text
stringlengths 14
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| meta
dict | __index_level_0__
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9.97k
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|---|---|---|
Rocca Di Cerere
2000 Oil
#C1178
This 2000 oil on canvas San Francisco abstracted cityscape entitled Rocca Di Cerere, is by San Francisco painter Jack Freeman (1938-2014). Freeman studied at the High Museum of Art in Atlanta, landscape and figure drawing with Oskar Kokoschka's School of Vision in Austria, and received his MFA from the San Francisco Art Institute in the late 1960s. Freeman's style varied between Abstract Expressionism, the Bay Area Figurative Movement, and plein air landscape painting.
16.5"x15" framed
Framed in a contemporary wood floater frame with a deep walnut finish. Signed and dated on the back. Excellent vintage condition.
SF Backyard Scene
20th Century Oil
#C0452 Jack Freeman $785
Modernist Figure Drawing
1976 Ink Wash
#94988 Jack Freeman Sold $995
Bold Seated Male Nude
20th Century Pastel
#A5027 Jack Freeman $285
Quiet Afternoon Scene
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,053
|
<?php
namespace W3TC;
/**
* W3 PgCache flushing
*/
class PgCache_Flush extends PgCache_ContentGrabber {
/**
* Array of urls to flush
*
* @var array
*/
private $queued_urls = array();
private $queued_groups = array();
private $flush_all_operation_requested = false;
/**
* PHP5 Constructor
*/
function __construct() {
parent::__construct();
}
/**
* Flushes all caches
*
* @return boolean
*/
function flush() {
$this->flush_all_operation_requested = true;
return true;
}
function flush_group( $group ) {
$this->queued_groups[$group] = '*';
}
/**
* Flushes post cache
*
* @param integer $post_id
* @return boolean
*/
function flush_post( $post_id = null ) {
if ( !$post_id ) {
$post_id = Util_Environment::detect_post_id();
}
if ( !$post_id )
return false;
$full_urls = array();
$post = get_post( $post_id );
$terms = array();
$feeds = $this->_config->get_array( 'pgcache.purge.feed.types' );
$limit_post_pages = $this->_config->get_integer( 'pgcache.purge.postpages_limit' );
if ( $this->_config->get_string( 'pgcache.rest' ) == 'cache' ) {
$this->flush_group( 'rest' );
}
if ( $this->_config->get_boolean( 'pgcache.purge.terms' ) ||
$this->_config->get_boolean( 'pgcache.purge.feed.terms' ) ) {
$taxonomies = get_post_taxonomies( $post_id );
$terms = wp_get_post_terms( $post_id, $taxonomies );
$terms = $this->_append_parent_terms( $terms, $terms );
}
$front_page = get_option( 'show_on_front' );
/**
* Home (Frontpage) URL
*/
if ( ( $this->_config->get_boolean( 'pgcache.purge.home' ) &&
$front_page == 'posts' ) ||
$this->_config->get_boolean( 'pgcache.purge.front_page' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_frontpage_urls( $limit_post_pages ) );
}
/**
* Home (Post page) URL
*/
if ( $this->_config->get_boolean( 'pgcache.purge.home' ) &&
$front_page != 'posts' ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_postpage_urls( $limit_post_pages ) );
}
/**
* Post URL
*/
if ( $this->_config->get_boolean( 'pgcache.purge.post' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_post_urls( $post_id ) );
}
/**
* Post comments URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.comments' ) &&
function_exists( 'get_comments_pagenum_link' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_post_comments_urls( $post_id ) );
}
/**
* Post author URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.author' ) && $post ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_post_author_urls( $post->post_author,
$limit_post_pages ) );
}
/**
* Post terms URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.terms' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_post_terms_urls( $terms, $limit_post_pages ) );
}
/**
* Daily archive URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.archive.daily' ) && $post ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_daily_archive_urls( $post, $limit_post_pages ) );
}
/**
* Monthly archive URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.archive.monthly' ) && $post ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_monthly_archive_urls( $post, $limit_post_pages ) );
}
/**
* Yearly archive URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.archive.yearly' ) && $post ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_yearly_archive_urls( $post, $limit_post_pages ) );
}
/**
* Feed URLs
*/
if ( $this->_config->get_boolean( 'pgcache.purge.feed.blog' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_feed_urls( $feeds, null ) );
}
if ( $this->_config->get_boolean( 'pgcache.purge.feed.comments' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_feed_comments_urls( $post_id, $feeds ) );
}
if ( $this->_config->get_boolean( 'pgcache.purge.feed.author' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_feed_author_urls( $post->post_author, $feeds ) );
}
if ( $this->_config->get_boolean( 'pgcache.purge.feed.terms' ) ) {
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_feed_terms_urls( $terms, $feeds ) );
}
/**
* Purge selected pages
*/
if ( $this->_config->get_array( 'pgcache.purge.pages' ) ) {
$pages = $this->_config->get_array( 'pgcache.purge.pages' );
$full_urls = array_merge( $full_urls,
Util_PageUrls::get_pages_urls( $pages ) );
}
// add mirror urls
$full_urls = Util_PageUrls::complement_with_mirror_urls( $full_urls );
$full_urls = apply_filters( 'pgcache_flush_post_queued_urls',
$full_urls );
/**
* Queue flush
*/
if ( count( $full_urls ) ) {
foreach ( $full_urls as $url )
$this->queued_urls[$url] = '*';
}
return true;
}
/**
* Flush a single url
*
* @param unknown $url
* @param unknown $cache
* @param unknown $mobile_groups
* @param unknown $referrer_groups
* @param unknown $encryptions
* @param unknown $compressions
*/
function _flush_url( $url, $cache, $mobile_groups, $referrer_groups,
$encryptions, $compressions ) {
foreach ( $mobile_groups as $mobile_group ) {
foreach ( $referrer_groups as $referrer_group ) {
foreach ( $encryptions as $encryption ) {
foreach ( $compressions as $compression ) {
$page_keys = array();
$page_keys[] = $this->_get_page_key( array(
'useragent' => $mobile_group,
'referrer' => $referrer_group,
'encryption' => $encryption,
'compression' => $compression ),
$url );
$page_keys = apply_filters(
'w3tc_pagecache_flush_url_keys', $page_keys );
foreach ( $page_keys as $page_key )
$cache->delete( $page_key );
}
}
}
}
}
/**
* Flush a single url
*
* @param unknown $url
*/
function flush_url( $url ) {
static $cache, $mobile_groups, $referrer_groups, $encryptions;
static $compressions;
if ( !isset( $cache ) )
$cache = $this->_get_cache();
if ( !isset( $mobile_groups ) )
$mobile_groups = $this->_get_mobile_groups();
if ( !isset( $referrer_groups ) )
$referrer_groups = $this->_get_referrer_groups();
if ( !isset( $encryptions ) )
$encryptions = $this->_get_encryptions();
if ( !isset( $compressions ) )
$compressions = $this->_get_compressions();
$this->_flush_url( $url, $cache, $mobile_groups, $referrer_groups,
$encryptions, $compressions );
}
/**
* Flushes global and repeated urls
*
* @return count of elements it has flushed
*/
function flush_post_cleanup() {
if ( $this->flush_all_operation_requested ) {
if ( $this->_config->get_boolean( 'pgcache.debug' ) ) {
self::log( 'flush all' );
}
$cache = $this->_get_cache();
$cache->flush();
$count = 999;
$this->flush_all_operation_requested = false;
$this->queued_urls = array();
} else {
if ( count( $this->queued_groups ) > 0 ) {
$cache = $this->_get_cache();
foreach ( $this->queued_groups as $group => $flag ) {
if ( $this->_config->get_boolean( 'pgcache.debug' ) ) {
self::log( 'pgcache flush "' . $group . '" group' );
}
$cache->flush( $group );
}
}
$count = count( $this->queued_urls );
if ( $count > 0 ) {
if ( $this->_config->get_boolean( 'pgcache.debug' ) ) {
self::log( 'pgcache flush ' . $count . ' urls' );
}
$cache = $this->_get_cache();
$mobile_groups = $this->_get_mobile_groups();
$referrer_groups = $this->_get_referrer_groups();
$encryptions = $this->_get_encryptions();
$compressions = $this->_get_compressions();
foreach ( $this->queued_urls as $url => $flag ) {
$this->_flush_url( $url, $cache, $mobile_groups,
$referrer_groups, $encryptions, $compressions );
}
// Purge sitemaps if a sitemap option has a regex
if ( $this->_config->get_string( 'pgcache.purge.sitemap_regex' ) ) {
$cache = $this->_get_cache();
$cache->flush( 'sitemaps' );
}
$this->queued_urls = array();
}
}
return $count;
}
/**
* Returns array of mobile groups
*
* @return array
*/
function _get_mobile_groups() {
$mobile_groups = array( '' );
if ( $this->_mobile ) {
$mobile_groups = array_merge( $mobile_groups, array_keys(
$this->_mobile->get_groups() ) );
}
return $mobile_groups;
}
/**
* Returns array of referrer groups
*
* @return array
*/
function _get_referrer_groups() {
$referrer_groups = array( '' );
if ( $this->_referrer ) {
$referrer_groups = array_merge( $referrer_groups, array_keys(
$this->_referrer->get_groups() ) );
}
return $referrer_groups;
}
/**
* Returns array of encryptions
*
* @return array
*/
function _get_encryptions() {
$is_https = ( substr( get_home_url(), 0, 5 ) == 'https' );
$encryptions = array();
if ( ! $is_https || $this->_config->get_boolean( 'pgcache.cache.ssl' ) )
$encryptions[] = '';
if ( $is_https || $this->_config->get_boolean( 'pgcache.cache.ssl' ) )
$encryptions[] = 'ssl';
return $encryptions;
}
private function _append_parent_terms( $terms, $terms_to_check_parents ) {
$terms_to_check_parents = $terms;
$ids = null;
for ( ;; ) {
$parent_ids = array();
$taxonomies = array();
foreach ( $terms_to_check_parents as $term ) {
if ( $term->parent ) {
$parent_ids[$term->parent] = '*';
$taxonomies[$term->taxonomy] = '*';
}
}
if ( empty( $parent_ids ) )
return $terms;
if ( is_null( $ids ) ) {
// build a map of ids for faster check
$ids = array();
foreach ( $terms as $term )
$ids[$term->term_id] = '*';
} else {
// append last new items to ids map
foreach ( $terms_to_check_parents as $term )
$ids[$term->term_id] = '*';
}
// build list to extract
$include_ids = array();
foreach ( $parent_ids as $id => $v ) {
if ( !isset( $ids[$id] ) )
$include_ids[] = $id;
}
if ( empty( $include_ids ) )
return $terms;
$new_terms = get_terms( array_keys( $taxonomies ),
array( 'include' => $include_ids ) );
$terms = array_merge( $terms, $new_terms );
$terms_to_check_parents = $new_terms;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,384
|
Footpaths on La Palma
La Palma has a network of well-marked footpaths, most of which are centuries old. As late as the 1960s, walking was still a major form of transport for the islanders, The whole network of hiking trails on the island comes to over 1,000 km, and between them they pass through just about every kind of scenery on the island: lava fields, pine forests, lush laurel forests, farmland and village centres….
The Coloured Waterfall (Cascada de colores)
A few years ago I hiked up into the Caldera de Taburiente to finally see the famous Coloured Waterfall. By my standards it was a long hike since it's 5.5 km each way, although it's pretty level. (That is, A Dutchman wouldn't consider it flat, but you only climb about 200 m in 4.5 km. If you carry on to the camp site, that's another 350 m climb in 3.2…
Los Tilos Waterfall
Canarian waterfalls aren't common. There are lots of temporary waterfalls after heavy rain, but they tend to be very short-lived. But La Palma has two, pretty much year round. This one is in the Los Tilos biosphere reserve. From the visitor centre, you follow either the ravine or the water channel upstream. (If you follow the ravine, be prepared for some scrambling. If you follow the channel, bring a torch…
Water Mines on La Palma
Although La Palma has more water than the other Canary Islands, many farmers used to be desperately poor and frequently hungry. The only water for irrigation was rainwater, and obviously they had no control over how much they got. Then somebody suggested digging into the hillside to find water. (If anybody knows who, please tell me.) The idea is that much of the rainwater seeps into the ground, and runs…
Cubo de la Galga
Cubo de la Galga is a very pretty walk along the bottom of the Galga ravine, between Puntallana and Los Sauces. By Palmeran standards, it's an easy walk. There is now a car park at the beginning of the walk, on the road at km 16. You're unlikely to get lost for the first kilometre or so, because the path's actually asphalted, never mind signposted. It's a matter of taste,…
Almond Blossom in Garaf??a and Puntagorda
The almond trees have blossomed early this year. Most of the almond trees are in Puntagorda and Garafía, although there's a good number in El Paso too. They produce small, rather bitter almonds which are out of fashion and don't fetch a good price, (although I enjoy a saucer of toasted almonds with a glass of wine) so that most people don't harvest the almonds, or only harvest for their…
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,672
|
{"url":"https:\/\/dml.cz\/handle\/10338.dmlcz\/126028","text":"Previous\u00a0| \u00a0Up\u00a0| \u00a0Next\n\n# Article\n\nFull entry | PDF \u00a0 (2.4 MB)\nKeywords:\ngeneralized linear differential equation; substitution method; variational stability; logarithmic prolongation; ordinary linear differential equation with a substitution\nSummary:\nThe generalized linear differential equation $dx=d[a(t)]x+df$ where $A,f\\in BV^{loc}_n(J)$ and the matrices $I-\\Delta^-\\ A(t), I+\\Delta^+\\ A(t)$ are regular, can be transformed $\\frac{dy}{ds}=B(s)y+g(s)$ using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.\nReferences:\n[F] Fra\u0148kov\u00e1 D.: A discontinuous substitution in the generalized Perron integral. (to appear in Mathematica Bohemica).\n[FS] Fra\u0148kov\u00e1 D., Schwabik \u0160.: Generalized Sturm-Liouville equations II. Czechoslovak Math. J. 38 (113) 1988, 531-553. MR\u00a00950307\n[K] Kurzweil J.: Ordinary differential equations. Studies in Applied Mechanics 13. Elsevier Amsterdam-Oxford-New York-Tokyo 1986. MR\u00a00929466 | Zbl\u00a00667.34002\n[S1] Schwabik \u0160.: Generalized differential equations. Fundamental results. Rozpravy \u010cSAV, Academia Praha 1985. MR\u00a00823224 | Zbl\u00a00594.34002\n[S2] Schwabik \u0160.: Variational stability for generalized ordinary differential equations. \u010casopis p\u011bst. mat. 109 (1984), Praha, 389-420. MR\u00a00774281 | Zbl\u00a00574.34034\n[STV] Schwabik \u0160., Tvrd\u00fd M., Vejvoda O.: Differential and integral equations. Boundary Value Problems and Adjoints. Academia Praha, Reidel Dordrecht, 1979. MR\u00a00542283\n\nPartner of","date":"2022-01-23 11:55:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9073536396026611, \"perplexity\": 7632.170763124849}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304261.85\/warc\/CC-MAIN-20220123111431-20220123141431-00613.warc.gz\"}"}
| null | null |
Rap-Up TV
2.1.2017 Live Performances
Watch D.R.A.M. Perform for NPR's Tiny Desk Concert
D.R.A.M. took over NPR as the latest star to perform on the "Tiny Desk" concert series. The Virginia native delivered a medley of tracks from his catalogue with slight alterations for a groovy in-office show.
He started off the set with a performance of "Cash Machine," making the dollar sounds on his own. "I recorded it back in June of 2015 when I got that first big check," he said. "Soon as I saw that joint, it was instant inspiration."
From there, Big Baby launched into "Cute." "Ladies, I hope you enjoy it," he said. "Definitely had y'all in mind for it."
"Sweet VA Breeze" followed, with a reflective D.R.A.M. "I come from a place called Hampton, Virginia," he said. "This song is really near and dear to me. It takes me back to the old days when things were a little more simpler."
Veering away from his own album, last year's Big Baby D.R.A.M., the rapper-singer performed "Special" from Chance the Rapper's Coloring Book.
"It feels good to be able to put a little motivational message out there in the world where there's a lot of fucked up shit going on, if we're going to be frank," he said. "It's just a little message, a little whisper in your ear, to get you through your day."
Finally, D.R.A.M. closed things out with "Broccoli." "Quiet as kept, we sold 4 million copies of this record," he said. "It's Grammy nominated. I'm very thankful to be going to the Grammys this year for that."
Watch the full performance below.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,011
|
\section{Discussion}
\label{sec:discussion}
\parh{Handling Real-World Attack Logs.}~Side channel observations (e.g.,
obtained in cross-virtual machine attacks~\cite{Zhang12}), are typically noisy.
\textsc{CacheQL}\xspace\ handles real attack logs by considering noise as non-determinism (see
\S~\ref{subsec:non-det}), thus quantifying leaked bits in those logs.
Nevertheless, we do not recommend localizing vulnerabilities using real attack
logs, since mapping these records back to program statements are challenging.
Pin is sufficient for developers to ``debug''.
\smallskip
\parh{Analyzing Media Data.}~\textsc{CacheQL}\xspace\ can smoothly quantify and localize
information leaks for media software. Unlike previous static-/trace-based tools,
which require re-implementing the pipeline to model floating-point instructions
for symbolic execution or abstract interpretation, \textsc{CacheQL}\xspace\ only needs the
compressor $\mathcal{R}$ to be changed. In addition, \textsc{CacheQL}\xspace\ is based on NN,
which facilitates extracting ``contents'' of media data to quantify leaks,
rather than simply comparing data byte differences. See evaluation results in
Appx.~\ref{appx:r-detail}.
\smallskip
\parh{Program Taking Public Inputs.}~We deem that different public inputs should
not largely influence our analysis over cryptographic and media libraries,
whose reasons are two-fold. First, for cryptosystems like OpenSSL, the public
inputs (i.e., plaintext or ciphertext) has a relatively minor impact on the
program execution flow. To our observation, public input values only influence a
few loop iterations and \texttt{if} conditions. Media libraries mainly process
private user inputs, which has no ``public inputs''. In practice, the influences
of public inputs (including other non-secret local variables) are treated as
non-determinism by \textsc{CacheQL}\xspace. That is, they are handled consistently as how \textsc{CacheQL}\xspace\
handles cryptographic blinding (\S~\ref{subsec:non-det}), because neither is related
to secret.
Given that said, configurations (e.g., cryptographic algorithm or image compression
mode) may notably change the execution and the logged execution traces. We view
that as an orthogonal factor. Moreover, modes of cryptographic algorithms and media
processing procedures are limited. Users of \textsc{CacheQL}\xspace\ are suggested to fix the mode
before launching an analysis with \textsc{CacheQL}\xspace, then use another mode, and so on.
\smallskip
\parh{Keystroke Templating Attacks.}~Quantifying and localizing the information
leaks that enable keystroke templating attacks should be feasible to explore.
With side channel traces logged by Intel Pin, machine learning is used to
predict the user's key press~\cite{wang2019unveiling}. Given sufficient data
logged in this scenario, \textsc{CacheQL}\xspace\ can be directly applied to quantify the leaked
information and localize leakage sites.
\smallskip
\parh{Large Software Monoliths.}~For analyzing complex software like browsers
and office products, our experience is that using Intel Pin to perform dynamic
instrumentation for production browsers is difficult. With this regard, we
anticipate adopting other dynamic instrumentors, if possible, to enable
localizing leaks in these software. With correctly logged execution trace,
\textsc{CacheQL}\xspace\ can quantify the leaked bits and attribute the bits to side channel
records.
\smallskip
\parh{Training Dataset Generalization.}~One may question to what degree traces
obtained from one program can be used as training set for detecting leaks in
another. In our current setup, we do not advocate such a ``transfer'' setting.
Holistically, \textsc{CacheQL}\xspace\ learns to compute PD by accurately distinguishing traces
produced when the software is processing different secret inputs. By learning
such distinguishability, \textsc{CacheQL}\xspace\ eliminates the need for users to label the
leakage bits of each training data sample. Nevertheless, knowledge learned for
distinguishability may differ between programs. It is intriguing to explore
training a ``general'' model that can quantify different side channel logs,
particularly when collecting traces for the target program is costly. To do so,
we expect to incorporate advanced training techniques (such as transfer
learning~\cite{pan2009survey}) into our pipeline.
\smallskip
\parh{Key Generation.}~In RSA evaluations, we feed cryptographic libraries with the key
in a file. That is, the cryptographic library execution does not involve ``key
generation'', which is \textit{not} due to ``limited coverage'' of \textsc{CacheQL}\xspace.
Previous works~\cite{moghimi2020copycat} have exploited RSA key generation.
With manual effort, we find that the key generation functions heavily use BIGNUM,
involving vulnerable BIGNUM initialization and computation functions already
localized by \textsc{CacheQL}\xspace\ in \S~\ref{subsec:eval-localization}, e.g.,
\texttt{BN_bin2bn} in Fig.~\ref{fig:openssl} and \texttt{BN_sub} (see our full
report~\cite{snapshot}).
\smallskip
\parh{BIGNUM Implementation.}~We also investigate other cryptosystems.
LibreSSL and BoringSSL are built on OpenSSL. Their BIGNUM
implementations and OpenSSL share similar vulnerable coding patterns (i.e., the
leading zero leak patterns; see \Type{C} of \S~\ref{subsec:eval-localization}).
We also find similar BIGNUM vulnerable patterns in Botan (see~\cite{botan-bn}).
In contrast, we find Intel IPP does not use an individual variable to
record \#leading zeros in BIGNUM (see~\cite{intel-ipp-bn}), hence it
is likely free of \Type{C}.
\section{Conclusion}
\label{sec:conclusion}
We present \textsc{CacheQL}\xspace\ to quantify cache side channel leakages via MI. We also
formulate secret leak as a cooperative game and enable localization via
Shapley value. Our evaluation shows that \textsc{CacheQL}\xspace\ overcomes typical hurdles (e.g.,
scalability, accuracy) of prior works, and computes information leaks in
real-world cryptographic and media software.
\section*{Acknowledgement}
We thank all anonymous reviewers and our shepherd for their valuable feedback.
We also thank Janos Follath, Matt Caswell, and developers of Libjpeg-turbo
for their prompt responses and comments on our reported vulnerabilities.
\section{Related Works \& Criteria}
\label{sec:requirement}
We propose eight criteria for a full-fledged detector. Accordingly, we review
related works in this field and assess their suitability.
\S~\ref{subsec:eval-localization} empirically compares them with \textsc{CacheQL}\xspace. Also,
many studies launch cache analysis on real-time systems and estimate worst-case
execution time
(WCET)~\cite{chu2016precise,li2003accurate,li2009timing,mitra2018time}; we omit
those studies as they are mainly for measurement, not for vulnerability
detection.
\smallskip
\parh{Execution Trace vs.~Cache Attack Logs.}~Most existing
detectors~\cite{wang2017cached,wang2019caches,brotzman2019casym} assume access
to execution traces. In addition to recording noise-free execution traces (e.g.,
via Intel Pin), considering real cache attack logs is equally important. Cryptosystem
developers often require evidence under real-world scenarios to issue patches.
For instance, OpenSSL by default only accepts side channel reports if they can
be successfully exploited in real-world scenarios~\cite{opensslpatch}. In sum,
we advocate that a side channel detector should~\Circ{1}~\textit{analyze both
execution traces and real-world cache attack logs}.
\smallskip
\parh{Deterministic vs.~Non-deterministic Observations.}~Deterministic
observations imply that, for a given secret, the observed side channel is fixed.
Decryption, however, may be non-deterministic due to various masking and
blinding schemes used in cryptosystems. Furthermore, techniques like
ORAM~\cite{goldreich1996software} can generate non-deterministic memory accesses
and prevent information leakage. Thus, memory accesses or executed branches may
differ between executions using one secret. Nearly all previous
works~\cite{wang2017cached,bao2021abacus,wang2019caches,brotzman2019casym} only
consider deterministic side channels, failing to analyze the protection offered
by blinding/ORAM and may overestimate leaks (not just keys, blinding/ORAM also
change side channel observations). We suggest that a detector
should~\Circ{2}~\textit{analyze both deterministic and non-deterministic
observations.} \textsc{CacheQL}\xspace\ uses statistics to quantify information leaks from
non-deterministic observations, as explained in \S~\ref{subsec:non-det}.
\smallskip
\parh{Analyze Source Code vs.~Binary.}~A detector should typically analyze
software in executable format. This allows the analysis of legacy code and
third-party libraries. More importantly, by analyzing assembly code, low-level
details like memory allocation can be precisely considered.
Studies~\cite{doychev2017rigorous,simon2018you} reveal that compiler
optimizations could introduce side channels not visible in high-level code
representations. Thus, we argue that detectors should~\Circ{3}~\textit{be able
to analyze program executables.}
\smallskip
\parh{Quantitative vs.~Qualitative.}~Qualitative detectors decide whether
software leaks information and pinpoint leakage program
points~\cite{wang2017cached,wang2019caches,brotzman2019casym}. Quantitative
detectors further quantify leakage from each software
execution~\cite{cacheaudit, chattopadhyay2019quantifying, doychev2017rigorous},
or at each vulnerable program point~\cite{bao2021abacus, jan2018microwalk}. We
argue that a detector should~\Circ{4}~\textit{deliver both qualitative and
quantitative analysis}. Developers are reluctant to fix certain vulnerabilities,
as they may believe those defects leak negligible secrets~\cite{bao2021abacus}.
However, identifying program points that leak large amounts of data can push
developers to prioritize fixing them. To clarify, though quantitative analysis
was previously deemed costly~\cite{jancar2021they}, \textsc{CacheQL}\xspace\ features efficient
quantification.
\smallskip
\parh{Localization.}~Along with determining information leaks, localizing
vulnerable program points is critical. Precise localization helps developers
debug and patch. Therefore, a detector should~\Circ{5}~\textit{localize
vulnerable program points leaking secrets.} Most static detectors struggle to
pinpoint leakage points~\cite{cacheaudit,doychev2017rigorous}, as they measure
the number of different cache statuses to quantify leakage. Trace-based analysis,
including \textsc{CacheQL}\xspace, can identify leakage instructions on the trace that can be
mapped back to vulnerable program points~\cite{wang2017cached,bao2021abacus}.
\smallskip
\parh{Key vs.~Private Media Data.}~Most detectors analyze cryptosystems to
detect key leakage~\cite{wang2017cached,cacheaudit,bao2021abacus}. Recent side
channel attacks have targeted media data~\cite{xu2015controlled,hahnel2017high}.
Media data like images used in medical diagnosis may jeopardize user privacy
once leaked. We thus advocate detectors to~\Circ{6}~\textit{analyze leakage of
secret keys and media data}. Modeling information leakage of high-dimensional
media data is often harder, because defining ``information'' contained in media
data like images may be ambiguous. \textsc{CacheQL}\xspace\ models image holistic content (rather
than pixel values) leakage using neural networks.
\smallskip
\parh{Scalability: Whole Program/Trace~vs.~Program/Trace Cuts.}~Some prior
trace-based analyses rely on expensive techniques (e.g., symbolic execution)
that are not scalable. Given that one execution trace logged from cryptosystems
can contain millions of instructions, existing
works~\cite{wang2017cached,bao2021abacus} require to first \textit{cut} a trace
and analyze only a few functions on the cutted trace.
Prior static analysis-based works may use abstract
interpretation~\cite{cacheaudit,wang2019caches}, a costly technique with limited
scalability. Only toy programs~\cite{cacheaudit} or a few sensitive functions
are analyzed~\cite{wang2019caches,doychev2017rigorous,brotzman2019casym}. This
explains why most existing works overlook ``non-deterministic'' factors like
blinding (criterion \Circ{2}), as blinding is applied \textit{prior to} executing
their analyzed program/trace cuts. Lacking whole-program/trace analysis limits
the study scope of prior works. \textsc{CacheQL}\xspace\ can analyze a whole trace logged by
executing production software, and as shown in \S~\ref{sec:evaluation}, \textsc{CacheQL}\xspace\
identifies unknown vulnerabilities in pre-processing modules of cryptographic libraries
that are not even covered by existing works due to scalability. In sum, we
advocate that a detector should be \Circ{7}~\textit{scalable for
whole-program/whole-trace analysis.}
\smallskip
\parh{Implicit and Explicit Information Flow.}~Explicit information flow
primarily denotes secret data flow propagation, whereas implicit information
flow models subtle propagation by using secrets as code pointers or branch
conditions~\cite{schwartz10}. Considering implicit information flow is
challenging for existing works based on static analysis due to scalability. They
thus do not \textit{fully} analyze implicit information
flow~\cite{wang2017cached,wang2019caches,brotzman2019casym,bao2021abacus}. We
argue a detector should \Circ{8}~\textit{consider both implicit and explicit
information flow} to comprehensively model potential information leaks. \textsc{CacheQL}\xspace\ delivers an
``end-to-end'' analysis and identifies changes in the trace due to either
implicit or explicit information flow propagation of secrets.
\smallskip
\parh{Comparing with Existing Detectors.}~Table~\ref{tab:detectors} compares
existing detectors and \textsc{CacheQL}\xspace\ to the criteria. Abacus and MicroWalk cannot
precisely quantify information leaks in many cases, due to either lacking
implicit information flow modeling or neglecting dependency among leakage sites
(hence repetitively counting leakage). CacheAudit only infers the upper bound of
leakage. Thus, they partially satisfy \Circ{4}.
MicroWalk quantifies per-instruction MI to localize vulnerable instructions,
whose quantified leakage per instruction, when added up, should not equal
quantification over the whole-trace MI (its another strategy) due to
program dependencies.
Also, MicroWalk cannot differ randomness (e.g., blinding) with secrets.
It thus partially satisfies \Circ{5}.
DATA~\cite{weiser2018data,weiser2020big} launches trace differentiation and
statistical hypothesis testing to decide secret-dependency of an execution
trace. Similar as \textsc{CacheQL}\xspace, DATA can also analyze non-deterministic traces.
Nevertheless, we find that DATA, by differentiating traces to detect leakage,
may manifest low precision, given it would neglect secret leakage if a cryptographic
module also uses blinding. It thus partially satisfies \Circ{2}.
More importantly, DATA does not deliver quantitative analysis.
Recent research attempts to reconstruct media data like private user photos from
side channels~\cite{yuan2022automated,kwon2020improving,wu2020remove}. In
Table~\ref{tab:detectors}, we compare \textsc{CacheQL}\xspace\ with
Manifold~\cite{yuan2022automated}, the latest work in this field. Manifold
leverages manifold learning to infer media data. Manifold learning is
\textit{not} applicable to infer secret keys (as admitted
in~\cite{yuan2022automated}): unlike media data which contain perception
constraints retained in manifold~\cite{hinton1994autoencoders}, each key bit is
sampled independently and uniformly from 0 or 1. It thus partially fulfills
\Circ{6}. \textsc{CacheQL}\xspace\ is the first to quantify information leaks over cryptographic
keys and media data.
Implicit information flow (\Circ{8}) is not tracked by most existing static
analyzers. Analyzing implicit information flow requires considering more program
statements and data/control flow propagations, which often largely increases the
code chunk to be analyzed. This introduces extra hurdles for static
analysis-based approaches. DATA and MicroWalk also do not systematically capture
implicit information flow. DATA/MicroWalk first \textit{align} traces and then
compare aligned segments, meaning that they can overlook holistic differences
(unalignment) on traces.
\textsc{CacheQL}\xspace\ satisfies \Circ{8} as it directly observes and analyzes changes in the
side channel traces. Given any information flow, either explicit or implicit,
can differ traces, \textsc{CacheQL}\xspace\ captures them in a unified manner. Nevertheless, it is
evident that the implicit information flow cannot be captured by \textsc{CacheQL}\xspace\ unless
it is covered in the dynamic traces.
\smallskip
\parh{Clarification.}~These criteria's importance may vary depending on the
situations. Having only some of these criteria implemented is not necessarily
``bad,'' which may suggest that the tool is targeted for specific use cases.
Analyzing private image leakage (\Circ{6}) may not be as important as others,
especially for cryptosystem developers. We consider image leakage because
recent works~\cite{xu2015controlled,hahnel2017high,yuan2022automated} consider
recovering private media data.
We present eight criteria for building full-fledged side-channel detectors. The
future development of detectors can refer to this paper and prioritize/select
certain criteria, according to their domain-specific need. Also, we clarify that
in parallel to research works that detect side channel leaks, another line of
approaches (i.e., static verification) aims at deriving precise, certified
guarantees~\cite{cacheaudit,daniel2020binsec}.
As clarified above, \textsc{CacheQL}\xspace\ can analyze real attack logs (\Circ{1}) and media
data (\Circ{6}). However, for the sake of presentation coherence, we discuss
them in \S~\ref{sec:discussion}. In the rest of the main paper, we explain
the design and findings of \textsc{CacheQL}\xspace\ using Pin-logged traces from cryptosystems.
\subsection{RQ3: Performance Comparison}
\label{subsec:eval-comparison}
To assess \textsc{CacheQL}\xspace's optimizations and re-formulations, we compare \textsc{CacheQL}\xspace\ with
previous tools on the speed, scalability, and capability of quantification and
localization.
\begin{table}[t]
\caption{Padded length of side channel traces collected using Intel Pin.
