounding, with a C++ wrapper~\cite{mpfrcpp} as an interface. Some numerical algorithms like differentiation and integration has been paralellized using the shared-memory API OpenMP~\cite{dagum1998openmp}.
Finally, the ANETO library has been released as Free Software under a GNU General Public License (GPL) and it can be found in~\cite{anetolib}, including source code and full documentation of the classes and functions available as well as examples of its functionalities. The library was originally developed due to need to go beyond the standard double precision for the study of certain problems associated with gravitational collapse in the context of General Relativity.
The type of problems we are interested in constitute an initial-boundary value problem, which consists in a system of Partial Differential Equations (PDEs) defined over a spatial domain ${\cal D} \in \mathbb{R}^{d}$, being $d$ the number of space dimensions, and for a time interval ${\cal T}\equiv [t_{i},t_{f}]\subset\mathbb{R}$:
\begin{eqnarray}
{\cal L}[\vec{u}](t,\vec{x}) = 0\,,
\quad
{\cal I}[\vec{u}](t_{i},\vec{x}) = 0\,,
\quad
{\cal B}[\vec{u}](t,\vec{W}(t,\vec{x})) = 0\,, \label{efes-general}
\end{eqnarray}
where $t\in{\cal T}\,,\, \vec{x}\in{\cal D}$, and where $u(t,\vec{x})$ denotes the vector of unknown variables; ${\cal L}$ is a given differential operator that determines the set of PDEs under consideration; ${\cal I}$ is another given operator representing the initial conditions of our evolution problem at $t=t_{i}$; and finally, ${\cal B}$ is the operator that determines the boundary conditions at a set of (timelike) hypersurfaces defined by a set of implicit equations $\vec{W}(t,\vec{x})=\vec{0}$. The operator ${\cal L}$ can be divided into two independent parts: One producing a set of hyperbolic evolution equations and the other hand producing a set of constraint equations that the initial data (expressed here in terms of the operator ${\cal I}$) has to satisfy and the evolution has to preserve.
In spectral methods the solution to this type of problems is approximated by using a spectral expansion of the form:
\begin{equation}
\vec{u}^{}_{N}(t,x) = \sum_{k=0}^{N} \vec{a}^{}_{k}(t)\, \phi^{}_k(x)\,, \label{chap_approx_spectral_repr}
\end{equation}
where $\phi_k$ are the basis functions and $\vec{a}_k$ are the vectors of (spectral) coefficients associated with the approximation to the vector of unknowns, $\vec{u}_{N}(t,x)$.
In this paper we choose the basis functions $\phi^{}_k$ to be Chebyshev polynomials:
\begin{equation}
T^{}_n(X) =\cos\left[n\cos^{-1}(X) \right] \quad (n=0,\ldots\,N; X \in [-1,1])\,.
\label{chap_chebyshev-polynomials}
\end{equation}
Notice that here, the coordinate $X$, which we call the {\em spectral} coordinate, will not be in general the same as the coordinate $x$ in Eq.~(\ref{chap_approx_spectral_repr}), which we call the {\em physical} coordinate. They will have in general different ranges and will be related by a one-to-one mapping\footnote{Although we are not considering it here, it is in principle possible to include a dependence on the time $t$ in this mapping.}, $x=x(X) \Leftrightarrow X=X(x)\,$.
The PSC method consists in finding the solution by demanding that our equations~(\ref{efes-general}) are exactly satisfied at a set of collocations points. In this work we consider only
the {\em Lobatto-Chebyshev} grid of collocation points:
\begin{eqnarray}
X^{}_i = -\cos\left(\frac{\pi\,i}{N}\right) \quad (i=0,\ldots,N)\,,
\label{chebyshevlobattogrid}
\end{eqnarray}
This particular choice, apart from minimizing the interpolation error, includes the boundary points, $X=\pm 1$, which allows us to directly impose boundary conditions there.
In the PSC method we have two representations of the approximation to the solution of our problem. The first one, given by Eq.~(\ref{chap_approx_spectral_repr}), is the standard one in spectral methods and hence we call it the {\em spectral} representation. The magnitude of the spectral coefficients decays exponentially with the degree, $n$, of the Chebyshev polynomial:
\begin{equation}
|a^{k}_n| \sim e^{- \alpha^{k}\, n} \quad (n=0,\ldots,N)\,,
\end{equation}
where $\alpha^{k}$ is a coefficient associated with the variable $u^{k}$. As a consequence, the discretization error due to the use of a finite number of points also decays exponentially. This behaviour is illustrated in Figure~\ref{fig_spec_coeffs} for numerical computations using different number of digits of precision.
The other type of representation is the {\em physical} representation, based on the use of the collocation values of our variables, $\vec{u}^{}_{i}$, and given by:
\begin{equation}
\vec{u}^{}_{N}(t,x) = \sum_{i=0}^N \vec{u}^{}_{i}(t)\, {\cal C}^{}_i(X)\,,
\label{physicalrepresentation}
\end{equation}
where $\vec{u}^{}_{i}(t)$ are the values of our variables $\vec{u}$ at the collocation point $X^{}_{i}$, and ${\cal C}^{}_i(X)$ are the {\em cardinal} functions~\cite{Boyd} associated with our basis functions and grid, obeying: ${\cal C}^{}_{i}(X^{}_{j}) = \delta^{}_{ij}$.
The two representation are related via a matrix transformation:
\begin{equation}
\vec{u}^{}_{i} = \sum_{n=0}^N \mathbb{M}^{}_{in} \vec{a}^{}_{n}\,,
\quad
\mathbb{M}^{}_{in} = T^{}_n(X^{}_{i}) \quad (i,n=0,\ldots,N) \,.
\label{matrix-transformation}
\end{equation}
Introducing a new coordinate via $X = \cos\theta$ ($\theta\in[0,\pi]$), the series in Chebyshev polynomials [Eq.~(\ref{chap_chebyshev-polynomials})] becomes a cosine series since $T^{}_n(\cos\theta) =\cos\left(n\theta \right)$. In the particular case of the Lobatto-Chebyshev grid [Eq.~(\ref{chebyshevlobattogrid})], the components of the matrix transformation are: $\mathbb{M}^{}_{in} = T^{}_n(X^{}_{i})= (-1)^{n}\cos(n\,i\,\pi/N)$. The resulting transformation becomes then a Discrete Cosine Transform (DCT) that can be computed by using a $2N$ Fast Fourier Transform (FFT) algorithm whose computation cost scales as $\sim N\ln N$ with the number of collocation points~\cite{canuto1988spectral,Boyd} in contrast with the $\sim N^{2}$ scaling of the matrix transformation.
The possibility of using the FFT algorithm to change between the physical and spectral representations is of particular relevance for the solution of time-dependent problems described by PDEs. The physical representation is useful to deal with non-linear terms since the collocation values associated with an arbitrary function of $\vec{u}$, say $\vec{f}(\vec{u})$, are simply given by $\vec{f}(\vec{u})^{}_{i} = \vec{f}(\vec{u}^{}_{i})$. In contrast, the spectral representation is better for other type of operators like differentiation and integration.
\begin{figure}[t]
\begin{center}
\resizebox{.6\textwidth}{!}
{
\includegraphics{Figure01-spec_coeffs.pdf}
}
\caption[Representation of the Spectral Coefficients]{
{\bf Representation of the Spectral Coefficients.} For a smooth function (we have used the function $f(x) = \exp(\tan(x))$) the modes in the spectral representation decay exponentially until they reach the precision error of our computations. We can see that with the use of arbitrary precision arithmetic we can control the level of the computer round-off error.\label{fig_spec_coeffs}}
\end{center}
\end{figure}
\subsection{Differentiation and Integration}\label{subsec_PSC_diff}
In the PSC method, differentiation can performed by means of the following schematic procedure:
\begin{eqnarray}
\partial^{}_{x} :\,\{\vec{u}^{}_i\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{\vec{a}^{}_n\} ~\stackrel{\tilde\partial^{}_{x}}{\longrightarrow}~
\{\vec{b}^{}_n\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{(\partial^{}_{x}{\vec{u}})^{}_i\}\,, \label{pscdifferentiation}
\end{eqnarray}
where $\tilde\partial^{}_{x}$ is the spectral-space derivative operator whose action is given by~\cite{Boyd,canuto1988spectral}
\begin{equation}
\left\{
\begin{array}{lcl}
\vec{b}^{}_{N} & = & 0\,, \\[1mm]
\vec{b}^{}_{N\m1} & = & 2 N \vec{a}^{}_{N}\,, \\[1mm]
\vec{b}^{}_{n} & = & \frac{1}{\bar{c}^{}_{n}}\left[ 2(n+1)\,\vec{a}^{}_{n+1}+\vec{b}^{}_{n+2} \right] \quad (n = N\m2\ldots 0)\,,
\end{array}
\right.
\label{spectralderivativeoperator}
\end{equation}
where the $\bar{c}^{}_{n}$ are such that: $\bar{c}^{}_{n}=2$ if $n=0$ or $n=N$ and $\bar{c}^{}_{n}=1$ otherwise.
In the case of integration, there are different relevant operators. First, let us consider the following simple first-order ODE
\begin{equation}
\frac{d f(x)}{d x} = g(x)\,,
\end{equation}
with a boundary condition at $x=x_{0}$, $f(x_{0}) = f_{0}$. The solution is simply given by
\begin{equation}
f(x) = f_{0} + \int_{x_0}^{x} dx'\,g(x') \,.
\end{equation}
This is what we call {\em integration from the left} because it incorporates a (boundary) condition at $X = X^{}_{-} = -1$ (assuming a mapping $x=x(X)$ so that $x_{0}=x(X^{}_{-})$):
\begin{equation}
I^{}_{\rm L}(x(X)) = I^{}_{\rm L}(x^{}_{0}) + \int^{X}_{X^{}_{-}}dX'\,\left(\frac{dx}{dX}\right)^{}_{X'}g(X') \,,
\label{integr_IL}
\end{equation}
where $I^{}_{\rm L}(x_{0})=f_{0}$. The scheme for the full left-integration process is:
\begin{eqnarray}
\begingroup\textstyle\int\endgroup^{X}_{X_{-}} :\,\{\vec{u}^{}_i\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{\vec{a}^{}_n\} ~\stackrel{\int^{}_{\rm L}}{\longrightarrow}~
\{\vec{b}^{\rm L}_n\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{(\int^{X}_{X_{-}}{\vec{u}})^{}_i\}\,,
\label{pscintegration-left}
\end{eqnarray}
where $\{\vec{b}^{\rm L}_{n}\}$ are the spectral coefficients corresponding to $I^{}_{\rm L}(x(X))$:
\begin{equation}
\left\{
\begin{array}{lcl}
\vec{b}^{\rm L}_{N} & = & \frac{\vec{a}^{}_{N\m1}}{2N}\,, \\[1mm]
\vec{b}^{\rm L}_{n} & = & \frac{1}{2n}\left(\bar{c}^{}_{n-1}\,\vec{a}^{}_{n\m1} - \vec{a}^{}_{n+1}\right) \quad \left(n = N-1,\ldots,1\right)\,, \\[1mm]
\vec{b}^{\rm L}_{0} & = & I(x(X^{}_{-})) - \sum^{N}_{n=1} \left ( -1\right)^n \vec{b}^{}_{n} \,.
\end{array}
\right.
\label{intspec2}
\end{equation}
In the same way, we can introduce the right integration (the boundary condition is imposed at the right boundary, i.e. at $X = X^{}_+ = +1$):
\begin{equation}
I^{}_{\rm R}(x(X)) = I^{}_{\rm R}(x^{}_{0}) + \int^{X^{}_{+}}_{X}dX'\,\left(\frac{dx}{dX}\right)^{}_{X'}g(X')\,.
\label{integr_IR}
\end{equation}
The scheme for the full right-integration process is:
\begin{eqnarray}
\begingroup\textstyle\int\endgroup^{X^{}_{+}}_{X} :\,\{\vec{u}^{}_i\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{\vec{a}^{}_n\} ~\stackrel{\int^{}_{\rm R}}{\longrightarrow}~
\{\vec{b}^{\rm R}_n\} ~\stackrel{\rm DCT}{\longrightarrow}~
\{(\int^{X^{}_{+}}_{X}{\vec{u}})^{}_i\}\,.
\label{pscintegration-right}
\end{eqnarray}
The action of the operator $\int^{}_{\rm R}$ is given by
\begin{equation}
\left\{
\begin{array}{lcl}
\vec{b}^{\rm R}_{N} &=& - \frac{1}{2 N} \vec{a}^{}_{N\m1}, \\[1mm]
\vec{b}^{\rm R}_{n} &=& - \frac{1}{2 n} \left ( c^{}_{n\m1} \, \vec{a}^{}_{n\m1} - \vec{a}^{}_{n\p1}\right) \quad \left(n = N-1,\ldots,1\right) \,,
\\[1mm]
\vec{b}^{\rm R}_{0} &=& I^{}_{\rm R}(x(X^{}_{+})) - \sum_{n=1}^{N} \vec{b}^{\rm R}_n \,.
\end{array}
\right.
\end{equation}
\subsection{The Multidomain PSC Method} \label{subsection_multidom}
In many problems, different regions of our computational domain require different degrees of spatial resolution and hence some form of grid refinement needs to be adopted. In the case of the PSC method, a simple choice is to use a multidomain scheme that distributes the subdomains so that regions where high resolution is required are covered by more subdomains than regions less demanding in terms of resolution. For the case of evolution problems in one spatial dimension we consider a decomposition of our physical computation domain $\Omega=[x^{}_{L},x^{}_{R}]$ in $D$ disjoint subdomains:
\begin{equation}
\Omega = \bigcup^{D-1}_{a=0} \Omega^{}_a\,, ~~\Omega^{}_a = \left[ x^{}_{a,L}, x^{}_{a,R}\right]\,,
\end{equation}
with the identification: $x^{}_{a,R}=x^{}_{a+1,L}$ ($a=0,\ldots,D-2$). Each subdomain $\Omega^{}_a$ is mapped into the spectral domain $[-1,1]$. For our computations we assume a simple linear mapping but other mappings are possible. Then, given the coordinate $x$ of a point in the physical domain $\Omega$, assuming it belongs to the subdomain $\Omega^{}_a$, it is mapped to a spectral coordinate $X^{}_{a}$ according to the following linear mapping:
\begin{eqnarray}
x \longrightarrow X^{}_a(x) = \frac{2x- x^{}_{a,L}- x^{}_{a,R}}{ x^{}_{a,R} - x^{}_{a,L} } \,. \label{lmap1}
\end{eqnarray}
whose inverse mapping is:
\begin{eqnarray}
X^{}_a \longrightarrow x(X^{}_a) = \frac{x^{}_{a,R}-x^{}_{a,L}}{2}X^{}_a + \frac{x^{}_{a,L}+ x^{}_{a,R}}{2}\,. \label{lmap2}
\end{eqnarray}
The Jacobian, ${d x}/{d X^{}_a} = (x^{}_{a,R}-x^{}_{a,L})/{2}$, is different from subdomain to subdomain unless all of them have the same physical coordinate size. Despite its simplicity, the linear mapping can be used for refinement adapting the length of each subdomain to the resolution needs of the problem under consideration. All what we need is a refinement criteria for the adaptation of the subdomain sizes.
The important point of using a multidomain scheme for solving PDEs is communication. In the case of elliptic equations we can just communicate variables by imposing continuity conditions. In the case of hyperbolic PDEs, assuming they are strongly hyperbolic~\cite{Courant:1989aa,John:1991fj}, which means the principal part operator has a complete set of eigenvectors and all the eigenvalues, the propagation speeds of the eigenvalue fields (known as characteristic fields) are real. Then, at the interface between two subdomains, the fields with positive speed are communicated from the left to the right subdomain, while the fields with negative speeds are communicated from the right to the left subdomain. This scheme can even accommodate discontinuities like those produced by the presence of point particles (see~\cite{Sopuerta:2005gz,Canizares:2010yx,Canizares:2011kw,Oltean:2018szc}). Finally, the characteristic variables are can also be used to impose boundary conditions at the global boundaries in a clear and simple manner, for instance for the case of in/outgoing boundary conditions.
\begin{figure}[t]
\begin{center}
\resizebox{.6\textwidth}{!}
{
\includegraphics{Figure02-domain_diagram_Y.pdf}
}
\caption[Communication between subdomains]{\textbf{Communication between subdomains.} Schematic representation of the communication between subdomains for hyperbolic PDEs. Characteristic variables, $Y_-$ and $Y_+$, with a well-defined speed of propagation are the key. We just need to copy the boundary values in the direction indicated by the arrows. \label{plot_diagram_U_V}}
\end{center}
\end{figure}
Although many operations generalize trivially to a PSC multidomain scheme in the sense that they can be performed independently at each subdomain, some others require some adaptation that has to do again with communication between subdomains. Differentiation is a good example. It is well-known that the computation of derivatives with the PSC method becomes noisy near the boundaries producing an accumulation of error there, which in some cases can be one or two orders of magnitude higher than in other more central regions. This phenomenon, known as Runge's phenomenon~\cite{Runge:1901cfr,Epperson:1987je}, gets worse as we increase the number of collocation points. We can take advantage of the multidomain scheme to reduce the error by using a dual grid scheme. The idea is to use a second grid constructed such that the boundaries of the subdomains coincide with the middle points of the subdomains of the main (original) grid (see Figure~\ref{fig_dualgrid_scheme}). As a consequence the dual grid has $D+1$ subdomains, $\left\{\bar{\Omega}^{}_{\bar{a}}\right\}^{}_{\bar{a}=0,\ldots,D}$, with the left boundary location of each subdomain, say $\bar{a}$ ($=0,\ldots, D$), being given by $\bar{x}^{}_{\bar{a},L} = (x^{}_{\bar{a}-1,L} + x^{}_{\bar{a}-1,R})/2$ (excepting for $\bar{a}=0$, which corresponds to the global left boundary point $x^{}_{L}$), and the right boundary location is given by $\bar{x}^{}_{\bar{a},R} = (x^{}_{\bar{a},L} + x^{}_{\bar{a},R})/2$ (excepting for $\bar{a}=D$, which corresponds to the global right boundary point $x^{}_{R}$). When we compute the derivative in the dual grid, the points where typically the error is the lowest coincide with the location where the error is typically the greatest in the subdomains of the main grid. Then, we compute the final derivative of our function by combining the derivatives computed separately in the main ($f'^{}_{\rm main}$) and dual ($f'_{\rm dual}$) grids. For a given point $x$ belonging to the subdomain $\Omega^{}_{a}$ of the main grid, the derivative is the following weighted sum of the derivatives from the main and the dual grids:
\begin{equation}
f'(x) = \pi^{}_{a}(x) f'^{}_{\rm main}(x) + \left(1-\pi^{}_{a}(x)\right) f'_{\rm dual}(x)
\quad
(x \in \Omega^{}_{a})\,,
\label{derivative_dual}
\end{equation}
where $\pi^{}_{a}(x)$ is a weighting function on the subdomain $\Omega^{}_{a}$ of the main grid (this function together with $1-\pi^{}_{a}(x)$ form a partition of unit associated with the subdomain $\Omega^{}_{a}$) that takes values between zero and one, being zero in the boundaries of the subdomains of the main grid and one at the boundaries of the dual grid. Moreover, it is smooth between the boundary and the centre of the subdomain. One example of a partition function is:
\begin{equation}
\pi^{}_{a}(x) = \left\{
\begin{array}{ll}
\frac{(x - x^{}_{a,L}) (x - x^{}_{a,R})}{(x - x^{}_{a,L}) (x - x^{}_{a,R}) + (x - \bar{x}^{}_{a,L}) (x - \bar{x}^{}_{a,R})}
& \mbox{if}~~x~\in~\bar{\Omega}^{}_{a} \,,\\[3mm]
\frac{(x - x^{}_{a,L}) (x - x^{}_{a,R})}{(x - x^{}_{a,L}) (x - x^{}_{a,R}) + (x - \bar{x}^{}_{a+1,L}) (x - \bar{x}^{}_{a+1,R})}
& \mbox{if}~~x~\in~\bar{\Omega}^{}_{a+1}\,.
\end{array}
\right.
\label{partition}
\end{equation}
This structure of $\pi^{}_{a}(x)$ is due to the fact that a given point of the subdomain $\Omega^{}_{a}$ can either be at the subdomain $\bar{\Omega}^{}_{a}$ or $\bar{\Omega}^{}_{a+1}$ of the dual grid (see Figure~\ref{fig_dualgrid_scheme}).
\begin{figure}[t]
\begin{center}
\resizebox{0.9\textwidth}{!}
{
\includegraphics{Figure03-dualgrid_scheme.pdf}
}
\caption[Dual Grid Structure]{{\bf Dual Grid Structure.} Schematic representation of the dual grid scheme (without showing the subdomains that have the global boundaries). The main grid (in purple) is composed of three subdomains and the dual grid (in green) is shifted and has four subdomains. The blue line shows the weighting function $\pi^{}_{a}(x)$ of Eq.~(\ref{partition}). \label{fig_dualgrid_scheme}
}
\end{center}
\end{figure}
We have carried out several numerical experiments to assess the use of the dual-grid structure by comparing with computations on a single grid. The results are summarized in Figure~\ref{fig_dualgrid_error}. In the top-left plot we show the error in the computation of a derivative using ten subdomains. It is easy to find in the plot the location of the boundaries between the subdomains since the error exhibits there a peak that in most cases is one order of magnitude higher than the average. The brown line of the bottom-left plot shows the error when we perform the same computation but using the dual-grid scheme. As we can see, the peaks have completely disappeared and the error in the derivative looks now quite flat. This improvement can be seen on the top-right plot of Figure~\ref{fig_dualgrid_error}, where we present the ratio between the differentiation errors with and without using the dual grid. Near the boundaries our computation has improved between one and two orders of magnitude in terms of the error. We have done the same comparison for the second-order derivative, which is shown in the bottom-right plot. As we can see, the improvement is even better than in the case of the first-order derivative, and the error at all the boundaries has been reduced in two/three orders of magnitude.
\begin{figure}[t]
\begin{center}
\resizebox{1\textwidth}{!}
{
\includegraphics{Figure04-dualgrid_error.pdf}
}
\caption[Differentiation using a Dual Grid]{\textbf{Differentiation using a Dual Grid.} The plots on the left show the error in the first-order derivative computed with the dual-grid setup (bottom) and without it (top). We see that at the boundaries across subdomains there is an efficient reduction of the error by eliminating the peaks. The magnitude of the improvement can be seen in the top right plot. The quotient of the error in both cases tells us that the improvement at the points near the boundaries is between one and two orders of magnitude. The bottom right plot shows the same for the second-order derivative where the improvement increases by two or three orders of magnitude.}
\label{fig_dualgrid_error}
\end{center}
\end{figure}
Let us now consider the computation of the integral of the simple function $f(x) = \cos(x)\,$, for different bit precisions and both for single and multidomain setups. We show the results in Figure~\ref{fig_err_integral}. The plot on the left shows that as long as one uses enough collocation points the error scales with the round-off error as expected. This is true even for a number of collocation points as low as $N=47$ up to $260$ bits of precision, when the error saturates because the discretization error becomes more important than the round-off error. We also see that it is enough to use $N=71$ collocation points to decrease the error up to $10^{-120}$. The same behaviour is shown on the right plot of Figure~\ref{fig_err_integral} for the case in which we use a multidomain scheme with only four subdomains ($D=4$) and $N=23$ and $N=47$ collocation points per subdomain. In the first case, the discretisation error is reached around $10^{-50}$ but, for the second one we can easily reach again $10^{-120}$. The horizontal lines in the figure show the round-off error for the single, double, and quadruple precisions respectively, which is obviously much higher than the error we can reach using arbitrary precision. Although the integral under consideration involved a simple function, we have checked that the same is true independently of the integrand, just restricting ourselves to smooth functions.
\begin{figure}[t]
\begin{center}
\resizebox{.475\textwidth}{!}
{
\includegraphics{Figure05a-integral_spectral.pdf}
}
\resizebox{.475\textwidth}{!}
{
\includegraphics{Figure05b-integral_multidom.pdf}
}
\caption[Maximum Error for Integration and different Bit Precision.]{\textbf{Maximum Error for Integration and different Bit Precision.} The left plot shows the error made in the integration on a single Lobatto-Chebyshev collocation grid. The plot on the right shows the same error but when using the multidomain PSC method. In both cases, the points show the errors made for two different number of collocation points with respect to the bit precision used. The horizontal lines indicate the error for single, double, and quadruple precision. \label{fig_err_integral} }
\end{center}
\end{figure}
The accuracy that PSC methods are capable to reach is even more clear when we compare it with other discretization methods. In the left plot of Figure~\ref{fig_err_integral_FD}, we compare the results obtained using the ANETO library to Finite-Differences Newton-Cotes (NC) formulae~\cite{Press:1992nr} of fourth and eleventh orders. In both cases, we set manually the points near the boundaries of the subdomain to avoid the problems associated with the borders. The NC formula of eleventh order seems to be good enough to achieve a reasonable accuracy but the polynomial dependence of the error makes it very difficult to reach accuracies beyond $10^{-50}$, while with the PSC method accuracies of the order of $10^{-300}$ are within reach by using just $100$ collocation points. Another relevant comparison is to look at the computational time required for a given accuracy for the two methods. This is shown in the right plot of Figure~\ref{fig_err_integral_FD}. In the case of the PSC method, we distinguish the case of using the matrix transformation versus the FFT algorithm for the transformation between the physical and the spectral representations. The plot clearly shows that the PSC method achieves high accuracies in much less time that the NC formulae. It also shows that the use of the FFT algorithm for the transformation between representations is also more efficient than the matrix transformation as expected. Going back to the comparison between the Finite Differences and PSC methods, it is clear that the development of a numerical code to implemented the NC formulae is always easier than the development of a PSC numerical code. However, the PSC code does not need to be modified in order to go to very high precisions while improving the Finite Differences code for taking advantage of the potential accuracy usually requires to increase the order of the Finite Differences algorithm. This last option can be very challenging and, for high accuracy, it will not beat the exponential convergence of the PSC method.
\begin{figure}[t]
\begin{center}
\resizebox{0.95\textwidth}{!}
{
\includegraphics{Figure06-psc_fd.pdf}
}
\caption[Comparison between the PSC method and Newton-Cotes formulae for numerical integrals.]{\textbf{Comparison between the PSC method and Newton-Cotes formulae for numerical integrals.} We compare the error made in the computation of the integral of our test function using the PSC method, both with the matrix transformation (yellow points) and the FFT transformation (red), and two different Newton-Cotes formulae, one of fourth order (dark blue) and another one of eleventh order (light blue). In the left plot we compare the error made with the different methods in terms of the number of points used. In the right plot we compare the computational time required by each method to achieve a certain level of accuracy.
\label{fig_err_integral_FD}}
\end{center}
\end{figure}
\subsection{Double versus Arbitrary Precision: Computational Time}
\label{subsec_comp_time}
Most of the available computers nowadays use $64$-bit processors that are highly optimised to work with double precision. Going beyond this precision usually requires a software implementation that slows down the computation. The question is how much slow down can be expected depending on the precision that we need to use.
To answer this question we use the test case consisting in the integral again of the simple function: $f(x) = \cos(x)\,$. We quantify the cost for four different grid configurations characterized by the pair $(D,N)$: (number of subdomains, number of collocation points per subdomain). The four grid configurations are: (i) $(D,N) = (14,64)$; (ii) $(D,N) = (14,128)$; (iii) $(D,N) = (50,64)$; and (iv) $(D,N) = (14,127)$. The results are presented in Figure~\ref{fig_time}, from where we can see how much slower is the use of arbitrary precision as compared with the standard double precision. In the range analysed, until $512$ bits, or around $150$ significant digits, the computational time seems to increase linearly with the number of bits, being a factor $150$-$300$ times slower than the double precision case. Moreover, we have to take into account that in order to take advantage of the additional significant digits we need to increase the number of subdomains and/or collocation points. Nevertheless, using the PSC method we do not need to change our numerical code, just the pair $(D,N)$. These considerations are important to estimate whether arbitrary precision can be a good solution for a given numerical problem.
Looking at Figure~\ref{fig_time}, it is also worth mentioning the jumps in the computational cost every time we cross a vertical line (corresponding to multiples of $64$ bits). Although the general behaviour is linear, at small scales the function is more or less flat, increasing in multiples of $64$ bits. This is not surprising because the tests have been done using a $64$-bit processor and are related to the way the library uses double precision numbers to store the arbitrary floating points variables.
\begin{figure}[t]
\begin{center}
\resizebox{.85\textwidth}{!}
{
\includegraphics{Figure07-time_comparison.pdf}
}
\caption[Computational Time versus Arbitrary Precision]{\textbf{Computational Time versus Arbitrary Precision.} The plot shows the computational time employed in the integration of $f(x)=\cos(x)$ for four different grid configurations characterized by the number of subdomains, $D$, and the number of collocation points per subdomain, $N$. The computational times shown are relative to the computational time corresponding to double precision. The computational time appears to increase linearly with the bit precision. The vertical lines just separate multiples of $64$ bits. The computational time experiences a jump at these boundaries, as expected, due to the CPU architecture. The runs with $N=64$ and $N=128$ use the FFT algorithm to transform between the physical and spectral representations, while the one with $N=127$ uses the matrix transformation. As a consequence, the computations for the grid configuration $(D,N)=(14,127)$ are much slower, in absolute computational time, than the ones for $(D,N)=(14,128)$.}
\label{fig_time}
\end{center}
\end{figure}
\subsection{Shared-memory parallelization of PSC computations using arbitrary precision}
\label{subsec_perf_openMP}
We have just seen that one of the drawbacks of using arbitrary precision is the loss of computational speed. Given that most current computer processors are designed for $64$-bit precision computations, arbitrary precision libraries usually emulate this using symbolic calculations or by means of a software layer that implements the arbitrary precision operations using $64$-bit data. Both options significantly slow down the computations that in our case can be of the order of $150-300$ times.
In order to alleviate this downside of the method we can resort to parallelization. Indeed, the use of parallel computing adapts perfectly to our PSC multidomain scheme because most of our computations are done independently within each of the subdomains. For instance, in the case of derivatives, they can be trivially parallelized because they are defined at each subdomain independently. Instead, in the case of global integrals (integrals over the whole computational domain) it is a bit more complex but they can be adapted without affecting the scaling with the number of operations. To see how we can parallelize integrals, let us look at the example of the indefinite integral of an arbitrary function $g(x)$ defined over the whole computational domain $\Omega$ and with a boundary condition at the left global boundary $x^{}_{L}$:
\begin{equation}
I^{}_{\rm L}(x) = I^{b}_{\rm L} + \int^{x}_{x^{}_{L}} dx' g(x')\,.
\end{equation}
where $I^{b}_{\rm L} = I^{}_{\rm L}(x^{}_{L})$ is the boundary condition. Since $x$ is an arbitrary point, it can belong to any subdomain $\Omega_{a}$ and then this integral may appear as a serial computation that is difficult to cast into a parallel one because the result depends on the integral for smaller values than $x^{}_{L}$. Nevertheless, we can parallelize the computation in the following way: First of all, let us assume $x$ belongs to the subdomain $\Omega^{}_{a}$. Then, we can divide the integral into the sum of partial integrals that can be computed individually at each subdomain:
\begin{equation}
I^{p}_{{\rm L},d} = \int^{x^{}_{d,R}}_{x^{}_{d,L}}dx\,g(x) = \int^{1}_{-1} dX\,\left(\frac{dx}{dX}\right)^{}_{X}g(x(X))
\quad (d=0,\ldots,a-1)\,,
\label{partial_integral_0a-1}
\end{equation}
and the integral of the subdomain of $x$ ($\Omega^{}_{a}$):
\begin{equation}
I^{}_{{\rm L},a} (x) = \int^{x}_{x^{}_{a,L}}dx'\,g(x') = \int^{X(x)}_{-1} dX'\,\left(\frac{dx}{dX}\right)^{}_{X'}g(x(X'))\,,
\label{partial_integral}
\end{equation}
so that the full integral can be computed as:
\begin{equation}
I^{}_{\rm L}(x) = I^{b}_{\rm L} + \sum_{d=0}^{a-1} I^{p}_{{\rm L},d} + I^{}_{{\rm L},a}(x) \quad (x \in \Omega^{}_{a})\label{full_integrall}\,.
\end{equation}
With this separation, each piece in the sum can be computed independently in each subdomain. To implement this in practice, we propose the use of shared-memory parallelization, more specifically the broadly used application programming interface OpenMP~\cite{dagum1998openmp}. This appears to be the simplest option to profit from the possibility of having independent computations in the different subdomains and adding the minimum possible communication overhead. In order to test this we have carried out a number of numerical experiments using OpenMP. The measure of the computational time speedup that we use, $S_p$, is a function of the number of cores employed, $p$, defined as:
\begin{equation}
S^{}_p = \frac{T}{T^{}_p} \,,
\label{speedup-definition}
\end{equation}
where $T$ is the computational time spent by a sequential computation and $T^{}_p$ is the time corresponding to a computation that uses $p$ cores. Of course, the ideal unreachable limit is $S^{}_p = p$. Within this framework we have performed tests for integration and differentiation with the parallel multidomain PSC method. This has been done both for double and quadruple ({\em float128}) precisions. The results for the computational speedup $S_p$ are shown in Figure~\ref{fig_openMP}. We can see that we are very close to the maximum speed-up which is indicated by a dashed line in the plot. It is also interesting to note that the differentiation computations are closer to full parallelism than the integration ones. This is expected from the fact that differentiation can be carried out fully independently at each subdomain while for integration we need to communicate the value of the partial integrals.
\begin{figure}[t]
\begin{center}
\resizebox{.85\textwidth}{!}
{
\includegraphics{Figure08-openMP_scalab.pdf}
}
\caption[Speed-up with OpenMP for multidomain PSC computations.]{\textbf{Speed-up with OpenMP for multidomain PSC computations.} This plot shows the computational speed-up, as defined in Eq.~(\ref{speedup-definition}), for integration and differentiation algorithms with both double and quadruple precision. Differentiation presents a better speed-up but both cases show that the multidomain method is a very good option for parallelization.}
\label{fig_openMP}
\end{center}
\end{figure}
Up to here it is clear that the subdomain is the minimum unit of parallelization. However, this approach does not address the principal reason for the slowdown, the emulation of the fundamental operations. This is a question currently under investigation in projects like~\cite{joldes:hal-01312858}, where arbitrary precision numbers are represented as an expansion of double precision numbers of different magnitudes and then the idea is to take advantage of parallel computations with graphics processing units (GPUs) to implement these basic operations.
\section{Gravitational Collapse in Asympotically-flat Spacetimes}\label{grav-collapse-Minkowski}
The first application of the numerical techniques we have just presented is the classical problem of gravitational collapse in a flat (Minkowski) spacetime (without cosmological constant, $\Lambda = 0$). The energy-momentum distribution of the matter collapsing corresponds to a massless real scalar field. For simplicity we assume the spacetime to be spherically symmetric, which means that the Einstein field equations become $1+1$ PDEs (in time and in the radial direction). Then, the setup of the gravitational dynamics to be followed numerically is quite simple: We consider initial states described by smooth initial data and such that the scalar field distribution is concentrated around a certain radial location. Then, there are only two possible end states for the evolution: \\
\noindent (i) {\em Collapse} of the scalar field and the formation of a Black Hole (BH). \\
\noindent (ii) {\em Dispersion} of the scalar field with flat spacetime as the end state of the evolution.\\
To which one of these two states will the evolution drive the system depends on the features of the initial scalar field configuration, in particular on its energy density. An interesting question is what separates these two very different outcomes of the evolution. M. Choptuik~\cite{Choptuik:1992jv} carried out a systematic numerical study of this question and found that the dependence of the final state on the initial data is through a single (arbitrary) parameter. Moreover, Choptuik found that in the threshold between collapse and dispersion there is a one-parameter family of critical solutions that exhibit a naked singularity. It was also found that the mass of the collapsed configurations near the threshold exhibits a scaling with a universal exponent. These unexpected results attracted a lot of attention to this problem and constituted a cornerstone in the development of Numerical Relativity. A detailed review of critical gravitational collapse for different types of matter fields and spacetime configurations in General Relativity can be found in~\cite{Gundlach:2002sx,Gundlach:2007gc}.
In this paper, like in the initial studies by Choptuik, we will restrict ourselves to problems in spherical symmetry. We do not make any further simplification of the problem apart from this one. The first step towards numerical simulations of gravitational collapse is to choose an adequate formulation of the Einstein field equations. We start by choosing what is called a characteristic approach to the problem (introduced in~\cite{Christodoulou:1986zr,Goldwirth:1987nu,Garfinkle:1994jb}). The main idea of this approach is to set initial data on a null (or light-like) slide instead of a constant time slide as in a Cauchy-based initial-value problem. The difference is that the normal to a null slide is a light-like one-form while the normal to a constant time slide is a space-like one-form, like in a standard initial-value Cauchy problem. The main advantage of the characteristic formulation is its ability to approach BH formation much more efficiently than a typical Cauchy one. It is worth mentioning that the initial Choptuik study~\cite{Choptuik:1992jv} was based on a Cauchy formulation and used adaptive mesh refinement to reach the necessary accuracy. Later, Garfinkle~\cite{Garfinkle:1994jb} revisited the problem using a characteristic approach and recovered some of the main results without the need of refinement, which shows the power of the characteristic formulation for the study of gravitational collapse. Our previous studies~\cite{Olivan:2015fmy,SantosOlivan:2016djn} confirmed the superior performance of the characteristic approach by evolving scalar fields, not in asymptotically-flat spacetimes but in Asymptotically-AdS (AAdS) spacetimes. However, the framework we set up in~\cite{Olivan:2015fmy,SantosOlivan:2016djn} is not suitable for the use of PSC methods, then we developed a Finite Differences numerical code. In this section we present a new and improved characteristic scheme that is adapted for the use of the PSC method.
The formulation of the characteristic problem goes as follows: Let us consider a self-gravitating massless scalar field, $\phi$. The set of PDEs that we need to solve are the coupled system formed by the Einstein field equations
\begin{equation}
R^{}_{\mu\nu} - \frac{1}{2} g^{}_{\mu\nu} R + \Lambda g^{}_{\mu\nu}= 2 \, T^{}_{\mu\nu}\,.
\label{efes}
\end{equation}
and the equations for the scalar field which come from the energy-momentum conservation equations
\begin{equation}
\nabla^{\nu}\left(R^{}_{\mu\nu} - \frac{1}{2} g^{}_{\mu\nu} R + \Lambda g^{}_{\mu\nu} \right) = 0\quad \Longrightarrow \quad
\nabla^{\nu}T^{}_{\mu\nu} = 0 \,. \label{divT}
\end{equation}
In these equations, $R^{}_{\mu\nu}$ is the Ricci Tensor associated with the metric tensor $g^{}_{\mu\nu}$; $R$ is the scalar of curvature; $\Lambda$ is the cosmological constant; and $T^{}_{\mu\nu}$ is the energy-momentum tensor, which for a real massless scalar field is given by
\begin{equation}
T^{}_{\mu\nu} = \phi^{}_{;\mu} \phi^{}_{;\nu} - 2\; g^{}_{\mu\nu} \phi^{}_{;\alpha}\phi^{;\alpha}\,,
\end{equation}
where the semicolon denotes covariant differentiation. Then, the resulting field equation for the scalar field is the well-known Klein-Gordon equation:
\begin{equation}
\Box \phi \equiv \phi_{;\mu}{}^{;\mu} =0\,.
\label{klein-gordon-equation}
\end{equation}
To introduce the characteristic formulation we first need an adapted coordinate system. We choose double-null coordinates for the time-radial section together with spherical coordinates for the spheres of symmetry of the problem. The metric tensor in those coordinates is:
\begin{equation}
ds^2 = g^{}_{\mu\nu}dx^{\mu}dx^{\nu} = - 2 f(u,v)\, r^{}_v(u,v) \, dudv + r^2(u,v)\, d\Omega^2\,,
\label{line-element}
\end{equation}
where $(u,v)$ are the double-null coordinates ($\partial/\partial u$ and $\partial/\partial v$ are light-like vectors) and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the line element of the unit 2-sphere. Moreover, $f$ and $r$ are two functions of $(u,v)$ and $r^{}_v$ is a shorthand for the partial derivative of $r$ with respect to the null coordinate $v$. In Figure~\ref{fig_doublenull_scheme} we show a representation of the characteristic grid for the case of an empty (flat) spacetime. In our case, the spacetime is curved by the presence of the massless scalar field $\phi$ and hence $(u,v)$ are not be perpendicular to the $(t,r)$ coordinate lines.
\begin{figure}[t]
\begin{center}
\resizebox{.75\textwidth}{!}
{
\includegraphics{Figure09-doublenull_scheme.pdf}
}
\caption[Scheme of an Evolution using Double-Null Coordinates]{\textbf{Scheme of an Evolution using Double-Null Coordinates.} The horizontal and vertical lines are the axes corresponding to the time and radial coordinates, $(t,r)$. In a characteristic formulation, we set our initial conditions on a $u=$const slide (purple thick line) and evolve each point in the direction indicated by the arrows.
Here, constant null coordinate lines form $45$ degrees with respect to the axes $(t,r)$ but this is just a simplification of the drawing. \label{fig_doublenull_scheme}}
\end{center}
\end{figure}
The set of PDEs for the components of the metric tensor, the metric functions $f(u,v)$ and $r(u,v)$, and the scalar field, $\phi(u,v)$, are obtained by introducing the metric in Eq.~(\ref{line-element}) into Eqs.~(\ref{efes}) and~(\ref{klein-gordon-equation}). In order to reduce the order of the equations, from second-order to first-order PDEs, and to decouple them we introduce new variables associated with the scalar field $\phi$:
\begin{eqnarray}
h & = & \frac{(r \phi)^{}_{v}}{r^{}_v}\,,\\
\bar{h} & = & \phi\,.
\end{eqnarray}
We also introduce a new metric variable:
\begin{equation}
\bar{f} = - 2 r^{}_u\,.
\label{def_fb}
\end{equation}
Then, the $vv$ and the $uv$ components of the Einstein field equations can be written as:
\begin{eqnarray}
f^{}_v &=& \frac{f \, r^{}_v}{r} \left(h - \bar h\right)^2 \,, \\
\bar{f}^{}_{v} &=& \frac{r^{}_v}{r} \left( f - \bar f \right) \,.
\label{equation_fv_fbv}
\end{eqnarray}
In this characteristic formulation of the initial-value problem we prescribe initial conditions for the variable $h$ on an initial $u=u_{i}=$const. null slide, that is $h(u_{i},v)\,$. We also need to prescribe $r(u_{i},v)$ and $r_{v}(u_{i},v)$. With this information we can obtain the rest of variables, at the same null slide $u=u_{i}$, using the equations above. The expressions for $h$, $f$, and $\bar{f}$ are:
\begin{eqnarray}
\bar{h}(u^{}_{i},v) &=& \frac{1}{r} \int_{v^{}_{o}(u^{}_{i})}^{v} d\tilde{v}\, h(u^{}_{i},\tilde{v}) \, r^{}_v(u^{}_{i},\tilde{v}) \,,
\label{char-eq-hbar} \\[1mm]
f(u^{}_{i},v) &=& f(u^{}_{i},v^{}_{o}(u^{}_{i}))\,\exp\left\{ \int_{v^{}_{o}(u^{}_{i})}^{v} \!\!\!\! d\tilde{v}\, \frac{r^{}_v(u^{}_{i},\tilde{v})}{r(u^{}_{i},\tilde{v})} \left[h(u^{}_{i},\tilde{v})-\bar{h}(u^{}_{i},\tilde{v})\right]^2 \right\}\,,
\label{char-eq-f} \\[1mm]
\bar{f}(u^{}_{i},v) &=& \frac{1}{r} \int_{v^{}_{o}(u^{}_{i})}^{v} d\tilde{v}\, f(u^{}_{i},v)\,r^{}_v(u^{}_{i},\tilde{v}) \,,
\label{char-eq-fbar}
\end{eqnarray}
where $v_o(u_{i})$ and $f(u^{}_{i},v^{}_{o}(u^{}_{i}))$ are the values of $v$ and $f(u,v)$ respectively, at the origin $r=0$ on the null slide $u=u_{i}$. By looking at these expressions we realize that we need to guarantee the regularity of the different quantities at the origin, which translates into imposing the conditions:
\begin{eqnarray}
\bar h(u^{}_{i},v^{}_{o}(u^{}_{i})) & = & h (u^{}_{i},v^{}_{o}(u^{}_{i}))\,, \\[1mm]
\bar f(u^{}_{i},v^{}_{o}(u^{}_{i})) & = & f (u^{}_{i},v^{}_{o}(u^{}_{i}))\,.
\end{eqnarray}
In this way, all the equations have a finite limit when we approach the origin $r=0$. However, for numerical purposes it is not convenient to have divisions where both numerator and denominator approach zero. This may be particularly problematic in the case of Eq.~(\ref{char-eq-fbar}), but we can transform it by using integration by parts and taking the right limits. The result is:
\begin{equation}
\bar{f}(u^{}_{i},v) = f(u^{}_{i},v) - \frac{1}{r} \int_{v^{}_{o}(u^{}_{i})}^{v} \!\!\!\! d\tilde{v}\, f(u^{}_{i},\tilde{v}) r^{}_v(u^{}_{i},\tilde{v}) \left[ h(u^{}_{i},\tilde{v}) -\bar{h}(u^{}_{i},\tilde{v})\right]^{2} \,.
\end{equation}
Once we have $(h,\bar{h},f,\bar{f})$ at the null slide $u=u_{i}$ we can evolve them to the next null slide by using the evolution equation for the scalar field, i.e. Eq.~(\ref{klein-gordon-equation}), which comes from the energy-momentum conservation equation~(\ref{divT}). Due to the spherical symmetry only the scalar field has true dynamics since the degrees of freedom of the gravitational field are not activated in spherical symmetry. The evolution equation to pass from a null slide to the next one is given by:
\begin{equation}
h^{}_{u} = \frac{1}{2\,r} \left(f - \bar f \right) \left(h- \bar h \right)\,.
\label{dncharevol}
\end{equation}
This equation is actually an ODE for each value of $v$. Indeed, let us consider a particular value of $v$, say $v_{\ast}$, then Eq.~(\ref{dncharevol}) takes the values of the variables at $(u_{i},v_{\ast})$ and gives us the value of $h$ at $(u_{i}+\Delta u,v_{\ast})$, being $\Delta u$ the time step used in the evolution.
In addition, from Eqs.~(\ref{def_fb}) and~(\ref{equation_fv_fbv}) we can obtain the evolution equations for $r$ and $r_{v}$:
\begin{eqnarray}
r^{}_u & = & - \frac{1}{2} \bar{f}\,, \label{eq_r_u} \\[1mm]
(r^{}_v)^{}_u & = & - \frac{1}{2} r^{}_v \left[ \frac{f - \bar{f}}{r} \right]\,. \label{eq_rv_u}
\end{eqnarray}
The only missing piece in this characteristic evolution scheme is the value of $f(u^{}_{i},v^{}_{o}(u^{}_{i}))$ that appears in Eq.~(\ref{char-eq-f}), which corresponds to the value of $f$ at the origin on the null slide $u=u_{i}$. This is a freely specifiable quantity that reflects the residual coordinate {\em gauge} freedom that we have in the choice of the null coordinate $v$. This, in turn, can be seen as the remaining gauge freedom in completely specifying the radial function $r$ in our characteristic formulation. In particular, it allows us to specify the location of the origin $r=0$ at the null slides $u=$const. Or in other words, the freedom in choosing the motion of the origin ($r=0$) as we evolve from one null slide to the next one. Indeed, since $r=r(u,v)$, and assuming that $v=v_{o}(u)$ corresponds to the location of the origin, i.e. $r(u,v_{o}(u))=0$, the equation of motion of the origin as we move through the spacetime foliation in null slides $u=$const. is given by:
\begin{equation}
\frac{dv^{}_{o}(u)}{du} = -\left.\frac{r^{}_{u}}{r^{}_{v}}\, \right|_{v=v^{}_{o}(u)} = \left. \frac{f^{}_{o}}{2\, r^{}_v}\, \right|^{}_{v=v^{}_{o}(u)}\,, \label{motion-of-origin}
\end{equation}
where we have used Eq.~(\ref{eq_r_u}). We can then use this freedom to make the origin move, for instance, with a uniform speed. To achieve this we just need to choose the freely specifiable quantity $f(u,v^{}_{o}(u))$ as:
\begin{equation}
f(u,v^{}_{o}(u)) = 2 \left. r^{}_v \right|^{}_{v=v^{}_{o}(u)} \quad \Longrightarrow \quad v^{}_o(u) = v^{}_o(u^{}_i) + u\,.
\end{equation}
In our formulation the formation of an apparent horizon (AH) happens when the following condition is fulfilled:
\begin{equation}
r^{}_v \longrightarrow 0\,, \quad \mbox{or equivalently,} \quad \frac{\bar{f}}{f} \longrightarrow 0\,.
\label{ah-formation-condition}
\end{equation}
This limit cannot be reached with our choice of system of coordinates (it corresponds to a coordinate singularity) although we can approach it as much as we want. Then, we assume that an AH has formed when the quantities in the AH condition above reach a value less than $10^{-8}$. At that point we stop the simulation.
Finally, we used s Gaussian packets as initial data for the scalar field:
\begin{equation}
h(u^{}_i,v) = \epsilon \; \exp\left\{ -\frac{(v - b)^{2}}{\omega^2} \right\}\,,
\label{initial-data}
\end{equation}
where the amplitude $\epsilon$, the width $\omega$, and the shift $b$ are the freely specifiable parameters of this 3-parameter family of initial data.
At this point we have presented all the necessary ingredients for the characteristic formulation of the problem. It is well-adapted for its implementation using the PSC method for the discretization in the $v$ coordinate. We evolve from one null slide $u=$const. to the next one by using a standard Runge-Kutta 4 (RK4) algorithm. We have implemented this formulation using the PSC method and arbitrary precision tools described in the previous sections. The error can be estimated from the absolute value of the last spectral coefficient. The value of the coefficients, $a_n$ ($n=0,\ldots,N$), decay exponentially, reaching or not round-off error. If round-off is not reached, the last spectral coefficient represents an estimation of the truncation error incurred in ignoring the rest of terms in the spectral series. Otherwise, the last coefficient represents the precision reached. In both cases, it can be used for a good estimation of the error.
We have evolved the initial data of Eq.~(\ref{initial-data}) with parameters: $\epsilon = 2.00$, $b= 0.15$ and $\omega = 0.05$. This is an example of initial configuration above the critical threshold so that it will collapse and form an AH. In Figure~\ref{fig_gravColl_error} we show the error in the location of the AH at the end of the evolution in terms of the number of collocation points per subdomain. Since we have different errors at each subdomain, we take the highest of all of them, which corresponds to the subdomain where the collapse takes place. In Figure~\ref{fig_gravColl_error} we only show the error associated with the function $r_v$, to monitor the first condition in Eq.~(\ref{ah-formation-condition}), because is the one that has the highest error of all of the three evolution variables.
\begin{figure}[t]
\begin{center}
\resizebox{.48\textwidth}{!}
{
\includegraphics{Figure10a-gravColl_error_prec.pdf}
}
\resizebox{.48\textwidth}{!}
{
\includegraphics{Figure10b-gravColl_error_domains.pdf}
}
\caption[Convergence in the estimation of the location of the formation of an Apparent Horizon (AH)]{\textbf{Convergence in the estimation of the location of the formation of an Apparent Horizon (AH):} The left plot shows the truncation error (estimated from the last spectral coefficient) at the moment of AH formation for several different grid configurations, all of them with $D=20$ subdomains. Each data set corresponds to simulations done with different bit precision. The error decays exponentially (spectral convergence) until the maximum precision is reached. On the right plot we show the impact of adding more subdomains. All data sets exhibit spectral convergence but the number of subdomains has an impact in the $\alpha$ factor of the exponential decay $e^{-\alpha N}$. These simulations use $300$ bit precision.
\label{fig_gravColl_error}}
\end{center}
\end{figure}
Moreover, on the left plot of Figure~\ref{fig_gravColl_error}, we use a setup with $D=20$ subdomains and change the number of collocation points for different bit precisions. In all the cases, the error has an exponential decay (spectral convergence) until we rearch the precision limit (round-off error) which, of course, improves as we increase the number of bits of our data types. The first one represents a $53$ bit precision, equivalent to the standard double precision, which allows us to obtain a maximum accuracy of $10^{{\rm -}10}$-$10^{{\rm -}11}$. Notice that this is few orders of magnitude above the theoretical limit of sixteen digits. This fact is not surprising since we have to consider that during the evolution the numerical noise piles up, reducing the maximum precision. In addition, the number of subdomains used in our test evolutions is not optimal. This is just a comparison of the same exact setup for several bit precisions. Increasing the number of significant bits we improve the maximum error and with $150$ bits (around $45$ significant digits) we easily decrease the error up to almost $10^{-40}$. On the right plot of Figure~\ref{fig_gravColl_error} we study the influence of the number of subdomains in the error as we change the number of collocation points per subdomain but keeping the number of subdomains constant. The error presents an exponential decay $|\Delta r_v| \approx \exp(-\alpha N)$. Varying the number of subdomains changes the factor of the exponential decay, $\alpha$. In this case we can reach the minimum error by adding subdomains with less collocation points. This can be a good idea considering that adding subdomains has, in general, a linear impact on the computational time while increasing the number of collocation points, $N$, has an impact of $\sim N \log N$ or $\sim N^2$ depending on whether the operations in the spectral domain are performed using a FFT transformation or a matrix transformation respectively.
\section{(In)Stability of Anti-de Sitter Spacetimes}\label{grav-collapse-anti-de-Sitter}
In this section we consider a new physical scenario for the application our hybrid PSC-arbitrary precision method. We present results of the evolution of (exact non-linear) ``perturbations'' in asymptotically Anti-de Sitter spacetimes using a Cauchy formulation of the initial-value problem. Cauchy-type evolutions in spherical symmetry were already done in the study of critical gravitational collapse by Choptuik~\cite{Choptuik:1992jv}. In the context of Anti-de Sitter spacetimes they were used recently to study also the problem of critical gravitational collapse in~\cite{Bizon:2011gg}. We adapted this formulation for the use of the PSC method in Refs.~\cite{Olivan:2015fmy,SantosOlivan:2016djn} and we found new physical features associated with the non-linear evolution. We now present results from a new adaptation of our numerical scheme to include arbitrary precision together with the PSC method.
Anti-de Sitter spacetimes have attracted a lot of attention in the last years, both for the interest in studying the non-linear (in)stability of AdS and for its relevance in the so-called AdS/CFT correspondence (also known as the gauge/gravity duality). The key feature of AAdS spacetimes is the presence of a boundary that light-like signals (light rays, massless fields, etc.) can reach in a finite time, but such that time-like signals (massive particles, massive fields, etc.) will take infinite proper time to reach. This property changes completely the landscape of gravitational collapse. As a consequence, the two-case scenario of asymptotically-flat spacetimes does not apply to AAdS spacetimes. Indeed, considering an initial profile like in Eq.~(\ref{initial-data}), we can also expect that the dynamics will make this configuration either to collapse [case (i)] or to disperse [case (ii)]. In the case the scalar field configuration collapses it will form a BH that eventually will settle down into a stationary state (a Schwarzschild-Anti-de Sitter BH). However, in the case the scalar field disperses we cannot expect this dispersion to proceed until we reach asymptotically AdS. The scalar field propagates locally at the speed of light and then, after some finite time, the scalar field profile will reach the AdS boundary, it will bounce back and will try to collapse again, only that the non-linear evolution will change the profile and we will be in the initial situation but with different initial conditions. Therefore, there will be again two possible outcomes, collapse to form an AH or dispersion until reaching the AdS boundary in a finite time and bounce back. This process will repeat itself until the scalar field configuration will have a profile dense enough to finally collapse forming an AH. Then, the possible states of the evolution of a scalar profile in spherically-symmetric AAdS spacetimes are: \\
\noindent (1) {\em Direct Collapse} of the scalar field and the formation of a Black Hole (BH). \\
\noindent (2) {\em Collapse} of the scalar field and BH formation after $1$ bounce off the AdS boundary. \\
$\vdots$\\
\noindent ($n_{c}$) {\em Collapse} of the scalar field and BH formation after $n_{c}-1$ bounces off the AdS boundary. \\
During the trip to the AdS boundary and back, the non-linear relativistic evolution induces a transfer of energy from low frequency (long wavelength) modes towards high frequency (short wavelength) modes, similar to what happens in the onset of turbulence. Due to this analogy, this process, that at some point will end in the collapse of the scalar field profile, has been named the {\it turbulent instability} of AdS spacetime. That is, no matter how small would be the amplitude of the initial profile (perturbation), the field will finally collapse and the end state would be an AdS-BH spacetime. Nevertheless, although this turbulent instability may appear to be a generic feature of the dynamics in AdS spacetime, there are indications of the existence of some islands of stability (see, e.g.~\cite{Choptuik:2018ptp} and references therein) that will not follow the channels just described. The reason is that some stable configurations have been found for some forms of initial configurations, but the exact extend of these ``stability islands'' in the parameter space of initial configurations is still under debate. In order to study this question we need extremely long and accurate evolutions. In conclusion, the end state of the evolution of perturbations in AdS spacetimes is an ideal testbed for numerical techniques that provide high accuracy, beyond the standard one, as the problem is highly demanding. In order to illustrate this, in this section we present a test case in which we evolve an initial massless scalar field configuration in AAdS spacetimes during two of these bounces and compare the accuracy using double precision with the accuracy using $300$-bit precision.
In order to solve the Einstein field equations [Eq.~(\ref{efes})] coupled to the massless scalar field equation [Eq.~(\ref{klein-gordon-equation})] in this new scenario we need a different coordinate system. This coordinate system has to be adapted to a Cauchy-type initial-value problem and, at the same time, it has to incorporate the AdS asymptotic structure of the spacetime. With this in mind, the form of the spacetime line element that we consider is~\cite{Bizon:2011gg}:
\begin{equation}
ds^2 = g^{}_{\mu\nu}dx^{\mu}dx^{\nu} = \frac{\ell^{2}}{\cos^{2}x}\left( - A {e}^{-2\delta}\,dt^2 +\frac{dx^{2}}{A} + \sin^2 x\, d\Omega^2 \right)\,, \label{aads_metric}
\end{equation}
where $A = A(t,x)$ and $\delta = \delta(t,x)$, $t$ is the time coordinate, and $x$ is a compactified radial coordinate in such a way that the AdS boundary is located at $x = \pi/2$ instead of at infinity. The overall factor contains $\ell$, the AdS length scale, which is related to the negative cosmological constant of the spacetime, $\Lambda<0$ [see Eq.~(\ref{efes})], by the expression: $\ell^2 = -3/\Lambda$. The time coordinate $t$ has an infinite range, i.e. $t\in$ ($-\infty$,$\infty$), whereas $x$, being compactified, goes from $x=0$ (center) to $\pi/2$ (AdS boundary). We can recover AdS spacetime by setting $A = 1$ and $\delta=0$.
The system of PDEs that be obtain, by choosing the right combination of variables, can be reduced to a first-order system of strongly hyperbolic PDEs. In addition, in order to use our multidomain scheme, we can further specialize our variables and take them to be the characteristic variables of the hyperbolic system~\cite{Courant:1989aa,SantosOlivan:2016djn}. The form of the characteristic variables associated with the scalar field that we adopt is:
\begin{eqnarray}
U & = & \frac{1}{\cos x} \left( \phi^{}_x - \frac{e^{\delta}}{A} \phi^{}_t \right) \,,
\label{Udef} \\
V & = & \frac{1}{\cos x} \left( \phi^{}_x + \frac{e^{\delta}}{A}\phi^{}_t \right) \,,
\label{Vdef}
\end{eqnarray}
where again, the $(t,x)$ subscripts denote partial differentiation with respect to these coordinates. Then, using the $(t,x)$ coordinates and the $(U,V)$ variables, the evolution problem is reduced to the following coupled system of PDEs:
\begin{eqnarray}
U^{}_t & = & - A e^{-\delta} U^{}_{x} - \frac{(3-2 \cos^2 x)}{\sin x \cos x} U \,e^{-\delta}\,(1-A)
- \frac{A e^{-\delta} }{\sin x \cos x} \left( U+V \right) \nonumber \\
& + & \frac{\sin x}{\cos x}U\,Ae^{-\delta} \,, \label{U-dot-eq} \\[1mm]
V^{}_t & = & + A e^{-\delta} V^{}_{x} + \frac{(3-2 \cos^2 x)}{\sin x \cos x} V \,e^{-\delta}\,(1-A)
+ \frac{A e^{-\delta} }{\sin x \cos x} \left( U+V \right) \nonumber \\
& - & \frac{\sin x}{\cos x}V\,Ae^{-\delta} \,.
\label{V-dot-eq}
\end{eqnarray}
It is also convenient to introduce the following normalized variable associated with the scalar field:
\begin{equation}
\psi = \frac{\phi}{\cos^{2}x} \,. \label{psi-def}
\end{equation}
This new scalar field variable satisfies both an evolution equation
\begin{eqnarray}
\psi^{}_t & = & \frac{A e^{-\delta}}{2\cos x} \left( V - U\right) \,,
\label{evol_psi}
\end{eqnarray}
and also a constraint equation (only containing spatial derivatives):
\begin{eqnarray}
\psi^{}_x = 2\,\frac{\sin x}{\cos x}\,\psi + \frac{1}{2}\, \frac{U + V}{\cos x} \,.
\label{psi_prime}
\end{eqnarray}
Then, we can solve for $\psi$ either by evolving Eq.~(\ref{evol_psi}) or by solving this constraint equation on a constant time slide. Regarding the metric functions, we do not expect them to satisfy hyperbolic equations since in spherical symmetry the true gravitational degrees of freedom are turned off. Then, we obtain constraint equations for $\delta$ and $A$:
\begin{eqnarray}
\delta^{}_x & = & - \textstyle{\frac{1}{2}} \sin x \cos^{3} x \left( V^2 + U^2 \right)\,,
\label{delta_prime} \\[1mm]
A^{}_x & = & \frac{1+2\sin^2 x}{\sin x\cos x}(1-A) - \frac{A}{2}\sin x\cos^{3}x\left( V^2 + U^2\right) \,,
\label{A_prime}
\end{eqnarray}
from which $\delta(t,x)$ and $A(t,x)$ can be obtained at a given time, once we have the solution for $(U,V)$ via the evolution equations, by performing the following integrals:
\begin{eqnarray}
\delta(t,x) & = & \int_x^{\frac{\pi}{2}}\!\! dy\, \sin y \cos^{3} y \left( \frac{U^2 + V^2}{2} \right)\,,
\label{eq-delta-cons}\\[1mm]
A(t,x) - 1 & = & -\frac{\cos^3 x\;e^{\delta}}{\sin x} \int_0^{x}\!\! dy\, e^{-\delta}\sin^{2}y \left( \frac{U^2 + V^2}{2} \right) \,,
\label{ads_integration_for_A}
\end{eqnarray}
where the boundary conditions are $A = 1$ both at $x=0$ and $x=\pi/2$ and we have chosen $\delta \left(\pi/2 \right) = 0$, fixing the time coordinate $t$ as the proper time at the AdS boundary. Then, the Cauchy evolution goes as follows: (i) We prescribe initial data on an initial Cauchy surface $t=t_{o}=$const. for $(U,V)$, i.e. $U(t_o, x)$ and $V(t_o, x)$. (ii) Using equations~(\ref{psi_prime}),~(\ref{eq-delta-cons}), and~(\ref{ads_integration_for_A}) we find $\psi(t_{o},x)$, $\delta(t_{o},x)$ and $A(t_{o},x)$. (iii) With this information we evolve $(U,V)$ from $t_{o}$ to $t_{o}+\Delta t$ using the evolution equations~(\ref{U-dot-eq}) and~(\ref{V-dot-eq}) and the boundary conditions. In our numerical simulations we have considered the following family of Cauchy initial data:
\begin{eqnarray}
U (t^{}_o, x) = \epsilon \exp \left\{ - \frac{ 4 \tan^2 x }{\pi^2 \sigma^{2}} \right\}\,,
\quad
V (t^{}_o, x) = - U (t^{}_{o}, x)\,,
\end{eqnarray}
where the freely specifiable parameters $\epsilon$ and $\sigma$ (amplitude and width of the Gaussian profile respectively) are chosen to be: $\epsilon = 2.0$ and $\sigma = 0.4$.
We set up a multidomain grid with $D=10$ subdomains and change the number of collocation points per subdomain to see how the error changes for both double precision and for $300$-bit precision. We evolve the initial ``perturbation'' for the time corresponding to two bounces off the AdS boundary. Since we need very high accuracy, we have used a sixth-order Runge-Kutta 10,6(7) (see Refs.~\cite{verner1978explicit,prince1981high} for details). This ODE solver uses ten intermediate steps to generate a sixth-order accurate in time integration with a seventh-order step that is used as an estimation of the error.
At any time step, we can compute the energy contained inside a sphere of a given compactified radius $x$, which is known as the mass function:
\begin{equation}
\mathcal{M}(t,x) = e^{\delta} \int_0^{x}\!\! dy\, e^{-\delta} \sin^{2} y \left( \frac{U^2 + V^2}{2}\right) \,,
\end{equation}
so that the total energy contained in the spacetime is $M(t) = \mathcal{M}(t,\pi/2)$. One can show that $M(t)$ should not depend on time, i.e. it is a conserved quantity. Then, we can monitor the conservation of $M$ as indicator of the accuracy of the simulation. To that end we introduce the following mass error function:
\begin{equation}
\Delta{M}(t) = \frac{| M(t) - M(t^{}_{o}) |}{M(t^{}_{o})}\,.
\label{mass-error-function}
\end{equation}
We have performed a set of numerical evolutions to assess the relevance of arbitrary precision in these computations. In Figure~\ref{fig_AdS_evol} we show the evolution of the mass error function of Eq.~(\ref{mass-error-function}). The purple line shows an evolution with a low number of collocation points ($N=12$) and with double precision. The error oscillates in the range $10^{- 12}-10^{- 11}$. Increasing the number of collocation points to $N=18$ (red line in Figure~\ref{fig_AdS_evol}) we can decrease the mass error down to around $10^{- 14}$. It is interesting to notice the different behaviour between these two lines. In the first case, there is plenty of oscillations and we estimate the error by taking the maximum value. This is due to the fact that here the error is determined by the discretization error in such a way that the total error can oscillate between the discretization and the round-off errors. In the second case we have reached by far the round-off error and the profile is quite flat. Then, it is obvious that it cannot be improved by using double precision. Just changing from double precision to $300$-bit precision but with the same number of collocation points per subdomain, the error drops more than two orders of magnitude and, again, it is determined by the discretization.
\begin{figure}[t]
\begin{center}
\resizebox{.9\textwidth}{!}
{
\includegraphics{Figure11-AdS_evol.pdf}
}
\caption[Evolution of the Mass Error Function in AAdS Spacetimes]{\textbf{Evolution of the Mass Error Function in AAdS Spacetimes.} We compare three different grid/precision configurations with $D=10$ subdomains. Using double precision, the use of $N=18$ collocation points per subdomain (red line) is enough to reach the round-off error. The same number of collocation points but with $300$-bit precision (turquoise line) allows us to reduce a few orders of magnitude the error during the evolution. It is interesting to notice how, when the error is determined by the discretization error, several fluctuations are present while in the case dominated by machine round-off the error remains almost flat during the time evolution. The grey vertical line to the right shows the instant of time at which we measure the error for the study presented in Figure~\ref{fig_AdS_error}.
\label{fig_AdS_evol}}
\end{center}
\end{figure}
Once the massless scalar field configuration has bounced twice off the AdS boundary and has come back to the initial location ($t \approx 2\pi$), we have studied not only the error mass function at that moment, but also the error in the characteristic fields $U$ and $V$. To that end, we have used the absolute value of the last spectral coefficient in the subdomain where the error is maximum. This is shown in Figure~\ref{fig_AdS_error} for configurations with $D=10$ subdomains and for different number of collocation points and numerical precisions. As expected, the error in the three quantities [$\Delta{M}$ (left plot), $U$ (center plot), and $V$ (right plot)] decays exponentially until we reach round-off error. For double precision (red points) this happens at values of the order of $\sim 10^{- 14}-10^{- 15}$. This is easily improved when we use higher-order precision, as in the case shown with turquoise dots in the same figure. This corresponds to $300$-bit precision and allows us to evolve the scalar field profile with an accuracy below $10^{\m24}$ with a number of collocation points per subdomain as small as $N=28$, for a total of $(N+1)\, D = 290$ collocation points.
\begin{figure}[t]
\begin{center}
\resizebox{\textwidth}{!}
{
\includegraphics{Figure12-AdS_error.pdf}
}
\caption[Convergence of the Truncation Error in Evolutions in AdS Spacetimes]{\textbf{Convergence of the Truncation Error in Evolutions in AdS Spacetimes.} The figure shows the error in an AdS spacetime evolution using double (red dots) and 300-bit (turquoise dots) precision computations. The plots represent the normalised mass error $\Delta{M}$ (left), the truncation error for $U$ (centre), and the truncation error for $V$ (right) with respect to the number of collocation points per subdomain $N$. This information is taken at the same time ($t=2\pi$) after the massless scalar field configuration has bounced twice off the AdS boundary. All the simulations use $D=10$ subdomains.