The above/below five rows are for SDA/SCB.}
\label{tab:trace-length}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
& Pre-processing & - & Pre-processing & - \\
& Decryption & Decryption & Decryption & Decryption \\
& Blinding & Blinding & - & - \\
\hline
OpenSSL 3.0.0 & $256 \times 256 \times 64$ & $256 \times 256 \times 15$ & $256 \times 256 \times 40$ & $256 \times 256 \times 9$\\
OpenSSL 0.9.7 & $256 \times 256 \times 20$ & $256 \times 256 \times 14$ & $256 \times 256 \times 14$ & $256 \times 256 \times 13$ \\
MbedTLS 3.0.0 & $256 \times 256 \times 16$ & $256 \times 256 \times 2$ & N/A & N/A\\
MbedTLS 2.15.0 & $256 \times 256 \times 14$ & $256 \times 256 \times 2$ & N/A & N/A \\
Libgcrypt 1.9.4 & $256 \times 256 \times 36$ & $256 \times 256 \times 22$ & $256 \times 256 \times 26$ & $256 \times 256 \times 25$ \\
Libgcrypt 1.6.1 & $256 \times 256 \times 5$ & $128 \times 128 \times 19$ & $256 \times 256 \times 11$ & $256 \times 256 \times 10$ \\
\hline
OpenSSL 3.0.0 & $256 \times 256 \times 40$ & $256 \times 256 \times 10$ & $256 \times 256 \times 24$ & $256 \times 256 \times 4$ \\
OpenSSL 0.9.7 & $256 \times 256 \times 8$ & $256 \times 256 \times 5$ & $256 \times 256 \times 5$ & $256 \times 256 \times 4$ \\
MbedTLS 3.0.0 & $256 \times 256 \times 6$ & $256 \times 256 \times 1$ & N/A & N/A\\
MbedTLS 2.15.0 & $256 \times 256 \times 6$ & $256 \times 256 \times 1$ & N//A & N/A \\
Libgcrypt 1.9.4 & $256 \times 256 \times 12$ & $256 \times 256 \times 8$ & $256 \times 256 \times 10$ & $256 \times 256 \times 9$ \\
Libgcrypt 1.6.1 & $256 \times 256 \times 3$ & $128 \times 128 \times 8$ & $256 \times 256 \times 6$ & $256 \times 256 \times 5$ \\
\hline
\end{tabular}
}
\end{table}
\smallskip
\parh{Trace Statistics.}~We report the lengths (after padding) of traces
collected using Pin in Table~\ref{tab:trace-length}. In short, all traces collected
from real-world cryptosystems are lengthy, imposing high challenge for analysis.
Nevertheless, \textsc{CacheQL}\xspace employs encoding module $\mathcal{S}$ and compressing module
$\mathcal{R}$ to effectively process lengthy and sparse traces, as noted in
\S~\ref{sec:framework}.
\smallskip
\parh{Impact of Re-Formulations/Optimizations.}~\textsc{CacheQL}\xspace\ casts MI as CP when
quantifying the leaks. This re-formulation is faster (see comparison below) and
more precise, because calculating MI via MP (as done in MicroWalk) cannot
distinguish blinding in traces. As reported in Table~\ref{tab:trace-length}, a
great number of records are related to blinding, and they lead to false
positives of MicroWalk. For localization, the unoptimized Shapley value has
$\mathcal{O}(2^N)$ computing cost. Given the trace length $N$ is often extremely
large (Table~\ref{tab:trace-length}), computing Shapley value is infeasible. With
our domain-specific optimizations, the cost is reduced as nearly constant.
\subsubsection{Time Cost and Scalability}
\label{subsubsec:scalability}
\begin{table}[t]
\caption{Scalability comparison of static- or trace-based tools.}
\label{tab:scalability}
\centering
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
& CacheD & Abacus & CacheS & CacheAudit \\
\hline
Technique & \multicolumn{2}{c|}{symbolic execution} & \multicolumn{2}{c}{abstract interpretation} \\
\hline
Libgcrypt & fail ($>$ 48h) & fail ($>$ 48h) & fail & fail \\
\hline
Libjpeg & fail & fail & fail & fail \\
\hline
\end{tabular}
}
\vspace{-5pt}
\end{table}
\parh{Scalability Issue of Static-/Trace-Based Tools.}~As noted in \Circ{7}
in \S~\ref{sec:requirement}, prior static- or trace-based analyses rely on
expensive and less scalable techniques. They, by default, primarily analyze a
program/trace cut and neglect those pre-processing functions in cryptographic
libraries. To faithfully assess their capabilities, we configure them to analyze
the entire trace/software (which needs some tweaks on their codebase). We
benchmark them on Libjpeg and RSA of Libgcrypt 1.9.4.
Abacus/CacheD/CacheS/CacheAudit can only analyze 32-bit x86 executable. We thus
compile 32-bit Libgcrypt and Libjpeg.
Results are in Table~\ref{tab:scalability}. CacheS and CacheAudit throw exceptions
of unhandled x86 instruction. Both tools, using rigorous albeit expensive
abstraction interpretation, appear to handle a subset of x86 instructions.
Fixing each unhandled instruction would presumably require defining a new
abstract operator~\cite{cousot1977abstract}, which is challenging on our end.
Abacus and CacheD can be configured to analyze the full trace of Libgcrypt.
Nevertheless, both of them fail (in front of unhandled x86 instructions) after
about 48h of processing. In contrast, \textsc{CacheQL}\xspace\ takes less than 17h to finish the
training and analysis of the Libcrypt case; see Table~\ref{tab:time}.
\begin{table}[t]
\caption{Training time of 50 epochs for the RSA cases.}
\label{tab:time}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c|cccc|cccc}
\hline
& \multicolumn{4}{c|}{SDA} & \multicolumn{4}{c}{SCB} \\
\hline
\multirow{3}{*}{Configuration} & Pre. & - & Pre. & - & Pre. & - & Pre. & - \\
& Dec. & Dec. & Dec. & Dec. & Dec. & Dec. & Dec. & Dec. \\
& Blind. & Blind. & - & - & Blind. & Blind. & - & - \\
\hline
OpenSSL 3.0.0 & $22$h & $5$h & $3$h & $50$min & $13$h & $3$h & $2.5$h & $20$min \\
\hline
OpenSSL 0.9.7 & $6.5$h & $5$h & $1$h & $1$h & $2.5$h & $1.5$h & $25$min & $20$min \\
\hline
MbedTLS 3.0.0 & $5$h & $40$min & N/A & N/A & $2.5$h & $20$min & N/A & N/A \\
\hline
MbedTLS 2.15.0 & $5$h & $40$min & N/A & N/A & $2.5$h & $20$min & N/A & N/A \\
\hline
Libgcrypt 1.9.4 & $12.5$h & $7.5$h & $1.5$h & $1.5$h & $4$h & $2.5$h & $50$min & $45$min \\
\hline
Libgcrypt 1.6.1 & $1.5$h & $1.5$h & $55$min & $50$min & $1$h & $40$min & $30$min & $25$min \\
\hline
\end{tabular}
}
\begin{tablenotes}
\footnotesize
\item 1. Due to the limited space, we use Pre., Dec., and Blind. to denote
Pre-processing, Decryption, and Blinding, respectively.
\item 2. Blind. has $\times 4$ training samples.
\end{tablenotes}
\vspace{-15pt}
\end{table}
\smallskip
\parh{Training/Analyzing Time of \textsc{CacheQL}\xspace.}~Table~\ref{tab:time} presents the RSA case training time, which is
calculated over 50 epochs (the maximal epochs required) on one Nvidia GeForce
RTX 2080 GPU. In practice, most cases can finish in less than 50 epochs.
For AES-128, training 50 epochs takes about 2 mins. Training 50 epochs for
Libjpeg/PathOHeap takes 2-3 hours. As discussed in \S~\ref{subsec:pdf-est},
since we transform computing MI as estimating CP, \textsc{CacheQL}\xspace\ only needs to be
trained (for estimating CP) once. Once trained, it can analyze 256 traces in
\textbf{1-2 seconds} on one Nvidia GeForce RTX 2080 GPU, and less than
\textbf{20 seconds} on Intel Xeon CPU E5-2683 of \textbf{4} cores.
In sum, \textsc{CacheQL}\xspace\ is much faster than existing trace-based/static tools. By using
CP, it principally reduces computing cost comparing with conventional dynamic
tools (see \S~\ref{sec:mi}). We also note that it is hard to make a fully fair
comparison: training \textsc{CacheQL}\xspace\ can use GPU while existing tools \textit{only}
support to use CPUs. Though \textsc{CacheQL}\xspace\ has smaller time cost on the GPU (Nvidia 2080
is \textit{not} very powerful), we do not claim \textsc{CacheQL}\xspace\ is faster than prior
dynamic tools. In contrast, we only aim to justify that \textsc{CacheQL}\xspace\ is \textit{not}
as heavyweight as audiences may expect. Enabled by our theoretical and
implementation-wise optimizations, \textsc{CacheQL}\xspace\ efficiently analyzes complex
production software.
\subsubsection{Capability of Quantification and Localization}
\label{subsubsec:small}
\parh{Small Programs and Trace cuts.}~As evaluated in
\S~\ref{subsubsec:scalability}, previous static-/trace-based tools are
incapable of analyzing the full side channel traces. Therefore, we compare
them with \textsc{CacheQL}\xspace\ using small program (e.g., AES) and trace cuts.
Overall, the speed of \textsc{CacheQL}\xspace\ (i.e., training + analyzing) still largely
outperforms static/trace-based methods. For instance, CacheD~\cite{wang2017cached},
a qualitative tool using symbolic execution, takes about 3.2 hours to analyze
only the decryption routine of RSA in Libgcrypt 1.6.1 without considering blinding.
\textsc{CacheQL}\xspace\ takes under one hour for this setting. In addition,
Abacus~\cite{bao2021abacus}, which performs quantitative analysis with
symbolic execution, requires 109 hours to process one trace of Libgcrypt 1.8.
Note that it only analyzes the decryption module (several caller/callee
functions) without considering the blinding, pre-rocessing functions, etc. In
contrast, \textsc{CacheQL}\xspace\ can finish the training within 2 hours (the trace length of
Libgcrypt 1.9 is about the same as ver. 1.8) in this setting. It's worth noting
that, \textsc{CacheQL}\xspace\ only needs to be trained for once, and it takes only several seconds
to analyze one trace. That is, when analyzing multiple traces, previous tools
has fold increase on the time cost whereas \textsc{CacheQL}\xspace\ only adds several seconds.
The quantification/localization precision of \textsc{CacheQL}\xspace\ is also much higher.
Abacus reports 413.6 bits of leakage for AES-128 (it neglects dependency among
leakage sites, such that the same secret bits can be repetitively counted at
different leakage sites), which is an overestimation since the key has only 128
bits. For RSA trace cuts, Abacus under-quantifies the leaked bits because it
misses many vulnerabilities due to implicit information-flow. When localizing
vulnerabilities in AES-128, we note that all static/trace-based have correct
results. For localization results of RSA trace cuts,
see~\S~\ref{subsubsec:categorization}. In short, none of the previous tools
can identify all the categories of vulnerabilities.
\smallskip
\parh{Dynamic Tools.}~Previous dynamic tools do not suffer from the scalability
issue and have comparable speed with \textsc{CacheQL}\xspace. Nevertheless, they require
re-launching their whole pipeline (e.g., sampling + analyzing) for each trace.
For quantification, MicroWalk over-quantifies the leaked bits of RSA as 1024
when blinding is enabled, since it regards random records as vulnerable.
Similarly, it reports that ORAM cases have all bits leaked despite they are
indeed secure. MicroWalk can correctly quantify the leaks of whole traces
for AES cases and constant-time implementations, because no randomness exists.
Nevertheless, since the same key bits are repeatedly reused on the trace,
its quantification results for single record, when summed up, are incorrectly
inconsistent with the result of whole trace.
For localization, as summarized in \S~\ref{subsubsec:categorization}, previous
dynamic tools are also either incapable of identifying all categories of
vulnerabilities or yields many false positive. For instance, MicroWalk can
regards all records related to blinding (over 1M in OpenSSL 3.0.0; see trace
statistics in Table~\ref{tab:trace-length}) as ``vulnerable''.
\begin{tcolorbox}[size=small]
\textbf{Answer to RQ3}: With domain-specific transformations and optimizations
applied, \textsc{CacheQL}\xspace\ addresses inherent challenges like non-determinism, and
features fast, scalable, and precise quantification/localization. Evaluations
show its advantage over previous tools.
\end{tcolorbox}
\vspace{-5pt}
\subsection{RQ2: Localizing Leakage Sites}
\label{subsec:eval-localization}
This section reports the leakage program points localized in RSA by \textsc{CacheQL}\xspace\ using
Shapley value. We report representative functions in Table~\ref{tab:func-cat}.
See~\cite{snapshot} for detailed reports.
When blinding is enabled, \textsc{CacheQL}\xspace localizes all previously-found
leak sites and hundreds of new ones.
\begin{table}[t]
\caption{Representative vulnerable func. and their categories.}
\label{tab:func-cat}
\centering
\resizebox{1.00\linewidth}{!}{
\begin{tabular}{c|c|c|c|c|c}
\hline
\textbf{OpenSSL 3.0.0 }& Type & \textbf{MbedTLS 3.0.0} & Type & \textbf{Libgcrypt 1.9.4} & Type\\
\hline
\multirow{2}{*}{\texttt{bn_expand2}} & \Type{A}, \Type{B}, & \multirow{2}{*}{\shortstack{\texttt{mbedtls_mpi}\\\texttt{_copy}}} & \Type{A}, \Type{B}, & \multirow{2}{*}{\texttt{mul_n_basecase}} & \Type{B}, \Type{C},\\
& \Type{C} & & \Type{C} & & \Type{D} \\
\hline
\multirow{2}{*}{\texttt{BN_bin2bn}} & \multirow{2}{*}{\Type{A}, \Type{C}} & \multirow{2}{*}{\shortstack{\texttt{mbedtls_mpi}\\\texttt{_read_binary}}} & \multirow{2}{*}{\Type{A}, \Type{C}} & \multirow{2}{*}{\texttt{do_vsexp_sscan}} & \multirow{2}{*}{\Type{A}, \Type{D}}\\
& & & & & \\
\hline
\multirow{2}{*}{\shortstack{\texttt{BN_mod_exp}\\\texttt{_mont}}} & \Type{B}, \Type{C} & \multirow{2}{*}{\texttt{mpi_montmul}} & \Type{B}, \Type{C}, & \multirow{2}{*}{\texttt{_gcry_mpih_mul}} & \Type{B}, \Type{C},\\
& \Type{D}, \Type{E} & & \Type{D} & & \Type{D}, \Type{E} \\
\hline
\end{tabular}
}
\end{table}
\smallskip
\parh{Clarification.}~Some leak sites localized by \textsc{CacheQL}\xspace\ are dependent, e.g.,
several memory accesses within a loop where only the loop condition depends on
secrets. To clarify, \textsc{CacheQL}\xspace\ does not distinguish dependent/independent leak
sites, because from the game theory perspective, those dependent leak sites
(i.e., players) collaboratively contribute to the leakage (i.e., the game).
Also, reporting all dependent/independent leak sites may be likely more
desirable, as it paints the complete picture of the software attack surface.
Overall, identifying \textit{independent} leak sites is challenging, and to our
best knowledge, prior works also do not consider this.
This would be an interesting future work to explore. On the other hand, vulnerabilities
identified by \textsc{CacheQL}\xspace\ are from \textit{hundreds of functions} that are not
reported by prior works, showing that the localized vulnerabilities spread
across the entire codebase, whose fixing may take considerable effort.
\subsubsection{Categorization of Vulnerabilities}
\label{subsubsec:categorization}
We list all identified vulnerabilities
in~\cite{snapshot}. Nevertheless, given the large number of (newly) identified
vulnerabilities, it is obviously infeasible to analyze each case in this paper.
To ease the comparison with existing tools that feature localization, we
categorize leak sites from different aspects. We first categorize the leak sites
according to their locations in the codebase (\Type{A} and \Type{B}). We then
use \Type{D} and \Type{E} to describe how secrets are propagated. Moreover,
since leaking-leading-zeros is less considered by previous work, we specifically
present such cases in \Type{C}.
\smallskip
\noindent \Type{A}~\ul{Leaking secrets in Pre-processing:}~Leak sites belonging
to \Type{A} occur when program parses the key and initializes relevant data
structures like BIGNUM. Note that this stage is rarely assessed by previous
static (trace-based) tools due to limited scalability; empirical results
are given in Table~\ref{tab:scalability}.
\smallskip
\noindent \Type{B}~\ul{Leaking secrets in Decryption:}~While \Type{B} is
primarily analyzed by prior static tools, in practice, they have to trade precision
for speed, omitting analysis of full implicit information flow (\Circ{8} in
Table~\ref{tab:detectors}). Therefore, their findings related to \Type{B} compose
only a small subset of \textsc{CacheQL}\xspace's findings. Also, prior dynamic tools, including
DATA and MicroWalk, are less capable of detecting \Type{B}. This is because
blinding is applied at Decryption (\Circ{2} in Table~\ref{tab:detectors}). DATA
likely neglects leak sites when blinding is enabled since it merely
differentiates logged side channel traces with key fixed/varied. MicroWalk
incorrectly regards data accesses/control branches influenced by blinding as
vulnerable. Blinding can introduce a great number of records (see
Table~\ref{tab:trace-length} for increased trace length), and MicroWalk
fails to correctly analyze all these cases.
\smallskip
\noindent \Type{C}~\ul{Leaking leading zeros:}~Besides \textsc{CacheQL}\xspace, findings
belonging to \Type{C} were only partially reported by DATA. Particularly, given
DATA is less applicable when facing blinding (noted in
\S~\ref{sec:requirement}), it finds \Type{C} only in Pre-processing, where
blinding is not enabled yet. Since \textsc{CacheQL}\xspace\ can precisely quantify (\Circ{4} in
Table~\ref{tab:detectors}) and apportion (\Circ{5}) leaked bits, it is capable of
identifying \Type{C} in Decryption; the same reason also holds for \Type{B}. At
this step, we manually inspected prior static tools and found they only
``taint'' the content of a BIGNUM, which is an array, if BIGNUM stores secrets.
The number of leading zeros, which has enabled exploitations (CVE-2018-0734 and
CVE-2018-0735~\cite{weiser2020big}) and is typically stored in a separate
variable (e.g., \texttt{top} in OpenSSL), is neglected.
\smallskip
\noindent \Type{D}~\ul{Leaking secrets via explicit information flow:}~Most
findings belonging to \Type{D} have been reported by existing static tools.
\textsc{CacheQL}\xspace\ re-discovers \textbf{all} of them despite it's dynamic.
We attribute the success to \textsc{CacheQL}\xspace's precise quantification, which recasts
MI as CP (\S~\ref{subsec:pdf-est}), and localization, where leaks are
re-formulated as a cooperative game (\S~\ref{sec:local}).
\smallskip
\noindent \Type{E}~\ul{Leaking secrets via implicit information flow:}~As
discussed above, prior static tools are incapable of fully detecting \Type{E}.
Also, many findings of \textsc{CacheQL}\xspace\ in \Type{E} overlap with that in \Type{B}. Since
DATA cannot handle blinding well (blinding is extensively used in Decryption), only
a small portion of \Type{E} were correctly identified by DATA. DATA also has the
same issue to neglect \textsc{CacheQL}\xspace's findings in \Type{D}.
In sum, static-/trace-based tools (CacheD, CacheS, Abacus) can detect \Type{B}
$\cap$ \Type{D} but cannot identify \Type{A} $\cup$ \Type{C} $\cup$ \Type{E}. As
noted in \S~\ref{sec:requirement}, MicroWalk cannot properly differ randomness
induced by blinding vs. keys, and is inaccurate for the RSA case with blinding
enabled. DATA pinpoints \Type{A} (accordingly include \Type{A} $\cap$ \Type{C})
and is less applicable for \Type{B}. \textsc{CacheQL}\xspace, due to its precise quantification,
localization, and scalability, can identify \Type{A} $\cup$ \Type{B} $\cup$
\Type{C} $\cup$ \Type{D} $\cup$ \Type{E}.
\subsubsection{Characteristics of Leakage Sites}
\label{subsubsec:characteristics}
The leakage sites exist in all stages of cryptosystems.
Below, we use case studies and the distribution of leaked bits to illustrate
their characteristics.
In short, the leaks start when parsing keys from files and initializing
secret-related BIGNUM, and persist during the whole life cycle of RSA execution.
\begin{figure}[!ht]
\vspace{-5pt}
\centering
\includegraphics[width=1.03\linewidth]{fig/libgcrypt2.pdf}
\vspace{-20pt}
\caption{Simplified vulnerable program points localized in Libgcrypt
1.9.4. This function has SCB directly depending on bits of the key.}
\vspace{-5pt}
\label{fig:libgcrypt}
\end{figure}
\begin{figure*}[!ht]
\centering
\includegraphics[width=1.0\linewidth]{fig/openssl4.pdf}
\vspace{-20pt}
\caption{Vulnerable program points localized in OpenSSL 3.0.0. We mark the
line numbers of \textcolor{pptred}{SDA vulnerabilities} and
\textcolor{pptgreen1}{SCB vulnerabilities} found by \textsc{CacheQL}\xspace. Secrets are
propagated from \Taint{p} to other
\textcolor{pptred}{variables} via explicit or implicit information flow, which
confirm each SDA/SCB vulnerability found by \textsc{CacheQL}\xspace.}
\label{fig:openssl}
\vspace{-5pt}
\end{figure*}
\parh{Case Study$_1$:}~Fig.~\ref{fig:libgcrypt} presents a case newly disclosed
by \textsc{CacheQL}\xspace, which is the key parsing implemented in Libgcrypt 1.6.1 and 1.9.4. As
discussed in \S~\ref{subsubsec:rsa} (see $\text{Case}_2$), this function has SCB
explicitly depending on the key read from files. It therefore contains
\Type{A} and \Type{D}. Similar leaks exist in other software. For instance, as
localized by \textsc{CacheQL}\xspace\ and DATA, the \texttt{EVP_DecodeUpdate} function in two
versions of OpenSSL have SDA via the lookup table \texttt{data_ascii2bin} when
decoding keys read from files.
\smallskip
\parh{Case Study$_2$:}~Fig.~\ref{fig:openssl} depicts the life-cycle of
BIGNUM in OpenSSL 3.0.0, including initialization and computations. We show how
secrets are leaked along the usage of BIGNUM.
\smallskip
\noindent \Boxnum{1} \texttt{BN_bin2bn}@\Line{39}{fig:openssl}: A BIGNUM is
initialized using \Taint{s} at \Line{40}{fig:openssl}, which is parsed
from the key file in the \texttt{.pem} format. A \texttt{for} loop at
\Line{42}{fig:openssl} \textit{skips leading zeros}, propagating \Taint{s} to
\Taint{len} via implicit information flow. Then, \Taint{len} is propagated to
\Taint{top} (\Line{49}{fig:openssl} or \Line{53}{fig:openssl}). Thus, future
usage of \Taint{top} clearly leaks secret.
\smallskip
\noindent \Boxnum{2} \texttt{BN_mod_exp_mont}@\Line{1}{fig:openssl}:
\texttt{BN_num_bits} is called to calculate \#bits (after excluding leading
zeros) of BIGNUM \Taint{p}. \texttt{BN_num_bits} further calls
\texttt{BN_num_bits_word} which we have discussed in \S~\ref{subsubsec:rsa}. \#bits
is stored in \Taint{bits} at \Line{5}{fig:openssl}. Later, \Taint{bits} is
propagated to \Taint{wstart} at \Line{7}{fig:openssl}.
\smallskip
\noindent \Boxnum{3} \texttt{BN_window_bits_for_exponent_size}@\Line{20}{fig:openssl}:
\Taint{w} is propagated from \Taint{bits} at \Line{6}{fig:openssl},
given control branches from \Line{21}{fig:openssl} to \Line{24}{fig:openssl}
directly depend on \Taint{b}.
\smallskip
\noindent \Boxnum{4} \texttt{BN_is_bit_set}@\Line{33}{fig:openssl}:
\Taint{top} of BIGNUM \Taint{p} directly decides the return value at
\Line{36}{fig:openssl}. Its content, namely array \Taint{d}, also sets the
return value at \Line{37}{fig:openssl}. Given \Taint{wvalue} and \Taint{wend} at
\Line{13}{fig:openssl} and \Line{15}{fig:openssl} are updated according to the
return value of \texttt{BN_is_bit_set}, they are thus implicitly propagated.
\smallskip
\noindent \Boxnum{5} \texttt{bn_mul_mont_fixed_top}@\Line{26}{fig:openssl}:
The access to array \texttt{val} at \Line{17}{fig:openssl} is indexed with
\Taint{wvalue}, and therefore, it induces SDA. Variable \Taint{b} at
\Line{27}{fig:openssl} is also propagated via \Taint{wvalue}, and the \texttt{if}
branch at \Line{28}{fig:openssl} thus introduces SCB.
Overall, \Boxnum{1} executes at Pre-processing and is only detected
by DATA and \textsc{CacheQL}\xspace. It has both explicit (\Line{47}{fig:openssl})
and implicit (\Line{42}{fig:openssl}) information flow. Thus, it has
\Type{A}~\Type{C}~\Type{D}~\Type{E}. Similarly, \Boxnum{2} contains
\Type{B}~\Type{C}~\Type{D}~\Type{E}. Both \Boxnum{3} and \Boxnum{4}
have \Type{B}~\Type{C}~\Type{D}. \Boxnum{5} only has \Type{B}~\Type{D}.
Among the leak sites discussed above, only five SCB at
\Line{21}{fig:openssl}-\Line{24}{fig:openssl} and
\Line{36}{fig:openssl} are detected by previous static tools; remaining ones
are newly reported by \textsc{CacheQL}\xspace.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.65\linewidth]{fig/mbedtls2.pdf}
\vspace{-10pt}
\caption{Vulnerable program points localized in MbedTLS 3.0.0. We mark the
line numbers of \textcolor{pptred}{SDA} and \textcolor{pptgreen1}{SCB}.}
\label{fig:mbedtls}
\vspace{-5pt}
\end{figure*}
\smallskip
\parh{Case Study$_3$:}~Fig.~\ref{fig:mbedtls} shows leaking sites disclosed by \textsc{CacheQL}\xspace\ in MbedTLS 3.0.0.
In short, MbedTLS has similar implementation of BIGNUM with OpenSSL, where the
variable \Taint{n} in BIGNUM stores the number of leading zeros. Later computations
rely on \Taint{n} for optimization, for instance, the SDA at \Line{8}{fig:mbedtls}
in \texttt{mbedtls_mpi_div_mpi}@\Line{1}{fig:mbedtls}. It's worth noting that
\texttt{mbedtls_mpi_copy}@\Line{12}{fig:mbedtls}
is extensively called within
the life cycles of all involved BIGNUM, contributing to notable leaks in the
whole pipeline. Similar leaks also exist in MbedTLS 2.15.0. See our
website~\cite{snapshot} for more details.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.04\linewidth]{fig/pie-mbedtls.pdf}
\vspace{-20pt}
\caption{Distribution of top-10 functions in MbedTLS leaking most bits via
either SDA or SCB vulnerabilities. Legend are in descending order.}
\label{fig:pie-mbedtls}
\end{figure}
\smallskip
\parh{Distribution.}~Fig.~\ref{fig:pie-mbedtls} reports the distribution of leaked bits among
top-10 vulnerable functions localized in MbedTLS. The two versions of MbedTLS
primarily leak bits in Pre-processing and have different
strategies when initializing BIGNUMs for CRT optimization. Thus, the
distributions of most vulnerable functions vary. For instance, the most
vulnerable functions in ver. 2.15.0 are for multiplication and division; they
are involved in calculating BIGNUMs for CRT. Notably, \texttt{mbedtls_mpi_copy}
is among the top-5 vulnerable functions on all four charts in
Fig.~\ref{fig:pie-mbedtls}. This function leaks the leading zeros of the input
BIGNUMs via both SDA and SCB. The \texttt{mbedtls_mpi_copy} function, as a
memory copy routine function, is frequently called (e.g., more than 1,000 times
in ver. 2.15.0). Though this function only leaks the leading zeros, given
that its input can be the private key or key-dependent intermediate value, the
accumulated leakage is substantial.
\begin{tcolorbox}[size=small]
\textbf{Answer to RQ2}: \textsc{CacheQL}\xspace\ confirms all known flaws and identifies many
new leakage sites, which span over the life cycle of cryptographic algorithms
and exhibit diverse patterns. Distributions of leaked bits among vulnerable
functions varies notably between software versions.
\end{tcolorbox}
\subsection{RQ1: Quantifying Side Channel Leakage}
\label{subsec:eval-quant}
We report quantitative results over Pin-logged traces.
Table~\ref{tab:quant-aes} and Fig.~\ref{fig:quant-rsa}
summarize the quantitative leakage results computed by \textsc{CacheQL}\xspace\ regarding
different software, where a large amount of secrets are leaked across all
settings. We discuss each case in the rest of this section. Quantitative
analyses of Libjpeg and \texttt{Prime+Probe}\ are presented in Appx.~\ref{appx:extended-eval}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.93\linewidth]{fig/rsa.pdf}
\vspace{-10pt}
\caption{Leaked bits of RSA in different settings. Blinding for Libgcrypt
1.6.1 refers to RSA optimized with CRT (see \S~\ref{subsubsec:rsa}). For cache line
side channels (\texttt{L} in the last row of legend), we present detailed breakdown:
enabling (\textcolor[RGB]{84,130,53}{\ding{51}}\ in the 4th row of legend) vs. disabling
(\textcolor[RGB]{176,35,24}{\ding{55}}\ in the 4th row of legend) blinding, and with (\textcolor[RGB]{84,130,53}{\ding{51}}\ in the 2nd row) vs. w/o
(\textcolor[RGB]{176,35,24}{\ding{55}}\ in the 2nd row) considering Pre-processing.}
\label{fig:quant-rsa}
\vspace{-10pt}
\end{figure*}
\subsubsection{AES}
\label{subsubsec:aes}
\begin{table}[t]
\caption{Leaked bits of AES/AES-NI in OpenSSL/MbedTLS.}
\label{tab:quant-aes}
\centering
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
& OpenSSL 3.0.0 & OpenSSL 0.9.7 & MbedTLS 3.0.0 & MbedTLS 2.15.0 \\
\hline
SCB & 0 & 0 & 0 & 0 \\
\hline
SDA & 128.0 & 128.0 & 0 & 0 \\
\hline
\end{tabular}
}
\end{table}
The side channels of AES collected from the in-house settings are deterministic. The
SDA of AES standard T-table version can leak all key bits, but this
implementation has no SCB~\cite{bao2021abacus,wang2017cached}. These facts
serve as the ground truth for verifying \textsc{CacheQL}\xspace's quantification and localization.
MbedTLS by default uses AES-NI, which has no SDA/SCB. As shown in
Table~\ref{tab:quant-aes}, \textsc{CacheQL}\xspace\ reports no leak in it.
\textsc{CacheQL}\xspace\ reports 128 bits SDA leakage in AES-128 of OpenSSL whereas the SCB
leakage in this implementation is zero. This shows that the quantification of
\textsc{CacheQL}\xspace\ is precise. We distribute the leaked bits to program points via Shapley
value. All 128 bits are apportioned evenly toward 16 memory accesses in function
\texttt{_x86_64_AES_encrypt_compact}. Manually, we find that these 16 memory
accesses are all key-dependent table lookups.
\subsubsection{Test Secure Implementations}
\label{subsubsec:quant-secure}
\textsc{CacheQL}\xspace\ also examines secure cryptographic implementations with no leakage. For these
cases, the quantification derived from \textsc{CacheQL}\xspace\ can also be seen as their correctness
assessments. Given that said, as a dynamic method, \textsc{CacheQL}\xspace\ is for bug detection,
\textit{not} for verification.
\parh{ORAM.}~PathOHeap yields non-deterministic side channels, by randomly
inserting dummy memory accesses to produce highly lengthy traces. Since it takes
several hours to process one logged trace of RSA, we apply PathOHeap on AES from
OpenSSL 3.0.0. Overall, PathOHeap delivers provable mitigation: memory access
traces, after being processed by PathOHeap, should not depend on secrets. \textsc{CacheQL}\xspace\
reports consistent and accurate findings to empirically verify PathOHeap, as the
leaked bit is quantified as \textit{zero}.
\parh{Constant-Time Implementations.}~13 constant-time utilities~\cite{ct-utils}
from Binsec/Rel~\cite{daniel2020binsec} are evaluated using \textsc{CacheQL}\xspace. Side channel
traces from these utilities are deterministic, whose quantified leaks are also
\textit{zero}. These results empirically show the correctness of \textsc{CacheQL}\xspace's
quantification.
\subsubsection{RSA}
\label{subsubsec:rsa}
RSA blinding is enabled by default in production cryptosystems. We quantify
the information leakage of RSA with/without blinding, such that the logged
traces are \textit{non-deterministic} when blinding is on. As noted in
\S~\ref{sec:requirement} (\Circ{7}), prior works mainly focus on the decryption
fragment of RSA due to limited scalability~\cite{wang2017cached,
doychev2017rigorous, wang2019caches, bao2021abacus, brotzman2019casym}. As will
be shown, this tradeoff neglects many vulnerabilities, primarily in the
pre-processing modules of cryptographic libraries, e.g., key parsing and BIGNUM
initialization.\footnote{For simplicity, we refer to the pre-processing
functions of cryptographic libraries as Pre-processing, whereas the following
decryption functions as Decryption.} \textsc{CacheQL}\xspace\ efficiently analyzes the whole
trace, covering Pre-processing and Decryption (see Fig.~\ref{fig:quant-rsa}). As
an ablation, \textsc{CacheQL}\xspace\ also analyzes only Decryption, e.g., the
\textcolor{pptgreen2}{green bar} in Fig.~\ref{fig:quant-rsa}.
\smallskip
\parh{Setup.}~Libgcrypt 1.9.4 uses blinding on both ciphertext and private keys.
We enable/disable them together. Libgcrypt 1.6.1 lacks blinding but implements
the standard RSA and another version using Chinese Remainder Theorem (CRT).
Libgcrypt 1.9.4 uses blinding in the CRT version and disables blinding in the
standard one. We evaluate these two RSA versions in Libgcrypt 1.6.1. MbedTLS
does not allow disabling blinding.
\smallskip
\parh{Results Overview.}~Fig.~\ref{fig:quant-rsa} shows the quantitative results.
Since cache bank only discards two least significant bits of the memory
addresses, it leaks more information than using cache line which discards six
bits. Blinding in modern cryptosystems notably reduces leakage: blinding
influences secret-dependent memory accesses, introducing non-determinism to
prevent attackers from inferring secrets.
Leakage varies across different software and variants of the same software.
Secrets are leaked via SCB and SDA to varying degrees. If blinding is disabled,
the total leak bits when considering only Decryption are close to the whole
pipeline's leakage. This is reasonable as they leak information from the same
source. With blinding enabled, leakage in Decryption is inhibited, and
Pre-processing contributes the most leakage. Though blinding minimizes leakage
in Decryption, Pre-processing remains highly vulnerable, and it is generally
overlooked previously.