\label{fig_AdS_error}}
\end{center}
\end{figure}
\section{Conclusions and Future Perspectives}
\label{conclusions-perspectives}
In this paper we have shown the potential of the combination of Pseudo-Spectral Collocation methods and arbitrary-precision arithmetic for the solution of ordinary/partial differential equations, and more specifically for hyperbolic problems related to the description of gravitational collapse in relativistic gravitation. The exponential convergence of the PSC method makes it a very suitable choice for reaching the maximum accuracy associated with a certain bit precision with a relatively low number of discretization (collocation) points as compared with other techniques. In this sense, we have seen that the power-law convergence of finite difference algorithms makes it unfeasible to reach the needed accuracy within a reasonable number of discretization points. In addition, the PSC method does not require relevant changes in the algorithms as we increase the number of precision bits, in contrast with finite difference algorithms, where we need to adapt the algorithm so that the error scales in a way that we can reach the level of accuracy allowed by the choice of precision arithmetic.
In Sec.~\ref{subsec_comp_time}, we have seen that the main problem of arbitrary precision arithmetic is that it is usually implemented via a software layer that slows down significantly the computations with respect to the speed of standard double precision arithmetic. In this sense, the PSC method helps since the number of collocation points required is relatively small and therefore, although the sparsity of the matrices involved in certain algorithms can be a drawback. Moreover, the multidomain scheme proposed in this paper allows for a simple parallelization of the computations, as we have shown with the use of OpenMP in our examples. We have also shown that the scalability of the multidomain scheme is close to the ideal case of full parallelism. Nevertheless, it would be desirable to explore improvements in the computation speed (and cost) based on an exploration of a more low-level approach to arbitrary precision arithmetic.
To illustrate the potential of these methods we have shown simulations in two problems in relativistic gravitational collapse: (i) The classical Choptuik collapse. Here we have seen that we can estimate with arbitrary precision the location of the apparent horizon. (ii) Collapse in asympotically anti-de Sitter spacetimes. In this example we have shown that arbitrary precision arithmetic allows us to preserve the total energy along the numerical evolution to a very high degree of precision. These numerical experiments have been carried out using a new library, the ANETO library~\cite{anetolib}, that we have developed in the course of our numerical studies of gravitational collapse in General Relativity. The current version has been released with a few basics tools to deal with evolution problems but it can be extended in the future to include other tools that the PSC methods offers. In this sense, one of the main possible improvements would be to add a solver for linear ODEs and also the incorporation of tools for non-linear systems. In addition, it would be interesting to add some type of Adapting Mesh Refinement to allow the grid to be more flexible under different conditions. At the moment, the library, like the systems analyzed in this work, can only deal with evolution problems in just one spatial dimension. It would be desirable to change in a future in order to incorporate tools to work on higher dimensional problems. Another aspect that we have not discussed much in this paper is the question of time integration. As the demand for accuracy increases, this aspect becomes more and more important and then, high-order integration algorithms would be required both in relation to the accuracy provided by the PSC method and to the one provided by arbitrary-precision arithmetic, otherwise we may be in a situation in which the evolution takes a considerably large number of time steps. An interesting solution to improve the accuracy of the evolutions with arbitrary precision arithmetic could be a spectral time integration like the one proposed in Refs.~\cite{Hennig:2008af,Hennig:2012zx,Macedo:2014bfa}.
\section*{Acknowledgements}
The authors acknowledge the high-performance computing resources provided by the Consorci de Serveis Universitaris de Catalunya (CSUC) and the Galicia Supercomputing Center (CESGA) under projects ICTS-CESGA-249 and ICTS-CESGA-266. They also acknowledge support from contracts ESP2015-67234-P, and ESP2017-90084-P (Spanish Ministry of Economy and Competitivity, MINECO), and from contract 2017-SGR-1469 from AGAUR (Catalan government). DS acknowledges support from a FPI doctoral contract BES-2012-057909 from MINECO. We also acknowledge networking support by the COST Action GWverse CA16104 (Horizon 2020 Framework Programme of the European Union).
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\section{Introduction}
It is a well established fact by now, that naive perturbation
theory breaks down at finite temperature, see refs.
\cite{K89,E90} for a recent overview. This is due the
modification of space-time symmetry in the presence of matter or
a heat bath \cite{BS75}, i.e. to the absence of stable
asymptotic states for the observable physical particles. To
overcome this problem, first of all one has to employ a
''doubling'' of the Hilbert space, leading to two-point Green
functions that are $2\times 2$ matrices.
The Schwinger-Keldysh \cite{SKM} or Closed-Time-Path method (CTP)
is such a model, which has been used to derive transport
equations, e.g. for nuclear phenomena, over several decades
\cite{KB62,DP91}. However, the physical interpretation of the
matrix structure of the propagator remains quite obscure in this
formalism, and no justification is obtained for a perturbation
expansion.
Another model is called Thermo Field Dynamics (TFD)
\cite{AU87,Ubook}. It differs from CTP in the important aspect,
that the doubling of the Hilbert space is due to two disjoint,
mutually (anti-) commuting representations of the canonical
commutation relations. Thus, in contrast to CTP, TFD has a
firm mathematical basis \cite{HHW67} and can be used perturbatively.
To this end, one has to sacrifice either the stability aspect or the
observability aspect of asymptotic states. The first
leads to a perturbation expansion in terms of unstable particles
\cite{L88}. It is the purpose of the present paper to
demonstrate, that the second way leads to a simpler and
straightforward approach.
To this end we show, that the thermal instability
of observable states
can be absorbed into a Bogoliubov transformation also
for interacting systems. This Bogoliubov transformation
then defines stable, albeit non-observable, quasi-particles
which serve as basis for a perturbation expansion \cite{HU92}.
Throughout this paper we term them, for historical more than
physical reasons, {\it statistical} quasi-particles \cite{BD71}.
\section{Thermal Bogoliubov transformation}
To establish the notation, we first discuss TFD for a single bosonic
quantum state. We introduce creation and annihilation operators
$a^\dagger$, $a$, $\wtilde{a}^\dagger$, $\wtilde{a}$ for the two representations,
with canonical commutation relations
\begin{equation}
\left[a,a^\dagger\right] = 1 \;\;\;\;\;\;
\left[\wtilde{a},\wtilde{a}^\dagger\right] =1\;\;\;\;\;\;
\left[a,\wtilde{a}^\dagger\right] =0
\;\end{equation}
(see ref. \cite{Ubook} for a complete discussion).
The $a$, $\wtilde{a}$ operators annihilate
the physical vacuum $\lds 0,0\rdb$and the two sets
are transformed into another by means of an anti-unitary
mapping, called the {\it tilde} conjugation (see \cite{EHUY91}
for the tilde conjugation rules). For brevity, we work in the
$\alpha=1$ representation, i.e., the thermal equilibrium state
at inverse temperature $\beta$ and chemical potential $\mu$
is described by two state vectors
\begin{equalign}
\lds O(\beta) \rdb& =&\exp\left( f a^\dagger\wtilde{a}^\dagger \right)\, \lds 0,0\rdb
\;\;\;\;\; f = \ee{-\beta(\omega-\mu)}\\
\ldb O(\beta) \rds& =&\ldb 0,0\rds\,\exp\left( a\wtilde{a} \right)
\; \end{equalign}
Within this framework, the ensemble average of an observable
${\cal E}[a,a^\dagger]$
is calculated as the expectation value
\begin{equation}\label{av3}
\Av{ {\cal E} } =
\frac{ \ldb O(\beta) \rds \;{\cal E} \;\lds O(\beta) \rdb}{\ldb O(\beta) \rds O(\beta)\rangle\mkern-4mu\rangle}
\;.\end{equation}
Obviously then the state vectors $\ldb O(\beta) \rds$ and $\lds O(\beta) \rdb$ are annihilated by
certain linear combinations of the
above operators,
\begin{equation}\label{tsc}
\xi\lds O(\beta) \rdb=0\;\;\;\;\wtilde{\xi}\lds O(\beta) \rdb=0 \;\;\;\;\ldb O(\beta) \rds\xi^{\ankh}=0 \;\;\;\;\ldb O(\beta) \rds\wtilde{\xi}^{\ankh}=0
\;, \end{equation}
obtained as
\begin{equation}\label{bdef}
\left(\array{r}\xi\\
\wtilde{\xi}^{\ankh}\endarray\right)=
{\cal B}
\left(\array{r}a\\ \wtilde{a}^\dagger\endarray\right)
\;\;\;\;
\left(\array{r}\xi^{\ankh}\\ -\wtilde{\xi}\endarray\right)^T=
\left(\array{r}a^\dagger\\-\wtilde{a}\endarray\right)^T
\B^{-1}
\;.\end{equation}
${\cal B}$ is a 2$\times$2 matrix with determinant 1. Since the $\xi$-operators
obey similar commutation relations as do the $a$-operators,
they define our quasi-particles.
It is crucial to realize, that these entities have nothing
in common with the ''usual'' definition of quasi-particles,
which refers to physical states with an almost pointlike mass
spectrum.
Eqn. \fmref{bdef} is essentially a Bogoliubov transformation
\cite{h90ber}. The most general form for the Bogoliubov matrix
compatible with our choice of state vectors is then
\begin{equation}\label{bform}
{\cal B} = \frac{1}{\sqrt{1- f}}\exp\left(s\tau_3\right)
\left(\array{cc}1 & -\;f\\
-1 & 1\endarray \right)
\;,\end{equation}
where $\tau_3=\mbox{diag}(1,-1)$ is a Pauli matrix, $s$ is
free parameter and $f$ is the statistical weight of a
physical single particle state within the ensemble. Therefore the
number density of the physical particles is
\begin{equation}\label{nd}
n = \frac{f}{1-f}
\;.\end{equation}
We find, that the following considerations
are extremely simplified by chosing
$s = 1/2\,\log(1+n) =-1/2\, \log(1-f)$,
since then the Bogoliubov transformation is linear in the density
parameter
\begin{equation}\label{lc}
{\cal B} =
\left(\array{cc}1+n & -n\\
-1 & 1\endarray\right)
\;.\end{equation}
\section{Interacting Boson field}
We now consider a fully interacting bosonic
quantum field. At finite temperature the irreducible
representations of the space-time symmetry group are characterized
by two rather than one continuous parameter \cite{BS75,L88}.
Hence, the {\it interacting} field can be expand into
modes with definite energy and momentum as
\begin{equation}
\phi(x) = \int\limits_0^\infty\!\! dE \,
\int\!\!\frac{d^3{\bf k}}{\sqrt{(2\pi)^3}}\,
\rho^{1/2}(E,{\bf k})\,\phi_{E,{\bf k}}(x)
\;\end{equation}
The commutation relation of these fields are, in general, not known.
However, we want to calculate only the two-point Green function
of the interacting field, i.e. it is sufficient to know the
expectation value of the commutator of these fields.
This expectation value in turn can be absorbed in the definition
of the weight function $\rho(E,{\bf k})$, i.e. for the field operators
we can define
\begin{equation}
\Av{ \left[ \phi_{E,{\bf k}}(t,{\bf x}),\partial_t
\phi_{E^\prime,{\bf k}^\prime}(t,{\bf x}^\prime) \right] }
= \ee{{\mbox i} {\bf k}({\bf x}-{\bf x}^\prime)}\,2E\,\delta(E-E^\prime)
\,\delta({\bf k}-{\bf k}^\prime)
\;.\end{equation}
In other words, for the calculation of bilinear expectation values
of interacting fields it is
sufficient to consider the $\phi_{E,{\bf k}}(x)$ as generalized
free fields \cite{L88}. The full information about the
single-particle spectrum of the theory is contained in the
weight function $\rho(E,{\bf k})$, and we require the
normalization
\begin{equation}\label{norm}
\int\!\!dE^2 \, \rho(E,{\bf k}) =1
\;.\end{equation}
Note, that the existence of this spectral
decomposition is only guaranteed in case the system is space-time
translation invariant, i.e., if it is in a thermal equilibrium
state.
We now apply the thermal quasi-particle concept to each
energy-momentum eigenmode of the system, i.e. we define
quasi-particle operators $\xi_{E,{\bf k}}$
associated with energy-momentum
eigenstates and annihilating the statistical state vectors as
in \fmref{tsc}. These are then Bogoliubov transformed
into physical particle operators for definite energy and momentum,
and those are summed with the above weight function to
operators $a_k(t)$ such that the interacting field
is
\begin{equation}
\phi(x) = \int\!\!\frac{d^3{\bf k}}{\sqrt{(2\pi)^3}}\,
\left(a^\dagger_k(t)\ee{-\mbox{\small i}{\bf k}{\bf x}}
+a_k(t)\ee{\mbox{\small i}{\bf k}{\bf x}}\right)
\; \end{equation}
(note the absorption of the usual energy normalization factor
into the operators). In this expansion
of course, the ''creation'' and ''annihilation'' operators
are quite complex objects. From the above reasoning, i.e.
their decomposition into the modes of definite energy and momentum,
we then obtain
\begin{equalign}\label{sdef}
\ldb O(\beta) \rds a_k^{\dagger(a)}(t) &a_{k^\prime}^{(b)}(t^\prime)\lds O(\beta) \rdb\\
=&\delta^3({\bf k}-{\bf k}^\prime)\,
\int\!\!dE\,\rho(E,{\bf k})\;
\left(\tau_3{\cal B}^T(n_{E{\bf k}})\tau_3\right)_{a2}
\left({\cal B}^T(n_{E{\bf k}})\right)^{-1}_{2b}
\ee{\mbox{\small i} E(t-t^\prime)}\\
\ldb O(\beta) \rds a_k^{(a)}(t) &a_{k^\prime}^{\dagger(b)}(t^\prime)\lds O(\beta) \rdb\\
=&\delta^3({\bf k}-{\bf k}^\prime)\,
\int\!\!dE\,\rho(E,{\bf k})\;
\left({\cal B}(n_{E{\bf k}})\right)^{-1}_{a1}
\left(\tau_3{\cal B}(n_{E{\bf k}})\tau_3\right)_{1b}
\ee{-\mbox{\small i} E(t-t^\prime)}
\;.\end{equalign}
The parameter $n_{E{\bf k}}$ defining the individual Bogoliubov
transformations thus appears under the energy integral.
While this result can be understood intuitively, i.e.
every mode is in thermal equilibrium with the rest of the
system, it can also be understood in a more formal way.
To this end on has to look at the time evolution of
initially free particle operators: it is highly nonlinear.
Therefore the Bogoliubov transformation in interacting systems
is non-linear, and this non-linearity is reflected in the above
energy integral. It is indeed possible to derive the above
equation without ever touching the concept of generalized
free fields \cite{HU92}.
Note, that in our notation the weight function $\rho$ has only support
for positive energy arguments. The retarded and advanced propagator are
in momentum space
\begin{equation}\label{rap}
D^{R,A}(E,{\bf k}) =
\int\!\!dE^\prime\;\rho(E^\prime,{\bf k})\;
\left(\frac{1}{E-E^\prime\pm{\mbox i}\epsilon}
-\frac{1}{E+E^\prime\pm{\mbox i}\epsilon} \right)
\;,\end{equation}
and the limit of free particles is recovered when
\begin{equation}\label{fb}
\rho(E,{\bf k}) \longrightarrow \delta(E^2-\omega_k^2)\Theta(E)
\;.\end{equation}
We then obtain for the propagator matrix
\begin{equalign}\label{fbp2}
D^{(ab)}(t,t^\prime;{\bf k})& = \;-{\mbox i}\int\!\!dE\,\rho(E,{\bf k})\\
\times& \left( ({\cal B}(n_{E{\bf k}}))^{-1}\;
\left(\!\!\!{\array{ll} \Theta(t-t^\prime) & \\
& -\Theta(t^\prime-t) \endarray}\!\!\!\right)\;
{\cal B}(n_{E{\bf k}})\tau_3 \,\ee{-\mbox{\small i} E(t-t^\prime)}\right.\\
+&
\left.\tau_3{\cal B}^T(n_{E{\bf k}})\;
\left(\!\!\!{\array{ll} \Theta(t^\prime-t) & \\
& -\Theta(t-t^\prime) \endarray}\!\!\!\right)\;
({\cal B}^T(n_{E{\bf k}}))^{-1} \,\ee{\mbox{\small i} E(t-t^\prime)}\right)
\;.\end{equalign}
This is a straightforward generalization of the result
from ref. \cite{UY92c}. In the free-particle limit,
the textbook result for the finite temperature boson propagator is
recovered.
The above expression is still somewhat unsatisfactory, since the
${\cal B}$-matrices are subject to an energy integration.
However, because of the special form \fmref{lc} we chose for the
parametrization, the integrand is {\it linear} in the
parameter $n_{E{\bf k}}$. Thus the integration can be carried out
if one defines
\begin{equalign}\label{nbdef}
\bar{N}(t,t^\prime) & =& \frac{1}{Z(t,t^\prime)}\,
\int\!\!dE\,\rho(E,{\bf k})\,n_{E{\bf k}}
\,\left(\ee{-\mbox{\small i} E(t-t^\prime)}+\ee{\mbox{\small i} E(t-t^\prime)}\right)\\
Z(t,t^\prime) & =& \int\!\!dE\,\rho(E,{\bf k})
\,\left(\ee{-\mbox{\small i} E(t-t^\prime)}+\ee{\mbox{\small i} E(t-t^\prime)}\right)
\;.\end{equalign}
Some elementary matrix operations then lead to the result
for the propapagator \cite{UY92c,HU92}
\begin{equalign}\label{bpfu}
&D^{(ab)}(t,t^\prime;{\bf k})\;\\
=&-{\mbox i}\,Z(t,t^\prime)\,({\cal B}(\bar{N}(t,t^\prime)))^{-1}\;
\left(\!\!\!{\array{ll} \Theta(t-t^\prime) & \\
& -\Theta(t^\prime-t) \endarray}\!\!\!\right)\;
{\cal B}(\bar{N}(t,t^\prime))\tau_3 \\
&-{\mbox i}\,Z^\star(t,t^\prime)\,\tau_3\,{\cal B}^T(\bar{N}(t,t^\prime))\;
\left(\!\!\!{\array{ll} \Theta(t^\prime-t) & \\
& -\Theta(t-t^\prime) \endarray}\!\!\!\right)\;
({\cal B}^T(\bar{N}(t,t^\prime)))^{-1}
\;.\end{equalign}
Here we have kept positive and negative energy states
separate: the inner propagator matrices are diagonal in this case.
One can, however, also
combine the two parts into one triangular
inner matrix, sandwiched between two Bogoliubov matrices:
\begin{equalign}\label{bpfu2}
&D^{(ab)}(t,t^\prime;{\bf k})\;\\
=&({\cal B}(\bar{N}(t,t^\prime)))^{-1}\;\times\\
&\left(\!\!\!{\array{rr}
-{\mbox i}\Theta(t-t^\prime)\left( Z(t,t^\prime)-Z^\star(t,t^\prime)\right)
&{\mbox i}\left(1+2\bar{N}(t,t^\prime)\right)Z^\star(t,t^\prime) \\
&{\mbox i}\Theta(t^\prime-t)\left( Z(t,t^\prime)-Z^\star(t,t^\prime)\right)
\endarray}\!\!\!\right)\\
&\times\; {\cal B}(\bar{N}(t,t^\prime))\tau_3
\;.\end{equalign}
The physical relevance of the function $\bar{N}(t,t^\prime)$
diagonalizing the propagator
becomes obvious, when we consider its equal time limit.
It approaches a constant then,
\begin{equation}\label{teq}
\lim_{t^\prime\rightarrow t} \bar{N}(t,t^\prime) = N^H_k
\;\;\;\;\;\;\;
\lim_{t^\prime\rightarrow t}
\frac{\partial}{\partial t} \bar{N}(t,t^\prime) = 0
\;.\end{equation}
Comparision to \fmref{sdef}
gives
\begin{equation}\label{mbh}
N^H_k = \frac{\displaystyle
\int\!\!dE\,\rho(E,{\bf k})\,n_{E{\bf k}}}{
\displaystyle \int\!\!dE\,\rho(E,{\bf k})}
=\frac{\displaystyle \lim_{t^\prime\rightarrow t}\int\!d^3{\bf k}\,
\ldb O(\beta) \rds a_k^\dagger(t) a_{k^\prime}(t^\prime)\lds O(\beta) \rdb}{
\displaystyle \int\!\!dE\,\rho(E,{\bf k})}
\;.\end{equation}
In other terms, $N^H_k$ is the time-independent equilibrium Heisenberg
density of the physical particles with momentum ${\bf k}$.
The quantity $\bar{N}(t,t^\prime)$ therefore is the {\it observable}
fluctuating particle number of these modes.
The
separate diagonality of the inner matrices of the propagator,
i.e. the requirement of unperturbed statistical
quasi-particle propagation, is therefore equivalent to chosing
the correct physical Bogoliubov parameter.
\section{Conclusions}
The concept of quasi-particles in statistical physics is known for some
time \cite{BD71}, and it is also known that a linear
relation between the matrix elements of an interacting
propagator can be used to bring it to a triangular $2\times 2$
matrix form \cite{RS86}. We have put these two things
together and introduced statistical quasi-particles into
Thermo Field Dynamics. This leads to a
diagonal propagator for nonrelativistic models \cite{YUNA92}.
When negative energy states are taken into account, the full
propagator can also be written as diagonal matrix sandwiched
among Bogoliubov matrices, but separately so for particle and
anti-particle states \cite{HU92}. Their combination then gives a
triangular inner propagator matrix, see eqn. \fmref{bpfu2}. We
find, that the Bogoliubov transformation necessary for this
diagonalization is given in terms of a physical parameter, the
observable fluctuating particle density $\bar{N}(t,t^\prime)$.
This diagonalization also defines stable physical modes with
fixed momentum and an average energy \cite{uh93}.
While this is clearly a conceptual advantage, the diagonalization
of the full propagator at finite temperature also has
a tremendous technical advantage over the CTP formalism.
In effect our method separates the statistical information
about the system (boundary conditions, thermal particle-hole
excitation etc.) from the purely spectral information contained in
the weight function $\rho(E,{\bf k})$. Let us note, that the
same separation can be achieved for non-equilibrium systems.
The application of the quasi-particle concept to a given system
not only demonstrates this technical advantage, but lends a
physical interpretation of the $2\times 2$ matrix structure of
thermal quantum theories \cite{K89,E90}. For brevity we only
state the results: Requiring, that the triangular propagator
\fmref{bpfu} solves the diagonal components of a Schwinger-Dyson
equation gives $\rho(E,{\bf k})$ as function of real and
imaginary part of a retarded self energy function.
The off-diagonal component of the Schwinger-Dyson equation
contains the statistical information,
i.e. it is a consistency criterion for the function
$\bar{N}(t,t^\prime)$. For the time-independent case considered
here, we were able to derive this consistency criterion as the
condition of global thermal equilibrium. For non-equilibrium
systems, where a similar separation of statistical and
''spectral'' information can be obtained, the diagonalization condition
for the propagator is nothing but a transport
equation \cite{HU92}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 655
|
\section{Introduction}
The thermal radiation of photons and dileptons from a quark-gluon plasma (QGP) is one of
the most promising signatures for the formation of a QGP in relativistic heavy ion collisions
\cite{Ruus}. Whereas real photons can be produced to lowest order $\alpha \alpha _s$ only via
the participation of a gluon, dileptons are produced to lowest order $\alpha ^2$ by
quark-antiquark annihilation into a virtual photon (Born term) \cite{Born}. However, in the
case of a small invariant photon mass $M^2=E^2-p^2$, where $E$ is the energy and $p=|{\bf p}|$
the momentum of the photon, radiative corrections of the order $\alpha ^2 \alpha _s$ become
increasingly important. These corrections have been considered
using perturbative QCD at finite temperature \cite{Corr,Alru}. In contrast to the production
of real photons the dilepton production rate turns out to be infrared finite in the case of a
vanishing quark mass. There is a cancellation of the infrared singularities of real and
virtual contributions, the latter appearing only for the production of virtual photons
\cite{Alru}. However, also medium effects leading to an effective quark mass $m_q^*$
by the interaction of the quark with the heat bath can screen these infrared divergences.
Altherr and Ruuskanen \cite{Alru} suggested that the effective quark mass simply replaces the
invariant photon mass as infrared cutoff if $M<m_q^*$.
Medium effects can be included consistently using the resummation technique of Braaten
and Pisarski \cite{Brpi}. This method has been applied to a number of interesting
quantities of the QGP leading to gauge invariant and infrared finite results that are complete
to leading order in the coupling constant \cite{Thom}. For example, the production rate
of energetic photons has been derived in this way, where a resummed quark propagator
containing the effective quark mass ${m_q^*}^2=g^2T^2/6$ has been used \cite{Kls,Bai,Tvt}.
Here we want to reconsider the result of Altherr and Ruuskanen \cite{Alru} performing
a complete calculation of the dilepton rate using the Braaten-Pisarski method analogously
to the photon case. In particular we will study in detail the role of $m_q^*$ and $M$ as
infrared cutoffs. Like Altherr and Ruuskanen, we will restrict ourselves
to small photon masses $M{\buildrel <\over \sim}T$, for which the $\alpha _s$-corrections
are of importance, and to large photon energies $E\gg T$.
\section{Dilepton production rate to order {\boldmath $\alpha ^2 \alpha
\mbox{\unboldmath $_s$}$ \unboldmath}}
The lowest order matrix elements for the production of virtual photons from the QGP are
shown in Fig.1. Besides the real contributions (Compton scattering and annihilation) of
Fig.1a,b, appearing also for the production of real photons, there are virtual
contributions, namely gluon absorption (Fig.1c) and the interference terms of the Born
term with radiative corrections (self energy insertion and vertex correction) in Fig.1d.
Due to phase space restrictions and kinematics ($E\gg T$) the gluon absorption
as well as the vertex correction can be neglected \cite{Alru}.
The matrix elements can be related to the imaginary part of the photon self energy
diagrams in Fig.2 \cite{Weld}. Here the processes of Fig.1a,b,c can be obtained by cutting
through the internal quark and gluon lines, while the interference term (Fig.1d)
corresponds to a cut through the quark lines only. In the latter case the quark
self energy becomes on-shell, leading to an infrared singularity which cancels the one
from the exchanged massless quark in the real contribution \cite{Alru}. This cancellation
can be considered as an example of the KLN-theorem at finite temperature \cite{Kln}.
The production rate for massless electron and muon pairs derived from the photon self energy is
given by \cite{Alru}
\begin{equation}
\frac {dR}{d^4xd^4p}=\frac {1}{6\pi ^4}\> \frac {\alpha }{M^2}\> \frac {1}{e^{E/T}-1}\>
Im \Pi _\mu ^\mu (P).
\label{e1}
\end{equation}
As in the case of the photon production rate \cite{Kls,Bai,Tvt}, we want to include
the effect of the medium by using a resummed quark propagator for soft momenta of the
exchanged quark. For this purpose we introduce a separation scale $k_c$ \cite{Bryu},
restricted by $gT\ll k_c\ll T$ in the weak coupling limit. For quark momenta smaller than
$k_c$ we will start from the photon self energy shown in Fig.3, where the blob denotes
the effective quark propagator, in which the quark self energy in the hard thermal loop
approximation has been resummed \cite{Bpy}. Owing to the high energy of the photon and
energy-momentum conservation we need to take into account only one effective quark
propagator and no effective vertices as opposed to the case of soft dileptons \cite{Bpy}.
For momenta of the exchanged quark larger than $k_c$, the diagrams of Fig.1 or Fig.2,
containing only bare propagators, will be considered. After adding up the
soft and hard contributions, the separation scale has to drop out, demonstrating
the completeness of the calculation \cite{Bryu}. In the case of the photon production
a covariant separation scale, i.e. $|K^2|=|\omega ^2-k^2|=k_c^2$ \cite{Kls}, and a
non-covariant one ($k=k_c$) \cite{Bai} have been used, both leading to the same result.
In the limit $M^2\ll ET$ the hard real contribution (Fig.1a,b) of the dilepton production rate
can be taken over from the photon rate \cite{Alru}, yielding in the case of a non-covariant
cutoff \cite{Bai}
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}_{hard}^{real}
=\frac {10}{27\pi ^3}\> \alpha ^2\alpha _s \> \frac
{T^2}{M^2}\> e^{-E/T}\> \left [\ln \frac {ET}{k_c^2}+\frac{3}{2}+\frac {\ln 2}{3}-\gamma
+\frac {\zeta '(2)}{\zeta (2)}\right ],
\label{e2}
\end{equation}
where $\gamma =0.57722$ and $\zeta '(2)/\zeta (2)=-0.56996$. In the case of a covariant cutoff
the constant in the square brackets is changed by adding a term $\ln 4-2$ \cite{Kls,Bai}.
The hard contribution to the photon rate follows from a momentum integration over the square
of the matrix elements of Fig.1a,b using distribution functions for the external partons.
Applying the Boltzmann approximation for the distributions of the incoming partons
and using $k_c\ll T$ the result (\ref{e2}) is found after some tedious manipulations.
It can be obtained more easily by calculating the photon absorption rate and applying
the principle of detailed balance \cite{tho1}.
The soft contribution to the dilepton rate can be computed by introducing spectral functions
for the resummed quark propagator \cite{Bpy} as in the case of the photon rate \cite{Kls}.
Following the arguments in the calculation of the photon rate we arrive at
\begin{eqnarray}
{\left (\frac {dR}{d^4xd^4p}\right )}_{soft}^{real} & = & \frac {40}{9\pi ^2}\> \frac {\alpha
^2}{M^2}\> \int \frac {d^3k}{(2\pi )^3}\> \int d\omega \> \int d\omega '\> \delta (E-\omega
-\omega ')\> n_F(\omega )\> n_F (\omega ')\nonumber \\
&& \biggl \{ \left (1+\hat {\bf q}\cdot \hat {\bf k}\right )\> \left [\rho _+(\omega ,k)\,
\delta (\omega '+q)+\rho _-(\omega ,k)\, \delta (\omega '-q)\right ]\nonumber \\
&&{}+
\left (1-\hat {\bf q}\cdot \hat {\bf k}\right )\> \left [\rho _+(\omega ,k)\,
\delta (\omega '-q)+\rho _-(\omega ,k)\, \delta (\omega '+q)\right ]\biggr \},
\label{e3}
\end{eqnarray}
where $n_F$ is the Fermi-Dirac distribution, ${\bf q}={\bf p}-{\bf k}$ the momentum of the
bare quark propagator, and
\begin{equation}
\rho _\pm (\omega, k)=\frac {\omega ^2-k^2}{2{m_q^*}^2}\> [\delta (\omega -\omega _\pm )+
\delta (\omega +\omega _\pm )]+\beta _\pm (\omega , k)\> \theta (k^2-\omega ^2)
\label{e4}
\end{equation}
the spectral functions with
\begin{equation}
\beta _\pm (\omega ,k)=-\frac {{m_q^*}^2}{2}\> \frac {\pm \omega -k}{\left [k\, (-\omega \pm k)
+{m_q^*}^2\, \left (\pm 1-\frac {\pm \omega -k}{2k}\, \ln \frac {k+\omega }{k-\omega } \right )
\right ]^2+\left [\frac {\pi}{2}\, {m_q^*}^2\, \frac {\pm \omega -k}{k}\right ]^2}.
\label{e5}
\end{equation}
The first term of the spectral functions (\ref{e4}), containing the dispersion relations
$\omega _\pm (k)$ (see Fig.4) of the collective quark modes \cite{Klwe} above the light
cone ($\omega >k$), gives rise to the virtual contribution of the dilepton rate. The real soft
contribution, on the other hand, follows from the second term below the light cone
($-k<\omega <k$), which comes from the imaginary part of the hard thermal loop quark self
energy. Using $E\gg T$ and assuming $\omega $ and $k$ to be soft \cite{Kls,Tvt}, the soft
real contribution reduces to
\begin{eqnarray}
{\left (\frac {dR}{d^4xd^4p}\right )}_{soft}^{real}
=\frac {5}{9\pi ^4}\> \frac {\alpha ^2}{M^2}\> e^{-E/T}\>
\int dk d\omega \> &&{} [(k-\omega +E-p)\> \beta _+(\omega ,k)\nonumber \\
&&{} +(k+\omega -E+p)\> \beta _-(\omega ,k)].
\label{e6}
\end{eqnarray}
The integration range, determined by $\omega =-k+E-p$, $\omega
=k$, and the separation scale $k=k_c$ or $k^2-\omega ^2=k_c^2$, respectively, are
shown in Fig.4. Compared to the photon case it is shifted by $E-p>0$, which renders
the integration in (\ref{e6}) more difficult.
The logarithmic dependence on the scale $k_c$ can be extracted by adopting the static limit
of the effective quark propagator \cite{Bai},
\begin{equation}
\beta _\pm (\omega =0,k)=\frac {{m_q^*}^2}{2}\> \frac{k}{(k^2+{m_q^*}^2)^2+(\pi {m_q^*}^2/2)^2}.
\label{e7}
\end{equation}
In the case of a non-covariant separation scale $k_c$, where the $\omega $-integration
is restricted by the limits $-k+E-p$ and $k$, while the $k$-integration ranges from
$(E-p)/2$ to $k_c$, we find to logarithmic approximation
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}_{soft}^{real}
=\frac {5}{9\pi ^4}\> \frac {\alpha ^2}{M^2}\> e^{-E/T}\> {m_q^*}^2\> \left (\ln
\frac {k_c^2}{{m_q^*}^2}+A\right ),
\label{e8}
\end{equation}
where we have used $k_c\gg gT$. In order to determine the function $A(E-p)$ beyond the
logarithm,
we cannot use the static approximation anymore, but have to solve (\ref{e6}) together with
(\ref{e5}) numerically. We restrict ourselves to $E-p\leq m_q^*$, which holds for realistic
values of the coupling constant $g{\buildrel > \over \sim }1$. Then we find that $A$ increases
from $A(E-p=0)=-1.31$, which corresponds to the photon result \cite{Bai}, to
$A(E-p=m_q^*)=-1.71$.
Adding the hard contribution (\ref{e2}) to (\ref{e8}) with ${m_q^*}^2=2\pi \alpha_s T^2/3$,
the arbitrary separation scale $k_c$ drops out
and we obtain an infrared finite result for the real contribution of the dilepton rate.
On the other hand, using a covariant separation scale, (\ref{e8}) is multiplied by a factor
$k_c^2/[(E-p)^2+k_c^2]$. Note that $E-p=M^2/(E+p)\ll T$ might be of the
same order as $k_c\ll T$ for $M\sim T$.
Hence the separation scale does not cancel in this case in contrast
to the real photon rate ($E=p$). This shows that we should demand that $\omega $
and $k$ are soft individually as it was also assumed in the derivation of the
Braaten-Pisarski method, which is based on the imaginary time formalism in euclidean space-time
\cite{Brpi}.
Finally, we consider the virtual part coming from the pole contribution to the imaginary
part of the photon self energy in (\ref{e1}). Now we have to use a covariant separation scale
since the infrared singularity in Fig.2 comes from the on-shell self energy at $K^2=0$.
The soft part follows from the pole contribution of the diagrams in Fig.3
corresponding to the first term of the spectral functions (\ref{e4}). Since
$\omega _\pm (k)$ lies always below $K^2=k_c^2\gg {m_q^*}^2$ (see Fig.4),
there is no hard contribution
to the virtual part. Thus the virtual contribution is determined solely by the imaginary
part of the self energy in Fig.3 coming from the pole of the effective quark propagator,
which gives a finite result when integrated over the entire momentum range.
Combining (\ref{e3}) and the first term of (\ref{e4}) we find
\begin{eqnarray}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}
& = & \frac {5}{9\pi ^4}\> \frac {\alpha ^2 }{M^2 {m_q^*}^2}\> \frac {1}{p}\>
\nonumber \\
\biggl \{ \! \! \! \! \! \! \! \! &&{} \int _{k_{min}^+}^{k_{max}^+} dk\>
n_F(\omega _+)\> n_F(E-\omega _+)\> (\omega _+^2-k^2)\> \left [ \frac {M^2}{2}-E\,
(\omega _+-k)+\frac {(\omega _+-k)^2}{2}\right ]\nonumber \\
& - & \int _{k_{min}^-}^{k_{max}^-} dk\> n_F(\omega _-)\> n_F(E-\omega _-)\>
(\omega _-^2-k^2)\> \left [ \frac {M^2}{2}-E\, (\omega _-+k)
+\frac {(\omega _--k)^2}{2}\right ] \biggr \},\nonumber \\
\label{e9}
\end{eqnarray}
where the limits $k_{max,min}^\pm$ are determined from the intersections of the
dispersion relations with $\omega =\pm k\pm (E-p)$ as shown in Fig.4. Note that
the virtual part vanishes for $M\rightarrow 0$ as $k_{min}^\pm $ tends to infinity.
In order to isolate the term proportional to $\alpha ^2\alpha _s$ and to investigate its
dependence on $M$, $m_q^*$, $E$, and $T$, we introduce again a separation scale
$gT\ll k_s\ll T$. For $k<k_s$ we may approximate the distribution functions by
$n_F(\omega _\pm)\simeq 1/2$ and $n_F(E-\omega _\pm)\simeq \exp(-E/T)$, whereas
for $k>k_s$ we may set $\omega _+\simeq k+{m_q^*}^2/k$ leading to $\omega _+^2-k^2
\simeq 2{m_q^*}^2$ and $\omega _-=k$ \cite{Pis}. Hence the plasmino branch $\omega _-(k)$
does not contribute for hard momenta as $\omega _-$ approaches $k$ exponentially
for $k\gg gT$ \cite{Pis}.
For the part of the integrals in (\ref{e9}) containing the terms with $M^2/2$ it is
sufficient to restrict to $k>k_s$ since the hard part is finite for $k_s \rightarrow 0$.
Considering $E\gg T$ the simplifications
$k_{max}^+\simeq (E+p)/2$, $k_{min}^+\simeq (E-p)/2$, $n_F(\omega _+)\simeq n_F(k)$, and
$n_F(E-\omega _+)\simeq n_F(E-k)$ can be assumed. Then this term reduces to the Born term,
which reads for $E\gg T$ and $E-p\ll T$
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}^{Born}=\frac {5}{9\pi ^4}\> \alpha ^2 \> e^{-E/T}.
\label{e10}
\end{equation}
The Born term is contained in the virtual contribution (\ref{e9}) because the effective
quark propagator includes the bare one.
Next we consider the terms under the integral in (\ref{e9}) proportional to $E(\omega _\pm
\pm k)$. Using the approximations discussed above, we find for $k<k_s$
\begin{eqnarray}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}_{soft} & = & \frac {5}{18\pi ^4}\> \frac
{\alpha ^2 }{M^2 {m_q^*}^2}\> \frac {1}{p}\> e^{-E/T}\> \nonumber \\
&& \left [-\int _{k_{min}^+}^{k_s} dk\> (\omega _+^2-k^2)\> (\omega _+-k)
+\int _{k_{min}^-}^{k_s} dk\> (\omega _-^2-k^2)\> (\omega _-+k)\right ].
\label{e11}
\end{eqnarray}
We proceed analogously to Baier et al. \cite{Bai} in the case of the photon rate, using
$(\omega _\pm^2-k^2)(\omega _\pm \mp k)/{m_q^*}^2=\omega _\pm -k(d\omega _\pm/dk)$. Then we obtain with $k_s\gg m_q^*$
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}_{soft}=-\frac {5}{9\pi ^4}\> \frac
{\alpha ^2 }{M^2}\> e^{-E/T}\> {m_q^*}^2\> \left (\ln \frac {k_s}{m_q+k_{min}^+} +B\right ).
\label{e12}
\end{equation}
The function $B(E-p)$ decreases from $B(E-p=0)=0$, where the virtual contribution vanishes,
to $B(E-p=m_q^*)=-0.66$, where $k_{min}^\pm=0$ (see Fig.4).
The hard part, $k>k_s$, using the approximations given above, reads
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}_{hard}=-\frac {10}{9\pi ^4}\> \frac
{\alpha ^2 }{M^2}\> {m_q^*}^2\> \int _{k_s}^E dk\> \frac {n_F(k)n_F(E-k)}{k}.
\label{e13}
\end{equation}
Considering $k_s\ll T\ll E$, we find
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}_{hard}=-\frac {5}{9\pi ^4}\> \frac
{\alpha ^2 }{M^2}\> e^{-E/T}\> {m_q^*}^2\> \ln \frac {E}{k_s}.
\label{e14}
\end{equation}
Adding up both the contributions (\ref{e12}) and (\ref{e14}) $k_s$ cancels and we arrive at
\begin{equation}
{\left (\frac {dR}{d^4xd^4p}\right )}^{virt}=-\frac {5}{9\pi ^4}\> \frac {\alpha ^2 }{M^2}
\> e^{-E/T}\> {m_q^*}^2\> \left (\ln \frac {E}{m_q^*+k_{min}^+}+B\right ).
\label{e15}
\end{equation}
We observe that the virtual contribution coming from an interference term is negative,
thus reducing the dilepton rate compared to the photon rate. Furthermore, the virtual
contribution vanishes for $m_q^*\rightarrow 0$ as it is also the case for the virtual
contribution in naive perturbation theory \cite{Alru}.
The part under the integral in (\ref{e9}) proportional to $(\omega _\pm -k)^2$ can be shown
to be suppressed by $m_q^*/E$ relative to (\ref{e15}).
Combining (\ref{e2}), (\ref{e8}), and (\ref{e15}) we end up with our final result for
the dilepton production rate
\begin{equation}
\frac {dR}{d^4xd^4p}=\frac {10}{27\pi ^3}\> \alpha ^2\alpha _s \frac {T^2}{M^2}\>
e^{-E/T}\> \left (\ln \frac {T(m_q^*+k_{min}^+)}{{m_q^*}^2}+C\right ),
\label{e16}
\end{equation}
where the function $C$ ranges from $C(E-p=0)=-0.73$ to $C(E-p=m_q^*)=-0.47$.
For giving an approximate analytic expression for $k_{min}^+$ we replace the exact
dispersion relation $\omega _+(k)$ by $\omega _+^2=k^2+{m_q^*}^2$, which deviates from
the exact one by less than 11\% over the entire momentum range. Then we obtain
$k_{min}^+=|E{m_q^*}^2/M^2-M^2/(4E)|$, restricted by $k_{min}^+\leq k_{max}^+\simeq (E+p)/2$.
\section{Discussion}
Now we will discuss our result (\ref{e16}) for various limits of $M$. For
$M\rightarrow 0$ the virtual contribution vanishes since $k_{min}^+$ tends to infinity
for $E-p\rightarrow 0$. Hence the dilepton rate is given by the real contribution which
agrees with the photon rate in this limit.
For $M\sim m_q^*$ we get $E-p=M^2/(2E)\ll m_q^*$. Hence $k_{min}^+$ is given by the
intersection
of $\omega \simeq k+{m_q^*}^2/k$ and $\omega=k+E-p$, i.e. $k_{min}^+={m_q^*}^2/(E-p)
\sim E$. Thus the result for the photon rate holds approximately, which agrees
with the result of Altherr and Ruuskanen \cite{Alru}, if we replace the invariant
photon mass in the logarithmic term of their formula (2.18) by the effective quark mass
in the case of $M<m_q^*$ as suggested in their paper.
For $M\sim T$, i.e. $E-p\sim T^2/E$, we have to distinguish two cases. First $E-p\sim m_q^*$,
which leads to $k_{min}^+\sim m_q^*$ (see Fig.4). Then the logarithm in (\ref{e16}) is given by
$\ln (T/m_q^*)$. Secondly, in the case $E-p\ll m_q$
we find from Fig.4 $k_{min}^+\gg m_q^*$. Now if $k_{min}^+\sim E$ we recover the photon result,
while for $k_{min}^+\sim T$ the logarithmic term is given by $\ln (T^2/{m_q^*}^2)$.
Finally we will comment on the extrapolation of our result to realistic values of the coupling
constant, $\alpha _s=0.2$-0.5, i.e. $g=1.5$-2.5. In any case the lowest order result
in $\alpha _s$ fails if the rate (\ref{e16}) becomes negative. (This unphysical
behaviour can originate from extracting the leading order contribution and extrapolating to
large
values of the coupling constant. A similar problem has been encountered in the case of transport
rates determining thermalization times and the viscosity of the QGP \cite{Thom,Tho2}.)
Thus one might expect
that for $M\sim T$ the result gets unphysical, since $m_q^*=T$ for $g=\sqrt {6}$.
However, for $E\gg T$ we find $E-p\ll T\sim m_q^*$ and $k_{min}^+\gg T$.
Hence we end up with a logarithmic term which is always positive. However, the constant
$C$ behind the logarithm is of the same order (typically half of the value of the logarithm),
indicating the necessity to go beyond the logarithmic approximation. (This is more important
for dileptons than for photons due to the negative virtual contribution.).
As an example we have chosen $\alpha _s=0.3$ and $T=300$ MeV. In Fig.5 the dilepton
production rate is shown as a function of $E$ for $M=300$ MeV and in Fig.6 as a function
of $M$ for $E=3$ GeV. Also shown is the result of Altherr and Ruuskanen \cite{Alru}.
(Here we replaced $M$ in their formula (2.18) by $m_q^*$ for $M<m_q^*$.)
For the chosen set of parameters both approaches give similar results, since for
$M<m_q^*=238$ MeV both rates are given by the photon result approximately (see Fig.6).
However, the dilepton rate found by Altherr and Ruuskanen breaks down at somewhat larger
values of $E$ as shown in Fig.5. In Fig.5 also the Born term (\ref{e10}) is depicted for
comparision.
Summarizing, we conclude that the complete calculation using a resummed quark propagator
according to the Braaten-Pisarski method leads to a finite result for the dilepton rate
to order $\alpha ^2\alpha _s$, where the effective quark mass $m_q^*$ cuts off the infrared
divergence. There is no need for using the invariant photon mass $M$ as an infrared
cutoff even for $M$ of the order of the temperature.
\acknowledgements
We would like to thank E. Braaten and V. Ruuskanen for helpful discussions.
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|
Nissan to make Ghana sales hub for W/Africa with assembly plant
November 7, 2018 November 7, 2018 nanaakua1Leave a Comment on Nissan to make Ghana sales hub for W/Africa with assembly plant
Ghana has signed a Memorandum of Understanding with Nissan Group of Africa for the establishment of automotive manufacturing industry in Ghana, which will make Ghana the hub for sales and marketing of Nissan in West Africa.
This was announced on Tuesday, 6th November 2018, when the Managing Director of Nissan Group of Africa, Mike Whitfield, paid a courtesy call on President Akufo-Addo.
According to Mr Whitfield, Nissan aims to be the first car-maker to assemble vehicles in Ghana, building on its market leadership in the country.
Nissan models, he explained, accounted for 32.8% of vehicle sales in Ghana last year, with the company's cars, pickups, and SUVs sold through a national network of six sales and service outlets.
"Nissan is the most popular auto brand in Ghana because the quality of our products and services has won the trust of our customers," Whitfield said.
He continued, "We want to build on our leadership by supporting the government to create the environment for a successful automotive manufacturing industry in the country. Building vehicles in Ghana will enable us to further improve the products and services we offer to our customers here and will have significant, long-term benefits for the economy in terms of jobs and growth."
On his part, President Akufo-Addo welcomed strongly the decision by Nissan to establish an automotive manufacturing industry in Ghana.
The President explained that his administration had embarked on a journey on moving the country away from being mere producers and exporters of raw materials, with a focus on value-addition and industrial activities.
One of the areas of focus, he stressed, was the automotive industry, the reason why his administration has spent the last 22 months strengthening the fundamentals of the Ghanaian economy to attract such investment.
"To have at A+ company like yours in Ghana is positive, and we welcome you strongly. We hope that the MoU that will be signed will not just remain an MoU, but will translate into concrete benefits for us all," President Akufo-Addo said.
The MoU seeks to unlock economic potential, promote the development of the automotive sector, and promote investor-friendly regulatory frameworks that encourage sustainable car manufacturing. The aim is to promote infrastructure development, job creation and skills development in Ghana.
The Minister for Trade, Alan Kyerematen, also praised Nissan's commitment to Ghana, saying, "we welcome this MOU and commit ourselves in turn to working with Nissan to create the necessary environment for the level of investment that will make Ghana's automotive sector a reality."
Industry-wide vehicle sales in Ghana have been growing steadily at an annual rate of about 10%, and now stand at about 9,150 vehicles a year.
Working closely with the government of Ghana and with other members of the African Association of Automotive Manufacturers, Nissan will provide its global expertise to establish a sustainable auto manufacturing industry in the country.
The agreement builds on Nissan's investment in Nigeria, where in 2013, the company became the first major automaker to assemble cars.
Nissan to set up assembly plant in Ghana – Bawumia
Nissan's plan to establish in Ghana was first announced by the Vice President, Dr Mahamudu Bawumia in September 2018.
German car manufacturer, Volkswagen and Sinotruk International from China have all served notice of setting up assembly plants in Ghana.
We're not scared of competition – Kantanka
Following announcement of international automobile companies setting up in Ghana, the Chief Executive Officer (CEO) of the Kantanka Automobile Company, Kwadwo Safo Jnr. says his company is not scared of competition from any international automobile firm seeking to set up in Ghana.
"Mr President. [The] Only thing we need is good policies to protect the automobile industry and help grow our local industries …To anyone that thinks @KantankaAuto is scared of competition. We are not. I simply want better policies for our auto industry," he stated.
Source:CitiNewsroom.com
Capital Bank collapse: Full truth will come out – Otabil
74 people including MPs to be surcharged for owing Agric Ministry
GRA bans selected items from being kept in warehouses
November 2, 2018 November 2, 2018 nanaakua1
Gov't justifies GH¢2bn support for 7 local banks
January 3, 2019 January 3, 2019 Samuel Yeboah
Otabil, ICGC, and 13 others sued over collapse of Capital Bank
|
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{"url":"https:\/\/www.esaral.com\/q\/the-height-of-an-equilateral-triangle-is-6-cm-find-its-area-50484\/","text":"The height of an equilateral triangle is 6 cm. Find its area.\nQuestion:\n\nThe height of an equilateral triangle is 6 cm. Find its area.\u00a0\u00a0$[$ Take $\\sqrt{3}=1.73]$\n\nSolution:\n\nLet the side of the equilateral triangle be\u00a0x\u00a0cm.\n\nAs, the area of an equilateral triangle $=\\frac{\\sqrt{3}}{4}(\\text { side })^{2}=\\frac{x^{2} \\sqrt{3}}{4}$\n\nAlso, the area of the triangle $=\\frac{1}{2} \\times$ Base $\\times$ Height $=\\frac{1}{2} \\times x \\times 6=3 x$\n\nSo, $\\frac{x^{2} \\sqrt{3}}{4}=3 x$\n\n$\\Rightarrow \\frac{x \\sqrt{3}}{4}=3$\n\n$\\Rightarrow x=\\frac{12}{\\sqrt{3}}$\n\n$\\Rightarrow x=\\frac{12}{\\sqrt{3}} \\times \\frac{\\sqrt{3}}{\\sqrt{3}}$\n\n$\\Rightarrow x=\\frac{12 \\sqrt{3}}{3}$\n\n$\\Rightarrow x=4 \\sqrt{3} \\mathrm{~cm}$\n\nNow, area of the equilateral triangle $=3 x$\n\n$=3 \\times 4 \\sqrt{3}$\n\n$=12 \\sqrt{3}$\n\n$=12 \\times 1.73$\n\n$=20.76 \\mathrm{~cm}^{2}$","date":"2022-06-29 00:05:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8557326793670654, \"perplexity\": 236.81886026587256}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103619185.32\/warc\/CC-MAIN-20220628233925-20220629023925-00398.warc.gz\"}"}
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DestinationsUnited StatesMinnesota
8 Fabulous Reasons Winter Is A Perfect Time To Visit The Brainerd, Minnesota Area
Shel Zolkewich
There is no shortage of trails in the Brainerd area to slide and glide on a pair of cross-country skis.
Photo credit: Explore Brainerd Lakes
Winter Travel
The landscape of northern Minnesota always makes me think of the classic flick The Great Outdoors, starring John Candy and Dan Aykroyd. Those towering spruces and wide-open spaces create the perfect setting for a much-needed getaway, although we can probably do without the hijinx of mischievous raccoons and bears with long memories.
While the movie takes place at the height of the summer season, winter is just as appealing, perhaps even more so, considering here in the heart of the North Woods, a blanket of white makes everything that much more magical. That's why we're headed to the Brainerd area — Minnesota's bonafide winter playground!
Brainerd, Minnesota, and its surrounding lakes and woods first caught my attention a couple of years ago while searching for places that were within a day's drive from home. At less than 6 hours from my Manitoba, Canada, home, Brainerd is a welcome addition to my list of 3-day weekend destinations! It was our last destination before the pandemic hit in March 2020 and our first after it ended in April 2022 when Canadians were able to cross the border.
Located between Minneapolis and the Canadian border, Brainerd shares the spotlight with its sister city, Baxter, together providing retail therapy, hotels, attractions, and dining options in an urban setting. It's in the surrounding areas, like Crow Wing State Park, Nisswa, and Crosslake, that you'll find even more to do, like meandering cross-country ski trails, posh lodges with outdoor hot tubs, and plenty of places to drop a line and try your hand at ice fishing.
Winter dining takes on a luxurious appeal with the igloos at Grand View Lodge.
Photo credit: Grand View Lodge
1. Grand View Lodge
Let's face it. Not everyone is into the exhilaration that comes from frosty cheeks and tingling fingers. But nearly everyone can get excited about a hot tub, especially ones that are outside so you can breathe that fresh air while luxuriating in the warm water.
One glance at the exterior of Grand View Lodge near Nisswa, Minnesota, makes it a must-visit. Built in 1916 to house overnight guests interested in buying property on Gull Lake, the historic pine structure remains the focal point for resort life. It's a beauty! But perhaps the best feature of Grand View Lodge is its giant outdoor hot tub, which stays open year round. It's part of the NorthPark complex that includes massive indoor and outdoor pools.
In addition to the handful of historic rooms still available in the main lodge, Grand View has cabins, townhouses, villas, cottages, and suites in clusters around the original building — including modern rooms in a new boutique hotel called North. The massive lobby invites guests to lounge by the fire and grab a coffee, beer, wine, or light snack from Brew, the café on site.
Pro Tip: The Grand Breakfast buffet in the Heritage Room of the main lodge is easily one of the most impressive and delicious breakfast offerings I've experienced. When you book, look for specials that include this morning treat.
Get ready to reel in bluegills and crappy, plus prized walleye and feisty northern pike.
2. Ice Fishing
For those who don't live in northern climes, it's tough to believe that those sparkling lakes of the summer season freeze solid and still attract die-hard anglers. Ice fishing is huge in Minnesota, proven on the big screen in 1993's Grumpy Old Men, when feuding next-door neighbors and retirees John Gustafson and Max Goldman take their battles to the ice in Wabasha, Minnesota.
Book an outing with one of the guides in the Brainerd area, who will supply licenses, rods, tackle, bait, shack, and cleaning services. Get ready to reel in bluegills and crappy, plus prized walleye and feisty northern pike. The Brainerd Jaycees Ice Fishing Extravaganza happens every January on Gull Lake and draws more than 10,000 participants who compete for more than $200,000 in prizes. It's the world's largest charitable ice fishing tournament.
Pro Tip: If the thought of dropping a line down a six-inch hole in the ice and waiting, waiting, waiting is not your thing, take a stroll on the ice through one of the ice fishing villages. They make great photography subjects, with crusty characters and creative shacks.
3. Cross-Country And Downhill Skiing
There is no shortage of trails in the Brainerd area to slide and glide on a pair of cross-country skis. The Brainerd Nordic Ski Club maintains and grooms three trail systems in the area, and rental shops can get you set up with what you need for a day's outing. Many of the hotels and lodges also have their own groomed trails along with rentals to make things ultra-convenient.
For those who feel the need for speed, head to Mount Ski Gull in nearby Nisswa. Seven downhill runs with lifts plus a tubing hill make it a winter blast for one and all. If you're a bit rusty, book a refresher lesson on a Saturday or Sunday. And if you're 65 or wiser, skiing is free at Mount Ski Gull!
Pro Tips: While you're in Nisswa, take an afternoon of retail therapy, then refuel at Main Street Ale House (order the pickle wings and smoked sea salt fries) and The Chocolate Ox (grab some sea salt caramels and gigantic peanut butter cups).
Paul Bunyan Waterpark inside Arrowwood Lodge is considerably large with slides, a treehouse, hot tubs, a zero-depth entry pool, and an arcade.
Photo credit: Arrowhead Lodge
4. Brainerd's Waterparks
After they've had a snow-filled day, your crew might be looking for a change of pace, and Brainerd's waterparks are just the ticket. Keep them busy at one of three area parks.
Rapid River Lodge & Waterpark has a heated pool, lazy river, and two large water slides inside the newly renovated hotel. Paul Bunyan Waterpark inside Arrowwood Lodge is considerably larger, with slides, a treehouse, hot tubs, a zero-depth entry pool, and an arcade. The waterpark is free for hotel guests and open to the public for a daily fee. Holiday Inn Express & Suites guests enjoy access to the Three Bear Water Park with its lazy river, gigantic splash bucket, and a 300-foot body slide that is very fast!
5. Crow Wing State Park
Just south of Brainerd is Crow Wing State Park, once site of a bustling community of 500 people along the Crow Wing and Mississippi rivers. The town died when the railroad chose to cross the river at Brainerd. The park has meandering self-guided trails and historic sites, including one building from the original settlement as well as a reconstructed boardwalk just like the one that ran across the front of several stores.
The entire park is open for snowshoeing and winter hiking with lots of interpretive signage that tells the story of this once-thriving community.
Northern Pacific Center Railway in Brainerd
Photo credit: Sydney218 / Shutterstock.com
6. Northern Pacific Center
A 47-acre Northern Pacific railyard that got its start in the late-1800s Brainerd is now the Northern Pacific Center, home to event venues, restaurants, and shops, all housed in stunning historic buildings. An area called The Shoppes hosts a variety of local vendors, while freestanding stores like The Smokestack, a barbecue supply store, attract barbecue enthusiasts with grills from Pit Boss and GMG as well as a huge selection of brines, sauces, and rubs.
Pro Tip: Alissa of Native in The Shoppes combines Pendleton blankets with leather and cowhide to create stylish modern pieces, including a crossbody bag that includes fringed details and a canvas lining for a one-of-a-kind look.
7. The Gull Lake Frozen Fore Winter Weekend
The highlight of the winter season around Brainerd is the end-of-February extravaganza called the Gull Lake Frozen Fore Winter Weekend. Polar plungers can sign up for the Frozen Flop, where brave, swimsuit-clad souls hop into the water through a hole in the ice, all in the name of charity. (A heated tent party follows with an ice bar, food, and live music.)
For duffers craving the summer sport, there's a winter version here, where tennis balls are used instead of golf balls and snowmobiles take the place of golf carts. It's a great way to see what nine different area courses look like during their winter sleep.
Five Rocks Distilling Company makes Cold North Vodka, Lizzy's Gin, and maple bourbon among others.
Photo credit: Five Rocks Distillery
8. Area Breweries And Distilleries
They say there's something in the water that makes the Brainerd area a hub for cozy tap rooms and classy cocktail bars. Take your pick from more than a half-dozen breweries including Big Axe Brewing Co., Snarky Loon Brewing Co., and 14 Lakes Craft Brewing Company.
Five Rocks Distilling Co, makes Cold North Vodka, Lizzy's Gin, and maple bourbon, for starters. Plus they have a cocktail room to serve up some innovative sips, including The Dirty Pickle and the grapefruit rosemary gimlet. Keep an eye out for special events where the distillery partners with local restaurants for cocktail dinners.
For more on the area, check out:
10 Adorable Small Towns To Visit Near Brainerd, Minnesota
13 Gorgeous Waterfalls To Explore In Minnesota
How To Spend A Perfect Long Weekend In Grand Marais, Minnesota
Shel Zolkewich View Full Profile
A journalist by trade and an adventurer at heart, Shel Zolkewich writes about travel, food, and the great outdoors from her not-so-fancy farmhouse in rural Canada. She's an avid forager, amateur naturalist, hunter, angler, gardener and soup maker, beekeeper, chicken lady and duck wrangler, bird nerd, and reader who hopes to one day write the great Canadian novel.
MinnesotaVisit Minnesota's Award-Winning Ice Castles This Winter — Everything You Need To Know
Minnesota8 Fabulous Reasons Winter Is A Perfect Time To Visit The Brainerd, Minnesota Area
Minnesota9 Local Shops For Perfect One-Of-A-Kind Gifts In The City Known For The Mayo Clinic
Minnesota10 Serene Duluth Vacation Rentals Near Lake Superior
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\section{Results}
In Fig.~\ref{fig:rho} we show normalized electrical resistance $R$ and heat capacity $C$ vs. temperature $T$ for several phosphorous concentrations $x$ (see Supplementary for magnetic susceptibility $\chi(T)$). The Kondo lattice behavior of the electrical resistance (i.e., non-monotonic temperature dependence below room temperature with a resistive peak near 80 K) for $T$ $\geq$ $T_0$ is unaffected by phosphorous substitution for $x$ $\leq$ 0.035 (Fig.~\ref{fig:rho}a), suggesting that the strength of the hybridization between the $f$- and conduction electron states does not change much in the range 0 $<$ $x$ $<$ 0.035. At lower temperatures, the hidden order transition temperature $T_0$ and the size of the anomalies in $R$ and $C$ associated with it are monotonically suppressed with increasing $x$ (Figs.~\ref{fig:rho}b,d). There is excellent agreement between the values of $T_0$ as extracted from $R$, $C/T$, and $\chi$, indicating that disorder effects are negligible. We find no evidence for hidden order in $x$ = 0.035 for $T$ $>$ 20 mK, showing that there is a quantum phase transition from hidden order to a paramagnetic correlated electron metal between 0.028 $<$ $x$ $<$ 0.035. However, the data does not preclude the possibility of a broad fluctuation regime around the quantum phase transition. The paramagnetic region subsequently extends up to $x$ $\approx$ 0.25 (a factor of ten larger $x$) where correlated electron antiferromagnetism appears.~\cite{note0} Unlike for other tuning strategies,~\cite{mcelfresh_87,amitsuka_99,jeffries_08,hassinger,kanchanavatee_11,kanchanavatee_14,das,amitsuka_88,dalichaouch_90,dalichaouch_89,bauer_05,butch_10} magnetism is distant from hidden order in this phase diagram.
The resistive superconducting transition temperature $T_{c,\rho}$ (Fig.~\ref{fig:rho}c) initially increases with $x$ and subsequently vanishes, with no evidence for bulk superconductivity above 20 mK for $x$ $>$ 0.02. While the value of $T_{c,C}$ extracted from heat capacity is in close agreement with $T_{c,\rho}$ for $x$ $\leq$ 0.01, these values separate for $x$ $=$ 0.02 where $T_{c,\rho}$ $>$ $T_{c,C}$. We note that a similar discrepancy between $T_{c,\rho}$ and $T_{c,C}$ is seen for high quality single crystal specimens of the correlated electron superconductor CeIrIn$_5$ and may be an intrinsic feature of the unconventional superconducting state.~\cite{petrovic} From both $\rho$ and $C/T$, we find that for $x$ $=$ 0.028 there is a transition into the hidden order state near 13.5 K, but no bulk superconductivity down to 20 mK. There is no evidence for superconductivity in $x$ $=$ 0.035 for $T$ $>$ 20 mK.
These results are shown in Fig.~\ref{fig:phase}a, where the superconducting region is enclosed by hidden order in the $T-x$ phase space. Over this concentration range, the ground state is mainly tuned by electronic variation, as indicated by the comparably small changes in other intrinsic and extrinsic factors. The lowest residual resistivity ratio ($RRR$ $\approx$ $\rho_{300K}/\rho_{0}$) for the specimens reported here is $RRR$ $=$ 10 (see supplementary), which is comparable to typical values for parent URu$_2$Si$_2$ where $T_0$ and $T_c$ depend weakly on $RRR$ in the range 10 - 500.~\cite{baumbach} The high crystal-chemical quality of these specimens is further highlighted by the observation of quantum oscillations in electrical transport measurements (see Supplementary), indicating that disorder effects are negligible. The unit cell volume and bond angles are also unchanged by phosphorous substitution (Fig.~\ref{fig:phase}e), in contrast to some previous studies.~\cite{kanchanavatee_11,kanchanavatee_14}
Having established the $T-x$ phase diagram (Fig.~\ref{fig:phase}a), we now discuss the region beneath the hidden order phase boundary, where unexpectedly rich behavior occurs. As evidenced by the jump size in $C_{5f}$/$T$ at $T_c$ ($\Delta$$C_{5f}$/$T_c$) (and the transition width), which evolve though a maximum (and a minimum) between 0.006 and 0.01, respectively (Figs.~\ref{fig:rho}e and ~\ref{fig:phase}b), phosphorous substitution non-monotonically enhances the thermodynamic signature of the superconductivity. Here, $C_{5f}$ refers to the heat capacity following subtraction of the nonmagnetic ThRu$_2$Si$_2$ lattice term, as described in the supplementary section. The non-monotonic behavior is reflected in the behavior of $S_{5f,Tc}$($x$) (Fig.~\ref{fig:phase}c), which goes through a maximum near $x$ $=$ 0.01. In contrast, for the hidden order $\Delta$$C$/$T_0$ and $S_{5f,T0}$ are monotonically suppressed with increasing $x$ (see Supplementary). Further evidence for non-monotonic evolution of the superconductivity is provided by the doping evolution of the ratio $\zeta$ $=$ $\Delta$$C_{5f}$/$\gamma$$T_c$($x$), (Fig.~\ref{fig:phase}b) which, for conventional superconductors, is a numeric constant $\zeta_{\rm{BCS}}$ $=$ 1.43. By using $C_{5f}/T$ at $T_c$ for the value of the normal state $\gamma$ we find that $\zeta(x)$ evolves non-monotonically through a maximum value of 1.2 at $x$ $=$ 0.01 (Fig.~\ref{fig:phase}b), where the $x$ $=$ 0 value is near 0.7 as previously reported.~\cite{lohneysen_2007_1,pfleiderer_2009_1} This suggests unconventional superconductivity where the coupling strength may evolve through a maximum.
Magnetoresistance data (Fig.~\ref{fig:RH}) show quantum oscillations, emphasizing the high quality of these specimens (see Supplementary). Similar to the parent compound, the upper critical field $H_{c2}$ (Fig.~\ref{fig:RH}d) is highly anisotropic at all $x$ and follows $H_{c2}(\theta)$ $\propto$ $1/\sqrt{g^2_c\cos^2\theta + g^2_a\sin^2\theta}$ dependence (where $\theta$ is measured from the $c$-axis), suggesting that the upper critical field is Pauli limited.~\cite{altarawneh_12} While there is little $x$ dependence in $g_c(x)$, the $a$-axis $g$-factor $g_a(x)$ significantly decreases before the superconductivity is destroyed near $x$ $\approx$ 0.028 (Fig.~\ref{fig:RH}d). We note that the actual value of the $g$-factor in the $a$-direction may differ significantly from the fitted $g_a(x)$ because of the increased importance of diamagnetic effects as the field rotates into the ab-plane. It remains to be seen whether these trends are consistent with recent theoretical proposals such as ref.~\cite{chandra}.
Magnetoresistance measurements further highlight the non-monotonic evolution with $x$ of the superconductivity and the underlying metallic state. Fig.~\ref{fig:RH}e demonstrates Kohler scaling for $H$ $<$ 9 T applied parallel to the $c$-axis at all dopings,~\cite{kohler} suggesting that the magnetotransport is controlled by the same (temperature dependent) relaxation time as the zero field resistivity. At each composition in the range 0 $<$ $x$ $<$ 0.028 the normalized magnetoresistance is described by a distinct $f_x(h)$ (where $h$ $=$ $H/\rho(0,T)$), which itself evolves with doping. Notably, the function $f_x(h)$ evolves non-monotonically with $x$ with a maximum near $x$ $=$ 0.006 (Fig.~\ref{fig:phase}d). The maximum in the value of $f_x(h)$ nearly coincides with the maximum in the thermodynamic signatures of the superconductivity inside the hidden order phase.
\section{Discussion}
Although much of the recent excitement surrounding URu$_2$Si$_2$ has focused on the uranium electronic structure and the symmetry of the hidden order phase, a more fundamental question is the degree to which the $f$-electrons can be treated as being localized and the role of quantum fluctuations. The continuity of experimental information extracted from well-developed applied pressure ($P$) and chemical substitution ($x$) series has proven essential to disentangle such effects in other correlated systems including high temperature superconducting cuprates, pnictides, and heavy fermion compounds. To some extent, URu$_2$Si$_2$ has also benefited from such studies. For example, pressure drives a first order phase transition from hidden order into antiferromagnetism near $P_c$ $=$ 0.5 GPa, with a simultaneous evolution of the Fermi surface,~\cite{mcelfresh_87,amitsuka_99,jeffries_08,hassinger} but the resulting insight is limited by the small number of pressure-cell compatible experimental probes. Ruthenium site substitution with Fe and Os produces $T-x$ phase diagrams that closely resemble the $T-P$ phase diagram,~\cite{kanchanavatee_11,kanchanavatee_14,das} but the information gained from these series is constrained by strong disorder. Moreover, ruthenium site substitution is particularly disruptive, as evidenced by the rather different phase diagrams resulting from Rh and Re substitution studies where the hidden order and superconductivity are rapidly destroyed.~\cite{amitsuka_88,dalichaouch_90,dalichaouch_89,bauer_05,butch_10} To understand the complex interplay between different phenomena in this compound a more ``gentle" tuning scheme has long been desired, which could provide access to the physics of URu$_2$Si$_2$ in clean single crystals at ambient pressure. In this context, ligand site substitution in URu$_2$Si$_2$ is an obvious target for investigation.