\smallskip
\parh{OpenSSL.}~OpenSSL 3.0.0 has higher SCB leakage in Pre-processing with
blinding enabled. As will be discussed in \S~\ref{subsec:eval-localization},
this leakage is primarily introduced by \texttt{BN_bin2bn} and
\texttt{bn_expand2} functions, which convert key from string into BIGNUM. The
issue persists with OpenSSL 0.9.7. Moreover, compared with ver. 3.0.0,
OpenSSL 0.9.7 has more SDA (but less SCB) leakage with blinding enabled. These
gaps are also primarily caused by the \texttt{BN_bin2bn} function in Pre-processing.
We find that OpenSSL 3.0.0 skips leading zeros when converting key from string into
BIGNUM, which introduces extra SCB leakage. In contrast, OpenSSL 0.9.7 first converts
the key with leading zeros into BIGNUM and then uses \texttt{bn_fix_top} to remove
those leading zeros, causing extra SDA leakage. Also, if blinding is disabled,
OpenSSL 0.9.7 leaks approximately twice as many bits as OpenSSL 3.0.0. According to
the localization results of \textsc{CacheQL}\xspace, OpenSSL 0.9.7 has memory accesses and branch
conditions that directly depend on keys, which are vulnerable and lead to over 800
bits of leakage. We manually check OpenSSL 3.0.0 and find that most of those vulnerable
functions have been re-implemented in a constant-time way.
\smallskip
\parh{MbedTLS.}~\textsc{CacheQL}\xspace\ finds many SDA in MbedTLS 3.0.0, which primarily occurs
in the \texttt{mbedtls_mpi_read_binary} and \texttt{mbedtls_mpi_copy} functions
during Pre-processing. The problem is not severe in ver. 2.15.0. We manually
compare the two versions' Pre-processing and find that the CRT initialization
routines differ. In short, MbedTLS 3.0.0 avoids computing DP, DQ and QP (parts
of the RSA private key in CRT) and instead reads them from the PKCS1 structure,
and therefore, \texttt{mbedtls_mpi_copy} function is called for several times.
The 2.15.0 version calculates DP, DQ and QP from the private key via BIGNUM
involved functions (e.g., \texttt{mbedtls_mpi_mul_mpi}). The
\texttt{mbedtls_mpi_copy} function leaks information via SDA and SCB, whereas
the BIGNUM computation in the 2.15.0 version mainly leaks via SCB. This
difference also explains why both versions have many SCB, which are dominated by
their Pre-processing.
\smallskip
\parh{Libgcrypt.}~Libcrypt 1.9.4 has most SCB leakage in Pre-processing with
blinding enabled. Nearly all leaked bits are from the \texttt{do_vsexp_sscan}
function, which parses the key from s-expression. Decryption only leaks
negligible bits. Manual studies show that in Libgcrypt 1.9.4, most
BIGNUM-involved functions in Decryption are constant time and safe.
Nevertheless, \textsc{CacheQL}\xspace\ identifies leaks in \texttt{do_vsexp_sscan}. This
illustrates that \textsc{CacheQL}\xspace\ comprehensively analyzes production cryptosystems,
whereas developers neglect patching all sensitive functions in a constant-time
manner, enabling subtle leakages. Also, both versions have SDA leakage primarily
in the \texttt{_gcry_mpi_powm} function; this is also noted in prior
works~\cite{wang2017cached,wang2019caches,brotzman2019casym}. As aforementioned,
Libgcrypt 1.9.4 uses the standard RSA without CRT when blinding is disabled. The
1.6.1 version does not offer blinding for both the standard RSA and the CRT
version. It's obvious that the standard version leaks more than the CRT version.
\smallskip
\parh{Correctness.}~It is challenging to obtain ground truth in our evaluation.
Aside from the AES cases and secure implementations in
\S~\ref{subsubsec:quant-secure} who have the ``ground truth'' (either leaking
128 bits or zero bits) to compare with, there are several cases in RSA whose
leaked bits can be calculated manually, facilitating to assess the correctness
of \textsc{CacheQL}\xspace's quantification.
\noindent \textbf{$\text{Case}_1$:}~\texttt{BN_num_bits_word} function in
OpenSSL 0.9.7, which is first identified by CacheD~\cite{wang2017cached} and
currently fixed in OpenSSL 3.0.0, has 256 different entries depending on
secrets. It leaks $-\log \frac{1}{256}$ = $8.0$ bits, in case entries are
accessed evenly (which should be true since key bits are generated independently
and uniformly). \textsc{CacheQL}\xspace\ reports the leakage as $7.4$ bits, denoting a close
quantification.
\noindent \textbf{$\text{Case}_2$:}~\texttt{do_vsexp_sscan} function (see
Fig.~\ref{fig:libgcrypt}) in both versions of Libgcrypt has control branches
depending on whether a secret is greater than 10. The SCB at
\Line{2}{fig:libgcrypt} of Fig.~\ref{fig:libgcrypt}, in theory, leaks $-\log
\frac{1}{16} + \log \frac{1}{10}$ = $0.68$ bits of information, as the possible key
values are reduced from 16 to 10 when \Line{2}{fig:libgcrypt} is executed.
Similarly, the SCB at \Line{4}{fig:libgcrypt} leaks $-\log \frac{1}{16} + \log
\frac{1}{6}$ = $1.42$ bits. When \textsc{CacheQL}\xspace\ analyzes one trace, it apportions around
$1$ bit to each of the two records corresponding to the SCB at
\Line{2}{fig:libgcrypt} and \Line{4}{fig:libgcrypt}. We interpret that \textsc{CacheQL}\xspace\
provides accurate quantification and apportionment for this case.
\begin{tcolorbox}[size=small]
\textbf{Answer to RQ1}: By quantifying leakage with \textsc{CacheQL}\xspace, we find that
information leaks are prevalent in cryptosystems, even when hardening
methods (e.g., blinding) are enabled. Most leaks reside in the pre-processing
stage neglected by existing research. For some cases, the development of
cryptosystems may increase the amount of leakage. The correctness of \textsc{CacheQL}\xspace\
is empirically validated using a total of 24 instances of known bit leakages.
\end{tcolorbox}
\section{Evaluation}
\label{sec:evaluation}
We evaluate \textsc{CacheQL}\xspace\ by answering the following research questions. \textbf{RQ1:}
What are the quantification results of \textsc{CacheQL}\xspace\ on production cryptosystems and
are they correct? \textbf{RQ2:} How does \textsc{CacheQL}\xspace\ perform on localizing side
channel vulnerabilities? What are the characteristics of these localized sites?
\textbf{RQ3:}~What are the impact of \textsc{CacheQL}\xspace's optimizations, and how does \textsc{CacheQL}\xspace\
outperform existing tools?
We first introduce evaluation setups below.
\subsection{Evaluation Setup}
\label{subsec:setup}
\noindent \textbf{Software.}~We evaluate T-table-AES and RSA in OpenSSL 3.0.0,
MbedTLS 3.0.0, and Libgcrypt 1.9.4. We consider an end-to-end pipeline where
cryptographic libraries load the private key from files and decrypt ciphertext
encrypted from ``hello world.'' We quantify input image leaks for Libjpeg-turbo
2.1.2. We use the \textit{latest} versions (by the time of writing) of all these
software. We also assess old versions, OpenSSL 0.9.7, MbedTLS 2.15.0 and
Libgcrypt 1.6.1, for a cross-version comparison. Some of them were also analyzed
by existing works~\cite{wang2017cached,wang2019caches,bao2021abacus}. We compile
software into 64-bit x86 executable using their default compilation settings.
Supporting executables on other architectures is feasible, because \textsc{CacheQL}\xspace's core
technique is platform-independent.
\noindent \textbf{Libjpeg \& \texttt{Prime+Probe}.}~For the sake of presentation coherence, we
focus on cryptosystems under the in-house setting (i.e., collecting execution
traces via Pin) in the evaluation. Experiments of Libjpeg (including quantified
leaks and localized vulnerabilities) and \texttt{Prime+Probe}\ are in Appx.~\ref{appx:extended-eval}.
\noindent \textbf{Data Preparing \& Training.}~When collecting the data for
training/analyzing, we fix the public input and randomly sample keys
to generate side-channel traces.
For AES, we use the Linux
\texttt{urandom} utility to generate 40K 128-bit keys for estimating CP using
their corresponding side channel traces (collected via Pin or \texttt{Prime+Probe}). We also
generate 10K extra keys and their side channel traces to de-bias non-determinism
induced by ORAM (\S~\ref{subsubsec:quant-secure}). The same keys are used for
benchmarking AES of all cryptosystems.
For RSA, we follow the same setting but generate 1024-bit private keys using
OpenSSL.
We have no particular requirements for training data (e.g. achieving certain
coverage) --- we observe that execution flows of cryptosystems are not largely
altered by different keys, except that key values may influence certain loop
iterations (e.g., due to zero bits). We find that the execution flows of
cryptosystems are relatively more ``regulated'' than general-purpose software,
which is also noted previously~\cite{wang2017cached}. If secrets could notably
alter the execution flow, it may indicate obvious issues like timing side
channels, which should have been primarily eliminated in modern cryptosystems.
\noindent \textbf{Trace Logging.}~Pin is configured to log program memory access
traces to detect cache side channels due to SDA. We primarily consider cache
side channels via cache lines and cache banks: for an accessed memory address
\texttt{addr}, we compute its cache line and bank index as $\texttt{addr} >> 6$
and $\texttt{addr} >> 2$, respectively. We also consider SCB, where Pin logs all
control transfer destinations. Cache line/bank indexes are computed in the same
way. We clarify that cache bank conflicts are inapplicable in recent Intel CPUs;
we use this model for easier empirical comparison with prior
works~\cite{wang2017cached,bao2021abacus,wang2019caches,yuan2020private,yuan2022automated}.
Trace statistics are presented in Table~\ref{tab:trace-length}.
\parh{Ground Truth.}~To clarify, \textsc{CacheQL}\xspace\ does \textit{not} require the ground truth
of leaked bits. Rather, as discussed in \S~\ref{subsubsec:cp}, \textsc{CacheQL}\xspace\ is trained to
\textit{distinguish} traces produced when the software is processing different secrets.
The ground truth is a one-bit variable denoting whether trace $o$ is generated when
the software processing secret $k$.
\parh{Non-Determinism.}~We quantify the leaks when enabling RSA blinding
(\S~\ref{subsubsec:rsa}). We also evaluate PathOHeap~\cite{shi2020path},
a popular ORAM protocol, and consider real attack logs.
\section{Background \& Motivating Example}
\label{sec:example}
\parh{Application Scope.}~\textsc{CacheQL}\xspace\ is designed as a \textit{bug detector}. It
shares the same design goal with previous detectors~\cite{wang2017cached,
cacheaudit, wang2019caches, weiser2018data, jan2018microwalk}, whose main
audiences are developers who aim to test and ``debug'' software. \textsc{CacheQL}\xspace\ is
incapable of synthesizing proof-of-concept (PoC) exploits and is hence incapable
of launching real attacks. In general, exploiting cache side channels in the
real world is often a multi-step procedure~\cite{liu2016cache} that involves
pre-knowledge of the target systems and manual efforts. It is challenging, if
not impossible, to fully automate the process. For instance, exploitability may
depend on the specific hardware
details~\cite{liu2014random,liu2016cache,yarom2017cachebleed}, and in cloud
computing, the success of co-residency attacks denotes a key pre-condition of
launching exploitations~\cite{zhang2011homealone}. These aspects are not
considered by \textsc{CacheQL}\xspace\ which performs software analysis.
Given that said, we evaluate \textsc{CacheQL}\xspace\ by quantifying information leaks over side
channel traces logged by standard \texttt{Prime+Probe}\ attack, and as a proof of concept, we
extend \textsc{CacheQL}\xspace\ to reconstruct secrets/images with reasonable quality over logged
traces (see details in Appx.~\ref{appx:reconstruction}). We believe these
demonstrations show the potential and extensibility of \textsc{CacheQL}\xspace\ in practice.
\smallskip
\parh{Threat Model.}~Aligned with prior works in this
field~\cite{bao2021abacus,cacheaudit,wang2017cached,wang2019caches,brotzman2019casym},
we assume attackers share the same hardware platforms with victim software.
Attackers can observe cache being accessed when victim software is running.
Attackers can log all cache lines (or other units) visited by the victim
software as a side channel trace~\cite{liu2016cache,yarom2017cachebleed}.
Given a program $g$, we define the attacker's observation, a side channel trace,
as $o \in O$ when $g$ is executing $k \in K$. $O$ and $K$ are sets of all
observations and secrets. $K$ can be cryptographic keys or user private inputs like
photos. We consider a ``debug'' scenario where developers measure leakage when
$g$ executes $k$. Aligned with prior
works~\cite{weiser2018data,jan2018microwalk}, we assume that developers can
obtain noise-free $o$, e.g., $o$ is execution trace logged by Pin. We also
assume developers are interested in assessing leaks under real attacks. Indeed,
OpenSSL by default only accepts side channel reports exploitable in real
scenarios~\cite{opensslpatch}. We thus also launch standard \texttt{Prime+Probe}\ attack to log
cache set accesses. We aim to quantify information in $k$ leaked via $o$. We
also analyze leakage distribution across program points to localize flaws.
Developers can prioritize patching vulnerabilities leaking more information.
\smallskip
\parh{Two Vulnerablities: Secret-Dependent Control Branch and Data Access.}~Our
threat model focuses on two popular vulnerability patterns that are analyzed and
exploited previously, namely, secret-dependent control branch (SCB) and
secret-dependent data access (SDA)~\cite{7163050, cacheaudit,
doychev2017rigorous, wang2017cached, wang2019caches, brotzman2019casym,
bao2021abacus}. SDA implies that memory access is influenced by secrets, and
therefore, monitoring which data cache unit is visited may likely reveal
secrets~\cite{yarom2017cachebleed}. SCB implies that program branches are
decided by secrets, and monitoring which branch is taken via cache may likely
reveal secrets~\cite{7163050}. \textsc{CacheQL}\xspace\ captures both SCB and SDA, and it models
secret information flow. That is, if a variable $v$ is influenced (``tainted'')
by secrets via either explicit or implicit information flow, then control flow
or data access that depends on $v$ are also treated as SCB and SDA. The
definition of SDA/SCB is standard and shared among previous
detectors~\cite{wang2017cached,cacheaudit,wang2019caches,weiser2018data,jan2018microwalk}.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.02\linewidth]{fig/demo-quant.pdf}
\vspace{-25pt}
\caption{Two pseudocode code of secret leakage. The secrets are 1024-bit keys.
$\texttt{s[i:j]}$ are bits between the $i$-th (included) and $j$-th bit
(excluded).}
\label{fig:demo-quant}
\end{figure}
\parh{Detecting SDA Using \textsc{CacheQL}\xspace.\footnote{SCB can be detected in the same way,
and is thus omitted here.}}~Consider two vulnerable programs depicted in
Fig.~\ref{fig:demo-quant}. In short, two program points in
Fig.~\hyperref[fig:demo-quant]{1(a)} have 128 (\Line{6}{fig:demo-quant}) and 512
(\Line{11}{fig:demo-quant}) memory accesses that are secret-dependent (i.e.,
SDA). Developer can use Pin to log one memory access trace $o$ when executing
Fig.~\hyperref[fig:demo-quant]{1(a)}, and by analyzing $o$, \textsc{CacheQL}\xspace\ reports a total
leakage of 768 bits.
\textsc{CacheQL}\xspace\ further apportions the SDA leaked bits as: 1) 2 bits for each of 128 memory accesses at
\Line{6}{fig:demo-quant}, and 2) 1 bit for each of 512 memory accesses at
\Line{11}{fig:demo-quant}.
For Fig.~\hyperref[fig:demo-quant]{1(b)}, two array lookups at
\Line{10}{fig:demo-quant} and \Line{12}{fig:demo-quant} depend on the secret.
Given a memory access trace $o$, \textsc{CacheQL}\xspace\ quantifies the leakage as 510 bits and
apportions 255 bits for each SDA. We discuss technical details of \textsc{CacheQL}\xspace\ in
\S~\ref{sec:mi}, \S~\ref{sec:framework}, and \S~\ref{sec:local}.
\smallskip
\parh{Comparison with Existing Quantitative Analysis.\footnote{We discuss their
analysis about SDA; SCB is conceptually the
same.}}~MicroWalk~\cite{jan2018microwalk} measures information leakage via
mutual information (MI). However, we find that its output is indeed mundane
Shannon entropy rather than MI over different program execution traces, since
both key and randomness like blinding can differ traces. MicroWalk has two
computing strategies: whole-trace and per-instruction. For
Fig.~\hyperref[fig:demo-quant]{1(b)}, MicroWalk reports 1024 leaked bits using the
whole-trace strategy. The per-instruction strategy localizes three leakage
program points, where each point leaks 1024 (\Line{6}{fig:demo-quant}), 255
(\Line{10}{fig:demo-quant}), and 255 (\Line{12}{fig:demo-quant}) bits,
respectively. However, it is clear that those 1024 memory accesses at
\Line{6}{fig:demo-quant} are decided by \textit{non-secret} randomness. Thus,
both quantification and localization are inaccurate.
Abacus~\cite{bao2021abacus} uses trace-based symbolic execution to measure
leakage at each SDA, by estimating number of different secrets (\texttt{s}) that
induces the access of different cache units. No implicit information flow is
modelled, thereby omitting to ``taint'' the memory access at
\Line{11}{fig:demo-quant} of Fig.~\hyperref[fig:demo-quant]{1(a)}. Abacus
quantifies leakage of Fig.~\hyperref[fig:demo-quant]{1(a)} over $o$ as 256 bits,
since it only finds SDA at \Line{6}{fig:demo-quant}.
Program points may have dependencies. For instance, one branch may have its
information leaked in its parent branch, and therefore, separately adding them
together largely over-estimates the leakage:
Abacus outputs a total leakage of 413.6 bits in AES-128, despite its 128-bit
key length.
\textsc{CacheQL}\xspace\ precisely calculates the leakage as 128.0 bits
(\S~\ref{subsubsec:aes}). Also, some static
analyses~\cite{cacheaudit,doychev2017rigorous,chattopadhyay2019quantifying} have
limited scalability due to heavyweight abstract interpretation or symbolic
execution. Real-world cryptosystems and media software are complex, with millions of
records per side channel trace. In addition, they are often unable to localize
vulnerable points.
\section{Key Properties of Shapley Values}
\label{appx:sv-properties}
Following \S~\ref{sec:local}, this Appendix shows several key properties of
Shapley value. We discuss why it is suitable for side channel analysis and how
the properties help our localization.
To clarify, the following theorems are \textit{not} proposed by this paper;
they are for reader's reference only.
\vspace{-5pt}
\begin{theorem}[Efficiency~\cite{shapley201617}]
\label{th:efficiency}
The sum of Shapley value of all participants (i.e., side channel records)
equals to the value of the grand coalition:
\begin{equation}
\sum_{i \in R^o} \pi_{i}(\phi) = \phi(o) - \phi(o_{\emptyset}).
\end{equation}
\end{theorem}
Theorem~\ref{th:efficiency} states that the assigned Shapley value for each side
channel record satisfies the apportionment defined in Definition~\ref{def:apportion}.
$\phi(o_{\emptyset}) = 0$ since an empty $o$ leaks no secret.
\vspace{-5pt}
\begin{theorem}[Symmetry~\cite{shapley201617}]
\label{th:symmetry}
If $\forall S \subseteq R^{o} \setminus \{i,j\}$, $i$- and $j$-th players
are equivalent, i.e., $\phi(o_{S \cup \{i\}}) = \phi(o_{S \cup \{j\}})$,
then $\pi_{i}(\phi) = \pi_{j}(\phi)$.
\end{theorem}
Theorem~\ref{th:symmetry} states records in $o$ contributing equally to
leakage have the same Shapley value. Divergent Shapley values suggest divergent
leakage on the relevant program points. It ensures that all contributions are
awarded Shapley values.
\begin{theorem}[Dummy Player~\cite{shapley201617}]
\label{th:dummy}
If the $i$-th participant is a dummy player, i.e.,
$\forall S \subseteq R^{o} \setminus \{i\}, \phi(o_{S \cup \{i\}}) - \phi(o_S)
= \phi(o_{\{i\}}) - \phi(o_{\emptyset})$, then
$\pi_{i}(\phi) = \phi(o_{\{i\}}) - \phi(o_{\emptyset})$.
\end{theorem}
Theorem~\ref{th:dummy}, dubbed as ``Dummy Player,'' states that the information
leakage in one program point is not distributed to non-correlated points. In
sum, Theorem~\ref{th:symmetry} and Theorem~\ref{th:dummy} guarantee that the Shapley
value apportionment is \textbf{accurate}. Further, if $\phi(o_{\{i\}}) =
\phi(o_{\emptyset})$, we have the following theorem.
\vspace{-5pt}
\begin{subtheorem}[Null player~\cite{shapley201617}]
\label{th:null}
If the $i$-th participant has no contribution to any grand coalition game $\phi$,
i.e., $\forall S \subseteq R^{o} \setminus \{i\}, \phi(o_{S \cup \{i\}}) = \phi(o_S)$,
then $\pi_{i}(\phi) = 0$.
\end{subtheorem}
Theorem~\ref{th:null} is one special case of Theorem~\ref{th:dummy}, and it guarantees
that the Shapley value has \textbf{no false negative}. That is, program points
assigned with a zero Shapley value is guaranteed to not contribute to
information leakage.
\vspace{-5pt}
\begin{theorem}[Linearity~\cite{shapley201617}]
\label{th:linearity}
If two coalition games, namely $\phi$ and $\psi$, are combined, then
$\pi_{i}(\phi + \psi) = \pi_{i}(\phi) + \pi_{i}(\psi)$ for
$\forall i \in R^{o}$.
\end{theorem}
Theorem~\ref{th:linearity} implies that if a secret has several independent
components, then the assigned Shapley value for each side channel record equals
to the linear sum of secrets leaked on this record from all components.
For instance, let $\pi(\phi_{g})$ and $\pi(\phi_{a})$ be the leaked information
by recovering two privacy-related properties, ``gender'' and ``age,'' over
portrait photos. Since these two properties are independent, when considering
both, the newly-computed leakage $\pi(\phi)$ must equal the sum of
$\pi(\phi_{g})$ and $\pi(\phi_{a})$ according to Theorem~\ref{th:linearity}. While
this property allows for fine-grained leakage analysis, we currently do not
separate a secret into independent components. We view this exploration as one
future work.
\vspace{-5pt}
\begin{theorem}[Uniqueness~\cite{shapley201617}]
\label{th:uniqueness}
The apportionment derived from Shapley value is the only one that
simultaneously satisfies Theorem~\ref{th:efficiency}, Theorem~\ref{th:symmetry},
Theorem~\ref{th:dummy} \& \ref{th:null}, and Theorem~\ref{th:linearity}.
\end{theorem}
\section{Responsible Disclosure}
\label{appx:confirmation}
\begin{table}[t]
\vspace{-5pt}
\caption{Statistics of reported vulnerabilities.}
\label{tab:report}
\centering
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{c|c|c|c}
\hline
& Reported flaws & Answered & Acknowledged flaws \\
\hline
OpenSSL & 10 (functions) & 10 & 10 \\
\hline
MbedTLS & 62 (leakage sites) & 62 & 62 \\
\hline
Libgcrypt & 5 (functions) & 0 & 0 \\
\hline
Libjpeg & 2 (functions) & 2 & 2 \\
\hline
\end{tabular}
}
\end{table}
We have reported our findings to developers. Since we find many vulnerabilities
in each cryptosystems and media software (see a complete list at~\cite{snapshot}),
we summarize representative cases and also clarify the leakage to developers to
seek prompt confirmation. Sometimes, as required, we further annotate the vulnerable
statements (e.g., for MbedTLS developers). Table~\ref{tab:report} lists the exact numbers
of vulnerable functions/program points we reported.
By the time of writing, OpenSSL developers have confirmed BIGNUM related
findings. MbedTLS developers also positively responded to our reports.
We are discussing with them the disclosure procedure. Libjpeg developers agreed
with our reported SDA/SCB cases but required PoC exploitations and patches to assess
cost vs. benefit. We did not receive response from Libgcrypt developers.
\section{Transformations in $\mathcal{R}$}
\label{appx:r-detail}
As stated in \S~\ref{sec:framework}, we adopt different mathematical
transformations in $\mathcal{R}$ for media data and cryptographic keys.
In what follows, we elaborate on compressing latent vectors of
media data and secret keys with different transformations in $\mathcal{R}$.
\parh{Media Data.}~We use image to demonstrate the cases for
media data; the conclusion can be extended to other media data
straightforward~\cite{yuan2022automated}. It is generally obscure to measure the
amount of information encoded in high-dimensional media data like
images~\cite{zhang2018unreasonable}. To extract information, an image, before
being fed into modern neural networks (e.g., classifier $\mathcal{C}$ in \textsc{CacheQL}\xspace),
is generally normalized into $[-1, 1]$ and represented as a
$channel \times width \times height$ matrix~\cite{krizhevsky2012imagenet}. Let
a private image be $k_m$ and the associated side channel log be $o_m$, we set
the output of $\mathcal{S}(o_{m})$ as a matrix of the same size. Accordingly,
our compressor $\mathcal{R}_{m}$ is implemented using the
$tanh: [-\infty, \infty] \rightarrow [-1, 1]$ function, facilitating
$H(o_m) \leq H(k_m)$. $tanh$ is parameter free
(i.e., $\theta_{\mathcal{R}} = \emptyset$), eliminating extra training cost.
Also, if certain properties in an image are particularly desirable by
attackers, e.g., the gender of portrait images, $k_m$ and the matrix from
$\mathcal{S}(o_{m})$ can be replaced with a vectorized representation for image
properties. We refer readers to~\cite{pumarola2018ganimation} for
vectorizing image attributes.
\parh{Cryptographic Keys.}~For a cryptographic key $k_c$ of length $L$, there
are total $2^{L}$ uniformly distributed key instances. The information in one
key instance is thus $H(k) = \log 2^{L} = L$ bits. Since a key only contains
binary values and neural networks are hard to be deployed with binary
parameters, for the associated side channel log $o_c$,
we need to transform the floating-point encoding output $\mathcal{S}(o_c)$,
which is accordingly a vector of length $L$, into binary bits. Existing
profiling-based side channel attacks map side channels to keys via $L$ bit-wise
classification tasks ~\cite{hospodar2011machine,kim2019make,hettwer2018profiled}.
In each task, a floating point is transformed to 1 if it is greater than a
threshold. Nevertheless, this transformation is not applicable in $\mathcal{R}$,
given that it is not differentiable, which impedes the optimization of $\theta$.
Inspired by the optimization for binary variables
~\cite{courbariaux2015binaryconnect,courbariaux2016binarized}, we design
$\mathcal{R}_{c}$ by joining two components: 1) a non-parametric $sigmoid:
[-\infty, \infty] \rightarrow [0, 1]$ function which generates $L$ independent
(as bits in $k_c$ are independent) probabilities, and 2) a parametric Bernoulli
distribution $Bern: \Pr(Bern(\cdot)=1) + \Pr(Bern(\cdot)=0) = 1$ which takes
its inputs, i.e., $sigmoid(\mathcal{S}(o_{c}))$, as parameters for optimization.
Thus, we have $\mathcal{R}_{c} = Bern \circ sigmoid$ and
$\theta_{\mathcal{R}_{c}} = sigmoid(\mathcal{S}(o_{c}))$.
See our codebase~\cite{snapshot} for details.
Recall as we discussed in \S~\ref{sec:framework}, $\mathcal{R}$ is designed for
shrinking maximal information in a side channel trace $o$, enabled by this
principle, \textsc{CacheQL}\xspace\ is ``forced'' to focus on secret-related information.
Outputs of $\mathcal{R}$ are shown in Fig.~\ref{fig:face}: it captures common
facial features, such as eyes and mouth, when quantifying information leaks
for human photos.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\linewidth]{fig/face.pdf}
\vspace{-5pt}
\caption{Outputs of $\mathcal{R}$ when quantifying leakage of images.}
\label{fig:face}
\end{figure}
\section{Reconstruct Secrets From Side Channels}
\label{appx:reconstruction}
Appx.~\ref{appx:r-detail} demonstrates that $\mathcal{R}$ captures facial attributes
when quantifying leakage of portrait photos. Readers may question whether \textsc{CacheQL}\xspace\
can directly reconstruct secrets from side channels.
To this end, we extend \textsc{CacheQL}\xspace\ by tweaking the optimization objectives (only adds
\textit{one} line of code; see our manuals~\cite{snapshot}), and present results in
Table~\ref{tab:recons}. We show reconstructed images on our
website~\cite{snapshot}. Overall, keys/images with reasonable quality are reconstructed using traces logged by
Intel Pin or \texttt{Prime+Probe}. In particular, reconstructed images have more vivid appearance
than Fig.~\ref{fig:face}; it's reasonable since quantifying leaks emphasizes on
the \textit{distinguishability}. These promising results illustrate the
superiority of \textsc{CacheQL}\xspace's design and extendability, and the exploitability of
localized program points. Having stated that, we clarify that aligned with prior
works in this field, \textsc{CacheQL}\xspace\ is a bug detector, \textit{not} for reconstructing
secret in real attacks. Also, as a proof of concept, the extended \textsc{CacheQL}\xspace\ shows
the feasibility of key/image reconstruction; the quality of reconstructed
keys/images may be further improved~\cite{yuan2022automated}, which we leave as
one future work.
\begin{table}[t]
\caption{Quantitative evaluation of the reconstruction.}
\label{tab:recons}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c|c|c|c}
\hline
& $^{*}$Libjpeg (Pin) & $^{\dagger}$OpenSSL 0.9.7 (Pin) & $^{*}$Libjpeg (Prime+Probe) \\
\hline
DA/D cache & 41.5\% & 58.2\% & 36.8\% \\
\hline
CB/I cache & 40.4\% & 57.7\% & 36.0\% \\
\hline
\end{tabular}
}
\begin{tablenotes}
\footnotesize
\item * We report percentage of reconstructed photos that are recognized as the
same faces with the reference photos. We use the criterion of~\cite{yuan2022automated}.
\item $\dagger$ Pre-processing + Decryption with blinding enabled.
We report percentage of correct bits. Random guess (baseline) should be 50\%.
\end{tablenotes}
\end{table}
\section{Extended Proof of Eq.~\ref{equ:appx}}
\label{appx:correctness}
This Appendix section proves Eq.~\ref{equ:appx} in a two-step approach; we refer
the following proof skeleton to~\cite{belghazi2018mutual,tsai2020neural}.
Since $c(k, o)$ is only decided by $\mathcal{F}_{\theta}$, for simplicity, we also
denote it as a parameterized function.
In particular, Lemma~\ref{lem:est} first shows that the empirically measured
$\hat{I}^{(n)}_{\theta^{\dagger}}(K; O)$ is consistent with
$\hat{I}_{\theta^{\dagger}}(K; O)$. Lemma~\ref{lem:appx} then proves that
$\hat{I}_{\theta^{\dagger}}(K; O)$ stays close to $I(K; O)$.
In all, Tsai et al.~\cite{tsai2020neural} has pointed out that the following two
prepositions hold for neural networks:
\begin{proposition}[Boundness]
\label{prop:boundness}
$\forall \hat{c}_{\theta}$ and $\forall k, o$, there exist two constant
bounds $B_l$ and $B_u$ such that $B_l \leq \log \hat{c}_{\theta}(k, o) \leq B_u$.
\end{proposition}
\begin{proposition}[$\log$-smoothness]
\label{prop:log}
$\forall k, o$ and $\forall \theta_1, \theta_2 \in \Theta$,
$\exists \alpha > 0$ such that
$| \log\hat{c}_{\theta_1}(k, o) - \log\hat{c}_{\theta_2}(k, o)| \leq \alpha \| \theta_1 - \theta_2 \|$.
\end{proposition}
Proposition~\ref{prop:boundness} states that outputs of a neural network are bounded
and Proposition~\ref{prop:log} clarifies that the output of a neural network will not
change too much if the parameter $\theta$ change slightly~\cite{tsai2020neural}.
By incorporating the bounded rate of uniform convergence on parameterized
functions~\cite{bartlett1998sample}, we have:
\begin{lemma}[Estimation]
\label{lem:est}
$\forall \epsilon > 0$,
\begin{equation*}
\begin{aligned}
&\Pr\limits_{\{\langle k, o \rangle \}^{(n)}} \left(
\sup\limits_{\hat{c}_{\theta^{\dagger}} \in \mathcal{C}} \left|
\hat{I}_{\theta^{\dagger}}^{(n)}(K; O) - \mathbb{E}_{P_{K \times O}} [
\log \hat{c}_{\theta^{\dagger}} (k, o)
]
\right| \geq \epsilon
\right) \\
&\leq 2 |\Theta| \exp \left(
\frac{- n \epsilon^2}{2(B_u - B_l)^2}
\right).
\end{aligned}
\end{equation*}
\end{lemma}
Lemma~\ref{lem:est} applies the classical consistency
theorem~\cite{geer2000empirical} for extremum estimators. Here, extremum
estimators denote parametric functions optimized via maximizing or minimizing
certain objectives; note that $\hat{c}_\theta$ is optimized to maximize the
binary cross-entropy in Eq.~\ref{equ:bce}. It illustrates that
$\hat{I}^{(n)}_{\theta^{\dagger}}(K; O)$ convergent to
$\hat{I}_{\theta^{\dagger}}(K; O)$ as $n$ grows. Future, based the universal
approximation theory of neural networks~\cite{hornik1989multilayer}, we have:
\begin{lemma}[Approximation]
\label{lem:appx}
$\forall \epsilon > 0$, there exists a family of neural networks
$N = \{ \hat{c}_{\theta}: \theta \in \Theta\}$ such that
\begin{equation*}
\inf\limits_{\hat{c}_{\theta} \in N}
|\mathbb{E}_{P_{K \times O}}[\log \hat{c}_{\theta}(k, o)] - I(K, O)|
\leq \epsilon.
\end{equation*}
\end{lemma}
Lemma~\ref{lem:appx} states that $\hat{I}_{\theta^{\dagger}}(K; O)$ can
approximate $I(K; O)$ with arbitrary accuracy. Therefore, Eq.~\ref{equ:appx} can
be derived from Lemma~\ref{lem:est} and Lemma~\ref{lem:appx} based on the
triangular inequality.
\section{Libjpeg and \texttt{Prime+Probe}\ Evaluation}
\label{appx:extended-eval}
\parh{Data Preparing.}~We choose the CelebA dataset~\cite{liu2015faceattributes}
as the image inputs of Libjpeg. The CelebA consists of 160,000 training and 10,000
validation images. We use the training images and their corresponding side channel
traces to estimate CP via $\mathcal{F}_{\theta}$\xspace. The validation images and their induced side
channels are adopted for de-biasing (in case there exists non-determinism).
\parh{Trace Logging.}~Following the same configuration of
\S~\ref{sec:evaluation}, we use Pin to log execution traces of Libjpeg.
\parh{Logging via \texttt{Prime+Probe}.}~Besides using Pin, we collect cache set access traces
(for both cryptosystems and Libjpeg) via
\texttt{Prime+Probe}~\cite{tromer2010efficient}, in userspace-only scenarios.
Following~\cite{yuan2022automated}, we use Mastik~\cite{yarom2016mastik}, a
micro-architectural side channel toolkit, to perform \texttt{Prime+Probe}\ and log victim's
access toward L1D and L1I cache. We use Linux \texttt{taskset} to pin victim
software and the spy process on the same CPU core. Scripts of \texttt{Prime+Probe}\
experiments are at~\cite{snapshot}.
\subsection{Libjpeg}
\label{appx:libjpeg}
\parh{Quantification.}~Libjpeg does not contain mitigations like blinding. Thus, side channels
collected by Pin are deterministic. Nevertheless, we argue that merely considering
the difference among SDA/SCB, which is a conventional setup for cryptosystems,
will \textit{over-estimate} the leakage. The reasons are two-fold: 1) Traces can
be largely divergent by tweaking trivial pixels in the input images, e.g.,
background color pixels. 2) Even if all pixels are equally sensitive, the
leakage may be insufficient to recover input images. For instance, attackers
only infer whether each pixel value is larger than a threshold (as how keys are
usually recovered) and obtain a binary image --- it's still infeasible to
recognize humans in such images.
To handle these, we extend the \textit{generalizability} consideration, which is
proposed to handle non-deterministic side channels derived from cryptosystems, to quantify deterministic side channels made by Libjpeg. Therefore,
our quantification is the same as for processing non-deterministic side channels
of cryptosystems. Due to the aforementioned issue, we deem trivial pixels
contain little information. As introduced in \S~\ref{subsec:non-det}, \textsc{CacheQL}\xspace\
should accordingly report a zero leakage since these non-sensitive factors are
not generalizable.
Given that the \#images is infinite, it's infeasible to decide the value of
$p(F)/p(T)$ in Eq.~\ref{equ:cond-prob} which should be a constant. We thus report
the leakage ratio for Libjpeg. As shown in Table~\ref{tab:quant-libjpeg}, leakage
ratios reach 100\% for SDA and SCB if we only consider the
\textit{distinguishability}, which indicates side channels can differ images.
Nevertheless, the leakage ratios are reduced to 93\% for SDA and 49\% for SCB
when we faithfully take \textit{generalizability} into account. To further prove
that \textsc{CacheQL}\xspace\ indeed captures the critical information in face photos, we present
the output of $\mathcal{R}$ in Fig.~\ref{fig:face}. The output manifests
common human face features like eyes and noses.
\begin{table}[t]
\caption{Leakage ratios (see Eq.~\ref{equ:non-deter}) of Libjpeg
without $\rightarrow$ with considering generalizability.}
\label{tab:quant-libjpeg}
\centering
\resizebox{0.7\linewidth}{!}{
\begin{tabular}{c|c|c}
\hline
& SDA & SCB \\
\hline
cache line & 100\% $\rightarrow$ 93\% & 100\% $\rightarrow$ 49\% \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[t]
\caption{Representative vulnerable functions
localized in Libjpeg and their types.}
\label{tab:loc-libjpeg}
\centering
\resizebox{0.75\linewidth}{!}{
\begin{tabular}{c|c}
\hline
Function & Type \\
\hline
\texttt{decode_mcu} & SDA, SCB \\
\hline
\texttt{jsimd_ycc_extbgrx_convert_avx2} & SDA \\
\hline
\texttt{jsimd_idct_islow_avx2} & SDA, SCB \\
\hline
\end{tabular}
}
\end{table}
\parh{Localization.}~Representative vulnerable functions of Libjpeg are given in
Table~\ref{tab:loc-libjpeg}. The leaked bits are spread across hundreds of program
points, mostly from the IDCT, YCC encoding, and MCU modules. Libjpeg converts
JPEG images into bitmaps, whose procedure has many SDA and SCB. We manually
checked our findings, which are \textit{aligned} with~\cite{yuan2022automated}.
Nevertheless, the YCC encoding-related functions are newly found by us. \textsc{CacheQL}\xspace\
also shows that SDA in IDCT leaks the most bits, whereas the MCU modules leak
more bits via SCB.~\cite{yuan2022automated} flags those issues without
quantification.
\begin{table}[t]
\caption{Leaks of side channels collected via \texttt{Prime+Probe}.}
\label{tab:quant-pp}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
& RSA D & RSA D & RSA D & RSA D \\
\hline
\#Repeating & 1 & 2 & 4 & 8 \\
\hline
Leakage & 19.3 (1.8\%) & 29.4 (2.8\%) & 34.9 (3.4\%) & 35.5 (3.4\%) \\
\hline
& RSA D & RSA I & Libjpeg D & Libjpeg I\\
\hline
\#Repeating & 16 & 8 & 8 & 8 \\
\hline
Leakage & 35.6 (3.4\%) & 21.6 (2.1\%) & 20.8\% & 12.9\% \\
\hline
\end{tabular}
}
\begin{tablenotes}
\footnotesize
\item 1. ``D'' denotes L1 D cache whereas ``I'' denotes L1 I cache.
\item 2. We also report leakage ratios for RSA cases to ease comparison.
\end{tablenotes}
\end{table}
\subsection{\texttt{Prime+Probe}}
\label{appx:pp}
Following previous setups~\cite{yuan2022automated}, a common \texttt{Prime+Probe}\ is launched on
the same core with the victim software and periodically probes L1 cache when the
victim software is executed. This mimics a practical and powerful attacker and
is also the default setup of the \texttt{Prime+Probe}\ toolkit~\cite{yarom2016mastik} leveraged
in our evaluation and relevant research in this field.
To prepare traces about Pre-processing, we halt the victim program after
the pre-processing stage.
Generally, launching \texttt{Prime+Probe}\ is costly. Without loss of generality, we use \texttt{Prime+Probe}\ to collect RSA
side channels from OpenSSL 0.9.7 and Libjpeg. Attackers often repeat launching
\texttt{Prime+Probe}~\cite{Zhang12}. As in Table~\ref{tab:quant-pp}, repeating an attack to collect
more logs does improve information, but only marginal. Compared to merely doing
\texttt{Prime+Probe}\ once, repeating $\times$4 yields more information. However, repeating more
times does not necessarily improve, since the information source is always the
same.
We also note that Pre-processing has more leaks: for two (RSA, \#Repeating=8) cases of
L1 D and I cache (i.e., the 2nd and 3rd columns in the last row), Pre-processing has
22.5 (total 35.5) and 14.1 (total 21.6) leaked bits.
Libjpeg leaks more information than RSA. Not like recovering keys where each key
bit needs to be analyzed, recovering every pixel is not necessary for inferring
images. As previously stated~\cite{yuan2022automated}, pixels conform to specific
constraints to form meaningful contents (e.g., a human face), which typically
have lower dimensions than pixel values. As a result, extracting these
constraints can give rich information already.
\section{Framework Design}
\label{sec:framework}
Fig.~\ref{fig:framework} shows the pipeline of \textsc{CacheQL}\xspace, including three components:
1) a sparse encoder $\mathcal{S}$ for converting side channel traces $o^*$ into
latent vectors, 2) a compressor $\mathcal{R}$ to shrink information in $o^*$,
and 3) a classifier $\mathcal{C}$ that fits the CP in Eq.~\ref{equ:cond-prob} via
binary classification.
We compute CP $p(T|k^*, o^*)$ using the following pipeline:
\vspace{-5pt}
\begin{equation}
\label{equ:pipeline}
{
\begin{aligned}
p(T|k^*, o^*)
= \mathcal{F}_{\theta}(k^*, o^*) =
\mathcal{C} (k^*, \mathcal{R}(\mathcal{S}(o^*))),
\end{aligned}
}
\end{equation}
\vspace{-10pt}
\noindent where parameters of these three components are jointly optimized,
i.e., $\theta = \theta_{\mathcal{S}} \cup \theta_{\mathcal{R}} \cup
\theta_{\mathcal{C}}$.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.0\linewidth]{fig/framework2.pdf}
\vspace{-23pt}
\caption{The framework of \textsc{CacheQL}\xspace. $\langle k^*, o^*\rangle$$\in$$P_{K \times O}$;
it is labeled as $T$. In contrast, $\langle k', o^*\rangle$$\in$$P_K P_O$ and is
labeled as $F$.}
\label{fig:framework}
\end{figure}
The framework takes a tuple $\langle k, o \rangle$ as input. As introduced in
\S~\ref{subsec:pdf-est}, we label a tuple $\langle k, o \rangle$ as positive if
$o$ is produced when the software is processing $k$. A tuple is otherwise
negative. In Fig.~\ref{fig:framework}, $\langle k^*, o^* \rangle$ is positive and
$\langle k', o^* \rangle$ is negative.
\smallskip
\parh{$\mathcal{S}$: Encoding Lengthy and Sparse Side Channel Traces.}~According
to our tentative experiments, naive neural networks perform poorly when analyzing
real-world software due to the \textit{highly lengthy} side channel traces. An
$o$, obtained via Pin or cache attacks, typically contains millions of records,
exceeding the capability of typical neural networks. ORAM can add dummy memory
accesses, often resulting in a tenfold increase of trace
length. Yuan et. al~\cite{yuan2022automated} found that side channel traces are generally
\textit{sparse}, with few secret-dependent records. It also has spatial
locality: adjacent records on a trace often come from the same or related
functions. Encoder $\mathcal{S}$ is inspired by~\cite{yuan2022automated}: to
approximate the locality, we fold the trace into a matrix (see configurations in
\S~\ref{sec:evaluation}). We employ the design in~\cite{yuan2022automated}
to construct $\mathcal{S}$ as a stack of convolutional NN layers. We find that our pipeline
effectively extracts informative features from $o$.
\smallskip
\parh{$\mathcal{R}$: Shrinking Maximal Information.}~A side channel trace $o^*$
frequently contains information unrelated to secret $k^*$. Our preliminary study
shows that directly bridging the latent vectors (outputs of $\mathcal{S}$) to
$\mathcal{C}$ is difficult to train. This stage thereby compresses
$\mathcal{S}$'s output in an information-dense manner. We propose that
\textit{information\footnote{Only secret-related information. Non-secret
variables (e.g., public inputs) that affect $o$ are regarded as randomness
and handled as in \S~\ref{subsec:non-det}.} in $o^*$, namely $H(o^*)$, should never exceed $H(k^*)$}.
Accordingly, we apply mathematical transformations $\mathcal{R}$ to confine the
value range of $\mathcal{S}$'s outputs. We propose various transformations for
media data and secret keys; details are in Appx.~\ref{appx:r-detail}. In short,
$\mathcal{R}$ aids in retrieving secret-related information from side channels.
As demonstrated in Appx.~\ref{appx:r-detail}, \textsc{CacheQL}\xspace\ effectively extracts facial
attributes when estimating leakage of human photos.
\smallskip
\parh{$\mathcal{C}$: Optimizing Parameters via Classification.}~Let the
parameter space be $\Theta$ and $\theta \in \Theta$. To train a
neural network, we search for parameter $\theta^{\dagger} \in \Theta$ to
maximize a pre-defined objective. As shown in Eq.~\ref{equ:cond-prob}, we recast
leakage estimation as approximating CP $p(T|k, o)$, which is further formed
as a classification task using $\mathcal{F}_{\theta}$\xspace. $\theta$ is updated by gradient-guided
search in $\Theta$ to maximize the following objective:
\begin{equation}
\label{equ:bce}
{
\begin{aligned}
\mathbb{E}_{P_{K \times O}} [\log \mathcal{F}_{\theta}(k, o)]
+ \mathbb{E}_{P_K P_O} [\log (1 - \mathcal{F}_{\theta}(k, o))],
\end{aligned}
}
\end{equation}
\noindent which is a standard binary cross-entropy loss over $P_{K \times O}$
and $P_K P_O$. Overall, this loss function compares the output of $\mathcal{F}_{\theta}$\xspace\ to the
ground truth, and it calculates the score that penalizes $\mathcal{F}_{\theta}$\xspace\ based on its
output distance from the expected value.
\parhs{$\bullet$ Example:}~Consider the program in Fig.~\ref{fig:demo-entropy}, in which we
have \colorbox{pptgreen}{$\langle \texttt{0}, \texttt{a} \rangle$} labeled as $T$.
To prepare $P_K P_O$, there is one
\colorbox{pptyellow}{$\langle \texttt{0}, \texttt{a} \rangle$} marked as $F$
when randomly combining $k$ and $o$ separated from pairs in $P_{K \times O}$.
Thus, $\mathcal{F}_{\theta}(\text{``\texttt{0}''}, a[0])$ is simultaneously
guided to output 1 and 0 with equal penalty. As expected, it eventually
yields 0.5 to minimize the global penalty, which outputs a leakage of 4 bits
(since $|K|=4$) following Eq.~\ref{equ:optimal-trace}.
\smallskip
\parh{Computing PD.}~Let the optimized parameter be $\theta^{\dagger}$, our definition of PD in
Eq.~\ref{equ:cond-prob} is re-expressed in the following way to compute
point-wise information leak of $k^*$ in its derived $o^*$:
\begin{equation}
\label{equ:optimal-trace}
{
c_{\theta^{\dagger}}(k^*; o^*) = |K| \frac{\mathcal{F}_{\theta^{\dagger}}(k^*, o^*)}
{1 - \mathcal{F}_{\theta^{\dagger}}(k^*, o^*)}
}
\end{equation}
\noindent Furthermore, we have the following program-level information
leak assessment over $K$ and $O$.
\begin{equation}
\label{equ:optimal-prog}
{
I(K; O) = \mathbb{E}_{P_{K \times O}}
\left[\log c_{\theta^{\dagger}} (k, o)
\right]
}
\end{equation}
\parh{Approximation and Correctness.}~Having access to all samples from a
distribution $P$ is difficult, if not impossible. As a common approximation, the
objective in Eq.~\ref{equ:bce} is instead optimized over the \textit{empirical}
distribution $P^{(n)}$ produced by $n$ samples drawn from $P$. Thus, the
estimated leakage becomes:
\begin{equation}
{
\hat{I}^{(n)}_{\theta^{\dagger}}(K; O) = \mathbb{E}_{P^{(n)}_{K \times O}}
[\log \hat{c}_{\theta^{\dagger}} (k, o)].
}
\end{equation}
Despite we estimate MI for side channels, the skeleton for analyzing
\textit{correctness} can be adopted from prior
works~\cite{tsai2020neural,belghazi2018mutual}, since all approaches involve
optimizing parameterized neural networks. In particular, we prove that $\exists
\theta^{\dagger} \in \Theta$,
\begin{equation}
\label{equ:appx}
{
|\hat{I}^{(n)}_{\theta^{\dagger}}(K; O) - I(K; O)| \leq \mathcal{O}(
\sqrt{\log (1 / \delta) / n}),
}
\end{equation}
\noindent with probability at least $1 - \delta$ where $0 < \delta < 1$. We
present detailed proofs in Appx.~\ref{appx:correctness}.
\section{Implementation}
\label{sec:implementation}
We implement \textsc{CacheQL}\xspace\ in PyTorch (ver. 1.4.0) with about 2,000 LOC. The
$\mathcal{C}$ of \textsc{CacheQL}\xspace\ uses convolutional neural networks for images and
fully-connected layers for keys; see details in~\cite{snapshot}.
We use Adam optimizer with learning rate $0.0002$ for all models. We find that
the learning rate does not largely affect the training process (unless it is
unreasonably large or small). Batch size is 256. We ran experiments on Intel
Xeon CPU E5-2683 with 256GB RAM and a Nvidia GeForce RTX 2080 GPU.
For experiments based on Pin-logged traces,
\S~\ref{subsubsec:scalability} presents the training time: \textsc{CacheQL}\xspace\ is generally comparable
or faster than prior tools. Experiments for \texttt{Prime+Probe}-logged traces
take 1--2 hours.
\section{Introduction}
\label{sec:introduction}
Cache side channels enable confidential data leakage through shared data and
instruction caches. Attackers can recover program secrets like secret keys and
user inputs by monitoring how victim software accesses cache units. Exploiting
cache side channels has been shown particularly effective for cryptographic
systems such as AES, RSA, and
ElGamal~\cite{goldreich1996software,tromer2010efficient}. Recent attacks show
that private user data including images and text can be
reconstructed~\cite{xu2015controlled,hahnel2017high,yuan2022automated}.
Both attackers and software developers are in demand to quantify and localize
software information leakage. It is also vital to precisely distribute
information leaks toward each vulnerable program point, given that exploiting
program points that leak more information can enhance an attacker's success
rate. Developers should also prioritize fixing the most vulnerable program
points. Additionally, cyber defenders are interested in assessing subtle
information leaks over cryptosystems already hardened by mitigation
techniques (e.g., blinding).
Nevertheless, most existing cache side channel detectors focus exclusively on
qualitative analysis, determining whether programs are vulnerable without
\textit{quantifying} information that these flaws may
leak~\cite{wang2017cached,wang2019caches,weiser2018data,brotzman2019casym,yuan2022automated}.
Given the complexity of real-world cryptosystems and media libraries, scalable,
automated, and precise vulnerability localization is lacking. As a result, developers
may be likely reluctant (or unaware) to remedy vulnerabilities discovered by existing
detectors. As shown in our evaluation (\S~\ref{sec:evaluation}), attack vectors
in production software are underestimated.
This work initializes a comprehensive view on detecting cache side-channel
vulnerabilities. We propose \textit{eight criteria} to design a full-fledged
detector. These criteria are carefully chosen by considering various important
aspects like scalability. Then, we propose \textsc{CacheQL}\xspace, an automated detector for
production software that meets all eight criteria. \textsc{CacheQL}\xspace\ quantifies information
leakage via mutual information (MI) between secrets and side channels. \textsc{CacheQL}\xspace\
recasts MI computation as evaluating conditional probability (CP),
characterizing \textit{distinguishability} of side channel traces induced by
different secrets. This re-formulation largely enhances computing efficiency and
ensures that \textsc{CacheQL}\xspace's quantification is more precise than existing works. It also
principally alleviates \textit{coverage issue} of conventional dynamic methods.
We also present a novel vulnerability localization method, by formulating
information leak via a side channel trace as \textit{a cooperative game} among
all records on the trace. Then, Shapley value~\cite{shapley201617}, a
well-established solution in cooperative game theory, helps to localize program
points leaking secrets. We rely on domain observations (e.g., side channel
traces are often sparse) to reduce the computing cost of Shapley value
from $\mathcal{O}(2^{N})$ to roughly constant with nearly no loss in
precision.\footnote{$N$, the length of a side channel trace, reaches 5M in
OpenSSL 3.0 RSA.}
\textsc{CacheQL}\xspace\ directly analyzes binary code, and captures both explicit and implicit
information flows. \textsc{CacheQL}\xspace\ analyzes \textit{entire} execution traces (existing
works require traces to be cut to reduce complexity) and overcomes
``non-determinism'' introduced by noises or hardening techniques (e.g.,
cryptographic blinding, ORAM~\cite{goldreich1996software}).
We evaluate \textsc{CacheQL}\xspace\ using production cryptosystems including the latest
versions (by the time of writing) of OpenSSL, Libgcrypt and MbedTLS. We also
evaluate Libjpeg by treating user inputs (images) as privacy. To mimic
debugging~\cite{wang2017cached}, we collect memory access traces of target software using Intel
Pin as inputs of \textsc{CacheQL}\xspace.\footnote{Using Intel Pin to log memory access
traces is a common setup in this line of works. \textsc{CacheQL}\xspace, however, is not specific
to Intel Pin~\cite{pin}.} We also mimic automated real attacks in userspace-only scenarios,
where highly noisy side channel logs are obtained via
\texttt{Prime+Probe}~\cite{tromer2010efficient} and fed to \textsc{CacheQL}\xspace. \textsc{CacheQL}\xspace\ analyzed 10,000 traces
in 6 minutes and found hundreds of bits of secret leaks per software.
These results confirm \textsc{CacheQL}\xspace's ability to pinpoint all known vulnerabilities
reported by existing works~\cite{wang2019caches,weiser2018data}
and quantify those leakages. \textsc{CacheQL}\xspace\ also discovers hundreds of unknown
vulnerable program points in these cryptosystems, spread across hundreds of
functions never reported by prior works. Developers promptly confirmed
representative findings of \textsc{CacheQL}\xspace. Particularly, despite the adoption of
constant-time paradigms to harden sensitive components, cryptographic software is not
fully constant-time, whose non-trivial secret leaks are found and quantified by
\textsc{CacheQL}\xspace. \textsc{CacheQL}\xspace\ reveals the \textit{pre-process} modules, such as key
encoding/decoding and BIGNUM initialization, can leak many secrets and affect
\textit{all} modern cryptosystems evaluated. In summary, we have the
following contributions:
\begin{itemize}[noitemsep,topsep=0pt]
\item We propose eight criteria for systematic cache side-channel
detectors, considering various objectives and restrictions. We design \textsc{CacheQL}\xspace,
satisfying all of them;
\item \textsc{CacheQL}\xspace\ reformulates mutual information (MI) with conditional probability
(CP), which reduces the computing error and cost efficiently. It then estimates
CP using neural network (NN). Our NN can properly handle lengthy side channel
traces and analyze secrets of various types. Moreover, it does \textit{not}
require manual annotations of leakage in training data;
\item \textsc{CacheQL}\xspace\ further uses Shapley value to localize program points leaking
secrets by simulating leakage as a cooperative game. With domain-specific
optimizations, Shapley value, which is computational infeasible, is
calculated with a nearly constant cost;
\item \textsc{CacheQL}\xspace\ identifies subtle leaks (even with RSA blinding enabled), and its
correctness has theoretical guarantee and empirical supports. \textsc{CacheQL}\xspace\ also
localizes all vulnerable program points reported by prior works and hundreds
of unknown flaws in the latest cryptosystems. Our representative findings
are confirmed by developers. It illustrates the general concern that BIGNUM
and pre-processing modules are largely leaking secrets and undermining recent
cryptographic libraries.
\end{itemize}
\noindent \textbf{Research Artifact.}~To support follow-up research, we release the
code, data, and all our findings at
\url{https://github.com/Yuanyuan-Yuan/CacheQL}~\cite{snapshot}.
\section{Apportioning Information Leakage}
\label{sec:local}
We analyze how leakage over $\langle o^*, k^* \rangle$ is apportioned among
program points. This section models information leakage as a \textit{cooperative
game} among players (i.e., program points). Accordingly, we use Shapley
value~\cite{shapley201617}, a well-developed game theory approach, to apportion
player contributions.
\parh{Overview.}~We use Shapley value (described below) to
\textit{automatically} flag certain records on a trace that contribute to leakage.
Those flagged records are \textit{automatically} mapped to assembly instructions
using Intel Pin, since Pin records the memory address of each executed instruction.
We then \textit{manually} identify corresponding vulnerable source code. We report
identified vulnerable source code to developers and have received timely confirmation
(see Appx.~\ref{appx:confirmation} for the disclosure details and their responses).
To clarify, this step is not specifically designed for Pin; users may replace Pin
with other dynamic instrumentors like Qemu~\cite{bellard2005qemu} or
Valgrind~\cite{nethercote2007valgrind}.
Shapley value decides the contribution (i.e., leaked bits) of each program point
covered on \textit{one} trace $o$. To compute the average leakage (as reported
in \S~\ref{subsec:eval-localization}), users can analyze multiple traces and
average the leaked bits at each program point.
We now formulate information leakage as a cooperative game and define leakage
apportionment as follows.
\begin{definition}[Leakage Apportionment]
\label{def:apportion}
Given total $n$ bits of leaked information and $m$ program points covered on
the Pin-logged trace, an apportionment scheme allocates each program point
$a_i$ bits such that $\sum_{i=1}^{m}a_i = n$.
\end{definition}
\parh{Shapley Value.}~We address the leakage apportionment via Shapley value.
Recall that each observation $o$ denotes a trace of logged
side channel records when target software is processing a secret $k$. Let
$\phi(o)$ be the leaked bits over one observation $o$, and let $R^o$ be the set
of indexes of records in $o$, i.e., $R^o = \{1, 2, \cdots, |o|\}$. For
\textit{all} $S \subseteq R^{o} \setminus \{i\}$, the Shapley value for the
$i$-th side channel record is formally defined as
\begin{equation}
\label{equ:shapley}
\pi_{i}(\phi) = \sum\limits_{S}
\frac{|S|!(|R^o| - |S| - 1)!}{|R^o|!}[\phi(o_{S \cup \{i\}}) - \phi(o_{S})],
\end{equation}
\noindent where $\pi_{i}(\phi)$ represents the information leakage contributed
by the $i$-th record in $o$. $o_{S}$ denotes that only records whose indexes in
$S$ serve as players in this cooperative game, and accordingly $o_{R^o} = o$.
Eq.~\ref{equ:shapley} is based on the intuition that contribution of a player
(i.e., its Shapley value) should be decided by its marginal contribution to all
$2^{|o| - 1}$ coalitions over the remaining players. All players cooperatively
form the overall leakage $\phi(o)$.
\subsection{Computation and Optimization}
\label{subsec:optimization-sv}
The conventional procedure of deciding each player's
contribution $\pi_{i}(\phi)$ for $\phi$ requires to generate a collection of
variants $V$ over $o$, where in each variant $o_v \in V$, some players
\texttt{involved} and others \texttt{removed}~\cite{shapley201617}. In our
scenario, it is infeasible to however remove a player when estimating
leakage---removing a side channel record requires a new $\phi$. Similar
to~\cite{lundberg2017unified}, we propose to involve or remove a side channel
record from $o$ as follows:
\vspace{-5pt}
\begin{definition}[\texttt{Involved}]
\label{def:involved}
The $i$-th record of $o$ gets involved in $\phi(o)$ if $o[i]$ is retained.
\end{definition}
\vspace{-5pt}
\begin{definition}[\texttt{Removed}]
\label{def:removed}
The $i$-th record of $o$ is removed from $\phi(o)$ if $o[i]$ is reset to
a constant, namely ``$base$''.
\end{definition}
The intuition is that, given $k$, if all records in its derived side channel
observation $o$, are set to the same constant, i.e., $o_{\emptyset} = [base,
\ldots, base]$, it's obvious that $o_{\emptyset}$ leaks no information of $k$,
namely $\phi(o_{\emptyset}) = 0$. Conversely, by gradually setting
$o_{\emptyset}[i] = o[i]$, which turns into $ o_{\{i\}}$, we finally have
$\phi(o_{R^o}) = \phi(o)$. The $base$ is $0$ in our setting for simplicity.
As stated in \S~\ref{sec:local}, computing Shapley value is costly, with
complexity $\mathcal{O}(2^{|o|})$. This is particularly challenging, since a side
channel trace $o$ frequently contains millions of records. We now propose
several simple yet highly effective optimizations which successfully reduce the
time complexity to (nearly) constant. These optimizations are based on domain
knowledge and observations about side channel traces.
\smallskip
\parh{Approximating All and Tuning Later.}~Shapley value given in
Eq.~\ref{equ:shapley} can be equivalently expressed as~\cite{castro2009polynomial}:
\begin{equation}
\label{equ:shapley-sampling}
\begin{aligned}
\pi_{i}(\phi) &= \sum\limits_{u \in {\rm Perm}(R^o)} \frac{1}{|R^o|!}
[\phi(o_{u^{i} \cup \{i\}}) - \phi(o_{u^{i}})] \\
&= \mathbb{E}_{{\rm Perm}(R^o)}
[\phi(o_{u^{i} \cup \{i\}}) - \phi(o_{u^{i}})],
\end{aligned}
\end{equation}
\noindent where $u$ is a permutation\footnote{A set of permuted indexes. Note that the permutation
does not exchange side channel records; it provides an order
for records to get \texttt{involved}.} that assigns each position $t$ a
player indexed with $u(t)$ and ${\rm Perm}(R^o)$ is the set of all permutations
over side channel records with indexes in $R^o$. $u^{i}$ is the set of all
predecessors of the $i$-th participant in permutation $u$, e.g., if $i = u(t)$,
then $u^{i} = \{u(1), \cdots, u(t-1)\}$.
This equation transforms the computation of Shapley values into calculating the
expectation over the distribution of $u$. Each time for a randomly selected $u$,
we can calculate $\phi(o_{\{u(1), \cdots, u(j)\}})$ and $\pi_{u(j)}(\phi)$ for
all $j$ by incrementally setting $o[u(j)]$ as \texttt{involved}. Each
$\pi_{u(j)}(\phi)$ is further updated as more permutations $u$ are sampled.
From the implementation side, Eq.~\ref{equ:shapley} iteratively calculates
accurate Shapley values for each record (but too slow), whereas
Eq.~\ref{equ:shapley-sampling} approximates Shapley values for all side channel
records and tunes the values in later iterations of updates. We point out that
Eq.~\ref{equ:shapley-sampling} is more desirable for side channels, because
\textit{not all side channel records are correlated}. That is, updating Shapley
value for one record may not affect the results of other records (i.e.,
``Dummy Players''; see Theorem~\ref{th:dummy} in Appx.~\ref{appx:sv-properties}).
Given that sample mean converges to the true expectation when \#samples
increases, $\pi_{u(j)}(\phi)$ reaches its true value when it gets convergent. As
a result, the calculation can be terminated early to reduce overhead, once the
Shapley values stay unchanged. Our empirical results show that the Shapley
values have negligible changes (i.e., the maximal difference of adjacent updates
is less than 0.5) after only tens of updates.
\parh{Pruning Non-Leaking Records Using Gradients.}~As discussed
in \S~\ref{sec:framework}, real-world software often generates \textit{lengthy}
and \textit{sparse} side channel records~\cite{{bao2021abacus,yuan2022automated,brotzman2019casym}}.
That is, usually only a few records in
a trace $o$ really contribute to inferring secrets, and most records are ``Null
Players'' (has no leak; see Theorem~\ref{th:null} in Appx.~\ref{appx:sv-properties}) in this cooperative game. Recall that the $\phi$
is formed by neural networks, whose gradients are typically informative. Here,
we use gradients to prune Null Players before computing the standard Shapley
values. In general, neural networks characterize the influence of one input
element (i.e., one record on $o$) via gradients, and the volumes of gradients over
inputs reflect how sensitive the output is to local perturbations on these input
elements: higher volumes suggest more important elements.
We first rank all records by gradient volumes. Then, starting with the top one, we
gradually set each record as \texttt{removed}. This way, we expose Null Players,
as they are the remaining ones left when the leakage is zero. We find that, by
setting at most a few hundred records as \texttt{removed} (which is far less
than $|o|$), the leakage can be reduced to zero.
\smallskip
\parh{Batch Computations.}~The above optimizations reduce cost from
$\mathcal{O}(2^{|o|})$ to hundreds of calls to $\phi$. Further, modern hardware
offers powerful parallel computing, allowing neural networks to accept a batch
of data as inputs. Therefore, we batch the computations formed in previous steps;
eventually, with \textit{one or two batched calls} to $\phi$, whose cost is
(nearly) constant and negligible, we obtain accurate Shapley values.
\smallskip
\noindent \textbf{Error Analysis.}~Our above approximation of Shapley value is
1) \textit{unbiased}: it arrives the ground truth value with enough iterations.
It is also 2) \textit{convergent}: such that we can finish iterating whenever
the approximated value unchanged.
Let the estimated and ground truth Shapley value be $\hat{\pi}$ and $\pi$.
Previous studies have pointed out that
$\hat{\pi} \sim \mathcal{N}(\pi, \frac{\sigma^2}{m})$ where $\mathcal{N}$
is the normal distribution and $m$ is \#iterations.
It is also proved that $\sigma^2 < \frac{(\pi_{\max} - \pi_{min})^2}{4}$ where
$\pi_{\max}$ and $\pi_{\min}$ are the maximum and minimal $\hat{\pi}$ during
all iterations~\cite{castro2009polynomial,castro2017improving}.
\smallskip
\parh{Accuracy.}~Shapley value is based on several important properties that
ensure the accuracy of localization~\cite{shapley201617}. In short,
\smallskip
\begin{tcolorbox}[size=small]
enabled by Shapley value, leakage localization, as a cooperative game,
is precise with nearly no false negatives.
\end{tcolorbox}
The standard Shapley value ensures no false negatives (see Theorem~\ref{th:null} in
Appx.~\ref{appx:sv-properties}). Nevertheless, since we trade accuracy for scalability to
handle lengthy $o$, our optimized Shapley value may have a few false negatives.
Empirically, we find that it is rare to miss a vulnerable program point, when
cross-comparing with findings of previous works~\cite{wang2019caches}.
\section{Quantifying Information Leakage}
\label{sec:mi}
\parh{Overview.}~This section discusses quantitative measurement of information leaks
over side channel observations. We start with preliminaries in \S~\ref{subsec:problem}.
The overview of our approach is illustrated in Fig.~\ref{fig:overview}.
\S~\ref{subsec:mi-pdf} introduces MI computation via Point-wise Dependence (PD).