While in many theoretical scenarios for hidden order in URu$_2$Si$_2$ the U-5$f$ electrons are treated as having mostly fixed valence in a particular atomic crystal field state,~\cite{chandra,blumberg} it is widely believed that they actually have a dual character: i.e., the dynamic nature of the U-5$f$ valence electrons allows for fluctuations between different configurations. However, measurements of the pure compound so far give no insight into the role of these fluctuations in producing hidden order and superconductivity. The rapid changes in the hidden order and superconductivity in our measurements confirm the importance of the itinerant electrons. This is further supported by the observation of weak observation of Kondo lattice physics \cite{hewson} (which tracks the hybridization strength between $f$- and conduction electrons) and strong evolution in the $g$-factor anisotropy (which is a marker for local moment character). Together these results point towards this series as a platform for unraveling the relationship between local and itinerant behavior in URu$_2$Si$_2$.
The stark contrast in the evolution of the hidden order and superconductivity in URu$_2$Si$_{2-x}$P$_x$ (monotonic vs. non-monotonic) further suggests that hidden order, although necessary, is not directly responsible for the superconducting pairing. Instead, the observation of a superconducting dome completely contained inside the hidden order region may indicate the presence of an independent collapsing phase boundary within the hidden order state, as is ubiquitous in other unconventional superconductors. This scenario is reinforced by the observed non-monotonic evolution of the normal state electrical transport, which is also common in correlated electron systems~\cite{lohneysen_2007_1,pfleiderer_2009_1,loram,walmsley} where the strongest deviation from Fermi liquid behavior is seen near the critical point. Alternatively, the independent evolution of hidden order and superconductivity may suggest several competing order parameters in the hidden order phase, as evidenced by electronic Raman ($A_{2g}$),~\cite{blumberg} elastoresistance ($B_{2g}$),~\cite{riggs} resonant ultrasound ($B_{1g}$),~\cite{yanagisawa,brad} and spectroscopic measurements ($E_{g}$).~\cite{wray} These studies should be extended into ligand site substituted URu$_2$Si$_2$.
Finally, the existing theoretical landscape focuses on $f$-electron physics with no guidance regarding the specificity of the transition metal ion. It is especially puzzling that hidden order and superconductivity are only observed in the U-Ru duo. Examination of silicon site substituted transition metal analogues (U$T_2$Si$_{2-x}$P$_x$, $T$ = transition metal), which can now be synthesized using molten metal flux growth,~\cite{note2} may be particularly illuminating in addressing the universality of hidden order and superconductivity in this fascinating uranium compound.
\section{Methods}
\textbf{Single crystal synthesis using molten indium flux} Single crystals of URu$_2$Si$_{2-x}$P$_x$ were grown from elements with purities $>99.9$\% in a molten In flux, as previously reported.\cite{baumbach} The reaction ampoules were prepared by loading the elements into a 5 cm$^3$ tantalum crucible in the ratio 1(U):2(Ru):2(Si):22(In). The crucible was then loaded into an alumina tube spanning the bore of a high temperature horizontal tube furnace. Argon gas was passed through the tube and a zirconium getter was placed in a pot before the tantalum crucible in order to purify the argon at high temperatures. The crucible was heated to 500 $^{\rm{o}}$C at 50 $^{\rm{o}}$C/hr, dwelled for 5 hours, heated to 600 $^{\rm{o}}$C at 50 $^{\rm{o}}$C/hr, dwelled for 5 hours, and heated to 1450 $^{\rm{o}}$C at 70 $^{\rm{o}}$C/hr. The dwells at intermediate temperature are intended to allow the phosphorous to completely dissolve into the indium flux without producing a dangerous high vapor pressure. The crucible was then cycled between 1450 - 1400 $^{\rm{o}}$C at 100 $^{\rm{o}}$C/hr ten times. Finally, the furnace was turned off and quickly cooled to room temperature. The indium flux was subsequently removed using hydrochloric acid, to which the URu$_2$Si$_{2-x}$P$_x$ crystals are insensitive. This technique produced single crystal platelets similar to the ones previously reported.
\textbf{Bulk thermodynamic and electrical transport measurements} Heat capacity measurements were performed for mosaics of single crystals using the He3 option in a Quantum Design Physical Properties Measurement System for temperatures 400 mK $<$ $T$ $<$ 20 K. Magnetization $M(T,H)$ measurements were carried out for mosaics of single crystals for temperatures $T$ $=$ 1.8 - 350 K under an applied magnetic field of $H$ $=$ 5 kOe applied parallel to the $c$-axis using a Quantum Design Magnetic Property Measurement System. Magnetic susceptibility $\chi$ is defined as the ratio $M/H$. Zero magnetic field electrical resistance $R$ was measured using the He3 option in Quantum Design Physical Properties Measurement System for temperatures 400 mK $<$ $T$ $<$ 300 K. Several individual crystals were measured for each concentration, which revealed a high degree of batch uniformity. The angular dependence of the superconducting upper critical field was measured using the superconducting magnet (SCM-1) dilution refrigerator system at the National High Magnetic Field Laboratory for $H$ $<$ 18 T and $T$ $=$ 20 mK. Additional magnetoresistance measurements were performed at the National High Magnetic Field Laboratory, Tallahassee, up to magnetic fields of 35 tesla and at $T$ $=$ 50 mK.
\section{Acknowledgements}
This work was performed at the National High Magnetic Field Laboratory (NHMFL), which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490, the State of Florida and the DOE." A portion of this work was supported by the NHMFL User Collaboration Grant Program (UCGP). TAS and SC acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Heavy Elements Chemistry Program, under Award Number DE-FG02-13ER16414.
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It's bluegrass as you haven't heard it before. Since a group of students at the University of North Carolina at Chapel Hill started Steep Canyon Rangers in 2000, the group has been expanding and redefining the parameters of bluegrass. The genre-defying band has brought elements of pop, country, folk and folk rock to music that is very much their own. In 2013, the group won a Grammy for Best Bluegrass album for their solo work, Nobody Knows You.
The band consists of Woody Platt on acoustic guitar and lead vocals; Graham Sharp on banjo and harmony vocals; Mike Guggino on mandolin and harmony vocals; Nicky Sanders on fiddle and vocals; Mike Ashworth on box kit, cajon and vocals; and Barrett Smith on upright bass. They began performing in festivals and recording. Their fame has spread since 2009, when they began collaborating with actor, comedian and banjo player Steve Martin, and their 2012 album with Martin, Rare Bird Alert, featuring guest appearances by Paul McCartney and the Dixie Chicks, was nominated for a Grammy. On July 4, 2011, Steep Canyon Rangers joined with Steve Martin for "A Capitol Fourth" on the West Lawn of the U.S. Capitol. That same year, the group and Martin were jointly named Entertainers of the Year at the International Bluegrass Music Association Awards. In 2017, the group and Martin, with Edie Brickell, recorded The Long Awaited Album and joined Martin and Brickell on tour.
A hallmark of Steep Canyon Rangers has been their ability to produce two distinct bodies of work, with Martin and on their own. They tour 150 dates a year and are considered one of the hardest working bands around. They have worked with a wide range of producers who have brought something new to the table each time the group steps into a studio. In 2017, they arrive at Fidelitorium Recordings in Kernersville, N.C., a facility owned by Mitch Easter, producer for R.E.M, among others. They were surprised to discover the plans their producer, three-time Grammy-winner Joe Henry, had for them. They were going to record Out in the Open in classic fashion, with all six members singing and playing in a room with no overdubs. In an interview, Platt admitted that they were nervous.
The process allows the group to record the album completely live, tracking a dozen songs in 3 ½ days. Tracks were jammed, rehearsed, played and recorded with no time wasted.
"There's no sleight of hand," Graham Sharp said. "It may not be straightforward, but it's honest." "What you lose in perfection, you gain in energy and authenticity," Platt added.
That pretty well sums up Street Canyon Rangers, too.
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"redpajama_set_name": "RedPajamaC4"
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**Contents**
_Title Page_
_Dedication_
_Epigraph_
_Eastern Europe—early seventeenth century_
_The Song_
**1.** Deep in the Woods
**2.** Slivovitz and Snow
**3.** The Suicide's Burial
**4.** The Goose
**5.** St. Andrew's Eve
**6.** The Dulcimer's Melody
**7.** Sheep and Wolves
**8.** The Shadow Queen
**9.** The Eternal Return
**10.** Refusal
**11.** Visitors
**12.** Closer
**13.** And Closer
**14.** Creeping
**15.** The Waters of Chust
**16.** Agnes
**17.** The Wedding of the Dead
**18.** At the Threshold
**19.** Turnings
**20.** Hands in the Dark
**21.** Threads
**22.** Calling
**23.** Things to Cover Our Dead
**24.** The Hut
**25.** The Winter King
**26.** Escape
**27.** The Island
**28.** The Dream of the Queen
**29.** Ancestors and Hostages
**30.** The Elders
**31.** Village Talk
**32.** Stillness
**33.** Tomas
**34.** The Camp
**35.** The Approach
**36.** Ordeal
**37.** The Sword
**38.** The Song of the Miorita
**39.** Resurrection
**40.** A Perfect Shade of Green
_Author's Note_
_About the Author_
_Also by Marcus Sedgwick_
_Copyright_
_For my father_
Once, they were as plentiful
as the blades of grass in the meadow
**Eastern Europe—early seventeenth century**
_The Land Beyond the Forests_
**The Song**
_There is a land beyond the forests. A land so beautiful that as you stand at the edge of the trees and gaze across the pastures to the snow-brushed mountains, you know that heaven is surely but a step away. From this land comes a song, and from the song comes a story. A story of murder._
_Down from the mountains one day came three shepherds. They had been in the high pastures for weeks with their flocks and were glad to be heading home, but in the minds of two of the shepherds, there was death._
_The third shepherd, the youngest of the three, and maybe the richest and maybe the most handsome, knew nothing of this, until one of his lambs, the smallest lamb, that he had saved from dying in a late spring snow, came to him and warned him of the plot to kill him._
_Now the shepherd looked sadly at his lamb, and said:_
_"If this is true, then I am doomed to die. But, my faithful creature, do this for me. When they have killed me, tell them to lay my bones somewhere close by, and bury my pipes with me, so that when the wind blows, it will play a tune and my sheep may come near, and my dogs too._
_"Tell them not that I am dead, but instead that I went to marry a princess from a distant land. Tell them how a star fell at my wedding, tell them how the sun and moon came down to hold my bride's crown, tell them how the trees were my guests, and the mountains my priests. How birds were my fiddlers, and stars my torchlight._
_"But if one day you meet a white-haired woman, my mother, my old mother, tell her simply that I went to marry a princess, there on Heaven's doorstep."_
_And so it all came to be._
**1**
**Deep in the Woods**
When he fell for the fifth time, when his face plunged into the deep snow, when his hands burnt from the cold but he didn't care, Radu the woodcutter knew he was going to die. Somewhere behind him in the darknesses of the forest he could hear the man who had attacked him. He was scared now, almost too scared to move, almost too cold to run anymore, but still he knew something was wrong. Something that should not be.
He got up and stumbled on desperately, sending snow flying in little spurts. Even here among the thickness of the trees it lay heavily on the ground, whisked and funneled by the east wind into strange hills and troughs, like white beasts lurking at the foot of the birches.
Radu looked behind him, but could see nothing. Nothing but the vast unfathomable forest. It was said you could ride from Poland to Turkey and never leave the trees behind, but he knew that wasn't true. Nothing could be that big! Not even the Mother Forest.
He stopped for a moment, listening hard, but all he could hear was his own panting as he sucked air into his painful chest. He no longer knew where he was, though the forest had been his home all his life. His hut and his village were far away. He looked around, straining to recognize anything, but all he saw were a hundred thousand silver birch trees.
A branch cracked, and with horror Radu's eyes snapped back to his pursuer. Now that Radu saw him again, he knew what was wrong.
"In the name of Jesus and the Forest..."
The words fell dead in the softness of the snow, but even as they did Radu turned and began to run, lurching wildly from tree to tree. His right hand left a smear of blood on the paper bark of a birch, but that wound was irrelevant now. It was such a short while since he'd been cutting wood with his axe. The axe that lay somewhere in the snow, its blade stained with blood, already frozen. His blood.
He hit another two trees, but barely noticed, and suddenly he realized where he was. Close to Chust, where his fellow woodcutter Tomas lived in a hut outside the village.
For a fleeting moment a flame of hope ignited in his heart. He had run fast, the village was only a short way through the trees, and he could no longer hear his attacker behind him.
But then Radu rounded a tree and ran straight into him.
The man was not tall, but he was fat. Bloated. His skin was as white as the trees around them. There was dried blood at the corners of his shriveled mouth. It had taken Radu all this time to recognize him.
Radu took a step backward, his fur boots brushing through the snow. He tripped over an unseen root, but kept his feet. He lifted a hand and pointed at the man.
"But Willem. You're dead!"
The man lunged forward and shoved his hand like a knife into Radu's chest, feeling for his heart.
"Not anymore," he said.
And now it was Radu who fell dead in the softness of the snow.
**2**
**Slivovitz and Snow**
Peter trudged behind his father toward Chust, shivering as he went. Their hut lay a little way behind them, outside the village itself. St. Andrew's Eve was still a few days off, and the snow was strong already. It would be a brutal winter. Through the cold Peter could smell his father; even the biting wind could not rid him of the constant reek of slivovitz and beer.
"Did you know Radu well, Father?" Peter said, simply so that there was something to say. His father didn't reply, and Peter knew the answer anyway. They didn't know anyone well—until they had come to Chust they had never stayed in one place long enough to know anyone at all. But Peter was aware that his father had helped Radu, the woodcutter from Koroceni, once or twice in the last year. Sometimes even the most solitary of woodcutters needed help felling a large tree.
The edge of the village was in front of them.
"Hurry," said Tomas. "They won't wait for us."
"They can't be starting," said Peter. "I can't hear the church bell."
His father spat into the snow, but didn't look around.
"There won't be bells at a suicide's funeral."
He walked into the village, through a small gate in the rickety birch paling that marked the boundary. The fence, no more than knee-high, and capped by a ragged thatch, was designed to stop chickens from wandering too far. It ran right around the settlement, marking its perimeter; apart from a few fields, everything beyond it was forest. In places along its length, as here, were gates with little roofs.
Peter hesitated at this gateway. It wasn't just that they weren't liked in the village; there was more to it than that.
"Suicide?"
Peter ran after his father, and caught up quickly. The ground was a mess of frozen mud and slush, and his father was unsteady on his feet, as usual.
"Be quiet," Tomas said, glaring at his son. He nodded at the huts, and Peter understood that he should not have spoken. That was all right. Peter was used to silence, used to keeping his own company; thanks to his taciturn father, most conversations Peter had took place in his own head.
Two sour-faced old women stood in the shadow of a low doorway. They spoke under their breath to each other and stared at Tomas and his son—a heavy man who looked older than he probably was, and his strong, young boy.
Peter knew they were not liked, and the village had little to offer. There was something bleak and unsettling about the place, something almost menacing, though Peter could not have put it into words, and yet for all that Tomas seemed content to stay. And in truth, Peter was happy to stay too. They finally seemed to have put down some roots after years and years of moving, and besides, there was Agnes.
They hurried on, down the slight hill that led to the area laughingly called "the square," as if this were some great city in the south and not a godforsaken village in the middle of nowhere. Chust was home to no more than two hundred people, but here in the center there were houses in place of huts; a few of them had two storeys. As they went, Peter kept an eye out for Agnes, but it was not a day to be abroad unless you had business to attend to. They passed the end of the street where she lived with her mother, still in mourning after her husband's death. Briefly Peter slowed his pace, hoping for some sign of Agnes, but there was none.
Slipping from the far corner of the square, a small track squeezed between two of the larger houses in the village, the priest's house and the feldsher's house. That way led to the church, which lay on a rise beyond. Peter could see its sagging wood-tiled roof with the onion-dome tower on its back, halfway along, like a boy riding a pig, but he was surprised when his father strode away toward the other side of the square.
He paused, and then understood. He should have known better. Radu was a suicide; there would be no bells, and there would be no holy ground for him either. Peter hurried on.
His father was nearly at the far side of the village.
"They think he killed himself?" Peter asked.
Tomas said nothing.
"Father?"
Tomas stopped for a moment and looked, not at his son, but somewhere away over his shoulder.
"He was found hanging from a tree by a rope round his neck. So he killed himself. Wouldn't be the first lonely woodcutter to have done that."
Something occurred to Peter.
"Why isn't he going to be buried in his own village?"
His father grunted.
"That type of death. They wanted nothing to do with him. Said he died on Chust land so we could deal with him."
"And we agreed?" asked Peter.
"Who is 'we'? There is no 'we' here," Tomas said abruptly. Then he sighed. "There was no choice. It was that, or leave him to the wolves. And anyway, the Elders commanded it."
They were out of the village now, and through the trees they could see a few people gathered in a small clearing.
Peter thought about Radu, about how he might have died. His father told him he was a dreamer, but Peter couldn't dream what might have happened to Radu. It was not the stuff of dreams, it was the stuff of nightmares.
"But Father," Peter whispered, "you said Radu's chest was burst, that his heart was pierced."
"What of it?"
"Well, he can't have done that to himself and then hung himself from a tree."
"So it must have happened afterward."
"You mean someone else did it to him after he was dead? Who would do that? Why?"
Tomas shrugged. "The wolves...?"
Peter was about to reply, but could tell his father was being deliberately obtuse.
"Listen, Peter. If a man is hanging from a tree by a rope, he killed himself. If Anna told them that's what happened, then that's what happened. Let it lie!"
Peter was not satisfied, but said no more. There was something troubling his father, he knew.
They made their way toward the meagre funeral party. There was Daniel, the priest; and Teodor, the feldsher—half doctor, half sorcerer. Radu might have been only a woodcutter, and not even from the village, but still two of its most important inhabitants had come to bury him. Peter wondered why. Why were both of them here? He knew they didn't always get on. People were as likely to visit Teodor with spiritual needs as Daniel, and just as likely to pray for their health with Daniel as visit the doctor. Each man knew he had to tolerate the other. An uneasy alliance.
A little way away stood the village sexton, an old man with strong arms but few teeth. It was clear he wanted nothing to do with the affair, and having struggled to dig a shallow hole in the frozen ground, he leant on the top of his tall spade, sucking his gums, peering out from under a wide-brimmed black felt hat.
Snow continued to trickle down around and about as Tomas nodded greetings to the others.
Outsiders were never welcome, even though this father and son had taken to their work well enough. They were a strange pair. The father was a drunk, everyone knew that, but there was an air about him. Something in the way he held himself. He was fat from drink, his face flushed and his eyes milky, but he still had a head of strong black hair.
The son was a young man, really, new to the game. He had even darker, thicker hair, and his skin was smooth and brown, as if he was from somewhere in the south. His eyes were rich and dark brown, like Turkish coffee, but he was nervous, for all his young strength, and there was something about him that made him seem more refined than his father. Few of the villagers had ever wondered what might have happened to the boy's mother, though it must have been from her that his refinement came.
Peter was absorbed in his own thoughts. Wolves couldn't have done that to Radu's chest after he hung himself from the tree. It didn't make sense. Someone must have stabbed him through the heart with great force, and then hung his body in the tree afterward.
But why? Most murderers tried to conceal their victims' bodies. Why display Radu's body instead?
To Peter, it seemed like a warning, a warning that death was walking in the woods.
And Peter was right.
**3**
**The Suicide's Burial**
Here came the body, uncovered on the back of Florin's cart, pulled by an ox rather than a horse. Peter had seen this in other villages, the locals believing that the horse was too pompous an animal to be trusted at funerals, and was flighty too, inclined to kick and bridle, disturbing the dead person's soul.
No one wanted that.
Besides, oxen were dependable, and noble in their own way.
Florin was a farmer, but there was little farming to be done in the winter and he had been told to fetch Radu's body. This didn't please him, but Anna had instructed him and he could not refuse. She was the most fearsome of the _kmetovi,_ the village elders, a terrifying lady of unknown age who commanded total obedience. She had assumed control of the village after her husband died, and no one had dared to dispute this state of affairs. Her husband might have been a ruthless man, but even so he was just a shadow of Anna. So Florin had made his way to the woodshed behind the church, where Radu had been put, though Anna herself was having nothing to do with the burial.
There was something that did not impress the villagers favorably. Radu's body had lain unguarded in the shed since he had been found. Anything could have happened to it. A cat might have jumped across it, and everyone knew what that could mean. But then Radu was already a suicide, so maybe there was no hope for him now anyway. His future was already in great danger.
Florin walked on one side of the head of his ox, and Magda, his old wife, walked on the other. Peter was surprised she had come, but not surprised to hear her singing. She sang the song that was always sung whenever anyone died, or was married, or, indeed, when anything important happened at all. Peter had heard it many times, in all the other places they had lived. It was called the Miorita. The Lamb.
_"By a rolling hill at Heaven's doorsill,_
_Where the trail descends to the plain and ends,_
_Here three shepherds keep their flocks of sheep."_
As Peter listened to the song, his mind began to drift. When he was a child he had been fascinated by the song's story—the little lamb that talks to its faithful master, the murderous shepherds, the princess. The mother, who will wait in vain for her son to return. Peter had never known his mother, but though he tried very hard to feel something of a life that never happened to him, nothing came. Later, as he grew up, he thought about the story in more detail, and came to think it baffling and stupid.
Peter's dreams were shaken from him by a snort from the ox.
Radu had arrived.
There was little ceremony. Peter's father helped Florin lift the body from the cart, Daniel mumbled some words from the Bible, and the sexton glared from underneath his hat. Peter watched, disturbed by the brevity of it all. Was there really so little to celebrate in a life that would soon be forgotten forever? He gazed at Radu's face. He had seen dead bodies before, everyone had, but the look that was literally frozen on Radu's face shook him. It was a mix of shock and horror—and incomprehension. Peter shuddered, and wished that he and his father were back in their hut by the stove.
There was no coffin. The men lowered the body into the hole. But then something strange happened. Now in the grave, Radu was turned over, so that he lay face down. This wasn't something Peter had seen before.
Florin had wheeled his ox around, and he and Magda began to trundle away, both riding in the cart. Peter turned to see Teodor step forward. Teodor untied a cloth bundle that he had been holding all the while, and a clutch of twigs fell into the snow. Just before the sexton started to pile clods of soil over Radu, Teodor placed the twigs on and around his body. They were short, but stout, thick with long sharp thorns. Peter knew they were hawthorn, and he glanced at his father for some explanation. But Tomas's lips were tightly drawn.
The funeral was over, and each party went their own way back to the village.
In the square, Tomas clutched his son's arm. It was only midway through the afternoon and already the light was failing. Peter's mind was full of questions, about the funeral, about why they had attended it at all. He'd been astonished when his father said they would go, but maybe it was the right thing to do. It was just that it was a long time since Tomas had done the right thing.
Tomas shook his shoulder.
"I'm tired. Let's go home, Son."
Peter smiled.
"Lean on me, Father."
Tomas draped his arm around his son.
"I think there's some slivovitz left."
The smile slipped from Peter's face, but as they made their way, he dutifully supported his father's weight.
"Why did they turn him face down?" Peter asked.
Tomas said nothing.
"What were the thorns for?"
"They're not for anything," Tomas snapped, pulling away from his son. "They're simple, superstitious people here. Don't take any notice of their foolishness."
"But—"
"But nothing, Peter. We have wood to cut. And plum brandy to drink."
And Peter knew, as so often, which of them would be cutting wood and which of them drinking brandy.
**4**
**The Goose**
Dusk fell with the snowflakes as father and son made their way home. Peter was as tall as his father now, and certainly as strong. Maybe Tomas had once been a powerful man, but Peter could not remember that time. Tomas did less and less work, and relied more and more on Peter to keep them fed. As far back as Peter could recall, Tomas had drunk. Once upon a time it must have been different. Peter's mother had died giving birth to him. Tomas had found a wet nurse, but had brought Peter up himself ever since the child could walk and talk. Maybe he hadn't been able to afford the nurse anymore, but it seemed he had wanted the woman out of their lives as soon as possible. Since then it had been just the two of them.
On Peter's fifth birthday Tomas had given him a clasp knife. Not a toy, but a well-made and useful tool.
"Time you learnt to use one," Tomas said.
Peter had watched, entranced, as his father took an off-cut of a branch and quickly carved a small bird for him. A goose.
"It's a good one," Tomas said. "Sharp."
"Yes, it's a good one," his little son had echoed, laughing, though it was not the knife he meant, but the slender little goose, the very image of the birds that he loved to gaze at as they flew overhead.
Later, Tomas taught him to read, and that wasn't the action of a drunkard, nor even a soldier, but once Tomas had belonged to a very different kind of family. Now the drink seemed to possess him, and it cost Peter a lot of effort chopping logs to buy a bottle of slivovitz or rakia.
As they came within sight of the hut, Peter could see the birch smoke trailing up from the chimney, gently twisting into ghostly shapes in the dusk, drifting away and spreading like mist through the treetops.
Peter smiled. The fire was still alight; the hut would be warm.
The hut stood in a strange position. The river Chust, from which the village took its name, forked in two here, as it snaked through the woods. With deep banks, the river had spent ten thousand years eating its way gently down into the thick, soft, dark forest soil. Its verges were moss-laden blankets that dripped leaf mold into the slow brown water. But at a certain point in its ancient history, the river had met some solid rock hidden in the soil, and had split in two. It was at the head of this fork that the hut stood.
Just over a year ago, in late autumn, Tomas and Peter had been traveling again when they'd heard there was a need for a woodcutter in Chust. They'd been moving from village to village, always heading as far from civilization as it seemed possible to go, and ever deeper into the vast forest. Tomas was pleased, and they took the job. There was a perfectly good, large hut on the edge of the village, but Tomas had insisted they build a new one of their own. Peter was used to such eccentricities, and he merely bent his back to the axe to cut the trees to make the planks for their new home.
They laid a rough bridge of two halved tree trunks to cross to the middle of the fork, and began to build.
Winter was coming on by the time they finished the hut, with a stable on one side and a toolshed on the other. Then Peter started to cut wood to earn their keep, but Tomas got his spade out.
"What's that for, Father?" Peter asked, but his father, as so often, replied only with actions.
He surveyed the hut from the very tip of the river fork. Then he strode around the sides of the hut's single storey, inspecting it from every angle.
Peter leant on his axe and watched his father from across the river, where they had decided to make their timber yard.
Tomas stood at a point twenty paces from the front of the hut, in the exact center between the two arms of the river. He swung his spade from his shoulder, thrust it into the spongy soil, and began to dig.
Peter shook his head and went back to work. They had promised the _kmetovi_ deliveries of chopped birch a week earlier, and they had already aroused suspicion by deciding to live outside the village. Father had tried to explain that it made more sense for them to live closer to their work, but that sort of logical explanation impressed no one in Chust.
After an hour, Peter straightened his back and looked across to his father. Tomas had by now dug a deep but narrow pit. Peter sat down and pulled his knife from his pocket, the same knife he'd been given on his fifth birthday. From another pocket he pulled a piece of plum wood he'd been working on, and began to shave curls of wood from the back of the little sheep he was carving.
Tomas was already up to his waist in the soil when Peter suddenly looked up to see his father's eyes on him.
"Get on with your work," Tomas called. "I've got enough to do here."
Peter muttered to himself, but did as he was told. His father was in a mood. A mood that told Peter to keep himself to himself. It seemed to Peter it had always been like that, the two of them living in the same single room, but like leaves that fall from the same tree, always spinning ever further apart.
Peter muttered again. There was always something.
Always something to do. Somewhere to go.
Something he was told to do. Something he was told not to do.
Something like the box his father owned, that Peter was never allowed to open.
After two days' digging, Tomas's hole had become a trench, and Peter began to have an inkling of what his father was doing. Two more days and the trench was four feet wide and stretched very nearly from one arm of the river to the other. Only a small gap of maybe three feet lay between the hole and the gurgling water at each end.
"Careful," Peter said, unable to keep quiet. "If you dig any closer the bank will give way."
Even as he said it he saw that that was just what his father wanted.
Tomas laughed, and swung his spade into the top of the last plug of earth. Water gushed into the trench, filling it more quickly than Peter would have believed possible. Tomas ran to the other arm of the river and breached the soil there too. He had dug the channel on a slight slant, so that water was already flowing in from the arm of the river nearest to the village, through the channel, and away to the other arm.
"I always wanted to live on an island!" Tomas, suddenly full of joy, and laughing like a young boy, called to his son. Soaked to the chest, he climbed out of the water and went inside to dry his clothes by the stove.
That night he got drunk on rakia, while outside the flowing water did a good job of cleaning and widening the trench, removing the last clods of soil from its two mouths. As he sat by the fire, his arms ached from the work, and through his tiredness something stirred within him. His muscles remembered working that hard. Years ago, he had swung his arms, but not with a spade.
Not with a spade.
Now Peter and his father made their way over the same bridge of trunks they had laid a year ago and onto their little triangular island.
Their horse, Sultan, whinnied softly as their footfalls sounded on the bridge. He pulled at his tether, a simple rope from his bridle to a tree stump.
"Put him in, Peter," Tomas said.
Peter nodded.
He patted Sultan's flank and led him into the tiny stable.
"Hay again, Sultan," he whispered. "One day I'll bring you some beet. You'd like that. One day soon, I promise."
Sultan flicked his head toward Peter, but it was a gentle gesture.
By the time Peter got inside, Tomas had already poured himself a mug of rakia.
"Have some?" he asked.
Peter shook his head.
"For God's sake!" his father shouted, without warning. "For God's sake have a drink with me for once!"
Peter stood, shaking a little, trying to stay calm and be friendly, as he always did at these moments, though his heart felt as if it were in a vice.
"I will, Father, I will."
He went over to sit by the stove with Tomas. The lamp glowed; a lone moth flitted about against the smoky glass. His father fumbled for another mug and poured a thick finger of rakia into it.
Peter forced the firewater down, trying not to shudder as it burnt its way into his belly. He knew that would irritate Tomas further. But his father seemed placated, and began to hum tunelessly. Peter looked at him, opened his mouth to speak, and closed it again. He could see that his father's eyes had glazed; his mind was elsewhere, miles away. Years away, maybe. Peter tried to think of something to say, something that despite the drink would reach out to his father, make a small bridge across to his island.
But it was Tomas who broke the silence.
"We haven't heard that tune for a while, have we?" he said.
Peter shook his head. "I never understood it anyway. Why does the shepherd let himself be murdered? Without trying to fight, or to argue? It's stupid."
"Ah," Tomas said. "Ah."
He began to sing, his eyes shut and his face turned to the roof beam.
_"By a rolling hill at Heaven's doorsill..."_
The moth tumbled onto the table, exhausted by its efforts to fly into the light. It lay on its back, struggling.
_"Where the trail descends to the plain and ends..."_
"Why does everyone sing it anyway?" Peter asked. "The Miorita?"
Tomas stopped and turned his gaze momentarily on his son, but he was distracted by the moth, which had flipped over onto its legs. From the strange milky skull-shape on its back they could see it was a death's-head moth.
"It makes no sense, but people sing it all the time," Peter went on.
Tomas slammed his hand down on the table and left it there. The moth had no chance of escaping. Peter winced, then looked away as his father lifted the squashed corpse from his palm, opened the door of the stove, and threw it in.
There was something wrong with killing even such a small thing for the sake of it. There was no point saying so, Peter knew that, unless he wanted a lecture about what ten years in the King's army did to your opinion on killing. Ten years in the army and four in jail. Enough to make any man violent.
Peter stood up and got the pot from the cupboard, to make soup.
"The song, the Miorita, makes sense to some people," Tomas said cantankerously, but Peter had turned his back on his father to chop vegetables, and couldn't tell what his father thought of it himself. Presumably he ranked it alongside all the other superstitious nonsense people spouted. His mood was thick and dour. The death of the moth seemed to have put a sudden end to his drunken good humor, and he sat by the open door of the stove for the rest of the evening, staring into the flames, until the bottle was empty and he staggered to his cot, ignoring the soup that Peter had set in front of him.
Peter finished his own meal, then sat by the fire, carving a miniature fir tree. Something about that appealed to him; it was almost like giving back to the forest, rather than just taking wood to sell and to burn all the time. Turning one small piece of wood back into a tree again was an offering to the Mother Forest, and Peter believed that was very important. It would never do to anger the great power that lurked all around them, every day of their lives. Peter finished the carving and put it on the shelf above his bed, along with all the others.
He sighed. He had never had his father's skill with his hands and the tree was clumsy. But it was his.
The snowy night hung thickly over the village, and the two arms of the river, and the trees. The forest stretched away in almost every direction for five hundred miles, unbroken except for the faint huddle of a village here and there.
The hut crouched on the island Tomas had made, as if waiting, a dim light shining weakly from the gaps in its two tiny shutters.
Away, across one of the river's arms, something watched the hut. It stirred. The figure of shadow moved slowly from cover and then sped like daybreak into the trees.
**5**
**St. Andrew's Eve**
A few days after Radu's funeral Peter went into Chust. His father and he were owed money by various people for deliveries of logs, and Peter knew it was better that he collect it than Tomas, who might spend half, or worse, in the inn before he got home.
It was a bright day, but the snow still lay defiant on the ground, with promise of more hanging like a gray blanket in the sky. Peter was passing the wooden well at the northern end of the village when he saw the first plumes of smoke curling up into the air. He was puzzled at first, but then remembered that it was St. Andrew's Eve.
As he rounded a corner there was the first bonfire. Slung over a fire pit, a huge iron cauldron was spewing coils of steam into the cold morning. Around the fire stood a paltry crowd of people, each waiting patiently with a wooden bucket in hand. They took no notice of Peter as he passed, on his way to his first call, the priest Daniel, at his house.
In the square was another bonfire, with an even larger cauldron, and even more people waiting, with their buckets. By the cauldron stood Teodor, the feldsher, and a fat man whose name Peter didn't know. Teodor seemed to be in charge of the fat man, who filled each bucket from the cauldron as it was presented to him. Occasionally Teodor would wave the man away and stand by the cauldron, muttering something over its steaming mouth. Whether the muttering was ill temper at a job badly done, or magic, Peter didn't know. If it was magic, Peter wondered what the priest would have to say about it. After a while, Teodor would stand back and nod for the work to begin again.
It was a messy, smelly job, and now Peter could see what he knew was in the cauldron, for there were thick black patches that had been spilled here and there in the snow-packed square. Tar.
All across Chust—and, Peter knew, probably all across the country—people would be doing the same thing. Daniel's house lay on the far side of the square, by the narrow alley that led to the church, and Peter could already see the priest, brush in hand, outside his house. Daniel dipped the fat, short, round hog's-hair brush into the tar, trying to be quick before it set again. He was working laboriously, painting the tar onto the windowsills of his house. People were working on their houses around the square, some slowly and methodically, others fast, but all around the village, every window frame was being coated with the thick black tar. It was tiring work; the tar cooled rapidly and got harder to use, and the clumsy hog's-hair brushes came apart all too easily.
Peter stood right behind Daniel, but the priest was so intent on what he was doing that he didn't notice. He had moved to his door, and after he had covered the frame with a good layer of the stuff he scraped the last from the sides of his bucket and daubed a large, untidy cross on the door itself. He stood back to inspect his work. He would have liked to have made a better-looking cross; he would have to get some more tar.
"Does the protection of the Lord need the help of tar, Father?" asked Peter.
The priest jumped, then turned to see Peter.
He scowled, dropping his empty bucket to the ground. His hands were sticky, and he tried to wipe them on his robes, but it was no use.
He was a tall man, balding, with a sharply pointed beard that mimicked the sharpness of his nose.
"St. Andrew's Eve, Peter," he said, as if that was an answer. "You and your father would be wise to take the same precautions. It's a long journey from here to St. George's Eve. And it can be an evil journey."
Peter agreed with that, at least, against his better judgment. He thought of the Miorita: it was when the shepherds had come down from the hills for the winter that the murder had happened. The whole dark winter lay before them, and the winter was a dangerous place to be. It was just that Peter didn't have much faith in tar for getting them through the long winter months. By spring, by St. George's Eve, flowers and holy sweet basil would be growing in the pastures, showing that God's power was increasing again.
In the towns they'd lived in, no man of religion would have abided such superstitious practices. But here, in the depths of the forest, it was different. Somewhere among the trees the path that led directly to God had gone astray. It had got lost among the folktales and superstitions and the hushed talk of the fireside.
Don't get involved, was what his father would have said, Peter knew that. He decided to take his father's advice for once.
"Father Daniel, I've come to collect money. We brought you two loads last week."
"You did bring me two loads, but then there was a funeral to pay for."
"What has that to do with us?"
"The woodcutter was your friend. Since there was no one else to pay for the funeral, I'm going to take it from what I owe you. I will pay you for one load of wood only."
"He was no more our friend than he was yours!" Peter said angrily. "Someone from Koroceni ought to pay."
"Well, you go and find someone from Koroceni and I'll happily take their money for the funeral."
"That was no funeral anyway," Peter said, knowing he was speaking rashly.
Daniel opened his door, then turned, pointing a long finger at Peter.
"Be careful what you say. He was lucky he got buried at all. We did our best for him. You should pray that it is enough!"
He made to go inside and was shutting the door when Peter stepped forward and stopped it from closing with one strong hand.
"Father," he said, as firmly as he could. "Our money."
Daniel glared at him.
"One load only."
Peter nodded. He could see he was not going to get any more from the crooked priest.
"Wait there," Daniel said, and Peter obeyed, but kept one foot inside the door. He and his father had been cheated too often for him to be careless about things like that.
The priest returned and grudgingly placed the few coins in Peter's hand.
"Take my bucket back to Teodor," he said. "Tell him I need a little more. Tell him we all need more tar."
Peter stepped back from the door and picked up the bucket from the snow. He shoved it into Daniel's hands.
"Tell him yourself. I have money to collect."
**6**
**The Dulcimer's Melody**
Peter stalked away across the square, immediately regretting his confrontation with the bad-tempered priest. It was the sort of thing that had kept them on the move all his life. Never settling anywhere, never belonging with others. Peter might not have liked Tomas's odd choice of a permanent home, but he was simply glad they had finally come to rest. What they needed now was to avoid trouble.
Peter heard them before he saw them. Their music drifted ahead of their caravans. Gypsies.
Three caravans and an open wagon rolled into the square, and people stopped what they were doing to watch. The caravans were brightly painted in yellows and reds—the canvas of their rounded roofs covered in strange, foreign swirls of color. Set against the dull gray-brown walls of the houses, the Gypsies' appearance became even more exotic, and the villagers were entranced.
The first caravan was pulled by a bay mare and driven by a tall, strong man with long black hair that he wore in a ponytail. On top of his head was a tiny round hat, and on his lips a smile, not a broad one, but one that looked as if he knew things.
The music came from the back of the open wagon, where four musicians played. There were two fiddlers and a man with a shallow drum. Next to the drummer sat a man with a dulcimer, which he played with miniature metal hammers. That was what Peter had heard first, an unearthly sound that spoke of other places and other times.
The four musicians sat in the four corners, but there was a fifth person in the wagon. She stood up as she began to sing, and her voice lifted gently above the music and floated around the square. Up to this moment, Peter had not recognized the tune, so exotic was the sound of the dulcimer and the drum, with the fiddles wavering on top. But there was no mistaking it as the girl's voice picked out the familiar melody of the Miorita.
_"Tell not a breath of how I met my death;_
_Say I could not tarry; I have gone to marry_
_A princess—my bride is the whole world's pride."_
A princess. Peter was transfixed. The girl was maybe just a few years older, but Peter knew she was very different from him. Even at this distance he could see her pride, her confidence. She stood tall, easily countering the rocking of the wagon with the sway of her hips, where she rested her hands, fingers splayed. Her head was up, her raven hair falling in ringlets across her shoulders.
Once again Peter struggled with the song. It made no sense to him. He still couldn't understand why the shepherd behaves the way he does. He hears he's going to be killed by the other shepherds, maybe jealous of his youth, or his handsome looks. So much, so easy. But then he does nothing. He doesn't run away, or hide. He doesn't fight. He accepts his death, and concocts that story for the lamb to tell his mother. That he married a princess from some distant land. Peter, who had never known his mother, could nonetheless understand wanting to protect a loved one from the painful truth. But he couldn't understand anyone's accepting his own murder so readily. Unless maybe it was the only way to such great beauty.
Such beauty as the cosmic princess from the song—
Suddenly Peter was aware that the Gypsy girl was looking straight at him, fixing him with a stare that was powerful, yet at the same time utterly devoid of emotion. He was unable to look away, and now the three caravans and the wagon pulled to a halt in the center of the square, and the musicians struck up a different, livelier tune, one that leapt to the beat of the drum. It was an instrumental piece, and the girl sat down in the wagon, no longer looking at Peter, though he could do nothing but look at her.
"I hope you don't think she's more beautiful than me?" said a voice behind him.
He turned to see Agnes looking up at him, smiling.
Even as he spoke he knew she had only been joking, yet some foolishness inside answered for him.
"No, no," he said quickly. "Of course not. I was just admiring the song, that's all."
Agnes stopped smiling.
"It was the song you were admiring, was it?"
Peter shuffled awkwardly. He looked down at Agnes, her short brown hair framing that pretty round face, those gray eyes and that little nose.
"How are you, Agnes? How's your mother? I haven't seen you for a while."
"That's because you come to the village only when it suits you."
"Agnes, I'd come more often," Peter stammered. "I'd come to see..."
He stopped; he didn't have the courage to say it.
"I know," Agnes said, and with a jolt in his heart Peter thought she had guessed what he was about to say. But she hadn't. "You'd come more often if only you weren't so busy, if only your father let you, if only you had the money."
"Don't, Agnes," Peter said. "That's not fair. You don't know what it's like."
He'd said the wrong thing.
"Don't I?" she cried, her voice high and uneven. "Father died less than a month ago, Mother's stayed in her bed ever since. I have all the work to do and I must look after her too, and you think I don't know what it's like?"
She turned and hurried away.
"Agnes," Peter called. "Wait! Please?"
People were staring; he ran after her for a few steps, then faltered.
"Agnes," he said quietly, but she had gone. He could tell the air what he wanted to say, but what was the point?
He turned and looked at the Gypsies again. A crowd had gathered, and some were even throwing a little money into a hat that a child was taking around.
Peter smiled bitterly. It would not be too strong to say he was unsettled by the Gypsies, but he felt some empathy with them. Here were the villagers, happy to listen to their music, happy even to pay for it; yet there was a contradiction. Peter knew that the Gypsies would not be allowed to stay in the village overnight but would have to pitch somewhere outside it when darkness fell. They were tolerated, not trusted.
He understood how that felt, and more besides. Something about the Gypsies spoke directly to Peter's heart.
Night was falling as he trudged toward home through the snow. As he walked, his pockets jingled with the money he had collected. Now all he had to do was keep Tomas from drinking it all, as well as explain why the priest had paid for only half his wood.
Should he tell his father there were Gypsies in the village? He thought better of it. He could see his father's look of indifference already, and besides, he felt something for them. If his father poured scorn on them the way he poured scorn on everything else, it would be one more thing to have happy dreams about that Peter would have lost. One more thing to push them apart.
Like the box. It was one of Peter's earliest memories, and it was a painful one. His father had a long wooden box that had always been with them, but Peter had never seen inside it. Wherever they had been, wherever they had lived, the box had always been there. Tomas always tucked it away out of sight under his mattress, and though Peter couldn't remember, he knew Tomas must once have told him never to open it.
As Peter had grown, so had his curiosity. One day it got the better of him. He'd been about to open the box when his father came into the room. Tomas thrashed Peter so hard that every night for weeks afterward he woke from the pain. But Tomas also did something worse.
On the shelf by Peter's bed sat the wooden goose Tomas had carved the day he gave his son his knife. Tomas snatched the carving from the shelf and threw it on the floor, then crushed it with his boot. Then he threw the pieces into the fire.
To this day, Peter resented it. What could be so important that Tomas had to keep it from him? The box was like his life, as far as Peter could see—something he had no control over, something shut away, not to be talked about, full of secrets and riches he must not explore.
Shutters barred every window as Peter walked out of Chust, but he could hear the sound of singing from every home he passed. Another form of protection, for everyone knew you should sing on St. Andrew's Eve to keep evil away.
Peter shrugged. It was the first night of the year, when evil was loosed on the world, and all the villagers had to protect themselves with were tar and singing.
Above his head he suddenly heard the beat of wings and then the honking of geese. He looked up to see the birds streaming their way across the sky like a living arrowhead.
"Very late," Peter whispered to them. "Very late to be heading south."
But at least the geese could leave; late or not, they could take flight away from the cold heart of winter.
For everyone else, it was a long journey indeed to the safety of spring.
**7**
**Sheep and Wolves**
For the next few days Peter worked hard, chopping and delivering as much wood as he could before the snows really bit deep. On about half the days he managed to get his father to help him. The rest of the time Tomas sat by the stove in the hut, drinking his way through a small cask of slivovitz that he'd bought with the money Peter brought back from his last trip to Chust.
Late one morning, as they were chopping logs from the lumber pile, Tomas dropped his axe. Not for the first time Peter noticed his father's hands shaking. Tomas bent to pick the axe up from the snow but dropped it twice more before he began to swing it again.
"Get on with your work, Peter," he said gruffly, seeing his son staring at him.
Peter didn't move.
"It's cold out here, isn't it?" Tomas said, pausing. "Can't keep my damn hands still."
"Yes, Father," Peter said. "The wind's cruel today."
But later, back in the warmth of the hut, Tomas's hands were still shaking.
Peter and Sultan made a dozen trips around the village, their battered cart laden and creaking through the snow. Most people had good stores of seasoned logs already, but no one would refuse another delivery; you could never be sure how hard the winter might be. The difficult thing was getting people to pay for the wood straightaway, but nevertheless Peter came home most days with coins to put in the tin under the loose stone in the corner of the hut.
One day, Peter came home with more than money. Stories were flying around the village, and Peter brought some of them with him too. He led Sultan over the bridge onto their island and hurriedly fed him. While the horse ate, he threw two blankets across the beast's back. He dragged a bucket through the channel that joined the two arms of the river, and poured water into Sultan's trough.
"Drink it before it freezes, boy," Peter said, shutting the stable door. He felt strange calling the horse "boy." Sultan was older than him, and somehow, Peter knew, much wiser, but that was what Tomas often called him and it had become a habit. "One day, I really will get you some beet."
Peter found his father inside, as usual. For once, though, there was no drink in sight, and there was a pot bubbling on the top of the stove.
"There's all sorts of commotion in the village," Peter said, before he had even closed out the cold.
"What?" Tomas asked, looking up from stirring the pot.
"Sheep have been attacked. In their sheds. Cattle in the pasture too."
"So the wolves are getting hungry," Tomas said. "What of it?"
"It's not wolves. Well, that's what they're saying in Chust."
"So what is it then?" Tomas asked.
"I think you know what they're saying," Peter said.
"Pah!" Tomas spat on the floor. "Idiots! And you're an idiot too for listening."
"I'm just telling you what I heard," said Peter. "That's all. You know the miller who died last month? Willem? His widow says he visits her in the night."
Tomas said nothing; he turned his attention back to the pot on the stove.
Peter kept going, for once seeing the chance to actually get his father to talk.
"She says he's been visiting her for a week now. She's very ill. Pale and won't eat."
"So what? She wouldn't be the first silly old woman to say that! That snooty girl of yours. Agnes."
"Yes?" said Peter angrily. "What about her?"
"They told me in the inn yesterday that her mother's been saying the same thing about her husband."
Peter looked at his father.
"They said what?"
"You heard," Tomas said.
"And you didn't tell me?"
Tomas whirled around, sending the pot of stew flying to the floor as he stormed over to Peter, who flinched, convinced his father was going to strike him.
"No," Tomas shouted right into his face, "I didn't tell you, because it's all nonsense!"
Peter stood, breathing heavily, trembling. The stew was spreading across the floor. He looked back at his father, determined to hold his gaze.
"I'm going to see Agnes," he said quietly. "I'm taking Sultan. I'll be back late."
Peter strode out of the hut and with a silent apology to Sultan put the horse's saddle and bridle on again. He galloped into Chust, fuming about his father as he rode, happy to let him have to clear up the spilled stew.
In the hut, Tomas stared at the mess he'd made.
He got on his hands and knees and tried to scrape what he could back into the pot, but all he managed to do was fill it with a muddy slop. He took the pot outside and threw its contents into the river, then swilled it out and went back indoors. He threw sawdust over what was left on the floor.
He stood, breathing in quick gulps of air. His eyes fixed on the barrel of slivovitz, but he forced them to move on, and found himself staring at his bed.
He glanced at the door, but he knew Peter was way into the village by now. Nonetheless he threw the bolt and went and knelt by his low cot, as if he were about to pray.
Instead, he rummaged with both hands under the mattress and pulled out a long, flat wooden case.
He let the mattress fall onto the bed again, and placed the box on top. He waited for a moment or two, catching his breath, as if scared of what he was about to do.
The case had a simple catch and no lock, and was rather plain, made of a dark-colored wood, so unlike the pine and birch of the Mother Forest. Tomas looked behind him once more, at the door, hesitating still. Then he took a deep breath and raised the lid, and from inside he lifted a strange and beautiful object up into the flickering orange lamplight.
It was a sword, and it was as frightening as it was beautiful, and as foreign as the sun in winter.
Its slim but lethal blade curved back halfway along from the hilt, widened out for its last third, then tapered to a fearsome point. The hilt itself was sheathed in horn, glossy, gray and mottled, and the crosspiece was an elegant brass creation.
The blade's surface was completely smooth, though a strange device was engraved in the steel by the hilt. Two triangles interlocked, forming a six-pointed star, between each arm of which was a small circle. In the middle of the star was a seventh circle, and around the outside ran two concentric circles, keeping everything in order.
Tomas held the sword, not by the hilt, but with its blade resting gently on his palms. He seemed hypnotized by it; even his breathing appeared to have stopped.
The only things that moved in the whole hut were the flames dancing in the stove and the tear that fell from Tomas's cheek onto the blade.
Memories flooded his brain, unbidden and uncaring. Suddenly he snapped from the reverie, roughly put the sword away in its case, and rammed the case under the bed, as if it were worthless, though nothing could have been further from the truth.
He grabbed a mug from the table, filled it to the brim with slivovitz, and began to drink, trying to wash the memories away.
Outside, there was a noise. Footsteps sounded on the log bridge to the island.
"Peter?" Tomas called, unsettled by something he could not place.
But Peter was by then knocking on the door of Agnes's house.
**8**
**The Shadow Queen**
"Go away, Peter!"
Agnes leant from the upstairs window, looking down at where he stood in the street, holding Sultan loosely by the reins. Dusk had fallen across the village. Agnes's father had been a well-to-do merchant, a draper, and the house was one of the very few with two floors.
"Let me in, Agnes," Peter called up to her, as quietly as he could. Here and there people came and went down the long street, and Peter was wary of them, wanting to avoid prying eyes. In truth, however, they were all hurrying home, eager to be out of the coming night. As so often, the streets of Chust seemed filled with a subtle menace that Peter could not have named.
"I will not," Agnes said, for the fourth time. "I told you. We have barricaded the doors. And the windows downstairs."
"Well, open them again," Peter said, exasperated now.
"No, Peter. Are you mad? It's getting dark. Go home."
As if in agreement, Sultan whinnied gently. Peter put his hand out and patted Sultan's neck to reassure him. There was little he could do. He had ridden to see Agnes, and now she wouldn't even let him in.
"Agnes," he tried again. "Agnes, you must tell me that you are all right. I've heard a story, that your—"
He stopped, waiting for an old man to hobble slowly by and out of earshot. In that little space of time Peter pondered what Tomas had told him. He didn't know that he believed what he'd heard, but he wasn't entirely sure that he didn't.
"What, Peter?"
"I heard that your mother said...that your...father...Your father has been back to visit her."
He whispered as loud as he dared, glancing up and down the street as he did so. Agnes's reply was almost inaudible.
"What of it?"
"So it's true?"
She glared down at him. Peter was getting cross as well as cold. Why couldn't she give him a straight answer? He couldn't believe she seemed so calm about it, but then an awful thought crossed his mind.
"Have you seen him, Agnes?"
For a moment her face softened. She looked away across the rooftops, toward, Peter thought, the church.
"No, I haven't," she said, quietly. Almost sadly. "I haven't seen him. And I don't know if Mother has, or if she's just..." She trailed off.
"Agnes, I'm sorry. I want to help you. Won't you let me in? Let me check that everything's all right. Can I bring you anything?"
"No, Peter. What could you do anyway? I can manage. I've blocked all the doors. I've protected the windows. We'll be all right. You should go away. It's not safe out there. In the dark. You know what people are saying, don't you?"
Her voice dropped to a whisper, so that Peter had to strain on tiptoe to catch the gentle words as they fell down to him.
"It's the Shadow Queen. People are saying she's back, that she's coming to make Chust her own. Some people even say they've seen her!"
With that Agnes seemed to have scared herself. With a wave of her hand, she indicated that the interview was over.
The Shadow Queen.
Peter knew what his father would say about that. All nonsense and tittle-tattle. Nevertheless he suddenly felt very exposed in the lonely village street, with no one but Sultan for company.
He swung his leg over Sultan's back and wearily headed for home again.
**9**
**The Eternal Return**
"Come on, Sultan."
Peter bent over Sultan's neck and whispered in his ear. "I'm tired too, but we should get back to Father."
That was true, but it was also true that, despite himself, Peter had been unsettled by Agnes.
Locking herself and her mother away every night seemed a desperate measure, and her talk of the Shadow Queen might just have been village gossip, but as he rode through the deserted streets, the darkness began to eat at him.
He steadied himself and rode on, but it was not long before he began to catch himself peering into the shadows that curled at the street corners. Then he'd snatch his eyes away again, like a frightened child. The darkness seemed to press in on him from all sides, ominously. What if it was true? What if the Shadow Queen was true, and was coming to take them all?
Peter and his father might not ever have seen her, but they had met plenty of people on their travels who said they had.
Was it last year? Or the year before? Peter couldn't remember, but once, he and his father had been passing through a district away to the southeast, nestled up against the Karpat Mountains. They had stopped in a village for the night. All evening, as they sat in the inn, there was talk of only one thing. The Shadow Queen. The locals spoke in hushed whispers, as if she was standing at the window of the inn, intent on catching anyone maligning her.
"She's a thousand years old!" someone said.
"Rubbish! She was born at the beginning of time. She has no age."
"Yes," someone else agreed. "And she's ten feet tall and has a hundred teeth! She can devour five children at once!"
"Ah!"
The audience grew fat on these morsels, while more beer was drunk and songs were sung. Peter found himself glancing over his shoulder, and after a while he moved closer to the fire.
The following day was a Sunday, and as it turned out, Palm Sunday, but Peter and Tomas were surprised to hear the locals call it Shadow Day. They were even more surprised when they learnt that they would be seeing the Shadow Queen herself later that day. After all the talk the previous night, it seemed absurd to hear the villagers discussing her imminent arrival.
Tomas announced that it was time to leave, but Peter was intrigued, and eventually he persuaded his father to stay for an hour more.
"Very well," Tomas said abruptly. "Maybe then you'll see what sort of superstitious buffoonery we are talking about."
They found a heavy oak, climbed to one of its massive lower branches, and watched.
They didn't have long to wait before the Shadow Queen arrived. All morning the villagers had been busy. Everyone had something to do or somewhere to be, but finally, just after noon, they made their way outside the village to a large field that led down to a wide, fast-flowing river. Here, on the grass, a large bonfire had been built, of birch logs on willow branches, kindled by hay from the village barns. Some people milled about, while others had much to do. Finally there was a sudden lull in all the hustle and bustle and a hush spread across the pasture.
Then, so quietly that at first Peter wondered if it was just the wind, came the voices of the village.
"The Shadow Queen! The Shadow Queen!"
Not a cry, but a thousand awed whispers that spread through the crowd. Now even Tomas sat up and shifted his position to get a better view. All eyes turned to the edge of the village, where a cart slowly trundled out to the field. It was pulled by a single white horse, driven by a young woman. And in the back of the cart sat what could only be the Shadow Queen.
Tomas began to laugh.
The Shadow Queen was made of straw. A simple effigy dressed, strangely, in a man's clothes. She was a life-size figure, though, and she lolled about as the cart rolled awkwardly out into the field.
"The Shadow Queen!" Tomas said mockingly, but Peter threw a twig at him and glared. It was never a good idea to make fun of strangers, they knew that well enough.
The cart reached the margin of the field, near the bonfire and the river. Tomas and Peter got down from their tree and went to watch the rest of the ceremony.
Solemnly, the Shadow Queen was sawn in half, and the two halves thrown onto the blazing bonfire, which snapped and cracked, sending blackened stalks of straw high into the warm spring air. Eventually the fire burned through, but there was one last ritual to observe. The ashes were gathered and cast into the river, where they sped away south, never to be seen again.
Peter tried to ask the villagers about it, but the answers he got only confused him more. Was that really the Shadow Queen he had seen? Who had been burnt? Was it just a straw dummy? Everyone he asked gave him a different answer, but it seemed that the locals knew it was just a straw figure, though somehow, at the same time, it was the real Shadow Queen too. In burning her, here, at the start of spring, they had sent her away, sent her underground for the spring, the summer and the autumn, so that she would plague them no more. At least until St. Andrew's Eve, and the start of winter. Then, as the long cold nights spread across the land, she would return, bringing illness, plague and pestilence with her once more. Evil would wash before her in a wave of malevolence.
Peter was unable to understand how the villagers made sense of it—the frightening figure of hideous power described in the inn the previous night was such a far cry from the laughable doll that had been sawn and burnt in the field.
As Peter got talking to more locals, there were those who claimed to have really seen her, up in the mountains, or in the depths of the forest, or lurking in the graveyard.
As he was being told that the clothes the figure wore were those of the most recent widow's husband, intended to keep him from "coming back," he noticed that Tomas was rolling out of the village on their own cart, having decided to waste no more time.
"Stop him from coming back?" Peter asked the man. "What do you mean, coming back?"
**10**
**Refusal**
As Peter rode through the murk on Sultan, his thoughts had drifted to a sunny field, a long time ago, and this should have done something to keep the power of the night at bay. In fact, it did nothing to make him less scared. There was a little starlight, but he knew he wouldn't be able to gallop Sultan once they were in the forest. Still, there was nothing to stop him from hurrying up the last street to the edge of the village.
He kicked Sultan on, suddenly feeling more terrified by the admission of his own fear, but just as they picked up speed something rushed into their path.
Sultan, usually so sure-footed, shied and reared. There was a scream. Peter fought for a moment to stay on Sultan's back, but lost the fight and hit the ground hard. In a moment's confusion, it seemed that Sultan was going to fall and crush him, but then he rolled beside Peter, struggled to his feet, and limped away, frightened.
Peter spun off his back and onto his front, worried that Sultan was going to bolt for home. Then he remembered the scream just before he fell.
"You nearly killed me!" cried a voice.
All Peter could see at first was hair, lots of it, coiling like small black snakes.
The figure moved into a sitting position and began to smooth her long skirts into place, checking that nothing was broken. Now he knew who it was. The Gypsy girl, the singer.
"You ride very badly!" she said, pointing a finger right at him.
"Me?" Peter spluttered. "It was your fault! What in Heaven's name were you doing? Running in front of a horse like that!"
She ignored Peter's anger, but with it, her own rage seemed to have vanished.
She smiled at him and tried to stand, but immediately shrieked.
"My back!" she cried, sinking to the ground. "Oh! I think it is broken!"
Peter doubted that very much, but nonetheless she appeared to be in pain.
"You must help me," she declared. "You nearly killed me! So get me out of this road."
Peter stood up slowly. He hurt too, but there was no point in protesting.
"Carry me. Over there."
She nodded toward the side of the road, and a low bank of grass.
Peter sighed and bent over her. For a moment he considered how best to pick her up; then he slid one arm under her legs and the other under her shoulders. She was light enough for him, he was used to carrying logs all day. But logs didn't wriggle, or complain, or hiss in pain, and he was glad when he had taken her the short distance and placed her on the soft grass, the start of a narrow strip that kept the forest away from the village.
They were just beyond the ragged edge of the huts here, with only the odd one or two dotted about, the street turning into nothing more than a snow-covered track that wound away into the trees. The puny thatched fence that marked the end of the village was defense against nothing, and yet being beyond it was disturbing. The Shadow Queen had already settled in the back of Peter's mind.
"Your back isn't broken," he said, looking down at the girl. "You couldn't move your legs if it were."
"My name's Sofia," she said. "What's yours?"
He sighed, looking around to see that Sultan was still close by.
"Peter," he said.
"I think my head is maybe hurt," Sofia announced.
Peter opened his mouth, then shut it again. She might sing beautifully, but he was finding her enormously irritating. Still, as he was carrying her, he hadn't stopped himself from noticing that her legs were long, and that her dress was cut very low. Nor had he stopped himself from looking at her brown skin, so different from that of everyone else in the village, and more like his own.
"My head hurts," Sofia said again, "Here. You must feel it. Come here!"
Peter stood where he was.
"Come!" she demanded, and reluctantly he knelt down beside her. She grabbed his hand nimbly and pushed it into her thick hair. "There's a bump. Yes? No?"
Peter gingerly moved his fingers through the girl's hair, but could feel nothing.
"I think you're fine." He pulled his hand away.
As he did, Sofia took his hand in hers and didn't let go.
"I think I'm lucky you didn't kill me," she said, but gently this time.
Awkwardly, Peter sat next to her. Still she didn't let go of his hand.
"What were you doing anyway?" he asked. "Out here, in the night? You shouldn't even be in the village after dark."
"Because of who we are?" Sofia said haughtily.
"Yes," Peter said. Then he added, "But I don't make the rules around here."
The girl laughed.
"No, I am sure of that."
Peter felt offended, at the same time wondering why Sofia was still clutching his hand. He realized that he didn't want her to let go.
"What do you want?" he asked. "It's dangerous out here."
"Let me tell you," she whispered, so quietly that despite himself Peter leant closer to her.
Peter was aware of the warmth from her body, and could smell her long raven locks. In that split second he wouldn't have cared if the Shadow Queen was right behind him.
"I want you to stay with me awhile," Sofia said.
Then she pulled his hand quickly, catching him off balance. He half fell on top of Sofia, who lifted herself high enough to plant a kiss on his lips.
Peter yelped as if he had been bitten by a dog and jumped to his feet.
She laughed.
"Peter!" she said, smiling.
He backed away and ran to Sultan.
"Peter!" Sofia called, this time more urgently. "Stay with me! My back hurts! I can't walk!"
But Peter wasn't fooled by Sofia's tricks anymore; his thoughts were full of Father and the hut, and Agnes. What would she say if she knew what the girl had done?
Sultan seemed sound enough after his fall, and Peter plunged into the forest, heedless of the danger of galloping over difficult ground in the dark. Behind him Sofia's cries grew fainter.
"Come back! Come back and help me. Peter!"
He rode.
**11**
**Visitors**
As Peter rode he saw neither trees nor snow, but instead a glorious vision of Sofia. The girl was arrogant for sure, but all he could see were the rich tresses of her hair, her welcoming brown eyes and dark skin. With a wrench he shook himself, and tried to push Agnes back into Sofia's place. He found Sofia floating into his mind again, and started to work on the image, lightening and shortening the hair, turning the brown eyes gray. Finally he watched as the brown skin grew paler, paler, paler. There, that was Agnes.
But no! He watched in horror, transfixed as Agnes's skin took on an evil whiteness, the whiteness of death, and became impossibly wrinkled and old. Her lips shriveled, her nose became pointed and thin, her hair grew lank and noisome. Her eyes flattened and widened, darkening and disappearing in shadow.
Shadow.
"No!" Peter cried into the air, then snatched himself away from the grotesque vision.
He let Sultan slow to a walk once Sofia was out of earshot. They followed the bank of the river Chust out to the hut. But Sultan was uneasy. He sensed something up ahead and now stopped completely.
For a while Peter urged him to walk on, and they managed to go a few more steps. Then once again Sultan stopped, this time for good.
"What's wrong, boy?" Peter whispered, his attention divided between the horse and whatever might be up ahead that was bothering him.
Sultan made no noise, but merely stood as still as any horse can.
"Well, you'll have to stay here."
Knowing what Tomas would say about leaving their most expensive possession alone in the forest in the night, he reluctantly tied Sultan's reins to a sturdy birch.
Peter turned around and all there was to see were the shadows of the night forest. Trees stretched off into the distance in every direction, becoming gray ghosts and then no more than suggestions of ghosts. In the gloom the river chugged softly somewhere away to his right, but there was just enough starlight to make his way, so he started off toward the hut.
As he went, Sultan gave one final snort, then was silent.
Peter knew Sultan well, knew that he was trustworthy, not the sort of horse that spooked easily. Sultan's refusal to go any closer to the hut was a sign that something was wrong. Peter slowed his walk to a crawl as he stepped as gently as he could along the riverbank, and was glad at least for the sound of the water rushing, hiding his quiet footfall.
There was the hut in front of him, across the log bridge. At first sight nothing seemed to be amiss, but Peter's heart froze as he made out the shapes of not one but two horses on the bank, just beyond the bridge. The horses were tethered, and alone.
He stared through the pricking darkness at the hut, but could see nothing, could hear nothing but the water. There was light coming from inside, flickering slightly, as if people were moving around.
Something was wrong. No one ever came to see them, certainly not late at night. He put a foot on the bridge, eyeing the horses as he did so. He didn't recognize them, but he noticed that strangely they bore no saddles. He turned his attention back to crossing the bridge without making a sound. He succeeded and stole a few hurried paces across the island to the hut, but instead of opening the door and walking straight in as he usually would, he slid close to the wall, crouching nervously beneath the shuttered window.
He could hear voices.
He raised himself on his knees, bringing his ear as close to the window as he dared. He knew that he could not be seen from inside, but still something made him desperate to keep hidden.
Now he could make out words.
"...you have no choice..."
A muffled reply. Peter knew it was his father's voice, but the words were not clear.
"Once, you would have spoken differently."
"You cannot refuse. There is no choice. The Shadow Queen has taken your choice away."
The Shadow Queen. Who was his father talking to in there? Now several voices all spoke at once, urgently.
"...the Shadow Queen is coming."
"...more hostages."
"...where is it, Tomas?"
"I don't have it."
"You will agree. You have to understand that."
"No!"
His father again, shouting this time.
There was silence for a short time, then quieter voices, indistinct but insistent nonetheless.
Peter was about to risk moving closer, when the door flew open on the far side of the hut. He dropped to the ground and crawled to the corner by Sultan's stable. Between the cracks in the planks of the stable, he saw four figures leaving, then crossing the bridge.
The light from the open door shone across the island and the bridge. Its glow was enough for Peter to see the identity of the visitors.
The Gypsies who had been with Sofia in the village.
**12**
**Closer**
Agnes closed the door to her mother's room and leant against the door frame for a moment, her eyes shut, running her hand through her hair. She had lost count of how many times she had been in to check on her through the day, and now the evening was thickening and the long night lay ahead. All day she had been trying to make some sense of her late father's business. People had come to collect orders that she knew nothing about; there had been arguments. She was exhausted.
She was still furious with Peter, but deep down she knew that was unfair. He had been trying to help. But he was tactless and certainly not as bold as she would have liked him to be. As she would have liked her future husband to be.
She blushed as she considered what she had told no one else, not even Peter himself. And he was poor too, she would never have dared tell her father of her desires. A draper's daughter does not marry a woodcutter's son.
Father, however, was gone. Though that was not what her mother said.
Agnes tried to push that thought away as she busied herself for bed. She slipped out of her clothes and into a nightdress, and began to brush her hair, but her fears would not stay away. Her hands began to tremble. She dropped the hairbrush clumsily on a table by the window, backing away from it uneasily. She knew the window was protected, but that didn't quell her fear.
What if Father had been coming back? To Mother, in the night? She did not doubt for a second that it was possible; everyone knew it. Cattle and sheep had been attacked in recent days too. And it was true that her mother did seem to be getting weaker with every night that passed. Weaker, and paler.
But he would not come in the house tonight, no one and nothing would; she had taken further precautions. There was still tar from St. Andrew's Eve on each window and door, and earlier in the day she had crushed five whole bulbs of garlic and smeared the paste on every window frame and doorsill.
There was no way in now. Or so she hoped.
She climbed into her own little bed and listened to the noises of the night.
In the street, outside Agnes's house, beneath her window, a large and bloated figure wavered, trying to come nearer. The figure, dressed in muddy, slightly torn clothes, sniffed the night air, which reeked of garlic.