Then, \S~\ref{subsec:pdf-est} recasts calculating PD into computing conditional
probability (CP). \textsc{CacheQL}\xspace\ employs parameterized neural networks $\mathcal{F}_{\theta}$
(see \S~\ref{sec:framework}) to estimate CP, which is carefully designed to quantify leakage of keys and private
images from extremely lengthy side channel traces. The error of estimating CP with
$\mathcal{F}_{\theta}$ is bounded by a negligible $\epsilon$. In contrast, prior
works use marginal probability (MP) to estimate MI. CP outperforms MP in terms of
lower cost, fewer errors, and better coverage, as compared in \S~\ref{subsec:pdf-est}.
In \S~\ref{subsec:non-det}, we extend the pipeline in Fig.~\ref{fig:overview} to handle
non-deterministic side channel traces.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.00\linewidth]{fig/overview.pdf}
\vspace{-20pt}
\caption{Overview. $|K|$ is the total number of possible keys. $|K|$ is
assumed as known to detectors by all existing quantification tools including
\textsc{CacheQL}\xspace.}
\label{fig:overview}
\end{figure}
\vspace{-10pt}
\subsection{Problem Setting}
\label{subsec:problem}
In general, side channel analysis aims to infer $k$ from $o$. The information
leak of $K$ in $O$ can be defined as their MI:
\begin{equation}
\label{equ:mi}
{
I(K;O) = H(K) - H(K|O),
}
\end{equation}
\noindent where $H(\cdot)$ denotes the entropy of an event.
According to Shannon's information theory, $I(K;O)$ describes how much
information about $K$ can be obtained by observing $O$.
Consider the program in Fig.~\hyperref[fig:demo-entropy]{3(a)}, where the
probability of correctly guessing each $k \in K$ (i.e., $\texttt{s} \in
\{\texttt{0}, \texttt{1}, \texttt{2}, \texttt{3}\}$), without any observation,
is $\frac{1}{4}$. Thus, $H(K)$ = $-\log\frac{1}{4}$ = $2$ bits.\footnote{Given
log base 2 is used by default, the unit of information is \textit{bit}.}
Nevertheless, the observation $o$ = $a[0]$ (\Line{6}{fig:demo-entropy})
indicates that $k$ must be ``\texttt{0}'' (i.e., the probability is $1$), thus,
$H(K|o$=$a[0])$ = $-\log 1$ = $0$. Therefore, $a[0]$ leaks 2 bits of
information.
Similarly, the memory access $b[0]$ (\Line{9}{fig:demo-entropy}) leaks $\log
\frac{4}{3}$ bits of information since $H(K|o$=$b[0])$ = $- \log \frac{1}{3}$ =
$\log 3$. Ideally, a secure program should have $H(K)$ = $H(K|O)$, indicating no
information in $K$ can be obtained from $O$. We continue discussing
Fig.~\hyperref[fig:demo-entropy]{3(b)} in \S~\ref{subsec:pdf-est}.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.00\linewidth]{fig/demo-entropy.pdf}
\vspace{-25pt}
\caption{Quantification of side channel leaks.}
\label{fig:demo-entropy}
\end{figure}
\subsection{Computing MI via PD}
\label{subsec:mi-pdf}
Following Eq.~\ref{equ:mi}, let $k$ and $o$ be random variables whose probability
density functions (PDF) are $p(k)$ and $p(o)$. The MI $I(K;O)$ can be
represented in the following way,
\begin{equation}
\label{equ:mi-int}
{
\begin{aligned}
&I(K;O) = \int\int_{K \times O} p(k,o) \log \frac{p(k, o)}{p(k)p(o)} \mathrm{d}k\mathrm{d}o \\
&= \mathbb{E}_{P_{K \times O}} \left[ \log \frac{p(k, o)}{p(k)p(o)} \right]
= \mathbb{E}_{P_{K \times O}} [\log c(k, o)],
\end{aligned}
}
\end{equation}
\noindent where $p(k,o)$ is the joint PDF of $K$ and
$O$, and $P_{K \times O}$ is the joint distribution.
$c(k, o) = \frac{p(k, o)}{p(k)p(o)}$ denotes \textit{point-wise dependency}
(PD), measuring discrepancy between the probability of $k$ and $o$'s
co-occurrence and the product of their independent occurrences.
Accordingly, $\log c(k, o)$ denotes the point-wise mutual
information (PMI).
The MI of $K$ and $O$, by definition, is the expectation of PMI. That is,
Eq.~\ref{equ:mi-int} measures the dependence retained in the joint distribution
(i.e., $\langle k, o \rangle \sim P_{K \times O}$) relative to the marginal
distribution of $K$ and $O$ under the assumption of independence
(i.e., $\langle k, o \rangle \sim P_K P_O$, where $P_K$ and $P_O$ are
marginal distributions of $K$ and $O$). When $K$ and $O$ are independent, we have
$p(k,o)$ = $p(k)$$\cdot$$p(o)$ and $\frac{p(k,o)}{p(k)p(o)}$ is $1$, thus, the leakage is
$\log 1 $=$ 0$. Nevertheless, whenever $o$ leaks $k$, $k$ and $o$ should co-occur
more often than their independent occurrences, and therefore, $c(k, o) > 1$
and $\log c(k,o) > 0$.
Eq.~\ref{equ:mi-int} illustrates two aspects for quantitatively computing
information leakage: 1) \textbf{PMI} $\log c(k$=$k^*, o$=$o^*)$, denoting
per trace leakage for a specific $k^*$ and its corresponding $o^*$, and 2)
\textbf{MI} $I(K;O)$, denoting program-level leakage over all possible secrets
$k \in K$. To compute $I(K;O)$, we average PMI over a collection of $\langle k,o
\rangle$, where sample-mean offers an unbiased estimation for the expectation
$\mathbb{E}(\cdot)$ of a distribution~\cite{castro2009polynomial}.
\smallskip
\parh{Comparison with Prior Works.}~Abacus~\cite{bao2021abacus} launches
symbolic execution on Pin-logged execution traces. It makes a strong assumption
that $k$ is uniformly distributed, i.e., $p(k$=$k^*)$ =
$\frac{1}{|K|}$.\footnote{``Uniform distribution'' does \textit{not}
hold for image pixel values~\cite{yuan2022automated}.} It also assumes that
each trace must be deterministic, such that $p(k$=$k^*, o$=$o^*)$ = $p(k$=$k^*)$
for a given $o^*$ and its corresponding $k^*$. This way, approximating MI in
Eq.~\ref{equ:mi-int} is recasted to estimating the marginal probability (MP)
$p(o$=$o^*)$. At a secret-dependent control transfer or data access point $l$,
Abacus finds all $k \in K'$ that cover $l$. The leakage at $l$ is computed as $-
\log p(o$=$o^*)$ = $-\log(\frac{|K'|}{|K|})$. Deciding $|K'|$ via constraint
solving is costly, and therefore, Abacus uses \textit{sampling} to approximate
$|K'|$. Nevertheless, estimating MP with sampling is unstable and error-prone
(see \S~\ref{subsec:pdf-est}). MicroWalk also samples $k$ to estimate MP; it
thereby has similar issues. CacheAudit quantifies program-wide leakage. Using
abstraction interpretation, it only analyzes small programs or code fragments,
and it infers only leak upper bound. \textsc{CacheQL}\xspace\ precisely computes PMI/MI via PD and
localizes flaws. We now introduce estimating PD.
\subsection{Estimating PD $c(k,o)$ via CP}
\label{subsec:pdf-est}
Because PD makes \textit{no} assumption on the secret's distribution, our
approach can infer different types of secrets (e.g., key or images). We denote
$k$ as a general representation of one secret, and for simplicity, we write $p(k
$=$ k^*)$ as $p(k^*)$ in followings. The same applies for $o$ and $o^*$. $o^*$ is
one side channel observation produced by $k^*$. However, $o^*$ may not be the only
one, given randomness like blinding can also induce different observations even
with a fixed $k^*$. We now recast computing PD over deterministic side channels
as estimating conditional probability (CP) via binary
classification~\cite{tsai2020neural}.
\subsubsection{Transforming PD to CP}
\label{subsubsec:cp}
Let $T$ depict that a pair $\langle k, o \rangle$ co-occurs (i.e.,
positive pair $\langle k, o \rangle \sim P_{K \times O}$). Let $F$ denote that
$k$ and $o$ in $\langle k, o \rangle$ occur independently (i.e., negative pair
$\langle k, o \rangle \sim P_K P_O$). Therefore, $p(k^*,o^*)$ and $p(k^*)
p(o^*)$ can be represented as the posterior PDF $p(k^*,o^* | T)$ and $p(k^*,o^*
| F)$, respectively.
According to Bayes' Theorem, PD $c(k^*,o^*)$ is re-expressed as
\vspace{-10pt}
\begin{equation}
\label{equ:cond-prob}
{
\text{PD} =
\frac{p(k^*, o^*)}{p(k^*)p(o^*)} = \frac{p(k^*,o^* | T)}{p(k^*,o^* | F)}
= \frac{p(F)}{p(T)} \frac{p(T | k^*,o^*)}{p(F | k^*,o^*)},
}
\end{equation}
\noindent where $p(T)$ and $p(F)$ are constants (decided by the analyzed
software). Given $P_K P_O$ is produced by separating each pair in $P_{K \times
O}$ and collecting random combinations of $k$ and $o$, $\frac{p(F)}{p(T)}$
equals to $|K|$. In practice, $P_{K \times O}$ is prepared by running the
analyzed software with each $k^*$ and collecting the corresponding $o^*$. For
the program in Fig.~\hyperref[fig:demo-entropy]{3(a)},
Fig.~\hyperref[fig:demo-entropy]{3(b)} colors $P_{K \times O}$ and $P_K P_O$ in
\colorbox{pptgreen}{green} and \colorbox{pptyellow}{yellow}.
Since $\frac{p(F)}{p(T)}$ is unrelated to $k^*$ or $o^*$, $c(k^*,o^*)$ ---
representing leaked $k^*$ from $o^*$ --- is only decided by CP $p(T|k^*,o^*)$. A
larger CP indicates that more information is leaked.
\parhs{$\bullet$ Example:}~We demonstrate this transformation using
Fig.~\ref{fig:demo-entropy}: for $k$=``$\texttt{0}$'' and $o$=$a[0]$, fetching a
block of $\langle \texttt{0}, \texttt{a} \rangle$ from
Fig.~\hyperref[fig:demo-entropy]{3(b)} has a $50\%$ chance of selecting the
\colorbox{pptgreen}{green} one (in the upper-left corner). That is, CP=$p(T |
\text{``\texttt{0}''}, a[0])$=$0.5$, and therefore, $p(T |
\text{``\texttt{0}''}, a[0])$= $p(F | \text{``\texttt{0}''}, a[0])$. Since
$\frac{p(F)}{p(T)}$=$4$, Eq.~\ref{equ:cond-prob} yields $4 \times \frac{0.5}{0.5}
= 4$, and therefore, $\log c(k, o)$ in Eq.~\ref{equ:mi-int} yields $\log 4 = 2$ bits,
equaling the leakage result computed in \S~\ref{subsec:problem}.
\subsubsection{Advantages of CP vs. Marginal Probability (MP)}
\label{subsubsec:advantage}
CP captures what factors make $o^*$,
which corresponds to $k^*$, distinguishable from other $o$. By observing both
dependent and independent $\langle k, o \rangle$ pairs, \textsc{CacheQL}\xspace\ measures the
leakage via describing how the distinguishability between different $o$ is
introduced by the corresponding $k$. It is principally distinct with existing
quantitative analysis~\cite{bao2021abacus,cacheaudit,jan2018microwalk}. Abacus
and MicroWalk approximate MP $p(o^*)$ via sampling, which has the following
three limitations compared with CP.
\smallskip
\parh{Computing Cost.}~Estimating CP is an one-time effort
over a collection of $\langle k, o \rangle$ pairs. Estimating MP, however, has to
\textit{re-perform} sampling for each $\langle k, o \rangle$. Note that the cost
for CP to estimate over the collection of $\langle k, o \rangle$ and each
re-sampling of MP is comparable. Thus, MP is much more costly.
\smallskip
\parh{Estimation Error.}~Recall that for a leakage program
point $l$, Abacus finds all $k \in K'$ that cover $l$
via constraint solving and denotes the leakage as $-\log(\frac{|K'|}{|K|})$.
Suppose it observes the first \texttt{for} loop in
Fig.~\hyperref[fig:demo-quant]{1(a)} has 128 consecutive accesses to $x[0]$. To
quantify the leakage, Abacus constructs the symbolic constraint
$(s[0$:$4]\%4==0) \land \ldots \land (s[508$:$512]\%4==0)$. Nevertheless,
sampling one key that satisfies this constraint has only an extremely low
probability of $(\frac{1}{4})^{128}$. That is, the MP can be presumably
underestimated when $|K'|$ is small. Thus, the leaked information can be largely
overestimated via $-\log(\frac{|K'|}{|K|})$.
MicroWalk observes $o^*$'s frequency by sampling different $k$; it thus has
similar issues. Worse, once $o^*$ is influenced by randomness like blinding, no
$o^*$ would be identical (non-replicability). Thus, it will incorrectly regard
$p(o^*)$ as $\frac{1}{|K|}$ and report $\log |K|$ leaked bits (i.e.,
equals to the key length).
In contrast, Eq.~\ref{equ:cond-prob} is free from this issue: even
$o^*$ is only produced by processing one or a few $k$, \textsc{CacheQL}\xspace\ directly
characterizes PD via CP $p(\cdot|k^*,o^*)$.
Overall, CP reflects: 1) the portion of records in $o^*$ affected by its
$k^*$~\cite{tsai2020neural}, and 2) to what extent $k^*$ affects each record in
$o^*$ (see \textit{\underline{Example}} below). Further, since any difference on
$o^*$, whether due to explicit or implicit information flows, contributes to
\textit{distinguishing} $o^*$ from the rest $o \in O$, \textsc{CacheQL}\xspace\ takes both
explicit and implicit flows into consideration.
\parhs{$\bullet$ Example:}~Consider the memory accesses at
\Line{10}{fig:demo-quant} and \Line{12}{fig:demo-quant} of the program in
Fig.~\hyperref[fig:demo-quant]{1(b)} and suppose $o^*$ is ``$y[0]$, $z[0]$''. To
estimate the leakage over $o^*$ via MP, it requires sampling keys where
$s[0$:$256]$ constitutes either 1 or 0, so do the second 256 bits, which results in a
total of 4 cases for $s[0$:$512]$. $s[0$:$512]$ has $2^{512}$ cases, denoting a
large search space. Nevertheless, CP can infer the leakage by only observing
that, the first record in $o$ increases ``1'' (i.e., distinguishable from other
$o$) whenever $s[0$:$256]$ increases 2 (with no need to simultaneously consider
$s[256$:$512]$). The same applies to the second 256 bits.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.03\linewidth]{fig/coverage.pdf}
\vspace{-25pt}
\caption{A schematic view of
how the coverage issue of dynamic methods is alleviated
via CP when \textit{quantifying} leaks.}
\label{fig:coverage}
\end{figure}
\parh{Coverage Issue.}~\textsc{CacheQL}\xspace, by using CP, principally alleviates
the coverage issue of conventional dynamic methods.
Consider the program in Fig.~\hyperref[fig:coverage]{4(c)}, \textsc{CacheQL}\xspace\ can quantify
the 8 SDA \textit{without} covering all paths, since CP captures how $o$ changes
with $k$. As shown in Fig.~\hyperref[fig:coverage]{4(b)}, covering a few cases is
sufficient to know that $o$ increases ``one'' (e.g., $a[0] \rightarrow a[1]$)
when $k$ increases one (e.g., $0 \rightarrow 1$), thus inferring program behavior
in Fig.~\hyperref[fig:coverage]{4(a)}. Prior dynamic methods need to fully cover all
paths to infer the program behavior and quantify the leaks, which is hardly
achievable in practice.
\subsubsection{Obtaining CP $p(T|k, o)$ via Binary Classification}
\label{subsubsec:obtain}
We show that
performing probabilistic classification can yield CP. In particular, we employ
neural networks $\mathcal{F}_{\theta}$\xspace\ (parameterized by $\theta$) to classify a pair of
$\langle k, o \rangle$, whose associated confidence score is $p(T|k, o)$.
Details of $\mathcal{F}_{\theta}$\xspace\ are in \S~\ref{sec:framework}.
Using neural networks (NN) to estimate MI is \textit{not} our
novelty~\cite{tsai2020neural,belghazi2018mutual}. However, we deem NN as
particularly suitable for our research context for three reasons:
1) non-parametric approaches, as in~\cite{gierlichs2008mutual}, suffer from
``curse of dimensionality''~\cite{bellman1966dynamic,bengio2013representation}.
They are thus infeasible, as even the AES-128 key is 128-dimensional. NN shows
encouraging capability of handling high-dimension data (e.g., images with
thousands of dimensions). 2) Recent works show that NN can effectively process
lengthy but sparse data, including side channel traces where only a few records
out of millions are informative and leaking program
secrets~\cite{yuan2022automated,kwon2020improving,wu2020remove}.
3) It's generally vague to define ``information'' in media data. For instance, a
$64 \times 64$ image may retain the same information as a $32 \times 32$ version
from human perspective since the content is unchanged. Recent
research~\cite{yuan2022automated} shows that high-dimensional media data have
perceptual constraints which implicitly encode image ``information.'' NNs are
currently widely used to process media data and extract critical information for
comprehension.
\subsection{Handling Non-determinism}
\label{subsec:non-det}
In practice, due to hardening schemes like RSA blinding and ORAM, side channel
observations can be non-deterministic, where memory access traces may vary during
different runs despite the same key is used. As discussed in Table~\ref{tab:detectors}
(i.e.,~\Circ{2}), however, non-determinism is not properly handled in previous
(quantitative) analysis.
\smallskip
\parh{Generalizability.}~For deterministic side channels, only $k$ induces
changes of $o$. Fitting $\mathcal{F}_{\theta}$\xspace\ on enough $\langle k, o \rangle$ pairs from $P_{K \times O}$ and $P_K
P_O$ can capture distinguishability for quantification. In contrast, for
non-deterministic side channels, the differences between $o$ may be due to
random factors, not only $k$. Therefore, in addition to
\textit{distinguishability} between $\langle k, o \rangle$ pairs, we also need
to consider \textit{generalizability} to alleviate over-estimation caused by
random differences.
In statistics, \textit{cross-validation} is used to test generalizability. Here,
we propose a simple yet effective method by using a \textit{de-bias} term with
cross-validation to prune non-determinism in the estimated PD. We first mix
$\langle k, o \rangle$ from $P_{K \times O}$ and $P_K P_O$ and split them into
non-overlapping groups. Then, we assess if the distinguishability over one group
applies to the others.
\smallskip
\parh{PD Estimation via De-biasing.}~We first extend the PD computation in
Eq.~\ref{equ:cond-prob} to handle non-determinism. In Eq.~\ref{equ:cond-prob}, $F$
and $T$ are finite sets, and $p(F)/p(T)$ equals to $|K|$. Here, we
conservatively assume that there exist infinite non-deterministic side channels.
That is, $F$ is a set with infinite elements.
We first require $m$ positive pairs $\langle k,o \rangle \sim P_{K \times O}$,
dubbed as $T^{(m)}$. We also construct $m' \leq m^2$ negative pairs (i.e.,
$\langle k,o \rangle \sim P_K P_O$), denoted as $F^{(m')}$, by replacing $o$ (or
$k$) of a pair from $P_{K \times O}$ with that of other random pairs. This way, PD defined in
Eq.~\ref{equ:cond-prob} is extended in the following form:
\begin{equation}
\label{equ:non-deter}
{
\text{PD} =
\frac{\log |K|}{\log (m'/m)} \log \frac{p(F^{(m')})}{p(T^{(m)})}
\frac{p(T|k^*, o^*)}{p(F|k^*, o^*)},
}
\end{equation}
\noindent where the $p(F^{(m')})/p(T^{(m)})$ works as a \textit{de-bias} term
to assess the generalizability for non-deterministic side channels. We denote
$p(T|k^*, o^*) / p(F|k^*, o^*)$ as the \textit{leakage ratio}: a 100\% ratio
implies that all bits of the key are leaked whereas 0\% ratio implies no
leakage. Consider the following two cases:
\noindent \underline{$\bullet$ $\textit{Case}_1$:}~In case the differences between samples from
$T^{(m)}$ and $F^{(m')}$ are all introduced by random noise (i.e., each $o^*$ is
independent of its $k^*$), the distinguishable factors should not be generalizable,
and the above formula yields a zero leakage. To understand this, let our neural
networks $\mathcal{F}_{\theta}$\xspace\ identify each pair based on random differences, which is indeed
equivalent to memorizing all pairs. This way, when it predicts the label of
$\langle k, o \rangle$, the output simply follows the frequency of $T^{(m)}$ and
$F^{(m')}$. Therefore, given an unseen pair $\langle k^*, o^* \rangle$, $\mathcal{F}_{\theta}$\xspace\
has $p(T|k^*, o^*) / p(F|k^*, o^*) = p(T^{(m)}) / p(F^{(m')})$, and the
estimated leakage is thus $\log
\frac{p(F^{(m')})}{p(T^{(m)})}\frac{p(T^{(m)})}{p(F^{(m')})} = \log 1 = 0$.
\noindent \underline{$\bullet$ $\textit{Case}_2$:}~If $o^*$ depends on $k^*$, $p(T|k^*, o^*)$
would not merely follow the distribution of $T^{(m)}$ and $F^{(m')}$, indicating
a non-zero leakage. More importantly, de-biased by $p(F^{(m)}) / p(T^{(m')})$,
quantifying leakage using Eq.~\ref{equ:non-deter} \textit{only} retains differences
related to $k$. This way, we precisely quantify leakage for non-deterministic side
channels.
\parh{Implementation Consideration.}~To alleviate randomness in each $o^*$, we collect four
observations $o_{i}^*$ by running software using $k^*$ for four times. By
classifying all $\langle k^*, o_{i}^* \rangle$ as positive pairs, $\mathcal{F}_{\theta}$\xspace\ is
guided to extract common characters shared by $o_{i}^*$ while neglecting
randomness in each $o_{i}^*$. Also, considering un-optimized neural networks
generally make prediction by chance (i.e., $p(T|k, o) = p(F|k, o)$), we let $m =
m'$.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
The Sylvester-Gallai theorem is a statement about configurations of points in $\R^d$ in which there is a certain structure of collinear triples.
\begin{thm}[Sylvester-Gallai]
Suppose $v_1,\ldots,v_n \in \R^d$ are such that for all $i \neq j \in [n]$ there is some $k \in [n] \setminus\{i,j\}$ for which $v_i,v_j,v_k$ are on a line. Then all the points $v_1,\ldots,v_n$ are on a single line.
\end{thm}
This theorem takes {\em local} information about dependencies between points and concludes {\em global} information about the entire configuration. For more on the history and generalizations of this theorem we refer to the survey \cite{BM90}. A complex variant of this theorem was proved by Kelly:
\begin{thm}[\cite{Kel86}]
Suppose $v_1,\ldots,v_n \in \C^d$ are such that for all $i \neq j \in [n]$ there is some $k \in [n] \setminus\{i,j\}$ for which $v_i,v_j,v_k$ are on a (complex) line. Then all the points $v_1,\ldots,v_n$ lie on a single (complex) plane.
\end{thm}
The global dimension bound given by Kelly's theorem is tight since, over the complex numbers, there are two-dimensional configurations of points satisfying the condition on triples.
In a recent work, Barak et. al. \cite{BDWY11} proved quantitative (or fractional) analogs of Kelly's theorem in which the condition `for all $i \neq j \in [n]$' is relaxed and we have information only on a large subset of the pairs of points for which there exists a third collinear point\footnote{The sets of points satisfying the conditions of the theorem were called $\delta$-SG configurations in \cite{BDWY11}.}.
\begin{thm}[\cite{BDWY11}]\label{thm-bdwy}
Suppose $v_1,\ldots,v_n \in \C^d$ are such that for all $i \in [n]$ there exist at least $\delta (n-1)$ values of $j \in [n]\setminus \{i\}$ for which there is $k \in [n] \setminus\{i,j\}$ such that $v_i,v_j,v_k$ are on a line. Then all the points $v_1,\ldots,v_n$ lie in an affine subspace of dimension $13/\delta^2$.
\end{thm}
A more recent work \cite{DSW12} improves the dimension upper bound obtained in the above theorem from $O(1/\delta^2)$ to the asymptotically tight $O(1/\delta)$ and also gives a new proof of Kelly's theorem (when $\delta=1$ one gets an upper bound of 2 on the dimension).
In this work we consider configurations of points in which there are many triples that are `almost' collinear, in the sense that there is a line close to all three points (in the usual Euclidean metric on $\C^d$). Equivalently, the points are contained in a narrow tube. Our goal is to prove stable analogs of the above theorems, where stable means that the conclusion of the theorem will not change significantly when perturbing the point set slightly. Clearly, in such settings one can only hope to prove that there is a low dimensional subspace that {\em approximates} the set of points. There are many technical issues to discuss when defining approximate collinearity and there are some non trivial examples showing that word-to-word generalizations of the above theorems do not hold in the approximate-collinearity setting (at least for some of the possible definitions). Nonetheless, we are able to prove several theorems of this flavor for configurations of points satisfying certain `niceness' conditions. We also study stable variants of error correcting codes (over the reals) which are locally correctable, in which such approximately collinear tuples of points naturally arise from the correcting procedure.
In \cite{BDWY11}, a connection was made between the Sylvester-Gallai theorem to a special kind of error correcting codes called Locally Correctable Codes (LCCs). In these codes, a receiver of a corrupted codeword can recover a single symbol of the codeword correctly, making only a small number of queries to the corrupted word. When studying linear LCCs over the real or complex numbers one encounters the same type of difficulties in trying to convert local dependencies into global dimension bounds. Building on this connection, and our ability to analyze `approximate' linear dependencies, we define the notion of {\em stable} LCC and show that these do not exist for constant query complexity. Stable LCCs correspond to configurations of points with many approximately dependent small subsets and so our techniques can be used to analyze them.
We note here that understanding the possible intersection structure of tubes in high dimensional real space comes up in connection to other geometric problems, most notably the Euclidean Kakeya problem \cite{Tao01} (we do not, however, see a direct connection between our results and this difficult problem).
Our proof techniques extend those of \cite{BDWY11,DSW12} and rely on high rank properties of sparse matrices whose support is a `design'. In this work we go a step further and, instead of relying on rank alone, we need to bound the number of small singular values of such matrices.
\paragraph{Organization:} In Section~\ref{sec-statement} we formally state our results for point configurations. The results are stated in several sub-sections, corresponding to different variants of the problem we consider. In Section~\ref{sec-resultslcc} we define stable LCCs and state our results in this scenario. The proofs are given in Sections~\ref{sec-pfgen} -- \ref{sec-apxlcc}.
\paragraph{Notations:} We use big `O' notation to suppress absolute constants only. For two complex vectors $u,v \in \C^d$ we denote their inner product by $\ip{u}{v} = \sum_{i=1}^d u_i \cdot \overline{v_i}$ and use $\|v\| = \sqrt{\ip{v}{v}}$ to denote the $\ell_2$ norm. For an $m \times n $ matrix $A$, we denote by $\|A\|$ the norm of $A$ as a vector of length $mn$ (i.e., the Forbenius norm). The {\em distance} between two points $u,v \in \C^d$ is defined to be $\|u-v\|$ and is denoted $\dist(u,v)$. For a set $S \subset \C^d$ and a point $v \in \C^d$ we denote $\dist(v,S) = \inf_{u \in S} \dist(u,v)$. We let $S^d \subset \C^{d+1}$ denote the $d$-dimensional unit sphere in complex $d+1$ dimensional space. By fixing a basis we can identify each $v \in S^d$ with a $d+1$ length complex vector of $\ell_2$-norm equal to one.
\section{Point configurations}\label{sec-statement}
In this section we state our results concerning point configurations. The first section, Section~\ref{sec-resultsaffine} deals with the most natural setting -- the affine setting -- in which we consider sets of points in $\C^d$ with many almost-collinear triples. In Section~\ref{sec-resultsproj} we consider the projective setting where the points are located on the sphere and collinearity is replaced with linear dependence. Section~\ref{sec-resultsgen} states a more general theorem from which both the affine and the projective results follow.
\subsection{The affine setting}\label{sec-resultsaffine}
We begin with the definition of an $\eps$-line.
\begin{define}[$\line$,$\line_\eps$]\label{def-line}
Let $u \neq v \in \C^d$. We define $\line(u,v) = \{ \alpha u + (1-\alpha) v \,|\, \alpha \in \C\}$ to be the complex line passing through $u,v$. We define $\line_\eps(u,v) = \{ w\in \C^d \,|\, \dist(w,\line(u,v)) \leq \eps \}$.
\end{define}
The following definition will be used to replace the notion of dimension with a more stable definition.
\begin{define}[\bf $dim_\eps$]
For a set of points $V \subset \C^d$ and $\eps>0$ we denote by $\dim_\eps(V)$ to be the minimal $k$ such that there exists a $k$-dimensional subspace\footnote{The difference of 1 between affine and linear dimension will not be significant in this paper and so we use a linear subspace in the definition.} $L \subset \C^d$ such that $\dist(v,L) \leq \eps$ for all $v \in V$.
\end{define}
To give an idea of the subtleties that arise when dealing with approximate collinearity, take an orthonormal basis $e_1,\ldots,e_d$ in $\C^d$ and consider the set $V = \{e_1,e_1',\ldots,e_d,e_d'\}$ with $e_i' = (1+\eps)e_i$. Clearly, there is no low dimensional subspace that approximates this set of points, even though there are many pairs for which there is a third $\eps$-collinear point ($e_i'$ is $\eps$-close to the line passing through $e_i$ and any other third point). An obvious solution to this problem is to require that the minimal distance between each pair of points is bounded from below (say by 1), so that the condition of $\eps$-collinearity is meaningful. We now describe another, less trivial, example which shows that this condition alone is not sufficient in general.
\begin{example}\label{ex-affine} Let $e_1,\ldots,e_d$ be an orthonormal basis in $\C^d$. Let $v_i = Be_i$, $u_i = (B-1)e_i$ for all $i \in [d]$ and let $V = \{e_i,u_i,v_i \,|\, i \in [d] \}$ be a set of $n = 3d$ points. Then for all $i,j \in [d]$ we have $u_i \in \line_\eps(v_i,e_j)$ and $v_i \in \line_\eps(u_i,e_j)$ with $\eps = 1/B$. Thus, there are many $\eps$-collinear triples in $V$ (as in the conditions of Theorem~\ref{thm-apxsgaffine} with $\delta = 1/3$). However, for any subspace $L$ of dimension $o(n)$, the distance of at least one of the point $v_i$ to $L$ must be at least $\Omega(B)$ (this can be shown, e.g., using Lemma~\ref{lem-pert}).
\end{example}
In this example, we had $\eps= 1/B$, where $B$ is roughly equal to the ratio between the smallest and the largest distance, or the `aspect ratio' of $V$. We will prevent this scenario by requiring that $\eps$ will be sufficiently smaller than $1/B$, where $B$ will be the aspect ratio. This motivates the following definition.
\begin{define}[$B$-balanced]\label{def-balanced}
A set $V \subset \C^d$ is said to be $B$-balanced if $1 \leq \dist(v,v') \leq B$ for all $v \neq v' \in V$.
\end{define}
The following theorem gives the most easy to state version of our results.
\begin{THM}\label{thm-apxsgaffinesimple}
Let $n, d > 0$ be integers and let $B, \eps > 0$ be real numbers with $ \eps < 1/16B$. Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be $B$-balanced and suppose that for every $i \neq j \in [n]$ there exists $k \in [n]\setminus \{i,j\}$ such that $v_k \in \line_\eps(v_i,v_j)$. Then, $\dim_{\eps'}(V) \leq O(B^{6})$ with $\eps' \leq O( \eps B^{2.5})$.
\end{THM}
Observe that a corollary of this theorem is that the number of points, $n$, is bounded from above by a function of $B$. A priori, we did not have this bound since a $B$-balanced configuration in $\C^d$ can have an unbounded number of points when $d$ grows.
Notice that our definition of $\eps$-collinearity is not symmetric in that it depends on the order of the triple. As is shown in Lemma~\ref{lem-arcreverseaffine}, this is not an issue for $B$-balanced configurations, as long as we are willing to replace $\eps$ with $\eps B$. For general (i.e., non balanced) configurations the situation can be more complicated and it is possible that using a stronger collinearity condition (e.g., requiring that any permutation of the triple satisfies our condition) is sufficient for obtaining a global dimension bound.
Theorem~\ref{thm-apxsgaffinesimple} will be a special case of the following, more general theorem, in which we only have information of a subset of the pairs $(i,j)$. Assuming $V$ has many $\eps$-collinear triples (for each point), we derive an upper bound on $\dim_{\eps'}(V)$ for $\eps'$ which depends on the other parameters. We also derive a better bound on $\eps'$ when restricting to a subset of the points.
\begin{THM}\label{thm-apxsgaffine}
Let $n, d > 0$ be integers. Let $B, \delta, \eps > 0$ be real numbers with $ \eps < 1/16B$. Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be $B$-balanced and suppose that for every $i\in [n]$ there are at least $\delta(n-1)$ values of $j \in [n]\setminus \{i\}$ for which there exists $k \in [n]\setminus \{i,j\}$ such that $v_k \in \line_\eps(v_i,v_j)$. Then
\begin{enumerate}
\item $\dim_{\eps'}(V) \leq O(B^{6}/\delta^2)$ with $\eps' \leq O( \eps B^{2.5}/\delta^{0.5})$.
\item There exists a subset $V' \subset V$ of size $\Omega(n)$ with $\dim_{\eps''}(V') \leq O(B^{6}/\delta^2)$ and $\eps'' \leq O(B\eps)$.
\end{enumerate}
\end{THM}
In both of the above theorems, the parameter $B$ appears in the resulting global dimension bound. We suspect that this dependence can be removed so that the bound on the dimension will be $O(1)$ in Theorem~\ref{thm-apxsgaffinesimple} and $O(1/\delta^2)$ (or even $O(1/\delta)$) in Theorem~\ref{thm-apxsgaffine}. The blowup in $\eps'$, compared to $\eps$ is also likely to be suboptimal.