**13**
**And Closer**
Again he sniffed the air. Now he cursed and moved down the street, shambling slightly. Something pushed him away from that house, the house he remembered, but he sensed there would be others.
It was to be even easier than that, however.
Two streets away, a young man called Stefan made a fatal mistake. In fact he had made several, each worse than the last. First, he had decided to spend the evening in the inn, where he had got very drunk with his friends. Second, he had played cards all night, and for some reason had lost every hand, and almost a week's wages—all the money he had in the world. Then he'd decided to stay in the inn when his friends left together, and to drink until his credit ran out.
Eventually the innkeeper had thrown him out. It was a cold night, but not snowing, and the ground was a mess of old snow and mud and footprints. Stefan had been shuffling home, too drunk still to be miserable about his evening, when he saw someone in front of him, no more than arm's length away.
Stefan puzzled for a moment to place who it was.
"Crista!" he announced, pleased he had remembered.
It was the draper, the one with the pretty daughter. What was her name? He couldn't remember at first, then it came to him.
"And how is little Agnes?"
The draper said nothing, and then slowly, very slowly, it occurred to Stefan that there was something strange about seeing Constantin Crista here. If only he could remember what—
Faster than a cat could blink, Stefan flew back against the wall. Crista leant in, pressing him back, holding the young man's head away to one side with one hand, while using his other arm to hold him fast. He leant his head in closer, his mouth nearing Stefan's neck.
His lips, now just a finger's breadth away, parted, and then Crista stuck his tongue out, straight through the skin, right into the artery.
For a moment Stefan struggled to realize he was dying.
**14**
**Creeping**
Daylight crept slowly over the mountains, and through the trees, and finally limped along the twisting streets of Chust. It was snowing, but softly. Agnes awoke, her heart feeling lighter than it had done for some time. She went in to see her mother, who smiled and even said she was feeling a little better. Agnes went back to her own room and dressed, then went downstairs to light a fire and make some porridge, picking her way past bolts of cloth stored at random in the hall.
Then she heard shouts from outside, and a scream.
She dropped the pot she was holding and frantically began to pull the barricade of chairs and tables away from the door.
**15**
**The Waters of Chust**
Deep in the forest, by the river, Sultan stood patiently. He snorted from time to time, blowing great clouds of steam into the frozen morning air. Nearby—the only other sound to be heard—was Tomas slowly sawing his way through a tree trunk that lay on the forest floor.
Tomas's mouth was a tight line as he tried to close his mind to everything except the saw and the tree. That was all he wanted to think about, but despite his hangover, and the exertion of sawing, images jostled in his head. He had been made to think about things he had sworn to forget. Who he was, thirty years ago. He paused in his work, exhausted, and glanced at Sultan. It was enough to make him remember another horse he had once owned. A huge stallion called Prince. How they had ridden! And how people had fled at the very sight of them! In his mind's eye now, Tomas could look to his right, and there was the King himself.
Mighty. How mighty.
For no more than a second, Tomas remembered glory; then he saw the glory turn sour, as it always had. Peter's face rose before him, and with it their argument from the night before. Then he remembered what he had done.
He bent to the saw again and worked until he collapsed over the carcass of the tree, fighting for breath, sobbing.
Sultan stamped his hooves in the snow.
**16**
**Agnes**
Peter woke late to find that his father had already left the hut. Pulling on his boots and coat, he stamped out into the snowy morning and looked around. The river flowed slowly by as usual, there was no sign of anything strange. No sign even of the hoofprints of the other horses from the previous night. He looked into the stable and saw that Sultan was gone too.
After watching the Gypsies leave, Peter had waited awhile, shivering in the stable. Then he'd gone back to find Sultan, who'd seemed perfectly happy to come home. He had stabled the horse, and gone in.
That was when the trouble began.
More arguments, more drink.
It seemed reasonable enough to Peter to ask why they, or rather why Tomas, had received a visit from Gypsies they had never met before. And what it was the Gypsies wanted, so late at night.
Tomas, however, was saying nothing.
He flung himself around the hut, jar of drink in hand, spilling most of it, drinking some. Peter had never seen him this bad, but for once he was not afraid of his father. He could see something was really wrong. Tomas was agitated as well as drunk, and Peter demanded to know why.
That was when Tomas hit him.
There had been no more talk after that. Peter had gone to bed.
Now he stood in the morning air. Where was his father? He felt the side of his head, where Tomas had struck him, but it didn't occur to him to feel sorry for himself, just as it didn't occur to him to be angry with his father.
Finally he thought to check the toolbox in the hut. Tomas's axe and the best saw were missing. So he had gone to work, that was something, though he was likely to be still drunk from the night before. Well, the cold and the work would sober him up soon enough.
There were sudden footsteps on the bridge, light and fast.
"Peter! Peter!"
Agnes.
She ran to him, right into his arms, without saying another word.
"Agnes! What is it? What's wrong?"
She said nothing, but trembled against him, her arms clutching him tightly.
"Have you run all the way from Chust?"
At last she lifted her head from his chest and stared up into his eyes. Her face was full of fright.
"There's—" She broke off and began to sob.
"What?" cried Peter, infected by her fear.
"Another death. Last night," Agnes wailed.
Peter grabbed her shoulders and held her away, needing to see her face, to see her speak, in order to understand, to believe.
"Another death?"
"Stefan," she cried. "You know Stefan? The miller's son? They found him in the street this morning. I saw—Oh, Peter." She stopped again and began to cry, burying her face in Peter's jacket.
"That's terrible," Peter said. It was all he could think of to say. Poor Stefan. But at least...
At least what? He was dreaming of a girl, and not the one who stood in front of him now. He forced himself to think clearly, to try to help Agnes.
She was mumbling now, almost incoherently, and Peter caught only two words.
"The blood!"
He steadied himself, knowing he needed to calm her down, though he felt far from calm.
"It's all right," he said. "You're safe. You're all right. It's terrible about Stefan, but you're safe. And I'll make sure it stays that way. I'll come to your house and stay through the night. Nothing will hurt you."
"No, Peter, no!" Agnes pushed herself from him, almost screaming. "You don't understand."
"What? What is it?"
"Stefan wasn't married."
"So?" asked Peter.
"Stefan wasn't married. There's to be a Wedding of the Dead."
"I know," said Peter. "I know, but that's normal—"
"But Peter," she cried, "I'm to be the bride!"
**17**
**The Wedding of the Dead**
Nunta Mortului. The Wedding of the Dead.
Stefan had been found in the street with his blood all around him. So that he did not have to suffer the fate of going into the ground as a bachelor, he would be married beside his open grave to a girl from the village.
Agnes was the oldest unmarried girl, and so had been chosen. It had been agreed upon by Anna and the other Elders, and that was that. There was no possibility of refusing.
And after the wedding service had been performed at the grave, Stefan would be buried, while Agnes, in order to serve the period of mourning, would be sent to a small hut at the edge of the forest, where she would see no other living soul for forty days.
Peter had done his best to console her, but what could he say? All he could do was assure her that he would see that her mother was all right, make sure she was looked after, that there was enough food in the house. As for the wedding, nothing could take away her fear of going through with it and of the forty days' isolation she must endure.
Forty days in a tiny hut, with all contact forbidden. Just within sight of the village, but outside it nonetheless, with the whole mass of the Mother Forest lurking at its back.
Stefan's was the second funeral Peter had attended in the village, but it was so unlike that of Radu, the woodcutter. Most of the village turned out, and besides, there was the added attraction of the wedding. There had not been a Nunta Mortului for several years, and the bride this time was particularly pretty, which moved the hearts of even the most cynical.
The bride was ready. She had been dressed not by her mother, who was too ill to stand, but by two women chosen by the Elders. She had arrived in the cemetery wearing a long, stiff wedding dress that had been found for her. The dress, however, was a sinister parody of its usual form, having been dyed black; it was to serve as wedding and mourning dress in one. It was completed by a high headdress and a heavy beaded veil, also black, which hid the bride's face totally. Peter could only guess that it was Agnes from her height and figure. He could see she was having trouble walking; her dress rustled like dead leaves with every uncertain step, and she held her hands clasped tightly in front of her. Maybe it was the weight of the clothes, maybe it was because she couldn't see, but in his heart Peter knew the real reason. She was scared stiff.
The groom had arrived too. He had been made ready at his home, where he had been dressed in his best clothes, suitable for church. A wedding. Or a funeral. His body had been rubbed with lovage. Protection. His coffin lay uncovered on trestles beside his grave.
The sexton had worked hard on this grave, harder than on Radu's, but then this was a proper burial, in the graveyard, overlooked by the sow-backed church with its wooden-tiled roof and sharply pointed onion dome. People crowded around, leaning on the fences, hemmed in between other graves, each of which had a wooden grave marker. Most of the markers were brightly painted crosses, set under small wooden roofs to shelter them from the worst of the weather. These little houses were painted too, and bore inscriptions concerning the occupant. A very few of the graves in the yard were stone, the resting places of the richest citizens of Chust.
Around the coffin stood the mourners, around them lay the graveyard, and outside the graveyard lay the village. Beyond all of this stood the endless silent forest, watching the Wedding of the Dead, seeing all, saying nothing.
Peter wrestled to get as close to Agnes as he could, but he was still far from her. Even so, he felt Agnes's loneliness from where he stood. It was as if her forty days' segregation had begun already.
As Daniel intoned the opening words of the wedding service, Peter saw that Agnes was trembling. At various points in the service, she had to make responses, but though Peter stood on tiptoe and craned his neck forward, he couldn't hear what she said. Maybe he was too far away, maybe her voice was too small. He could only guess at what she was having to say, agreeing to marry a dead man. As for the groom, he was excused from having to make his responses, being in no state to do so.
As well as Agnes and Daniel there was the familiar figure of Teodor, the feldsher, who stood nearby but took no part in the ceremony. Old Anna stood next to him, her cruel, aged face glowering at anyone who dared look in her direction.
The wedding was soon over, and the burial began. As Stefan's coffin lid was lowered onto the box, Peter saw Teodor step forward. Daniel reached out and put a hand on his arm, as if trying to stop him from approaching the coffin. Though Peter couldn't hear what they said, he could tell there was some argument between them. People began to grow agitated; they shifted uneasily, muttering. At last Daniel appeared to relent. Teodor stepped forward and placed various items inside the coffin, along with the body. A net, some whitethorn, and small figures like a child's dolls. Then the lid was hammered into place and the whole thing put in the ground. As it went, the mourners began to sing, spontaneously, of one accord. They sang the Miorita.
At first their singing was quiet, but as the verses told of the shepherd's fanciful version of events, of his marriage to the princess of the stars, their voices grew louder and more rousing, until Peter found that despite his skepticism, there were tears in his eyes.
_"At my wedding, tell how a bright star fell,_
_Sun and moon came down to hold my bridal crown."_
As the singing reached its climax, a single image was left in Peter's mind. The princess from the stars. The young shepherd had found his magical bride, even in death.
Peter woke from his dream of the princess. The burial was over and he began to push through the crowd toward Agnes. He was cursed for his lack of manners, and pressed in on all sides by the crowd swarming through the graveyard. Looking to see where Agnes was, he saw with alarm that she was being led away by Anna and the other Elders.
"Agnes!" he called, but it was no use. She was too far away, and the Elders were taking her straight to the hut. There she would begin her mourning. Peter, imagining her dread, watched her disappear. It was said that she should speak to no one while she was in mourning for her husband. In this way, after forty days, it would be understood that she had mourned her husband for a lifetime, and she could adopt the position of a young, unwed maiden once more.
Desperately Peter made one last effort to push through the crowd. He managed to fight to within a few feet of Agnes, but here his way was barred. The Elders formed a procession around and behind Agnes, a cortege to guide her to the hut. Angry faces turned on him as he tried to force his way through.
"Agnes!" he called, and at last she heard. He saw her turn and begin to pull at her veil, desperate to see him.
"Get away from her, boy!" someone shouted sternly.
"But she's my—"
"She's nothing to you anymore. Not now! She's married someone else!"
Peter wrestled, trying to protest, but a fist struck him in the back, and then another in his side, near the kidney.
He collapsed, gasping for air. As he fell he caught a single glimpse of Agnes. She had succeeded in wrenching the veil from her face, a face that was now wreathed in horror alone.
**18**
**At the Threshold**
Peter limped wearily home from the wedding.
He had decided that Sultan needed a rest, and had walked all the way, along the forest path that led home. The trees crowded in on him, silent but strong, and once again Peter had the sense of being watched. He shook his head free from the feeling; he had more pressing things to worry about. His side and back still hurt from the blows he'd taken.
He staggered across the bridge, and let Sultan find his own way to his stable.
As soon as he crossed the threshold he knew things were wrong. Tomas lay on the floor of the hut, his eyes open.
"Father!"
Peter rushed to him.
"What happened?"
He smelt the drink that clung to his father's clothes, to his breath. A smashed stone jar of slivovitz lay nearby, its dregs oozing into the earthen floor.
"I can't move my arm," Tomas said, "or my leg."
He nodded his head at his left side, on which he was lying. His eyes looked at Peter wildly, like those of a frightened dog.
Peter was scared, and what scared him the most was seeing that his father was afraid. It was not something he had thought possible.
"Help me up," Tomas said.
His father was very heavy, and his being a dead weight, unable to move two of his limbs, made it hard to lift him properly. Despite his strength, it was all Peter could manage to drag his father to his bed and haul him onto it.
"The drink," Peter said as gently as he could, though he felt angry inside. "The drink did that to you."
"Nonsense," Tomas spluttered out. "I had a fall."
Peter said nothing. Tomas did not have falls. But then, his hands never used to shake either.
He didn't believe his father, but he didn't want to fight him. He needed to keep things simple. Practical.
"Are you in pain?" he asked.
"A little," Tomas said. "Nothing serious. Just can't move my damn arm."
Peter pulled the covers from under his father, and put them over him. Then he went and stoked the stove, and made some soup. By the time he had done that, Tomas seemed slightly better.
"I think I can move my fingers," Tomas said. "Yes? Are they moving?"
Peter wondered why Tomas couldn't tell for himself. He didn't want to think about what it meant. He looked at his father's hands, but could see no movement at all.
"Yes, Father," he said, "I think they are moving."
With that, Tomas had exhausted himself. He fell asleep, but even in his sleep he tried to move his fingers, as if to close them around something, something like the hilt of a sword.
Dreams rode like wild horses through Tomas's sleep, dreams in which he himself was riding, and riding hard.
Riding out for a reason, for a cause.
A good cause.
**19**
**Turnings**
The days passed.
Tomas recovered, slowly at first. He had taken soup from Peter after waking from the accident, and had seemed more lucid. Peter was surprised that though Tomas had refused to attend the Nunta Mortului, he had asked about the wedding, and how Agnes was.
On the third day, Tomas got out of bed for an hour or so, moving his arm and leg freely once more. He even went out to talk to Sultan for a while.
Peter was worn out, for it fell to him to do all the work he could, as well as nurse his father and make two trips into Chust to deliver logs, collect money, and buy food. In the village he tried to inquire after Agnes, but no one would even meet his eye, let alone talk to him. But then, what was there to say? Apart from a basket of food that was left silently on the windowsill of the hut every morning, no one went near her. No one had spoken to her, no one was willing to talk about her; no one was allowed close.
There was an ominous mood in Chust. Something had changed, and a place that was dour at the best of times had become even more cold and unwelcoming. Peter knew why: fear. More cattle had been attacked—two were found with dried blood caked on their forequarters—and a couple of ewes had been killed, drained of blood. It was not the work of wolves, but no one would say more than that.
When he went to talk to Agnes's neighbors to see how her mother was, they refused to speak to him at all, merely indicating that they would look after everything. Peter felt the village excluding him, felt barriers that he couldn't break. One shrewish old woman bluntly told him that he and Tomas weren't wanted. They had always been outsiders, Peter knew that. Now that darkness had descended on the village, anything strange, anyone foreign was a target.
Had Agnes's mother received more visits in the night from Agnes's father? Again, no one would tell him.
As time wore on, Peter grew anxious and restless. Tomas, as he got better, needed his son's help less and less, so Peter was free to worry about other things.
Agnes had been shut up in the hut for six days. The hut lay beyond the little thatched fence, beyond the threshold, beyond the safety of the village, and was no place for a young woman even in normal times. And now something evil was happening around Chust. Tomas dismissed all talk of the Shadow Queen as nonsense, but the villagers believed in her, and whether or not she was real, the result was the same: fear and suspicion had crept into Chust like an outbreak of plague. Soon everyone would be infected, and Tomas and Peter would have to move on again, back to their old nomadic life. Running, always running, though Peter still didn't know what it was they were running from.
By the time Tomas swung an axe again, Peter had made up his mind.
He couldn't go near the hut in daylight—it was just visible through the trees from the edge of the village, and he knew he couldn't take that risk. If he got caught, the very least that would happen would be that Agnes would have to start her mourning all over again. But he was going to visit Agnes.
On that first day out with Tomas he worked hard enough, but reserved his energy as much as he could. Tomas didn't seem to notice. Peter had sensed a change in his father: he seemed to have retreated into himself. He was quieter than usual, and was even drinking a little less. Peter wondered how much, if anything, it had to do with the accident, or with the visit from the Gypsies, but Tomas wasn't telling.
That evening, Tomas drank and Peter ate stew, and they both stared into the fire in the potbellied stove, thinking their own fireside thoughts. Then they went to bed, and while Tomas was soon snoring heavily, Peter lay awake, thinking, and waiting.
When he was sure Tomas was sound asleep, Peter swung his legs out of bed, and by the faint glow coming from the stove, slipped his boots on and left the hut. Outside, he pulled the door to again and waited for a moment, listening; but he need not have worried, his father was still snoring just as loudly as before.
Once again, he left Sultan where he was; the noise of getting the horse from the stable might be enough to wake Tomas.
The bridge lay picked out by faint starlight, and Peter cautiously slipped across the planks, the pure water gurgling past underneath.
It took him a while to reach the village, but he wasn't going in tonight. Instead he chose a path that ran along the eastern edge of Chust, and set off around it. As the boundary fence curved here and there, so did the path, and Peter didn't hesitate. It wouldn't do to hesitate. If he had stopped to think who or what might be out in the night forest, he would never have left the safety of the hut. In the last few days there had been two murders, and wolves were the least terrifying of the possible culprits. Peter hurried on, and pushed thoughts of anything and anyone else to the back of his mind.
Another few minutes and he saw the shape of the church hunkered in the darkness. He went on past it, and then slowed. Somewhere soon, he knew, he would meet another, smaller path running out from the village and up to the hut where Agnes had been left.
As his pace slowed, he began to wonder if there was something wrong with his eyes, because suddenly it seemed he could see much less than before in the dark. He lifted his face to the sky, trying to see what light there was, and felt snowflakes brush his face. Snow clouds had moved in, taking almost all the light away.
Just as he began to doubt he would be able to find the path, he heard the crunch of grit and pebbles underfoot.
He turned to the left and hurried up the path, which rose steeply, toward the edge of the trees. There, a stone's throw back inside the forest, was the hut.
**20**
**Hands in the Dark**
"Agnes," Peter called, as loudly as he dared.
Nothing.
Peter stood at the door of the hut and wondered if he should try to open it and go in. But maybe better not; it might startle her.
"Agnes," he tried again, a fraction louder this time.
"Peter? Is that you?"
She was awake.
"Agnes, it's me, Peter," he said, feeling stupid. "Can I come in?"
He heard movement inside and felt the door shift slightly as Agnes leant against it on the other side.
"I can't let you in, Peter," she said. She sounded miserable. "You know I have to stay here by myself. And anyway, the door's locked. They locked me in."
He tried the door. Shut tight.
"Come around to the window, Peter. We can talk there."
He crept around the side of the hut, feeling his way by running his hand along the rough wall. He heard the creak of a wooden hinge as she opened the shutter. Suddenly her voice was right above him.
"Peter! Here!"
The window was small and quite high up, about head height. It had a wide windowsill, where Agnes's baskets of food were left. With an effort, she could have climbed out of the window and escaped, but the Elders knew what they were doing. Locking the door was merely symbolic; they knew she could not escape even if it stood wide open. She had nowhere to go, and would not be allowed to return to village life until she had completed her mourning properly.
"Peter," Agnes said, sounding a little calmer. "Can you feel my hand?"
He felt around in the darkness, and there was her arm. He slipped his fingers along her sleeve until he felt her hand.
"Agnes, let me in. I can't stand the thought of you alone in there. It's not safe."
"No, Peter," Agnes said, but her voice wavered. "You know what will happen if we're caught. And I'll have to begin all over again."
"But it's not fair. Why did _you_ have to marry Stefan? Why not someone else?"
"Because Anna chose me. When you've lived here a bit longer, you'll understand that that's how it is."
Peter said nothing. He had lived in Chust long enough to understand that the old woman's word was as good as law.
"At least you feel a little warmer tonight," Agnes said.
He froze.
"What did you say?"
"Your hand," Agnes said, innocently. "It feels warmer than last night."
Peter suddenly let Agnes's hand drop from his, as if it were something dangerous.
"Peter, what is it?"
He hesitated, then spoke quickly, his words catching in his throat.
"Are you saying I came here to see you, last night?"
"Yes, you did. You asked me to—Oh, Peter! It wasn't you?"
Suddenly he felt behind his back the huge darkness of the forest, in which myriad horrors might be tracking him. It surrounded him with almost unbearable menace, a vast world that ruled and ran his life, seeing everything that passed beneath its branches, yet giving away no secrets.
"Agnes. You have to let me in."
He spoke with a quiet strength, but with fear mingled in, and it scared her into agreement.
"Yes," she said. "Oh God! Yes. It wasn't you? Yes. But the door, the door!"
"Never mind. Stand away from the window."
He reached up and groped around, swinging the shutter open and gauging how wide the window was. Small, but he could make it. Putting a foot against the irregular log wall of the hut, he found a foothold and half jumped, half pulled himself though the gap. Then he wriggled and pulled and fell headfirst into the hut, spraining his hand as he landed.
"Oh!" cried Agnes. "Are you all right?"
"Don't you have any light?" asked Peter, standing up. He rubbed his hand, but it wasn't bad.
"No. I'm not allowed. Tell me you're joking, Peter. That you just said that to get in here with me."
Peter said nothing in reply.
"Who was it?" she whispered in horror. "He said he was you! He asked to come in."
"You didn't—?"
"No!" Agnes said quickly. "I wouldn't let you...him...in."
"Thank God for that."
"But who is it?"
Peter shook his head in the dark.
"I don't know."
He went back to the window. From the stillness outside he could sense that it was still snowing, though he couldn't actually see it. The shutter was banging against the outside of the hut. Somewhere there was an iron handle to pull it shut. The last thing in the world he wanted to do was put his hand back out into the night, but he had no choice. Expecting his wrist to be grabbed at any moment, he felt out through the window, found the handle, and pulled the shutter inward, swinging the bolt into place. He turned to Agnes.
"I don't know," he repeated, "but something is wrong around here. Tell me exactly what happened."
"I told you. You...someone came to the window last night. He asked to come in and I said no. He asked again and I said no again and..."
She stopped.
"Oh!" she said.
"What, Agnes?" Peter felt for her in the dark and put his arms around her. "What?"
"When I wouldn't let him in, he asked for a kiss."
"You didn't do it?"
"Peter, I thought it was you. I've been so scared. Anyway I said no, but I let you...him kiss my hand."
Peter swore.
"I thought it was you," Agnes said.
"I know. I know."
Peter felt her tense in his arms. Her head jerked up toward his in the blackness.
"Oh God and the Forest!"
"Agnes? Agnes?"
"He said he'd come back again tonight."
**21**
**Threads**
For a long time neither of them moved, as if expecting to hear a voice at the window at any time. When they were finally convinced they could hear nothing, they began to breathe again.
"Sit down," Agnes said, guiding Peter to the small bunk where she had been sleeping. They sat on the edge of the bed, neither willing to voice their fears.
Peter cursed himself for being so naïve. He could have brought his axe with him. He had tried to believe Tomas, that this was all village superstition, but deep down he had known something evil was afoot.
"Have you been all right here?"
"Yes," said Agnes simply. "But I'm worried about Mother. I've been thinking about her. And about Father."
"Your mother's fine," Peter said quickly. "I saw her yesterday. I spoke to the widow Caterina next door. She was very reassuring."
It was all lies, but Agnes didn't need more to worry about, and as far as he knew, her mother was all right.
"But what have you been doing? It's been a week!" he said.
"Spinning," Agnes said. She laughed. "If you could see the floor of this place. There's enough wool to dry up the river in here. They said I might as well make myself useful. And I started after a couple of days. I was too angry at first. But then I began to get bored and I was grateful for something to do. I must have spun a mile of it by now!"
"And someone brings you food every day?"
"Yes, one of the Elders, I think, but the person doesn't speak. I just hear the basket being left on the windowsill. It's such a small window. I can only see a few branches and a little bit of sky. But at night, I can see the stars in the heavens...." She sighed.
"I'm going to find out what's going on, Agnes. Trust me. I've got an idea."
"But what about...him? If he comes again. He said he would."
"That's my idea. We'll find out what's happening. How much wool have you spun? Really?"
Agnes and Peter waited. Peter had explained to Agnes what he wanted her to do. She wasn't happy, though eventually she had agreed. They waited, and though Peter had often longed to be alone with Agnes, now that it was happening he didn't know what to say or do. Surely there were a thousand things he wanted to ask her? Surely she wanted to talk and talk to him, to hold him and maybe kiss him? But somehow they sat next to each other as mute as stone. Peter wondered if it was because they were both scared out of their wits, but he began to suspect there was another reason. A reason that shocked him at first, but once he had picked it up and looked at it and turned it over in his mind, a reason that he calmly accepted as something approaching the truth.
The truth. That maybe he didn't love her.
For much of the time they sat in silence on the bed, in reach of each other, but miles apart. After a while Peter found his mind playing tricks on him. He saw tiny pinpricks of light but decided it was his imagination. Nevertheless he felt he could have been anywhere; his enforced blindness seemed to remove the walls from the hut, and even the presence of the forest itself receded until he felt utterly alone.
Hours passed, and Peter was just about to ask Agnes if she had any food, when he heard a noise outside. It was clear from the way Agnes shifted next to him that she had heard it too.
Silence for a moment, then: "Agnes? Agnes? Are you there, pretty one?"
Peter's heart pounded. He reached across and nudged Agnes, wordlessly urging her to answer.
"Yes," called Agnes up to the window. "Yes, I'm here."
Her voice was frail and nervous, and Peter thought it was too obvious, but whoever was outside didn't seem to have noticed.
"Let me in, pretty Agnes!" came the voice.
"Who is it?" Agnes replied.
"It's me," the voice said. "Peter."
Agnes sat dumbly next to Peter, the sheer terror of the moment paralyzing her, but Peter nudged her again, willing her to go over to the window. He strained to see in the blackness, all his senses going wild but telling him nothing.
Still she refused to move. He pushed her to her feet, shoving the end of the spun wool into her fingers as she went. He squeezed her hand.
"I can't let you in, Peter. You know that."
"Let me in, pretty one. I'm so cold!"
"I can't let you in."
"I'm so cold. Feel my hand. Open the shutter and feel my hand."
There was silence, and Peter could imagine Agnes rooted to the spot from terror. In his mind he tried to force her to move, to stick to his plan.
"Open the shutter, Agnes, pretty one. You felt my hand last night."
After a long, long pause, Peter heard Agnes move up to the window and unbolt the shutter.
"Here," she said bravely.
"See how cold I am?" said the voice. Peter marveled at it. It didn't sound like him, but it was so quiet that he couldn't have said that it wasn't his own voice either.
"Touch me," said the voice. "Let me in."
"I won't let you in, Peter."
"Then kiss me."
There was another terrible pause, as Agnes steeled herself, trying to be calm enough to go through with what she and Peter had agreed.
"Very well," she said finally, in a tiny voice. "I will kiss you. Wait a moment."
Agnes moved and found the small stool she sat on to work. She pulled it to the window.
Peter waited in an agony of fear, paralyzed by inaction. All he could do was pray to the Forest to protect her, if that was who he should be praying to.
He heard Agnes climb onto the stool. Then she leant through the window. He heard the faint noise of the thread starting to slip out from the huge winding of wool on the floor, and silently he prayed that his idea would work.
There was a moment of total silence, and Peter tried not to think of what was happening. He couldn't hear the kiss.
Then Agnes shrieked.
"You're so cold!"
"Come here!" said the voice, suddenly loud, angry and vicious. "Let me in, pretty bitch!"
There was the sound of a struggle and thuds fell against the wall outside. Agnes screamed and fell back into the hut. Peter now dared to stand and pull the shutter back into place.
"I'll be back," said the voice, shrieking in rage. "I'll be back tomorrow night!"
Silence.
**22**
**Calling**
For a long time, neither Agnes nor Peter dared move. Eventually Peter crawled over and found her huddled on the floor. He held her gently and then realized he could hear something.
The wool was being pulled out slowly from the winding.
"You did it!" Peter cried. "Well done!"
Agnes was silent.
"You did it."
Peter went over to the shutter, and felt the wool paying out through the gap between the shutter and frame. It was not moving fast, or even that steadily, but it was moving.
Being careful not to snag the wool, he opened the shutter again, and saw that the snow had stopped. The sky had cleared and there was enough starlight to see the outlines of the trees. He spent a long time looking for the terrible visitor, but could see no sign.
Faint light was spilling onto the floor of the hut now and he checked the pile of wool. Agnes had been busy; there was enough wool to stretch to Turkey, as far as he could tell. Making sure it could move freely from the skein that Agnes had coiled from her spinning, he turned to her.
"Agnes. It's time for me to go. Stay here. I'll be back soon."
He lifted Agnes up and placed her on the bed again, pulling the blanket up to her neck.
She turned to face him.
"Don't go," she said, her voice small and still.
"I have to. This is what we agreed. You've done your part. Now I must do mine."
He took her chin in his hands and tilted her face up to his.
Agnes shivered.
"I kissed him, Peter."
"You did what you had to. You fixed the wool. That's all that matters."
"He was so...cold. So..."
But she couldn't explain what she had felt.
"Stay here," Peter said, and leant down, kissing her forehead. "It'll be light soon. That will make you feel better. Close the shutter when I've gone."
He got up and, without another word, set the stool upright by the window once more and climbed out, slightly more easily than when he had entered, earlier in the night.
Once outside, it was easy to tell that dawn was still far off, and it was hard to see clearly. But Peter smiled to himself. He didn't need to see, he just had to follow. Agnes had done her job well. The distaff that she had been spinning with had a metal clip on it. They'd broken the clip off and tied the wool to it. In that awful moment as she leant out of the window, she had fixed the clip onto the back of the jacket of her nocturnal visitor.
Now all Peter had to do was follow the wool, and he would find the culprit. He tried to tell himself it was probably just some young fool from the village who had a desire for Agnes, but nonetheless he wished he had his axe with him.
Peter followed the wool, threading his way through the trees.
Agnes lay on the bed in the hut, unable to move. Overwhelmed by fear, she blinked in the gloom for a long time, powerless to get up and close the shutter as Peter had told her to. Her mind was occupied with a single thought: she had kissed the thing at the window. She could still taste something on her lips, something foul. At last she made a small movement and wiped her lips with the back of her hand. It felt no better, so she did it again. And again, and again, and then frantically she began to scratch at her face, desperate to rid herself of whatever disgusting coldness it was that clung to her.
She rolled onto the floor and crawled to find her jug of water, wasting it all trying to wash the taste from her lips. Then she heard a noise at the window.
She lifted her head as she knelt on all fours, like a dog getting a scent.
"Peter?" she called. And then panicked. It had to be Peter. Who else could it be?
"Peter! Come in and help me! Come in!"
**23**
**Things to Cover Our Dead**
Peter stopped, to check the wool. It lay slack.
So. Whoever it was, was back home, and Peter knew that every step he took now was a step nearer the mysterious visitor.
He checked the sky. If only dawn were closer. The promise of light struck at his heart. He longed to see the sun, for what evil can occur by daylight?
Nonetheless, by starlight he could see the village in front of him, and now he could even see the wool stretching away toward the village. His breath quickened. It would be soon.
Picking up the pace once again, he hurried on, letting the wool run freely through his hand.
He came to the first houses and saw that the wool ran away up a small alley that he had never noticed before. He must never have made a delivery there, but it didn't matter. He didn't need to know where he was going, he just needed to follow the wool. Once again he praised himself for his quick thinking in the hut, and thought of Agnes. At least she would be safe for the time being. Her assailant was somewhere out there ahead of him, presumably climbing angrily back into bed. Well, he would be angrier still when Peter had finished with him.
He followed the wool up the alley, moving more slowly than before, taking care not to make a sound. He was in luck. The snow that had been falling through the night had been gentle but persistent, and enough had fallen to recarpet the streets with a blanket that hid any noise he might have made.
Something bothered him as he padded through the snow, but he couldn't place it. A few more steps and he turned around. Behind him he could see his footsteps in the fresh snow. He looked forward again, and there was the wool running in front of him.
So why couldn't he see any footsteps from the man he was following?
The wool turned a corner into a wider street that he knew well. He'd been convinced that it was going to lead to one of the houses he'd already passed, but the trail showed no sign of ending. Ahead lay the back of the priest's house, but the wool ran on beyond that, and around another corner.
He hurried up the street, glancing at the tarred windows of Daniel's house as he did so, then turned the corner.
He stopped dead.
The wool led away. There were no houses left. There was only the church before him, but that was not where the wool was taking him.
In the half-light he could now see the grayish line snake out across the purer whiteness of the snow. The wool caught on a stone here, and on a fence there, but it was unbroken as it led the way, surely and utterly, straight into the graveyard.
Now, moving as if in a nightmare, Peter's feet stepped unwillingly forward. The wool felt like wire in his hands. Maybe it was just that it had been frozen in the snow, but it seemed to cut into his skin like metal.
He came to the gate of the graveyard. There could be no doubt. The wool ran over the fence next to the gate, as if his quarry had sailed clean over it. Dumbly, he gripped it, as if it were a lifeline leading him to safety, when in reality it was leading him toward death itself.
The wool wound its way between this grave and that, snagging on crosses, trailing on the ground. At last, his eyes wide open in horror, Peter saw its destination.
There, no more than five feet away, was Stefan's grave. The wool not only went right up to the grave, but disappeared into the soil itself. Then Peter saw that though there was snow all over the graveyard, and on the other graves, Stefan's was, for some reason, free of it.
An awful self-destructive curiosity pulled Peter closer. Unable to stop himself, he got down on hands and knees and crawled the final few inches toward the grave. As he approached, something else caught his attention. There was a hole in the soil at the head of the grave, near the cross. The hole was about the size of a small fist, and it was perfectly circular, like a rat hole in a riverbank.
Peter leant over it.
He looked in.
There was just enough light to see inside the hole.
At the bottom he saw an eye.
It was open, seemingly lifeless, though looking straight at him.
Then it blinked.
Peter screamed and ran as if the Devil himself were chasing him.
**24**
**The Hut**
At first he ran blindly, not thinking where he was going. Not thinking at all. He blundered out of the graveyard to the edge of the village once more, and then he knew where he had to go.
Only once did he stop and look behind, but he couldn't see anything, and neither could he hear anything. And that was some comfort. But what comfort could there be for what he had seen at the bottom of the dreadful hole? That cold, dead eye.
Was that Stefan in there? Dead? Or, even worse, maybe, alive?
Agnes! He had to get to Agnes and warn her. Get her to leave the hut.
It didn't take long for him to stumble through the trees, retracing his steps around the edge of the forest and to the hut.
He ran straight to the window.
_Silly girl,_ he thought, seeing the shutter hanging open. But then a worse thought pushed into his mind.
He jumped up at the window, once again landing uncomfortably halfway over the sill.
"Agnes! Agnes!"
But already he knew she had gone.
"No!" he shouted. "Agnes, where are you?"
He dropped inside the hut, frantic, praying that she lay horror-struck in a corner; but she was not there.
Overwhelmed by fear, and tired, he suddenly felt utterly powerless. He forced himself to stay calm. He had to find her. She had gone. Or maybe she had been taken....
Whatever had happened, he had to find her.
Yet again he made ready to climb from the window of the hut, and then he saw something that froze his blood.
No more than twenty feet from the hut, and heading straight toward it, was the figure of someone he knew to be dead. Radu, the woodcutter. So it was not just Stefan who was out there. How many were there?
Peter gasped, and dropped back into the hut, terrified.
There were noises on the roof. It took him a moment to realize the thumps were footsteps. There was another of them on the roof too!
He looked to the shutter. Getting to his feet, he waded clear of his terror and made it to the window. He saw Radu nearly at the hut. Suddenly a face appeared, upside down, in front of him.
"Help me!"
It was a face he was glad to see. Sofia, the Gypsy.
He put his arms out and pulled her, dragging her through the window. They collapsed in a heap.
"Quick," Peter shouted, "the shutter!"
"Wait," Sofia cried, and before Peter could do anything, she snatched something from a bag around her waist and flung it out of the window. Only then did she tug the shutter closed and bolt it tight.
"All right," she said. "I hope."
Once again it was dark in the hut, and Peter had no idea what she was talking about, or even what she was doing here.
"I made a circle of it. Right around the hut."
"Of what?" said Peter, at a loss.
"Millet seed," said Sofia simply. "We'll be safe for a while. Just pray the sun gets here soon."
"It's at least two hours till sunrise," Peter said, "and I don't see that millet will save us from anyone."
"Really?" Sofia said. "So have a look for yourself."
Peter didn't move.
"Go on, have a look!"
Peter crept to the window and peered through a crack in the shutter. Whether it was starlight, or the moon showing at last, he didn't know, but there was enough of a silver-gray light outside to illuminate a mysterious scene. There, on the snow-covered ground, he could see thousands of millet seeds, forming a circle around the hut, just as she had said.
Sofia talked to him as he peered through the crack.
"One of them's been in here once already tonight, I think. That's why I used the seed."
"But what are you doing here?"
"Looking after you," Sofia said.
"What?"
"I saw you coming up from the village, and then I saw him." She nodded toward the window. "I climbed a tree, dropped onto the roof, and got as much of the stuff around the hut as I could before he got here."
Through his spy hole Peter watched, wide-eyed in horror, as Radu knelt in the snow, picking the seeds up, one by one, placing them in his pockets. Every now and then he glanced up in Peter's direction, and although Peter knew Radu probably couldn't see him, the look of malevolence on his pale face terrified Peter even more.
Sofia, in contrast, seemed calm.
"He can't come in till he's picked them all up."
"And what if he does?"
She didn't reply.
"And what if he does?" Peter cried, turning away from the crack.
Radu was out of sight somewhere, randomly working his way round the hut. It was even more frightening to Peter to know he was out there but not be able to see him, and he could bear it no more.
"He's dead, Sofia! That man is dead. I went to his funeral!"
"I know," she said, frankly but gently.
"That's no answer! I don't understand. How can he be out there when he's dead?"
"I don't know either. But he is. We call people like him 'hostages.' He is dead and he is out there. And he is trying to get in here."
"But it's not possible."
"Did you not see him with your own eyes?"
"Yes, but—"
"Then, Peter, you must understand that it is possible."
Peter turned back to the window, to the crack.
Radu was in sight again, still slowly working his way through the seed. His fingers were swollen and clumsy, and he was making heavy going of it. His skin was blue, in places almost black.
"And if the sun comes up before he finishes, then we're safe?"
"Yes," said Sofia, "for the time being."
Peter wheeled around on his heel like a trapped animal looking for an escape, but there was none.
"And you? What do you mean, you're looking after me? I don't understand."
"Be still, Peter. We must be calm."
"Calm? How can you be calm?"
Sofia put her arms out wide, a gesture of submission.
"Peter, you think this is easy for me? You think I am not scared enough to drop dead right here? Because I am. I am. But I have something you do not. I have knowledge. I have done this before, many times. But trust me, you will need all your wits about you. You will need to be calm, in order to live. Do you understand me?"
Peter shook his head in disbelief, but he understood.
"But what about Agnes?" he said. "I must find her."
"I think it is probably too late for your friend."
"How can you say that?" Peter cried. "What do you know?"
"I know that she has been taken from here. Given what you have seen, you should understand. We can do nothing. For the moment we are trapped. If we can get out, then that is a start. That might be of some help."
It was almost too much for Peter.
"What do you mean?"
"If we can get out of this, so good. But there is a greater evil at work. There are bigger battles to be fought."
Something clicked inside Peter.
"You mean the Shadow Queen, don't you. But I don't understand."
"No," said Sofia. "I know. Your father has spent your whole life stopping you from understanding."
"What do you know of my father?"
Sofia was silent for a moment.
"More than you do, I suspect."
**25**
**The Winter King**
It was strange. Even as it was happening, Peter knew it was strange. It was like sitting in the center of a hurricane. Outside, a man whom he had seen buried was prowling around, intent on doing them harm, and prevented from doing so only by millet seed. Inside the hut, in relative safety, he sat quietly, though not peacefully, with a girl he barely knew, as she told him the story of his father.
"Have you heard of the Winter King, Peter?" Sofia asked.
"Yes," he answered. "It's a story. The king who'll save us all from every evil. He was supposed to have saved the land from the Turks. Everyone knows that story. But it's just a story that the peasants tell each other."
"The peasants? That's not you talking. That's your father. It's more than a story. Your father could tell you that the Winter King is real. Or was. Your father fought with him."
Peter laughed.
"Don't be foolish," he said. "My father fought with King Michael. They fought the Turks."
"That's right, Peter. King Michael was the Winter King. That was thirty years ago, no more. But memories are short when lives are short. Already the King has become a legend.
"The Turks were greater in number, but the forest in winter is a treacherous place for the unwary. They were overcome by King Michael's men. Massacred. Some escaped and slipped away into the depths of the forest, never to be heard of again. The Mother Forest dealt with them. When her anger is aroused she takes no prisoners, but it wouldn't have happened without the Winter King."
Peter nodded his head. He understood what she meant about the forest, and thought about why he made his little carvings, to give something back. It would never do to betray the forest's generosity; Peter believed that those who thought the forest was simply a gathering of trees were foolish, unwise, and that there was something else that gathered among those trees.
"The Winter King," Sofia said, "who will save us from all evil. Now he must save us from the Shadow Queen. His greatest battle ever."
"But he's dead. King Michael is dead."
"Yes. He died and the new king was weak. He let the country crumble into factions, no longer unified. In the chaos that followed many bad things happened. Fighting between men who had been allies. And your father was put in jail."
"How do you know this?"
"I know because your father fought alongside my father."
"A Gypsy? Fought with King Michael?"
"A Gypsy, yes, Peter." Sofia glared at him. "What is so wrong with that? There is more to some of us than there might seem."
"And your father is here now? He spoke to my father that night when...?" Peter stopped.
"My father is dead. He died in jail when I was just three years old. My uncle leads us. My uncle, Milosh. And yes, he went to speak to your father that night, when we met on the road."
Peter remembered it all too clearly. He hated himself, with Agnes missing, but he couldn't help remembering what had happened. What he had felt. Blood rushed to his face as he remembered how he had carried Sofia, cradled her arms and long slender legs, and how she had held his hand.
"I'm sorry about your father," Peter said, but Sofia merely held his gaze, a sorrowful look on her face.
"That night," Peter said, quietly.
"What of it?"
"That night, when you—"
Sofia interrupted him.
"Don't think anything," she said flatly. "I was sent to delay you from returning home, so that my uncle could speak to your father alone. I did what I had to do. My uncle has been following your father for years and he didn't want anyone to get in his way."
Now Peter was angry. With himself. With Sofia. Too angry even to ask why her uncle had been hunting Tomas. Was that why they'd always lived on the move? Always keeping to the edges of the civilized world? He was not surprised at what she said, and yet he knew he was disappointed too. An image of Agnes flickered into his mind, and though he tried to push it away, he could not do so entirely.
Frustrated, he turned back to the crack in the shutter.
Still Radu crawled around. Peter tried to work out if more than half the seed was left on the ground, but Radu's feet had turned the snow to mush, making it hard to see anything clearly. He wondered if it was his imagination, but it seemed to him that Radu was moving faster than before.
He turned back to Sofia.
"And now?"
"What do you mean?"
"So the Shadow Queen is coming. Making dead people walk again. To make us like them? But the Winter King is dead. How can he save us now?"
"King Michael is dead. But the Winter King lives. In us. In your father. In us, the Gypsies. Even in you, Peter. We all belong to one another, to the ancestors, and we can fight. We have fought for as long as I can remember. Moving, traveling, fighting. We live the life of Gypsies, but we fight the fight of the Winter King."
She stopped.
Peter shook his head, sighing. It was too much. He didn't want to be here, didn't want to believe what he was hearing—a story coming to life.
"Peter. You must join the Winter King. We need you. We need your father."
"Why?"
"Because your father was the finest warrior in King Michael's army. He was famed for it. And he had something else. A sword. A Turkish sword. He found it on a campaign far into Turkish territory in the summer before that final battle in the winter forest."
"A sword? What sword?"
"A fine sword, Peter. One that is perfectly balanced. It is as light as the wind, yet as hard as the winter. But there is more to it than that. It stops them."
"What do you mean?" Peter asked, still confused by everything Sofia was revealing to him.
"People like him. Outside. The woodcutter. The sword stops them. Returns them to the soil, for good. It was forged in a land often plagued by such people. There they call them _vrykolakoi._ Here we call them _nosferatu,_ or _moroii._ It is all the same. They are all hostages. And once, it is said, they were as common as the blades of grass in the meadow, or berries in a pail. In every land they have a thousand names. It doesn't matter; it is up to us to stop them."
"Us?"
"Us," said Sofia. "The ancestors. All across the land there are groups of us. Some are Gypsies, some are soldiers, some are common people, some are priests. It doesn't matter. Those who fight the hostages are all ancestors—my uncle, my father, your father. Even you, Peter."
Peter shook his head, incredulous.
"We need your father's sword," Sofia said.
"My father was no warrior. My father has no sword," Peter answered. "I would have seen it."
But even as he said the words, he thought of the box his father had kept from him all these years. Was it possible? Was there really a sword inside?
"Perhaps," said Sofia. "But wherever it is, we must find it."
"My father is no hero," Peter said bitterly, still refusing to believe. "He fought with Michael, that is true, but he is not a great soldier. My father is a drunkard."
Sofia stared at him, but Peter could not fathom what she was thinking.
Now something else struck him. Saying nothing, he moved yet again to the crack in the shutter.
He stepped back, as if he had been stung.
"Sofia," he said, "look!"
She came and pressed her face to the hole, and gasped.
Outside, Radu was busy picking up the seeds. But Peter's fears had proved true: Radu was moving more quickly. His hands flew through the snow, sending small flurries all around him.
It was not possible, but it seemed that he grew faster with every passing minute, until he whirled around the hut like a dervish, faster and faster.
The remaining seeds grew fewer, and fewer. Very soon, there would be none left at all.
**26**
**Escape**
"What are we going to do?" Peter cried.
"I don't know," Sofia said quietly. "I'm thinking."
"There's no sign of dawn. He'll be finished long before!"
"Listen to me. How far is it from here to your hut?"
"Not far," said Peter, "but far enough. Why?"
"I've got a little of the millet left. If we throw it behind us as we run, he'll have to stop and collect it."
"This is madness!"
"Can you break the door down?"
Peter looked at the door and nodded.
"I think so. How much seed do you have left?"
Sofia showed him the remnants in the bottom of the little sackcloth bag. He didn't like what he saw.
"Very well," he said. "I'll break the door down as soon as he's on the far side. I'll lead the way, you throw the seed. And I warn you, I am a fast runner."
"So am I," said Sofia, tipping her chin up. "A kiss for luck?"
But Peter was in no mood to play her games.
Sofia moved to the shuttered window, her hand raised, waiting till she saw Radu move away.
"Oh!" she whispered. "He's finished!"
"Then it's time to go!"
Peter ran at the door and jumped, hitting it squarely with both boots near the lock. The wood was in fact quite thin; splinters flew in all directions, and the door smashed open with a loud crack.
Peter landed in the snow amid the wreckage of the door, and scrambled to his feet. He heard footsteps immediately behind him and for a second thought it was Radu, but Sofia passed him in an instant, flinging a few grains of millet as she went.
"Come!" was all she had time to yell, and Peter followed.
Within a few strides he had caught her, and grabbed her hand, pulling her on. He threw a glance behind. What he saw made his limbs want to seize and stop, but he forced himself to run. Hardly more than a breath behind them, Radu followed, lurching through the snow.
Sofia flung another handful of seed, which struck Radu in the face. He howled, hurling himself to the ground, scrabbling to pick it all up.
Now at last they put some distance between themselves and their pursuer. They did not dare slow down, and sickeningly, as Peter looked back once more, he could see Radu closing on them again with shocking speed. If he could run that fast, Peter thought, what in the name of God was his strength like?
Fleetingly, Peter wondered how they would be any safer in his father's hut than in the one they had left, but they were running too hard to gasp a single word to each other, and Peter decided that if they could reach the hut, he could perhaps get to one of their axes before Radu got to them. And his father? Maybe he could help them. Was there really a sword in that box? If what Sofia said about him was true...
Even as he ran, Peter knew that was ludicrous. His father was a drunkard, who must have malingered his way through his years under King Michael. He was no use.
Sofia shouted at him.
_Stop shouting,_ Peter thought. _Just run!_ He bounded a few more paces through the trees before realizing what she had said.
"It's all gone!"
They were on their own now, with only their legs to keep them from harm. Radu groped in the snow, then stood. He had all the seeds.
Sofia shrieked and, for a few steps, overtook Peter.
"There!" he shouted. "The hut!"
Had he turned he would have seen Radu close behind, his hands clawing out toward them, inches away.
A dozen more paces would see them over the bridge. But Radu was on them. Seeing that Sofia was the easier target, Radu flung himself at her. Sensing the attack, she dodged to the side, but she had misjudged her distance from the river itself, and the bank gave way under her foot.
She slipped and tumbled into the water with an almighty splash.
Peter stopped, only feet from the bridge. He hesitated, seeing Radu standing on the bank, looking at Sofia in the water. Then Radu spun around and saw Peter, and made for him.
"Go!" shouted Sofia from the water. The current was taking her the wrong way, away from the hut and the island on which it stood, but she seemed to be swimming toward the island.
Peter didn't need telling twice.
He ran over the bridge, screaming.
"Father! Father! Wake up!"
Peter made the hut, not daring once to look back, desperate to find his axe; but as he burst through the door, it suddenly occurred to him that he had heard no footsteps on the bridge behind him.
He turned in the doorway, and saw Radu on the far side of the bridge, shaking his fists at him but making no effort to cross. Peter watched, confused and relieved in equal measure, as Sofia reached the island.
"Help me out!" she called to Peter, angrily.
He ran over to her and, wrapping his hands around her wrists, pulled her in one long motion from the water and up the steep bank.
"What in the devil is going on?"
Tomas staggered from the hut, a lamp in his hand, pushing the hair back from his eyes with the other. He had been dragged from a bottomless sleep, and was not amused.
"Father!" Peter cried. "It was after us! Look there!"
But when Peter pointed to where Radu had stood at the edge of the bridge, there was nothing but the rustle of bare branches in the half-light.
He had gone.
**27**
**The Island**
Peter stood, panting heavily. He began to shake and for a while was unaware of his father shouting.
Sofia was wet through with icy river water. She moved to Peter, who acknowledged her with a lifeless smile.
"What in God's name?" Tomas said. He grabbed Peter by the scruff of his neck. "What are you playing at? Who's that?"
Sofia stepped right up to Tomas, ignoring his rage.
"You know me, Tomas!" she declared. "I am Sofia, Caspar's daughter."
For a moment Peter thought he saw a glimmer in his father's eyes, but then it was gone.
"I don't know what you're talking about," Tomas said, deliberately. "I don't know anyone called Caspar."
Sofia fell back suddenly, as though he had struck her.
"Liar," she said.
Tomas lifted his hand in fury, but Peter stepped between them.
Tomas tried to push him aside, but Peter stood firm, though his legs shook.
"Father," he said, "why are you angry? If this girl is nothing to you? Or do you know her?"
Tomas spun away.
"Get her off here!" he spat.
Peter pulled Tomas back and was surprised by how easy it was. He looked at his father's face as if for the first time. Tomas's face was ruddy and swollen from drinking, his nose pockmarked; broken veins showed in his cheeks. He stank of drink. He was old.
As they stood facing each other, Peter became aware that daybreak had come. A few low streaks of sunlight pushed weakly through the trees, gules dappling the roof of the hut here and there.
"Father," he said again, more quietly this time, "Sofia says you knew her father, that you fought with him, for King Michael. Is that true?"
Tomas stared at his son.
"She says you were put in jail after the war," Peter went on. "With her father. And she says that you have a sword. A sword that stops these people who have come back from their graves."
Tomas blinked, then walked away, still mute.
Peter wouldn't give up.
"What is it? What are these people, who won't stay dead? Father?"
"Nonsense," Tomas said over his shoulder. He moved toward the door. "All nonsense, and Gypsy tales."
"No!" Sofia cried. "No. Look at me! I am soaked to the skin. We were chased by a dead man. He chased us here!"
"Nonsense," Tomas said again.
"No!"
Sofia, seething, stepped toward Tomas, but now Peter stopped her, grabbing her soaking-wet sleeve firmly at the elbow.
"Sofia," he said gently, "don't."
"What, Peter? You too? Do you think this is a Gypsy tale? Your father knows it's true. Ask him! Ask him why he built a house on an island if it's all nonsense!"
Peter's hand dropped from her arm.
"What do you mean?"
"Do you need to ask? You saw the dead woodcutter stop at the water. You saw it with your own eyes. They cannot cross running water, which is why your father put himself on an island in a river. Ask him!"
Peter was cold and tired, shaking violently now, and yet his heart had just been chilled still further.
"Is that why you did it, Father?" he said. "Is that why you dug the channel?"
Nothing.
Then Tomas turned back from the doorway.
"Get her off here," he said, almost too quietly to hear.
"Father, we can't do that. She's wet to the—"
"Get her away! Go!"
Thrown into a rage, Tomas spat the words, his eyes wild. Just as suddenly tears welled in the old man's eyes, as he stood in the doorway, defeated.
Peter looked at his father and his shame was almost too much to endure.
He turned to Sofia.
"It's all right," she said, before he could speak. "I'll go."
"You can't," Peter said, but she was already crossing the bridge. "It's not safe." Peter lifted his hand to Sofia, but in friendship.
"It's safe enough," she said. "The sun is almost here. There can be no evil by daylight. I must go back to my people."
"Wait!" Peter said. "You'll freeze before you get there."
He was weighing something in his mind.
"Take Sultan," he said at last. "He'll give you some warmth and you'll be home quickly. I'll come for him later."
Sofia nodded.
"Thank you. You must not worry. I'll look after him."
Peter smiled and said, "When Father finds out..."
Sofia returned the smile.
They fetched Sultan from his stall. He seemed pleased to see Peter. He snorted steam into the cold morning air.
Sofia swung herself easily into the saddle.
"What will you do?" she asked.
"I'm going to look for Agnes. I must."
"Peter, you should know—"
"Don't say it," Peter said, interrupting her. "I must try to find her. She...I..."
He hesitated. He couldn't say what he was thinking, and anyway, he didn't even know if it was true. Had there ever been anything between them?
"I understand," Sofia said. "But be careful." She leant down in the saddle and, taking Peter by surprise, planted a quick kiss on his cheek.
"For luck," she explained, kicking Sultan into life. She laughed. "You should have let me do it before—we might have had an easier time of it!"
Peter watched her go, and then heard her begin to sing. She sang the Miorita, of course, and Peter smiled in spite of himself.
_"Let it just be said I have gone to wed_
_A princess so great, at Heaven's gate."_
Peter watched her go, and without even meaning to, raised a hand to his cheek, feeling the wetness of her lips with his fingertips.
As soon as she was out of sight Peter suddenly realized how bitterly cold he was. He went into the hut, and saw his father poking the fire, trying to coax it into life after its quiet slumber through the long night.
"Father," Peter said.
Tomas lifted his head.
"Has she gone?" he asked, still shaking from his outburst, but Peter didn't answer. Through his mind ran a series of pictures, each more evil than the last, culminating with the awful sight of Stefan's eye staring from inside his grave.
"Son?"
Exhausted, freezing, and scared, Peter's body gave up, and the world faded as he collapsed onto the floor.
**28**
**The Dream of the Queen**
In the dreamworld through which Peter struggled, everything was shadow. As he lay unconscious, he knew nothing, saw nothing, yet somewhere nearby a presence closed in on him.
Out of the darkness, a white spectre floated toward him. As it came closer he saw a pale face, disembodied and deathly. It was the face of an ancient but powerful woman, with strong nose and eyebrows, and vicious eyes. Now the face pressed right against his own, and he saw that though the face was ghost-white, there was a shadow across it, from eyes to lips, a strange five-sided shadow, like an inverted pentagon hanging from the brow and pointing at the lips.
The face drifted away, and fortunately for Peter, when he woke, he remembered nothing of his nightmare.
**29**
**Ancestors and Hostages**
When Peter woke, it was to the sound of singing. Someone was singing the Miorita, but as he opened his eyes he realized that it was he. Had he been singing in his sleep?
_Tell my murderers_
_To let my bones lie somewhere close by,_
_By the sheepfold here so my flocks are near,_
_On the open ground so I'll hear my hound...._
_Tell not a breath of how I met my death,_
_Say I could not tarry; I have gone to marry_
_A princess—my bride is the whole world's pride._
That stupid song! It was even in his dreams now.
Peter opened his eyes and found he was lying in bed. He swung his legs to the floor and sat up, rubbing his head.
Suddenly he knew what it was about that song that annoyed him so much. It was the weakness of it. The meekness. The way the shepherd gives in, without even trying to fight his murderers. Peter couldn't understand it, giving in to fate, to death, without even trying to stop it. Surely you had to be stronger than that, to survive? To live?
Tomas was nowhere in sight. The shutter was open and Peter saw bright, burning daylight beyond, though he had no idea what time it might be. Daylight. How he had longed for it! How he wished it would never grow dark again! What had Sofia said?
"There can be no evil by daylight."
He stood up, unsteady on his feet at first, unable to get the Miorita out of his head. He thought about the end of the song, where the shepherd marries the princess from the stars. That's the story he tells his lamb to pass on to his mother. To stop her from grieving, from being hurt. Peter understood that. If only he hadn't hurt his own mother. It had been his first act in the world. His birth, her death. If only he could have saved her from harm! And though he knew he was guiltless, the guilt still came.
The full meaning of the ending was lost to him—a cloud he could not penetrate. Nonetheless there was something about the story that was pulling him in. The princess. A wedding to the cosmos. A place and a purpose in life, even in death.
No.
He killed his thoughts, tired of it all.
Peter put his hand above the stove. Still warm. Gradually everything that had happened came back to him, right up to the moment when he had collapsed. Someone, presumably Father, had put him into bed. But how long had he slept? His belly ached with hunger, so maybe it had been a long time.
He felt awful. He was hungry, his head hurt, his legs ached, but he had to ignore all that, because there was something he had to do. The something was to look for Agnes, and now he remembered the shock of finding her prison hut empty when he returned from the graveyard.
"By the Forest!" Peter said aloud. "What is happening here?"
He needed his father. He checked the toolshed and found that his father's axe was missing. Had he actually gone to work? Without Sultan?
His father was useless as a source of physical help, but Peter instinctively knew that what Sofia had told him about Tomas was all true. If only Tomas would admit it, then maybe he could help Peter to understand the things he'd seen. In the hut, in the graveyard, in the forest...
All he really wanted to do was harness Sultan to their cart, put Tomas and everything they owned onto it, and ride far, far away. Peter had once heard there was a country to the west by the sea, a warm country where grapes as large as apples hung from endless vines. Maybe they could just ride and ride until they found it.
But Tomas was out somewhere, Peter had lent Sultan to the Gypsy girl, and there was Agnes to find. If anything had happened to her...
He closed that thought because the end of it puzzled him, and was not what he wanted to feel.
Then he remembered something else.
The sword.
Sofia had talked about a sword and now, after all these years, Peter knew what was in his father's box without even opening it. He looked around the room until his eyes fell on his father's mattress. That was where it was.
He took a step toward the bed, then hesitated, thinking about a small wooden goose, and the tears he had shed when Tomas had destroyed it.
But no.
There should be no more secrets.
Guiltily, he stepped forward and lifted the mattress, feeling with his other hand for the box.
There was nothing there.
**30**
**The Elders**
Peter walked to Chust. As he went he chewed on some rye bread he'd found in the jar, trying to quell the ache in his belly and find some strength. By the time he reached the village, the bread was all gone, but his hunger remained.
"That will have to do," he said.
He had no plan, but as he walked down the main street he suddenly thought that maybe he should start at the hut at the edge of the forest. That was where he had last seen Agnes. Maybe daylight would give some clue as to where she had gone. Maybe some tracks.
The thought of daylight made him look to the sky. The earlier morning sun had vanished behind a high and thick bank of cloud. But it seemed light enough, and he had no choice. He would go back to the hut.
He retraced his steps back up the main street. As he went, his thoughts were invaded by the events of the night. He had seen things that were not possible, or rather that his father had told him were not possible. All his life Tomas had told him to ignore the stories they heard, as they moved from one town to the next. Now, in the smallest, most God-forgotten place they had ever lived, it had all come true. It had all come to life, just as Radu and Stefan seemed to have done.
He was passing underneath a high window when his attention was caught by raised voices.
He might have walked on, were it not for two words.
"...Shadow Queen..."
He paused, but could make out no more, because of the babble of voices. Deciding he was wasting time, he hurried on toward the hut.
It looked so different by daylight. What had been a place of living terror a few hours before was now simply lifeless—cold and empty. By day, though still not welcoming, it held none of the horrors of the night.
Peter hunted around, but found nothing. Enough snow had fallen in the night to obscure even the frantic marks Radu had made scrabbling for the millet. The wood from the shattered door lay cast around, almost hidden but for one or two spikes of timber.
And in the freshly fallen snow there was not the slightest sign of a footprint, or anything else that might have given Peter a clue.
Inside, his search was just as useless. There was nothing there but the bed, the stool, and piles of unspun wool.
The only thing he learned from his visit was that it had been real. Everything he had thought he had seen, all the awfulness, had really happened.
He sat on the stool, wondering what to do. In truth, he knew there was only one answer, but he didn't like it. He must walk back to Chust, find an Elder, tell what had happened, and ask for help looking for Agnes.
The Elders. Old Anna, taciturn but fearsome. He certainly didn't want to face them. And then he realized—those voices floating down to him from the window had come from Anna's house.
An irrational anger seized him, and he stormed back into Chust.
**31**
**Village Talk**
He didn't even knock.
As he'd thundered toward Anna's house, it had occurred to him that it was all her fault. She was the one who ran things in the village, she had ordered that Agnes should be the bride at the Nunta Mortului. She must know by now that Agnes was missing, that the door of her prison lay splintered in the snow. She should have organized a search party.
He burst into Anna's house and, following his instincts, went up a low flight of stairs. There! He could hear the voices again. He flung open a door, striding into the room, all sorts of accusations on his lips.
What he saw took the words away.
"How dare you!" Anna was the first to recover from the shock of Peter's entrance.
She was surrounded by a motley group. Other Elders, as well as Daniel, the priest, and Teodor, the feldsher, stood arranged on one side of Anna. On the other Peter was amazed to see a party of the Gypsies. Sofia was not there, but Peter recognized Milosh, her uncle, at their head.
Peter suddenly doubted himself. Feeling like a small and stupid boy, he wanted to run from the room, but he forced himself to speak.
"Agnes!" he blurted out.
"What?"
Anna barked the word at him, and even that was enough to unsettle him. She was an alarming figure, very tall for an old woman. Her face was sharp and her nose sharper. She had eyebrows like a man's that seemed fixed perpetually in a scowl. It was no wonder she ordered everyone else around, controlling this wretched little kingdom with ease.
Peter tried again, desperately trying to make some sense.
"Agnes! You put her in the hut, but she's been taken! By those things!"
Anna took several steps toward him, and despite himself, Peter retreated slightly.
"People are coming back from the grave!" he yelled. "You know it. I heard you talking about the Shadow Queen. And they know it!" He pointed at the Gypsies. "They've come to try to stop it, but Agnes is missing! You have to do something! Help me find her."
Peter stopped. The silence in the room was even more terrifying than Anna herself.
"This is not your place, boy," Anna said, when she was sure he had finished. "You do not belong in this village. You and your useless father! I have tolerated you. Now I find out there is more to you than at first appeared."
Almost imperceptibly she glanced toward the Gypsies. They must have told her about Tomas. The sword.
"You should understand this, boy. Chust is my concern. Do not trespass on my patience. I am aware of everything, not just in Chust, but all around it. I have been discussing the threat posed by the Shadow Queen with those assembled here. These people, from the village and outside it, who are wise enough and powerful enough to act. And yet you dare to break in here and insult us all!"
She stopped for effect, and Peter took the opportunity.
"But Agnes," he gasped, "you've as good as killed her! She was taken. Why don't you—"
"Be quiet!" Anna shrieked, with such intent that the room seemed to darken. "You know nothing. Yes. Agnes is no longer where she should be. In the hut. But she was not taken. She left herself. You helped her! She has disgraced us all by breaking her honor in this way."
"No," said Peter. "That's not true. She's missing."
"Enough!" Anna declared. "Remove him. We have no time."
The men closed around Peter, and though he struggled, they forced him from the room easily, and dragged him back down the stairs.
In a moment he found himself sitting in the street in the snow.
"But Agnes!" he cried. "We must find her and help her."
One of the Elders paused and considered Peter.
"You are a foolish boy. Agnes is at home. With her mother. She has disgraced herself, and you helped her do it. Count yourself fortunate we don't punish you and your father for the shame of it all."
"At home?" Peter could scarcely believe what the man had said. "At home?"
"Go and see for yourself."
The man spat at Peter's feet, and shut the door.
Peter stood up. He looked down the street that led to Agnes's house.
He ran all the way there, skidding in the snow and ice.
He didn't even have to get as far as her house.
There she was, up ahead of him, looking just as she always did, though Peter saw with a shiver that she was still dressed in mourning weeds. Why hadn't she changed to her own clothes? The forty days had been broken after all. Was there still a need to dress for them?
She was crossing the street, toward her front door.
"Agnes!" he called, breaking into a run again as he saw her unlock the door.
He saw her turn and look at him, but the relief he felt rapidly turned to confusion as she saw him, then deliberately looked away.
She opened the door, and while Peter was still yards away, she slid inside.
Peter was in time to hear the door being bolted from inside.
"Agnes!" he called through the door.
No answer. He tried again, this time slamming the palm of his hand against the wood.
"Agnes! What is it? What's wrong?"
"Go away, Peter."
Her voice came through the wood, muffled and faint.
"What?" Peter cried. "What do you mean? Are you all right? I've been looking for you since—What happened to you?"
"Go away, Peter." Once again, her voice was dull and flat.
"Why are you being like this, Agnes? What's wrong?"
Peter strained to hear her, pressing his ear to the door to catch her words.
"Go away. I left the hut and now I'm in disgrace. My whole family. Hah! What's left of it. Go away, Peter. I want nothing to do with you. I never did want you. You were never good enough for me. Now you are less than useless."
"Agnes!"
"I'm well, Peter. Does that make you happy? Now go away."
Peter stepped back from the door, looking stupidly at the wood, trying but failing to understand.
Agnes was right.
What was he good for?
He walked away.
As he went he passed again by Anna's house, but this time could hear nothing.
Neither did he see Old Anna looking down at him, a wide smile slowly spreading across her face.
"The sword."
She mouthed the words silently.
"The sword!"
**32**
**Stillness**
Peter brooded. Tomas drank. For days neither of them stirred from their own little island.
Something had changed.
For a year or so, Tomas and his son had enjoyed a period of relative comfort and simplicity. They had stopped running and found a place to live, with plenty of work to be done, and for some of that time Tomas had even been sober enough to do some of the work.
Not anymore. Everything was closing in around them, the way snow clouds sometimes enveloped the mountains and the forest. The flakes that fell from the clouds were the purest white, but the clouds themselves were darker than confusion, darker than death.
Tomas had spoken only once. He'd been staring out into the forest from the door of the hut, when without warning he said, "We may have to move on, Peter."
That was all, and he would say no more, despite Peter's questions and pleading. Peter was left running over in his mind everything that had happened, again and again, struggling for the answers he desperately craved.
The day after Peter had been thrown from Anna's house, the day he had seen Agnes, they had a visitor.
Peter was stirred from his mood by the sound of hoof-beats on the bridge.
He went outside to find Sofia leading Sultan home.
"You didn't come for him," Sofia said.
Peter shrugged.
"I had things to do," he said.
"There," she said, smiling. "I looked after him. As I promised."
Peter took Sultan's reins willingly enough, but didn't speak.
Sofia watched him stable the horse and come back to the front of the hut. She tried again.
"It was kind of you. To lend him to me," she said. She hesitated. "It was good of you to...trust me."
Peter turned to her.
"I did trust you, Sofia," he said. "But then I found your uncle telling the village Elders all about my father. My father just wants to be left alone. You had no right to do that."
"I am not responsible for what my uncle does," Sofia snapped. "But that is not the point. It is hardly important what anyone knows about you and your father. My people went to talk to the Elders about the threat from the Shadow Queen. They went to offer their services in the name of the Winter King. You can't hide on your little island forever, Peter."
Peter waited for her to finish, then went back inside.
"Thank you for returning Sultan to us," he said quietly as he entered the hut.
Through the door he heard Sofia.
"The Miorita, Peter. You should understand it."
He heard her gentle footsteps retreat across the bridge, the bridge to their little island.
_Damn her!_ Peter thought. What did she mean by that? The Miorita? What had that to do with anything? And yet, it was not only the Gypsy girl who had got under his skin. That song had too.
"You should _understand it._ "
What did it mean?
After that brief encounter, Peter had spent the hours lying on his bed, ignoring Tomas as he opened jar after jar of rakia, thinking about Agnes, about the forest, and Radu and Stefan. About Sofia.
And yes, about the Miorita too.
After three days, Peter's body rebelled. His mind might have been drifting rudderless like a raft on the open sea, but his body was used to hard work and he was restless. Finally, on the third morning, he practically threw himself out of bed and pulled his boots on so violently that even Tomas raised an eyebrow.
"What are you doing?" Tomas asked.
"Going to work," Peter said. "It's all I know."
He grabbed his axe and put Sultan into the harness of the cart, and they lurched off into the depths of the snowy forest.
Peter didn't particularly care where they went, but at the back of his mind was a tree that he and Tomas had been going to fell some weeks before. It was a huge old birch and it would take days to saw and chop it all, but Peter just wanted to see it fall, and smash to the ground. His body cried out for it. And he wanted this wood to fall, not to carve but to burn.
After an hour or so they found the tree. They were far into the depths of the forest, but it was a sunny morning, and for a short while it was possible to believe that midwinter was more than a few weeks away. Peter tethered Sultan to a tree some way from the birch, more from habit than necessity. His horse was by far the most reliable thing in his life. That, and possibly the forest, though recent events had made him begin to doubt that the forest was always benign.
Peter sized the tree. Even from the ground it looked vast, and he had learnt in his career as a woodcutter that no matter how big a tree looked in the air, it would be twice as big when it was on the ground. He tried to circle its girth with his arms, and could only just brush his fingertips against each other.
He stood back, made a silent prayer of thanks to the forest, and then swung his axe as if his life depended on it.
Wood chips rained around him, and around his feet the snow was rapidly covered with the spoil from his axe.
Something possessed him as the axe flew through the air faster and faster with each stroke. He formed a perfect undercut in less than twenty strokes, and freed the opposite side of the tree from its sheath of bark. Then he began the real work, making the cut that would bring the monster to the ground, exactly where he wanted it.
Still the blows from the axe fell, and nothing could have stood in its way, not twenty men, and least of all a tree, even one that would keep a family warm for a whole winter. A vision of his father thirty years ago came into his mind—in King Michael's army, fighting the Turks. And maybe other, more deadly enemies.
Peter's axe fell. Tomas's sword swung.
Both cut their foe to the ground, blow after blow after blow.
Suddenly Peter stopped. He had been so hypnotized by the swing of his axe that he'd barely noticed how far he had cut. The trunk where he'd been chopping gave a deafening crack, as if lightning had struck nearby. The tree moved. It had begun to go.
Peter stood back, knowing he had done enough. How slowly it moved at first, its motion barely perceptible as it inched its way from the sky! There was another crack as the timber split under its own weight, and then the tree came with a rush, leaning into the air, finding nothing to support it, and accelerating downward till it hammered into the snowy floor of the forest.
The ground shook.
Sultan whinnied and Peter looked over at him.
"That's all for today," he said. On any other day he would have begun the process of sawing logs short enough for Sultan to drag home. The cart was empty and waiting, but Peter wanted to do no more work. He had escaped from the torpor of the hut, and felt his body come alive once more. More than that, he had been in control, and it felt good.
Peter never knew how it happened, but suddenly he saw something glinting in the snow. Looking closer, he saw it was an axe, and immediately, instinctively, he knew whose it was. It had belonged to Radu.
Suddenly he was filled with dread, seeing the axe as an omen.
His exhilaration at felling the tree evaporated, for he was certain, as certain as he had ever been of anything, that his father was in trouble. At that very moment.
Even as his blows had struck the tree.
He freed Sultan from the harness, and leaving the cart and the fallen tree where they were, galloped home.
**33**
**Tomas**
As soon as Peter saw the hut, he knew that whatever it was that had told him of the danger to his father had not lied.
Peter rode Sultan straight over the bridge, threw himself from the horse's back, then froze. The place had been turned upside down.
There had obviously been a fight; the sawhorse lay on its side, the stable door was swinging open, the log pile by the hut had collapsed.
Then Peter saw blood in the snow. An irregular, smeary trail of it leading across the small triangle that was their island to the ditch that Tomas had dug. With bile rising in his throat, Peter followed the trail, dreading what he might find. He looked over the lip of the bank and saw a body face down in the water, snagged by a tree root.
Peter recognized the clothes. It was one of the Gypsies. He turned and ran to the hut.
"Father! Father?"
There was Tomas, lying beside his bed as if he'd been trying to get there before collapsing on the floor. Peter crouched beside him.
"I thought you were...," Peter began, but couldn't bring himself to say it.
Tomas smiled, but he was obviously weak. Next to him on the floor was his axe. Peter saw blood on the blade, and he knew whose it was.
"Are you hurt?"
Tomas shook his head.
"Fit as a flea. Help me up, will you?"
Peter tried to lift his father onto the bed, but couldn't manage it. He was so heavy, now, it was hard to believe.
With a grunt Tomas sank back to the floor, and Peter knew he had been lying about being hurt.
"What did they do?"
"Nothing," Tomas smiled. "I wouldn't let them."
"Wait," Peter said, and dragged the covers from the bed. "Lie on these instead, till you're ready to get up."
"Get me a drink, would you?" Tomas said, wincing as he rolled onto the makeshift bed.
Peter didn't know what to say, but that in itself was enough to irritate Tomas.
"I've just been attacked by four men. I've killed one of them. I need a damn drink, Peter!"
Peter nodded.
"Sorry. Yes, yes."
He fumbled around with a bottle, trying to find a mug.
"Give me that." Tomas snatched it from Peter and drank deeply. Slivovitz dribbled down his chin and dripped onto his shirt.
Peter knelt by his father again.
"What did they want with you, Father? Why did they do this?"
Tomas took another drink, then looked into Peter's face.
"I'm sorry, Peter," he said. "I've lied to you."
Peter shook his head, putting his hand on his father's shoulder.
"Listen to me," his father said. "I've lied to you. About so many things."
"I don't care," Peter said gently. "I don't care about that. Why did they do this? What did they want with you?"
"It's not me they want. Well, not anymore. Not now I'm like this."
Peter wasn't sure whether he meant hurt, or something more, that he was a useless drunk.
"It's not me they want. It's that."
Tomas nodded up, behind Peter.
"There," he said, pointing. "Up in the eaves, behind the beam."
Peter followed Tomas's shaking hand to the top of the wall. He stood on a stool and felt around, and there, tucked into the crook between the joists and the roof, was the box.
"Take it down, Peter. Take it down."
Peter had seen the box before, but now, even before he opened it, he knew what was inside. And if the sword was true, then it was all true.
His father, a hero.
"The sword?" Peter asked.
Tomas nodded.
"Have a look if you like."
Peter's hands trembled as he lifted the lid and gazed upon the blade inside. He didn't dare touch it.
"But why?" he asked, shaking his head. "It's just a sword."
Tomas laughed, then winced again.
"Sit down, Peter. Listen to me. I've lied to you. That thing you see there. It's so much more than a sword. It has power over those who return. Return from the grave. You understand?"
"Sofia told me. What I don't understand is why you've denied them all these years. Why?"
Tomas took another drink, then a deep breath. He looked across the room to the fire.
"Thirty years ago, I fought with the King. They call him the Winter King now, but then he was just King Michael. The Turks had fought their way far from home, right up as far as Poland. For years we'd been powerless to stop them; the noble voivods who ruled each region too busy arguing with each other to unite. Michael changed all that, and got each voivod to swear allegiance to him. In that way he formed a mighty army that pushed the Turks back as far as the Danube. The river ran red! And then we pushed even further. I was with him as we headed far into Turkish territory. It was there that I found the sword.
"And there that I learnt of something worse than the Turks. I had heard of the _vrykolakoi_ before, in fireside tales. Everyone has. But in that strange land I found myself fighting them as well as the Turks. The sword was made in a land where these terrors were common, and it has the power to destroy them for good, with a single stroke."
Peter nodded, but it still didn't make sense.
"But why are the Gypsies fighting you for it? Why have you never told me about any of this?"
"Wait. A story has its purpose and its path. It must be told correctly for it to be understood. Remember that, Peter.
"Well. The wars ended, but not before the King died. Not from the sword, but from some disease that found him on our forays onto foreign soil. It ate him from inside and it was terrible to see. It was then, in the disbanded armies that were making their way home, that I met Caspar. Sofia's father.
"From him I learnt all about the ways of those who return from death. They are to be found in every land, he said, and I found out how true that was. He had heard about my fighting, about my sword, and we spent the years that followed hunting them down and putting them back to eternal rest.
"For they are like a disease too, Peter. They infect the living and make us like them. Once an outbreak starts, it is like an epidemic. It can be hard to stop. Sometimes a great many people die before it is brought to an end."
"And the Shadow Queen?"
Tomas shook his head.
"That I do not know. I thought she was no more than a story. But if she is real, I don't know how she is involved in all this."
Tomas paused, staring at the floor, breathing heavily.
"Sofia said you were put in jail after the wars, because of the chaos when King Michael died," Peter said. "Is that not true?"
Tomas shook his head.
"No. It was Caspar's fault."
"But he died in jail too! Or is that a lie, as well?"
"No, that much is true. After years of hunting the dead, I had had enough. Things were getting more dangerous for us, simply because we were living in a time of peace. Think about it. Think of what we used to do. We would prowl around at night, hunting through graveyards, digging up graves. Why? To stop them, send them back. Kill them, if you like. During a time of war and strife, no one gives much care to their dead. People are lucky if they get to bury them at all. But in times of peace, men who desecrate graves are not well liked. I wanted to stop. Caspar had married and had a baby girl. I wanted to do the same."
_A baby girl,_ Peter thought. _Sofia._
"But Caspar convinced me to continue; he said it was too important. Soon we were arrested, but I had hidden the sword. We were tried and both ended up in jail for desecrating the grave of a nobleman. A nobleman who was returning from the soil every night to attack young girls. It didn't matter. The local voivod locked us away for life, and I would have stayed there forever had he himself not been deposed. That came too late for Caspar, and I vowed that when I got out I would have no more to do with any of it.
"That was years ago. I met your mother a year later. She died giving birth...to you, Peter. And I renewed my vow to fight no more, to look after you. I just haven't done a very good job, that's all."
"No, Father. That's not true. I didn't understand things, about the box, about why we had to keep moving all the time. But I understand one thing. You think it was my fault that my mother died—"
"No!" Tomas cried, sitting up, grimacing with pain. "No. I never thought that."
"Didn't you?" Peter said quietly. "Didn't you?"
They fell silent.
Peter thought about his father's old life. He had fought with the King, and the King had died. He had fought with his friend to protect people from the hostages, and instead of being rewarded, they had been thrown in jail. His friend had died there. Turning his back on this warrior life, he had found a wife, and then had seen her die giving birth to his only son. He should have been proud of his son. But he had turned away from him. Was it simply too much, to see a reminder of his wife's death every day?
So finally, as soon as Peter was old enough to fend for himself, Tomas had turned to the one thing in life that had never let him down. Drink.
Peter looked again at the sword.
_To stop me seeing this, you broke my toy,_ he thought. _My little wooden goose._
But he said nothing.
Though his heart had been damaged by his father's story, there was one small seed of hope. Tomas had finally told him everything. There were no more secrets between them, none of the secrets that had kept them apart all Peter's life.
They could act.
"Tomas. My father," Peter said. "Why did they do this to you? You were on the same side, once."
"There are no sides here. I vowed not to fight anymore, and I will not. Look at me! One scrap and I'm all but done for. Another one would kill me."
"So you refused to join them? And they wanted the sword instead? So why not just give it to them? Give it to them, and let's get out of here. Go far away."
"And go where? We've been running all our lives. The hostages, those who return, are everywhere. The ancestors, those who fight them, are everywhere too. But the sword is mine and I will not give it up. They'll be back soon, and I suppose next time they'll send more than four weaklings to get it. I cannot help that."
"No, but it doesn't have to be this way. If you won't help them, then just give them the sword. It makes no sense to keep it. Then they can try to stop this epidemic."
Tomas grew suddenly angry.
"I told you! It's not my fight anymore! It's not our business. We're just woodcutters. I want to live quietly on this island. Bother no one, and be bothered by no one!"
Peter stood and stared at his father.
"How can you be so selfish!" he shouted. "Help them! Give them the sword at least. They need you. I need you!"
"Everything was fine until they arrived."
"If that's true, then why do you drink? It's killing you, and yet you will not stop! And why would you need to drink but to stop yourself from seeing, from thinking?"
In answer, Tomas kicked out and knocked a chair flying across the room.
Peter jumped back and watched, horrified, as Tomas lifted the bottle of slivovitz to his lips.
When Peter closed the door of the hut behind him, Tomas had still not stopped drinking.
**34**
**The Camp**
Peter had not been to the Gypsy camp before, but he knew it was somewhere away to the west of Chust. He'd heard that the Gypsies had settled in a clearing in the trees. Sultan moved easily through the great forest, still willing to do his master's bidding despite their fruitless logging trip.
Peter's mood was grim, and though rage boiled inside him, his face was nothing but a mask of determination. As he rode he kept one hand on the reins, the other on the shaft of his axe. The world had gone crazy, turned itself upside down. His father had killed someone, and he was riding to confront the victim's family. He might just need his horse and axe to make it out alive.
And if he didn't make it out again? At that moment, he didn't much care. Wasn't that what the Miorita was telling him? To accept your fate, meekly, with no resistance, no struggle. If that was the case, then he would go to the Gypsy camp without fear, and get them to leave Tomas alone, to fight their own battles.
And as for Sofia...
He kicked Sultan in the ribs unnecessarily hard. The old horse broke into a canter, but shook his head to show he wasn't happy.
The clearing was ahead, and even at this distance, through the trees, Peter could see the yellows and reds of the caravans, and wood smoke twisting up into the sky.
He pulled Sultan to a halt and tethered him to a tree.
"Wait here," he said. "I'll be back soon."
Peter wished he were as confident of that as he sounded, but sliding the axe from Sultan's saddle, he knew he had no choice but to go through with it. The Gypsies would regroup, and be back for the sword.
At first he walked boldy, upright, making no attempt to hide himself as he neared the clearing. He could see the camp clearly now. There were five caravans and two open carts. The caravans were arranged in a circle with their doorways facing a large campfire, over which hung a cooking pot. Horses, tethered to stakes and tree stumps, chomped on the contents of hay bags. Peter saw a series of stakes planted in the ground, in a circle just outside the camp itself, about halfway to where the trees began. From the top of each stake hung a cluster of something white and bulbous. It took a moment for him to realize they were strings of garlic bulbs. Protection.
Now Peter saw someone jump from the low step of one of the caravans, and as he watched the Gypsy crossing the circle, something caught his eye. He dropped to a crouch, and crept a little closer.
Sitting against a large birch trunk on the far side of the clearing was Sofia. She was alone, in the snow, with her legs out straight in front of her, and her arms by her sides.
Seeing her there, and puzzled by it, Peter forgot all about what he had come for, and his anger with her. He crept forward nearly to the edge of the trees, then began to circle round toward her.
He was used to moving through the winter forest, and he made no sound as he deepened his arc slightly to approach Sofia from behind. For a while he thought he had lost sight of the tree where she was sitting, but there it was again, ahead of him.
Now he understood.
Ropes were tied tightly around the trunk and around Sofia. The others had bound her to a tree, and outside the circle of garlic.
"Sofia!" he whispered.
There was no reply, but then, she was on the far side of a thick trunk, unable to move.
As he crept closer, his dexterity deserted him. The head of the axe caught against the trunk of a dead sapling, which cracked loudly. He glanced ahead and saw Sofia's hair flick out—she had turned her head.
Fearing she might call for help, he rushed the last few paces until he was right against her tree trunk.
"Sofia! It's me! Peter."
Nothing for a second, but then he heard: "Peter!" It was no more than a whisper. "Thank God! Set me free!"
"Why did they do this to you?" he asked.
"Not now! Set me free."
Peter nodded. He pulled his knife from his pocket and began to saw through the thick hemp binding her. As he did so it crossed his mind that maybe she had been tied to the tree for a good reason, maybe she had even been—
No. It was daylight. She couldn't be one of them and out in the daylight, he reminded himself, and kept on sawing.
"Quick!" Sofia said. "They might come out at any time."
"There!" Peter said, and loosened the rope.
Sofia stayed motionless, waiting to be sure that she was unobserved, then flung the rope away and spun around the tree into Peter's arms.
"Thank you!" she cried. "Let's go away from here."
"I can't," said Peter. "I've come to stop them from attacking my father. They'll have to listen to me."
"No!" Sofia cried. "They'll kill you. Nothing is going to stop them. They tied me to the tree because I tried to stop them from stealing the sword from your father. I told them to leave him alone, and they did this to me! One of their kind!"
"They would have left you out in the night?"
"They threatened to. I think it was just meant to scare me. I think. But you can't stop them."
"I have to try, Sofia."
"Peter! Listen to me! My own uncle tied me up. Imagine what they will do to you and your father! Come. Come away."
She pulled Peter's hands, dragging him deeper into the wood, and he knew she was right.
He shook himself.
"This way," he said. "I've got Sultan with me."
They ran.
**35**
**The Approach**
It didn't take long to reach Sultan, but Sofia was shattered by the time they found him. She had been sitting on the frozen ground all morning and her legs would barely move at first, but Peter urged her on. He had to lift her onto Sultan's back. Then he swung up behind her. Very soon, with the warmth from horse and boy, Sofia began to feel better.
The trotted through the trees, aiming nowhere until they were convinced that there was no pursuit from the camp. Peter slowed Sultan to a walk, mindful of the double cargo the horse was carrying. As he rode with his arms around Sofia, he felt her leaning back against his chest. But Sofia had other things on her mind.
"It's up to us," she said.
"What do you mean?"
"This is all going to end badly, Peter."
Peter grunted.
"My uncle went to talk to the Elders. The woman called Anna?"
"Yes," Peter said.
"He tried to warn them of the danger, but they ignored him. She is a difficult woman, and thinks she knows best how to run her village. We know better than she about hostages, but she refused to listen. After hours of talking, they did nothing."
"But there are other ways of killing the...hostages?"
"The hostages, yes. They did not want to become what they are. It is like a disease, it makes them become that way. It is not a question of killing them, but of returning them to the ground. Forever. And it can be done without the sword. But the sword is easier. A single cut from it is enough. And they fear it. It is as if the power of the Winter King is inside it. Inside the blade."
"But what can we do?" Peter cried. "You and me. A boy and a girl."
"Peter. There is nothing else now. The villagers are too frightened to know what to do. Your father refuses to help, or even give the sword. My people may kill him for it now, as he has killed one of us."
"Sofia, I'm sorry. I wasn't there. I might have stopped it."
"And you might be dead too," Sofia said. "Don't worry. I am not surprised it is Georg who is dead. He was always first to anger. I heard he rushed at your father with his knife. Your father defended himself, and the others fled back to my uncle. But listen, Peter, it won't be long. Once they've licked their wounds they'll be back. If you go to stop them you'll be hurt too. And the epidemic is growing worse. Very soon there will be more hostages in the village. If it spreads, it will become impossible to control."
Peter thought about what she'd said, about Tomas, and her uncle, and the hostages. But one thing she had said stood out: "You might be dead too."
She cared about him. And that small spark was enough to kindle something in Peter. He couldn't lie down and die, like the meek shepherd. He was going to fight.
"Sofia, I will help you. What can we do?"
"We are not powerless. We can act. We need to show the villagers that they must act, despite their fear. And we need to do this before my people hurt your father."
"How?"
Sofia hesitated. Sultan hesitated too, and came to a standstill. Peter barely noticed. Sofia twisted around so that she could see Peter's face. Around them the snow-laden branches of the bare trees hung heavily, pointing their twig fingers toward the couple on horseback. There was total silence across the face of the earth, and the silence centered on this small universe in the trees.
"We have Sultan," Sofia said slowly. "There is an old way of finding hostages in their own ground."
"You mean, in their graves?" Peter's mouth twisted with fear as he spoke.
"Yes. We must start there. If a virgin tries to ride a horse over a grave where a hostage lies, the horse will know, and will refuse to cross. We must find the graves of the hostages, all of them, and prevent them from leaving the ground."
"How?"
"There are ways. We could stake them in. That holds them to the ground. Or we could put nets into the graves. That works like the millet; they have to unpick every knot before they can leave again. But we have no nets."
"I've heard things like that too," Peter said. "My father said they were fireside tales, but now..."
"You know it's true. We could put charcoal in the graves. They must write with the charcoal, and that keeps them from returning until the charcoal runs out."
"Or we can use buckthorn," Peter said. "That's what they did at Radu's funeral."
"Yes. The thorns are like little stakes," Sofia said, nodding. "They pierce the skin. The hostages cannot move through the thorns. But if that's what they put into Radu's grave, and he still came out..."
"What does it mean?"
"My uncle knew about this. It is why we are so afraid. It's never been like this before; it is as if there is something more powerful happening. Something giving them greater power."
Peter shook his head.
"I don't know. Is there nothing else we can do?"
"Yes. You could sever the head with your axe, and place it at the hostage's feet. Only two things work better than that. Fire. And your father's sword."
Peter stared straight through Sofia.
He knew there was no way he could sever a head, even the head of a corpse.
"If you're too afraid, then give me your horse at least. I'll try on my own."
"No!" Peter cried. "I'm not afraid. I'll help you."
He kicked Sultan into a walk again.
"We'll need a spade," he said. "We can steal one from the sexton's shed. I know where that is."
Sofia laughed.
"Excellent! And we can cut some buckthorn on the way."
"But..."
"What?" asked Sofia.
"You said we need a virgin to ride the horse."
Sofia twisted in the saddle and slapped Peter's cheek.
"That's me, you pig!"
Peter burst out laughing.
"I'm joking," he cried, rubbing his face in mock pain, and now Sofia laughed too.
"Be careful, Peter," she said, but with a warm smile on her face.
"Let's go, Sultan!" he said, grinning.
Holding not only the reins but Sofia too, he spurred the horse into life and they thundered toward the village.
**36**
**Ordeal**
"Remember. It is day. We have an hour or so before there is any danger to us. No matter what you see, remember there is no danger."
Peter nodded.
It was a bitter afternoon. It had begun to snow heavily as they rode around the outskirts of the village, and more than this, the snow was being driven by a nasty wind from the east, straight off the mountains. It had come from nowhere, quite suddenly, and now the storm was at its most furious. On any other day, Peter might have cursed it all, but today was different. He and Sofia were glad of the appalling weather because it meant no one else was about. They seemed to have the village to themselves, which was just as well, given what they were about to do. Tomas and Caspar had both been locked up for doing the same.
They had cut buckthorn from a large bush, and Peter had broken the lock on the sexton's shed with one swing of his axe.
Now they stood at the edge of the graveyard.
The snow hurled itself out of the sky, tearing around their heads, making it hard to see more than a few feet, never mind to the far side, where the wooden church hunkered into the slight hillside, as if trying to escape the storm.
"Ready?" Sofia called.
Peter smiled.
"Go on."
He made a stirrup with his hands to help Sofia into the saddle. She smiled and, allowing him this indulgence, settled herself. She guided Sultan to the gateway, and Peter followed, axe in one hand, spade in the other.
With the back of his wrist he brushed at the snow clogging his eyebrows, and then opened the gate for Sultan and Sofia.
"Where?" he asked.
By way of answer Sofia steered Sultan over to the path that ran down the middle of the graveyard, to the first grave in the first row. That made sense. To start at the beginning.
Peter and Sofia exchanged one last look, and Sultan walked forward.
It was slow progress. At first Sultan seemed unsure of what he was supposed to do, but then he understood. He stepped over the first grave, passing the wooden cross, and to the other side.
Nothing.
He had moved as calmly as if he had been walking in a summer's hay meadow.
Peter looked at Sofia, but she didn't look back, urging Sultan to the next grave.
Nothing.
Again Sultan moved happily across the ground.
The third grave approached.
Nothing.
Sofia urged him on.
"Sofia," Peter called. "It's not—"
He didn't finish what he was saying.
Sultan reared so suddenly and violently that he threw Sofia before she could do anything about it.
Peter ran to her side as Sultan shied into the snow, becoming a gray ghost in the gloom.
"We've found one," Sofia said.
"Are you—?"
"I'm all right," Sofia said. "Hurry. We have to try." She got to her feet. "Come on!"
It was so hard. What they were doing was so hard, and the ferocity of the snowstorm only made it harder.
Peter picked the spade up, his hands numb already, and began to dig. His first efforts cleared the snow, and then he hit the ground. The winter had frozen the soil solid but he didn't give up, driving the spade down with his boot. He prized up a huge sod and flung it to one side, and with that achieved, his work became much easier.
In a short time he had dug a hole halfway along the grave, going deeper with each blow.
Suddenly the spade tip struck something, something other than soil.
"Wood!" he called to Sofia. "I've found it."
She nodded.
"Now what?" he asked.
She grabbed the axe and seemed to be about to swing it, when Peter stopped her.
"That's my job," he said, taking the axe from her, "and anyway, I've got an idea. Go and fetch Sultan back."
Sofia stalked away into the swirling snow, to Sultan, who had been too scared to come any closer and too scared to leave altogether. She coaxed him back toward the graveside, in time to see Peter swing the axe at the surface of the coffin. He had made two blows already and shattered the lid. He left the axe sticking out of the wood, and came over to Sultan. From the horse's saddle he took a rope and tied it first to the axe, and then to Sultan's saddle.
Immediately Sofia understood his intentions and they both began to walk Sultan away from the grave. He was only too happy to oblige, and pulled.
There was an earsplitting crack and half the lid of the coffin flew up into the air to land on the snow beside them.
Now that they had done it, they realized that the worst bit was still to come.
They stood motionless, not even daring to look at each other, but staring at the lip of the hole they had made. Peter couldn't move. Then Sultan whinnied and seemed to goad Sofia into life. She sprang forward, giving herself as little time to think about what she was doing as possible.
Shamed, Peter rushed to her side and saw what she had already seen.
"It's empty!" he cried. "Sultan was wrong."
Sofia was silent, as she peered deeper into the coffin.
"Sultan was wrong," Peter repeated, "this isn't going to work."
"No," she said. "Sultan wasn't wrong. Look!"
She pulled Peter down beside her.
As they leant into the hole, they were for a moment oblivious to everything else around them. They no longer noticed the snow, and they didn't hear Sultan snorting. They didn't see the snow shifting strangely on top of the graves that lay behind them.
"Look," Sofia said again.
Peter didn't need telling. He remembered what Sofia had told him about the things you could bury with a hostage to stop them from walking. Charcoal. That was the one. Charcoal.
Peter stared at the inside of the coffin they had uncovered. Every inch of wood was covered in writing, in charcoal. It was scrawled, as if written in a fury, or a great hurry, but nevertheless it was legible. And with every word that Peter read his heart grew colder and his hair became a little whiter.
"What does it say?" Sofia asked. "I can't read. What does it say?"
Peter shook his head. He wished Tomas had never taught him to read, because then he wouldn't have had to read what was written on the inside of the coffin. Words of such anger, and malevolence, and hatred. Toward the living. Descriptions. Statements of intent. Jealous rantings. All the disgusting horror to be perpetrated on those still above ground.
"If I tell you," Peter said, "you'll wish I never had."
Sofia looked away from the writing, and then they both heard something slither behind them.
They turned, and Sofia screamed. Sultan bolted, and this time nothing was going to keep him in the graveyard. It was all Peter could do to keep from screaming. They watched with a deep and mortal fear as snow slid from the top of graves on every side.
Sofia whirled around.
"Peter!"
He followed her movement and saw the same thing happening behind them. What they saw next was even more terrible. All around them, squares of snow lifted bizarrely into the air. The graves were opening. Snow slipped from the squares to reveal coffin lids being pushed upright, being pushed aside.
Then came a hand, grasping for something to hold on to. Desperately Peter and Sofia looked around. Sultan was long gone, and every glance showed them something worse than the last.
They came out.
In front, behind, to the left and right, they came out, and it was clear they knew Peter and Sofia were there.
"This can't be happening!" Peter cried. "It's daylight!"
"I know," Sofia shouted.
"Run!"
They ran, heading for the side of the church, aiming for a gap between the hostages. But there were dozens of them now, all running for them, all with a simple, deadly intent.
"Quickly!" Peter shouted, and pulled Sofia onward.
But she stumbled in the snow, and fell awkwardly against a gravestone. Peter reached to help her, but got only halfway when one last grave opened before him. The lid flew off as if blown apart by gunpowder. Snow fountained into the air, then settled, and Peter gasped.
Not one, but two figures rose from the grave hole. The first was Stefan, and with him was a girl. Agnes.
**37**
**The Sword**
Peter and Sofia backed away from the menace on all sides. Radu was there, and Willem, who had made Radu a hostage. Stefan and Agnes joined the others. Agnes's mourning dress wafted around her, making her look like a black ghost against the snow.
Instinctively perhaps, Peter and Sofia had edged backward to the church until they found themselves pressed up against its northern wall, and they were trapped.
Peter knew what would happen next. He had heard enough stories in his time. His father had told him that those stories were nonsense, but now he knew they were not. Very soon they would become hostages too, and would rage against those still living like a murderous pestilence.
He felt Sofia beside him, and could hear her breath coming in short, stifled gasps. He put his hand out to his side and found hers, squeezing it tightly.
"If we're going to go, we may as well fight," he whispered. "I don't care what the song says."
In another two paces, dozens of pairs of hands would be on them.
He leapt forward with a roar, hitting out with his fists, kicking with his feet, but it was no good. Their strength was beyond human strength, and they brought him to the ground with ease. He heard Sofia scream; he knew they had got her too. Hands held him fast on all sides. Hands with swollen, blotched fingers and long, yellowing fingernails. The skin was bruised, blue-black and dry, like the faces that harried him. Peter saw the face nearest to his own. This had been a man, once, but from the dried blood crusting its mouth, he was sure there was nothing human left in it.
"Go on, then!" he cried, shutting his eyes.
But nothing happened. He flailed some more, then opened his eyes, and found he was being lifted to his feet.
No killer blow came. No clawlike hand. No savage bite.
He twisted furiously in his captor's grip and saw that Sofia was nearby, likewise held, but unharmed.
"What is it?" he called to her. "What are they doing?"
"I don't know!" she cried. "They shouldn't be able to do this. It's still daylight!"
It was true. It was a weak, miserable, gray afternoon, with no sun in sight, but it was still daylight.
Something occurred to Peter.
"Is it the Shadow Queen?"
Sofia didn't answer, but she didn't need to. As Peter mentioned the name of the Shadow Queen, the hostages all around seemed to shiver, and some began to wail, wordlessly.
It was true, then. It was her power that was changing things.
The hostages gripped Peter and Sofia more tightly and forced them to walk. They made their way out of the graveyard.
Peter had been sure they would head for the forest, but he was wrong. In full daylight, they walked into the village square, heedless that they might be seen.
And now they had been seen. An old woman, venturing out to her woodshed, saw the unholy procession. Screaming, she crossed herself, and fled back indoors.
At this other doors and windows opened, and Peter watched aghast as other people saw them, then flung their doors and windows shut, leaving them to their fate.
They were forced on, faster and faster, through the village.
"Where are they taking us?" Sofia called to Peter. "What do they want?"
It was slowly dawning on Peter what was happening, but he didn't dare voice his fear. They were moving, almost at a running pace, and the edge of the village came in sight.
They moved on, and now Peter was sure. They were being taken to his very own house, and he knew why.
As they passed the village gate, many of the hostages hung back, letting Peter and Sofia be taken by eight, four of them holding each. Stefan was among those holding Peter, and Peter saw Agnes pushing Sofia along. He tried to pull free once more, but the hostages' strength was far too great, and their speed increased still further.
The hostages began to run, and Peter and Sofia found their feet lifted from the ground. They skimmed along, inches above the snow.
There.
The hut came into view.
"Father!" Peter called out, trying to warn Tomas, but it was useless.
They had arrived.
Peter knew why they had come. The hostages had taken some hostages of their own. There was only one thing they feared, and they had come to barter for it. Peter and Sofia would be held against the surrender of the sword.
Before, when Radu had chased them from Agnes's hut, the river water was enough to stop him from crossing to the island. Peter wondered if, under the Shadow Queen's power, they could now cross the bridge, but then he realized they wouldn't need to.
They would draw Tomas out.
"Father!" Peter cried, and before the words had reached the hut, he found himself held high in the air by his throat. Stefan had not been a strong young man when alive, but now he could have snapped Peter's neck in an instant.
One of the other hostages, an older man with a swollen belly and bloodied eyes, stepped forward. His hair was long and filthy, uncut for years. He raised a hand and pointed at the hut.
"Come out!"
Peter shuddered at the sound of the man's voice. He fought to breathe in Stefan's grip, trying to lift himself on Stefan's arm, to find some air.
"Come out!" The voice was quiet, but commanding. It was indeed dead, but it carried across the safety of the water to the hut.
Sickeningly, Peter heard the door on the far side of the hut, hidden from view. There was an awful pause, during which Peter fought to see.
Tomas appeared, and Peter's heart sank.
Tomas was drunk.
He staggered uneasily, swaying from one foot to the other, stopping to try to balance every now and again. In one hand he held a stone bottle of rakia, in the other, the sword, naked and dangerous.
"No!"
Peter tried to cry out to his father, but the warning was crushed by Stefan's grip.
Tomas came toward the bridge. He waved the sword around, holding it loosely, heedless of the risk of cutting himself. The bottle he held much more tightly, and pushing it to his lips, he tilted his head back for a long swig.
At the sight of the sword, the hostages murmured restlessly. It was this they had come for.
"The sword. For your boy."
That was it. Peter had guessed right. He and Sofia were the trade-off for the sword, and as if to prove the point, Stefan squeezed Peter's throat, choking him, taking his air completely. Peter kicked and struggled, but Stefan might as well have been made of rock.
He knew he didn't have long left, but in his heart he still prayed that his father would stay on his side of the water, perhaps safe. Perhaps.
Tomas stepped forward and put his foot on the bridge. He staggered over the water, and somehow managed not to fall in. As he reached the other side the hostages began to wail with delight, while their leader withdrew a little, pointing at Peter, warning Tomas to come no nearer.
"The sword."
Tomas tried to focus. He turned his head one way and the other, as if utterly failing to comprehend what was happening. But it was clear he knew what was expected of him.
He took another half step away from the safety of the bridge, and then, turning the sword around, threw it onto the snowy ground, just in front of the leader's feet.
There was a shriek of ecstasy from the hostages, and their leader moved to pick up the sword.
Peter had almost passed out, but he had enough life in him yet to be amazed.
And then, Tomas changed.
"Now!" he shouted.
There was a rustle in the sky, and a dozen Gypsies dropped from the trees onto the hostages' backs. The surprise was enough to make Stefan let Peter fall to the ground. Peter rolled away, in time to see his father fly forward, beating the hostage to the sword where it lay. In a single motion, Tomas rolled on his back, picked up the sword, and slid it into the man's chest.
Once the initial surprise was over, the Gypsies were no match for the strength of the hostages. They were thrown to the ground. All seemed lost, but Tomas had left his first victim in the snow. He walked steadily into the heart of the fight, and the sword swung around him so fast that Peter could barely see it moving.
Now Peter saw the power of the sword. All it took was a single cut from its blade. Hostage after hostage fell, unmoving. Only three remained, Agnes and two men.
Peter scrambled over to where Sofia lay in the snow, to see if she was all right.
Sofia cried out and Peter turned to see Agnes right behind them, reaching ice-cold hands toward his throat.
Behind her, Tomas made a mistake. The final two hostages closed on him at the same time, and he hesitated, trying to decide which to attack first. One of them punched him in the stomach, so hard that he was sent spinning to the ground.
In that endless second he recovered himself. With a single sweep of the sword he made an inch-deep cut in two necks. The hostages fell to the ground, at rest once more.
Tomas squirmed in the snow, the pain in his stomach enough to prevent him from standing.
Seven. There had been a girl, too, and craning his neck he saw her, with her hands on Peter's throat.
In desperation he threw the sword at her, but it missed, landing short in the snow.
Sofia grabbed Agnes's arms, trying to break her grip, unaware that the sword lay just behind her, but her efforts were meaningless; the hostage was stronger than love, stronger than hate.
Sofia let go of Agnes's arms.
She took a step back.
Then she began to speak. To call it singing would be a lie. She mouthed the words at first, and seeing Agnes look away from Peter for a second, she felt hope rise in her.
She gave the silent words a voice, still not singing, but whispering.
Agnes's hands dropped to her sides, and Peter gasped air back into his lungs. He stared at Sofia, in wonder at what he was seeing.
But now Sofia was singing.
She sang the Miorita, and finally Peter understood the meaning of the song.
He tried to join in, but at first his damaged throat would give no voice.
Sofia sang louder, and Agnes backed away from her, floundering through the snow, yet somehow transfixed by the song. And at last Peter found his voice.
Together they sang the Miorita, and as they reached the end of the song, Agnes lay down in the snow, as still as all those who had been touched by the sword. Sofia fetched Tomas's sword from the snow and handed it to Peter.
He hesitated, but Sofia held his hand.
"A single cut?" he asked, and Sofia nodded.
He moved the tip toward Agnes, surprised by the sword's weight, then made a mark so small on her neck that it might have been a pinprick.
"Sorry," he whispered, so faintly that even Sofia, a step behind him, didn't hear.
He gazed at the girl he had once thought he loved lying in the snow, dead.
**38**
**The Song of the Miorita**
Peter scrambled to his feet and rushed to Tomas, who had managed to sit up.
"How did you do that?" Peter cried. "You were so drunk!"
Tomas smiled.
"Haven't touched a drop all afternoon," he said, "but it worked."
Sofia came over. Her uncle was with her.
"Tomas," Milosh said. "Are you hurt?"
Tomas chose not to answer this.
"How are your people?" he asked.
"Some are hurt. Some badly. But all are alive. Thanks to you. I am glad you came to your senses at last."
"I didn't come willingly," Tomas said.
"What happened?" Peter asked. He turned to Milosh. "You came to take the sword by force?"
"Yes," said Milosh, "but when we got here we found Tomas waiting for us. It seems something had changed his mind. Someone."
Peter glanced at his father, then looked away at the ground, his heart pounding.
"We knew they would come for the sword, so we waited. We didn't expect them to bring you, but it played into our hands."
Sofia looked at Peter, smiling, but Peter was looking at his father on the ground.
"Are you hurt? What happened?"
Milosh knelt down beside Tomas, feeling for the wound.
"One of them got a blow in. Here."
Tomas winced as Milosh pressed his stomach. Pulling back Tomas's clothes, they could see a huge discolored swelling already forming.
"You're bleeding. We must get you inside."
"No," said Tomas.
"No?" said Milosh.
"Peter. What did you see in the village? There are others from the graveyard, aren't there?"
Peter nodded.
"Dozens," he said.
"That's not possible!" Milosh cried.
"It's true, Uncle. I saw them," Sofia said. "And they won't stop now."
Tomas sat up straighter. "We have to act. We have learned a lot today. We have seen hostages walk in daylight. It seems Agnes was among them. There may be others. I have never heard of that before. So the Shadow Queen's power is growing. Some of them may even have been living among us, biding their time. And the hostages have learned about the sword, and they want it. Well, they will get it, but we have learned something else too."
Tomas turned to Sofia.
"What you did...How did you know?"
"I...I didn't," Sofia said slowly. "I just believed."
"There!" Tomas said. "And what did you believe?"
"I'm not sure..."
She shook her head, puzzled, and Peter laughed.
"I know!" he said. "I believe it too. I understand the song."
"And what does the song teach us?" Tomas said. "Does it teach us to go to our deaths, without fighting? To accept our fate?"
"No," said Peter. "No. It teaches us to embrace death while we live, to understand it, so that when we do finally come to die, we may accept it without fear. And that way we can live free of fear, believing in ourselves."
"That is it," said Milosh. "That is it. Death is part of life. They are inseparable. You cannot have one without the other. The song teaches us that if we accept a wedding with death, we can go to our graves content. It is people's failure to understand this that makes them prey to the Shadow Queen."
"How?" asked Peter.
"She can feel the discontent of the dead, those who were not content, those who had not understood the Miorita. These people are open to her power, and so she brings them back from the grave."
"And now we have another weapon!" Peter cried. "A song!"
Peter could feel this faith within him, as a presence that he had failed to see until now. Belief in the song, and true understanding of its message, that was it. Enough to give power, enough to lay the hostages to rest.
"Yes," said Tomas. "And a great confrontation is upon us. Help me to my feet."
"No, Father!" Peter cried. "You must rest. Let someone else take the sword."
"Your son is right," Milosh said. "Give the sword to me. I am not as skilled as you, but I will do my best. You are hurt."
Seeing no help from anyone, Tomas rolled onto his side, then scrambled to all fours. He raised his head, and trunk, and knelt. He put one foot flat in the snow, and pushed for all he was worth. He stood.
"No, Milosh. I am not hurt," he said. "I am dying. But my swordhand is singing. I will take the sword into the village, and put an end to it."
Milosh dropped his head, unable to meet Tomas's stare.
"Please, Father," Peter said. "Please don't."
Tomas turned to his son, his face pale with pain.
He spoke softly, so that only Peter could hear.
"I have been a bad father to you. Please give me the chance to be a good one."
Tears welled in Peter's eyes, and he wiped them away with the back of his hand, but he looked his father in the face, and nodded.
"We will help you," he said.
"The song!" Sofia cried. "We can help you with the song!"
Tomas nodded.
"Then let's be ready," he said. "We have the sword. Milosh, you have six men here. More in your camp. Peter, my son. Sofia. And we have the song! If only I had a horse. Peter, you should have seen me with King Michael! When we rode our warhorses into battle, the ground itself shook with fear!"
"But look!" said Sofia. "You do have a horse! Sultan!"
They all turned and saw the old white horse walking serenely through the trees toward them, his head nodding as he came.
Tomas laughed.
"What do you say, Sultan? Can you manage it?"
Sultan snorted.
**39**
**Resurrection**
They made an extraordinary sight, but there was no one to see them as they made their way through the forest, toward the stricken village.
At their head, a fat, red-cheeked man rode a stocky white horse. The rider and horse formed the point of an arrow, as behind and to each side walked his friends. His son. His dead comrade's brother and daughter. Others of their kind, maybe twenty in all.
They saw no one, and no one saw them.
They reached the village.
They walked down the long main street that led to the square and still they saw no one. Not a word was spoken, and the silence in the streets was absolute.
They arrived in the square, and stopped.
And now they came.
From every alley and street, they came. Those whom the Shadow Queen had brought from the ground.
They did not come slowly. They ran, they hurtled toward the man and his horse.
"Sing!" he shouted.
They sang, twenty voices in unison, with full lungs and loud voices. The hostages began to falter and hesitate, slowing in their great number. But still they came on. And there were scores of them.
The rider knew the moment had come.
He looked down to his son, and smiled.
He kicked the horse into action.
"Sultan!" he cried. "Hah!"
Away he rode into the fight. Behind him, the singing voices lifted higher and higher, reaching out to protect him as he darted this way and that through the crowd, the sword flashing in the failing light in the square.
Bodies began to pile all around him, bodies that lay still, that did not wish to leave the ground anymore, and as he fought on through the grappling hands and the clawing fingers, he saw that he would die.
There was nothing for Tomas now.
Not the singing.
Not the square.
Not the dead.
Not even Sultan.
Just the sword, which flew so fast that the air itself was cut in two.
But the hands grasped and grappled and there were too many. He was pulled from Sultan's back, landing clumsily in the mud.
From a seemingly vast distance, he heard a cry.
"Father!"
Peter. It was his son, sprinting to be beside him in a moment. Dimly, Tomas saw Peter snatch the sword from the ground and begin to swing it wildly about him. The hostages faltered, shocked by the fluid energy of the boy, by his strength.
Tomas's eyes were closed now, but in his mind he could see Peter twisting and stroking the blade from side to side.
"That's it," he whispered. "That's it. Feel it!"
In his heart, he heard Peter's reply.
"Yes, Father. My swordhand is singing."
Tomas found himself staring at nothing but a bright white light that seemed to open in the sky above him, pouring down onto the blade, bathing him in joy.
Joy that he had been good, one last time.
That he had given.
That he was a good father, with a good son. It was the joy of completeness.
Even as Peter swung the sword for the last time, and gave rest to the last of the hostages, and fell to his knees by his father, the joy was irrepressible.
As Tomas died, his heart was singing, and a smile spread across his face.
**40**
**A Perfect Shade of Green**
Days pass, whether you want them to or not. For Peter the days passed slowly, but nonetheless, one day winter had gone.
The Gypsies stayed on through the winter, living in the clearing just as they had before. Every day, Sofia would visit Peter and Sultan on their little island.
Peter, his mind drifting, seldom spoke on these occasions, but Sofia would talk to him anyway, tell him of news from the camp, and from the village. She told him that when St. George's Day arrived, they would be on their way once more. This was a piece of news that Peter had taken in.
One day, as Sofia cooked some soup on the stove in the hut, Peter got to his feet abruptly.
Startled, she looked at him.
"What is it?"
"Come with me," he said.
He took her by her lovely brown hand and gently led her out, and around the side of the hut, to the toolshed.
"Look," he said. "That was the first thing I found."
Sofia shook her head, not understanding what he meant.
All she saw was a row of tools. Saws, chisels, hammers, gouges, laid out on a bench in a neat line.
"He was never like that," Peter said. "The tools were always a mess. Whenever he finished with something he'd leave it where it fell. If he put things back in here at all, he'd leave them all over the place. But when we came back from the village—that last day—this was the first thing I found. At some point, when he and Milosh and the others were waiting for the hostages' attack, he came out here and tidied up the tools."
He hesitated, then asked, "Why did he do that?"
Sofia shrugged.
"He knew," she said. "He'd already decided what he was going to do. He did it for you, because you brought him back to himself."
What happened after Tomas fell from the horse was a blur to Peter. He knew he had rushed to his father, that he had taken his father's sword and fought for him, but he couldn't remember the details.
He knew they had won. He remembered that the villagers came rushing out from their houses, among them Teodor and Daniel, who fell on their knees in thankfulness before Peter and the Gypsies.
And then there was Anna, old Anna, whom everyone feared. Even she came and begged forgiveness from them all.
"What can I do to thank you?" she wailed.
Milosh had told her.
"Stand up," he said.
She did so, a puzzled look on her face.
"Turn around," he said, and the old woman complied.
Before she had turned back, Milosh had snatched the sword from the ground beside Tomas, and with a single stroke had cut Anna to the ground.
There had been shock and outrage. But only at first, as Milosh gently rolled Anna's body face down.
There he pointed out what no one else had seen, or at least understood. The old woman's back was covered in sawdust.
"From a coffin," he explained. "That was why she didn't want to help us. She was one of them! She was guiding them, infected perhaps by the Shadow Queen herself.... I would have understood sooner, but we didn't know they could move by daylight."
Milosh had turned to give the sword to Peter.
"This is yours now," he said, but Peter shook his head.
"No," he said. "I don't want it. It's what my father didn't want to be. Let me give it to you. You will need it. You can use it."
Milosh nodded.
They buried Tomas in the very graveyard from which the epidemic had first sprung, now truly a place of final rest, thanks to their efforts.
Finally, St. George's Day came, and with it, Sofia came to see Peter for the last time.
It was a beautiful spring morning, full of bursting hope. The trees were heavy with leaves, and flowers had leapt into life in the meadow. Peter was sitting on a tree stump on his island, thinking how the early green of spring was the most perfect of the year, when into his vision walked Sofia.
"May I come over?" she called.
Peter waved and she crossed the bridge.
Beyond her, Peter saw the Gypsy caravan on the path through the trees, and knew the day had come.
"Don't say anything," he said as she came close.
The smile softened his words, but Sofia still ignored him.
"Why don't you come with us?" she said.
"I can't," Peter said. "You know that as well as I do. This is my home. I belong here, in the home my father and I built. I want to stay."
Sofia hung her head.
"Besides," he went on, "who'd cut their stupid wood for them?"
He nodded toward the village.
Despite herself, Sofia laughed.
"Where are you going?" he asked.
"I don't know," she said. "Wherever we need to. Wherever we hear of hostages that need to be freed. But first we are going to roll through the forest, and eventually we will find the meadows that lie beneath the mountains. They will be full of flowers and bees, and the rivers will be full of fish. We'll stop there for a while and rest. We'll sing, and make music."
"It sounds wonderful," said Peter.
"It is."
"God bless you."
"And you, Peter."
She stepped forward, and finally closed the gap there had always been between them. She kissed him, and then they both heard laughter from the caravan.
She pulled away, blushing, and without another word, walked back to her future.
As she went, she sang, and Peter heard the final verse of the Miorita float into the air, where the shepherd marries a princess from the heavens.
He watched as she climbed aboard the cart and sat next to her uncle, and they disappeared into the trees.
Peter wandered back to the hut, but he found his eyes pricking and his head full of sorrow. Deciding he needed to be busy, he walked around to the toolshed, intending to sharpen the axes. There on the bench, as before, everything lay in neat and tidy rows, but suddenly he saw something he had missed before, a small rag, twisted into a ball. He picked it up and a tiny object inside the cloth fell into the sawdust at his feet.
He bent down and picked it up.
It was a carving, a small wooden carving.
Of a goose.
When his father had carved it, Peter didn't know, but he knew it was for him. Tomas must have left it with the tools as he and Milosh and the others waited.
Peter knew what it meant.
His mind drifted back to the first day when he'd seen Sofia arrive in the square. Something had tugged at his heart that day, but he had not known what it was. Now, suddenly, he knew. It was right in front of him, in Sofia, in his hand, but until this moment he just hadn't seen it.
It was their life, their nomadic life. He thought he was tired of traveling all his life, always on the move with his father. But now he saw that it was the only life he knew. It was the life he wanted. He looked again at the carving, identical to the one his father had made for him on his fifth birthday. He saw it not only as an apology but as a message, and knew that it was time to fly away again, like the geese.
And just like the shepherd in the song, he had a princess waiting for him.
Gently, he tucked the goose into his pocket.
"Sultan!" he cried, running to the stable, and Sultan came.
He flung himself onto the horse, and they hammered away over the bridge.
"Wait!" he called. "Wait! I'm coming with you!"
**Author's Note**
Most people are familiar with the whys and wherefores of the vampire, but few realize how far a journey this nightmare figure has made. Today we can recognize a vampire in film or book by pointed canine teeth, a cape maybe, or an accompanying bat. The suave, sometimes overtly attractive vampire of modern myth is very far from the original revenants of the folklore where these creatures originated. In fact those first vampires are more like zombies with a bloodlust—either horrific bloated corpses returned from the earth, or beings indistinguishable from their former living selves (and what a dangerous thing that would be). Sometimes even the bloodlust is absent; one vampire's preferred sustenance was noted to be milk! In many instances it is impossible to distinguish between the vampire and what we would know as the werewolf.
In writing this book I sought to capture the flavor of the early reports of vampirism, from the well-known case of the Shoemaker of Silesia in 1591 to the unnatural dealings of Peter Plogojowitz in 1725, via a myriad of less popular stories collected from various Eastern European countries: Pëtr Bogatyrëv's studies of the Subcarpathian Rus and Alan Dundes' unsurpassed anthropological collection _The Vampire: A Casebook_ repaid their reading many times over. I also urge you, if interested, to read Paul Barber's _Vampires, Burial, and Death,_ which presents a well-argued theory using forensic pathology on the possible biochemical origin of many of the traits of the vampire.
To make a coherent story I had to pick and choose from among hundreds of stories, many of which flatly contradicted one another—for there are almost as many types of vampire as there are vampire stories. One example would be the vampires' reaction to light: in some stories they may appear only at night, in others they are immune from any potentially destructive power of this force for good. Even giving vampires a name is not a simple thing; here are just a few of them: _krvoijac, vukodlak, wilkolak, varcolac, vurvolak, liderc nadaly, liougat, kulkutha, moroii, strigoii, murony, streghoi, vrykolakoi, upir, dschuma, velku dlaka, nachzehrer, zaloznye, nosferatu_ —this last, quite familiar to us, is the vampire's name in that most unholy of vampire lands, Transylvania, literally the Land Beyond the Forest. Transylvania is in fact a beautiful place, with mountains, pastures, and forest just as described in this book. And it is here that the stories of the Miorita, the Wedding of the Dead and the Shadow Queen would be familiar to local people, though again I have had to take certain liberties for the sake of the story. Nowadays we know all these fabulous stories of the undead to be myth, though it might be wise to remember that there are still some people who do not agree with this conclusion. Even in the first few years of this new century, stories have emerged from Romania of modern-day belief in vampires; in 2004 the relatives of a Romanian man were prosecuted for exhuming his corpse, burning his heart and drinking the ashes in water because they believed he had been visiting them in the night....
**About the Author**
Since _Floodland_ won the Branford Boase Award for the best first children's novel of 2000, Marcus Sedgwick's books have been short-listed for many awards, including the _Guardian_ Children's Fiction Award, the Blue Peter Book Award, the Carnegie Medal, and the Edgar Allan Poe Award.
By day he works in children's publishing, and by night he is the drummer in a rock band in Brighton. He lives in Sussex with his wife, Pippa, and has a daughter, Alice.
About _My Swordhand Is Singing_ he says: "It was fascinating to discover the original folklore that gave birth to the vampire legend. No snowy graveyard is left unvisited, no corpse undisturbed, no spell unspoken, no date with destiny unmet. But it's not all gloom; there are misery and horror, too."
ALSO BY MARCUS SEDGWICK
_The Book of Dead Days_
_The Dark Flight Down_
_The Dark Horse_
_Floodland_
_The Foreshadowing_
_Witch Hill_
Published by Wendy Lamb Books
an imprint of Random House Children's Books
a division of Random House, Inc.
New York
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
Text and illustrations copyright © 2006 by Marcus Sedgwick
Originally published in Great Britain in 2006 by Orion Children's Books
All rights reserved.
WENDY LAMB BOOKS and colophon are trademarks of Random House, Inc.
www.randomhouse.com/teens
Educators and librarians, for a variety of teaching tools, visit us at www.randomhouse.com/teachers
_Library of Congress Cataloging-in-Publication Data_
Sedgwick, Marcus.
My swordhand is singing / Marcus Sedgwick.—1st ed.
p. cm.
Summary: In the dangerous dark of winter in an Eastern European village during the early seventeenth century, Peter learns from a gypsy girl that the Shadow Queen is behind the recent murders and reanimations, and his father's secret past may hold the key to stopping her.
[1. Supernatural—Fiction. 2. Murder—Fiction. 3. Fathers and sons—Fiction. 4. Vampires—Fiction. 5. Romanies—Fiction. 6. Villages—Fiction. 7. Superstition—Fiction. 8. Europe, Eastern—History—17th century—Fiction.] I. Title.
PZ7.S4484My 2007
[Fic}—dc22 2007007051
eISBN: 978-0-375-89084-0
v3.0
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Q: Python : Mock a module that raises an exception I need to test a function in a module that import another module which raises an exception when imported.
#a.py
raise ValueError("hello")
my_const = 'SOMETHING'
#b.py
from a import my_const
def foo():
# do something with my_const
return "expected_result"
#test_foo.py
def test_foo():
from b import foo
assert foo() == "expected_result"
Here when I import foo in test_foo.py, a.py get imported in b.py, an exception is raised and the import is never completed so my_const is not available in b.py.
I'm not allowed to modify neither a.py or b.py. Also, using unittest.patch and @patch('a', 'my_const') does import a.py so it doens't work.
It is possible create the module dynamically with the import lib and add it to sys.modules, but is there another solution that doesn't require importlib ?
A: As far as I know, you can create and importe the module dynamically. Here is a code inspired from the
"Approximating importlib.import_module()" section in the import lib documentation
from importlib.util import module_from_spec, find_spec
import sys
def patched_import(name, **kwargs):
spec = find_spec(name)
m = module_from_spec(spec)
for k in kwargs:
setattr(m, k, kwargs[k])
sys.modules[name] = m
Edit: My solution should be ok for a mock-up but be careful as manipulation of referential can have side effects.
To use it, just do:
patched_import('a', my_const='stuff')
Before importing b.py.
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Produced by Irma Spehar and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive/Canadian Libraries)
A LECTURE ON THE STUDY OF HISTORY
MACMILLAN AND CO., LIMITED
LONDON . BOMBAY . CALCUTTA
MELBOURNE
THE MACMILLAN COMPANY
NEW YORK . BOSTON . CHICAGO
ATLANTA . SAN FRANCISCO
THE MACMILLAN CO. OF CANADA, LTD.
TORONTO
A LECTURE
ON
THE STUDY OF HISTORY
_DELIVERED AT CAMBRIDGE,
JUNE 11, 1895_
BY
LORD ACTON
LL.D., D.C.L.
REGIUS PROFESSOR OF MODERN HISTORY
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
1911
RICHARD CLAY AND SONS, LIMITED,
BRUNSWICK STREET, STAMFORD STREET, S. E.,
AND BUNGAY, SUFFOLK
_First Edition, October, 1895.
Second Edition, January, 1896.
Reprinted, 1905, 1911._
FELLOW STUDENTS,
I look back to-day to a time before the middle of the century, when I
was reading at Edinburgh, and fervently wishing to come to this
University. At three colleges I applied for admission, and, as things
then were, I was refused by all. Here, from the first, I vainly fixed
my hopes, and here, in a happier hour, after five-and-forty years,
they are at last fulfilled.
[Sidenote: UNITY OF MODERN HISTORY]
I desire first to speak to you of that which I may reasonably call the
Unity of Modern History, as an easy approach to questions necessary
to be met on the threshold by any one occupying this place, which my
predecessor has made so formidable to me by the reflected lustre of
his name.
You have often heard it said that Modern History is a subject to which
neither beginning nor end can be assigned. No beginning, because the
dense web of the fortunes of man is woven without a void; because, in
society as in nature, the structure is continuous, and we can trace
things back uninterruptedly, until we dimly descry the Declaration of
Independence in the forests of Germany. No end, because, on the same
principle, history made and history making are scientifically
inseparable and separately unmeaning.
[Sidenote: LINK BETWEEN HISTORY AND POLITICS]
"Politics," said Sir John Seeley, "are vulgar when they are not
liberalised by history, and history fades into mere literature when
it loses sight of its relation to practical politics." Everybody
perceives the sense in which this is true. For the science of politics
is the one science that is deposited by the stream of history, like
grains of gold in the sand of a river; and the knowledge of the past,
the record of truths revealed by experience, is eminently practical,
as an instrument of action, and a power that goes to the making of the
future.[1] In France, such is the weight attached to the study of our
own time, that there is an appointed course of contemporary history,
with appropriate textbooks.[2] That is a chair which, in the
progressive division of labour by which both science and government
prosper,[3] may some day be founded in this country. Meantime, we do
well to acknowledge the points at which the two epochs diverge. For
the contemporary differs from the modern in this, that many of its
facts cannot by us be definitely ascertained. The living do not give
up their secrets with the candour of the dead; one key is always
excepted, and a generation passes before we can ensure accuracy.
Common report and outward seeming are bad copies of the reality, as
the initiated know it. Even of a thing so memorable as the war of
1870, the true cause is still obscure; much that we believed has been
scattered to the winds in the last six months, and further revelations
by important witnesses are about to appear. The use of history turns
far more on certainty than on abundance of acquired information.
Beyond the question of certainty is the question of detachment. The
process by which principles are discovered and appropriated is other
than that by which, in practice, they are applied; and our most sacred
and disinterested convictions ought to take shape in the tranquil
regions of the air, above the tumult and the tempest of active
life.[4] For a man is justly despised who has one opinion in history
and another in politics, one for abroad and another at home, one for
opposition and another for office. History compels us to fasten on
abiding issues, and rescues us from the temporary and transient.
Politics and history are interwoven, but are not commensurate. Ours is
a domain that reaches farther than affairs of state, and is not
subject to the jurisdiction of governments. It is our function to keep
in view and to command the movement of ideas, which are not the
effect but the cause of public events;[5] and even to allow some
priority to ecclesiastical history over civil, since, by reason of the
graver issues concerned, and the vital consequences of error, it
opened the way in research, and was the first to be treated by close
reasoners and scholars of the higher rank.[6]
[Sidenote: NOT GOVERNED BY NATIONAL CAUSES]
In the same manner, there is wisdom and depth in the philosophy which
always considers the origin and the germ, and glories in history as
one consistent epic.[7] Yet every student ought to know that mastery
is acquired by resolved limitation. And confusion ensues from the
theory of Montesquieu and of his school, who, adapting the same term
to things unlike, insist that freedom is the primitive condition of
the race from which we are sprung.[8] If we are to account mind not
matter, ideas not force, the spiritual property that gives dignity,
and grace, and intellectual value to history, and its action on the
ascending life of man, then we shall not be prone to explain the
universal by the national, and civilisation by custom.[9] A speech of
Antigone, a single sentence of Socrates, a few lines that were
inscribed on an Indian rock before the Second Punic War, the footsteps
of a silent yet prophetic people who dwelt by the Dead Sea, and
perished in the fall of Jerusalem, come nearer to our lives than the
ancestral wisdom of barbarians who fed their swine on the Hercynian
acorns.
[Sidenote: MEDIAEVAL LIMIT OF MODERN HISTORY]
For our present purpose, then, I describe as modern history that which
begins four hundred years ago, which is marked off by an evident and
intelligible line from the time immediately preceding, and displays
in its course specific and distinctive characteristics of its own.[10]
The modern age did not proceed from the mediaeval by normal succession,
with outward tokens of legitimate descent. Unheralded, it founded a
new order of things, under a law of innovation, sapping the ancient
reign of continuity. In those days Columbus subverted the notions of
the world, and reversed the conditions of production, wealth and
power; in those days, Machiavelli released government from the
restraint of law; Erasmus diverted the current of ancient learning
from profane into Christian channels; Luther broke the chain of
authority and tradition at the strongest link; and Copernicus erected
an invincible power that set for ever the mark of progress upon the
time that was to come. There is the same unbound originality and
disregard for inherited sanctions in the rare philosophers as in the
discovery of Divine Right, and the intruding Imperialism of Rome. The
like effects are visible everywhere, and one generation beheld them
all. It was an awakening of new life; the world revolved in a
different orbit, determined by influences unknown before. After many
ages persuaded of the headlong decline and impending dissolution of
society,[11] and governed by usage and the will of masters who were in
their graves, the sixteenth century went forth armed for untried
experience, and ready to watch with hopefulness a prospect of
incalculable change.
[Sidenote: INFLUENCE OF KNOWLEDGE ON MODERN HISTORY]
That forward movement divides it broadly from the older world; and the
unity of the new is manifest in the universal spirit of investigation
and discovery which did not cease to operate, and withstood the
recurring efforts of reaction, until, by the advent of the reign of
general ideas which we call the Revolution, it at length
prevailed.[12] This successive deliverance and gradual passage, for
good and evil, from subordination to independence is a phenomenon of
primary import to us, because historical science has been one of its
instruments.[13] If the Past has been an obstacle and a burden,
knowledge of the Past is the safest and the surest emancipation. And
the earnest search for it is one of the signs that distinguish the
four centuries of which I speak from those that went before. The
middle ages, which possessed good writers of contemporary narrative,
were careless and impatient of older fact. They became content to be
deceived, to live in a twilight of fiction, under clouds of false
witness, inventing according to convenience, and glad to welcome the
forger and the cheat.[14] As time went on, the atmosphere of
accredited mendacity thickened, until, in the Renaissance, the art of
exposing falsehood dawned upon keen Italian minds. It was then that
history as we understand it began to be understood, and the
illustrious dynasty of scholars arose to whom we still look both for
method and material. Unlike the dreaming prehistoric world, ours knows
the need and the duty to make itself master of the earlier times, and
to forfeit nothing of their wisdom or their warnings,[15] and has
devoted its best energy and treasure to the sovereign purpose of
detecting error and vindicating entrusted truth.[16]
[Sidenote: INTERNATIONAL IDEAS]
[Sidenote: MEMORABLE MEN]
[Sidenote: INDEPENDENT MINDS]
In this epoch of full-grown history men have not acquiesced in the
given conditions of their lives. Taking little for granted they have
sought to know the ground they stand on, and the road they travel, and
the reason why. Over them, therefore, the historian has obtained an
increasing ascendancy.[17] The law of stability was overcome by the
power of ideas, constantly varied and rapidly renewed;[18] ideas that
give life and motion, that take wing and traverse seas and frontiers,
making it futile to pursue the consecutive order of events in the
seclusion of a separate nationality.[19] They compel us to share the
existence of societies wider than our own, to be familiar with distant
and exotic types, to hold our march upon the loftier summits, along
the central range, to live in the company of heroes, and saints, and
men of genius, that no single country could produce. We cannot afford
wantonly to lose sight of great men and memorable lives, and are bound
to store up objects for admiration as far as may be;[20] for the
effect of implacable research is constantly to reduce their number. No
intellectual exercise, for instance, can be more invigorating than to
watch the working of the mind of Napoleon, the most entirely known as
well as the ablest of historic men. In another sphere, it is the
vision of a higher world to be intimate with the character of Fenelon,
the cherished model of politicians, ecclesiastics, and men of letters,
the witness against one century and precursor of another, the advocate
of the poor against oppression, of liberty in an age of arbitrary
power, of tolerance in an age of persecution, of the humane virtues
among men accustomed to sacrifice them to authority, the man of whom
one enemy says that his cleverness was enough to strike terror, and
another, that genius poured in torrents from his eyes. For the minds
that are greatest and best alone furnish the instructive examples. A
man of ordinary proportion or inferior metal knows not how to think
out the rounded circle of his thought, how to divest his will of its
surroundings and to rise above the pressure of time and race and
circumstance,[21] to choose the star that guides his course, to
correct, and test, and assay his convictions by the light within,[22]
and, with a resolute conscience and ideal courage, to re-model and
reconstitute the character which birth and education gave him.[23]
[Sidenote: FOREIGN CONSTITUTIONS]
For ourselves, if it were not the quest of the higher level and the
extended horizon, international history would be imposed by the
exclusive and insular reason that parliamentary reporting is younger
than parliaments. The foreigner has no mystic fabric in his
government, and no _arcanum imperii_. For him, the foundations have
been laid bare; every motive and function of the mechanism is
accounted for as distinctly as the works of a watch. But with our
indigenous constitution, not made with hands or written upon paper,
but claiming to develope by a law of organic growth; with our
disbelief in the virtue of definitions and general principles and our
reliance on relative truths, we can have nothing equivalent to the
vivid and prolonged debates in which other communities have displayed
the inmost secrets of political science to every man who can read.
And the discussions of constituent assemblies, at Philadelphia,
Versailles and Paris, at Cadiz and Brussels, at Geneva, Frankfort and
Berlin, above nearly all, those of the most enlightened States in the
American Union, when they have recast their institutions, are
paramount in the literature of politics, and proffer treasures which
at home we have never enjoyed.