A stronger definition of collinearity, for which Example~\ref{ex-affine} fails, is to require that each point in the triple is $\eps$-close to the line spanned by the other two points. Let us call such triples {\em strongly} $\eps$-collinear triples. It is easy to see that, in Example~\ref{ex-affine}, the triples do not satisfy this stronger definition. Thus, it is possible that one could prove analogs of Theorem~\ref{thm-apxsgaffine} for configurations that are not $B$-balanced using this stronger definition of approximate collinearity.
We conclude this discussion with yet another example showing that, even for the case $\delta=1$ (i.e, the original Sylvester-Gallai condition) the weak definition of $\eps$-collinearity requires some balancedness condition (though potentially weaker).
\begin{example}
Fix some large $B > 0$. Take an orthonormal basis $e_1,\ldots,e_d \in \C^d$ and define $V = \{0\} \cup \bigcup_{i\in [d]}\left\{ B^{i-1}e_i, (B^{i-1}+1)e_i \right\}$. One can verify by induction that for every $u,v \in V$ there is a third point inside $\line_{\eps}(u,v)$ with $\eps \approx 1/B$. There is also no low dimensional subspace that approximates $V$ (similar to the previous examples).
\end{example}
\subsection{The projective setting}\label{sec-resultsproj}
Since the definition of $\eps$-collinearity (that is, $v_k \in \line_\eps(v_i,v_j)$) is sensitive to scaling, a projective statement of Theorem~\ref{thm-apxsgaffine}, in which these scaling issues do not arise, seems natural. In this setting we consider points on a sphere and lines are replaced by circles (two dimensional subspaces intersected with $S^d$).
\begin{define}[$\arc$,$\arc_\eps$]\label{def-arc}
Let $u,v \in S^d$. We define $\arc(u,v) = \span\{u,v\} \cap S^d$. We define $\arc_\eps(u,v) = \{ w\in S^d \,|\, \dist(w,\arc(u,v)) \leq \eps \}$.
\end{define}
An instructive example in the projective case is the following:
\begin{example}\label{ex-proj}
Take $V$ to be a maximal set in $S^d$ with pairwise distances at least $\mu >0$ (so that $n \approx (c/\mu)^{d}$ with $c$ a constant). Since every point in $S^d$ is of distance at most $\mu$ from one of the points in $V$ (otherwise we could add it) we get that each set $\arc_\mu(v_i,v_j)$ contains at least $\Omega(1/\mu) > 2$ points from $V$. On the other hand, for any low dimensional subspace $L$ (say, with dimension $d'$ independent of $n$) almost all points in $V$ will have distance at least $1/100$ from $L$.
\end{example}
From this example we see that there needs to be some upper bound on $\eps$ as a function of the minimal distance in the set. We will use the following definition to replace $B$-balancedness.
\begin{define}[$\mu$-separated]\label{def-sep}
A set $V \subset S^d$ is said to be $\mu$-separated if for every $u \neq v \in V$ we have $\min\{ \dist(u,v), \dist(u,-v)\} \geq \mu$.
\end{define}
We now state our theorem for points on a sphere.
\begin{THM}\label{thm-apxsg}
Let $n, d > 0$ be integers and let $\delta,\mu,\eps > 0$ be real numbers with $ \eps < \mu^2/32$. Let $V = \{v_1,\ldots,v_n\} \subset S^d$ be $\mu$-separated and suppose that for every $i\in [n]$ there are at least $\delta(n-1)$ values of $j \in [n]\setminus \{i\}$ for which there exists $k \in [n]\setminus \{i,j\}$ such that $v_k \in \arc_\eps(v_i,v_j)$. Then
\begin{enumerate}
\item $\dim_{\eps'}(V) \leq O(1/\delta^2\mu^6)$ with $\eps' \leq O( \eps/\delta^{0.5}\mu^{2.5})$.
\item There exists a subset $V' \subset V$ of size $\Omega(n)$ with $\dim_{\eps''}(V') \leq O(1/\delta^2\mu^6)$ and $\eps'' \leq O(\eps/\mu)$.
\end{enumerate}
\end{THM}
Notice that, when compared with Theorem~\ref{thm-apxsg}, the parameters $\mu$ corresponds to $1/B$. However, the condition on $\eps < \mu^2/32$ is more restrictive in this case. We do not know whether this condition can be improved to $\eps \leq O(\mu)$.
As is the case with Theorem~\ref{thm-apxsg}, we do not expect the dependency in the dimension bound and in $\eps'$ to be tight.
\subsection{The general statement}\label{sec-resultsgen}
Both Theorem~\ref{thm-apxsgaffine} and Theorem~\ref{thm-apxsg} will follow from a more general statement requiring a set of points with a family of $\eps$-dependent triples satisfying certain conditions.
\begin{define}[$(\eps,\mu)$-dependent]
We say that a triple of points $u,v,w \in \C^d$ is $(\eps,\mu)$-dependent if there exists complex numbers $\alpha,\beta,\gamma$ with $|\alpha|,|\beta|,|\gamma| \in [\mu,1]$ such that $$\| \alpha u + \beta v + \gamma w \| \leq \eps.$$
\end{define}
\begin{define}[$(p,g)$-design]
Let $T \subset {[n] \choose 3}$ be a family of triples in $[n]$. We say that $T$ is a $(p,g)$-design if
\begin{enumerate}
\item For all $i \in [n]$ there are at least $p$ triples in $T$ that contain $i$.
\item For all $i \neq j \in [n]$ there are at most $g$ triples in $T$
containing both $i$ and $j$.
\end{enumerate}
\end{define}
The following theorem gives a low dimensional subspace that approximates all points in a configuration in which there is a design of triples that are $(\eps,\mu)$-dependent. Below we will also prove a slightly more refined statement (see Theorem~\ref{thm-apxsggenav}) giving better distance from $L$ for {\em many} points in the configuration.
\begin{THM}\label{thm-apxsggen}
Let $n, d > 0$ be integers and $p,g,\delta,\mu,\eps > 0$ be real numbers. Let $V = \{v_1,\ldots,v_n\} \subset \C^d$, $T \subset {[n] \choose 3}$ be such that $T$ is $(p,g)$-design, and for every $\{i,j,k\} \in T$ the triple $v_i,v_j,v_k$ is $(\eps,\mu)$-dependent. Then, $$\dim_{\eps'}(V) \leq \frac{2n^2 g^2}{p^2 \mu^4}$$ with $$\eps' \leq \frac{5\eps \sqrt{g|T|}}{p\mu^2}.$$
\end{THM}
A setting of the parameters which will be most relevant to us is when $|T|$ is quadratic in $n$, $p$ is linear in $n$ and $g$ and $\mu$ are constants. In this case we get a constant upper bound on the dimension $\dim_{\eps'}(V)$ with $\eps' = O(\eps)$.
The proof of Theorem~\ref{thm-apxsggen} is given in the next section with the proofs of Theorems~\ref{thm-apxsgaffine} and ~\ref{thm-apxsg} in Sections~\ref{sec-pfaffine} and \ref{sec-pfproj} respectively. We give a high level overview of the proof below.
\paragraph{Proof overview:}
We place the points $v_1,\ldots,v_n$ as rows in a matrix $A$. We then use the triple family $T$ to construct a matrix $M$ such that
\begin{itemize}
\item $M$ is a $|T| \times n$ matrix whose support is determined by $T$. More precisely, the non zero coordinates of the $t$'th row of $M$, with $t \in T$, will be the three elements in $t$.
\item The values of the entries of $M$ will be in absolute value between $\mu$ and $1$.
\item The product $M\cdot A$ will have small Forbenius norm.
\end{itemize}
We then observe that the matrix $X = M^{*} M$ is diagonal dominant (its diagonal elements are much larger than its off-diagonal elements). This implies, using the Hoffman-Wielandt inequality, that $M$ has only a few small singular values. From this we get that the columns of $A$ must have small distance (on average) to the span of the small singular vectors of $M$ and so can be approximated well by a low dimensional space. We then show that the same statement holds when one replaces the columns of $A$ with the rows of $A$ (a fact which generalizes the simple fact that the row rank is equal to the column rank). Using the bound on the average distance of rows we argue that there is a large subset that is approximated well by a low dimensional subspace. We then extend this to {\em all} points using interpolation.
\section{Stable Locally Correctable Codes}\label{sec-resultslcc}
Before discussing local correction, we briefly mention the exciting recent developments regarding `standard' (non-local) error correcting codes over the reals. Like in the analogous theory over finite fields, one would like to encode (typically via a linear transformation) a vector of entries from a given field $\F$ by a longer one, such that the original message can be decoded even when some entries of the codeword are corrupted. The breakthrough of `compressed sensing' by Donoho and Candes-Tao, and subsequent developments (see e.g. \cite{CT05, RV05, Donoho06, KT07, DMT07, GLW09}) has lead to an understanding of codes over the reals that is almost as good as in the finite-field case. In particular, there are real-valued codes which achieve the gold-standard of coding theory of constant rate linear codes with efficient encoding and decoding algorithms from a linear number of errors of arbitrary magnitude. Moreover, these codes have {\em stable} versions which can recover a vector close to the original message even if small errors affect {\em all} coordinates of the encoding. Our local variant may be viewed as one local analog of such stable codes.
Informally, Locally Correctable Codes (LCCs) are error correcting codes that allow the transmission of information over a noisy channel so that the symbols of the transmitted words have many local dependencies between them. The most general definition requires that one can reconstruct (w.h.p) {\em any} coordinate in a possibly corrupted codeword, using a small number of (randomly chosen) queries to the other coordinates. The noise model is adversarial, meaning that the corrupted positions are arbitrary (and not random) and one only has a bound on the total number of errors (which is usually assumed to be a small constant fraction). LCCs are closely related to another type of codes - Locally Decodable Codes (LDCs)-- whose study was initiated in a work of Katz and Trevisan \cite{KTldc}. We refer the interested reader to \cite{Y_now} for the relevant background on LDCs and LCCs and their applications in computer science.
The connection between LCCs and the Sylvester-Gallai theorem was first observed in \cite{BDWY11}. When studying the special case of {\em linear} LCCs (i.e., LCCs that are given by linear mappings over a field) one can easily show that LCCs are equivalent to point configurations with many linearly dependent small subsets. The general definition of linear LCCs is as follows (we fix the field to be $\C$ but the same definition works for any field). We use $w(v)$ to denote the number of non zero elements in a vector $v \in \C^n$.
\begin{define}[Linear LCC -- first definition]\label{def-lcc1}
A $(q,\delta)$-LCC over $\C$ is a linear subspace $U \subset \C^m$ such that there exists a randomized
decoding procedure $D : \C^m \times [m] \mapsto \C$ with the following properties:
\begin{enumerate}
\item For all $x \in U$, for all $i \in [m]$ and for all $v \in \C^m$ with $w(v) \leq \delta m$ we have that $D\left( x + v, i\right) = x_i$ with probability at least $3/4$ (the probability is taken only over the internal randomness of $D$).
\item For every $y \in \C^m$ and $i \in [m]$, the decoder $D(y,i)$ reads at most $q$ positions in $y$.
\end{enumerate}
The {\em dimension} of an LCC is simply its dimension as a subspace of $\C^m$.
\end{define}
It is shown in \cite{BDWY11} that, w.l.o.g. the decoding procedure is {\em linear}, in the sense that it first picks a set of at most $q$ coordinates to read and then outputs a linear combination of them (with coefficients in $\C$). This linearity of the decoder implies that, for each coordinate in the code, there are many small subsets of the other coordinates that span it. Since each coordinate corresponds to a row of the generating matrix of the code, we obtain a configuration of points with many dependent small subsets. We will make this formal in the next definition, which is equivalent to the first definition, if one replaces $\delta$ with the slightly worse bound of $\delta/q$ (when $q$ is constant this change is negligible).
\begin{define}[Linear LCC -- second definition]\label{def-lcc2}
We say that a finite set $V = \{v_1,\ldots,v_n\} \subset \C^d$ is a $(q,\delta)$-{\em LCC} if for every $i \in [n]$ and every set $S \subset [n]$ of size $|S| \leq \delta n$ there exists a set $J \subset [n]\setminus S$ with $|J| \leq q$ such that $v_i \in \span(v_j\,|\, j \in J)$.
\end{define}
The main open problem regarding LCCs is to determine the maximum dimension (as a function of $n$) when we fix $q,\delta$ to be constants. Intuitively, the larger $d$ is, the more `information' we can transmit using the code (the {\em rate} of the code if $d/n$). While the case of $q=2$ is understood quite well ($d$ is at most logarithmic over finite fields and constant over characteristic zero \cite{BDWY11,BDSS11}), it is an open problem to determine the maximum dimension of a $q$-query LCC when $q > 2$. There are exponential gaps between the known lower and upper bound. For example, when $q=3$, the best upper bound is $d \leq O(\sqrt{n})$ \cite{Woodruff, KdW} while the best constructions give poly-logarithmic $d$ over finite fields and constant $d$ over characteristic zero. We refer the reader to the survey article \cite{Dvir-survey} for more background on LCCs and for an overview of the known constructions.
Due to their roots in coding theory, LCCs were traditionally studied exclusively over finite fields. The study of LCCs over arbitrary fields was initiated in \cite{BDWY11} and was motivated by its connection to the Sylvester-Gallai theorem. Another motivation comes from a work connecting LCCs with an approach for constructing {\em rigid matrices} over infinite fields \cite{Dvi10}. We note here that for $q>2$, the best upper bounds on the dimensions of LCCs are the same, no matter what the field is. This also motivates the study of LCC's over infinite fields as a potentially easier scenario to tackle first, before proceeding to codes over finite fields (where we have fewer techniques).
Our methods enable us to prove strong upper bounds on the dimension of codes that we call {\em stable LCCs}. Before discussing the relation between stable and non-stable LCCs we give the formal definition.
\begin{define}[$\span_B$]
Let $v,u_1,\ldots,u_m \in \C^d$. We say that $v \in \span_B(u_1,\ldots,u_m)$ if there exist $a_1,\ldots,a_m \in \C$ with $|a_i| \leq B$ for all $i$ and $v = \sum_{i=1}^m a_i u_i$.
\end{define}
\begin{define}[Stable LCC]\label{def-stablelcc}
We say that a finite set $V = \{v_1,\ldots,v_n\} \subset \C^d$ is a $(q,\delta,B,\eps)$-{\em stable LCC} if for every $i \in [n]$ and every set $S \subset [n]$ of size $|S| \leq \delta n$ there exists a set $J \subset [n]\setminus S$ with $|J| \leq q$ such that $\dist(v_i, \span_B(v_j\,|\, j \in J)) \leq \eps$.
\end{define}
Notice that this definition is incomparable to Definition~\ref{def-lcc2}: On the one hand, we restrict the linear dependencies to use only coefficients of bounded magnitude. On the other hand, we allow the linear combinations to result in an `approximate' vector, instead of the exact one. To see why the bound on the coefficients is natural (once you allow approximate recovery), notice that the decoder can handle small perturbations {\em even in the `correct positions'}. Stated in the scenario of Definition~\ref{def-lcc1}, suppose that in a received codeword at most $\delta$ fraction of the positions are completely changed (to arbitrary values) and, in addition, all other coordinates are perturbed by some small $\alpha$ in Euclidean distance. Then, the decoder can still recover (approximately) the value of a given codeword coordinate by reading at most $q$ other positions, as long as $\alpha \ll \eps/qB$. Since each of the read coordinates is multiplied by a coefficient that can be as large as $B$ and the errors sum over $q$ positions, we get at most $\alpha\cdot qB$ resulting error in the output of the decoder.\footnote{One can potentially define stable LCCs in this sense (as in Definition~\ref{def-lcc1}) and then prove (similarly to \cite{BDWY11}) that, up to constants, it is equivalent to Definition~\ref{def-stablelcc} (we did not verify the details).}
The next simple claim shows that Definition~\ref{def-stablelcc} is also stable in the sense that, perturbing the elements in a stable LCC gives another stable LCC (with slightly worse parameters).
\begin{claim}
Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be a $(q,\delta,B,\eps)$-{\em stable LCC} and let $V = \{v'_1,\ldots,v'_n\} \subset \C^d$ be such that $\dist(v_i,v_i') \leq \alpha$ for all $i \in [n]$. Then $V'$ is a $(q,\delta,B,\eps')$-{\em stable LCC} with $\eps' \leq \eps + (qB+1)\alpha$.
\end{claim}
\begin{proof}
Take some $v_i \in V$ and a set $J \subset [n]$ of size $|J| \leq q$ such that $\dist(v_i, \span_B(v_j\,|\, j \in J)) \leq \eps$. Then, there exist coefficients $b_j, j \in J$ with $|b_j| \leq B$ and such that $$ \left\| v_i - \sum_{j \in J}b_j v_j \right\| \leq \eps.$$ Replacing $v_i$ with $v_i'$ we get that $$ \left\|v_i' - \sum_{j \in J}b_j v_j' \right\| \leq \eps + \|v_i - v_i'\| + \sum_{j \in J}b_j\|v_j - v_j'\| \leq \eps + (qB+1)\alpha.$$
\end{proof}
Notice that, if we didn't have the bound on the coefficients in the span, the small perturbations would have resulted in large errors in the linear combinations. Intuitively, if $u$ is not in $\span_B(u_1,\ldots,u_m)$ then a small perturbation to the $u_i$'s may result in $u$ being very far from $\span(u_1,\ldots,u_m)$. This explains the need for two separate stability parameters, $\eps$ and $B$.
Our main result regarding stable LCC's is the following theorem:
\begin{THM}\label{thm-apxlcc}
Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be a $(q,\delta,B,\eps)$-{\em stable LCC}. Then, $$\dim_{\eps'}(V) \leq O((qB/\delta)^4)$$ with $$\eps' = O( q^2 B \eps / \delta^{1.5}).$$
\end{THM}
In particular, when $q$ is a constant and $B$ and $\delta$ are fixed, the upper bound on $\dim_{\eps'}$ can be interpreted as saying that there {\em do not exist} stable $q$-query LCCs, where `do not exist' means that the amount of information one can transmit is constant, regardless of the codeword length. The proof of Theorem~\ref{thm-apxlcc}, which follows the same lines as the proof of the Sylvester-Gallai type theorems, works also for the more general setting where $V$ is allowed to be an ordered multiset (i.e., when different $v_i$'s can repeat several times).
If one sets $\eps=0$ the definition of stable LCC changes into a definition of an LCC with bounded coefficients. That is, the linear dependencies are required to be exact (as in the usual definition of an LCC) and, in addition, need to use bounded coefficients. Applying Theorem~\ref{thm-apxlcc} to this special case one gets $\eps'=0$ and so obtains the stronger conclusion that the set $V$ is actually {\em contained} in a low dimensional space. Stated more formally, we have:
\begin{cor}
Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be a $(q,\delta,B,0)$-{\em stable LCC}. Then, $$\dim(V) \leq O((qB/\delta)^4)$$.
\end{cor}
\section{Proof of Theorem~\ref{thm-apxsggen}}\label{sec-pfgen}
We will derive Theorem~\ref{thm-apxsggen} from the following, more refined, statement.
\begin{thm}\label{thm-apxsggenav}
Under the same conditions as in Theorem~\ref{thm-apxsggen}, there exists a subspace $L \subset \C^d$ with $$\dim(L) \leq \frac{2n^2 g^2}{p^2 \mu^4}$$ and such that $$ \sum_{i=1}^n \dist( v_i, L)^2 \leq \frac{4|T| \eps^2}{\mu^2 p}.$$
\end{thm}
\begin{proof}
First, observe that, for convenience, we can take $d=n$ so that the vectors $v_i$ are in $\C^n$. The case $d > n$ is not interesting since we can restrict our attention to the span of the $n$ vectors. The case $d < n$ can be similarly handled by padding each vector with zeros.
Let $m = |T|$. We use $T$ to construct an $m \times n$ matrix $M$ so that there is a one-to-one correspondence between rows of $M$ and elements of $T$. By our assumptions, for each triple $t = \{i,j,k\} \in T$ there are complex numbers $\alpha,\beta,\gamma$ such that $\|\alpha v_i + \beta v_j + \gamma v_k\| \leq \eps$ and s.t $\mu \leq |\alpha|,|\beta|,|\gamma| \leq 1$. Let $s_t$ denote the row vector in $\C^n$ with the value $\alpha$ in position $i$, the value $\beta$ in position $j$, the value $\gamma$ in position $k$ and zeros everywhere else. We define $M$ to be the matrix with rows $s_t$ where $t$ goes over all triples in $T$ (in some order).
Next, let $A$ be a complex $n \times n$ matrix whose $i$'th row is the vector $v_i$. Then, from our definition of the rows of $M$, we have that the rows of the $m \times n$ matrix
\begin{equation}\label{eq-E=MA}
E = M A
\end{equation}
all have norm at most $\eps$.
The next claim summarizes some of the properties of $M$ that we will use. All three items follow immediately from the fact that $T$ is a $(p,g)$-design and the bounds on the entries of $M$.
\begin{claim}\label{cla-propM}
Let $M$ be as above and let $M_j \in \C^m$, $j \in [n]$ denote the $j$'th column of $M$. Then
\begin{enumerate}
\item Each entry of $M$ has absolute value at least $\mu$ and at most $1$.
\item For each $j \in [n]$, $\|M_j\|^2 \geq p \mu^2 $.
\item For each $j \neq j' \in [n]$, $\left|\ip{M_j}{M_{j'}}\right| \leq g$.
\end{enumerate}
\end{claim}
The main technical ingredient in the proof is the following simple observation regarding the eigenvalues of {\em diagonal dominant} matrices, i.e., matrices in which the diagonal elements are much larger than the off-diagonal elements. This lemma can be viewed as an extension of a folklore result regarding the {\em rank} of such matrices (see, e.g., \cite{Alo09}). The proof is a simple application of the Hoffman-Wielandt inequality.
\begin{lem}\label{lem-pert}
Let $X = (X_{ij})_{i,j \in [n]}$ be an $n \times n$ complex Hermitian matrix with eigenvalues $ \lambda_1, \ldots,\lambda_n$. Suppose that for all $i \in [n]$ we have $X_{ii} \geq K$, where $K$ is some positive real number. Then,
$$ \left| \left\{ i \in [n]\,\,\,|\,\,\, \lambda_i \leq K/4 \right\} \right| \leq \frac{2}{K^2} \sum_{i \neq j} |X_{ij}|^2.$$
\end{lem}
\begin{proof}
Let $D$ be an $n \times n$ diagonal matrix with $D_{ii} = X_{ii}$ for all $i \in [n]$. Clearly, the eigenvalues of $D$ are $D_{11},\ldots,D_{nn}$. The Hoffman-Wielandt inequality \cite{HW53} states that, under some ordering of the eigenvalues of $X$ (w.l.o.g the one we have chosen) we have
$$ \sum_{i \in [n]} | \lambda_i - D_{ii} |^2 \leq ||X - D||^2 = \sum_{i \neq j} |X_{ij}|^2. $$
Using the fact that all $D_{ii}$'s are at least $K$ we get the required bound.
\end{proof}
Let $\sigma_1,\ldots,\sigma_n$ be the singular values of the matrix $M$ (recall that these are the square roots of the eigenvalues of the PSD matrix $M^* M$). Let $r_1,\ldots,r_n$ be the corresponding right singular vectors (i.e., the corresponding eigenvectors of $M^*M$). We thus have
\begin{enumerate}
\item $r_1,\ldots,r_n$ form an orthonormal basis of $\C^n$.
\item For each $j \in [n]$, $\|Mr_j\| = \sigma_j$.
\item The vectors $ Mr_1, \ldots, Mr_n$ are orthogonal (i.e., $\ip{ Mr_i}{ Mr_j} = 0$ for $i \neq j$).
\end{enumerate}
Let $$J = \{ j \in [n] \,|\, \sigma_j \leq \mu \sqrt{p}/2 \}$$ and let $$ L = \span\{ r_j \,|\, j \in J \}.$$ We will now show that $L$ is of small dimension and that most columns of $A$ are close to $L$. We start by bounding the dimension of $L$.
\begin{claim}\label{cla-dimL}
Let $L$ be as above. Then $|J| = \dim(L) \leq \frac{2n^2 g^2}{p^2 \mu^4}$.
\end{claim}
\begin{proof}
Consider the $n \times n$ matrix $X = M^* M$ with eigenvalues $\sigma_1^2,\ldots,\sigma_n^2$. By Claim~\ref{cla-propM} the diagonal elements of $X$ are all lower-bounded by $ p\mu^2 $ and the off-diagonal elements of $X$ are all upper bounded by $g$ in absolute value. Using Lemma~\ref{lem-pert}, and these bounds on the entries of $X$, we get that
$$ \left| \left\{ i \in [n] \,\,|\,\, \sigma_i^2 \leq p \mu^2/4 \right\}\right| \leq \frac{2n^2 g^2}{p^2 \mu^4}. $$ Taking square roots completes the proof.
\end{proof}
Let $u_1,\ldots,u_n$ denote the columns of $A$. We can write each $ u_j$ in the orthonormal basis $r_1,\ldots,r_n$ in a unique way as $$ u_j = \sum_{k=1}^n \alpha_{jk} r_k.$$ Observe that
\begin{equation}\label{eq-distuL}
\dist( u_j, L)^2 = \sum_{k \not\in J} |\alpha_{jk}|^2
\end{equation}
Denote the rows of the matrix $E = M A$ by $e_i, i \in [m]$ so that $\|e_i\| \leq \eps$ for all $i \in [m]$. Let $f_1,\ldots,f_n$ be the columns of $E$ and observe that
\begin{equation}\label{eq-sumnormf}
\sum_{j \in [n]}\| f_j\|^2 = \sum_{i \in [m] } \| e_i\|^2 \leq m\eps^2
\end{equation}
The next claim bounds the sum of distances of the vectors $ u_j$ to the subspace $L$.
\begin{claim}\label{cla-sumofdist}
With the above notations, we have
$$ \sum_{j=1}^n \dist( u_j, L)^2 \leq \frac{4m \eps^2}{\mu^2 p}.$$
\end{claim}
\begin{proof}
Using (\ref{eq-distuL}), (\ref{eq-sumnormf}), the orthogonality of the $ Mr_j$'s and the fact that $\sigma_j > \frac{\mu \sqrt{p}}{2}$ for all $j \not\in J$, we have
\begin{eqnarray*}
m\eps^2 &\geq& \sum_{j \in [n]} \|f_j\|^2 = \sum_{j \in [n]} \| M u_j \|^2 \\
&=& \sum_{j \in [n]}\left\| \sum_{k \in [n]} \alpha_{jk} Mr_k \right\|^2 \\
&=& \sum_{j \in [n]} \sum_{k \in [n]} |\alpha_{jk}|^2 \sigma_k^2 \\
&\geq& \frac{\mu^2 p}{4} \sum_{j \in [n]} \sum_{k \not\in J} |\alpha_{jk}|^2 \\
&=& \frac{\mu^2 p}{4} \sum_{j \in [n]} \dist( u_j, L)^2.
\end{eqnarray*}
This proves the claim.
\end{proof}
We now use Claim~\ref{cla-sumofdist} to deduce that many {\em rows} of $ A$ are close to a low dimensional subspace.
\begin{claim}\label{cla-sumofdistrows}
There exists a subspace $L' \subset \C^n$ with $\dim(L') \leq \frac{2n^2 g^2}{p^2 \mu^4}$ and s.t
$$ \sum_{j=1}^n \dist( v_j, L')^2 \leq \frac{4m \eps^2}{\mu^2 p}.$$
\end{claim}
\begin{proof}
Let $Y$ be an $n\times n$ matrix such that the $j$'th column of $Y$ is the element of $L$ closest to $ u_j$. If we let $L'$ be the span of the {\em rows} of $Y$ we have $\dim(L') \leq \dim(L)$ and, using Claim~\ref{cla-sumofdist},
$$ \sum_{j \in [n]}\dist( v_j,L')^2 \leq \| Y - A\|^2 = \sum_{j \in [n]}\dist( u_j,L)^2 \leq \frac{4m \eps^2}{\mu^2 p}.$$
\end{proof}
This claim completes the proof of Theorem~\ref{thm-apxsggenav}.
\end{proof}
\subsection*{Proof of Theorem~\ref{thm-apxsggen} using Theorem~\ref{thm-apxsggenav}}
From Theorem~\ref{thm-apxsggenav} we can get a large subset of $V$ that is $\eps'$-close to a low dimensional subspace $L$. To derive the conclusion of Theorem~\ref{thm-apxsggen}, we will show that the rest of the points in $V$ are also close to $L$, though with a slightly worse bound on the distance. This will follow by showing that, for every point $v \in V$, there are two points $u,w \in V$ that are close to $L$ and s.t $v$ is close to the line passing through them. This will imply that $v$ is also close to $L$. The details follow.
First, apply Theorem~\ref{thm-apxsggenav} to get a subspace $L$ so that $$\dim(L) \leq \frac{2n^2 g^2}{p^2 \mu^4}$$ and such that $$ \sum_{i=1}^n \dist( v_i, L)^2 \leq \frac{4m \eps^2}{\mu^2 p}.$$
Let $$I = \left\{i \in [n] \, \left|\, \dist(v_i,L)^2 > \frac{4gm \eps^2}{\mu^2 p^2} \right. \right\}$$ and observe that $|I| < p/g$. Our final step is to argue that the points $v_i, i \in I$ are also close to $L'$ since they are close to the span of two points $v_j,v_k$ with $j,k \not\in I$ (using the design properties of $T$).
\begin{claim}\label{cla-therest}
For each $i \in I$ there are indices $j,k \in [n] \setminus I$ such that $\{i,j,k\} \in T$.
\end{claim}
\begin{proof}
Fix some $i \in I$. If the claim is false then every triple in $T$ that contains $i$ must have some other element in $I$. By a pigeon hole argument, there must be an element $j \in I \setminus \{i\}$ and at least $p/|I| > g$ triples containing both $i$ and $j$, contradicting the design property of $T$.
\end{proof}
We will need the following simple lemma:
\begin{lem}\label{lem-trianglesubsp}
Let $u,v,w \in \C^d$ be an $(\eps,\mu)$-dependent triple. Let $L \subset \C^d$ be a subspace with $\dist(v,L),\dist(u,L) \leq \rho$ for some $\rho >0$. Then $\dist(w,L) \leq (\eps + 2\rho)/\mu$.
\end{lem}
\begin{proof}
Let $\alpha,\beta,\gamma$ be such that $|\alpha|,|\beta|,|\gamma| \in [\mu,1]$ and $\| \alpha u + \beta v + \gamma w \| \leq \eps$. Let $v',u' \in L$ be s.t $\| v - v'\|,\|u-u' \| \leq \rho$. Then
\begin{eqnarray*}
\dist(w,L) &\leq& \| w + (\alpha/\gamma)v' + (\beta/\gamma)u' \| \\
&\leq& \| w + (\alpha/\gamma)v + (\beta/\gamma)u \| + \| (\alpha/\gamma)v - (\alpha/\gamma)v'\| + \| (\beta/\gamma)u - (\beta/\gamma)u'\| \\
&\leq& \eps/|\gamma| + |\alpha/\gamma|\rho + |\beta/\gamma|\rho \\
&\leq& (\eps + 2\rho)/\mu.
\end{eqnarray*}
\end{proof}
Combining Claim~\ref{cla-therest} with Lemma~\ref{lem-trianglesubsp} we have that each $v_i, i \in [n]$ is $\eps'$ close to $L$ with $ \eps' \leq (\eps + 2\rho)/\mu$, where $\rho = \frac{2\eps \sqrt{gm}}{p\mu}$. Simplifying, we get $$\eps' \leq \frac{5\eps \sqrt{gm}}{p\mu^2}$$ as was required. This completes the proof of Theorem~\ref{thm-apxsggen}.\qed
\section{Proof of Theorem~\ref{thm-apxsgaffine}}\label{sec-pfaffine}
We start with some preliminary lemmas.
\begin{lem}\label{lem-triplecoefaffine}
Let $\{u,v,w\} \in \C^d$ be $B$-balanced. If $w \in \line_\eps(u,v)$ with $\eps < 1/2$ then the triple $u,v,w$ is $(\eps,1/4B)$-dependent. Furthermore, there exists a complex $\alpha$ with $|\alpha| \geq 1/4B$ such that $\| w - \alpha u - (1-\alpha)v \| \leq \eps$.
\end{lem}
\begin{proof}
By shifting $w$ to zero we can assume that both $u$ and $v$ have norm bounded by $B$.
By definition, there exists $\alpha \in \C$ such that $\| w - \alpha u - (1-\alpha)v \| \leq \eps$ and so we only need to show that $|\alpha| \geq 1/4B$ (the same argument will apply to $1-\alpha$ by symmetry). Observe that
\begin{eqnarray*}
1 &\leq& \|w - v \| \\
&\leq& \|w - \alpha u - (1-\alpha) v \| + \|\alpha u\| + \|\alpha v\| \\
&\leq& \eps + 2\alpha B,
\end{eqnarray*}
which proves the lemma.
\end{proof}
\begin{lem}\label{lem-arcreverseaffine}
Let $\{u,v,w\} \in \C^d$ be $B$-balanced and let $0 < \eps \leq 1/2 $ be a real number such that $w \in \line_{\eps}(u,v)$. Then $v \in \line_{\eps'}(w,u)$ with $\eps' = 4\eps B$.
\end{lem}
\begin{proof}
By Lemma~\ref{lem-triplecoefaffine} there exists a complex $\alpha$ with $|\alpha| \geq 1/4B$ such that $$ \|w - \alpha v - (1-\alpha) u \| \leq \eps.$$ Then $$ \|v - (1/\alpha) w + (1/\alpha-1) v\| \leq \eps/\alpha \leq 4\eps B.$$ This completes the proof.
\end{proof}
\begin{lem}\label{lem-arcdensityaffine}
Let $u,v \in \C^d$ be two distinct points. Let $k$ be the maximum size of a $B$-balanced set contained in $\line_\eps(u,v)$. If $\eps < 1/4$ then $k \leq 5B$.
\end{lem}
\begin{proof}
Suppose $k > 5B$ and let $V = \{v_1,\ldots,v_k\}$ be a $B$-balanced set contained in $\line_\eps(u,v)$. For each $v_i$ let $u_i \in \line(u,v)$ be a point of distance at most $\eps$ from it. Since the $k$ points $u_1,\ldots,u_k$ are all on a line segment of length at most $2B$ we can apply a pigeon hole argument to conclude that there must be $i \neq j$ with $\dist(u_i,u_j) \leq 2B/(k-1)$. This implies $\dist(v_i,v_j) \leq 2\eps + 2B/(k-1) < 1$, a contradiction.
\end{proof}
\subsection*{Proof of Theorem~\ref{thm-apxsgaffine}}
We define $T \subset {[n] \choose 3}$ to be the set of triples $\{i,j,k\} \subset [n]$ (with three distinct indices) for which $v_k \in \line_\eps(v_i,v_j)$. By Lemma~\ref{lem-triplecoefaffine} we have that for each triple $\{i,j,k\}$ in $T$, the corresponding triple $v_i,v_j,v_k \in \C^d$ is $(\eps,1/4B)$-dependent.
\begin{claim}\label{cla-Tdesignaffine}
$T$ as defined above is a $(p,g)$ design with $p = \delta(n-1)$ and $g < 5B$.
\end{claim}
\begin{proof}
By the conditions of the theorem, each $v_i$ is contained in at least $\delta(n-1)$ triples that are in $T$ and so the bound on $p$ holds. To prove the bound on $g$, fix $i \neq j \in [n]$. If the triple $\{i,j,k\}$ appears in $T$. Then either $v_k \in \line_\eps(v_i,v_j)$, $v_i \in \line_\eps(v_j,v_k)$ or $v_j \in \line_\eps(v_i,v_k)$. In all three cases, we have, using Lemma~\ref{lem-arcreverseaffine}, that $v_k \in \line_{\eps'}(v_i,v_j)$ with $\eps' = 4\eps B$. Since $\eps < 1/16B$ we have $\eps' < 1/4$ and we can apply Lemma~\ref{lem-arcdensityaffine} to conclude that there could be at most $5B$ such triples.
\end{proof}
Observe that we can discard some of the triples in $T$ so that $|T| \leq \delta n^2$ and so that $T$ is still a $(p,g)$-design (simply keep for each $i$ only $\delta (n-1)$ dependent triples).
Plugging the bounds obtained in the above claims and the bound $|T| \leq \delta n^2$ into Theorem~\ref{thm-apxsggen} we get a subspace $L$ with $\dim(L) \leq O(B^6/\delta^2)$ and such that $\dist(v_i,L) \leq O(\eps B^{2.5}/\sqrt{\delta})$ for all $i \in [n]$. The second part of the theorem follows from applying Theorem~\ref{thm-apxsggenav}.
\section{Proof of Theorem~\ref{thm-apxsg}}\label{sec-pfproj}
We first prove some preliminary lemmas.
\begin{lem}\label{lem-distance}
Suppose $u,v \in S^d$ are s.t $\min\{\dist(u,v),\dist(u,-v)\} = \mu$. Then, for all complex $\beta$, $\dist(u,\beta v) \geq \mu/4$.
\end{lem}
\begin{proof}
Suppose w.l.o.g $\dist(u,v) = \mu \leq \sqrt{2}$. We have $$ \mu = \sqrt{ \ip{u-v}{{u-v}}} = \sqrt{ 2 - 2\ip{u}{ v}},$$ which gives
$ \ip{u}{ v} = 1 - \mu^2/2$. Since $\dist(u, \gamma v)$ is minimized for $\gamma = \ip{u}{ v}$ we have $\dist(u,\beta v) \geq \dist(u,(1-\mu^2/2)v) = || u -v + (\mu^2/2)v|| \geq ||u-v|| - ||(\mu^2/2)v|| \geq \mu - \mu^2/2 \geq \mu/4$ (for $\mu \leq \sqrt{2}$).
\end{proof}
\begin{lem}\label{lem-triplecoef1}
Let $u,v,w \in S^d$ be distinct and let $\eps,\mu>0$ be real numbers s.t $\eps < \mu/8$. Suppose $\|w - \alpha u - \beta v \| \leq \eps$ for some complex numbers $\alpha,\beta$. If $\min\{ \dist(w,v),\dist(w,-v) \} \geq \mu$ then $|\alpha| > \mu/8$.
\end{lem}
\begin{proof}
By the triangle inequality $$ \|w - \beta v \| \leq \| \alpha u\| + \eps = |\alpha| + \eps. $$ Using Lemma~\ref{lem-distance} we have $\dist(w,\beta v) \geq \mu/4$ which gives $|\alpha| \geq \mu/4 - \eps \geq \mu/8.$
\end{proof}
\begin{lem}\label{lem-triplecoef}
Let $u,v,w \in S^d$ be $\mu$-separated and suppose $\eps < \mu/8$. Suppose $w \in \arc_\eps(u,v)$. Then, there exist complex numbers $\alpha,\beta,\gamma$ with $\| \alpha u + \beta v + \gamma w \| \leq \eps$ and s.t $\mu/8 \leq |\alpha|,|\beta|,|\gamma| \leq 1$.
\end{lem}
\begin{proof}
By the assumption, there are $\alpha',\beta'$ with $\|w - \alpha' u - \beta' v\| \leq \eps$. If $|\alpha'|$ and $|\beta'|$ are at most 1 then we are done using Lemma~\ref{lem-triplecoef1}. If not, suppose $|\alpha'| = \max\{|\alpha'|,|\beta'|\} > 1$ and divide the equation by $\alpha'$ to obtain $\|(1/\alpha')w - u - (\beta'/\alpha')v\| \leq \eps/|\alpha'| < \eps$. Now, all three coefficients are at most 1 in absolute value and, using Lemma~\ref{lem-triplecoef1}, we have the lower bound $\mu/8$ on $|1/\alpha'|, |\beta'/\alpha'|$.
\end{proof}
\begin{lem}\label{lem-arcreverse}
Let $u,v,w \in S^d$ be distinct. Let $\eps,\mu > 0$ be real numbers such that $\eps < \mu/8$. Suppose $w \in \arc_{\eps}(u,v)$ and $\min\{\dist(w,v),\dist(w,-v)\} \geq \mu$. Then $u \in \arc_{\eps'}(w,v)$ with $\eps' = 8\eps/\mu$.
\end{lem}
\begin{proof}
By our assumption, there exist complex numbers $\alpha,\beta$ such that $$ \|w - \alpha u - \beta v \| \leq \eps.$$ By Lemma~\ref{lem-triplecoef1} we have $|\alpha| > \mu/8$ and so $$ \|u - (1/\alpha) w + (\beta/\alpha) v\| \leq 8\eps/\mu.$$ This implies $u \in \arc_{\eps'}(w,v)$ as was required.
\end{proof}
\begin{lem}\label{lem-arcdensity}
Let $u,v \in S^d$ be two distinct points. Let $k$ be the maximum size of a $\mu$-separated set contained in $\arc_\eps(u,v)$. If $\eps < \mu/4$ then $k \leq 8/\mu$.
\end{lem}
\begin{proof}
Suppose $k > 8/\mu$ and let $V = \{v_1,\ldots,v_k\}$ be a $\mu$-separated set contained in $\arc_\eps(u,v)$. For each $v_i$ let $u_i \in \arc(u,v)$ be a point of distance at most $\eps$ from it. By a pigeon hole argument, there must be $i \neq j$ with $\min\{\dist(u_i,u_j),\dist(u_i,-u_j)\} \leq \pi/k \leq \mu/2$. This implies $\min\{\dist(v_i,v_j),\dist(v_i,-v_j)\} \leq 2\eps + \mu/2 < \mu$, a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-apxsg}]
To reduce to Theorem~\ref{thm-apxsggen} we will define $T \subset {[n] \choose 3}$ to be the set of triples $\{i,j,k\} \subset [n]$ for which $v_k \in \arc_\eps(v_i,v_j)$.
\begin{claim}\label{cla-Tdepsphere}
Let $\{i,j,k\} \in T$. Then the triple $v_i,v_j,v_k \in \C^d$ is $(\eps,\mu/8)$-dependent.
\end{claim}
\begin{proof}
This is immediate from Lemma~\ref{lem-triplecoef}.
\end{proof}
\begin{claim}\label{cla-Tdesignsphere}
$T$ as defined above is a $(p,g)$ design with $p = \delta(n-1)$ and $g < 8/\mu$.
\end{claim}
\begin{proof}
By the conditions of the theorem, each $v_i$ is contained in at least $\delta(n-1)$ triples that are in $T$ and so the bound on $p$ holds. To prove the bound on $g$, fix $i \neq j \in [n]$. If the triple $\{i,j,k\}$ appears in $T$, then either $v_k \in \arc_\eps(v_i,v_j)$, $v_i \in \arc_\eps(v_j,v_k)$ or $v_j \in \arc_\eps(v_i,v_k)$. In all three cases, we have, using Lemma~\ref{lem-arcreverse}, that $v_k \in \arc_{\eps'}(v_i,v_j)$ with $\eps' = 8\eps/\mu$. Since $\eps < \mu^2/32$ we have $\eps' < \mu/4$ and we can apply Lemma~\ref{lem-arcdensity} to conclude that there could be at most $8/\mu$ such triples.
\end{proof}
Plugging the bounds obtained in the above claims and the bound $|T| \leq \delta n^2$ (which can be obtained by discarding some of the triples in $T$, as before) into Theorem~\ref{thm-apxsggen} and into Theorem~\ref{thm-apxsggenav} completes the proof.
\end{proof}
\section{Proof of Theorem~\ref{thm-apxlcc}}\label{sec-apxlcc}
Since the proof follows the same lines as the proof of Theorem~\ref{thm-apxsggen}, we will assume familiarity with the proof of that theorem and only give details where the proofs differ.
We will use the following definition:
\begin{define}[LCC-matrix]\label{def-lccmat}
Let $M$ be an $nk \times n$ matrix over $\C$ and let $M_1,\ldots,M_n$ be $k \times n$ matrices so that $M$ is the concatenation of the blocks $M_1,\ldots,M_n$ placed on top of each other (so $M_\ell$ contains the rows of $M$ numbered $k(\ell-1) + 1, \ldots,k \ell$). We say that $M$ is a $(k,q)$-LCC matrix if, for each $i \in [n]$ the block $M_i$ satisfies the following conditions:
\begin{itemize}
\item Each row of $M_i$ has support size at most $q+1$.
\item All rows in $M_i$ have the value $1$ in position $i$.
\item The supports of two distinct rows in $M_i$ intersect only in position $i$.
\end{itemize}
\end{define}
Let $V = \{v_1,\ldots,v_n\} \subset \C^d$ be a $(q,\delta,B,\eps)$-stable LCC and assume w.l.o.g that $d=n$ (that is, pad the vectors $v_i$ with zeros so that we can think of them as vectors in $\C^n$). Let $A$ be the $n \times n$ matrix with rows $v_i$.
\begin{claim}\label{cla-lccM}
There exists a $(k,q)$-LCC matrix $M$ with dimensions $nk \times n$ and with $k = \Omega(\delta n/q)$ such that all entries of $M$ have absolute values at most $B$ and such that $$ ||M A ||^2 \leq n^2 \eps^2.$$
\end{claim}
\begin{proof}
We will show how to construct the $k \times n$ block $M_i$ of $M$ (see Definition~\ref{def-lccmat}) row by row. Using the definition of stable LCC, there exists a family $Q_i$ of $k = \Omega(\delta n/q)$ disjoint $q$-tuples of elements of $V$ such that, for each $q$-tuple $J \in Q_i$, we have $\dist(v_i,\span_B(J)) \leq \eps$. Each of these $q$-tuples, $J$, defines a row vector $w_J$ with $1$ in the $i$'th position, $B$-bounded entries in positions indexed by $J$, and zeros everywhere else in the following manner: Suppose $v_i = \sum_{j \in J}b_jv_j + e$ with $|b_j| \leq B$ for all $j \in J$ and $||e|| \leq \eps$. Then we define $w_j$ to have $1$ in position $i$ and values $-b_j$ in positions $j \in J$ (with zeros in all other positions). Then, we have $||w_J A|| = ||e|| \leq \eps$. Taking all these row vectors to construct $M_i$ we get the required bound on $||MA||^2$.
\end{proof}
Let $E = MA$ so that $||E||^2 \leq n^2 \eps^2$. We now construct another $nk \times n$ matrix $R$ so that $R^T M$ will be diagonal dominant. $R$ will be comprised of $n$ blocks, $R_1,\ldots,R_n$, each of dimensions $k \times n$ so that $R_i$ has $1$'s in the $i$'th column and zeros everywhere else. Notice that, the $i$'th row of $R^T M$ is the sum of the rows in the block $M_i$ of $M$.
Let $\hat M = R^TM$ and $\hat E = R^TE$ so that $\hat E = \hat M A$. An application of the Cauchy-Schwarz inequality shows that $$|| R^TE ||^2 \leq n ||E||^2 \leq n^3 \eps^2.$$ Observe that the diagonal elements of $\hat M$ are all equal to $k$ and that the off-diagonal elements of $\hat M$ are all of absolute value at most $B$ (since the supports of rows in $M_i$ are disjoint except for the $i$'th coordinate).
We proceed with analyzing the spectrum of $\hat M$. Let $r_1,\ldots,r_n$ be the right singular vectors and $\sigma_1,\ldots,\sigma_n$ the corresponding singular values. If we take $X =\hat M^* \hat M$ then the diagonal elements of $X$ are all at least $K^2 \geq k^2$ and the off diagonal elements can be bounded by $2kB + nB^2 \leq O(nB^2)$. If we define $$ L = \span\{ r_j \,|\, \sigma_j < K/2\}$$ we get that, using Lemma~\ref{lem-pert}, $$\dim(L) \leq O(n^4 B^4/K^4) = O((qB/\delta)^4).$$
As in the proof of Theorem~\ref{thm-apxsggen}, we consider the columns $u_1,\ldots,u_n$ of $A$ and obtain the bound
$$ \sum_{j=1}^n \dist(u_j,L)^2 \leq 4||\hat E||^2/K^2 = O(n^3 \eps^2/K^2). $$ This means that there is a subspace $L'$ with the same dimension as $L$ such that
$$ \sum_{i=1}^n \dist(v_j,L')^2 \leq O(n^3 \eps^2/K^2). $$
Thus, there is a set $V' \subset V$ of size $n' \geq (1 - \delta/2)n$ such that for all $ v' \in V'$ we have $\dist(v', L')^2 \leq O(n^2 \eps^2/\delta K^2) = O(q^2 \eps^2/\delta^3)$. To finish the proof we observe that, using the definition of a stable LCC, for every $v \in V$ there is a $q$-tuple $ J \subset V'$ with $\dist(v_i, \span_B(J)) \leq \eps$. Using the bound on the distances of elements of $V'$ to $L'$ and the bound $B$ on the coefficients in the linear combinations in $\span_B(J)$, we get that $\dist(v,L') \leq \eps + O( qB \cdot (q \eps/\delta^{1.5}) ) = O( q^2 B \eps / \delta^{1.5})$. This completes the proof of Theorem~\ref{thm-apxlcc}.
\ignore{
\section{Error-correction over infinite fields}\label{sec-codesreal}
While coding theory deals mainly with codes over finite fields, the need in numerous applications to transmit real-valued signals in a way which is noise-tolerant has lead to the study of such codes over the reals.
Recent breakthrough in understanding such codes arose from the new area of compressed sensing, initiated by Donoho and Candes-Tao. We give a quick survey of real-valued codes and how stability questions naturally arise in this setting. Given this understanding, exploring the local variants, well motivated and studied for finite fields, such as locally testable, decodable and correctable codes, naturally suggests itself.
The notion of a code over the Reals is defined in complete analogy with the definition over finite fields: A $w$-error-correcting code of dimension $d$ and block length
$n$ over the reals is given by a linear
map $A : \R^d \rightarrow \R^n$, such that for each $f \in \R^d$, $f
\neq 0$, $\| Af \|_0 > 2w$. (Thus the Hamming distance of the code is at least $2w+1$).
The rate of the code is the ratio
$d/n$. $A$ may be viewed as an $n\times d$ matrix which is called the generating matrix of the code.
Given a received word $y = A f + e$ with $\|e\|_0 \le w$, one
can recover the message $f$ as the solution $x$ to optimization
problem:
\[ \min_{x \in \R^d} \|y - A x \|_0 \ . \]
In the Compressed Sensing view (which in coding theory is called `syndrome decoding'), the sparse vector $e$ is actually considered the (length $n$) message, and upon receiving the vector $g=A^{\perp} y = A^{\perp} e$ we must find the sparsest element $z$ satisfying $z= A^{\perp} y$, which in a code like above would be $e$. Note that $g$ can be easily computed from $y$. Here $A^{\perp}$ is the parity-check matrix of the code.
The above non-convex optimization problem is NP-hard to solve in
general. Quite remarkably, if the code $A$ meets certain conditions (which are met with high probability e.g. by a random matrix whose entries are standard independent Gaussians or even random signs),
one can recover $f$ by solving the linear program
\[ \min_{x \in \R^d} \|y - A x \|_1 \ . \] (The above
$\ell_1$-minimization task, which is easily written as a linear
program, is often called {\em basis pursuit} in the literature.) Note
that we are not restricting the magnitude of erroneous entries in $e$,
only that their number is at most $w$.
However the above is clearly a very stylized model, as when transmitting real values one can expect inaccuracies and noise besides the adversarially changed $w$ positions of the output. Noise and round-off can affect all coordinates of the input signal $f$, the output signal $y$, and even the entries of the matrix $A$ when they are sampled from e.g. the Gaussian distribution.
As it happens, almost nothing is lost when allowing these errors.
By the virtue of their probabilistic constriction and analysis these random codes are stable under small changes in $A$, and moreover since all $A$'s entries are bounded input errors can be captured in output errors. Thus one focuses on the more realistic model in which the output error $e$ is composed of two parts, $e=e' + e''$, where $e'$ is a $w$-sparse error as before, whose entries are completely arbitrary, and $e''$ is a global error vector, which may be non-zero in all coordinates, but has small norm. In this case one cannot recover the message $f$ exactly, and we would like the difference of the decoded message from the original one $f$ should have small norm as well.
Many papers, e.g. \cite{CT05, RV05, KT07, DMT07, GLW09} give different decoding algorithms which tolerate such errors in different norms. The upshot of all is that there are such stable real-valued codes which achieve the gold-standard of coding theory of constant rate codes with efficient encoding and decoding algorithms from a linear number of errors of arbitrary magnitude. Their stability means that small global error of norm $\alpha$ translates to distance $O(\alpha)$ in the decoding, which can be achieved both for the $L_1$ and $L_2$ norms.
The notion of stable locally correctable codes over the reals (and complexes), which we have introduced above, is a natural extension of this line of work. In our notation the rows of the generating matrix $A$ are the set of points $v_1, v_2, \cdots v_n \in \R^d$. The `stability' of the codes guarantees that small perturbations in the codeword (in Euclidean norm) have small effect on the decoding process.
}
\bibliographystyle{alpha}
|
{
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| 568
|
Corpus Christi, Texas Area Real Estate :: Kristen Gilstrap Team | Serving your real estate needs in the Corpus Christi, Texas Area!
The Kristen Gilstrap of Keller Willams Realty has helped hundreds of people buy, sell and invest all over the Corpus Christi, TX area. From Port Aransas, Padre Island, Calallen and Portland. Kristen Gilstrap and her team are committed to the Coastal Bend and has been among the top 20 agents since 2010. Our team of experienced, full-time Corpus Christi real estate agents is ready to walk you through the entire process of buying the perfect home, selling your existing house or breaking into the investment market. The Kristen Gilstrap Team is committed to helping you achieve your Real Estate Goals!
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A member account that you manage.
Hassle-free assistance. We are here if you need us.
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Professional interior and exterior photos.
Customized web pages with photo and video tours.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 9,607
|
{"url":"http:\/\/universeinproblems.com\/index.php?title=Particles%27_motion_in_general_black_hole_spacetimes&diff=prev&oldid=1422","text":"# Difference between revisions of \"Particles' motion in general black hole spacetimes\"\n\nIn this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.","date":"2020-10-25 03:15:22","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\": 0.9542105793952942, \"perplexity\": 1281.865647037049}, \"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-2020-45\/segments\/1603107885126.36\/warc\/CC-MAIN-20201025012538-20201025042538-00293.warc.gz\"}"}
| null | null |
describe('RestService', function() {
var restService;
var $httpBackend;
/*
Fake Data
*/
var fakeUser = "testuser";
var fakePass = "testpass";
var fakeKeystoneId = "123";
var fakeSessionId = "abc123";
var fakeAccessToken = "5s5s5s";
var fakeUdrMeters = { meter: "bla" };
var fakeMeterName = "net";
var fakeFrom = "2015-01-01 12:00:00";
var fakeTo = "2015-01-02 12:00:00";
var fakeRateQuery = "?resourcename="+fakeMeterName+"&from="+fakeFrom+"&to="+fakeTo;
var fakeChargeQuery = "?userid="+fakeUser+"&from="+fakeFrom+"&to="+fakeTo;
var fakeAccessQuery = "?access_token=" + fakeAccessToken;
var fakeSessionQuery = "?session_id=" + fakeSessionId;
var fakeUserIdQuery = "?user_id=" + fakeUser;
var fakeBillQuery = "?user_id=" + fakeUser + "&from=" + fakeFrom + "&to=" + fakeTo;
var fakePolicyConfig = { rate_policy:'dynamic' };
var fakeAdmins = ['test1', 'user2'];
var fakeUpdatedAdmins = {
'admins': fakeAdmins,
'sessionId': fakeSessionId
};
var fakeBillItems = {
"cpu_util": {
price: 5
}
};
var fakeBill = {
userId: fakeUser,
from: fakeFrom,
to: fakeTo,
items: fakeBillItems
};
/*
Test setup
*/
beforeEach(function(){
resetAllMocks();
/*
Load module
*/
module('dashboard.services');
sessionServiceMock.getAccessToken.and.returnValue(fakeAccessToken);
module(function($provide) {
$provide.value('sessionService', sessionServiceMock);
});
/*
Inject dependencies and configure mocks
*/
inject(function(_restService_, _$httpBackend_) {
restService = _restService_;
$httpBackend = _$httpBackend_;
});
$httpBackend.whenPOST("/dashboard/rest/usage").respond(200);
$httpBackend.whenGET("/dashboard/rest/rate" + fakeRateQuery).respond(200);
$httpBackend.whenGET("/dashboard/rest/rate/status").respond(200);
$httpBackend.whenPOST("/dashboard/rest/rate/status").respond(200);
$httpBackend.whenGET("/dashboard/rest/charge" + fakeChargeQuery).respond(200);
$httpBackend.whenPOST("/dashboard/rest/keystone").respond(200);
$httpBackend.whenPOST("/dashboard/rest/login").respond(200);
$httpBackend.whenPOST("/dashboard/rest/session").respond(200);
$httpBackend.whenPUT("/dashboard/rest/keystone").respond(200);
$httpBackend.whenGET("/dashboard/rest/tokeninfo" + fakeAccessQuery).respond(200);
$httpBackend.whenGET("/dashboard/rest/keystonemeters").respond(200);
$httpBackend.whenGET("/dashboard/rest/udrmeters").respond(200);
$httpBackend.whenPOST("/dashboard/rest/udrmeters").respond(200);
$httpBackend.whenGET("/dashboard/rest/udrmeters/externalids" + fakeUserIdQuery).respond(200);
$httpBackend.whenPOST("/dashboard/rest/udrmeters/externalids").respond(200);
$httpBackend.whenPOST("/dashboard/rest/udrmeters/externalsources").respond(200);
$httpBackend.whenGET("/dashboard/rest/users" + fakeSessionQuery).respond(200);
$httpBackend.whenGET("/dashboard/rest/admins" + fakeSessionQuery).respond(200);
$httpBackend.whenPUT("/dashboard/rest/admins").respond(200);
$httpBackend.whenGET("/dashboard/rest/billing" + fakeChargeQuery).respond(200);
$httpBackend.whenGET("/dashboard/rest/billing/bills" + fakeUserIdQuery).respond(200);
$httpBackend.whenPOST("/dashboard/rest/billing/bills/pdf").respond(200);
$httpBackend.whenGET("/dashboard/rest/billing/bills/pdf" + fakeBillQuery).respond(200);
$httpBackend.whenGET("/dashboard/rest/users/" + fakeUser + fakeSessionQuery).respond(200);
});
/*
Test teardown
*/
afterEach(function() {
$httpBackend.verifyNoOutstandingExpectation();
$httpBackend.verifyNoOutstandingRequest();
});
/*
Tests
*/
describe('getUdrData', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/usage", {
'keystoneId': fakeKeystoneId,
'from': fakeFrom,
'to': fakeTo
});
restService.getUdrData(fakeKeystoneId, fakeFrom, fakeTo);
$httpBackend.flush();
});
});
describe('sendKeystoneAuthRequest', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/keystone", {
'username': fakeUser,
'password': fakePass
});
restService.sendKeystoneAuthRequest(fakeUser, fakePass);
$httpBackend.flush();
});
});
describe('sendLoginRequest', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/login", {
'username': fakeUser,
'password': fakePass
});
restService.sendLoginRequest(fakeUser, fakePass);
$httpBackend.flush();
});
});
describe('requestSessionToken', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/session", {
'username': fakeUser,
'password': fakePass
});
restService.requestSessionToken(fakeUser, fakePass);
$httpBackend.flush();
});
});
describe('storeKeystoneId', function() {
it('should send complete PUT request', function() {
$httpBackend.expectPUT("/dashboard/rest/keystone", {
'username': fakeUser,
'sessionId': fakeSessionId,
'keystoneId': fakeKeystoneId
});
restService.storeKeystoneId(fakeUser, fakeKeystoneId, fakeSessionId);
$httpBackend.flush();
});
});
describe('getTokenInfo', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/tokeninfo" + fakeAccessQuery);
restService.getTokenInfo(fakeAccessToken);
$httpBackend.flush();
});
});
describe('getKeystoneMeters', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/keystonemeters");
restService.getKeystoneMeters();
$httpBackend.flush();
});
});
describe('getAdminGroupInfo', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/admins" + fakeSessionQuery);
restService.getAdminGroupInfo(fakeSessionId);
$httpBackend.flush();
});
});
describe('getAllUsers', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/users" + fakeSessionQuery);
restService.getAllUsers(fakeSessionId);
$httpBackend.flush();
});
});
describe('getUdrMeters', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/udrmeters");
restService.getUdrMeters();
$httpBackend.flush();
});
});
describe('updateUdrMeters', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/udrmeters", fakeUdrMeters);
restService.updateUdrMeters(fakeUdrMeters);
$httpBackend.flush();
});
});
describe('getRateForMeter', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/rate" + fakeRateQuery);
restService.getRateForMeter(fakeMeterName, fakeFrom, fakeTo);
$httpBackend.flush();
});
});
describe('getChargeForUser', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/charge" + fakeChargeQuery);
restService.getChargeForUser(fakeUser, fakeFrom, fakeTo);
$httpBackend.flush();
});
});
describe('getActiveRatePolicy', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/rate/status");
restService.getActiveRatePolicy();
$httpBackend.flush();
});
});
describe('setActiveRatePolicy', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/rate/status", fakePolicyConfig);
restService.setActiveRatePolicy(fakePolicyConfig);
$httpBackend.flush();
});
});
describe('updateAdmins', function() {
it('should send complete PUT request', function() {
$httpBackend.expectPUT("/dashboard/rest/admins", {
'admins': fakeAdmins,
'sessionId': fakeSessionId
});
restService.updateAdmins(fakeAdmins, fakeSessionId);
$httpBackend.flush();
});
});
describe('getUserInfo', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/users/" + fakeUser + fakeSessionQuery);
restService.getUserInfo(fakeUser, fakeSessionId);
$httpBackend.flush();
});
});
describe('createBillPDF', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/billing/bills/pdf", fakeBill);
restService.createBillPDF(fakeBill);
$httpBackend.flush();
});
});
describe('getBillPDF', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/billing/bills/pdf" + fakeBillQuery);
restService.getBillPDF(fakeUser, fakeFrom, fakeTo);
$httpBackend.flush();
});
});
describe('getBills', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/billing/bills" + fakeUserIdQuery);
restService.getBills(fakeUser);
$httpBackend.flush();
});
});
describe('getBillingInformation', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/billing" + fakeChargeQuery);
restService.getBillingInformation(fakeUser, fakeFrom, fakeTo);
$httpBackend.flush();
});
});
describe('getExternalUserIds', function() {
it('should send complete GET request', function() {
$httpBackend.expectGET("/dashboard/rest/udrmeters/externalids" + fakeUserIdQuery);
restService.getExternalUserIds(fakeUser);
$httpBackend.flush();
});
});
describe('updateExternalUserIds', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/udrmeters/externalids", {
userId: fakeUser,
externalIds: [1]
});
restService.updateExternalUserIds(fakeUser, [1]);
$httpBackend.flush();
});
});
describe('addExternalMeterSource', function() {
it('should send complete POST request', function() {
$httpBackend.expectPOST("/dashboard/rest/udrmeters/externalsources", {
source: "test"
});
restService.addExternalMeterSource("test");
$httpBackend.flush();
});
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,505
|
<HTML>
<HEAD>
<meta charset="UTF-8">
<title>MyArea.<init> - </title>
<link rel="stylesheet" href="../../style.css">
</HEAD>
<BODY>
<a href="../index.html">com.epimorphics.android.myrivers.data</a> / <a href="index.html">MyArea</a> / <a href="."><init></a><br/>
<br/>
<h1><init></h1>
<code><span class="identifier">MyArea</span><span class="symbol">(</span><span class="symbol">)</span></code>
<p>Data constructor which initialises all properties to null</p>
</BODY>
</HTML>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,195
|
Q: XML reading is not working in my ASP CLASSIC code - Load file returns FALSE In my outlook 2013 I created a function that saves the sender, subject and domain name in variables. These variables are then transferred/created into an XML file which all works.
Though the next thing I want to achieve is to read the XML files variables and use them on my web page. The reading however does not work, it returns nothing.
The Outlook 2013 macro creates the XML file on a server, and it also ends up there (checked it).
The ASP file is located on the same server of the XML file in the same folder.
My code is as follows:
The XML file that is created (with a xml extension)
<Mail2Memo>
<Receiver>Nicolas</Receiver>
<Domain>hotmail.com</Domain>
<Subject>I am awesome</Subject>
</Mail2Memo>
The ASP code that should read, and output the XML values. (did the output in a table for now)
<!DOCTYPE html>
<HTML>
<HEAD>
<meta charset="utf-8">
<TITLE>Send mails</TITLE>
</HEAD>
<BODY>
<%
Set xmlDOM = Server.CreateObject("MSXML2.DOMDocument")
xmlDOM.async = False
xmlDOM.setProperty "ServerHTTPRequest", True
xmlDOM.Load("MyXMLFile.xml")
Test = xmlDOM.Load("MyXMLFile.xml")
response.write (Test)
Set itemList = XMLDom.SelectNodes("Mail2Memo")
For Each itemAttrib In itemList
newReceiver =itemAttrib.SelectSingleNode("Receiver").text
newDomain =itemAttrib.SelectSingleNode("Domain").text
newSubject =itemAttrib.SelectSingleNode("Subject").text
%>
<tr>
<td><%=newReceiver%></td>
<td><%=newDomain%></td>
<td><%=newSubject%></td>
</tr>
<%
Next
Set xmlDOM = Nothing
Set itemList = Nothing
%>
</BODY>
</HTML>
This code just returns an empty web page for some reason..
The test variable that I have build in (to see what goes wrong) returns the value FALSE, so I am guessing the XML file isn't picked up?
By the way, in the Outlook VBA I added the Microsoft XML, 3.0 as reference
Anyone has a clue what I am doing wrong here?
UPDATE XML Generating Code
The code below is being executed in outlook itself.
'----------------------------------------------------'
'-----------------WRITE TO XML FILE------------------'
'----------------------------------------------------'
Dim objDom As DOMDocument
Dim objXMLRootelement As IXMLDOMElement
Dim objXMLelement As IXMLDOMElement
Dim objXMLattr As IXMLDOMAttribute
Dim objMemberElem As IXMLDOMElement
Set objDom = New DOMDocument
'~~> Creates root element
Set objXMLRootelement = objDom.createElement("Mail2Memo")
objDom.appendChild objXMLRootelement
'~~> Create Melder element
Set objXMLelement = objDom.createElement("Melder")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = MelderNaam
'~~> Create Bedrijf element
Set objXMLelement = objDom.createElement("Bedrijf")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = MelderDomein
'~~> Creates Onderwerp element
Set objXMLelement = objDom.createElement("Onderwerp")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Onderwerp
'~~> Creates Omschrijving element
Set objXMLelement = objDom.createElement("Omschrijving")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Omschrijving
'~~> Creates Datum element
Set objXMLelement = objDom.createElement("Datum")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = OntvangstDatum
'~~> Creates Ontvanger element
Set objXMLelement = objDom.createElement("Receiver")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Receiver
'~~> Creates EindDatum element
Set objXMLelement = objDom.createElement("EindDatum")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = EindDatum
'~~> Creates Status element
Set objXMLelement = objDom.createElement("Status")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Status
'~~> Creates Debiteur element
Set objXMLelement = objDom.createElement("Debiteur")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Debiteur
'~~> Creates Soort element
Set objXMLelement = objDom.createElement("Soort")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Soort
'~~> Creates Becode element
Set objXMLelement = objDom.createElement("Becode")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = Becode
'~~> Creates Email element
Set objXMLelement = objDom.createElement("MelderEmail")
objXMLRootelement.appendChild objXMLelement
objXMLelement.Text = MelderEmail
'~~> Saves XML data to a file
Select Case Receiver
Case "DAVE"
objDom.Save ("\\SERVERADRESS\DaveMailFile.xml")
Case "RICHARD"
objDom.Save ("\\SERVERADRESS\RichardMailFile.xml")
Case "NICK"
objDom.Save ("\\SERVERADRESS\NickMailFile.xml")
End Select
A: You can try this:
Set xmlDOM = Server.CreateObject("msxml2.DOMDocument.3.0")
xmlDOM.async = false
xmlDOM.setProperty "ServerHTTPRequest", True
' DO NOT validate XML
xmlDOM.validateOnParse = false
'xmlDOM.preserveWhiteSpace = false ' if you want to remove spaces
Test = xmlDOM.Load("MyXMLFile.xml")
If Not Test Then
Response.write "File is empty" 'check the path
Else
set itemList = xmlDOM.getElementsByTagName("Mail2Memo")
For Each objNode In objNodeList
'...