[Sidenote: RESOURCES OF MODERN HISTORY]
[Sidenote: BEGINNING OF THE DOCUMENTARY AGE]
To historians the later part of their enormous subject is precious
because it is inexhaustible. It is the best to know because it is the
best known and the most explicit. Earlier scenes stand out from a
background of obscurity. We soon reach the sphere of hopeless
ignorance and unprofitable doubt. But hundreds and even thousands of
the moderns have borne testimony against themselves, and may be
studied in their private correspondence and sentenced on their own
confession. Their deeds are done in the daylight. Every country opens
its archives and invites us to penetrate the mysteries of State. When
Hallam wrote his chapter on James II., France was the only Power whose
reports were available. Rome followed, and the Hague; and then came
the stores of the Italian States, and at last the Prussian and the
Austrian papers, and partly those of Spain. Where Hallam and Lingard
were dependent on Barillon, their successors consult the diplomacy of
ten governments. The topics indeed are few on which the resources have
been so employed that we can be content with the work done for us, and
never wish it to be done over again. Part of the lives of Luther and
Frederic, a little of the Thirty Years' War, much of the American
Revolution and the French Restoration, the early years of Richelieu
and Mazarin, and a few volumes of Mr. Gardiner, show here and there
like Pacific islands in the ocean. I should not even venture to claim
for Ranke, the real originator of the heroic study of records, and the
most prompt and fortunate of European pathfinders, that there is one
of his seventy volumes that has not been overtaken and in part
surpassed. It is through his accelerating influence mainly that our
branch of study has become progressive, so that the best master is
quickly distanced by the better pupil.[24] The Vatican archives alone,
now made accessible to the world, filled 3,239 cases when they were
sent to France; and they are not the richest. We are still at the
beginning of the documentary age, which will tend to make history
independent of historians, to develope learning at the expense of
writing, and to accomplish a revolution in other sciences as well.[25]
[Sidenote: MODERN HISTORY]
To men in general I would justify the stress I am laying on modern
history, neither by urging its varied wealth, nor the rupture with
precedent, nor the perpetuity of change and increase of pace, nor the
growing predominance of opinion over belief, and of knowledge over
opinion, but by the argument that it is a narrative told of ourselves,
the record of a life which is our own, of efforts not yet abandoned to
repose, of problems that still entangle the feet and vex the hearts of
men. Every part of it is weighty with inestimable lessons that we
must learn by experience and at a great price, if we know not how to
profit by the example and teaching of those who have gone before us,
in a society largely resembling the one we live in.[26] Its study
fulfils its purpose even if it only makes us wiser, without producing
books, and gives us the gift of historical thinking, which is better
than historical learning.[27] It is a most powerful ingredient in the
formation of character and the training of talent, and our historical
judgments have as much to do with hopes of heaven as public or private
conduct. Convictions that have been strained through the instances and
the comparisons of modern times differ immeasurably in solidity and
force from those which every new fact perturbs, and which are often
little better than illusions or unsifted prejudice.[28]
[Sidenote: A SCHOOL OF OPINION]
[Sidenote: INFLUENCE OF THE RELIGIOUS ELEMENT]
The first of human concerns is religion, and it is the salient feature
of the modern centuries. They are signalised as the scene of
Protestant developments. Starting from a time of extreme indifference,
ignorance, and decline, they were at once occupied with that conflict
which was to rage so long, and of which no man could imagine the
infinite consequences. Dogmatic conviction--for I shun to speak of
faith in connection with many characters of those days--dogmatic
conviction rose to be the centre of universal interest, and remained
down to Cromwell the supreme influence and motive of public policy. A
time came when the intensity of prolonged conflict, when even the
energy of antagonistic assurance, abated somewhat, and the
controversial spirit began to make room for the scientific; and as the
storm subsided, and the area of settled questions emerged, much of
the dispute was abandoned to the serene and soothing touch of
historians, invested as they are with the prerogative of redeeming the
cause of religion from many unjust reproaches, and from the graver
evil of reproaches that are just. Ranke used to say that Church
interests prevailed in politics until the Seven Years' War, and marked
a phase of society that ended when the hosts of Brandenburg went into
action at Leuthen, chanting their Lutheran hymns.[29] That bold
proposition would be disputed even if applied to the present age.
After Sir Robert Peel had broken up his party, the leaders who
followed him declared that no-popery was the only basis on which it
could be reconstructed.[30] On the other side may be urged that, in
July 1870, at the outbreak of the French war, the only government
that insisted on the abolition of the temporal power was Austria; and
since then we have witnessed the fall of Castelar, because he
attempted to reconcile Spain with Rome.
[Sidenote: RELIGION]
Soon after 1850 several of the most intelligent men in France, struck
by the arrested increase of their own population and by the telling
statistics from Further Britain, foretold the coming preponderance of
the English race. They did not foretell, what none could then foresee,
the still more sudden growth of Prussia, or that the three most
important countries of the globe would, by the end of the century, be
those that chiefly belonged to the conquests of the Reformation. So
that in Religion, as in so many things, the product of these
centuries has favoured the new elements; and the centre of gravity,
moving from the Mediterranean nations to the Oceanic, from the Latin
to the Teuton, has also passed from the Catholic to the
Protestant.[31]
[Sidenote: THE CAUSE OF LIBERTY]
[Sidenote: REVOLUTION]
Out of these controversies proceeded political as well as historical
science. It was in the Puritan phase, before the restoration of the
Stuarts, that theology, blending with politics, effected a fundamental
change. The essentially English reformation of the seventeenth century
was less a struggle between churches than between sects, often
subdivided by questions of discipline and self-regulation rather than
by dogma. The sectaries cherished no purpose or prospect of prevailing
over the nations; and they were concerned with the individual more
than with the congregation, with conventicles, not with
state-churches. Their view was narrowed, but their sight was
sharpened. It appeared to them that governments and institutions are
made to pass away, like things of earth, whilst souls are immortal;
that there is no more proportion between liberty and power than
between eternity and time; that, therefore, the sphere of enforced
command ought to be restricted within fixed limits, and that which had
been done by authority, and outward discipline, and organised
violence, should be attempted by division of power, and committed to
the intellect and the conscience of free men.[32] Thus was exchanged
the dominion of will over will for the dominion of reason over reason.
The true apostles of toleration are not those who sought protection
for their own beliefs, or who had none to protect; but men to whom,
irrespective of their cause, it was a political, a moral, and a
theological dogma, a question of conscience, involving both religion
and policy.[33] Such a man was Socinus; and others arose in the
smaller sects--the Independent founder of the colony of Rhode Island,
and the Quaker patriarch of Pennsylvania. Much of the energy and zeal
which had laboured for authority of doctrine was employed for liberty
of prophesying. The air was filled with the enthusiasm of a new cry;
but the cause was still the same. It became a boast that religion was
the mother of freedom, that freedom was the lawful off spring of
religion; and this transmutation, this subversion of established forms
of political life by the development of religious thought, brings us
to the heart of my subject, to the significant and central feature of
the historic cycle before us. Beginning with the strongest religious
movement and the most refined despotism ever known, it has led to the
superiority of politics over divinity in the life of nations, and
terminates in the equal claim of every man to be unhindered by man in
the fulfilment of duty to God[34]--a doctrine laden with storm and
havoc, which is the secret essence of the Rights of Man, and the
indestructible soul of Revolution.
[Sidenote: THE MODE OF LIBERTY]
[Sidenote: PROGRESS]
[Sidenote: THE MARK OF PROVIDENCE]
When we consider what the adverse forces were, their sustained
resistance, their frequent recovery, the critical moments when the
struggle seemed for ever desperate, in 1685, in 1772, in 1808, it is
no hyperbole to say that the progress of the world towards
self-government would have been arrested but for the strength afforded
by the religious motive in the seventeenth century. And this
constancy of progress, of progress in the direction of organised and
assured freedom, is the characteristic fact of modern history, and its
tribute to the theory, of Providence.[35] Many persons, I am well
assured, would detect that this is a very old story, and a trivial
commonplace, and would challenge proof that the world is making
progress in aught but intellect, that it is gaining in freedom, or
that increase in freedom is either a progress or a gain. Ranke, who
was my own master, rejected the view that I have stated;[36] Comte,
the master of better men, believed that we drag a lengthening chain
under the gathered weight of the dead hand;[37] and many of our recent
classics, Carlyle, Newman, Froude, were persuaded that there is no
progress justifying the ways of God to man, and that the mere
consolidation of liberty is like the motion of creatures whose advance
is in the direction of their tails. They deem that anxious precaution
against bad government is an obstruction to good, and degrades
morality and mind by placing the capable at the mercy of the
incapable, dethroning enlightened virtue for the benefit of the
average man. They hold that great and salutary things are done for
mankind by power concentrated, not by power balanced and cancelled and
dispersed, and that the whig theory, sprung from decomposing sects,
the theory that authority is legitimate only by virtue of its checks,
and that the sovereign is dependent on the subject, is rebellion
against the divine will manifested all down the stream of time.
[Sidenote: CERTAINTY]
[Sidenote: DEPENDENT ON RESERVE]
I state the objection not that we may plunge into the crucial
controversy of a science that is not identical with ours, but in order
to make my drift clear by the defining aid of express contradiction.
No political dogma is as serviceable to my purpose here as the
historian's maxim to do the best he can for the other side, and to
avoid pertinacity or emphasis on his own. Like the economic precept
_Laissez-faire_[38] which the eighteenth century derived from Colbert,
it has been an important, if not a final step in the making of method.
The strongest and most impressive personalities, it is true, like
Macaulay, Thiers, and the two greatest of living writers, Mommsen and
Treitschke, project their own broad shadow upon their pages. This is a
practice proper to great men, and a great man may be worth several
immaculate historians. Otherwise there is virtue in the saying that a
historian is seen at his best when he does not appear.[39] Better for
us is the example of the Bishop of Oxford, who never lets us know what
he thinks of anything but the matter before him; and of his
illustrious French rival, Fustel de Coulanges, who said to an excited
audience: "Do not imagine you are listening to me; it is history
itself that speaks."[40] We can found no philosophy on the observation
of four hundred years, excluding three thousand. It would be an
imperfect and a fallacious induction. But I hope that even this narrow
and disedifying section of history will aid you to see that the action
of Christ who is risen on mankind whom he redeemed fails not, but
increases;[41] that the wisdom of divine rule appears not in the
perfection but in the improvement of the world;[42] and that achieved
liberty is the one ethical result that rests on the converging and
combined conditions of advancing civilisation.[43] Then you will
understand what a famous philosopher said, that History is the true
demonstration of Religion.[44]
[Sidenote: MEANING OF LIBERTY]
But what do people mean who proclaim that liberty is the palm, and the
prize, and the crown, seeing that it is an idea of which there are two
hundred definitions, and that this wealth of interpretation has caused
more bloodshed than anything, except theology? Is it Democracy as in
France, or Federalism as in America, or the national independence
which bounds the Italian view, or the reign of the fittest, which is
the ideal of Germans?[45] I know not whether it will ever fall within
my sphere of duty to trace the slow progress of that idea through the
chequered scenes of our history, and to describe how subtle
speculations touching the nature of conscience promoted a nobler and
more spiritual conception of the liberty that protects it,[46] until
the guardian of rights developed into the guardian of duties which are
the cause of rights,[47] and that which had been prized as the
material safeguard for treasures of earth became sacred as security
for things that are divine. All that we require is a workday key to
history, and our present need can be supplied without pausing to
satisfy philosophers. Without inquiring how far Sarasa or Butler, Kant
or Vinet, is right as to the infallible voice of God in man, we may
easily agree in this, that where absolutism reigned, by irresistible
arms, concentrated possessions, auxiliary churches, and inhuman laws,
it reigns no more; that commerce having risen against land, labour
against wealth, the state against the forces dominant in society,[48]
the division of power against the state, the thought of individuals
against the practice of ages, neither authorities, nor minorities, nor
majorities can command implicit obedience; and, where there has been
long and arduous experience, a rampart of tried conviction and
accumulated knowledge,[49] where there is a fair level of general
morality, education, courage, and self-restraint, there, if there
only, a society may be found that exhibits the condition of life
towards which, by elimination of failures, the world has been moving
through the allotted space.[50] You will know it by outward signs:
Representation, the extinction of slavery, the reign of opinion, and
the like; better still by less apparent evidences: the security of the
weaker groups[51] and the liberty of conscience, which, effectually
secured, secures the rest.
[Sidenote: THE GROWTH OF REVOLUTION]
[Sidenote: RENOVATION OF HISTORY BY REVOLUTION]
Here we reach a point at which my argument threatens to abut on a
contradiction. If the supreme conquests of society are won more often
by violence than by lenient arts, if the trend and drift of things is
towards convulsions and catastrophes,[52] if the world owes religious
liberty to the Dutch Revolution, constitutional government to the
English, federal republicanism to the American, political equality to
the French and its successors,[53] what is to become of us, docile and
attentive students of the absorbing Past? The triumph of the
Revolutionist annuls the historian.[54] By its authentic exponents,
Jefferson and Sieyes, the Revolution of the last century repudiates
history. Their followers renounced acquaintance with it, and were
ready to destroy its records and to abolish its inoffensive
professors. But the unexpected truth, stranger than fiction, is that
this was not the ruin but the renovation of history. Directly and
indirectly, by process of development and by process of reaction, an
impulse was given which made it infinitely more effectual as a factor
of civilisation than ever before, and a movement began in the world of
minds which was deeper and more serious than the revival of ancient
learning.[55] The dispensation under which we live and labour consists
first in the recoil from the negative spirit that rejected the law of
growth, and partly in the endeavour to classify and adjust the
revolution, and to account for it by the natural working of historic
causes. The Conservative line of writers, under the name of the
Romantic or Historical School, had its seat in Germany, looked upon
the Revolution as an alien episode, the error of an age, a disease to
be treated by the investigation of its origin, and strove to unite the
broken threads and to restore the normal conditions of organic
evolution. The Liberal School, whose home was France, explained and
justified the Revolution as a true development, and the ripened fruit
of all history.[56] These are the two main arguments of the generation
to which we owe the notion and the scientific methods that make
history so unlike what it was to the survivors of the last century.
Severally, the innovators were not superior to the men of old.
Muratori was as widely read, Tillemont as accurate, Leibniz as able,
Freret as acute, Gibbon as masterly in the craft of composite
construction. Nevertheless, in the second quarter of this century, a
new era began for historians.
[Sidenote: USE OF UNPUBLISHED SOURCES]
[Sidenote: INSUFFICIENCY OF BOOKS]
I would point to three things in particular, out of many, which
constitute the amended order. Of the incessant deluge of new and
unsuspected matter I need say little. For some years, the secret
archives of the papacy were accessible at Paris; but the time was not
ripe, and almost the only man whom they availed was the archivist
himself.[57] Towards 1830 the documentary studies began on a large
scale, Austria leading the way. Michelet, who claims, towards 1836,
to have been the pioneer,[58] was preceded by such rivals as
Mackintosh, Bucholtz, and Mignet. A new and more productive period
began thirty years later, when the war of 1859 laid open the spoils of
Italy. Every country in succession has now allowed the exploration of
its records, and there is more fear of drowning than of drought. The
result has been that a lifetime spent in the largest collection of
printed books would not suffice to train a real master of modern
history. After he had turned from literature to sources, from Burnet
to Pocock, from Macaulay to Madame Campana, from Thiers to the
interminable correspondence of the Bonapartes, he would still feel
instant need of inquiry at Venice or Naples, in the Ossuna library or
at the Hermitage.[59]
[Sidenote: HISTORY RENEWED BY CRITICISM]
These matters do not now concern us. For our purpose, the main thing
to learn is not the art of accumulating material, but the sublimer art
of investigating it, of discerning truth from falsehood, and certainty
from doubt. It is by solidity of criticism more than by the plenitude
of erudition, that the study of history strengthens, and straightens,
and extends the mind.[60] And the accession of the critic in the place
of the indefatigable compiler, of the artist in narrative,
the skilled limner of character, the persuasive advocate of good, or
other, causes, amounts to a transfer of government, to a change of
dynasty, in the historic realm. For the critic is one who, when he
lights on an interesting statement, begins by suspecting it. He
remains in suspense until he has subjected his authority to three
operations. First, he asks whether he has read the passage as the
author wrote it. For the transcriber, and the editor, and the official
or officious censor on the top of the editor, have played strange
tricks, and have much to answer for. And if they are not to blame, it
may turn out that the author wrote his book twice over, that you can
discover the first jet, the progressive variations, things added, and
things struck out. Next is the question where the writer got his
information. If from a previous writer, it can be ascertained, and the
inquiry has to be repeated. If from unpublished papers, they must be
traced, and when the fountain head is reached, or the track
disappears, the question of veracity arises. The responsible writer's
character, his position, antecedents, and probable motives have to be
examined into; and this is what, in a different and adapted sense of
the word, may be called the higher criticism, in comparison with the
servile and often mechanical work of pursuing statements to their
root. For a historian has to be treated as a witness, and not believed
unless his sincerity is established.[61] The maxim that a man must be
presumed to be innocent until his guilt is proved, was not made for
him.
[Sidenote: CRITICAL STUDY OF EARLIER TIMES]
For us then the estimate of authorities, the weighing of testimony, is
more meritorious than the potential discovery of new matter.[62] And
modern history, which is the widest field of application, is not the
best to learn our business in; for it is too wide, and the harvest has
not been winnowed as in antiquity, and further on to the Crusades. It
is better to examine what has been done for questions that are
compact and circumscribed, such as the sources of Plutarch's
_Pericles_, the two tracts on Athenian government, the origin of the
epistle to Diognetus, the date of the life of St. Antony; and to learn
from Schwegler how this analytical work began. More satisfying because
more decisive has been the critical treatment of the mediaeval writers,
parallel with the new editions, on which incredible labour has been
lavished, and of which we have no better examples than the prefaces of
Bishop Stubbs. An important event in this series was the attack on
Dino Compagni, which, for the sake of Dante, roused the best Italian
scholars to a not unequal contest. When we are told that England is
behind the Continent in critical faculty, we must admit that this is
true as to quantity, not as to quality of work. As they are no longer
living, I will say of two Cambridge professors, Lightfoot and Hort,
that they were critical scholars whom neither Frenchman nor German has
surpassed.
[Sidenote: DEGREES OF IMPARTIALITY]
The third distinctive note of the generation of writers who dug so
deep a trench between history as known to our grandfathers and as it
appears to us, is their dogma of impartiality. To an ordinary man the
word means no more than justice. He considers that he may proclaim the
merits of his own religion, of his prosperous and enlightened country,
of his political persuasion, whether democracy, or liberal monarchy,
or historic conservatism, without transgression or offence, so long as
he is fair to the relative, though inferior merits of others, and
never treats men as saints or as rogues for the side they take. There
is no impartiality, he would say, like that of a hanging judge. The
men who, with the compass of criticism in their hands, sailed the
uncharted sea of original research, proposed a different view.
History, to be above evasion or dispute, must stand on documents, not
on opinions. They had their own notion of truthfulness, based on the
exceeding difficulty of finding truth, and the still greater
difficulty of impressing it when found. They thought it possible to
write, with so much scruple, and simplicity, and insight, as to carry
along with them every man of good will, and, whatever his feelings, to
compel his assent. Ideas which, in religion and in politics, are
truths, in history are forces. They must be respected; they must not
be affirmed. By dint of a supreme reserve, by much self-control, by a
timely and discreet indifference, by secrecy in the matter of the
black cap, history might be lifted above contention, and made an
accepted tribunal, and the same for all.[63] If men were truly
sincere, and delivered judgment by no canons but those of evident
morality, then Julian would be described in the same terms by
Christian and pagan, Luther by Catholic and Protestant, Washington by
Whig and Tory, Napoleon by patriotic Frenchman and patriotic
German.[64]
[Sidenote: MORALITY THE SOLE RULE OF JUDGMENT]
I speak of this school with reverence, for the good it has done, by
the assertion of historic truth and of its legitimate authority over
the minds of men. It provides a discipline which every one of us does
well to undergo, and perhaps also well to relinquish. For it is not
the whole truth. Lanfrey's essay on Carnot, Chuquet's wars of the
Revolution, Ropes's military histories, Roget's Geneva in the time of
Calvin, will supply you with examples of a more robust impartiality
than I have described. Renan calls it the luxury of an opulent and
aristocratic society, doomed to vanish in an age of fierce and sordid
striving. In our universities it has a magnificent and appointed
refuge; and to serve its cause, which is sacred, because it is the
cause of truth and honour, we may import a profitable lesson from the
highly unscientific region of public life. There a man does not take
long to find out that he is opposed by some who are abler and better
than himself. And, in order to understand the cosmic force and the
true connection of ideas, it is a source of power, and an excellent
school of principle, not to rest until, by excluding the fallacies,
the prejudices, the exaggerations which perpetual contention and the
consequent precautions breed, we have made out for our opponents a
stronger and more impressive case than they present themselves.[65]
Excepting one to which we are coming before I release you, there is no
precept less faithfully observed by historians.
[Sidenote: EXAMPLE OF RANKE]
Ranke is the representative of the age which instituted the modern
study of history. He taught it to be critical, to be colourless, and
to be new. We meet him at every step, and he has done more for us than
any other man. There are stronger books than any one of his, and some
may have surpassed him in political, religious, philosophic insight,
in vividness of the creative imagination, in originality, elevation,
and depth of thought; but by the extent of important work well
executed, by his influence on able men, and by the amount of knowledge
which mankind receives and employs with the stamp of his mind upon it,
he stands without a rival. I saw him last in 1877, when he was feeble,
sunken, and almost blind, and scarcely able to read or write. He
uttered his farewell with kindly emotion, and I feared that the next I
should hear of him would be the news of his death. Two years later he
began a Universal History which is not without traces of weakness, but
which, composed after the age of eighty-three, and carried, in
seventeen volumes, far into the Middle Ages, brings to a close the
most astonishing career in literature.
[Sidenote: SUPPRESSION OF OPINION]
His course had been determined, in early life, by _Quentin Durward_.
The shock of the discovery that Scott's Lewis the Eleventh was
inconsistent with the original in Commynes made him resolve that his
object thenceforth should be above all things to follow, without
swerving, and in stern subordination and surrender, the lead of his
authorities. He decided effectually to repress the poet, the patriot,
the religious or political partisan, to sustain no cause, to banish
himself from his books, and to write nothing that would gratify his
own feelings or disclose his private convictions.[66] When a strenuous
divine who, like him, had written on the Reformation, hailed him as a
comrade, Ranke repelled his advances. "You," he said, "are in the
first place a Christian: I am in the first place a historian. There is
a gulf between us."[67] He was the first eminent writer who exhibited
what Michelet calls _le desinteressement des morts_. It was a moral
triumph for him when he could refrain from judging, show that much
might be said on both sides, and leave the rest to Providence.[68] He
would have felt sympathy with the two famous London physicians of our
day, of whom it is told that they could not make up their minds on a
case and reported dubiously. The head of the family insisted on a
positive opinion. They answered that they were unable to give one, but
he might easily find fifty doctors who could.
[Sidenote: CRITICISM OF MODERN SOURCES]
Niebuhr had pointed out that chroniclers who wrote before the
invention of printing generally copied one predecessor at a time, and
knew little about sifting or combining authorities. The suggestion
became luminous in Ranke's hands, and with his light and dexterous
touch he scrutinised and dissected the principal historians, from
Machiavelli to the _Memoires d'un Homme d'Etat_, with a rigour never
before applied to moderns. But whilst Niebuhr dismissed the
traditional story, replacing it with a construction of his own, it was
Ranke's mission to preserve, not to undermine, and to set up masters
whom, in their proper sphere, he could obey. The many excellent
dissertations in which he displayed this art, though his successors in
the next generation matched his skill and did still more thorough
work, are the best introduction from which we can learn the technical
process by which within living memory the study of modern history has
been renewed. Ranke's contemporaries, weary of his neutrality and
suspense, and of the useful but subordinate work that was done by
beginners who borrowed his wand, thought that too much was made of
these obscure preliminaries which a man may accomplish for himself, in
the silence of his chamber, with less demand on the attention of the
public.[69] That may be reasonable in men who are practised in these
fundamental technicalities. We who have to learn them, must immerse
ourselves in the study of the great examples.
[Sidenote: METHOD TO BE LEARNT FROM SCIENCES]
Apart from what is technical, method is only the reduplication of
common sense, and is best acquired by observing its use by the ablest
men in every variety of intellectual employment.[70] Bentham
acknowledged that he learned less from his own profession than from
writers like Linnaeus and Cullen; and Brougham advised the student of
Law to begin with Dante. Liebig described his _Organic Chemistry_ as
an application of ideas found in Mill's _Logic_, and a distinguished
physician, not to be named lest he should overhear me, read three
books to enlarge his medical mind; and they were Gibbon, Grote, and
Mill. He goes on to say, "An educated man cannot become so on one
study alone, but must be brought under the influence of natural,
civil, and moral modes of thought."[71] I quote my colleague's golden
words in order to reciprocate them. If men of science owe anything to
us, we may learn much from them that is essential.[72] For they can
show how to test proof, how to secure fulness and soundness in
induction, how to restrain and to employ with safety hypothesis and
analogy. It is they who hold the secret of the mysterious property of
the mind by which error ministers to truth, and truth slowly but
irrevocably prevails.[73] Theirs is the logic of discovery,[74] the
demonstration of the advance of knowledge and the development of
ideas, which as the earthly wants and passions of men remain almost
unchanged, are the charter of progress, and the vital spark in
history. And they often give us invaluable counsel when they attend to
their own subjects and address their own people. Remember Darwin,
taking note only of those passages that raised difficulties in his
way; the French philosopher complaining that his work stood still,
because he found no more contradicting facts; Baer, who thinks error
treated thoroughly, nearly as remunerative as truth, by the discovery
of new objections; for, as Sir Robert Ball warns us, it is by
considering objections that we often learn.[75] Faraday declares that
"in knowledge, that man only is to be condemned and despised who is
not in a state of transition." And John Hunter spoke for all of us,
when he said: "Never ask me what I have said or what I have written;
but if you will ask me what my present opinions are, I will tell you."
[Sidenote: ALL ADOPT THE HISTORIC METHOD]
From the first years of the century we have been quickened and
enriched by contributors from every quarter. The jurists brought us
that law of continuous growth which has transformed history from a
chronicle of casual occurrences into the likeness of something
organic.[76] Towards 1820 divines began to recast their doctrines on
the lines of development, of which Newman said, long after, that
evolution had come to confirm it.[77] Even the Economists, who were
practical men, dissolved their science into liquid history, affirming
that it is not an auxiliary, but the actual subject-matter of their
inquiry.[78] Philosophers claim that, as early as 1804, they began to
bow the metaphysical neck beneath the historical yoke. They taught
that philosophy is only the amended sum of all philosophies, that
systems pass with the age whose impress they bear,[79] that the
problem is to focus the rays of wandering but extant truth, and that
history is the source of philosophy, if not quite a substitute for
it.[80] Comte begins a volume with the words that the preponderance of
history over philosophy was the characteristic of the time he lived
in.[81] Since Cuvier first recognised the conjunction between the
course of inductive discovery and the course of civilization,[82]
science had its share in saturating the age with historic ways of
thought, and subjecting all things to that influence for which the
depressing names historicism and historical-mindedness have been
devised.
[Sidenote: DANGER OF OBLIVION]
[Sidenote: PROPHECY OF PITT]
There are certain faults which are corrigible mental defects on which
I ought to say a few denouncing words, because they are common to us
all. First: the want of an energetic understanding of the sequence and
real significance of events, which would be fatal to a practical
politician, is ruin to a student of history who is the politician with
his face turned backwards.[83] It is playing at study, to see nothing
but the unmeaning and unsuggestive surface, as we generally do. Then
we have a curious proclivity to neglect, and by degrees to forget,
what has been certainly known. An instance or two will explain my
idea. The most popular English writer relates how it happened in his
presence that the title of Tory was conferred upon the Conservative
party. For it was an opprobrious name at the time, applied to men for
whom the Irish Government offered head-money; so that if I have made
too sure of progress, I may at least complacently point to this
instance of our mended manners. One day, Titus Oates lost his temper
with the men who refused to believe him, and after looking about for a
scorching imprecation, he began to call them Tories.[84] The name
remained; but its origin, attested by Defoe, dropped out of common
memory, as if one party were ashamed of their godfather, and the other
did not care to be identified with his cause and character. You all
know, I am sure, the story of the news of Trafalgar, and how, two
days after it had arrived, Mr. Pitt, drawn by an enthusiastic crowd,
went to dine in the city. When they drank the health of the minister
who had saved his country, he declined the praise. "England," he said,
"has saved herself by her own energy; and I hope that after having
saved herself by her energy, she will save Europe by her example." In
1814, when this hope had been realised, the last speech of the great
orator was remembered, and a medal was struck upon which the whole
sentence was engraved, in four words of compressed Latin: "_Seipsam
virtute, Europam exemplo._" Now it was just at the time of his last
appearance in public that Mr. Pitt heard of the overwhelming success
of the French in Germany, and of the Austrian surrender at Ulm. His
friends concluded that the contest on land was hopeless, and that it
was time to abandon the Continent to the conqueror, and to fall back
upon our new empire of the sea. Pitt did not agree with them. He said
that Napoleon would meet with a check whenever he encountered a
national resistance; and he declared that Spain was the place for it,
and that then England would intervene.[85] General Wellesley, fresh
from India, was present. Ten years later, when he had accomplished
that which Pitt had seen in the lucid prescience of his last days, he
related at Paris what I scarcely hesitate to call the most astounding
and profound prediction in all political history, where such things
have not been rare.
[Sidenote: RULES FOR THE STUDY OF HISTORY]
I shall never again enjoy the opportunity of speaking my thoughts to
such an audience as this, and on so privileged an occasion a lecturer
may well be tempted to bethink himself whether he knows of any
neglected truth, any cardinal proposition, that might serve as his
selected epigraph, as a last signal, perhaps even as a target. I am
not thinking of those shining precepts which are the registered
property of every school; that is to say--Learn as much by writing as
by reading; be not content with the best book; seek sidelights from
the others; have no favourites; keep men and things apart; guard
against the prestige of great names;[86] see that your judgments are
your own, and do not shrink from disagreement; no trusting without
testing; be more severe to ideas than to actions;[87] do not overlook
the strength of the bad cause or the weakness of the good;[88] never
be surprised by the crumbling of an idol or the disclosure of a
skeleton; judge talent at its best and character at its worst; suspect
power more than vice,[89] and study problems in preference to periods;
for instance: the derivation of Luther, the scientific influence of
Bacon, the predecessors of Adam Smith, the mediaeval masters of
Rousseau, the consistency of Burke, the identity of the first Whig.
Most of this, I suppose, is undisputed, and calls for no enlargement.
But the weight of opinion is against me when I exhort you never to
debase the moral currency or to lower the standard of rectitude, but
to try others by the final maxim that governs your own lives, and to
suffer no man and no cause to escape the undying penalty which history
has the power to inflict on wrong.[90] The plea in extenuation of
guilt and mitigation of punishment is perpetual. At every step we are
met by arguments which go to excuse, to palliate, to confound right
and wrong, and reduce the just man to the level of the reprobate. The
men who plot to baffle and resist us are, first of all, those who made
history what it has become. They set up the principle that only a
foolish Conservative judges the present time with the ideas of the
Past; that only a foolish Liberal judges the Past with the ideas of
the Present.[91]
[Sidenote: JUSTIFICATION OF THE PAST]
The mission of that school was to make distant times, and especially
the middle ages, then most distant of all, intelligible and acceptable
to a society issuing from the eighteenth century. There were
difficulties in the way; and among others this, that, in the first
fervour of the Crusades, the men who took the Cross, after receiving
communion, heartily devoted the day to the extermination of Jews. To
judge them by a fixed standard, to call them sacrilegious fanatics or
furious hypocrites, was to yield a gratuitous victory to Voltaire. It
became a rule of policy to praise the spirit when you could not defend
the deed. So that we have no common code; our moral notions are always
fluid; and you must consider the times, the class from which men
sprang, the surrounding influences, the masters in their schools, the
preachers in their pulpits, the movement they obscurely obeyed, and so
on, until responsibility is merged in numbers, and not a culprit is
left for execution.[92] A murderer was no criminal if he followed
local custom, if neighbours approved, if he was encouraged by
official advisers or prompted by just authority, if he acted for the
reason of state or the pure love of religion, or if he sheltered
himself behind the complicity of the Law. The depression of morality
was flagrant; but the motives were those which have enabled us to
contemplate with distressing complacency the secret of unhallowed
lives. The code that is greatly modified by time and place, will vary
according to the cause. The amnesty is an artifice that enables us to
make exceptions, to tamper with weights and measures, to deal unequal
justice to friends and enemies.
[Sidenote: PHILOSOPHIES OF HISTORY]
It is associated with that philosophy which Cato attributes to the
gods. For we have a theory which justifies Providence by the event,
and holds nothing so deserving as success, to which there can be no
victory in a bad cause, prescription and duration legitimate,[93] and
whatever exists is right and reasonable; and as God manifests His will
by that which He tolerates, we must conform to the divine decree by
living to shape the Future after the ratified image of the Past.[94]
Another theory, less confidently urged, regards History as our guide,
as much by showing errors to evade as examples to pursue. It is
suspicious of illusions in success, and, though there may be hope of
ultimate triumph for what is true, if not by its own attraction, by
the gradual exhaustion of error, it admits no corresponding promise
for what is ethically right. It deems the canonisation of the historic
Past more perilous than ignorance or denial, because it would
perpetuate the reign of sin and acknowledge the sovereignty of wrong,
and conceives it the part of real greatness to know how to stand and
fall alone, stemming, for a lifetime, the contemporary flood.[95]
[Sidenote: DEBASING THE CURRENCY]
Ranke relates, without adornment, that William III. ordered the
extirpation of a Catholic clan, and scouts the faltering excuse of his
defenders. But when he comes to the death and character of the
international deliverer, Glencoe is forgotten, the imputation of
murder drops, like a thing unworthy of notice.[96] Johannes Mueller, a
great Swiss celebrity, writes that the British Constitution occurred
to somebody, perhaps to Halifax. This artless statement might not be
approved by rigid lawyers as a faithful and felicitous indication of
the manner of that mysterious growth of ages, from occult beginnings,
that was never profaned by the invading wit of man;[97] but it is
less grotesque than it appears. Lord Halifax was the most original
writer of political tracts in the pamphleteering crowd between
Harrington and Bolingbroke; and in the Exclusion struggle he produced
a scheme of limitations which, in substance, if not in form,
foreshadowed the position of the monarchy in the later Hanoverian
reigns. Although Halifax did not believe in the Plot,[98] he insisted
that innocent victims should be sacrificed to content the multitude.
Sir William Temple writes:--"We only disagreed in one point, which was
the leaving some priests to the law upon the accusation of being
priests only, as the House of Commons had desired; which I thought
wholly unjust. Upon this point Lord Halifax and I had so sharp a
debate at Lord Sunderland's lodgings, that he told me, if I would not
concur in points which were so necessary for the people's
satisfaction, he would tell everybody I was a <DW7>. And upon his
affirming that the plot must be handled as if it were true, whether it
were so or no, in those points that were so generally believed." In
spite of this accusing passage Macaulay, who prefers Halifax to all
the statesmen of his age, praises him for his mercy: "His dislike of
extremes, and a forgiving and compassionate temper which seems to have
been natural to him, preserved him from all participation in the worst
crimes of his time."
[Sidenote: SINFULNESS OF HISTORY]
[Sidenote: SOVEREIGNTY OF THE MORAL CODE]
If, in our uncertainty, we must often err, it may be sometimes better
to risk excess in rigour than in indulgence, for then at least we do
no injury by loss of principle. As Bayle has said, it is more probable
that the secret motives of an indifferent action are bad than
good;[99] and this discouraging conclusion does not depend upon
theology, for James Mozley supports the sceptic from the other flank,
with all the artillery of Tractarian Oxford. "A Christian," he says,
"is bound by his very creed to suspect evil, and cannot release
himself.... He sees it where others do not; his instinct is divinely
strengthened; his eye is supernaturally keen; he has a spiritual
insight, and senses exercised to discern.... He owns the doctrine of
original sin; that doctrine puts him necessarily on his guard against
appearances, sustains his apprehension under perplexity, and prepares
him for recognising anywhere what he knows to be everywhere."[100]
There is a popular saying of Madame de Stael, that we forgive whatever
we really understand. The paradox has been judiciously pruned by her
descendant, the Duke de Broglie, in the words: "Beware of too much
explaining, lest we end by too much excusing."[101] History, says
Froude, does teach that right and wrong are real distinctions.
Opinions alter, manners change, creeds rise and fall, but the moral
law is written on the tablets of eternity.[102] And if there are
moments when we may resist the teaching of Froude, we have seldom the
chance of resisting when he is supported by Mr. Goldwin Smith: "A
sound historical morality will sanction strong measures in evil times;
selfish ambition, treachery, murder, perjury, it will never sanction
in the worst of times, for these are the things that make times
evil.--Justice has been justice, mercy has been mercy, honour has been
honour, good faith has been good faith, truthfulness has been
truthfulness from the beginning." The doctrine that, as Sir Thomas
Browne says, morality is not ambulatory,[103] is expressed as follows
by Burke, who, when true to himself, is the most intelligent of our
instructors: "My principles enable me to form my judgment upon men and
actions in history, just as they do in common life; and are not formed
out of events and characters, either present or past. History is a
preceptor of prudence, not of principles. The principles of true
politics are those of morality enlarged; and I neither now do, nor
ever will admit of any other."[104]
[Sidenote: HISTORY AND CHARACTER]
Whatever a man's notions of these later centuries are, such, in the
main, the man himself will be. Under the name of History, they cover
the articles of his philosophic, his religious, and his political
creed.[105] They give his measure; they denote his character: and, as
praise is the shipwreck of historians, his preferences betray him more
than his aversions. Modern history touches us so nearly, it is so deep
a question of life and death, that we are bound to find our own way
through it, and to owe our insight to ourselves. The historians of
former ages, unapproachable for us in knowledge and in talent, cannot
be our limit. We have the power to be more rigidly impersonal,
disinterested and just than they; and to learn from undisguised and
genuine records to look with remorse upon the past, and to the future
with assured hope of better things; bearing this in mind, that if we
lower our standard in history, we cannot uphold it in Church or
State.
NOTES
[1] No political conclusions of any value for practice can be
arrived at by direct experience. All true political science is, in one
sense of the phrase, _a priori_, being deduced from the tendencies of
things, tendencies known either through our general experience of
human nature, or as the result of an analysis of the course of
history, considered as a progressive evolution.--MILL, _Inaugural
Address_, 51.
[2] Contemporary history is, in Dr. Arnold's opinion, more
important than either ancient or modern; and in fact superior to it by
all the superiority of the end to the means.--SEELEY, _Lectures and
Essays_, 306.
[3] The law of all progress is one and the same, the
evolution of the simple into the complex by successive
differentiations.--_Edinburgh Review_, clvii. 428. Die Entwickelung
der Voelker vollzieht sich nach zwei Gesetzen. Das erste Gesetz ist das
der Differenzierung. Die primitiven Einrichtungen sind einfach und
einheitlich, die der Civilisation zusammengesetzt und geteilt, und die
Arbeitsteilung nimmt bestaendig zu.--SICKEL, _Goettingen Gelehrte
Anzeigen_, 1890, 563.
[4] Nous risquons toujours d'etre influences par les
prejuges de notre epoque; mais nous sommes libres des prejuges
particuliers aux epoques anterieures.--E. NAVILLE, _Christianisme de
Fenelon_, 9.
[5] La nature n'est qu'un echo de l'esprit. L'idee est la
mere du fait, elle faconne graduellement le monde a son
image.--FEUCHTERSLEBEN, _in_ CARO, _Nouvelles Etudes Morales_, 132. Il
n'est pas d'etude morale qui vaille l'histoire d'une idee.--LABOULAYE,
_Liberte Religieuse_, 25.
[6] Il y a des savants qui raillent le sentiment religieux.
Ils ne savent pas que c'est a ce sentiment, et par son moyen, que la
science historique doit d'avoir pu sortir de l'enfance.... Depuis des
siecles les ames independantes discutaient les textes et les
traditions de l'eglise, quand les lettres n'avaient pas encore eu
l'idee de porter un regard critique sur les textes de l'antiquite
mondaine.--_La France Protestante_, ii. 17.
[7] In our own history, above all, every step in advance has
been at the same time a step backwards. It has often been shown how
our latest constitution is, amidst all external differences,
essentially the same as our earliest, how every struggle for right and
freedom, from the thirteenth century onwards, has simply been a
struggle for recovering something old.--FREEMAN, _Historical Essays_,
iv. 253. Nothing but a thorough knowledge of the social system, based
upon a regular study of its growth, can give us the power we require
to affect it.--HARRISON, _Meaning of History_, 19. Eine Sache wird nur
voellig auf dem Wege verstanden, wie sie selbst entsteht.--In dem
genetischen Verfahren sind die Gruende der Sache, auch die Gruende des
Erkennens.--TRENDELENBURG, _Logische Untersuchungen_, ii. 395, 388.
[8] Une telle liberte ... n'a rien de commun avec le savant
systeme de garanties qui fait libres les peuples modernes.--BOUTMY,
_Annales des Sciences Politiques_, i. 157. Les trois grandes reformes
qui ont renouvele l'Angleterre, la liberte religieuse, la reforme
parlementaire, et la liberte economique, ont ete obtenues sous la
pression des organisations extra-constitutionnelles.--OSTROGORSKI,
_Revue Historique_, lii. 272.
[9] The question which is at the bottom of all constitutional
struggles, the question between the national will and the national
law.--GARDINER, _Documents_, xviii. Religion, considered simply as the
principle which balances the power of human opinion, which takes man
out of the grasp of custom and fashion, and teaches him to refer
himself to a higher tribunal, is an infinite aid to moral strength and
elevation.--CHANNING, _Works_, iv. 83. Je tiens que le passe ne suffit
jamais au present. Personne n'est plus dispose que moi a profiter de
ses lecons; mais en meme temps, je le demande, le present ne
fournit-il pas toujours les indications qui lui sont propres?--MOLE,
_in_ FALLOUX, _Etudes et Souvenirs_, 130. Admirons la sagesse de nos
peres, et tachons de l'imiter, en faisant ce qui convient a notre
siecle.--GALIANI, _Dialogues_, 40.
[10] Ceterum in legendis Historiis malim te ductum animi,
quam anxias leges sequi. Nullae sunt, quae non magnas habeant
utilitates; et melius haerent, quae libenter legimus. In universum
tamen, non incipere ab antiquissimis, sed ab his, quae nostris
temporibus nostraeque notitiae propius cohaerent, ac paulatim deinde
in remotiora eniti, magis e re arbitror.--GROTIUS, _Epistolae_, 18.
[11] The older idea of a law of degeneracy, of a "fatal drift
towards the worse," is as obsolete as astrology or the belief in
witchcraft. The human race has become hopeful, sanguine.--SEELEY,
_Rede Lecture_, 1887. _Fortnightly Review_, July, 1887, 124.
[12] Formuler des idees generales, c'est changer le salpetre
en poudre.--A. DE MUSSET, _Confessions d'un Enfant du Siecle_, 15. Les
revolutions c'est l'avenement des idees liberales. C'est presque
toujours par les revolutions qu'elles prevalent et se fondent, et
quand les idees liberales en sont veritablement le principe et le but,
quand elles leur ont donne naissance, et quand elles les couronnent a
leur dernier jour, alors ces revolutions sont legitimes.--REMUSAT,
1839, in _Revue des Deux Mondes_, 1875, vi. 335. Il y a meme des
personnes de piete qui prouvent par raison qu'il faut renoncer a la
raison; que ce n'est point la lumiere, mais la foi seule qui doit nous
conduire, et que l'obeissance aveugle est la principale vertu des
chretiens. La paresse des inferieurs et leur esprit flatteur
s'accommode souvent de cette vertu pretendue, et l'orgueil de ceux qui
commandent en est toujours tres content. De sorte qu'il se trouvera
peut-etre des gens qui seront scandalises que je fasse cet honneur a
la raison, de l'elever au-dessus de toutes les puissances, et qui
s'imagineront que je me revolte contre les autorites legitimes a cause
que je prends son parti et que je soutiens que c'est a elle a decider
et a regner.--MALEBRANCHE, _Morale_, i. 2, 13. That great statesman
(Mr. Pitt) distinctly avowed that the application of philosophy to
politics was at that time an innovation, and that it was an innovation
worthy to be adopted. He was ready to make the same avowal in the
present day which Mr. Pitt had made in 1792.--CANNING, June 1, 1827.
_Parliamentary Review_, 1828, 71. American history knows but one
avenue of success in American legislation, freedom from ancient
prejudice. The best lawgivers in our colonies first became as little
children.--BANCROFT, _History of the United States_, i. 494.--Every
American, from Jefferson and Gallatin down to the poorest squatter,
seemed to nourish an idea that he was doing what he could to overthrow
the tyranny which the past had fastened on the human mind.--ADAMS,
_History of the United States_, i. 175.
[13] The greatest changes of which we have had experience as
yet are due to our increasing knowledge of history and nature. They
have been produced by a few minds appearing in three or four favoured
nations, in comparatively a short period of time. May we be allowed to
imagine the minds of men everywhere working together during many ages
for the completion of our knowledge? May not the increase of knowledge
transfigure the world?--JOWETT, _Plato_, i. 414. Nothing, I believe,
is so likely to beget in us a spirit of enlightened liberality, of
Christian forbearance, of large-hearted moderation, as the careful study
of the history of doctrine and the history of interpretation.--PEROWNE,
_Psalms_, i. p. xxxi.
[14] Ce n'est guere avant la seconde moitie du XVIIe siecle
qu'il devint impossible de soutenir l'authenticite des fausses
decretales, des Constitutions apostoliques, des Recognitions
Clementines, du faux Ignace, du pseudo-Dionys, et de l'immense fatras
d'oeuvres anonymes ou pseudonymes qui grossissait souvent du tiers
ou de la moitie l'heritage litteraire des auteurs les plus
considerables.--DUCHESNE, _Temoins anteniceens de la Trinite_, 1883,
36.
[15] A man who does not know what has been thought by those
who have gone before him is sure to set an undue value upon his own
ideas.--M. PATTISON, _Memoirs_, 78.
[16] Travailler a discerner, dans cette discipline, le solide
d'avec le frivole, le vrai d'avec le vraisemblable, la science d'avec
l'opinion, ce qui forme le jugement d'avec ce qui ne fait que charger
la memoire.--LAMY, _Connoissance de soi-meme_, v. 459.
[17] All our hopes of the future depend on a sound
understanding of the past.--HARRISON, _The Meaning of History_, 6.
[18] The real history of mankind is that of the slow advance
of resolved deed following laboriously just thought; and all the
greatest men live in their purpose and effort more than it is possible
for them to live in reality.--The things that actually happened were
of small consequence--the thoughts that were developed are of infinite
consequence.--RUSKIN. Facts are the mere dross of history. It is from
the abstract truth which interpenetrates them, and lies latent among
them like gold in the ore, that the mass derives its value.--MACAULAY,
_Works_, v. 131.
[19] Die Gesetze der Geschichte sind eben die Gesetze der
ganzen Menschheit, gehen nicht in die Geschicke eines Volkes, einer
Generation oder gar eines Einzelnen auf. Individuen und Geschlechter,
Staaten und Nationen, koennen zerstaeuben, die Menschheit bleibt.--A.
SCHMIDT, _Zuericher Monatschrift_. i. 45.
[20] Le grand peril des ages democratiques, soyez-en sur,
c'est la destruction ou l'affaiblissement excessif des parties du
corps social en presence du tout. Tout ce qui releve de nos jours
l'idee de l'individu est sain.--TOCQUEVILLE, Jan. 3, 1840,
_OEuvres_, vii. 97. En France, il n'y a plus d'hommes. On a
systematiquement tue l'homme au profit du peuple, des masses, comme
disent nos legislateurs ecerveles. Puis un beau jour, on s'est apercu
que ce peuple n'avait jamais existe qu'en projet, que ces masses
etaient un troupeau mi-partie de moutons et de tigres. C'est une
triste histoire. Nous avons a relever l'ame humaine contre l'aveugle
et brutale tyrannie des multitudes.--LANFREY, March 23, 1855. M. DU
CAMP, _Souvenirs Litteraires_, ii. 273. C'est le propre de la vertu
d'etre invisible, meme dans l'histoire, a tout autre oeil que celui
de la conscience.--VACHEROT, _Comptes Rendus de l'Institut_, lxix.
319. Dans l'histoire ou la bonte est la perle rare, qui a ete bon
passe presque avant qui a ete grand.--V. HUGO, _Les Miserables_, vii.
46. Grosser Maenner Leben und Tod der Wahrheit gemaess mit Liebe zu
schildern, ist zu allen Zeiten herzerhebend; am meisten aber dann,
wenn im Kreislauf der irdischen Dinge die Sterne wieder aehnlich
stehen wie damals als sie unter uns lebten.--LASAULX, _Sokrates_, 3.
Instead of saying that the history of mankind is the history of the
masses, it would be much more true to say that the history of mankind
is the history of its great men.--KINGSLEY, _Lectures_, 329.
[21] Le genie n'est que la plus complete emancipation de
toutes les influences de temps, de moeurs, et de pays.--NISARD,
_Souvenirs_, ii. 43.
[22] Meine kritische Richtung zieht mich in der Wissenschaft
durchaus zur Kritik meiner eigenen Gedanken hin, nicht zu der der
Gedanken Anderer.--ROTHE, _Ethik_, i., p. xi.
[23] When you are in young years the whole mind is, as it
were, fluid, and is capable of forming itself into any shape that the
owner of the mind pleases to order it to form itself into.--CARLYLE,
_On the Choice of Books_, 131. Nach allem erscheint es somit
unzweifelhaft als eine der psychologischen Voraussetzungen des
Strafrechts, ohne welche der Zurechnungsbegriff nicht haltbar waere,
dass der Mensch fuer seinen Charakter verantwortlich ist und ihn muss
abaendern koennen.--RUeMELIN, _Reden und Aufsaetze_, ii., 60. An der
tiefen und verborgenen Quelle, woraus der Wille entspringt, an diesem
Punkt, nur hier steht die Freiheit, und fuehrt das Steuer und lenkt den
Willen. Wer nicht bis zu dieser Tiefe in sich einkehren und seinen
natuerlichen Charakter von hier aus bemeistern kann, der hat nicht den
Gebrauch seiner Freiheit, der ist nicht frei, sondern unterworfen dem
Triebwerk seiner Interessen, und dadurch in der Gewalt des Weltlaufs,
worin jede Begebenheit und jede Handlung eine nothwendige Folge ist
aller vorhergehenden.--FISCHER, _Problem der Freiheit_, 27.
[24] I must regard the main duty of a Professor to consist,
not simply in communicating information, but in doing this in such a
manner, and with such an accompaniment of subsidiary means, that the
information he conveys may be the occasion of awakening his pupils to
a vigorous and varied exertion of their faculties.--SIR W. HAMILTON,
_Lectures_, i. 14. No great man really does his work by imposing his
maxims on his disciples, he evokes their life. The pupil may become
much wiser than his instructor, he may not accept his conclusions, but
he will own, "You awakened me to be myself, for that I thank
you."--MAURICE, _The Conscience_, 7, 8.
[25] Ich sehe die Zeit kommen, wo wir die neuere Geschichte
nicht mehr auf die Berichte selbst nicht der gleichzeitigen
Historiker, ausser in so weit ihnen neue originale Kenntniss
beiwohnte, geschweige denn auf die weiter abgeleiteten Bearbeitungen
zu gruenden haben, sondern aus den Relationen der Augenzeugen und der
aechten und unmittelbarsten Urkunden aufbauen werden.--RANKE,
_Reformation_, _Preface_, 1838. Ce qu'on a trouve et mis en oeuvre
est considerable en soi: c'est peu de chose au prix de ce qui reste a
trouver et a mettre en oeuvre.--AULARD, _Etudes sur la Revolution_,
21.
[26] N'attendez donc pas les lecons de l'experience; elles
coutent trop cher aux nations.--O. BARROT, _Memoires_, ii. 435. Il y a
des lecons dans tous les temps, pour tous les temps; et celles qu'on
emprunte a des ennemis ne sont pas les moins precieuses.--LANFREY,
_Napoleon_, v. p. ii. Old facts may always be fresh, and may give out
a fresh meaning for each generation.--MAURICE, _Lectures_, 62. The
object is to lead the student to attend to them; to make him take
interest in history not as a mere narrative, but as a chain of causes
and effects still unwinding itself before our eyes, and full of
momentous consequences to himself and his descendants--an unremitting
conflict between good and evil powers, of which every act done by any
one of us, insignificant as we are, forms one of the incidents; a
conflict in which even the smallest of us cannot escape from taking
part, in which whoever does not help the right side is helping the
wrong.--MILL, _Inaugural Address_, 59.
[27] I hold that the degree in which Poets dwell in sympathy
with the Past, marks exactly the degree of their poetical
faculty.--WORDSWORTH in C. FOX, _Memoirs_, June, 1842. In all
political, all social, all human questions whatever, history is the
main resource of the inquirer.--HARRISON, _Meaning of History_, 15.
There are no truths which more readily gain the assent of mankind, or
are more firmly retained by them, than those of an historical nature,
depending upon the testimony of others.--PRIESTLEY, _Letters to French
Philosophers_, 9. Improvement consists in bringing our opinions into
nearer agreement with facts; and we shall not be likely to do this
while we look at facts only through glasses by those very
opinions.--MILL, _Inaugural Address_, 25.
[28] He who has learnt to understand the true character and
tendency of many succeeding ages is not likely to go very far wrong in
estimating his own.--LECKY, _Value of History_, 21. C'est a l'histoire
qu'il faut se prendre, c'est le fait que nous devons interroger, quand
l'idee vacille et fuit a nos yeux.--MICHELET, _Disc. d'Ouverture_,
263. C'est la loi des faits telle qu'elle se manifeste dans leur
succession. C'est la regle de conduite donnee par la nature humaine et
indiquee par l'histoire. C'est la logique, mais cette logique qui ne
fait qu'un avec l'enchainement des choses. C'est l'enseignement de
l'experience.--SCHERER, _Melanges_, 558. Wer seine Vergangenheit nicht
als seine Geschichte hat und weiss wird und ist characterlos Wem ein
Ereigniss sein Sonst ploetzlich abreisst von seinem Jetzt wird leicht
wurzellos.--KLIEFOTH, _Rheinwalds Repertorium_, xliv. 20. La politique
est une des meilleures ecoles pour l'esprit. Elle force a chercher la
raison de toutes choses, et ne permet pas cependant de la chercher
hors des faits.--REMUSAT, _Le Temps Passe_, i. 31. It is an unsafe
partition that divides opinions without principle from unprincipled
opinions.--COLERIDGE, _Lay Sermon_, 373.
Wer nicht von drei tausend Jahren sich weiss Rechenschaft zu geben,
Bleib' im Dunkeln unerfahren, mag von Tag zu Tage leben!
GOETHE.
What can be rationally required of the student of philosophy is not a
preliminary and absolute, but a gradual and progressive, abrogation of
prejudices.--SIR W. HAMILTON, _Lectures_, iv. 92.
[29] Die Schlacht bei Leuthen ist wohl die letzte, in welcher
diese religioesen Gegensaetze entscheidend eingewirkt haben.--RANKE,
_Allgemeine Deutsche Biographie_, vii. 70.
[30] The only real cry in the country is the proper and just
old No Popery cry.--_Major Beresford_, July, 1847. Unfortunately the
strongest bond of union amongst them is an apprehension of
Popery.--_Stanley_, September 12, 1847. The great Protectionist party
having degenerated into a No Popery, No Jew Party, I am still more
unfit now than I was in 1846 to lead it.--_G. Bentinck_, December 26,
1847. _Croker's Memoirs_, iii. 116, 132, 157.
[31] In the case of Protestantism, this constitutional
instability is now a simple matter of fact, which has become too plain
to be denied. The system is not fixed, but in motion; and the motion
is for the time in the direction of complete self-dissolution.--We
take it for a transitory scheme, whose breaking up is to make room in
due time for another and far more perfect state of the Church.--The
new order in which Protestantism is to become thus complete cannot be
reached without the co-operation and help of Romanism.--NEVIN,
_Mercersburg Review_, iv. 48.
[32] Diese Heiligen waren es, die aus dem unmittelbaren
Glaubensleben und den Grundgedanken der christlichen Freiheit zuerst
die Idee allgemeiner Menschenrechte abgeleitet und rein von
Selbstsucht vertheidigt haben.--WEINGARTEN, _Revolutionskirchen_, 447.
Wie selbst die Idee allgemeiner Menschenrechte, die in dem gemeinsamen
Character der Ebenbildlichkeit Gottes gegruendet sind, erst durch das
Christenthum zum Bewusstsein gebracht werden, waehrend jeder andere
Eifer fuer politische Freiheit als ein mehr oder weniger
selbstsuechtiger und beschraenkter sich erwiesen hat.--NEANDER, _Pref.
to Uhden's Wilberforce_, p. v. The rights of individuals and the
justice due to them are as dear and precious as those of states;
indeed the latter are founded on the former, and the great end and
object of them must be to secure and support the rights of
individuals, or else vain is government.--CUSHING in CONWAY, _Life of
Paine_, i. 217. As it is owned the whole scheme of Scripture is not
yet understood; so, if it ever comes to be understood, before the
restitution of all things, and without miraculous interpositions, it
must be in the same way as natural knowledge is come at--by the
continuance and progress of learning and liberty.--BUTLER, _Analogy_,
ii. 3.
[33] Comme les lois elles-memes sont faillibles, et qu'il
peut y avoir une autre justice que la justice ecrite, les societes
modernes ont voulu garantir les droits de la conscience a la poursuite
d'une justice meilleure que celle qui existe; et la est le fondement
de ce qu'on appelle liberte de conscience, liberte d'ecrire, liberte
de pensee.--JANET, _Philosophie Contemporaine_, 308. Si la force
materielle a toujours fini par ceder a l'opinion, combien plus ne
sera-t-elle pas contrainte de ceder a la conscience? Car la
conscience, c'est l'opinion renforcee par le sentiment de
l'obligation.--VINET, _Liberte Religieuse_, 3.
[34] Apres la volonte d'un homme, la raison d'etat; apres la
raison d'etat, la religion; apres la religion, la liberte. Voila toute
la philosophie de l'histoire.--FLOTTES, _La Souverainete du Peuple_,
1851, 192. La repartition plus egale des biens et des droits dans ce
monde est le plus grand objet que doivent se proposer ceux qui menent
les affaires humaines. Je veux seulement que l'egalite en politique
consiste a etre egalement libre.--TOCQUEVILLE, September 10, 1856.
_Mme. Swetchine_, i. 455. On peut concevoir une legislation tres
simple, lorsqu'on voudra en ecarter tout ce qui est arbitraire, ne
consulter que les deux premieres lois de la liberte et de la
propriete, et ne point admettre de lois positives qui ne tirent leur
raison de ces deux lois souveraines de la justice essentielle et
absolue.--LETROSNE, _Vues sur la Justice Criminelle_, 16. Summa enim
libertas est, ad optimum recta ratione cogi.--Nemo optat sibi hanc
libertatem, volendi quae velit, sed potius volendi optima.--LEIBNIZ,
_De Fato_. TRENDELENBURG, _Beitraege zur Philosophie_, ii. 190.
[35] All the world is, by the very law of its creation, in
eternal progress; and the cause of all the evils of the world may be
traced to that natural, but most deadly error of human indolence and
corruption, that our business is to preserve and not to
improve.--ARNOLD, _Life_, i. 259. In whatever state of knowledge we
may conceive man to be placed, his progress towards a yet higher state
need never fear a check, but must continue till the last existence of
society.--HERSCHEL, _Prel. Dis._, 360. It is in the development of
thought as in every other development; the present suffers from the
past, and the future struggles hard in escaping from the
present.--MAX MUeLLER, _Science of Thought_, 617. Most of the great
positive evils of the world are in themselves removable, and will, if
human affairs continue to improve, be in the end reduced within narrow
limits. Poverty in any sense implying suffering may be completely
extinguished by the wisdom of society combined with the good sense and
providence of individuals.--All the grand sources, in short, of human
suffering are in a great degree, many of them almost entirely,
conquerable by human care and effort.--J. S. MILL, _Utilitarianism_,
21, 22. The ultimate standard of worth is personal worth, and the only
progress that is worth striving after, the only acquisition that is
truly good and enduring, is the growth of the soul.--BIXBY, _Crisis of
Morals_, 210. La science, et l'industrie qu'elle produit, ont, parmi
tous les autres enfants du genie de l'homme, ce privilege particulier,
que leur vol non-seulement ne peut pas s'interrompre, mais qu'il
s'accelere sans cesse.--CUVIER, _Discours sur la Marche des Sciences_,
24 Avril, 1816. Aucune idee parmi celles qui se referent a l'ordre des
faits naturels, ne tient de plus pres a la famille des idees
religieuses que l'idee du progres, et n'est plus propre a devenir le
principe d'une sorte de foi religieuse pour ceux qui n'en ont pas
d'autres. Elle a, comme la foi religieuse, la vertu de relever les
ames et les caracteres.--COURNOT, _Marche des Idees_, ii. 425. Dans le
spectacle de l'humanite errante, souffrante et travaillant toujours a
mieux voir, a mieux penser, a mieux agir, a diminuer l'infirmite de
l'etre humain, a apaiser l'inquietude de son coeur, la science
decouvre une direction et un progres.--A. SOREL, _Discours de
Reception_, 14. Le jeune homme qui commence son education quinze ans
apres son pere, a une epoque ou celui-ci, engage dans une profession
speciale et active, ne peut que suivre les anciens principes, acquiert
une superiorite theorique dont on doit tenir compte dans la hierarchie
sociale. Le plus souvent le pere n'est-il pas penetre de l'esprit de
routine, tandis que le fils represente et defend la science
progressive? En diminuant l'ecart qui existait entre l'influence des
jeunes generations et celle de la vieillesse ou de l'age mur, les
peuples modernes n'auraient donc fait que reproduire dans leur ordre
social un changement de rapports qui s'etait deja accompli dans la
nature intime des choses.--BOUTMY, _Revue Nationale_, xxi. 393. Il y a
dans l'homme individuel des principes de progres viager; il y a, en
toute societe, des causes constantes qui transforment ce progres
viager en progres hereditaire. Une societe quelconque tend a
progresser tant que les circonstances ne touchent pas aux causes de
progres que nous avons reconnues, l'imitation des devanciers par les
successeurs, des etrangers par les indigenes.--LACOMBE, _L'Histoire
comme Science_, 292. Veram creatae mentis beatitudinem consistere in
non impedito progressu ad bona majora.--LEIBNIZ to WOLF, February 21,
1705. In cumulum etiam pulchritudinis perfectionisque universalis
operum divinorum progressus quidam perpetuus liberrimusque totius
universi est agnoscendus, ita ut ad majorem semper cultum
procedat.--LEIBNIZ ed. Erdmann, 150_a_. Der Creaturen und also auch
unsere Vollkommenheit bestehet in einem ungehinderten starken
Forttrieb zu neuen und neuen Vollkommenheiten.--LEIBNIZ, _Deutsche
Schriften_, ii. 36. Hegel, welcher annahm, der Fortschritt der Neuzeit
gegen das Mittelalter sei dieser, dass die Principien der Tugend und
des Christenthums, welche im Mittelalter sich allein im Privatleben
und der Kirche zur Geltung gebracht haetten, nun auch anfingen, das
politische Leben zu durchdringen.--FORTLAGE, _Allg. Monatschrift_,
1853, 777. Wir Slawen wissen, dass die Geister einzelner Menschen und
ganzer Voelker sich nur durch die Stufe ihrer Entwicklung
unterscheiden.--MICKIEWICZ, _Slawische Literatur_, ii. 436. Le progres
ne disparait jamais, mais il se deplace souvent. Il va des gouvernants
aux gouvernes. La tendance des revolutions est de le ramener toujours
parmi les gouvernants. Lorsqu'il est a la tete des societes, il marche
hardiment, car il conduit. Lorsqu'il est dans la masse, il marche a
pas lents, car il lutte.--NAPOLEON III., _Des Idees Napoleoniennes_.
La loi du progres avait jadis l'inexorable rigueur du destin; elle
prend maintenant de jour en jour la douce puissance de la Providence.
C'est l'erreur, c'est l'iniquite, c'est le vice, que la civilisation
tend a emporter dans sa marche irresistible; mais la vie des individus
et des peuples est devenue pour elle une chose sacree. Elle transforme
plutot qu'elle ne detruit les choses qui s'opposent a son
developpement; elle procede par absorption graduelle plutot que par
brusque execution; elle aime a conquerir par l'influence des idees
plutot que par la force des armes, un peuple, une classe, une
institution qui resiste au progres.--VACHEROT, _Essais de Philosophie
Critique_, 443. Peu a peu l'homme intellectuel finit par effacer
l'homme physique.--QUETELET, _De l'Homme_, ii. 285. In dem Fortschritt
der ethischen Anschauungen liegt daher der Kern des geschichtlichen
Fortschritts ueberhaupt.--SCHAeFER, _Arbeitsgebiet der Geschichte_, 24.
Si l'homme a plus de devoirs a mesure qu'il avance en age, ce qui est
melancolique, mais ce qui est vrai, de meme aussi l'humanite est tenue
d'avoir une morale plus severe a mesure qu'elle prend plus de
siecles.--FAGUET, _Revue des Deux Mondes_, 1894, iii. 871. Si donc il
y a une loi de progres, elle se confond avec la loi morale, et la
condition fondamentale du progres, c'est la pratique de cette
loi.--CARRAU, _Ib._, 1875, v. 585. L'idee du progres, du
developpement, me parait etre l'idee fondamentale contenue sous le mot
de civilisation.--GUIZOT, _Cours d'Histoire_, 1828, 15. Le progres
n'est sous un autre nom, que la liberte en action.--BROGLIE, _Journal
des Debats_, January 28, 1869. Le progres social est continu. Il a ses
periodes de fievre ou d'atonie, de surexcitation ou de lethargie; il a
ses soubresauts et ses haltes, mais il avance toujours.--DE DECKER,
_La Providence_, 174. Ce n'est pas au bonheur seul, c'est au
perfectionnement que notre destin nous appelle; et la liberte
politique est le plus puissant, le plus energique moyen de
perfectionnement que le ciel nous ait donne.--B. CONSTANT, _Cours de
Politique_, ii. 559. To explode error, on whichever side it lies, is
certainly to secure progress.--MARTINEAU, _Essays_, i. 114. Die
saemmtlichen Freiheitsrechte, welche der heutigen Menschheit so theuer sind,
sind im Grunde nur Anwendungen des Rechts der Entwickelung.--BLUNTSCHLI,
_Kleine Schriften_, i. 51. Geistiges Leben ist auf Freiheit beruhende
Entwicklung, mit Freiheit vollzogene That und geschichtlicher
Fortschritt.--_Muenchner Gel. Anzeigen_ 1849, ii. 83. Wie das Denken
erst nach und nach reift, so wird auch der freie Wille nicht fertig
geboren, sondern in der Entwickelung erworben.--TRENDELENBURG,
_Logische Untersuchungen_, ii. 94. Das Liberum Arbitrium im vollen
Sinne (die vollstaendig aktuelle Macht der Selbstbestimmung) laesst sich
seinem Begriff zufolge schlechterdings nicht unmittelbar geben; es
kann nur erworben werden durch das Subjekt selbst, in sich moralisch
hervorgebracht werden kraft seiner eigenen Entwickelung.--ROTHE,
_Ethik_, i. 360. So gewaltig sei der Andrang der Erfindungen und
Entdeckungen, dass "Entwicklungsperioden, die in frueheren Zeiten erst
in Jahrhunderten durchlaufen wurden, die im Beginn unserer Zeitperiode
noch der Jahrzehnte bedurften, sich heute in Jahren volienden, haeufig
schon in voller Ausbildung ins Dasein treten."--PHILIPPOVICH,
_Fortschritt und Kulturentwicklung_, 1892, i. quoting SIEMENS, 1886.
Wir erkennen dass dem Menschen die schwere koerperliche Arbeit, von der
er in seinem Kampfe um's Dasein stets schwer niedergedrueckt war und
grossenteils noch ist, mehr und mehr durch die wachsende Benutzung der
Naturkraefte zur mechanischen Arbeitsleistung abgenommen wird, dass
die ihm zufallende Arbeit immer mehr eine intellektuelle
wird.--SIEMENS, 1886, _Ib._ 6.
[36] Once, however, he wrote:--Darin koennte man den idealen
Kern der Geschichte des menschlichen Geschlechtes ueberhaupt sehen,
dass in den Kaempfen, die sich in den gegenseitigen Interessen der
Staaten und Voelker vollziehen, doch immer hoehere Potenzen emporkommen,
die das Allgemeine demgemaess umgestalten und ihm wieder einen anderen
Charakter verleihen.--RANKE, _Weltgeschichte_, iii. 1, 6.
[37] Toujours et partout, les hommes furent de plus en plus
domines par l'ensemble de leurs predecesseurs, dont ils purent
seulement modifier l'empire necessaire.--COMTE, _Politique Positive_,
iii. 621.
[38] La liberte est l'ame du commerce.--Il faut laisser faire
les hommes qui s'appliquent sans peine a ce qui convient le mieux;
c'est ce qui apporte le plus d'avantage.--COLBERT, in _Comptes Rendus
de l'Institut_, xxxix. 93.
[39] Il n'y a que les choses humaines exposees dans leur
verite, c'est-a-dire avec leur grandeur, leur variete, leur
inepuisable fecondite, qui aient le droit de retenir le lecteur et qui
le retiennent en effet. Si l'ecrivain parait une fois, il ennuie ou
fait sourire de pitie les lecteurs serieux.--THIERS to STE. BEUVE,
_Lundis_, iii. 195. Comme l'a dit Taine, la disparition du style,
c'est la perfection du style.--FAGUET, _Revue Politique_, lii. 67.
[40] Ne m'applaudissez pas; ce n'est pas moi qui vous parle;
c'est l'histoire qui parle par ma bouche.--_Revue Historique_, xli.
278.
[41] Das Evangelium trat als Geschichte in die Welt, nicht
als Dogma--wurde als Geschichte in der christlichen Kirche
deponirt.--ROTHE, _Kirchengeschichte_, ii. p. x. Das Christenthum ist
nicht der Herr Christus, sondern dieser macht es. Es ist sein Werk,
und zwar ein Werk das er stets unter der Arbeit hat.--Er selbst,
Christus der Herr, bleibt der er ist in alle Zukunft, dagegen liegt es
ausdruecklich im Begriffe seines Werks, des Christenthums, dass es
nicht so bleibt wie es anhebt.--ROTHE, _Allgemeine kirchliche
Zeitschrift_, 1864, 299. Diess Werk, weil es dem Wesen der Geschichte
zufolge eine Entwickelung ist, muss ueber Stufen hinweggehen, die
einander abloesen, und von denen jede folgende neue immer nur unter der
Zertruemmerung der ihr vorangehenden Platz greifen kann.--ROTHE, _Ib._
April 19, 1865. Je groesser ein geschichtliches Princip ist, desto
langsamer und ueber mehr Stufen hinweg entfaltet es seinen Gehalt;
desto langlebiger ist es aber ebendeshalb auch in diesen seinen
unaufhoerlichen Abwandelungen.--ROTHE, _Stille Stunden_, 301. Der
christliche Glaube geht nicht von der Anerkennung abstracter
Lehrwahrheiten aus, sondern von der Anerkennung einer Reihe von
Thatsachen, die in der Erscheinung Jesu ihren Mittelpunkt
haben.--NITZSCH, _Dogmengeschichte_, i. 17. Der Gedankengang der
evangelischen Erzaehlung gibt darum auch eine vollstaendige Darstellung
der christlichen Lehre in ihren wesentlichen Grundzuegen; aber er gibt
sie im allseitigen lebendigen Zusammenhange mit der Geschichte der
christlichen Offenbarung, und nicht in einer theoretisch
zusammenhaengenden Folgenreihe von ethischen und dogmatischen
Lehrsaetzen.--DEUTINGER, _Reich Gottes_, i. p. v.
[42] L'Univers ne doit pas estre considere seulement dans ce
qu'il est; pour le bien connoitre, il faut le voir aussi dans ce qu'il
doit estre. C'est cet avenir surtout qui a ete le grand objet de Dieu
dans la creation, et c'est pour cet avenir seul que le present
existe.--D'HOUTEVILLE, _Essai sur la Providence_, 273. La Providence
emploie les siecles a elever toujours un plus grand nombre de familles
et d'individus a ces biens de la liberte et de l'egalite legitimes
que, dans l'enfance des societes, la force avait rendus le privilege
de quelques-uns.--GUIZOT, _Gouvernement de la France_, 1820, 9. La
marche de la Providence n'est pas assujettie a d'etroites limites;
elle ne s'inquiete pas de tirer aujourd'hui la consequence du principe
qu'elle a pose hier; elle la tirera dans des siecles, quand l'heure
sera venue; et pour raisonner lentement selon nous, sa logique n'est
pas moins sure.--GUIZOT, _Histoire de la Civilisation_, 20. Der Keim
fortschreitender Entwicklung ist, auch auf goettlichem Geheisse, der
Menschheit eingepflanzt. Die Weltgeschichte ist der blosse Ausdruck
einer vorbestimmten Entwicklung.--A. HUMBOLDT, January 2, 1842, _Im
Neuen Reich_, 1872, i. 197. Das historisch grosse ist religioes gross;
es ist die Gottheit selbst, die sich offenbart.--RAUMER, April 1807,
_Erinnerungen_, i. 85.
[43] Je suis arrive a l'age ou je suis, a travers bien des
evenements differents, mais avec une seule cause, celle de la liberte
reguliere.--TOCQUEVILLE, May 1, 1852, _OEuvres Inedites_, ii. 185.
Me trouvant dans un pays ou la religion et le liberalisme sont
d'accord, j'avais respire.--J'exprimais ce sentiment, il y a plus de
vingt ans, dans l'avant-propos de la _Democratie_. Je l'eprouve
aujourd'hui aussi vivement que si j'etais encore jeune, et je ne sais
s'il y a une seule pensee qui ait ete plus constamment presente a mon
esprit.--August 5, 1857, _OEuvres_, vi. 395. Il n'y a que la liberte
(j'entends la moderee et la reguliere) et la religion, qui, par un
effort combine, puissent soulever les hommes au-dessus du bourbier ou
l'egalite democratique les plonge naturellement.--December 1, 1852,
_OEuvres_, vii. 295. L'un de mes reves, le principal en entrant dans
la vie politique, etait de travailler a concilier l'esprit liberal et
l'esprit de religion, la societe nouvelle et l'eglise.--November 15,
1843, _OEuvres Inedites_, ii. 121. La veritable grandeur de l'homme
n'est que dans l'accord du sentiment liberal et du sentiment
religieux.--September 17, 1853, _OEuvres Inedites_, ii. 228. Qui
cherche dans la liberte autre chose qu'elle-meme est fait pour
servir.--_Ancien Regime_, 248. Je regarde, ainsi que je l'ai toujours
fait, la liberte comme le premier des biens; je vois toujours en elle
l'une des sources les plus fecondes des vertus males et des actions
grandes. Il n'y a pas de tranquillite ni de bien-etre qui puisse me
tenir lieu d'elle.--January 7, 1856, _Mme. Swetchine_, i. 452. La
liberte a un faux air d'aristocratie; en donnant pleine carriere aux
facultes humaines, en encourageant le travail et l'economie, elle fait
ressortir les superiorites naturelles ou acquises.--LABOULAYE, _L'Etat
et ses Limites_, 154. Dire que la liberte n'est point par elle-meme,
qu'elle depend d'une situation, d'une opportunite, c'est lui assigner
une valeur negative. La liberte n'est pas des qu'on la subordonne.
Elle n'est pas un principe purement negatif, un simple element de
controle et de critique. Elle est le principe actif, createur
organisateur par excellence. Elle est le moteur et la regle, la source
de toute vie, et le principe de l'ordre. Elle est, en un mot, le nom
que prend la conscience souveraine, lorsque, se posant en face du
monde social et politique, elle emerge du moi pour modeler les
societes sur les donnees de la raison.--BRISSON, _Revue Nationale_,
xxiii. 214. Le droit, dans l'histoire, est le developpement progressif
de la liberte, sous la loi de la raison.--LERMINIER, _Philosophie du
Droit_, i. 211. En prouvant par les lecons de l'histoire que la
liberte fait vivre les peuples et que le despotisme les tue, en
montrant que l'expiation suit la faute et que la fortune finit
d'ordinaire par se ranger du cote de la vertu, Montesquieu n'est ni
moins moral ni moins religieux que Bossuet.--LABOULAYE, _OEuvres de
Montesquieu_, ii. 109. Je ne comprendrais pas qu'une nation ne placat
pas les libertes politiques au premier rang, parce que c'est des
libertes politiques que doivent decouler toutes les autres.--THIERS,
_Discours_, x. 8, _March_ 28, 1865. Nous sommes arrives a une epoque
ou la liberte est le but serieux de tous, ou le reste n'est plus
qu'une question de moyens.--J. LEBEAU, _Observations sur le Pouvoir
Royal_: Liege, 1830, p. 10. Le liberalisme, ayant la pretention de se
fonder uniquement sur les principes de la raison, croit d'ordinaire
n'avoir pas besoin de tradition. La est son erreur. L'erreur de
l'ecole liberale est d'avoir trop cru qu'il est facile de creer la
liberte par la reflexion, et de n'avoir pas vu qu'un etablissement
n'est solide que quand il a des racines historiques.--RENAN, 1858,
_Nouvelle Revue_, lxxix. 596. Le respect des individus et des droits
existants est autant au-dessus du bonheur de tous, qu'un interet moral
surpasse un interet purement temporel.--RENAN, 1858, _Ib._ lxxix. 597.
Die Rechte gelten nichts, wo es sich handelt um das Recht, und das
Recht der Freiheit kann nie verjaehren, weil es die Quelle alles
Rechtes selbst ist.--C. FRANTZ, _Ueber die Freiheit_, 110. Wir
erfahren hienieden nie die ganze Wahrheit: wir geniessen nie die ganze
Freiheit.--REUSS, _Reden_, 56. Le gouvernement constitutionnel, comme
tout gouvernement libre, presente et doit presenter un etat de lutte
permanent. La liberte est la perpetuite de la lutte.--DE SERRE.
BROGLIE, _Nouvelles Etudes_, 243. The experiment of free government is
not one which can be tried once for all. Every generation must try it
for itself. As each new generation starts up to the responsibilities
of manhood, there is, as it were, a new launch of Liberty, and its
voyage of experiment begins afresh.--WINTHROP, _Addresses_, 163.
L'histoire perd son veritable caractere du moment que la liberte en a
disparu; elle devient une sorte de physique sociale. C'est l'element
personnel de l'histoire qui en fait la realite.--VACHEROT, _Revue des
Deux Mondes_, 1869, iv. 215. Demander la liberte pour soi et la
refuser aux autres, c'est la definition du despotisme.--LABOULAYE,
December 4, 1874. Les causes justes profitent de tout, des bonnes
intentions comme des mauvaises, des calculs personnels comme des
devouemens courageux, de la demence, enfin, comme de la raison.--B.
CONSTANT, _Les Cent Jours_, ii. 29. Sie ist die Kunst, das Gute der
schon weit gediehenen Civilisation zu sichern.--BALTISCH, _Politische
Freiheit_, 9. In einem Volke, welches sich zur buergerlichen
Gesellschaft, ueberhaupt zum Bewusstseyn der Unendlichkeit des
Freien--entwickelt hat, ist nur die constitutionelle Monarchie
moeglich.--HEGEL'S _Philosophie des Rechts_, Sec. 137, _Hegel und
Preussen_, 1841, 31. Freiheit ist das hoechste Gut. Alles andere ist
nur das Mittel dazu: gut falls es ein Mittel dazu ist, uebel falls es
dieselbe hemmt.--FICHTE, _Werke_, iv. 403. You are not to inquire how
your trade may be increased, nor how you are to become a great and
powerful people, but how your liberties can be secured. For liberty
ought to be the direct end of your government.--PATRICK HENRY, 1788.
WIRT, _Life of Henry_, 272.
[44] Historiae ipsius praeter delectationem utilitas nulla est,
quam ut religionis Christianae veritas demonstretur, quod aliter quam
per historiam fieri non potest.--LEIBNIZ, _Opera_, ed. Dutens, vi.
297. The study of Modern History is, next to Theology itself, and only
next in so far as Theology rests on a divine revelation, the most
thoroughly religious training that the mind can receive. It is no
paradox to say that Modern History, including Medieval History in the
term, is co-extensive in its field of view, in its habits of
criticism, in the persons of its most famous students, with
Ecclesiastical History.--STUBBS, _Lectures_, 9. Je regarde donc
l'etude de l'histoire comme l'etude de la providence.--L'histoire est
vraiment une seconde philosophie.--Si Dieu ne parle pas toujours, il
agit toujours en Dieu.--D'AGUESSEAU, _OEuvres_, xv. 34, 31, 35. Fuer
diejenigen, welche das Wesen der menschlichen Freiheit erkannt haben,
bildet die denkende Betrachtung der Weltgeschichte, besonders des
christlichen Weltalters, die hoechste, und umfassendste Theodicee.--VATKE,
_Die Menschliche Freiheit_, 1841, 516. La theologie, que l'on regarde
volontiers comme la plus etroite et la plus sterile des sciences, en
est, au contraire, la plus etendue et la plus feconde. Elle confine a
toutes les etudes et touche a toutes les questions. Elle renferme tous
les elements d'une instruction liberale.--SCHERER, _Melanges_, 522. The
belief that the course of events and the agency of man are subject to
the laws of a divine order, which it is alike impossible for any one
either fully to comprehend or effectually to resist--this belief is the
ground of all our hope for the future destinies of mankind.--THIRLWALL,
_Remains_, iii. 282. A true religion must consist of ideas and facts
both; not of ideas alone without facts, for then it would be mere
philosophy; nor of facts alone without ideas, of which those facts are
the symbols, or out of which they are grounded; for then it would be
mere history.--COLERIDGE, _Table Talk_, 144. It certainly appears
strange that the men most conversant with the order of the visible
universe should soonest suspect it empty of directing mind; and, on
the other hand, that humanistic, moral and historical studies--which
first open the terrible problems of suffering and grief, and contain
all the reputed provocatives of denial and despair--should confirm, and
enlarge rather than disturb, the prepossessions of natural
piety.--MARTINEAU, _Essays_, i. 122. Die Religion hat nur dann eine
Bedeutung fuer den Menschen, wenn er in der Geschichte einen Punkt
findet, dem er sich voellig unbedingt hingeben kann.--STEFFENS,
_Christliche Religionsphilosophie_, 440, 1839. Wir erkennen darin nur
eine Thaetigkeit des zu seinem aechten und wahren Leben, zu seinem
verlornen, objectiven Selbstverstaendnisse sich zuruecksehnenden
christlichen Geistes unserer Zeit, einen Ausdruck fuer das Beduerfniss
desselben, sich aus den unwahren und unaechten Verkleidungen, womit ihn
der moderne, subjective Geschmack der letzten Entwicklungsphase des
theologischen Bewusstseyns umhuellt hat, zu seiner historischen allein
wahren und urspruenglichen Gestalt wiederzugebaeren, zu derjenigen
Bedeutung zurueckzukehren, die ihm in dem Bewusstseyn der Geschichte
allein zukommt und deren Verstaendniss in dem wogenden luxurioesen Leben
der modernen Theologie laengst untergegangen ist.--GEORGII, _Zeitschrift
fuer Hist. Theologie_, ix. 5, 1839.
[45] Liberty, in fact, means just so far as it is realised,
the right man in the right place.--SEELEY, _Lectures and Essays_,
109.
[46] In diesem Sinne ist Freiheit und sich entwickelnde
moralische Vernunft und Gewissen gleichbedeutend. In diesem Sinne ist
der Mensch frei, sobald sich das Gewissen in ihm entwickelt.--SCHEIDLER,
_Ersch und Gruber_, xlix. 20. Aus der unendlichen und ewigen Geltung
der menschlichen Persoenlichkeit vor Gott, aus der Vorstellung von der
in Gott freien Persoenlichkeit, folgt auch der Anspruch auf das Recht
derselben in der weltlichen Sphaere, auf buergerliche und politische
Freiheit, auf Gewissen und Religionsfreiheit, auf freie
wissenschaftliche Forschung u.s.w., und namentlich die Forderung dass
niemand lediglich zum Mittel fuer andere diene.--MARTENSEN, _Christliche
Ethik_, i. 50.
[47] Es giebt angeborne Menschenrechte, weil es angeborne
Menschenpflichten giebt.--WOLFF, _Naturrecht_; LOEPER, _Einleitung
zu Faust_, lvii.
[48] La constitution de l'etat reste jusqu'a un certain point
a notre discretion. La constitution de la societe ne depend pas de
nous; elle est donnee par la force des choses, et si l'on veut elever
le langage, elle est l'oeuvre de la Providence.--REMUSAT, _Revue des
Deux Mondes_, 1861, v. 795.
[49] Die Freiheit ist bekanntlich kein Geschenk der Goetter,
sondern ein Gut das jedes Volk sich selbst verdankt und das nur bei
dem erforderlichen Mass moralischer Kraft und Wuerdigkeit
gedeiht.--IHERING, _Geist des Roemischen Rechts_, ii. 290. Liberty, in
the very nature of it, absolutely requires and even supposes, that
people be able to govern themselves in those respects in which they
are free; otherwise their wickedness will be in proportion to their
liberty, and this greatest of blessings will become a curse.--BUTLER,
_Sermons_, 331. In each degree and each variety of public development
there are corresponding institutions, best answering the public needs;
and what is meat to one is poison to another. Freedom is for those who
are fit for it.--PARKMAN, _Canada_, 396. Die Freiheit ist die Wurzel
einer neuen Schoepfung in der Schoepfung.--SEDERHOLM, _Die ewigen
Thatsachen_, 86.
[50] La liberte politique, qui n'est qu'une complexite plus
grande, de plus en plus grande, dans le gouvernement d'un peuple, a
mesure que le peuple lui-meme contient un plus grand nombre de forces
diverses ayant droit et de vivre et de participer a la chose publique,
est un fait de civilisation qui s'impose lentement a une societe
organisee, mais qui n'apparait point comme un principe a une societe
qui s'organise.--FAGUET, _Revue des Deux Mondes_, 1889, ii. 942.
[51] Il y a bien un droit du plus sage, mais non pas un droit
du plus fort.--La justice est le droit du plus faible.--JOUBERT,
_Pensees_, i. 355, 358.
[52] Nicht durch ein pflanzenaehnliches Wachsthum, nicht aus
den dunklen Gruenden der Volksempfindung, sondern durch den maennlichen
Willen, durch die Ueberzeugung, durch die That, durch den Kampf
entsteht, behauptet, entwickelt sich das Recht. Sein historisches
Werden ist ein bewusstes, im hellen Mittagslicht der Erkenntniss und
der Gesetzgebung.--_Rundschau_, Nov. 1893, 313. Nicht das Normale,
Zahme, sondern das Abnorme, Wilde, bildet ueberall die Grundlage und
den Anfang einer neuen Ordnung.--LASAULX, _Philosophie der
Geschichte_, 143.
[53] Um den Sieg zu vervollstaendigen, eruebrigte das zweite
Stadium oder die Aufgabe: die Berechtigung der Mehrheit nach allen
Seiten hin zur gleichen Berechtigung aller zu erweitern, d.h. bis zur
Gleichstellung aller Bekenntnisse im Kirchenrecht, aller Voelker im
Voelkerrecht, aller Staatsbuerger im Staatsrecht und aller socialen
Interessen im Gesellschaftsrecht fortzufuehren.--A. SCHMIDT, _Zuericher
Monatschrift_, i. 68.
[54] Notre histoire ne nous enseignait nullement la liberte.
Le jour ou la France voulut etre libre, elle eut tout a creer, tout a
inventer dans cet ordre de faits.--Cependant il faut marcher, l'avenir
appelle les peuples. Quand on n'a point pour cela l'impulsion du
passe, il faut bien se confier a la raison.--DUPONT WHITE, _Revue des
Deux Mondes_, 1861, vi. 191. Le peuple francais a peu de gout pour le
developpement graduel des institutions. Il ignore son histoire, il ne
s'y reconnait pas, elle n'a pas laisse de trace dans sa
conscience.--SCHERER, _Etudes Critiques_, i. 100. Durch die Revolution
befreiten sich die Franzosen von ihrer Geschichte.--ROSENKRANZ, _Aus
einem Tagebuch_, 199.
[55] The discovery of the comparative method in philology, in
mythology--let me add in politics and history and the whole range of
human thought--marks a stage in the progress of the human mind at
least as great and memorable as the revival of Greek and Latin
learning.--FREEMAN, _Historical Essays_, iv. 301. The diffusion of a
critical spirit in history and literature is affecting the criticism
of the Bible in our own day in a manner not unlike the burst of
intellectual life in the fifteenth and sixteenth centuries.--JOWETT,
_Essays and Reviews_, 346. As the revival of literature in the
sixteenth century produced the Reformation, so the growth of the
critical spirit, and the change that has come over mental science, and
the mere increase of knowledge of all kinds, threaten now a revolution
less external but not less profound.--HADDAN, _Replies_, 348.
[56] In his just contempt and detestation of the crimes and
follies of the Revolutionists, he suffers himself to forget that the
revolution itself is a process of the Divine Providence, and that as
the folly of men is the wisdom of God, so are their iniquities
instruments of His goodness.--COLERIDGE, _Biographia Literaria_, ii.
240. In other parts of the world, the idea of revolutions in
government is, by a mournful and indissoluble association, connected
with the idea of wars, and all the calamities attendant on wars. But
happy experience teaches us to view such revolutions in a very
different light--to consider them only as progressive steps in
improving the knowledge of government, and increasing the happiness of
society and mankind.--J. WILSON, November 26, 1787, _Works_, iii. 293.
La Revolution, c'est-a-dire l'oeuvre des siecles, ou, si vous
voulez, le renouvellement progressif de la societe, ou encore, sa
nouvelle constitution.--REMUSAT, _Correspondance_, October 11, 1818. A
ses yeux loin d'avoir rompu le cours naturel des evenements, ni la
Revolution d'Angleterre, ni la notre, n'ont rien dit, rien fait, qui
n'eut ete dit, souhaite, fait, ou tente cent fois avant leur
explosion. "Il faut en ceci," dit-il, "tout accorder a leurs
adversaires, les surpasser meme en severite, ne regarder a leurs
accusations que pour y ajouter, s'ils en oublient; et puis les sommer
de dresser, a leur tour, le compte des erreurs, des crimes, et des
maux de ces temps et de ces pouvoirs qu'ils ont pris sous leur
garde."--_Revue de Paris_, xvi. 303, on Guizot. Quant aux nouveautes
mises en oeuvre par la Revolution Francaise on les retrouve une a
une, en remontant d'age en age, chez les philosophes du XVIIIe
siecle, chez les grands penseurs du XVIe, chez certains Peres
d'Eglise et jusque dans la Republique de Platon.--En presence de cette
belle continuite de l'histoire, qui ne fait pas plus de sauts que la
nature, devant cette solidarite necessaire des revolutions avec le
passe qu'elles brisent.--KRANTZ, _Revue Politique_, xxxiii. 264.
L'esprit du XIXe siecle est de comprendre et de juger les choses du
passe. Notre oeuvre est d'expliquer ce que le XVIIIe siecle avait
mission de nier.--VACHEROT, _De la Democratie_, pref., 28.
[57] La commission recherchera, dans toutes les parties des
archives pontificales, les pieces relatives a l'abus que les papes ont
fait de leur ministere spirituel contre l'autorite des souverains et
la tranquillite des peuples.--DAUNOU, _Instructions_, Jan. 3, 1811.
LABORDE, _Inventaires_, p. cxii.
[58] Aucun des historiens remarquables de cette epoque
n'avait senti encore le besoin de chercher les faits hors des livres
imprimes, aux sources primitives, la plupart inedites alors, aux
manuscrits de nos bibliotheques, aux documents de nos archives.--MICHELET,
_Histoire de France_, 1869, i. 2.
[59] Doch besteht eine Grenze, wo die Geschichte aufhoert und
das Archiv anfaengt, und die von der Geschichtschreibung nicht
ueberschritten werden sollte. _Unsere Zeit_, 1866, ii. 635. Il faut
avertir nos jeunes historiens a la fois de la necessite ineluctable du
document et, d'autre part, du danger qu'il presente.--M. HANOTAUX.
[60] This process consists in determining with documentary
proofs, and by minute investigations duly set forth, the literal,
precise, and positive inferences to be drawn at the present day from
every authentic statement, without regard to commonly received
notions, to sweeping generalities, or to possible consequences.--HARRISSE,
_Discovery of America_, 1892, p. vi. Perhaps the time has not yet come
for synthetic labours in the sphere of History. It may be that the
student of the Past must still content himself with critical
inquiries.--_Ib._ p. v. Few scholars are critics, few critics are
philosophers, and few philosophers look with equal care on both sides
of a question.--W. S. LANDOR in HOLYOAKE'S _Agitator's Life_, ii. 15.
Introduire dans l'histoire, et sans tenir compte des passions
politiques et religieuses, le doute methodique que Descartes, le
premier, appliqua a l'etude de la philosophie, n'est-ce pas la une
excellente methode? n'est-ce pas meme la meilleure?--CHANTELAUZE,
_Correspondant_, 1883, i. 129. La critique historique ne sera jamais
populaire. Comme elle est de toutes les sciences la plus delicate, la plus
deliee, elle n'a de credit qu'aupres des esprits cultives.--CHERBULIEZ,
_Revue des Deux Mondes_, xcvii. 517. Nun liefert aber die Kritik, wenn
sie rechter Art ist, immer nur einzelne Data, gleichsam die Atome des
Thatbestandes, und jede Kombination, jede Zusammenfassung und
Schlussfolgerung, ohne die es doch einmal nicht abgeht, ist ein
subjektiver Akt des Forschers. Demnach blieb Waitz, bei der eigenen
Arbeit wie bei jener der anderen, immer hoechst mistrauisch gegen jedes
Resume, jede Definition, jedes abschliessende Wort.--SYBEL,
_Historische Zeitschrift_, lvi. 484. Mit blosser Kritik wird darin
nichts ausgerichtet, denn die ist nur eine Vorarbeit, welche da
aufhoert wo die echte historische Kunst anfaengt.--LASAULX, _Philosophie
der Kuenste_, 212.
[61] The only case in which such extraneous matters can be
fairly called in is when facts are stated resting on testimony; then
it is not only just, but it is necessary for the sake of truth, to
inquire into the habits of mind of him by whom they are
adduced.--BABBAGE, _Bridgewater Treatise_, p. xiv.
[62] There is no part of our knowledge which it is more
useful to obtain at first hand--to go to the fountain-head for--than
our knowledge of History.--J. S. MILL, _Inaugural Address_, 34. The
only sound intellects are those which, in the first instance, set
their standard of proof high.--J. S. MILL, _Examination of Hamilton's
Philosophy_, 525.
[63] There are so few men mentally capable of seeing both
sides of a question; so few with consciences sensitively alive to the
obligation of seeing both sides; so few placed under conditions either
of circumstance or temper, which admit of their seeing both
sides.--GREG, _Political Problems_, 1870, 173. Il n'y a que les
Allemands qui sachent etre aussi completement objectifs. Ils se
dedoublent, pour ainsi dire, en deux hommes, l'un qui a des principes
tres arretes et des passions tres vives, l'autre qui sait voir et
observer comme s'il n'en avait point.--LAVELEYE, _Revue des Deux
Mondes_, 1868, i. 431. L'ecrivain qui penche trop dans le sens ou il
incline, et qui ne se defie pas de ses qualites presque autant que ses
defauts, cet ecrivain tourne a la maniere.--SCHERER, _Melanges_, 484.
Il faut faire volte-face, et vivement, franchement, tourner le dos au
moyen age, a ce passe morbide, qui, meme quand il n'agit pas, influe
terriblement par la contagion de la mort. Il ne faut ni combattre, ni
critiquer, mais oublier. Oublions et marchons!--MICHELET, _La Bible de
l'Humanite_, 483. It has excited surprise that Thucydides should speak
of Antiphon, the traitor to the democracy, and the employer of assassins,
as "a man inferior in virtue to none of his contemporaries." But
neither here nor elsewhere does Thucydides pass moral judgments.--JOWETT,
_Thucydides_, ii. 501.
[64] Non theologi provinciam suscepimus; scimus enim quantum hoc
ingenii nostri tenuitatem superet: ideo sufficit nobis [Greek: to
hoti] fideliter ex antiquis auctoribus retulisse.--MORINUS, _De
Poenitentia_, ix. 10.--Il faut avouer que la religion chretienne a
quelque chose d'etonnant! C'est parce que vous y etes ne, dira-t-on.
Tant s'en faut, je me roidis contre par cette raison-la meme, de peur
que cette prevention ne me suborne.--PASCAL, _Pensees_, XVI., 7.--I
was fond of Fleury for a reason which I express in the advertisement;
because it presented a sort of photograph of ecclesiastical history
without any comment upon it. In the event, that simple representation
of the early centuries had a good deal to do with unsettling
me.--NEWMAN, _Apologia_, 152.--Nur was sich vor dem Richterstuhl einer
aechten, unbefangenen, nicht durch die Brille einer philosophischen
oder dogmatischen Schule stehenden Wissenschaft als wahr bewaehrt, kann
zur Erbauung, Belehrung und Warnung tuechtig seyn.--NEANDER,
_Kirchengeschichte_, i. p. vii. Wie weit bei katholischen Publicisten
bei der Annahme der Ansicht von der Staatsanstalt apologetische
Gesichtspunkte massgebend gewesen sind, mag dahingestellt bleiben. Der
Historiker darf sich jedoch nie durch apologetische Zwecke leiten
lassen; sein einziges Ziel soll die Ergruendungder Wahrheit
sein.--PASTOR, _Geschichte der Paebste_, ii. 545. Church history
falsely written is a school of vainglory, hatred, and uncharitableness;
truly written, it is a discipline of humility, of charity, of mutual
love.--SIR W. HAMILTON, _Discussions_, 506. The more trophies and
crowns of honour the Church of former ages can be shown to have won in
the service of her adorable head, the more tokens her history can be
brought to furnish of his powerful presence in her midst, the more
will we be pleased and rejoice, Protestant though we be.--NEVIN,
_Mercersburg Review_, 1851, 168. S'il est une chose a laquelle j'ai
donne tous mes soins, c'est a ne pas laisser influencer mes jugements
par les opinions politiques ou religieuses; que si j'ai quelquefois
peche par quelque exces, c'est par la bienveillance pour les oeuvres
de ceux qui pensent autrement que moi.--MONOD, _R. Hist._, xvi. 184.
Nous n'avons nul interet a faire parler l'histoire en faveur de nos
propres opinions. C'est son droit imprescriptible que le narrateur
reproduise tous les faits sans aucune reticence et range toutes les
evolutions dans leur ordre naturel. Notre recit restera completement
en dehors des preoccupations de la dogmatique et des declamations de
la polemique. Plus les questions auxquelles nous aurons a toucher
agitent et passionnent de nos jours les esprits, plus il est du devoir
de l'historien de s'effacer devant les faits qu'il veut faire
connaitre.--REUSS, _Nouvelle Revue de Theologie_, vi. 193, 1860. To
love truth for truth's sake is the principal part of human perfection
in this world, and the seed plot of all other virtues.--LOCKE, _Letter
to Collins_. Il n'est plus possible aujourd'hui a l'historien d'etre
national dans le sens etroit du mot. Son patriotisme a lui c'est
l'amour de la verite. Il n'est pas l'homme d'une race ou d'un pays, il
est l'homme de tous les pays, il parle au nom de la civilisation
generale.--LANFREY, _Hist. de Nap._, iii. 2, 1870. Juger avec les
parties de soi-meme qui sont le moins des formes du temperament, et le
plus des facultes penetrees et modelees par l'experience, par l'etude,
par l'investigation, par le non-moi.--FAGUET, _R. de Paris_, i. 151.
Aucun critique n'est aussi impersonnel que lui, aussi libre de parti
pris et d'opinions preconcues, aussi objectif.--Il ne mele ou parait
meler a ses appreciations ni inclinations personnelles de gout ou
d'humeur, ou theories d'aucune sorte.--G. MONOD, of Faguet, _Revue
Historique_, xlii. 417. On dirait qu'il a peur, en generalisant ses
observations, en systematisant ses connaissances, de meler de lui-meme
aux choses.--Je lis tout un volume de M. Faguet, sans penser une fois
a M. Faguet: je ne vois que les originaux qu'il montre.--J'envisage
toujours une realite objective, jamais l'idee de M. Faguet, jamais la
doctrine de M. Faguet.--LANSON, _Revue Politique_, 1894, i. 98.
[65] It should teach us to disentangle principles first from
parties, and again from one another; first of all as showing how
imperfectly all parties represent their own principles, and then how
the principles themselves are a mingled tissue.--ARNOLD, _Modern
History_, 184. I find it a good rule, when I am contemplating a person
from whom I want to learn, always to look out for his strength, being
confident that the weakness will discover itself.--MAURICE, _Essays_,
305. We may seek for agreement somewhere with our neighbours, using
that as a point of departure for the sake of argument. It is this
latter course that I wish here to explain and defend. The method is
simple enough, though not yet very familiar.--It aims at conciliation;
it proceeds by making the best of our opponent's case, instead of
taking him at his worst.--The most interesting part of every disputed
question only begins to appear when the rival ideals admit each
other's right to exist.--A. SIDGWICK, _Distinction and the Criticism
of Beliefs_, 1892, 211. That cruel reticence in the breasts of wise
men which makes them always hide their deeper thought.--RUSKIN,
_Sesame and Lilies_, i. 16. Je offener wir die einzelnen Wahrheiten
des Sozialismus anerkennen, desto erfolgreicher koennen wir seine
fundamentalen Unwahrheiten widerlegen.--ROSCHER, _Deutsche
Vierteljahrschrift_, 1849, i. 177.
[66] Dann habe ihn die Wahrnehmung, dass manche Angaben in
den historischen Romanen Walter Scott's, mit den gleichzeitigen
Quellen im Widerspruch standen, "mit Erstaunen" erfuellt, und ihn zu
dem Entschlusse gebracht, auf das Gewissenhafteste an der
Ueberlieferung der Quellen festzuhalten.--SYBEL, _Gedaechtnissrede auf
Ranke_. _Akad. der Wissenschaften_, 1887, p. 6. Sich frei zu halten
von allem Widerschein der Gegenwart, sogar, soweit das menschenmoeglich,
von dem der eignen subjectiven Meinung in den Dingen des Staates, der
Kirche und der Gesellschaft.--A. DOVE, _Im Neuen Reich_, 1875, ii. 967.
Wir sind durchaus nicht fuer die leblose und schemenartige
Darstellungsweise der Ranke'schen Schule eingenommen; es wird uns
immer kuehl bis ans Herz heran, wenn wir derartige Schilderungen der
Reformation und der Revolution lesen, welche so ganz im kuehlen Element
des Pragmatismus sich bewegen und dabei so ganz Undinenhaft sind und
keine Seele haben.--Wir lassen es uns lieber gefallen, dass die Maenner
der Geschichte hier und dort gehofmeistert werden, als dass sie uns
mit Glasaugen ansehen, so meisterhaft immer die Kunst sein mag die sie
ihnen eingesetzt hat.--GOTTSCHALL, _Unsere Zeit_, 1866, ii. 636, 637. A
vivre avec des diplomates, il leur a pris des qualites qui sont un
defaut chez un historien. L'historien n'est pas un temoin, c'est un
juge; c'est a lui d'accuser et de condamner au nom du passe opprime et
dans l'interet de l'avenir.--LABOULAYE on RANKE. _Debats_, January 12,
1852.
[67] Un theologien qui a compose une eloquente histoire de la
Reformation, rencontrant a Berlin un illustre historien qui, lui
aussi, a raconte Luther et le XVIe siecle, l'embrassa avec effusion
en le traitant de confrere. "Ah! permettez," lui repondit l'autre en
se degageant, "il y a une grande difference entre nous: vous etes
avant tout chretien, et je suis avant tout historien."--CHERBULIEZ,
_Revue des Deux Mondes_, 1872, i. 537.
[68] Nackte Wahrheit ohne allen Schmuck; gruendliche
Erforschung des Einzelnen; das Uebrige, Gott befohlen.--_Werke_,
xxxiv. 24. Ce ne sont pas les theories qui doivent nous servir de base
dans la recherche des faits, mais ce sont les faits qui doivent nous
servir de base pour la composition des theories.--VINCENT, _Nouvelle
Revue de Theologie_, 1859, ii. 252.
[69] Die zwanglose Anordnungs--die leichte und leise
Andeutungskunst des grossen Historikers voll zu wuerdigen, hinderte ihn
in frueherer Zeit sein Beduerfniss nach scharfer begrifflicher Ordnung
und Ausfuehrung, spaeter, und in immer zunehmenden Grade, sein Sinn fuer
strenge Sachlichkeit, und genaue Erforschung der ursaechlichen
Zusammenhaenge, noch mehr aber regte sich seine geradherzige Offenheit
seine maennliche Ehrlichkeit, wenn er hinter den fein verstrichenen
Farben der Rankeschen Erzaehlungsbilder die gedeckte Haltung des klugen
Diplomaten zu entdecken glaubte.--HAYM, _Duncker's Leben_, 437. The
ground of criticism is indeed, in my opinion, nothing else but
distinct attention, which every reader should endeavour to be master
of.--HARE, _Dec._, 1736, _Warburton's Works_, xiv. 98. Wenn die
Quellenkritik so verstanden wird, als sei sie der Nachweis, wie ein
Autor den andern benutzt hat, so ist das nur ein gelegentliches
Mittel--eins unter anderen--ihre Aufgabe, den Nachweis der Richtigkeit
zu loesen oder vorzubereiten.--DROYSEN, _Historik_, 18.
[70] L'esprit scientifique n'est autre en soi que l'instinct
du travail et de la patience, le sentiment de l'ordre, de la realite
et de la mesure.--PAPILLON, _R. des Deux Mondes_, 1873, v. 704. Non
seulement les sciences, mais toutes les institutions humaines s'organisent
de meme, et sous l'empire des memes idees regulatrices.--COURNOT,
_Idees Fondamentales_, i. 4. There is no branch of human work whose
constant laws have not close analogy with those which govern every
other mode of man's exertion. But more than this, exactly as we reduce
to greater simplicity and surety any one group of these practical
laws, we shall find them passing the mere condition of connection or
analogy, and becoming the actual expression of some ultimate nerve or
fibre of the mighty laws which govern the moral world.--RUSKIN, _Seven
Lamps_, 4. The sum total of all intellectual excellence is good sense
and method. When these have passed into the instinctive readiness of
habit, when the wheel revolves so rapidly that we cannot see it
revolve at all, then we call the combination genius. But in all modes
alike, and in all professions, the two sole component parts, even of
genius, are good sense and method.--COLERIDGE, _June_, 1814, _Mem. of
Coleorton_, ii. 172. Si l'exercice d'un art nous empeche d'en
apprendre un autre, il n'en est pas ainsi dans les sciences: la
connoissance d'une verite nous aide a en decouvrir une autre.--Toutes
les sciences sont tellement liees ensemble qu'il est bien plus facile
de les apprendre toutes a la fois que d'en apprendre une seule en la
detachant des autres.--Il ne doit songer qu'a augmenter les lumieres
naturelles de sa raison, non pour resoudre telle ou telle difficulte
de l'ecole, mais pour que dans chaque circonstance de la vie son
intelligence montre d'avance a sa volonte le parti qu'elle doit
prendre.--DESCARTES, _OEuvres Choisies_, 300, 301. _Regles pour la
Direction de l'Esprit._ La connaissance de la methode qui a guide
l'homme de genie n'est pas moins utile au progres de la science et
meme a sa propre gloire, que ses decouvertes.--LAPLACE, _Systeme du
Monde_, ii. 371. On ne fait rien sans idees preconcues, il faut avoir
seulement la sagesse de ne croire a leurs deductions qu'autant que
l'experience les confirme. Les idees preconcues, soumises au controle
severe de l'experimentation, sont la flamme vivante des sciences
d'observation; les idees fixes en sont le danger.--PASTEUR, in
_Histoire d'un Savant_, 284. Douter des verites humaines, c'est ouvrir
la porte aux decouvertes; en faire des articles de foi, c'est la
fermer.--DUMAS, _Discours_, i. 123.
[71] We should not only become familiar with the laws of
phenomena within our own pursuit, but also with the modes of thought
of men engaged in other discussions and researches, and even with the
laws of knowledge itself, that highest philosophy.--Above all things,
know that we call you not here to run your minds into our moulds. We
call you here on an excursion, on an adventure, on a voyage of
discovery into space as yet uncharted.--ALLBUTT, _Introductory Address
at St. George's_, October 1889. Consistency in regard to opinions is
the slow poison of intellectual life.--DAVY, _Memoirs_, 68.
[72] Ce sont vous autres physiologistes des corps vivants,
qui avez appris a nous autres physiologistes de la societe (qui est
aussi un corps vivant) la maniere de l'observer et de tirer des
consequences de nos observations.--J. B. SAY to DE CANDOLLE, June 1,
1827.--DE CANDOLLE, _Memoires_, 567.
[73] Success is certain to the pure and true: success to
falsehood and corruption, tyranny and aggression, is only the prelude
to a greater and an irremediable fall.--STUBBS, _Seventeen Lectures_,
20. The Carlylean faith, that the cause we fight for, so far as it is
true, is sure of victory, is the necessary basis of all effective
activity for good.--CAIRD, _Evolution of Religion_, ii. 43. It is the
property of truth to be fearless, and to prove victorious over every
adversary. Sound reasoning and truth, when adequately communicated,
must always be victorious over error.--GODWIN, _Political Justice_
(Conclusion). Vice was obliged to retire and give place to virtue.
This will always be the consequence when truth has fair play.
Falsehood only dreads the attack, and cries out for auxiliaries. Truth
never fears the encounter; she scorns the aid of the secular arm, and
triumphs by her natural strength.--FRANKLIN, _Works_, ii. 292. It is a
condition of our race that we must ever wade through error in our
advance towards truth: and it may even be said that in many cases we
exhaust almost every variety of error before we attain the desired
goal.--BABBAGE, _Bridgewater Treatise_, 27. Les hommes ne peuvent, en
quelque genre que ce soit, arriver a quelque chose de raisonnable
qu'apres avoir, en ce meme genre, epuise toutes les sottises
imaginables. Que de sottises ne dirions-nous pas maintenant, si les
anciens ne les avaient pas deja dites avant nous, et ne nous les
avaient, pour ainsi dire, enlevees!--FONTENELLE. Without premature
generalisations the true generalisation would never be arrived at.--H.
SPENCER, _Essays_, ii. 57. The more important the subject of
difference, the greater, not the less, will be the indulgence of him
who has learned to trace the sources of human error,--of error, that
has its origin not in our weakness and imperfection merely, but often
in the most virtuous affections of the heart.--BROWN, _Philosophy of
the Human Mind_, i. 48, 1824. Parmi les chatiments du crime qui ne lui
manquent jamais, a cote de celui que lui inflige la conscience,
l'histoire lui en inflige un autre encore, eclatant et manifeste,
l'impuissance.--COUSIN, _Phil. Mod._ ii. 24. L'avenir de la science
est garanti; car dans le grand livre scientifique tout s'ajoute et
rien ne se perd. L'erreur ne fonde pas; aucune erreur ne dure tres
longtemps.--RENAN, _Feuilles Detachees_, xiii. Toutes les fois que
deux hommes sont d'un avis contraire sur la meme chose, a coup sur,
l'un ou l'autre se trompe; bien plus, aucun ne semble posseder la
verite; car si les raisons de l'un etoient certaines et evidentes, il
pourroit les exposer a l'autre de telle maniere qu'il finiroit par le
convaincre egalement.--DESCARTES, _Regles: OEuvres Choisies_, 302.
Le premier principe de la critique est qu'une doctrine ne captive ses
adherents que par ce qu'elle a de legitime.--RENAN, _Essais de
Morale_, 184. Was dem Wahn solche Macht giebt ist wirklich nicht er
selbst, sondern die ihm zu Grunde liegende und darin nur verzerrte
Wahrheit.--FRANTZ, _Schelling's Philosophie_, i. 62. Quand les hommes
ont vu une fois la verite dans son eclat, ils ne peuvent plus
l'oublier. Elle reste debout, et tot ou tard elle triomphe, parce
qu'elle est la pensee de Dieu et le besoin du monde.--MIGNET,
_Portraits_, ii. 295. C'est toujours le sens commun inapercu qui fait
la fortune des hypotheses auxquelles il se mele.--COUSIN, _Fragments
Phil._ i. 51. Preface of 1826. Wer da sieht wie der Irrthum selbst ein
Traeger mannigfaltigen und bleibenden Fortschritts wird, der wird auch
nicht so leicht aus dem thatsaechlichen Fortschritt der Gegenwart auf
Unumstoesslichkeit unserer Hypothesen schliessen.--Das richtigste
Resultat der geschichtlichen Betrachtung ist die akademische Ruhe, mit
welcher unsere Hypothesen und Theorieen ohne Feindschaft und ohne
Glauben als das betrachtet werden was sie sind; als Stufen in jener
unendlichen Annaeherung an die Wahrheit, welche die Bestimmung unserer
intellectuellen Entwicklung zu sein scheint.--LANGE, _Geschichte des
Materialismus_, 502, 503. Hominum errores divina providentia reguntur,
ita ut saepe male jacta bene cadant.--LEIBNIZ, ed. Klopp, i., p. lii.
Sainte-Beuve n'etait meme pas de la race des liberaux, c'est-a-dire
de ceux qui croient que, tout compte fait, et dans un etat de
civilisation donne, le bien triomphe du mal a armes egales, et la
verite de l'erreur.--D'HAUSSONVILLE, _Revue des Deux Mondes_, 1875, i.
567. In the progress of the human mind, a period of controversy
amongst the cultivators of any branch of science must necessarily
precede the period of unanimity.--TORRENS, _Essay on the Production of
Wealth_, 1821, p. xiii. Even the spread of an error is part of the
wide-world process by which we stumble into mere approximations to
truth.--L. STEPHEN, _Apology of an Agnostic_, 81. Errors, to be
dangerous, must have a great deal of truth mingled with them; it is
only from this alliance that they can ever obtain an extensive
circulation.--S. SMITH, _Moral Philosophy_, 7. The admission of the
few errors of Newton himself is at least of as much importance to his
followers in science as the history of the progress of his real
discoveries.--YOUNG, _Works_, iii. 621. Error is almost always partial
truth, and so consists in the exaggeration or distortion of one verity
by the suppression of another, which qualifies and modifies the
former.--MIVART, _Genesis of Species_, 3. The attainment of scientific
truth has been effected, to a great extent, by the help of scientific
errors.--HUXLEY: WARD, _Reign of Victoria_, ii. 337. Jede neue tief
eingreifende Wahrheit hat meiner Ansicht nach erst das Stadium der
Einseitigkeit durchzumachen.--IHERING, _Geist des R. Rechts_, ii. 22.
The more readily we admit the possibility of our own cherished
convictions being mixed with error, the more vital and helpful
whatever is right in them will become.--RUSKIN, _Ethics of the Dust_,
225. They hardly grasp the plain truth unless they examine the error
which it cancels.--CORY, _Modern English History_, 1880, i. 109. Nur
durch Irrthum kommen wir, der eine kuerzeren und gluecklicheren
Schrittes, als der andere, zur Wahrheit; und die Geschichte darf
nirgends diese Verirrungen uebergehen, wenn sie Lehrerin und Warnerin
fuer die nachfolgenden Geschlechter werden will.--_Muenchner Gel.
Anzeigen_, 1840, i. 737.
[74] Wie die Weltgeschichte das Weltgericht ist, so kann in
noch allgemeinerem Sinne gesagt werden, dass das gerechte Gericht,
d.h. die wahre Kritik einer Sache, nur in ihrer Geschichte liegen
kann. Insbesondere in der Hinsicht lehrt die Geschichte denjenigen,
der ihr folgt, ihre eigene Methode, dass ihr Fortschritt niemals ein
reines Vernichten, sondern nur ein Aufheben im philosophischen Sinne
ist.--STRAUSS, _Hallische Jahrbuecher_, 1839, 120.
[75] Dans tous les livres qu'il lit, et il en devore des
quantites, Darwin ne note que les passages qui contrarient ses idees
systematiques.--Il collectionne les difficultes, les cas epineux, les
critiques possibles.--VERNIER, _Le Temps_, 6 Decembre, 1887. Je
demandais a un savant celebre ou il en etait de ses recherches. "Cela
ne marche plus," me dit-il, "je ne trouve plus de faits
contradictoires." Ainsi le savant cherche a se contredire lui-meme
pour faire avancer sa pensee.--JANET, _Journal des Savants_, 1892, 20.
Ein Umstand, der uns die Selbstaendigkeit des Ganges der Wissenschaft
anschaulich machen kann, ist auch der: dass der Irrthum, wenn er nur
gruendlich behandelt wird, fast ebenso foerdernd ist als das Finden der
Wahrheit, denn er erzeugt fortgesetzten Widerspruch.--BAER, _Blicke
auf die Entwicklung der Wissenschaft_, 120. It is only by virtue of
the opposition which it has surmounted that any truth can stand in the
human mind.--BISHOP TEMPLE; KINGLAKE, _Crimea_, _Winter Troubles_,
app. 104. I have for many years found it expedient to lay down a rule
for my own practice, to confine my reading mainly to those journals
the general line of opinions in which is adverse to my own.--HARE,
_Means of Unity_, i. 19. Kant had a harder struggle with himself than
he could possibly have had with any critic or opponent of his
philosophy.--CAIRD, _Philosophy of Kant_, 1889, i. p. ix.
[76] The social body is no more liable to arbitrary changes
than the individual body.--A full perception of the truth that society
is not a mere aggregate, but an organic growth, that it forms a whole
the laws of whose growth can be studied apart from those of the
individual atom, supplies the most characteristic postulate of modern
speculation.--L. STEPHEN, _Science of Ethics_, 31. Wie in dem Leben
des Einzelnen Menschen kein Augenblick eines vollkommenen Stillstandes
wahrgenommen wird, sondern stete organische Entwicklung, so verhaelt es
sich auch in dem Leben der Voelker, und in jedem einzelnen Element,
woraus dieses Gesammtleben besteht. So finden wir in der Sprache stete
Fortbildung und Entwicklung, und auf gleiche Weise in dem Recht. Und
auch diese Fortbildung steht unter demselben Gesetz der Erzeugung aus
innerer Kraft und Nothwendigkeit, unabhaengig von Zufall und
individueller Willkuer, wie die urspruengliche Entstehung.--SAVIGNY,
_System_, i. 16, 17. Seine eigene Entdeckung, dass auch die geistige
Produktion, bis in einem gewissen Punkte wenigstens, unter dem Gesetze
der Kausalitaet steht, dass jedeiner nur geben kann was er hat, nur hat
was er irgendwoher bekommen, muss auch fuer ihn selber gelten.--BEKKER,
_Das Recht des Besitzes bei den Roemern_, 3, 1880. Die geschichtliche
Wandlung des Rechts, in welcher vergangene Jahrhunderte halb ein Spiel
des Zufalls und halb ein Werk vernuenftelnder Willkuer sahen, als
gesetzmaessige Entwickelung zu begreifen, war das unsterbliche
Verdienst der von Maennern wie Savigny, Eichhorn und Jacob Grimm
gefuehrten historischen Rechtsschule.--GIERKE, _Rundschau_, xviii.
205.
[77] The only effective way of studying what is called the
philosophy of religion, or the philosophical criticism of religion, is
to study the history of religion. The true science of war is the
history of war, the true science of religion is, I believe, the
history of religion.--M. MUeLLER, _Theosophy_, 3, 4. La theologie ne
doit plus etre que l'histoire des efforts spontanes tentes pour
resoudre le probleme divin. L'histoire, en effet, est la forme
necessaire de la science de tout ce qui est soumis aux lois de la vie
changeante et successive. La science de l'esprit humain, c'est de
meme, l'histoire de l'esprit humain.--RENAN, _Averroes_, Pref. vi.
[78] Political economy is not a science, in any strict sense,
but a body of systematic knowledge gathered from the study of common
processes, which have been practised all down the history of the human
race in the production and distribution of wealth.--BONAMY PRICE,
_Social Science Congress_, 1878. Such a study is in harmony with the
best intellectual tendencies of our age, which is, more than anything
else, characterized by the universal supremacy of the historical
spirit. To such a degree has this spirit permeated all our modes of
thinking, that with respect to every branch of knowledge, no less than
with respect to every institution and every form of human activity, we
almost instinctively ask, not merely what is its existing condition,
but what were its earliest discoverable germs, and what has been the
course of its development.--INGRAM, _History of Political Economy_, 2.
Wir dagegen stehen keinen Augenblick an, die Nationaloekonomie fuer eine
reine Erfahrungswissenschaft zu erklaeren, und die Geschichte ist uns
daher nicht Huelfsmittel, sondern Gegenstand selber.--ROSCHER,
_Deutsche Vierteljahrschrift_, 1849, i. 182. Der bei weitem groesste
Theil menschlicher Irrthuemer beruhet darauf, dass man zeitlich und
oertlich Wahres oder Heilsames fuer absolut wahr oder heilsam ausgiebt.
Fuer jede Stufe der Volksentwickelung passt eine besondere
Staatsverfassung, die mit allen uebrigen Verhaeltnissen des Volks als
Ursache und Wirkung auf's Innigste verbunden ist; so passt auch fuer jede
Entwickelungsstufe eine besondere Landwirthschaftsverfassung.--ROSCHER,
_Archiv f. p. Oek._, viii., 2 Heft 1845. Seitdem vor allen Roscher,
Hildebrand und Knies den Werth, die Berechtigung und die
Nothwendigkeit derselben unwiderleglich dargethan, hat sich immer
allgemeiner der Gedanke Bahn gebrochen dass diese Wissenschaft, die
bis dahin nur auf die Gegenwart, auf die Erkenntniss der bestehenden
Verhaeltnisse und die in ihnen sichtbaren Gesetze den Blick gerichtet
hatte, auch in die Vergangenheit, in die Erforschung der bereits
hinter uns liegenden wirthschaftlichen Entwicklung der Voelker sich
vertiefen muesse.--SCHOeNBERG, _Jahrbuecher f. Nationaloekonomie und
Statistik_, Neue Folge, 1867, i. 1. Schmoller, moins dogmatique et
mettant comme une sorte de coquetterie a etre incertain, demontre, par
les faits, la faussete ou l'arbitraire de tous ces postulats, et
laisse l'economie politique se dissoudre dans l'histoire.--BRETON, _R.
de Paris_, ix. 67. Wer die politische Oekonomie Feuerlands unter
dieselben Gesetze bringen wollte mit der des heutigen Englands, wuerde
damit augenscheinlich nichts zu Tage foerdern als den allerbanalsten
Gemeinplatz. Die politische Oekonomie ist somit wesentlich eine
historische Wissenschaft. Sie behandelt einen geschichtlichen, das
heisst einen stets wechselnden Stoff. Sie untersucht zunaechst die
besondern Gesetze jeder einzelnen Entwicklungsstufe der Produktion und
des Austausches, und wird erst am Schluss dieser Untersuchung die
wenigen, fuer Produktion und Austausch ueberhaupt geltenden, ganz
allgemeinen Gesetze aufstellen koennen.--ENGELS, _Duehrings Umwaelzung der
Wissenschaft_, 1878, 121.
[79] History preserves the student from being led astray by a
too servile adherence to any system.--WOLOWSKI. No system can be
anything more than a history, not in the order of impression, but in
the order of arrangement by analogy.--DAVY, _Memoirs_, 68. Avec des
materiaux si nombreux et si importants, il fallait bien du courage
pour resister a la tentation de faire un systeme. De Saussure eut ce
courage, et nous en ferons le dernier trait et le trait principal de
son eloge.--CUVIER, _Eloge de Saussure_, 1810.
[80] C'etait, en 1804, une idee heureuse et nouvelle,
d'appeler l'histoire au secours de la science, d'interroger les deux
grandes ecoles rivales au profit de la verite.--COUSIN, _Fragments
Litteraires_, 1843, 95, on Degerando. No branch of philosophical
doctrine, indeed, can be fairly investigated or apprehended apart from
its history. All our systems of politics, morals, and metaphysics
would be different if we knew exactly how they grew up, and what
transformations they have undergone; if we knew, in short, the true
history of human ideas.--CLIFFE LESLIE, _Essays in Political and Moral
Philosophy_, 1879, 149. The history of philosophy must be rational and
philosophic. It must be philosophy itself, with all its elements, in
all their relations, and under all their laws represented in striking
characters by the hands of time and of history, in the manifested
progress of the human mind.--SIR WILLIAM HAMILTON, _Edin. Rev._ l.
200, 1829. Il n'est point d'etude plus instructive, plus utile que
l'etude de l'histoire de la philosophie; car on y apprend a se
desabuser des philosophes, et l'on y desapprend la fausse science de
leurs systemes.--ROYER COLLARD, _OEuvres de Reid_, iv. 426. On ne
peut guere echapper a la conviction que toutes les solutions des
questions philosophiques n'aient ete developpees ou indiquees avant le
commencement du dix-neuvieme siecle, et que par consequent il ne soit
tres difficile, pour ne pas dire impossible, de tomber, en pareille
matiere, sur une idee neuve de quelque importance. Or si cette
conviction est fondee, il s'ensuit que la science est faite.--JOUFFROY,
in DAMIRON, _Philosophie du XIXe Siecle_, 363. Le but dernier de tous
mes efforts, l'ame de mes ecrits et de tout mon enseignement, c'est
l'identite de la philosophie et de son histoire.--COUSIN, _Cours de
1829_. Ma route est historique, il est vrai, mais mon but est
dogmatique; je tends a une theorie, et cette theorie je la demande a
l'histoire.--COUSIN, _Ph. du XVIIIe Siecle_, 15. L'histoire de la
philosophie est contrainte d'emprunter d'abord a la philosophie la
lumiere qu'elle doit lui rendre un jour avec usure.--COUSIN, _Du Vrai_,
1855, 14. M. Cousin, durant tout son professorat de 1816 a 1829, a
pense que l'histoire de la philosophie etait la source de la
philosophie meme. Nous ne croyons pas exagerer en lui pretant cette
opinion.--B. ST. HILAIRE, _Victor Cousin_, i. 302. Il se hata de
convertir le fait en loi, et proclama que la philosophie, etant
identique a son histoire, ne pouvait avoir une loi differente, et
etait vouee a jamais a l'evolution fatale des quatre systemes, se
contredisant toujours, mais se limitant, et se moderant, par cela meme
de maniere a maintenir l'equilibre, sinon l'harmonie de la pensee
humaine.--VACHEROT, _Revue des Deux Mondes_, 1868, iii. 957. Er hat
ueberhaupt das unvergaengliche Verdienst, zuerst in Frankreich zu der
Erkenntniss gelangt zu sein, dass die menschliche Vernunft nur durch
das Studium des Gesetzes ihrer Entwickelungen begriffen werden
kann.--LAUSER, _Unsere Zeit_, 1868, i. 459. Le philosophe en quete du
vrai en soi, n'est plus reduit a ses conceptions individuelles; il est
riche du tresor amasse par l'humanite.--BOUTROUX, _Revue Politique_,
xxxvii. 802. L'histoire, je veux dire l'histoire de l'esprit humain,
est en ce sens la vraie philosophie de notre temps.--RENAN, _Etudes de
Morale_, 83. Die Philosophie wurde eine hoechst bedeutende
Huelfswissenschaft der Geschichte, sie hat ihre Richtung auf das
Allgemeine gefoerdert, ihren Blick fuer dasselbe geschaerft, und sie,
wenigstens durch ihre Vermittlung, mit Gesichtspuncten, Ideen,
bereichert die sie aus ihrem eigenen Schoosse sobald noch nicht
erzeugt haben wuerde. Weit die fruchtbarste darunter war die aus der
Naturwissenschaft geschoepfte Idee des organischen Lebens, dieselbe auf
der die neueste Philosophie selbst beruht. Die seit zwei bis drei
Jahrzehnten in der Behandlung der Geschichte eingetretene
durchgreifende Veraenderung, wie die voellige Umgestaltung so mancher
anderen Wissenschaft ... ist der Hauptsache nach ihr Werk.--HAUG,
_Allgemeine Geschichte_, 1841, i. 22. Eine Geschichte der Philosophie
in eigentlichen Sinne wurde erst moeglich als man an die Stelle der
Philosophen deren Systeme setzte, den inneren Zusammenhang zwischen
diesen feststellte und--wie Dilthey sagt--mitten in Wechsel der
Philosophien ein siegreiches Fortschreiten zur Wahrheit nachwies. Die
Gesammtheit der Philosophie stellt sich also dar als eine
geschichtliche Einheit.--SAUL, _Rundschau_, Feb. 1894, 307. Warum die
Philosophie eine Geschichte habe und haben muesse, blieb uneroertert, ja
ungeahnt, dass die Philosophie am meisten von allen Wissenschaften
historisch sei, denn man hatte in der Geschichte den Begriff der
Entwicklung nicht entdeckt.--MARBACH, _Griechische Philosophie_, 15.
Was bei oberflaechlicher Betrachtung nur ein Gewirre einzelner Personen
und Meinungen zu sein schien, zeigt sich bei genauerer und gruendlicherer
Untersuchung als eine geschichtliche Entwicklung, in der alles, bald
naeher, bald entfernter, mit allem anderen zusammenhaengt.--ZELLER,
_Rundschau_, Feb. 1894, 307. Nur die Philosophie, die an die
geschichtliche Entwickelung anknuepft kann auf bleibenden Erfolg auch
fuer die Zukunft rechnen und fortschreiten zu dem, was in der
bisherigen philosophischen Entwickelung nur erst unvollkommen erreicht
oder angestrebt worden ist. Kann sich doch die Philosophie ueberhaupt
und insbesondere die Metaphysik ihrer eigenen geschichtlichen
Entwickelung nicht entschlagen, sondern hat eine Geschichte der
Philosophie als eigene und zwar zugleich historische und spekulative
Disziplin, in deren geschichtlichen Entwickelungsphasen und
geschichtlich aufeinanderfolgenden Systemen der Philosophen die neuere
Spekulation seit Schelling and Hegel zugleich die Philosophie selbst
als ein die verschiedenen geschichtlichen Systeme umfassendes ganzes
in seiner dialektischen Gliederung erkannt hat.--GLOATZ, _Spekulative
Theologie_, i. 23. Die heutige Philosophie fuehrt uns auf einen
Standpunkt von dem aus die philosophische Idee als das innere Wesen
der Geschichte selbst erscheint. So trat an die Stelle einer abstrakt
philosophischen Richtung, welche das Geschichtliche verneinte, eine
abstrakt geschichtliche Richtung welche das Philosophische
verlaeugnete. Beide Richtungen sind als ueberschrittene und besiegte zu
betrachten.--BERNER, _Strafrecht_, 75. Die Geschichte der Philosophie
hat uns fast schon die Wissenschaft der Philosophie selbst
ersetzt.--HERMANN, _Phil. Monatshefte_, ii. 198, 1889.
[81] Le siecle actuel sera principalement caracterise par
l'irrevocable preponderance de l'histoire, en philosophie, en
politique, et meme en poesie.--COMTE, _Politique Positive_, iii. 1.
[82] The historical or comparative method has revolutionized
not only the sciences of law, mythology, and language, of anthropology
and sociology, but it has forced its way even into the domain of
philosophy and natural science. For what is the theory of evolution
itself, with all its far-reaching consequences, but the achievement of
the historical method?--PROTHERO, _Inaugural_. _National Review_,
_Dec._ 1894, 461. To facilitate the advancement of all the branches of
useful science, two things seem to be principally requisite. The first
is, an historical account of their rise, progress, and present state.
Without the former of these helps, a person every way qualified for
extending the bounds of science labours under great disadvantages;
wanting the lights which have been struck out by others, and
perpetually running the risk of losing his labour, and finding himself
anticipated.--PRIESTLEY, _History of Vision_, 1772, i. Pref. i.
Cuvier se proposait de montrer l'enchainement scientifique des
decouvertes, leurs relations avec les grands evenements historiques,
et leur influence sur les progres et le developpement de la
civilisation.--DARESTE, _Biographie Generale_, xii. 685. Dans ses
eloquentes lecons, l'histoire des sciences est devenue l'histoire meme
de l'esprit humain; car, remontant aux causes de leurs progres et de
leurs erreurs, c'est toujours dans les bonnes ou mauvaises routes
suivies par l'esprit humain, qu'il trouve ces causes.--FLOURENS,
_Eloge de Cuvier_, xxxi. Wie keine fortlaufende Entwickelungsreihe von
nur Einem Punkte aus vollkommen aufzufassen ist, so wird auch keine
lebendige Wissenschaft nur aus der Gegenwart begriffen werden
koennen.--Deswegen ist aber eine solche Darstellung doch noch nicht der
gesammten Wissenschaft adaequat, und sie birgt, wenn sie damit
verwechselt wird, starke Gefahren der Einseitigkeit, des Dogmatismus
und damit der Stagnation in sich. Diesen Gefahren kann wirksam nur
begegnet werden durch die verstaendige Betrachtung der Geschichte der
Wissenschaften, welche diese selbst in stetem Flusse zeigt und die
Tendenz ihres Fortschreitens in offenbarer und sicherer Weise
klarlegt.--ROSENBERGER, _Geschichte der Physik_, iii., p. vi. Die
Continuitaet in der Ausbildung aller Auffassungen tritt um so
deutlicher hervor, je vollstaendiger man sich damit, wie sie zu
verschiedenen Zeiten waren, vertraut macht.--KOPP, _Entwickelung der
Chemie_, 814.
[83] Die Geschichte und die Politik sind Ein und derselbe
Janus mit dem Doppelgesicht, das in der Geschichte in die
Vergangenheit, in der Politik in die Zukunft hinschaut.--GUeGLER'S
_Leben_, ii. 59.
[84] The papers inclosed, which give an account of the
killing of two men in the county of Londonderry; if they prove to be
Tories, 'tis very well they are gone.--I think it will not only be
necessary to grant those a pardon who killed them, but also that they
have some reward for their own and others' encouragement.--ESSEX,
_Letters_, 10, _Jan._ 10, 1675. The author of this happened to be
present. There was a meeting of some honest people in the city, upon
the occasion of the discovery of some attempt to stifle the evidence
of the witnesses.--Bedloe said he had letters from Ireland, that there
were some Tories to be brought over hither, who were privately to
murder Dr. Oates and the said Bedloe. The doctor, whose zeal was very
hot, could never after this hear any man talk against the plot, or
against the witnesses, but he thought he was one of these Tories, and
called almost every man a Tory that opposed him in discourse; till at
last the word Tory became popular.--DEFOE, _Edinburgh Review_, l.
403.
[85] La Espana sera el primer pueblo en donde se encendera
esta guerra patriotica que solo puede libertar a Europa.--Hemos oido
esto en Inglaterra a varios de los que estaban alli presentes. Muchas
veces ha oido lo mismo al duque de Wellington el general Don Miguel de
Alava, y dicho duque refirio el suceso en una comida diplomatica que
dio en Paris el duque de Richelieu en 1816.--TORENO, _Historia del
Levantamiento de Espana_, 1838, i. 508.
[86] Nunquam propter auctoritatem illorum, quamvis magni sint
nominis (supponimus scilicet semper nos cum eo agere qui scientiam
historicam vult consequi), sententias quas secuti sunt ipse tamquam
certas admittet, sed solummodo ob vim testimoniorum et argumentorum
quibus eas confirmarunt.--DE SMEDT, _Introductio ad historiam critice
tractandam_, 1866, i. 5.
[87] Hundert schwere Verbrechen wiegen nicht so schwer in der
Schale der Unsittlichkeit, als ein unsittliches Princip.--_Hallische
Jahrbuecher_, 1839, 308. Il faut fletrir les crimes; mais il faut
aussi, et surtout, fletrir les doctrines et les systemes qui tendent a
les justifier.--MORTIMER TERNAUX, _Histoire de la Terreur_.
[88] We see how good and evil mingle in the best of men and
in the best of causes; we learn to see with patience the men whom we
like best often in the wrong, and the repulsive men often in the
right; we learn to bear with patience the knowledge that the cause
which we love best has suffered, from the awkwardness of its
defenders, so great disparagement, as in strict equity to justify the
men who were assaulting it.--STUBBS, _Seventeen Lectures_, 97.
[89] Caeteris paribus, on trouvera tousjours que ceux qui ont plus de
puissance sont sujets a pecher davantage; et il n'y a point de theoreme
de geometrie qui soit plus asseure que cette proposition.--LEIBNIZ,
1688, ed. Rommel, ii. 197. Il y a toujours eu de la malignite dans la
grandeur, et de l'opposition a l'esprit de l'Evangile; mais maintenant
il y en a plus que jamais, et il semble que comme le monde va a sa
fin, celui qui est dans l'elevation fait tous ses efforts pour dominer
avec plus de tyrannie, et pour etouffer les maximes du Christianisme et
le regne de Jesus-Christ, voiant qu'il s'approche.--GODEAU, _Lettres_,
423, March 27, 1667. There is, in fact, an unconquerable tendency in
all power, save that of knowledge, acting by and through knowledge, to
injure the mind of him by whom that power is exercised.--WORDSWORTH,
June 22, 1817. _Letters of Lake Poets_, 369.
[90] I cieli han messo sulla terra due giudici delle umane
azioni, la coscienza e la storia.--COLLETTA. Wenn gerade die edelsten
Maenner um des Nachruhmes willen gearbeitet haben, so soll die
Geschichte ihre Belohnung sein, sie auch die Strafe fuer die
Schlechten.--LASAULX, _Philosophie der Kuenste_, 211. Pour juger ce qui
est bon et juste dans la vie actuelle ou passee, il faut posseder un
criterium, qui ne soit pas tire du passe ou du present, mais de la
nature humaine.--AHRENS, _Cours de Droit Naturel_, i. 67.
[91] L'homme de notre temps! La conscience moderne! Voila
encore de ces termes qui nous ramenent la pretendue philosophie de
l'histoire et la doctrine du progres, quand il s'agit de la justice,
c'est-a-dire de la conscience pure et de l'homme rationnel, que
d'autres siecles encore que le notre ont connu.--RENOUVIER, _Crit.
Phil._ 1873, ii. 55.
[92] Il faut pardonner aux grands hommes le marchepied de
leur grandeur.--COUSIN, in J. SIMON, _Nos Hommes d'Etat_, 1887, 55.
L'esprit du XVIIIe siecle n'a pas besoin d'apologie: l'apologie d'un
siecle est dans son existence.--COUSIN, _Fragments_, iii. 1826.
Suspendus aux levres eloquentes de M. Cousin, nous l'entendimes
s'ecrier que la meilleure cause l'emportait toujours, que c'etait la
loi de l'histoire, le rhythme immuable du progres.--GASPARIN, _La
Liberte Morale_, ii. 63. Cousin verurtheilen heisst darum nichts
Anderes als jenen Geist historischer Betrachtung verdammen, durch
welchen das 19 Jahrhundert die revolutionaere Kritik des 18
Jahrhunderts ergaenzt, durch welchen insbesondere Deutschland die
geistigen Wohlthaten vergolten hat, welche es im Zeitalter der
Aufklaerung von seinen westlichen Nachbarn empfangen.--IODL, _Gesch.
der Ethik_, ii. 295. Der Gang der Weltgeschichte steht ausserhalb der
Tugend, des Lasters, und der Gerechtigkeit.--HEGEL, _Werke_, viii.
425. Die Vermischung des Zufaelligen im Individuum mit dem an ihm
Historischen fuehrt zu unzaehligen falschen Ansichten und Urtheilen.
Hierzu gehoert namentlich alles Absprechen ueber die moralische
Tuechtigkeit der Individuen, und die Verwunderung, welche his zur
Verzweiflung an goettlicher Gerechtigkeit sich steigert, dass
historisch grosse Individuen moralisch nichtswuerdig erscheinen koennen.
Die moralische Tuechtigkeit besteht in der Unterordnung alles dessen
was zufaellig am Einzelnen unter das an ihm dem Allgemeinen
Angehoerige.--MARBACH, _Geschichte der Griechischen Philosophie_, 7.
Das Sittliche der Neuseelaender, der Mexikaner ist vielmehr ebenso
sittlich, wie das der Griechen, der Roemer; und das Sittliche der
Christen des Mittelalters ist ebenso sittlich, wie das der
Gegenwart.--KIRCHMANN, _Grundbegriffe des Rechts_, 194. Die
Geschichtswissenschaft als solche kennt nur ein zeitliches und mithin
auch nur ein relatives Maass der Dinge. Alle Werthbeurtheilung der
Geschichte kann daher nur relativ und aus zeitlichen Momenten
fliessen, und wer sich nicht selbst taeuschen und den Dingen nicht
Gewalt anthun will, muss ein fuer allemal in dieser Wissenschaft auf
absolute Werthe verzichten.--LORENZ, _Schlosser_, 80. Only according
to his faith is each man judged. Committed as this deed has been by a
pure-minded, pious youth, it is a beautiful sign of the time.--DE
WETTE to Sand's Mother, CHEYNE, _Founders of Criticism_, 44. The men
of each age must be judged by the ideal of their own age and country,
and not by the ideal of ours.--LECKY, _Value of History_, 50.
[93] La duree ici-bas, c'est le droit, c'est la sanction de
Dieu.--GUIRAUD, _Philosophie Catholique de l'Histoire_.
[94] Ceux qui ne sont pas contens de l'ordre des choses ne
scauroient se vanter d'aimer Dieu comme il faut.--Il faut toujours
estre content de l'ordre du passe, parce qu'il est conforme a la
volonte de Dieu absolue, qu'on connoit par l'evenement. Il faut tacher
de rendre l'avenir, autant qu'il depend de nous, conforme a la volonte
de Dieu presomptive.--LEIBNIZ, _Werke_, ed. Gerhardt, ii. 136. Ich
habe damals bekannt und bekenne jetzt, dass die politische Wahrheit
aus denselben Quellen zu schoepfen ist, wie alle anderen, aus dem
goettlichen Willen und dessen Kundgebung in der Geschichte des
Menschengeschlechts.--RADOWITZ, _Neue Gespraeche_, 65.
[95] A man is great as he contends best with the
circumstances of his age.--FROUDE, _Short Studies_ i. 388. La
persuasion que l'homme est avant tout une personne morale et libre, et
qu'ayant concu seul, dans sa conscience et devant Dieu, la regle de sa
conduite, il doit s'employer tout entier a l'appliquer en lui, hors de
lui, absolument, obstinement, inflexiblement, par une resistance
perpetuelle opposee aux autres; et par une contrainte perpetuelle
exercee sur soi, voila la grande idee anglaise.--TAINE; SOREL,
_Discours de Reception_, 24. In jeder Zeit des Christenthums hat es
einzelne Maenner gegeben, die ueber ihrer Zeit standen und von ihren
Gegensaetzen nicht beruehrt wurden.--BACHMANN, _Hengstenberg_, i. 160.
Eorum enim qui de iisdem rebus mecum aliquid ediderunt, aut solus
insanio ego, aut solus non insanio; tertium enim non est, nisi (quod
dicet forte aliquis) insaniamus omnes.--HOBBES, quoted by DE MORGAN,
June 3, 1858, _Life of Sir W. R. Hamilton_, iii. 552.
[96] I have now to exhibit a rare combination of good
qualities, and a steady perseverance in good conduct, which raised an
individual to be an object of admiration and love to all his
contemporaries, and have made him to be regarded by succeeding
generations as a model of public and private virtue.--The evidence
shows that upon this occasion he was not only under the influence of
the most vulgar credulity, but that he violated the plainest rules of
justice, and that he really was the murderer of two innocent
women.--Hale's motives were most laudable.--CAMPBELL'S _Lives of the
Chief Justices_, i. 512, 561, 566. It was not to be expected of the
colonists of New England that they should be the first to see through
a delusion which befooled the whole civilized world, and the gravest
and most knowing persons in it.--The people of New England believed
what the wisest men of the world believed at the end of the
seventeenth century.--PALFREY, _New England_, iv. 127, 129 (also
speaking of witchcraft). Il est donc bien etrange que sa severite
tardive s'exerce aujourd'hui sur un homme auquel elle n'a d'autre
reproche a faire que d'avoir trop bien servi l'etat par des mesures
politiques, injustes peut-etre, violentes, mais qui, en aucune
maniere, n'avaient l'interet personnel du coupable pour objet.--M.
Hastings peut sans doute paraitre reprehensible aux yeux des
etrangers, des particuliers meme, mais il est assez extraordinaire
qu'une nation usurpatrice d'une partie de l'Indostan veuille meler les
regles de la morale a celles d'une administration forcee, injuste et
violente par essence, et a laquelle il faudrait renoncer a jamais pour
etre consequent.--MALLET DU PAN, _Memories_, ed. Sayous, i. 102.
[97] On parle volontiers de la stabilite de la constitution
anglaise. La verite est que cette constitution est toujours en
mouvement et en oscillation et qu'elle se prete merveilleusement au
jeu de ses differentes parties. Sa solidite vient de sa souplesse;
elle plie et ne rompt pas.--BOUTMY, _Nouvelle Revue_, 1878, 49.
[98] This is not an age for a man to follow the strict
morality of better times, yet sure mankind is not yet so debased but
that there will ever be found some few men who will scorn to join
concert with the public voice when it is not well grounded.--_Savile
Correspondence_, 173.
[99] Cette proposition: L'homme est incomparablement plus
porte au mal qu'au bien, et il se fait dans le monde incomparablement
plus de mauvaises actions que de bonnes--est aussi certaine qu'aucun
principe de metaphysique. Il est donc incomparablement plus probable
qu'une action faite par un homme, est mauvaise, qu'il n'est probable
qu'elle soit bonne. Il est incomparablement plus probable que ces
secrets ressorts qui l'ont produite sont corrompus, qu'il n'est
probable qu'ils soient honnetes. Je vous avertis que je parle d'une
action qui n'est point mauvaise exterieurement.--BAYLE, _OEuvres_,
ii. 248.
[100] A Christian is bound by his very creed to suspect evil,
and cannot release himself.--His religion has brought evil to light in
a way in which it never was before; it has shown its depth, subtlety,
ubiquity; and a revelation, full of mercy on the one hand, is terrible
in its exposure of the world's real state on the other. The Gospel
fastens the sense of evil upon the mind; a Christian is enlightened,
hardened, sharpened, as to evil; he sees it where others do
not.--MOZLEY, _Essays_, i. 308. All satirists, of course, work in the
direction of Christian doctrine, by the support they give to the
doctrine of original sin, making a sort of meanness and badness a law
of society.--MOZLEY, _Letters_, 333. Les critiques, meme malveillants,
sont plus pres de la verite derniere que les admirateurs.--NISARD,
_Lit. fr._, Conclusion. Les hommes superieurs doivent necessairement
passer pour mechants. Ou les autres ne voient ni un defaut, ni un
ridicule, ni un vice, leur implacable oeil l'apercoit.--BARBEY
D'AUREVILLY, _Figaro_, March 31, 1888.
[101] Prenons garde de ne pas trop expliquer, pour ne pas
fournir des arguments a ceux qui veulent tout excuser.--BROGLIE,
_Reception de Sorel_, 46.
[102] The eternal truths and rights of things exist,
fortunately, independent of our thoughts or wishes, fixed as
mathematics, inherent in the nature of man and the world. They are no
more to be trifled with than gravitation.--FROUDE, _Inaugural Lecture
at St. Andrews_, 1869, 41. What have men to do with interests? There
is a right way and a wrong way. That is all we need think
about.--CARLYLE to FROUDE, _Longman's Magazine_, Dec. 1892, 151. As to
History, it is full of indirect but very effective moral teaching. It
is not only, as Bolingbroke called it, "Philosophy teaching by
examples," but it is morality teaching by examples.--It is essentially
the study which best helps the student to conceive large thoughts.--It
is impossible to overvalue the moral teaching of History.--FITCH,
_Lectures on Teaching_, 432. Judging from the past history of our
race, in ninety-nine cases out of a hundred, war is a folly and a
crime.--Where it is so, it is the saddest and the wildest of all
follies, and the most heinous of all crimes.--GREG, _Essays on
Political and Social Science_, 1853, i. 562. La volonte de tout un
peuple ne peut rendre juste ce qui est injuste: les representants
d'une nation n'ont pas le droit de faire ce que la nation n'a pas le
droit de faire elle-meme.--B. CONSTANT, _Principes de Politique_, i.
15.
[103] Think not that morality is ambulatory; that vices in
one age are not vices in another, or that virtues, which are under the
everlasting seal of right reason, may be stamped by opinion.--SIR
THOMAS BROWNE, _Works_, iv. 64.
[104] Osons croire qu'il seroit plus a propos de mettre de
cote ces traditions, ces usages, et ces coutumes souvent si
imparfaites, si contradictoires, si incoherentes, ou de ne les
consulter que pour saisir les inconveniens et les eviter; et qu'il
faudroit chercher non-seulement les elements d'une nouvelle
legislation, mais meme ses derniers details dans une etude approfondie
de la morale.--LETROSNE, _Reflexions sur la Legislation Criminelle_,
137. M. Renan appartient a cette famille d'esprits qui ne croient pas
en realite la raison, la conscience, le droit applicables a la
direction des societes humaines, et qui demandent a l'histoire, a la
tradition, non a la morale, les regles de la politique. Ces esprits
sont atteints de la maladie du siecle, le scepticisme moral.--PILLON,
_Critique Philosophique_, i. 49.
[105] The subject of modern history is of all others, to my
mind, the most interesting, inasmuch as it includes all questions of
the deepest interest relating not to human things only, but to
divine.--ARNOLD, _Modern History_, 311.
End of Project Gutenberg's A Lecture on the Study of History, by Lord Acton
***
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Most famous Companies from Lebanon
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Najjadeh Party
By the name 'the rescuers' or 'the helpers' (Arabic: حزب النجادة | An-Najjadah, Najjadah, Najjadeh or Najjada) is known a Lebanese nationalist party of Fascist trend that appeared in Lebanon during the 1930s. Lebanon in the 1930s witnessed the emergence of two paramilitary youth sport organizations of sectarian cast with clear fascist tendencies in Beirut and other Lebanese cities, the Lebanese Phalanges led by Pierre Gemayel and the Najjadah. The latter began its existence in 1933-34 as a Sunni Muslim boy-scouts organization founded and led by Muhi al-Din al-Nasuli, the editor of the influential pan-Arabist Muslim newspaper, "Beirut" (Arabic: Bayrut), with the purpose of protecting the Muslim community and to act as a counterweight to the Phalangists. A keen admirer of Adolf Hitler and Benito Mussolini – whom were viewed at the time in both Syria and Lebanon as role models of strong statebuilders – al-Nasuli's newspaper (among others) was involved since 1933 in publishing Hitler's speeches and excerpts from Mein Kampf. He often criticized the "moral chaos" in public life and adopted the supremacist motto "Arabism Above All" on his own newspaper's masthead. Al-Nasuli's Bayrut also
Ministry of Information
Future Movement
Future Movement (Arabic: تيار المستقبل, Tayyar Al-Mustaqbal) (FM) is a Lebanese political movement, led by MP Saad Hariri, the younger son of the assassinated former Prime Minister of Lebanon, Rafik Hariri. The movement is the largest member of the March 14 Alliance, which won a majority of the seats in the 2009 parliamentary elections. The Party was officially founded in August 2007, yet it was only declared on April 5, 2009 in a convention held at the BIEL convention center in Beirut. After the killing of Ahmad Abdel-Wahid, the Future Movement called on Prime Minister Najib Mikati to immediately resign, claiming his cabinet had shown incapability to maintain the country's security. Future Movement is part of the March 14 Alliance that includes, amongst many groups, the Christians associated with the Lebanese Forces and Kataeb parties (main 2 allies of FM). Most of its base is made up of Sunni Muslims. The main opponent of Future Movement is the March 8 Alliance, most important parts being Free Patriotic Movement (FPM) led by General Michel Aoun and the Shia Hezbollah and Amal Movements. There are also other opponents that are neither with March 8 Alliance nor March 14 Alliance
New Lebanese Movement
The New Lebanese Party {Hizb Al-Lubnaniyin Al-Judud) is a Lebanese political party established in 2006. Its main targets are changing the political system in Lebanon into a more stable, less dependent on confessionalism system and leading Lebanon to become a major regional power in terms of culture, economics & diplomacy.
Lebanese Option Party
Lebanese Option Gathering (LOG) (Arabic: تيار الإنتماء اللبناني French Courant de l'Option libanaise ) is a Lebanese predominantly Shia political movement established in 2007. It is headed by Ahmad al-Asaad the second (Arabic: أحمد الأسعد), the son of the former speaker of the Lebanese Parliament Kamel al-Asaad and strongly protests the political hegemony of the two movements Hezbollah and Amal on the Shi'ite community in Lebanon. It's platform is more in line with the Lebanese majority March 14 Alliance and greatly opposed to mainstream Shi'ite movements allied with the March 8 Alliance, namely Hezbollah and Amal Movements, but the Lebanese Option Gathering is not an official part of the March 14 Alliance and keeps an independent secular status.
Razkari Party
Established in 1975 by Faissal Fakhro, Riz Kari is a Lebanese Kurdish group dissatisfied with the leadership of the Kurdish Democratic Party (Lebanon). Riz Kari supported the Kurdish forces fighting against the Iraqi regime. For a brief period during the 1975 Civil War, however, Riz Kari joined forces with the Kurdish Democratic Party to form the Progressive Kurdish Front in an effort to eliminate differences in the ranks of Lebanese Kurds. Riz Kari was weakened in the mid-1970s by the defection of part of its organization, which called itself the Leftist Riz Kari, or Riz Kari II. This organization, led by Abdi Ibrahim, a staunch ally of Syria, rejected the formation of the Progressive Kurdish Front because it included the "right-wing" leadership of Jamil Meho.
Democratic Left Movement
The Democratic Left Movement (DLM, Arabic: حركة اليسار الديمقراطي Haraket Al-Yassar Al-Dimuqratiy, Arabic acronym HYD) is a leftist political party with seats in the Lebanese Parliament. It was founded in September 2004 by left-wing intellectuals and activists who previously split from the Lebanese Communist Party (LCP). The DLM affirms a European-style social democracy—but is open to all forms of leftism—and encourages the development of a secular state. The party operates under a decentralized framework that emphasizes diversity of thought. It participated in the 2005 Cedar Revolution, a wave of demonstrations against the Syrian occupation of Lebanon, and calls for correcting imbalanced relations with Syria. The DLM won its first parliamentary seat in Lebanon's 2005 elections representing the Tripoli district. On June 2, 2005, amid election rounds, Samir Kassir, a founder of the movement, was assassinated in a car bombing. Less than one month later, George Hawi, a former secretary general of the Lebanese Communist Party and an ally of the DLM, was killed in a similar car bombing in Beirut. In the 2009 elections, the party again won a single seat, instead representing the West
Marada Movement
The Marada Movement (Arabic: تيار المردة | Tayyar Al-Marada) is a Lebanese political party and a former militia active during the Lebanese civil war, named after the legendary Syriac Marada or Mardaites warriors of the early Middle Ages. Designated the Marada Brigade (Arabic: Katibatun al-Marada), they were the personal militia of Suleiman Franjieh, president of Lebanon at the outbreak of the war, otherwise known as the Zgharta Liberation Army – ZLA (Arabic: Zgharta Jayish al-Tahrir) or Armée de Liberation de Zgharta (ALZ) in French, after Franjieh's home town of Zgharta in northern Lebanon. El Marada: The modernity of heritage The evolving legacy Clarity of purpose Firm attitude Pride, glory, potency, depth The Sword: Symbol of justice Lighting: Creativity and sharpness Red Color: Symbol of Sacrifice Green Color: Cedar of Lebanon Blue Color: Blue Horizon Pi: Unity of purpose, Depth in justice, Core values, Perseverance through adversity, Resilient stands, Circle: Unshakable loyalty, Evolving dynamism, Genuine relations, Eternity of being, Compass: Right direction, Clear decision, Safety value, Genuine legacy Green Color: Eternity of life, Versatility of nature, Promise of
Shuraya Party
Shuraya party (Syriac: ܫܘܖܝܐ) is an Assyrian political organisation established on July 25, 1978 in Lebanon, when the country was in the middle of its civil war. It is composed of "free philosophers" of different Eastern Church origin, who see themselves as Assyrians. Shuraya insists it is not bound by any Church, but resists against the reproach to be anti-Christian. It stress out to represent all oriental Christians and to maintain a coined Christian Lebanon. Since its foundation, it has worked closely together with the founder and leader of the Lebanese Forces militia party Bachir Gemayel, in which it see itself connected in the struggle and destiny for all Lebanese Christians. A party established its own magazine in Lebanon in the 1980s, dubbed "Shuraya". The party also opened its own radio show named my voice, to the wave of FM-radio. Shuraya supports a formation of an Assyrian State in the Assyrian homeland, in what is today Northern Iraq.
Party of Socialist Revolution
The Party of Socialist Revolution (Arabic: حزب الثورة الإشتراكية | Hizb al-Thawra al-Ishtirakiyya) was a communist party in Lebanon, emerging as a seemingly pro-Chinese split from the Lebanese Communist Party. The party was founded in 1964. The formation of the party was announced in early September 1964, with Youssef Moubarek as party chairman and Moustafa Chaker as general secretary. The group published Ila amam ('Forward'). The group did not fully support the line of Mao Zedong, but notably did not support the Soviet Union in their dispute with China. The group argued that the mainstream of the Lebanese Communist Party had become too supportive of Nasser, and accused Khalid Bakdash of 'revisionism'. The group argued that solely through people's war could Israel be defeated. Their best-known recruit was the Syrian General Afif al-Bizri.
Christian Democratic Union
The Christian Democratic Union of Lebanon is a right wing party currently headed by Lebanese MP Neemtallah Abi Nasr. and part of March 8 Alliance
Lebanese National Bloc
Lebanese National Bloc (Arabic: الكتلة الوطنية اللبنانية; al-Kutlah al-wataniyyah al-lubnaniyah), is a Lebanese conservative political party founded in 1936. The party's founder, Émile Eddé became president the same year under French protectorate. His son, Raymond Eddé, succeeded him as head of the party. In 1968, the National Bloc joined the Helf Alliance, a coalition which included former President Camille Chamoun's National Liberal Party and Pierre Gemayel's Kataeb Party. The alliance was relatively successful in Parliamentary elections held the same year, winning 30 seats (out of 99). Nevertheless, in 1969, the Bloc left the alliance after the Cairo Agreement. During the Lebanese civil war, although mainly Christian, the party refused to rejoin the Lebanese Front, formed by his former allies. During the war years, the party refused to arm a militia and kept a moderate line consisting mainly in defending the independence and unity of Lebanon aligning itself with the positions of the Sunni bourgeoisie, represented by Rashid Karami and Saeb Salam. The exile (in 1976) and the death (in 2000) of its leader, combined with the rise of the Lebanese Forces and Aounist current,
National Liberal Party
The National Liberal Party (NLP, Arabic: حزب الوطنيين الأحرار, literally Hizb Al-Waṭaniyīn Al-Aḥrār) is a center-right political party in Lebanon, established by President Camille Chamoun in 1958. It is now under the leadership of Dory Chamoun, his son. The party has adopted a hard line in regard to the preservation of Lebanese independence, and to the safeguard of the distinctive liberal practices in Lebanon with respect to freedom of expression and opinion and religious freedoms. Like most Lebanese political organization, it has a sectarian basis; the NLP is mainly supported by Christians. (For more information on this, see Demographics of Lebanon) In 1968, the party joined The Helf Alliance formed with the two other big mainly Christian parties in Lebanon: the Kataeb of Pierre Gemayel, and National Bloc of Raymond Eddé. During the Lebanese Civil War of 1975-90, the NLP was aligned with the mainly Maronite Christian alliance who fought the Lebanese National Movement (LNM). It had its own armed militia, the Tigers. In 1976, the NLP joined with the Kataeb Party (Phalange) and the Lebanese Renewal Party (LRP) to form the Lebanese Front, a political coalition. This was paralleled by
Arab Democratic Party
The Arab Democratic Party – ADP (Arabic: الحزب العربي الديمقراطي | al-Hizb al-'Arabi al-Dimuqrati) or Parti Démocratique Arabe (PDA) in French, is a Lebanese party, based in Tripoli. Its current leader is Rifaat Eid. The ADP traced back its origins to an earlier leftist students' organization called the Alawite Youth Movement – AYM (Arabic: حركة الشباب العلوي | Harakat al-Shabab al-Alawiyya) or Mouvement de la Jeunesse Alaouite (MJA) in French, originally formed in 1972 at Tripoli by Ali Eid, a chemistry teacher. As its name implies, the AYM drew its support from the Shia Alawite sect minority of Lebanon, even receiving the personal backing of Rifa'at al-Assad, Syria's vice-president at the time and himself a member of that sect. During the early war years, the AYM kept itself outside the LNM-PLO alliance, but in 1977-78 the movement joined the Patriotic Opposition Front – POF, a pro-Syrian multiconfessional coalition of Lebanese notables and activists founded in Tripoli by the MP Talal El-Merhebi (elected in 1972), Souhale Hamadah, Rashid Al-Muadim, George Mourani, and Nassib Al-Khatib, with Ali Eid being elected vice-president of the new formation. However, internal
Toilers League
The Toilers League (Arabic: رابطة الشغيلة| Rabitat al-Shaghila) or Ligue des Travailleurs (LT) in French, is a Lebanese left-wing political party founded in Lebanon at the late 1960s and currently led by former Shouf MP Zaher el-Khatib. The Toilers League originated from a previous socialist students association formed at the American University of Beirut (AUB) in 1968 by the then student activist and Progressive Socialist Party (PSP) militant Zaher el-Khatib. In 1974 the group broke away from the PSP and re-emerged as a separated political party under Khatib's leadership, who succeeded to be elected to the Lebanese Parliament as the socialist deputy for the Iqlim al-Kharrub district of the Shouf. Marxist-Leninist and Pan-Arab nationalist in ideology, the League joined Kamal Jumblatt's Lebanese National Movement (LNM) in early 1975, even raising a militia named the Zafer el-Khatib Forces – ZKF (Arabic: Al-Quwwat Zafer el-Khatib), also known as Les Forces de Zafer el-Khatib (FZK) in French. After the collapse of the LNM alliance in 1982, the WL/ZKF switched their alligence to Syria and established a close relationship with the Shia Amal Movement. During the 1975-76 war the ZKF's
Arab Socialist Union
The Arab Socialist Union (Arabic: الاتحاد الاشتراكى العربى, Al-Ittiḥād Al-Ištirākī Al-ʿArabī) was an Egyptian political party based on the principles of Nasserist Arab socialism. The Arab Socialist Union was founded in Egypt in December 1962 by Gamal Abdel Nasser as the country's sole political party. The ASU grew out of the Free Officers Movement of the Egyptian Revolution of 1952. The party's formation was just one part in Nasser's National Charter. The Charter set out an agenda of nationalisation, agrarian reform, and constitutional reform, which formed the basis of ASU policy. The programme of nationalisation under Nasser saw seven billion Egyptian pounds of private assets transferred into the public sector. Banks, insurance companies, many large shipping companies, major heavy industries and major basic industries were converted to public control. Land reforms saw the maximum area of private land ownership successively reduced from 200 to 100 feddans. A 90% top rate of income tax was levied on income over ten thousand Egyptian pounds. Boards of directors were required to have a minimum number of workers, and workers and peasants were guaranteed at least half of the seats in
Green Party of Lebanon
The Green Party of Lebanon (Arabic: حزب الخضر اللبناني ĥizb-al-khodor-al-lubnanī) is a Lebanese green political party. Founded in August 2008 by Philip Skaf, the green party advocates environmental protection, sustainable development, and human rights in Lebanon. It is the first Lebanese party to focus primarily on Green politics. Environmental concerns in Lebanon have been overshadowed by the sectarian nepotist political system and consigned to the bottom of the political agenda. This is despite the fact that Lebanon's tourism industry - a key part of its economy - relies heavily on the green spaces and woodlands that distinguish Lebanon from neighboring countries. The Green Party of Lebanon was founded on 20 August 2008 at a conference in Beirut's Monroe Hotel. The three hour conference, which assembled 65 members of the Lebanese elite, presented the party's charter and political and economic work plan followed by the election of its 20 member executive board. The executive board elected Philip Skaf, chief executive and creative officer at Grey Worldwide MENA, an advertising agency, as party president. In March 2009, the Green Party declared a state of environmental emergency
Syriac Union of Lebanon
Syrian–Lebanese Communist Party
Syrian-Lebanese Communist Party (Arabic: الحزب الشيوعي السوري اللبناني, al-hizb al-shuyū'ī al-sūrī al-lubnānī), a communist political party operating in Syria and Lebanon founded in 1924 by the Lebanese Egyptian Fu'ad al-Shimali, the Lebanese Yusuf Yazbek and the Armenian Artin Madoyan. Its general secretary was Khalid Bakdash. The party was represented at the 6th Congress of the Communist International in 1928 by Fu'ad al-Shimali. Under the French Mandate it was an underground organization, then legalized in 1936–1939 by the French Front Populaire government, and again in 1941. The party took a new option of collaboration with the nationalist movement and playing down its socialist themes in 1936, in accordance with the 7th Congress of the Communist International in 1935. Later the party was divided into the Syrian Communist Party and the Lebanese Communist Party, but the decision, taken at the end of 1943, was only implemented in 1964. In between, common central committee and political bureau were maintained.
Deprived Movement
The Movement of the Deprived (harakat al-mahrūmin) (Arabic:حركة المحرومين) was founded in 1969 By the Imam Musa al-Sadr. It called upon peace between all Lebanese confessions and religions. The movement aimed at having no more "Deprived" people or regions in Lebanon. It had support from all confessions, but mainly the Shia confession. The movement was absorbed in 1975 into was it now called Amal movement.
Lebanese Communist Party
The Lebanese Communist Party – LCP (Arabic: الـحـزب الشـيـوعـي اللبـنـانـي | Hizb ash-shuy'uī al-lubnānī) or Parti communiste libanais (PCL) in French, is a communist political party in Lebanon. It was founded in 1924 by the Lebanese intellectual, writer and reporter Youssef Ibrahim Yazbek and Fou'ad al-Shmeli, a tobacco worker from Bikfaya. The Lebanese Communist Party was officially founded on October 24, 1924, in the Lebanese town of Hadath, south of Beirut. The first meeting was made up of union workers, who formed independent unions for the first time in Lebanon (Previously, labor unions were controlled by the French). The meeting was also attended by scholars, academics, writers and journalists who were active in promoting the ideas of the French Revolution, and who were familiar with the writings of Karl Marx and Friedrich Engels. The party was founded to cover the area held under the French mandate, which is now Syria and Lebanon. Initially, the party's name was "Lebanese People Party", in an attempt to evade the French ban on "Bolshevik" activities. The party was declared illegal at first, but the ban was relaxed during World War II. For about twenty years, the LCP
Socialist Lebanon
Socialist Lebanon (Arabic: لبنان الاشتراكي) was a small Arab nationalist group in Lebanon. The group was formed in 1965 by people like Ahmed Beydoun, Waddah Sharara and Fawwaz Trabulsi. In 1970 the group merged with the Organization of Lebanese Socialists, and formed the Communist Action Organization in Lebanon.
Islamic Unification Movement
The Islamic Unification Movement – IUM (Arabic: حركة التوحيد الإسلامي | Harakat al-Tawhid al-Islami), also named Islamic Unity Movement or Mouvement de Unification Islamique (MUI) in French, but best known as 'Al-Tawhid', 'At-Tawhid', or 'Tawheed', is a Lebanese Sunni Muslim fundamentalist political party. It plays an active role in Lebanese internal politics since the Lebanese Civil War in the early 1980s. The IUM was founded in Tripoli in 1982 from a splinter faction of the Lebanese Islamic Group led by Sheikh Said Shaaban, one of Lebanon's Islamist movements' few charismatic Sunni religious leaders. A hardliner who believed that force was a good solution in politics, the radical Shaaban broke away from the Islamic Group soon after the June 1982 Israeli invasion of Lebanon, in protest for that Party's leadership decision of adopting a non-violent, moderate political line in the early 1980s. Nevertheless, the two organizations have always maintained a good relationship, especially with Sheikh Fathi Yakan, founder and Secretary-general of the Islamic Group. At the height of its power in 1985, the IUM splintered, when dissident leaders Khalil Akkawi and Kanaan Naji left the
Kurdish Democratic Party
Jamil Mihhu established the Kurdish Democratic Party in 1960, but it was not licensed until 1970. Mihhu, however, supported the Iraqi government against Kurdish rebels fighting in that country, and he was captured and imprisoned by the Kurdish resistance in Iraq. Consequently, the leadership of the party passed to Jamil's son, Riyad. Another son, Muhammad, disagreed with his family's position on several issues and therefore in 1977 started his own movement, the Kurdish Democratic Party--Temporary Leadership.
24 October Movement
The 24 October Movement (Harakat 24 Techrin) (Arabic: حركة ٢٤ تشرين الأول ديمقراطية الاشتراكية) is a leftist progressist party known as the 24 October socialist progressist movement in Lebanon, founded in the year of 1969 by Farouk el-Moukaddem. The party main goals was to represent the oppressed citizens that were constantly sinking into poverty, unable to face the continuous growth of inflation. On 19 December 1973, during a wide protest organized by the movement, gun clashes erupted between pro-governmental armed men and the student branch of the party known as "Al itihad el sawri" (Arabic:الاتحاد الثوري), where multiple injuries occurred. This incident prompted a major military retaliation led by Farouk el Moukaddem against all armed men responsible for the attack which forced the army to intervene and conduct a raid on the party headquarters located at Farouk Moukaddem street near Al tall street and arrest the prominent young leader with 18 of his companions. The response was a full scale closure of the Northern city of Tripoli in protest for the arrest which rapidly extended to become a major demonstration in the capital city of Lebanon, Beirut, to reach out the Southern
Ba'ath Party
The Arab Socialist Ba'ath Party (Arabic: حزب البعث العربي الاشتراكي Hizb Al-Ba'ath Al-'Arabi Al-Ishtiraki) was a political party founded in Syria by Michel Aflaq, Salah al-Din al-Bitar and associates of Zaki al-Arsuzi. The party espoused Ba'athism, an ideology mixing Arab nationalist, pan-Arabism, Arab socialist and anti-imperialist interests. Ba'athism calls for the renaissance or resurrection and unification of the Arab world into a single state. Its motto—"Unity, Liberty, Socialism" (wahda, hurriya, ishtirakiya)—refers to Arab unity, and freedom from non-Arab control and interference. The party was founded by the merger of the Arab Ba'ath Movement, led by Aflaq and al-Bitar, and the Arab Ba'ath, led by al-Arsuzi, on 7 April 1947 as the Arab Ba'ath Party. The party quickly established branches in other Arab countries, although it would only hold power in Iraq and Syria. The Arab Socialist Ba'ath Party was merged with the Arab Socialist Party led by Akram al-Hawrani in 1952 to form the Arab Socialist Ba'ath Party. The newly-formed party was a relative success, and became the second-largest party in the Syrian parliament in the 1954 parliamentary election. This, coupled with the
General Confederation of Lebanese Workers
The General Confederation of Lebanese Workers (CGTL) (in French Confédération Générale des Travailleurs Libanais (CGTL), in Arabic الإتحاد العمالي العام في لبنان) is a loosely organized national trade union center in Lebanon. It was founded in 1958, and has a membership of 200,000.
Syrian Social Nationalist Party
The Syrian Social Nationalist Party (SSNP) (Arabic: الحزب السوري القومي الاجتماعي, transliterated: al-Ḥizb as-Sūrī al-Qawmī al-'Ijtimāʕī, often referred to in French as Parti Populaire Syrien or Parti Social Nationaliste Syrien), is a secular nationalist political party operating in Lebanon, Syria and Jordan. It advocates the establishment of a Syrian nation state spanning the Fertile Crescent, including present day Syria, Lebanon, Iraq, Jordan, the Palestinian Territories, Israel, Cyprus, Kuwait, Sinai, southeastern Turkey and southwestern Iran. It is the largest political group in Syria after the Arab Socialist Ba'ath Party, with over 100,000 members. In Lebanon, it is part of the March 8 Alliance. Founded in Beirut in 1932 as a national liberation organization hostile to French colonialism, the party played a significant role in Lebanese politics and was involved in attempted coup d'etats in 1949 and 1961 following which it was thoroughly repressed. It was active in the resistance against the Israeli invasion of Lebanon from 1982 to 2000 while continuously supporting the Syrian presence in Lebanon. In Syria, the SSNP became a major political force in the early 1950s, but was
Four Mothers
Guardians of the Cedars
The Guardians of the Cedars – GoC (Arabic: حراس الأرز; Ḥurrās al-Arz), also designated Gardiens du Cedre or Gardiens des Cèdres (GdC) in French, are a far-right ultranationalist Lebanese party and former militia in Lebanon. It was formed by Étienne Saqr (also known with the kunya or nom de guerre "Abu Arz" or "Father of the Cedars") and others along with the Lebanese Renewal Party in the early 1970s. It operated in the Lebanese Civil War under the slogan: Lebanon, at your service. The Guardians of the Cedars started to form a militia in the years leading up to the Lebanese Civil War and commenced military operations in April 1975. In September 1975, Communiqué No. 1 was issued to denounce advocates of the partition of Lebanon. The second communiqué contained a bitter attack on the Palestinians. The third articulated the party's attitude on the issue of Lebanese identity: Lebanon should dissociate itself from Arabism. The party spread its messages by means of graffiti in East Beirut, including slogans against Syria, the "Palestinian Resistance", and Pan-Arabism, sometimes with violent anti-Palestinian tones, as in the slogan على كل لبناني ان يقتل فلسطينياً ("It is a duty for each
Communist Action Organization in Lebanon
The Communist Action Organization in Lebanon (Arabic: منظمة العمل الشيوعي في لبنان, munaẓẓamah al-'amal al-shuyū'ī fī lubnān, French: Organisation de l'Action Communiste du Liban, abbreviation OACL) is a Marxist-Leninist political party and former militia group in Lebanon. The OACL was one of Lebanon's few multi-sectarian parties, with Christian, Muslim and Druze members, but its main base was among Shi'a Muslims. OACL played a major role in the political radicalization of the Shi'a community during the 1970s. In the 1980s, it had a membership of about 2000. The OACL was formed around 1970 through the merger of the Organization of Lebanese Socialists and Socialist Lebanon. The Organization of Lebanese Socialists was led by Muhsin Ibrahim and Muhammed Kishli. It had its roots in the Lebanese branch of the Arab Nationalist Movement (ANM), a radical pan-Arab movement. During the 1960s Ibrahim was a leading figure in the leftist tendency with the ANM. This tendency, led by Naif Hawatmeh, argued that the ANM ought to adopt a Marxist outlook. This was opposed by the top ANM leader George Habash who, although being open to introducing Marxist concepts like anti-imperialism into the
Lebanese Forces
The Lebanese Forces (LF) (Arabic: القوات اللبنانية al-quwāt al-lubnāniyah, Syriac: ܚܝܠܘܬܐ ܠܒܢܢܝܐ ḥailaoṯe lebnonoye) may refer to The organization was created by the Gemayels, Camille Chamoun, and other party leaders during the Lebanese Civil War. It was initially a conglomerate of the various right-wing party militias, placed under the control of a council composed of various party representatives. The Kataeb Regulatory Forces provided the largest share of fighters and the Kataeb had the largest share on the council. Despite its original creation from party militias, the Lebanese Forces accepted new recruits without any specific party allegiance. The movement fought as the main militia within the Christian-dominated Lebanese Front. During the civil war, the LF fought different opponents at different times: The Palestinian Liberation Organization, the LNM, the LNRF, the Syrian Army, the Druze PSP in the Chouf, and the Lebanese Army loyal to General Aoun. In In the mid-1980s, political friction within the Lebanese Front resulted in growing distance between the Kataeb militants and the rest of the Lebanese Forces. In the end the Lebanese Forces and Kataeb became two separate forces
Democratic Socialist Party
The Democratic Socialist Party, is a small Lebanese party founded and led by former lebanese speaker of parliament Kamel Al-Assad. The party is hostile to Hezbollah and Amal movement, and enjoys most of its support from the Lebanese Shi'a community. However, support for it is tiny.
Democratic Renewal
The Democratic Renewal Movement (or Tajaddod) is a reformist, social liberal, secular political party in Lebanon. At the last legislative elections, in May and June 2005, the party was allied to the anti-Syrian March 14 Alliance, led by Future Movement of late Prime minister Rafic Hariri, that won these elections. The Democratic Renewal was founded in 2001 by a group of 50 Lebanese political figures, intellectuals and businessmen. It is headed by Nassib Lahoud(1944-2012), former Presidential aspirant, deputy of the Metn region from 1991 until 2005. The Democratic Renewal has one member in the Parliament, Misbah Ahdab of Tripoli in North Lebanon.
Independence Movement
The Independence Movement (Harakat Al-Istiqlal known also as Al Haraka) (Arabic:حركة اللإستقلال) is a neoconservative and secularist Lebanese political party based in Zgharta (Lebanon), founded in 2006 by Michel René Moawad, son of slain Lebanese President René Moawad and MP and former first lady Nayla Moawad. The movement is part of the anti-Syrian Qornet Shehwan Gathering and the March 14 Alliance. In the 2005-2009 it had 3 Maronite MPs for the Zgharta District in the Lebanese Parliament, Nayla Moawad, Jawad Simon Boulos and Samir Frangieh. Since 2009, the party has been led by Michel René Moawad, Jawad Simon Boulos and Youssef Bahaa El Douaihy.
Al-Ahbash
Al-Ahbash (Arabic: الأحباش / al-aḥbash / English: The Ethiopians), also known as the Association of Islamic Charitable Projects (Arabic: جمعية المشاريع الخيرية الإسلام / jam'iyyat al-mashari' al-khayriyya al-islamiyya) is a Sufi religious movement which was founded in the mid 1980s. The group follow the teachings of Ethiopian scholar Abdullah al-Harari. The Association of Islamic Charitable Projects was founded in the 1930s by Ahmad al-Ajuz, According to Gary Gambill the AICP arrived in the Lebanon in the 1950's were he says "they blended Sunni and Shi'a theology with Sufi spiritualism into a doctrinal eclecticism that preached nonviolence and political quietism". The AICP remained without a leader until the 1980s when Abdullah al-Harari became the nominal head of the organization. and was taken over by Al-Ahbash in 1983. Al-Ahbash was founded in the suburb of Bourj Abu Haidar in Beirut and from there spread throughout the Lebanon to Tripoli, Akkar and Iqlim Al-Kharrub where they founded educational and religious institutions. Beginning in the 90's Ahbash propelled from a minority group to the largest Sunni movement in Lebanon mainly due to Syrian government backing. In 1995
Liberty Front
After Syrian withdrawal, Fouad Abou Nader restarted his public activities in launching, with his former companions, the "Lebanese Forces veterans" group. He decided to return to the Kataeb Social Democratic Party in the hope of initiating the necessary changes to avoid the repetition of the mistakes of the past. These necessary changes were: making the party more democratic to avoid fratricidal struggles for power and redefining the Cause. Quickly, he clashed with the direction of the party who refused any change about the feudal, hereditary and therefore anti-democratic structures. Consequently, Fouad Abou Nader decided with his companions who come from the Kataeb Social Democratic Party but also from other parties and movements to launch the "Liberty Front" on April 2007. The Liberty Front is a Lebanese political movement, social-democratic & defender of Lebanon sovereignty, independence & freedom, heir of the "Front for Freedom & Man" founded in 1975 by Dr. Charles Malik who became the "Lebanese Front" and the political offspring of the Resistance of the Front's parties & movements fighters who cooperated together in the "Lebanese Forces Command Council" since 1976 before
Progressive Socialist Party
The Progressive Socialist Party or PSP (Arabic: الحزب التقدمي الاشتراكي, al-hizb al-taqadummi al-ishtiraki), also known as Parti Socialiste Progressiste in French, is a political party in Lebanon. Its current leader is Walid Jumblatt. It is ideologically secular and officially non-sectarian, but in practice is led and supported mostly by followers of the Druze faith. The party was founded on 5 January 1949, and registered on 17 March the same year, under notification N°789. The founders comprised six individuals, all of different backgrounds. The most notable of these was Kamal Jumblatt (Walid Jumblatt's father). The others were Farid Joubran, Albert Adeeb, Abdallah Alayli, Fouad Rizk, and George Hanna. The PSP held in Beirut the first conference for the Socialist Arab Parties in Lebanon, Syria, Egypt and Iraq in 1951. From 1951 through 1972 the party had between three and six deputies in parliament. Under Kamal Jumblatt's leadership, the PSP was a major element in the Lebanese National Movement (LNM) which supported Lebanon's Arab identity and sympathised with the Palestinians. Despite Jumblatt's initial reluctance to engage in paramilitarism, it built a powerful private army,
Lebanese Democratic Party
The Lebanese Democratic Party (Hizb al-democraty al-lubnany; Arabic:الحزب الديمقراطي اللبناني) is a political party in Lebanon established by Prince Talal Arslan in 2001. Prince Talal is the son of Druze leader L'Emir Magid Arslan and has presided the party ever since its establishment. The Lebanese Democratic Party is officially secular and has members from all Lebanese sects, but most of its support comes from the Druze, who support the Arslan family. It is part of the March 8 Alliance. The party was represented in the Lebanese parliament in 2000 and in 2009. Emir Talal Arslan won a seat in the Parliament representing the Aley district, and three others representing the Baabda district. In July 2011, Emir Talal appointed his brother-in-law Marwan Kheireddine as minister of state to represent him and the party in the newly formed cabinet.
Amal Movement
The Amal Movement (or Hope Movement in English) is a Lebanese political party associated with Lebanon's Shia community. It was founded as the "Movement of the Dispossessed" in 1974. The Amal Movement is, by a small margin, the largest Shia party in parliament, having thirteen representatives to Hezbollah's tweleve. Amal is currently in an alliance which includes the Aounists, Hezbollah, and the Progressive Socialist Party. The movement's current name was originally used by the Movement of the Dispossessed militia, the "Lebanese Resistance Regiments", Arabic: أفواج المقاومة اللبنانية. This name, when abbreviated, created the acronym "Amal", which means "Hope" in Arabic.. The Amal militia was founded in 1975 as the militant wing of the Movement of the Disinherited, a Shi'a political movement founded by Musa al-Sadr and Hussein el-Husseini a year earlier. It became one of the most important Shi'a Muslim militias during the Lebanese Civil War. Amal grew strong with the support of, and through its ties with, Syria and the 300,000 Shi'a internal refugees from southern Lebanon after the Israeli bombings in the early 1980s. Amal's practical objectives were to gain greater respect for
Free Patriotic Movement
The Free Patriotic Movement (FPM) (Arabic: التيار الوطني الحر, al-tayyar al-waṭani al-ħur), also known as the Aounist Movement (Arabic: التيار العوني, al-tayyar al-ʕaouni), is a Lebanese political party, led by Michel Aoun. It is the second largest party in Lebanon's parliament (after the Future Movement) and the largest party in the Christian half of the parliament. It has 18 out of the 128 seats in parliament (of which 64 seats represent Christians). The FPM is the main party of the March 8 Alliance, which includes Amal (13 seats), Hezbollah (12 seats), and the Progressive Socialist Party (7 seats), as well as seven other minor parliamentary parties (who between them have 16 seats). The FPM party promotes secularism, the rights of Lebanese expatriates, proportional representation, and a relatively high minimum wage. The party's support base is overwhelmingly from Lebanon's Christian community, but includes a small number of Shia Muslims. For many years, while Aoun was exiled in Paris, he led the FPM from abroad. He returned to Lebanon on May 7, 2005 after the Cedar Revolution forced the withdrawal of the Syrian forces, and then contested the legislative elections held in late May
Hezbollah (Arabic: حزب الله ḥizbu-llāh, literally "Party of God") is a Shi'a Muslim militant group and political party based in Lebanon. It receives financial and political support from Iran and Syria, and its paramilitary wing is regarded as a resistance movement throughout much of the Arab and Muslim worlds. The United States, the Netherlands, the United Kingdom, Australia, Canada and Israel classify Hezbollah as a terrorist organization, in whole or in part. Hezbollah first emerged in response to the 1982 Israeli invasion of Lebanon, during the Lebanese civil war. Its leaders were inspired by Ayatollah Khomeini, and its forces were trained and organized by a contingent of Iranian Revolutionary Guards. Hezbollah's 1985 manifesto listed its four main goals as "Israel's final departure from Lebanon as a prelude to its final obliteration," ending "any imperialist power in Lebanon," submission of the Phalangists to "just rule" and bringing them to trial for their crimes, and giving the people the chance to choose "with full freedom the system of government they want," while not hiding its commitment to the rule of Islam. Hezbollah leaders have also made numerous statements calling
Islamic Group
The Islamic Group (Arabic: الجماعة الإسلامية Al-Jama'ah Al-Islamiyah) is a Sunni Islamist group or gathering in Lebanon. Jamaa Islamiya was founded in 1952 as the Lebanese branch of the Muslim Brotherhood. Its leader is Faisal Mawlawi. The party has a military wing known as the al-Fajr Forces. Lately, it entered the Lebanese general election, 2009 beside Future Movement in Beirut 3 district. Currently they have one seat in the Lebanese Parliament.
Kataeb Party
The Lebanese Phalanges (Arabic: حزب الكتائب اللبنانية, Hizb Al-Kata'eb Al-Loubnaniyya), better known in English as the Phalange (Arabic: Kata'eb), is a traditional right-wing political-paramilitary organization. Although it is officially secular, it is mainly supported by Maronite Christians. The party played a major role in the Lebanese War (1975–90). In decline in the late 1980s and 1990s, the party slowly re-emerged since the early 2000s (decade). It is now part of the March 14 Alliance, opposed to the March 8 Alliance, led by Hezbollah, and the Free Patriotic Movement. The Lebanese Social Democratic Party is also known as Phalanges Libanaises in French and either Kataeb (الكتائب اللبنانية Al-Kata'eb Al-Lubnaniyya) or 'Phalangist Party' (Hezb al-Kata'eb al-Lubnaniyya) in Arabic. Kataeb is the plural of Katiba which is a translation into Arabic of the Greek word phalanx ("battalion") which is also the origin of the Spanish term Falange. The Kataeb party was Formed in 1936 as a Maronite paramilitary youth organization by Pierre Gemayel who modeled the party after Spanish Falange and Italian Fascist parties he had observed as an Olympic athlete during the 1936 Summer Olympics held
Popular Nasserist Organization
The Popular Nasserist Organization – PNO (Arabic: التنظيم الشعبي الناصري | Al-Tanzim al-Sha'aby al-Nassery) or Organisation Populaire Nassérienne (OPN) in French, is a Sidon-based Nasserist party originally formed in 1973 by Maarouf Saad, a Sunni Pan-Arab politician and member of Parliament (MP) later killed by the Lebanese Army during a February 1975 dock strike held in that port city. The PNO's military wing, the National Liberation Army – NLA (Arabic: Jayish al-Tahrir al-Watani) or Armée de Liberation Nationale (ALN) was first raised in March 1975 at Sidon by Mustafa Saad, son of the late Maarouf. Trained and armed by Fatah, the NLA was initially financed by Yasser Arafat's organization and Libya, later replaced in the mid-1980s by the Sidon-born Saudi-Lebanese millionaire Rafic Hariri. A small but disciplinated fighting force, the NLA comprised some 500-1000 uniformed Male and Female fighters organized into conventional 'Commando', Infantry, Signals, and Military Police branches. It fielded a 'mechanized' corps provided with a single UR-416 armoured car seized from the Lebanese Forces in 1985, plus 40 all-terrain vehicles (Land-Rover, Toyota Land Cruiser and GMC light pickups)
Al Badil Al Taharouri
Al Badil Al Taharouri (The Anarchist Alternative) is a Lebanese anarchist organization. It is linked to the French anarchist group Alternative libertaire. Another name commonly used is Al Badil Al Chouyouii Al Taharouri (The Anarcho-Communist Alternative) or Alternative Communiste Libertaire.
Nasserist Unionists Movement
The Nasserist Unionists Movement – NUM or Nasserite Unification Movement (Arabic: حركة الوحدويين الناصريين; Al-Harakat Al-Tawhidiya Al-Nassiriya) is a minor Lebanese political party headed by Samir Sabbagh. It was founded in 1982 out from a splinter faction of the INM/Al-Murabitun, originally under the label Movement of Unionist Nasserites – MUN (Arabic: Harakat al-Wihdawiyin al-Nasiriyin). The NUM aims to unify all Lebanese Nasserite parties under one leadership and is currently a member of the pro-Syrian March 8 Alliance.
Organization of Lebanese Socialists
The Organization of Lebanese Socialists (Arabic: منظمة الاشتراكيين اللبنانيين) was a political organization in Lebanon. The organization was led by Muhsin Ibrahim and Muhammed Kishli. It had its roots in the Lebanese branch of the Arab Nationalist Movement (ANM), a radical pan-Arab movement. During the 1960s Ibrahim was a leading figure in the leftist tendency with the ANM. This tendency, led by Naif Hawatmeh, argued that the ANM ought to adopt a Marxist outlook. This was opposed by the top ANM leader George Habash who, although being open to introducing Marxist concepts like imperialism into the discourse of the ANM, wanted to retain the anti-Communist character of the organization. As the central leadership of ANM had shifted to Damascus, the Lebanese branch began to function more autonomously. The official ANM organ al-Hurriya ('Freedom'), of which Ibrahim had become editor in 1960, became a de facto mouthpiece for the Marxist sector. In 1968 the Lebanese branch of ANM broke its links to the mother organization, and renamed itself as the Organization of Lebanese Socialists. The viewpoint of the Organization of Lebanese Socialists on the split were formulated in the pamphlet
Tags: most, companies, from, famous, lebanon
Discuss Most famous Companies from Lebanon
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AviationHeadlinesTransport
Flight delayed, passengers stranded as traffic controllers protest poor working conditions.
By maritimemag November 24, 2021
Several passengers were yesterday stranded at various airports across the country as flights were delayed and suspended due to three hours flow control declared by the National Association of Air Traffic Controllers (NATCA) over demise of one of its members, Aniekan Inuk Effiong in Abuja on Monday.
Recall that NATCA had declared flow control over poor current working environment and conditions of Controllers.
It was gathered that the flow control by the ATCs which commenced from 0600UTC to 0900UTC on Tuesday was to register their displeasure at the current working environment of controllers affected airlines who put out statements to alert the public on their predicament.
However, the Nigerian Airspace Management Agency (NAMA), said normal flight operations have been restored after NATCA suspended their strike action.
But, despite the strike suspension, airlines such as Air Peace, Dana, Green Africa were all affected by the flow control action embarked upon by NATCA as flights in and out of all the nation's airports were disrupted.
For instance, Air Peace in a statement told passengers that due to the ATC strike, all its domestic flights would experience delays while some will be cancelled due to sunset limitations, which was beyond the airline's control.
It stressed that the industrial action is slowing down traffic flow as departure is paced by 20 minutes on the domestic flight.
Dana on its part, said it would do its best to alleviate the delays and it understands the negative impact the unannounced strike might have caused its passenger's schedule while apologizing for the havoc it has wreaked
Green Africa Airways flight scheduled to depart Lagos for Akure at 1.34 pm was first cancelled, then rescheduled to 4.45 pm and then another notice of delay was sent to customers.
"Flight 318 from QOW to AKR scheduled for 23 Nov 2021 04:45:00 has been cancelled. This is due to an ongoing industrial action by Air Traffic Controllers. We apologize for any inconvenience this might cause you.", the message to passengers read.
"Please be notified that your flight 307 from AKR to LOS will be delayed due to an industrial action by Air Traffic Controllers at some major Airports in the country today.", another message reads.
The Nigerian Airspace Management Agency (NAMA), however, said normal flight has been restored after the NATCA suspended their strike action.
NAMA in a press statement signed by the Managing Director, Fola Akinkoutu and a copy made available to newsmen, said the suspension of the strike action will pave way for a follow-up meeting between NAMA's management, Director General of the Nigerian Civil Aviation Authority (NCAA) and the association, scheduled to hold later on Tuesday.
The statement reads, "NAMA hereby wishes to inform the general public that following the intervention of NAMA management, normalcy has been restored, as the Flow Control, earlier embarked upon by members of the Nigerian Air Traffic Controllers Association today, the 23rd of November, 2021 has been suspended.
"The above suspension is to pave way for a follow-up meeting between NAMA management, Director General of the Nigerian Civil Aviation Authority (NCAA) and the association, scheduled to hold later today."
"Meanwhile, NAMA hereby wishes to reassure airspace users and the general public that the Nigerian airspace remains safe for seamless and economic air travel. We also regret any inconveniences caused earlier today."
Clearing agents lament Customs arbitrary increase of exchange rate
Piracy: NIMASA, Korea collaborate on maritime security, safety
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Cannot prevent POST form from executing PHP code and image processing UPDATED
CONTEXT:
I have a form that is editing event data: event photo, title, description, etc. When the user opens the edit form, all of the data is in the form fields. The user can choose to edit the data and update, or not to.
PROBLEM:
If the user does not change the photo and presses 'submit', the photo disappears.
DESIRED RESULT:
I want the photo to be updated if the user chooses to change the photo, and I want the existing photo to remain if the user doesn't change the photo.
** UPDATE**
I previously thought this was a PHP isset() / empty() check but it's not. It's a javascript problem, which I'm not experienced with. I believe the submit button and submit id is causing the javascript to fire regardless of the php.
CODE of HTML image input form.
<div class="row">
<div class="col-md-12">
<div id="event_picture" style="width:350px"></div>
</div>
<div class="col-md-12" style="padding-top:30px;">
<strong>Select Image:</strong>
<br/>
<img width='100' src="images/<?php echo $e['event_picture'];?>" alt="">
<input type="file" id="upload" class="form">
<br/>
<input type="hidden" name="event_picture" id="event_picture_base64">
</div>
</div>
SUBMIT BUTTON
<input type="submit" name="submit" class="upload-result" value="Update Event" class="btn btn-primary btn-lg">
JAVASCRIPT
<script type="text/javascript">
$uploadCrop = $('#event_picture').croppie({
enableExif: true,
viewport: {
width: 300,
height: 300,
type: 'square'
},
boundary: {
width: 350,
height: 350
}
});
$('#upload').on('change', function () {
var reader = new FileReader();
reader.onload = function (e) {
$uploadCrop.croppie('bind', {
url: e.target.result
}).then(function(){
console.log('jQuery bind complete');
});
}
reader.readAsDataURL(this.files[0]);
});
$('.upload-result').on('click', function (ev) {
$uploadCrop.croppie('result', {
type: 'canvas',
size: 'viewport'
}).then(function (resp) {
document.getElementById("event_picture_base64").value = resp;
});
});
</script>
I think the problem is I need a conditional to prevent the javascript from firing when I press submit button. I'm searching online now, but I really don't know how to write that conditional.
Any help is appreciated.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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Why certainly. I believe I can dig up another steak.
What a gentleman you are Bushy. It's darn nice of you to invite MY GIRL for breakfast.
Ginger ... uh ...."beef" maybe?
Circumstances decide what you will and will not eat, not preferences.
So true Lester. As my dear old Dad used to say...you get hungry enough, you'll eat sh1te if it's cooked properly.
I think the vast majority of Americans are the most pathetic bunch sissy assed morons in the world. They don't have a clue what its like in the rest of the world. Maybe they should add another year to the current lame education they get and put them in the middle of some third world county with a book of matches and a blanky for a year of discovery, those that survive graduate and get to come home. Or, even better re-institute the draft for everybody no exceptions and survival training is a mandatory part of service. Too soft too long, pampered beyond reason.
There was an old western comedy that had an 'Indian' wanting "dog meat roasted over buffalo chips" tooo funny Eating my dog or horse or coyote would be a last ditch event for my very survival.
Coyote has to be better than some of the things I've eaten. I know cougar is darn good, snake isn't bad and insects are edible - most of them.
A starving man eats only a gourmet meal.
So true Ron. Even the scrappiest piece of meat would taste like filet steak when you are starving.
While we're on the subject of different types of meat. What does kangaroo taste like?
Kangaroo is not unpleasant, very flavoursome with a unique distinctive flavour. It lends itself well to the BBQ adrift her steaks or kebabs.
I'm not sure if kangaroo meat is imported into the US, but you can buy it in the supermarket out here in OZ.
Can't say what coyote tastes like-I killed a bunch,but have'nt put a fork to one.I have eaten banana rat when in gtmo-dam things live in trees,look like woodchucks/groundhogs,eat only tree vegitation and have prehensile tails for climbing etc.,but I have killed many a groundhog/rockchuck out west here for crop protection-and have'nt eaten a one.Dam things got too much "aroma" for my taste!I might have been hungry enough if obama was still in office-not now!!!Now I've got hereford,angus,deer and elk venison,occasionally a sheep(for food only)you south pacific guys!!!
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"redpajama_set_name": "RedPajamaC4"
}
| 7,684
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www.kidney.org
Social Media Ambassador Team
Ankit Sakhuja
Diana Mahbod
Edgar Lerma
Juan Carlos Velez
Kam Kalantar
Melissa Prest
Nathaniel Reisinger
Teri Browne
Dr. Ankit Sakhuja is a nephrologist and cardiac intensivist in Morgantown, West Virginia and is affiliated with West Virginia University Hospitals. He received his medical degree from All India Institute of Medical Sciences and has been in practice between 0-5 years. He is one of 3 doctors at West Virginia University Hospitals who specialize in cardiac critical care and nephrology.
@ansakhuja
Diana (Mina) Mahbod is a private practice nephrologist with Dallas Renal Group in Dallas, Texas. Since discovering the robust online and social media nephrology community, she has become an enthusiast for online medical education and collaboration. She was part of the 2018 class of the Nephrology Social Media Collective (NSMC) and has contributed to NephMadness, Renal Fellow Network, and the AJKD blog. She has hosted NephJC chats and is co-creator of LandmarkNephrology.com, an online educational initiative. Diana is on the executive committee of the Nephrology Business and Leadership University (NBLU) and is a member of the ASN and Women in Nephrology. She is also a Fellow of the National Kidney Foundation (FNKF) and has served on the Young Professional Board of the NKF of Northern California, as well as being featured as a guest in the "Life as a Nephrologist" podcast series. Diana hopes to provide the best care to her patients as a community nephrologist while staying connected to educational initiatives, mentorship opportunities, and the collaboration that is now possible across the world via the online nephrology community.
@DiMiRenalMD
Edgar Lerma, MD, earned his Doctor of Medicine from the University of Santo Tomas Faculty of Medicine and Surgery in Manila, Philippines. He completed Residency Training in Internal Medicine at UIC/Mercy Hospital and Medical Center, where he also served as Chief Resident. He completed a Fellowship in Nephrology and Hypertension at Northwestern Memorial Hospital, the Feinberg School of Medicine at Northwestern University, and the Veterans Administration Lakeside Medical Center in Chicago, Illinois.
Dr. Lerma is a Diplomate in the Subspecialty of Nephrology with the American Board of Internal Medicine. He has authored more than 100 peer-reviewed publications and presentations; notable publications include Current Diagnosis & Treatment: Nephrology & Hypertension, Nephrology Secrets, and Henrich's Principles and Practice of Dialysis. He has peer reviewed and served on editorial boards for numerous journals, and currently is Associate Editor for International Urology and Nephrology, Clinical Reviews in Bone and Mineral Metabolism, and Journal of Clinical Lipidology.
At present, he holds the rank of Clinical Professor of Medicine with the Section of Nephrology at University of Illinois at Chicago. He serves as the Educational Coordinator for Nephrology with UIC/Advocate Christ Medical Center.
In recognition of his clinical work and expertise, Dr. Lerma has been elected to Fellowship of the American College of Physicians, Fellowship of the American Society of Nephrology, Fellowship of the American Heart Association/ Council on Kidney in Cardiovascular Disease, Fellowship of the American Society of Hypertension, Fellowship of the National Lipid Association, Fellowship of the National Kidney Foundation, and Fellowship of the American Society of Diagnostic and Interventional Nephrology. He has also been recognized for teaching and has been the recipient of "Subspecialty Teaching Attending Physician of the Year" in 2006 and "Physician Recognition Award for Excellence in Academics" in 2011 from UIC/Advocate Christ Medical Center Internal Medicine Residency Training Program.
Dr. Lerma's research interests include CKD, hypertension, bone and mineral disorders, and dyslipidemias in CKD.
@edgarvlermamd
Dr. Juan Carlos Q. Velez earned a medical degree from Universidad Peruana Cayetano Heredia in Lima, Peru. He completed internship and residency in Internal Medicine at Advocate Illinois Masonic Medical Center, serving as Chief Medical Resident. Subsequently, he completed a clinical and research fellowship in Nephrology at Emory University School of Medicine.
Following his training, he joined the Division of Nephrology at the Medical University of South Carolina (MUSC). After spending 11 years of intense clinical, educational and investigational efforts at MUSC, he was recruited by the Department of Nephrology at the Ochsner Clinic Foundation in 2016 to enhance the academic and clinical research programs of the department.
Dr. Velez has conducted experimental research in the area of the intrarenal renin-angiotensin system that resulted in federal funding and peer-reviewed publications. Clinically, his areas of expertise and ongoing investigation include hepatorenal acute kidney injury, hyponatremia, glomerular diseases and renal syndromes associated with exposure to antimicrobials.
Dr. Velez is certified in Nephrology by the American Board of Internal Medicine, in Kidney Ultrasonography by the American Society of Diagnostic and Interventional Nephrology, and as a Hypertension Specialist by the American Society of Hypertension, and he is a member of the Southern Society of Clinical Investigation. In addition, he serves in the American Society of Nephrology / National Board of Medical Examiners Fellows In-Training Examination Development Committee and has a track record of mentoring young trainees.
@VelezNephHepato
Kamyar Kalantar-Zadeh, (a.k.a. Kam Kalantar) is a triple board-certified physician, who studied medicine at the Universities of Bonn and Nuremberg in Germany (MD degree) and received MPH and PhD degrees in Epidemiology from University of California Berkeley, School of Public Health. Dr. Kalantar's postgraduate training includes an internship training in Nuremberg, Germany, residency training in Internal Medicine and also in Pediatrics in the State University of New York (SUNY, 1993-97), and a nephrology fellowship at University of California San Francisco (UCSF, 1997-2000). During 2000-2012 Dr. Kalantar was a full-time faculty at Harbor-UCLA including Professor of Medicine, Pediatrics and Epidemiology at UCLA. Since 2012 he has served as a tenured Professor and HEAD of the Division of Nephrology and Hypertension at University of California Irvine (UC Irvine) School of Medicine, in Orange and Irvine, CA, and a Professor of Medicine, Pediatrics, and Public Health at UC Irvine and attending nephrologist at UC Irvine as well as Long Beach Veterans Affairs (VA) Hospital. Dr. Kalantar-Zadeh has been recognized by several prestigious top/best physician directories in the USA including SUPER DOCTORS™ and TOP PHYSICIANS™ Castle-Connelly. In 2014 he was ranked among top 10 experts in ESRD and Chronic Kidney Failure by Expertscape™. He serves as the immediate past President of the International Society of Renal Nutrition & Metabolism. President elect of the International Federation of Kidney Foundations and the International Steering Committee of the World Kidney Day. Prof. Kalantar-Zadeh has published over 600 scientific articles, authored many chapters, and presented numerous grand rounds and other lectures in national and international conferences. He is an editor of nephrology and nutrition textbooks including the "Nutritional Management of Renal Disease", 3rd edition 2013. Dr. Kalantar is an Associate Editor or member of the editorial board of several top journals.
@kamkalantar
Melissa Prest completed her doctorate in clinical nutrition from Rutgers University and her master's degree in clinical nutrition from the University of Medicine and Dentistry of New Jersey. Dr. Prest has presented both locally and nationally on her research and other healthcare related topics. She has authored product updates for the Journal on Renal Nutrition, and mobile application reviews for the Renal Nutrition Forum.
Dr. Prest has worked in a variety of practice settings with a focus on clinical nutrition. For the past eleven years she has provided medical nutrition therapy to patients with chronic kidney disease receiving maintenance hemodialysis. Dr. Prest is also the owner of a private practice and health coaching firm, Kidney Nutrition Specialists, based in Chicago, IL. In 2014, Dr. Prest was awarded with the Emerging Dietetic Leader award for her leadership in the Chicago Academy of Nutrition and Dietetics, the Illinois Academy of Nutrition and Dietetics, and the Illinois Council on Renal Nutrition.
@windycityRD
Nathaniel Reisinger is a nephrologist and assistant professor of medicine at Cooper University Hospital in Camden, New Jersey. He graduated medical school at the University of Texas Southwestern, finished internal medicine residency at Columbia, and completed nephrology fellowship and clinical ultrasound fellowship at Penn. He is interested in novel technologies to enhance the patient experience in nephrology and is an active researcher and educator in point-of-care ultrasound. Nathaniel serves with the Cardio Renal Society of America as the Creator and Director of the Point-of-Care Ultrasound Immersion Course for Cardio Renal University and has helped organize ultrasound point-of-care ultrasound lectures and courses for the American Society of Nephrology's Kidney Week, National Kidney Foundation's Spring Clinical Meeting, Nephrology Business Leadership University, KIDNEYcon, and the Penn Division of Critical Care Medicine. He has a keen interest in the use of social media in nephrology, having completed an internship with the Nephrology Social Media Collective 2017. He is a past contributor to AJKDblog and intern with the American Society of Nephrology Advocacy and Public Policy Committee.
@nephrothaniel
Dr. Teri Browne is an Associate Professor and Associate Dean of Faculty and Research at the University of South Carolina College of Social Work in Columbia, SC. Dr. Browne is the co-director of interprofessional health for the health sciences at the University of South Carolina. She earned her MSW at the State University of New York at Buffalo and her PhD at the University of Chicago. Dr. Browne worked as a nephrology social worker for 13 years in NY, CA and IL and was the national chairperson of Council of Nephrology Social Work. Dr. Browne is the editor-in-chief of National Kidney Foundation's Journal of Nephrology Social Work. She is currently the Chairperson of the End Stage Renal Disease Network of the South Atlantic Grievance Committee and a member of the Network's Divisional Board. Dr. Browne is a social work field instructor for the National Kidney Foundation of South Carolina and is a member of the American Association of Kidney Patient's Board of Directors. She is also the co-editor of the Handbook of Health Social Work.
@TeriBrowne
About SCM19
Educational Stipends
Disclosure Policy & Expectation of Presenters
Obtaining Employer Support to Attend
About NKF
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,452
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Moe preparing cabinet for fall legislative session
Brendan Ellis CTVNewsRegina.ca Digital Content Producer
@BrendanEllisCTV Contact
Published Tuesday, October 27, 2020 1:39PM CST Last Updated Tuesday, October 27, 2020 5:49PM CST
REGINA -- Scott Moe will be announcing his cabinet in the coming days, in preparation for a fall legislative session.
At a press conference on Tuesday afternoon, Moe reiterated that his party would keep its campaign commitments and would be looking to get to work on them in the house this year.
"As I said last night, we are going to get to work right away," Moe said. "I will be meeting with our new caucus, we'll be putting together a new cabinet, and we will be starting to plan precisely what is going to go in our speech from the throne for this fall, as well as a fall session."
Moe said the first priority for his party would be a home renovation tax credit that was promised on the campaign trail.
"That will be bill one when we open the house here this fall and there may be some other pieces of legislation that are pertinent to some of the campaign commitments that we had made," Moe said.
The premier-elect would not commit to any additional funding for health care or education in the province, heading into a fall legislative sitting.
Moe emphasized the funding announcements his government made before the campaign began.
"Our health care system is going to be funded here in the province. We're in some unprecedented times," Moe said. "We did, I believe, before we went on the campaign trail, pass a number of special warrants to ensure that the health care system did have the funding that they would be required to manage for an unspecified period of time."
He noted that education funding still remains available to school divisions upon request.
UNDECLARED SEATS
With many constituencies still up in the air and mail in ballots yet to be counted, Moe said he has reached out to his candidates that are still in tight races across the province.
"It's hard to change the trajectory, it is hard to turn seats around, but in saying that there are tens of thousands of ballots out there, there are some close seats," Moe said.
"This is a little different than other years where we don't have this lag effect, at least to this level where the mail-in ballots are as large as they are."
Elections Saskatchewan is expecting to have final vote counts by Nov. 7.
Saskatchewan Party Leader Scott Moe makes his victory speech to media at the party's campaign event in Saskatoon, Sask., on Oct. 26, 2020. (Liam Richards / THE CANADIAN PRESS)
Four Moe Years: Scott Moe and Saskatchewan Party defeat NDP for 4th straight majority
|
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| 4,119
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