Next
End if
Note the part
xmlDOM.validateOnParse = false
to understand if the problem is about the XML file format.
A: You shouldn't automate Outlook on the server from any unattended application or service.
Microsoft does not currently recommend, and does not support, Automation of Microsoft Office applications from any unattended, non-interactive client application or component (including ASP, ASP.NET, DCOM, and NT Services), because Office may exhibit unstable behavior and/or deadlock when Office is run in this environment.
If you are building a solution that runs in a server-side context, you should try to use components that have been made safe for unattended execution. Or, you should try to find alternatives that allow at least part of the code to run client-side. If you use an Office application from a server-side solution, the application will lack many of the necessary capabilities to run successfully. Additionally, you will be taking risks with the stability of your overall solution.
Read more about this in the Considerations for server-side Automation of Office article.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 543
|
Der Mölmeshof ist ein Ortsteil der Gemeinde Gerstungen im Wartburgkreis in Thüringen.
Lage
Der Mölmeshof befindet sich etwa 1,8 Kilometer westlich der Ortslage von Marksuhl und etwa 12 Kilometer (Luftlinie) von der Kreisstadt Bad Salzungen entfernt. Er liegt in waldreicher Umgebung am Mölmesbach, einem linken Zufluss der Suhl. In Höhe der Ortslage befindet sich ein Fischweiher.
Geschichte
Der Mölmeshof gehört zu einer Gruppe von Höfen und Kleinsiedlungen (Baueshof, Clausberg, Lutzberg, Kriegersberg, Hütschhof, Frommeshof, Rangenhof, Meileshof, Lindigshof, Josthof und andere), die im Herrschaftsbereich der Frankensteiner Grafen und ihres Hausklosters Frauensee seit dem Hochmittelalter im Buntsandstein-Hügelland bei Marksuhl gegründet wurden. Mitte des 19. Jahrhunderts verfügte der Baueshof über 5 Wohnhäuser und 27 Einwohner. Der Ort war stets nach Marksuhl eingepfarrt und eingeschult gewesen.
Unweit westlich der Ortslage befinden sich nahe einer Quelle Überreste einer spätmittelalterlichen Glashütte. Die älteste urkundliche Erwähnung des Mölmeshofes erfolgte 1531.
Seit den 1990er Jahren wurde das villenartige Gutsgebäude aufwändig saniert und dient seit 1997 als Schulungs- und Tagungsort.
Verkehr
Der Mölmeshof liegt an der Ortsverbindungsstraße Baueshof – Mölmeshof – Josthof, an der Landstraße 1023. Zum Mölmeshof verkehrt keine direkte Busverbindung. Die nächstgelegene Haltestelle befindet sich am Baueshof, sie gehört zur Buslinie L-52 auf der Strecke Eisenach – Förtha – Marksuhl – Berka/Werra – Dankmarshausen – Großensee der Verkehrsgesellschaft Wartburgkreis.
Kultur und Sehenswürdigkeiten
Der Mölmeshof befindet sich am Verlauf des Ökumenischen Pilgerweges im Abschnitt von Eisenach über Frauensee, Vacha nach Bad Hersfeld und Fulda. Das liebevoll restaurierte Gutsgebäude kann als Pilgerherberge genutzt werden.
Literatur
W. Döpel: Geschichte von Marksuhl, Druck- und Verlag der Hofbuchdruckerei Eisenach H. Kahle, Eisenach 1909
Weblinks
Pilgerherberge Mölmeshof
Einzelnachweise
Geographie (Gerstungen)
Ort im Wartburgkreis
Ort im Naturpark Thüringer Wald
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,746
|
package org.kaazing.gateway.management.jmx;
import javax.management.ObjectName;
import org.kaazing.gateway.management.system.CpuListManagementBean;
public class CpuListMXBeanImpl implements CpuListMXBean {
/*
* The management bean this MBean is wrapping.
*/
private CpuListManagementBean cpuListManagementBean;
/*
* Storing the session's name only so we can retrieve it during shutdown,
* when we need to have it to unregister it.
*/
private final ObjectName objectName;
public CpuListMXBeanImpl(ObjectName objectName, CpuListManagementBean cpuListManagementBean) {
this.objectName = objectName;
this.cpuListManagementBean = cpuListManagementBean;
}
@Override
public ObjectName getObjectName() {
return objectName;
}
@Override
public int getNumCpus() {
return cpuListManagementBean.getNumCpus();
}
@Override
public String getSummaryDataFields() {
return cpuListManagementBean.getSummaryDataFields();
}
@Override
public String getSummaryData() {
return cpuListManagementBean.getSummaryData();
}
@Override
public int getSummaryDataGatherInterval() {
return cpuListManagementBean.getSummaryDataGatherInterval();
}
@Override
public void setSummaryDataGatherInterval(int interval) {
cpuListManagementBean.setSummaryDataGatherInterval(interval);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,910
|
{"url":"http:\/\/clay6.com\/qa\/47642\/the-number-of-different-ways-of-distributing-10-marks-among-3-questions-eac","text":"Comment\nShare\nQ)\n\n# The number of different ways of distributing 10 marks among 3 questions,each questions carrying atleast 1 mark,is\n\n$\\begin{array}{1 1}(A)\\;72\\\\(B)\\;71\\\\(C)\\;36\\\\(D)\\;\\text{None of these}\\end{array}$\n\n\u2022 $C(n,r)=\\large\\frac{n!}{r!(n-r)!}$\n$\\Rightarrow 9C_7=9C_2$\n$9C_2=\\large\\frac{9!}{2!7!}=\\frac{9\\times 8\\times 7!}{2\\times 7!}$\n$\\Rightarrow 36$","date":"2019-11-14 23:30:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4666675329208374, \"perplexity\": 3851.204554440401}, \"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\/1573496668544.32\/warc\/CC-MAIN-20191114232502-20191115020502-00194.warc.gz\"}"}
| null | null |
require 'helper'
class TestGraphiti < MiniTest::Unit::TestCase
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,733
|
Europamästerskapen i badminton 1968 anordnades den 19-21 april i Bochum, Tyskland.
Medaljsummering
Resultat
Referenser
1968 i Västtyskland
Sport i Bochum
Sportevenemang i Tyskland
Sportåret 1968
1968
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,484
|
Q: Debug Haskell composition chain without converting to point-full representation I have learned that point-free style is preferred in the Haskell community, and I often write expressions like this:
naive = (slugifyUnicode . T.take maxFilenameSize . T.pack . stripHtmlTags . T.unpack . plainToHtml) sfld
However, while debugging, I find myself repeatedly converting expressions like this into chains of $ operators in order to use some variant of trace, and I am starting to think it is preferable to forgo the point-free style and just write lots of $s to start with, in order to make debugging less cumbersome. After all, it ends up being about the same number of characters.
Does anyone know of a way to debug long chains of composed functions without de-composing them?
And more generally, any comments on this inconvenience are very welcome.
A: For any intermediate value that has a Show instance you can just use traceShowId inline:
naive = (slugifyUnicode . T.take maxFilenameSize . traceShowId . T.pack . stripHtmlTags . T.unpack .plainToHtml) sfld
If the intermediary value is a String you can use traceId instead.
For anything that doesn't have a Show instance you'd have to define a helper:
data CustomType = CustomType String
traceHelper :: CustomType -> CustomType
traceHelper s@(CustomType c) = trace c $ s
-- arbitrary functions we want to compose
a :: aIn -> CustomType
b :: CustomType -> bOut
c :: aIn -> bOut
c = b . traceHelper . a
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 567
|
\section{Introduction}
\label{Introduction}
Problems dealing with quantum many-body systems in lattices appear very often in different branches of Physics and Chemistry. They typically correspond to discretized versions of first-principle continuum models, like in high-energy physics, atomic physics, or quantum chemistry, or provide a phenomenological description of a complex system, as in condensed matter physics. They are characterized in terms of a lattice Hamiltonian, $H$, which describes the motion, as well as the interactions among the different constituents. Apart from generating the dynamics via the Schr\"odinger equation, the Hamiltonian defines the quantum state of the system in thermal equilibrium through the Gibbs density operator,
\begin{equation}
\label{rho}
\rho = \frac{e^{-\beta H}}{Z} = \frac{e^{-\beta H}}{\mathrm{tr}\left[e^{-\beta H}\right]},
\end{equation}
where $Z$ is the partition function and $\beta=1/\kappa_B T$ is the inverse temperature (we set the Bolzmann constant $\kappa_B=1$). This operator encodes all the (statical) physical properties of our systems. Extracting that information becomes a hard problem, even for systems consisting of very few particles. The reason is that, in order to determine expectation values of observables, we have to express $\rho$ in a basis of the corresponding Hilbert space, and the dimension of the latter grows exponentially with the number of lattice sites, $N$ (i.e. volume) of the lattice. This fact is ultimately related to the tensor product structure inherent in quantum mechanical problems dealing with composite objects, and thus ubiquitous in several branches of science.
There exist different ways around that problem, at least in some specific situations. For instance, one can employ sampling techniques in certain models (not suffering from the sign problem), to accurately determine the physical properties of a system in thermal equilibrium. Alternatively, one can restrict oneself to simple tractable families of states depending on few parameters, which can then be determined by variational techniques. This last approach typically requires a good intuition to select which family will encompass all the physical properties that one has to describe, and can easily lead to either wrong or inaccurate results.
Yet another approach is that of quantum simulation, where the
Hamiltonian of interest is implemented on a different system on which
one has enough control \cite{Cirac2012a}.
Strictly speaking, the exponential scaling of the dimension of the Hilbert space with the size of the lattice should not be the ultimate reason for the difficulty of quantum many-body problems, at least for the ones that naturally appear in nature. For instance, if $H$ is the sum of terms acting non-trivially only on at most $x$ lattice sites, then we can characterize all possible Hamiltonians with a number of parameters that scales only polynomially with $N$. If those terms are local, meaning that the distance between the sites on which term of $H$ acts is bounded by a constant, this scale is even linear in $N$. Thus, for all those problems, $\rho$ itself only depends on few parameters. One says that the states can only explore a very small ``corner'' of the Hilbert space \cite{Cirac2009}. Consequently, it may be possible to utilize this fact to find families of states that {\em describe all possible many-body lattice problems with $x$-body interactions in thermal equilibrium, and that depend on a number of parameters that only grows polynomially with} $N$. Thus, a central problem in this context is to find and characterize such a family of states. A first and fundamental step would be to solve that problem for local Hamiltonians, on which we will concentrate in the following.
Matrix Product States (MPS) \cite{Fannes1992,Perez-Garcia2007} provide the answer for one dimensional models at zero temperature for both, gapped \cite{Hastings2007,Landau2013} and critical models \cite{Verstraete2006}. Specifically, if $\Psi_0$ is the ground state of such a Hamiltonian there exists a MPS of bond dimension $D$, $\Psi_{\mathrm{MPS}}$, such that $\|\Psi_0 -\Psi_{\mathrm{MPS}}\|<\epsilon$ with $D=O[poly(N/\epsilon)]$.
Note that, in turn, the number of parameters to characterize the MPS scales polynomially with $D$. This result is strongly connected to the area law \cite{Srednicki1993,Eisert2010}, which is fulfilled (or only slightly violated) for those models and MPS. In higher dimensions and still at zero temperature, it is conjectured (and proven under certain assumptions \cite{Masanes2009,Hamza2009b}), that the area law still holds (with logarithmic corrections for certain critical models \cite{Wolf2006,Gioev2006}). In that case, one would expect that the Projected Entangled-Pair States (PEPS) \cite{Verstraete2004,Verstraete2004c}, which extend MPS to higher dimensions, would provide us with the efficient description of that corner of the Hilbert space \cite{Cirac2009}. Moreover, for any finite temperature (independent of $N$), an area law has been proven \cite{Wolf2008} both for Gibbs states (\ref{rho}), as well as for Projected Entangled-Pair Operators (PEPO), the extension of PEPS to mixed sates. This also suggests that PEPOs can efficiently describe Gibbs states of local Hamiltonians.
From the physics point of view, this is actually the relevant question, as any extended system can only be cooled down to a certain temperature independent of the system size.
Hasting \cite{Hastings2006} has already derived some remarkable results addressing that question. He has shown that in $d$ spatial dimensions, one can build a PEPO, $\rho_{\rm PEPO}$, such that $||\rho-\rho_{\rm PEPO}||_1< \epsilon$ with bond dimension scaling as
\begin{equation}
D=e^{O(\beta\log(N/\epsilon)^{d})}.
\end{equation}
This gives a polynomial scaling for one dimension, and a sub-exponential (although superpolynomial) one for higher ones. This result also implies a bound for the approximation of the ground state. In fact, if $H$ is gapped and the density of states for a fixed energy only grows as poly$(N)$, then choosing $\beta=O(\log N)$ in (\ref{rho}) we obtain a state that is as close as we want to the ground state \cite{Hastings2007b}. This means that, under those conditions, we can find a PEPS approximation of the ground state with
\begin{equation}
D=e^{O(\log(N/\epsilon)^{d+1})}.
\end{equation}
In the present paper we derive the following results. First, we use a novel method to obtain a bound for $\beta \leq O(\log(N))$ independent of the dimension (although still superpolynomial in $N$),
\begin{equation}
\label{Trotterbound1}
D=e^{O(\log^2(N/\epsilon))}.
\end{equation}
Under the same condition on the density of states as before, we also obtain that the ground state can be approximated with
\begin{equation}
\label{Trotterbound2}
D=e^{O(\log^2(N/\epsilon))},
\end{equation}
independent of the dimension.
Finally, using Hastings' construction of the PEPO (see also \cite{Kliesch2013}), we show that it is possible to have a polynomial scaling for any temperature, i.e.
\begin{equation}
\label{Poly}
D=(N/\epsilon)^{O ( \beta)}.
\end{equation}
The paper is organized as follows. In section \ref{problem} we define the problem we are addressing in this work. Section \ref{sec:trotter} derives the bounds (\ref{Trotterbound1}) and (\ref{Trotterbound2}) using a technique based on the Trotter expansion. In Section \ref{sec:Hastings} we use a different encoding of the PEPO based on Hastings' construction to obtain the polynomial bound (\ref{Poly}).
In all these sections we quote the results and explain how we have proven them. In the appendix we give details of the proofs.
\section{Problem}
\label{problem}
We consider a growing sequence of finite spin systems, $S_n$, with two-body interactions. To every system, $S_n$, we assign a graph, $\mathcal{G}_n = (\mathcal{V}_n, \mathcal{E}_n)$, where the vertices $\mathcal{V}_n$ correspond to the individual spins and the edges $\mathcal{E}_n$ to interactions. The Hamiltonian is such that only the connected points interact:
\begin{equation}
H_n = \sum_{e \in \mathcal{E}_n} h_e,
\end{equation}
where $h_e$ acts non-trivially on spins $v$ and $w$ if $e = ( v, w)$. Even though for simplicity we have considered only nearest neighbor interactions, the results generalize to local more-body interactions. We will assume that the (operator) norm of all the terms in the Hamiltonians is bounded by 1, i.e., $\norm{h_i}\leq 1$. If the norm of the Hamiltonians were bounded by $J$ instead of $1$, this factor could be included into the definition of the temperature.
We assume that all graphs are connected, and that their degree is uniformly bounded. That is, the number of edges starting from a given point is smaller than some constant $z$. This implies that $2|\mathcal{E}_n|/z < |\mathcal{V}_n| \leq |\mathcal{E}_n| + 1$. Thus, we can equally characterize the size of the system by the number of spins or interactions, $N=|\mathcal{V}_n|$ and $|\mathcal{E}_n|$, respectively. For convenience we will denote $|\mathcal{E}_n|$ by $K$ and omit the index $n$ in the following.
We also assume that there is a uniformly bounded lattice growth constant. This means that there is a universal constant, $\gamma$, such that for any given $e \in E$ and all $l \in \mathbb{Z}^+$
\begin{equation}\label{eq:latticegrowth}
\big| \left\{ \mathcal{I} \subseteq \mathcal{E}| \mathcal{I} \ \text{connected}, \ e \in \mathcal{I}, \ | \mathcal{I} | = l \right\}\big| \leq \gamma^l.
\end{equation}
That is, the number of connected regions having $l$ edges that include a specific edge, $e$, grows at most exponentially with $l$. In particular, this is the case if $\mathcal{G}_n$ is a regular lattice in any spatial dimension \cite{Klarner1967}. Thus, our treatment includes all those cases.
We consider the Gibbs state corresponding to $H$ given by (\ref{rho}). We will construct a PEPO, $\tilde{\rho}$, of bond dimension $D$, that is close to that state. In particular, for any $\varepsilon>0$,
\begin{equation}
\left\| e^{-\beta H} - \tilde{\rho} \right\|_1 \le \varepsilon \|e^{-\beta H}\|_1,
\end{equation}
where $\| x \|_p=\left[{\rm tr} (x^\dagger x)^{p/2}\right]^{1/p}$ stands for the Schatten-p-norm ($\| x \|=\|x\|_\infty$ for the operator norm). We will be interested in how $D$ scales with $N$ (or equivalently, with $K$) and $\varepsilon$.
By a PEPO on a graph $\mathcal{G} = ( \mathcal{E}, \mathcal{V})$ we mean that the operator
$\tilde{\rho}$ admits the following form:
\begin{equation}
\tilde{\rho} = \sum_{\alpha : \mathcal{E} \rightarrow \{ 1 \ldots D \}} \bigotimes_{v
\in \mathcal{V}} X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}. \label{eq:mpo}
\end{equation}
Here, $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$ are operators acting on the vertex $v$ alone, $z( v)$ is the degree of $v$, and $e^v_1, \ldots e^v_{z ( v)}$ are the edges going through $v$. This definition is the straightforward generalization of PEPS \cite{Verstraete2004} for operators \cite{Verstraete2004b,Zwolak2004}. One can readily see \cite{Cirac2009} that this operator can be written as a tensor network on the graph $\mathcal{G}$, where the bond dimension is $D$.
\section{Construction based on a Trotter expansion and compression}\label{sec:trotter}
In this section we use a Trotter expansion combined with a compression method to approximate the Gibbs state. The intuition about why this expansion should give rise to a PEPO description is the following (see also \cite{Hastings2010}). Let us assume that the operators $h_e$ commute with each other. Then, the Gibbs state (\ref{rho}) is proportional to a product of exponentials, each of them of the form $e^{-\beta h_e}$. One can easily show that each term in that product creates a link in the PEPO \cite{Wolf2008}. The bond dimension, $D_0$, is simply the maximum number of singular values of $h_e$, when decomposed in terms of the vertices it connects, and thus it is independent of $K$ and the temperature. In the general case where the $h_e$ do not commute with each other, we can still perform a Trotter expansion and approximate $\rho$ (up to a constant factor) by $(\tau^\dagger \tau)^M$ where
\begin{equation}
\tau = \prod_{i=1}^K e^{-\beta h_i/2M} .\label{eq:tau}
\end{equation}
The integer $M$ has to be chosen such that the approximation is good, i.e.
\begin{equation}
\label{Trotterapp}
\|e^{-\beta H} -(\tau^\dagger \tau)^M\|_1\leq \varepsilon \|e^{-\beta H}\|_1
\end{equation}
for some $\varepsilon>0$. Now, if we use the same argument we see that each time we apply $\tau$, we create a bond between each pair of vertices that are connected in the graph. That is, we multiply the bond dimension by $D_0$. Thus, naively, the final bond dimension will be $D_0^{2M}$, and since $M$ has to grow polynomially with $K$, we get a very bad bound. However, for large $M$ each of the terms in $\tau$ is close to the identity operator. Thus, this operator creates very little entanglement and it should be possible to compress the information that is contained in the bond variables for any pair of connected vertices, and therefore to decrease the bond dimension. In fact, in the case of commuting Hamiltonians one can reduce it to $D_0$, independent of $M$. This is, in fact, what we do in this section: we first find $M$ such that (\ref{Trotterapp}) holds, and then we compress the bond to get a better scaling of the bond dimension with $K$.
More specifically, we write $e^{-\beta h_i/2M}={\openone} + (e^{-\beta h_i/2M}-{\openone})$, then, after collecting the $K$ terms of $\tau$ and $\tau^\dagger$ into one product of $2K$ terms, we obtain
\begin{equation}
(\tau^\dagger \tau)^M=\prod_{j = 1}^M \prod_{i = 1}^{2K} e^{- \beta \tilde{h}_i / 2 M} = \prod_{j = 1}^M \prod_{i = 1}^{2K} ( 1 + x_i),
\end{equation}
where $\tilde{h_i}$ denotes $h_{K+1-i}$ if $i\leq K$, and $h_{i-K}$ otherwise, and $x_i=e^{-\beta \tilde{h}_i/2M}-{\openone}$. After expanding the product, this operator takes the form
\begin{equation}\label{eq:expansion}
(\tau^\dagger \tau)^M= \sum_{\lambda
\in \mathcal{M}_{M,2K}^b} \prod^M_{j = 1} \prod_{i = 1}^{2 K}
x_i^{\lambda_{i, j}}.
\end{equation}
The sum runs over all $M\times2K$ matrices with entries $0$ or $1$, denoted by $\mathcal{M}_{M,2K}^b$.
From this sum we only keep those terms in which any given $x_i$ appears at most $L$ times in (\ref{eq:expansion}). In Section \ref{sec:compression} we show that the resulting operator $\tilde{\rho}$ is a good approximation to $(\tau^\dagger \tau)^M$ if $L \approx \log K$.
In section \ref{sec:coding} we show then that the resulting operator can be written as a PEPO, in the sense of (\ref{eq:mpo}), with bond dimension $M^{O ( L)}$. The reason why this operator admits a PEPO form can be understood as follows. First we identify each particular term in the expansion of $(\tau^\dagger \tau)^M$ with the help of indices defined on the edges. This can be done by specifying at every edge, $i$, the position where $x_{K+1-i}$ and/or $x_{K+i}$ appear out of the $M$ possibilities. Once a term is identified, we proceed with the Schmidt decomposition of that term in order to build the local operators $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$. Let us notice that the latter only depends on the order in which the operators $x_{e_1^v}$, $x_{e_2^v}$, ... $x_{e_{z(v)}^v}$ appear in the given term, where $e_1^v,\dots e_{z(v)}^v$ are the edges starting from point $v$. This order can be obtained locally from the edges that surround $v$, which contain information about the $x$ involved in each of them.
As a result of that, at every edge we have to specify ${M \choose L}^2\approx M^{2L}$ natural numbers. As $M=poly(K)$ and $L=O(\log K)$, this gives a bond dimension $K^{O(\log K)}$ for the approximating operator. Therefore, as $N\leq 2K/z$, we obtain a bond dimension that scales like $N^{O(\log N)}$.
\subsection{Trotter expansion}\label{sec:trotterproof}
We know that $(\tau^\dagger \tau)^M$ ($\tau$ as in equation (\ref{eq:tau})) tends to $e^{-\beta H}$ if $M\to \infty$. The question is how big M has to be chosen such that we obtain a good approximation in one-norm. Here we prove that setting $M=poly(K)$ is enough (see also \cite{Berry2006}).
We present the proof in two steps. First we show that $\|e^{- \beta H}-(\tau^\dagger \tau)^M\|_1$ is small compared to $\|e^{-\beta H}\|_1$ as long as $\|\eta-\tau\|_{2M}$ is small compared to $\|\eta\|_{2M}$ where $\eta=e^{- \beta H / 2 M}$. Second, we show that $\|\eta-\tau\|_{2M}$ is small compared to $\|e^{-\beta H/2M}\|_{2M}$. The key point is that both $e^{-\beta H}$ and $(\tau^\dagger\tau)$ are close to $({\openone}-\beta H/M)^M$.
We state the first step as a proposition:
\begin{proposition}
\label{prop:T} If $\varepsilon<1/3$ and
\begin{equation}
\| \eta - \tau \|_{2 M} \leq \frac{\varepsilon}{M} \| \eta \|_{2M},
\end{equation}
then
\begin{equation}
\| \eta^{2 M} - ( \tau^{\dagger} \tau)^M \|_1 \leq
9 \varepsilon \| \eta^{2 M} \|_1.
\end{equation}
\end{proposition}
The proof combines the identity $a^m-b^m=\sum_i a^i (a-b) b^{m-i-1}$ with the H\"older inequality for matrices \cite{Bhatia1997} and it is presented in the Appendix. We state the second statement (that $\eta$ is close to $\tau$) as a lemma.
\begin{lemma} \label{lem:trotterproof}
If $M > 36 \beta^2 K^2/\epsilon$ and $\epsilon < 1$, then
\[ \| \eta - \tau \|_{2 M} \leq \frac{\epsilon}{M} \| \eta \|_{2 M}. \]
\end{lemma}
The main idea is that it is enough to prove the statement for the operator norm, as $\|\eta -\tau\|_{2M}$ is bounded by the H\"older inequality
$$\|\eta-\tau\|_{2M} = \|\eta^{-1}\eta(\eta-\tau)\|_{2M}\leq \|\eta^{-1}\| \|\eta\|_{2M}\|\eta-\tau\|,$$
and $\|\eta^{-1}\|$ is not too big as $\eta$ is close to the identity operator. In order to show that $\|\eta-\tau\|$ is close to zero, by a simple series expansion we obtain that $\|\eta-{\openone}+\beta H/M\|$ is small and so is $\|\tau-{\openone}+\beta H/M\|$. The statement then follows from the triangle inequality. The detailed proof is presented in the appendix.
Putting together Proposition (\ref{prop:T}) and Lemma (\ref{lem:trotterproof}), we obtain that the Trotter approximation is $\varepsilon$-close (in one-norm) if the trotter steps are chosen to be $M>360\beta^2 K^2/\varepsilon$.
\subsection{Compression}\label{sec:compression}
We approximate now $( \tau^{\dagger} \tau)^M$ by an operator $\tilde{\rho}$ starting from Eq. (\ref{eq:expansion}). This expansion can be pictured as follows. We can think of the resulting operator as a sum:
\begin{equation}
( \tau^{\dagger} \tau)^M = \sum_{\text{all fillings}}
\begin{array}{|c||c|c|c|c|}
\hline
& x_1 & x_2 & \ldots & x_{2 K}\\
\hline \hline
1 & X & & & \\
\hline
2 & & & X & \\
\hline
\vdots & & X & & \\
\hline
M & X & & & X\\
\hline
\end{array}\ ,
\end{equation}
where the table can be understood as follows. We begin to read from the upper-left
corner, from left to right, row-by-row. Whenever we meet an $X$ in the actual
cell, we write down the corresponding operator $x_i$ (according to the column), and
otherwise the identity operator. The value assigned to a given table is then the product of those operators. We finally have to sum up the resulting operators for all possible fillings
of the table.
The approximating operator $\tilde{\rho}$ can be thought of in the same way, just
limiting the number of $X$'s in each of the columns.
\begin{equation}
\tilde{\rho} = \sum_{\tmscript{\begin{array}{c}
\text{filling per}\\
\text{column} \leq L
\end{array}}} \begin{array}{|c||c|c|c|c|}
\hline
& x_1 & x_2 & \ldots & x_{2 K}\\
\hline\hline
1 & X & & & \\
\hline
2 & & & X & \\
\hline
\vdots & & X & & \\
\hline
M & X & & & X\\
\hline
\end{array}\ .
\end{equation}
We want to prove that this is a good approximation: $\| ( \tau^{\dagger}
\tau)^M - \tilde{\rho} \|_1 \leq \varepsilon \| \rho \|_1$ if the maximal number of $X$'s per column, $L$, is chosen big enough. We will show that $L = O ( \log K)$ is enough.
Let us first explain the main idea of the proof. Given a set of columns $\mathcal{I}\subseteq \{1,2\dots K\}$, define $S(\mathcal{I})$ to be the sum of all tables containing more than $L$ $X$'s in all columns $i\in \mathcal{I}$, but with no restriction for the columns not belonging to $\mathcal{I}$. Formally, let $\mathcal{Q}(\mathcal{I})$ denote the set of these tables:
\[ \mathcal{Q} ( \mathcal{I}) = \left\{ \lambda \in \mathcal{M}_{M,2K}^b \mid i \in \mathcal{I} \Rightarrow \sum_j \lambda_{i, j} > L \right\} , \]
then $S(\mathcal{I})$ is the sum
\begin{equation}
S ( \mathcal{I}) = \sum_{\lambda \in \mathcal{Q} ( \mathcal{I})} \prod^M_{j = 1} \prod_{i = 1}^{2 K}
x_i^{\lambda_{i, j}}.
\end{equation}
In any column that has no restriction, the sum can be evaluated, giving back $e^{-\beta \tilde{h}_i/2M}$ in every row of that column. By evaluating those sums we arrive to a sum containing only a few terms. In these remaining terms still a large number of $X$'s appear, therefore the norm of each such term is small. Thus the one-norm of $S(\mathcal{I})$ can be bounded. We will express $\tilde{\rho}$ with the help of the sums $S(\mathcal{I})$ in order to be able to bound its norm.
We use this observation in order to upper bound the one-norm of $(\tau^\dagger \tau)^M - \tilde{\rho}$. That difference contains one or more columns where there are more than $L$ appearances of $X$. We regroup the tables as follows. First, given a set of columns, $\mathcal{I}$, we sum up all tables that have more than $L$ appearances of $X$ in the columns $i\in\mathcal{I}$, albeit at most $L$ in all columns $i\notin \mathcal{I}$. This set of tables is the following set:
\[ \mathcal{T} ( \mathcal{I}) = \left\{ \lambda \in \mathcal{M}_{M,2K}^b \mid \sum_j \lambda_{i, j} > L
\Leftrightarrow i \in \mathcal{I} \right\} . \]
The sum of these tables will be called $R(\mathcal{I})$:
\begin{equation}
R ( \mathcal{I}) = \sum_{\lambda \in \mathcal{T} ( \mathcal{I})} \prod^M_{j = 1} \prod_{i = 1}^{2 K}
x_i^{\lambda_{i, j}}.
\end{equation}
Note that the operator $\tilde{\rho}$ is expressed by $R(\emptyset)$, as $\tilde{\rho}$ is the sum of tables that in each column contain at most $L$ $X$'s.
We can express the sum $S(\mathcal{I})$ with the help of $R(\mathcal{I})$:
\begin{equation}
S ( \mathcal{I}) = \sum_{\mathcal{J} \supseteq \mathcal{I}} R ( \mathcal{J}), \label{eq:S}
\end{equation}
because in any table in $S(\mathcal{I})$, the columns containing more than $L$ $X$'s form a set $\mathcal{J}\supseteq\mathcal{I}$.
Note that $(\tau^\dagger \tau)^M=S(\emptyset)$, as $( \tau^{\dagger} \tau)^M$ contains all tables, with no restriction on the number of $X$'s in any column.
The difference $(\tau^\dagger \tau)^M - \tilde{\rho}$ is then
\begin{equation}\label{eq:S-R}
( \tau^{\dagger} \tau)^M - \tilde{\rho} = S ( \emptyset) - R ( \emptyset).
\end{equation}
To bound the norm of this difference, we need to express $R(\emptyset)$ with the help of the $S(\mathcal{I})$'s; that is, we need the inverse relation of Eq. (\ref{eq:S}). This inverse relation is given by the M{\"o}bius inversion formula, which is used, for example, in the context of the Kirkwood-Salzburg equations, for a cluster expansion for the partition function \cite{Kotecky1986,Griffiths1980}. The statement of the M{\"o}bius inversion is the following.
Let $\mathcal{A}$ be a finite set, $\mathcal{P} ( \mathcal{A})$ the set of all its subsets, and $V$ a vector space. Given a function, $f:\mathcal{P} ( \mathcal{A}) \rightarrow V$, we define the following transformations:
\begin{eqnarray}
\hat{f} ( \mathcal{I}) & : = & \sum_{\mathcal{J} : \mathcal{A} \supseteq \mathcal{J} \supseteq \mathcal{I}} f ( \mathcal{J}) \label{eq:hat}\\
\check{f} ( \mathcal{I}) & : = & \sum_{\mathcal{J} : \mathcal{A} \supseteq \mathcal{J} \supseteq \mathcal{I}} ( - 1)^{| \mathcal{J} \backslash \mathcal{I} |} f ( \mathcal{J}) \label{eq:check} .
\end{eqnarray}
\begin{lemma}[M{\"o}bius inversion] \label{lem:Mobius}
\[ \hat{\check{f}} = \check{\hat{f}} = f \]
\end{lemma}
This lemma just expresses that the second transformation is the inverse of the first one. The proof is presented in the Appendix. We will use the lemma by setting $\mathcal{A}$ to be the set of columns, and $f = R$. Thus, comparing the definitions (\ref{eq:S}) and (\ref{eq:hat}) we deduce that $\hat{f} = S$. Applying the lemma we obtain the desired relation
$$R(\emptyset)=\sum_\mathcal{I} ( - 1)^{| \mathcal{I} |} S ( \mathcal{I}),$$
and thus substituting back to Eq. (\ref{eq:S-R})
\begin{equation}
( \tau^{\dagger} \tau)^M - \tilde{\rho} = S ( \emptyset) - \sum_\mathcal{I} ( - 1)^{| \mathcal{I} |} S ( \mathcal{I}) ,
\end{equation}
therefore
\begin{equation}
( \tau^{\dagger} \tau)^M - \tilde{\rho} = - \sum_{\mathcal{I} \neq \emptyset} (
- 1)^{| \mathcal{I} |} S ( \mathcal{I}).
\end{equation}
The one-norm of the difference can be bounded by the triangle inequality:
\begin{equation}\label{eq:compression_bound}
\| ( \tau^{\dagger} \tau)^M - \tilde{\rho} \|_1 \leq \sum_{m = 1}^{2 K} \binom{2 K}{m} \max_{\mathcal{I}:|\mathcal{I}| = m} \|S (\mathcal{I})\|_1.
\end{equation}
We obtained this form by counting the number of subsets $\mathcal{I}$ of the $2K$ columns that have $|\mathcal{I}|=m$. Now, we need to bound the one-norm of $S (\mathcal{I})$. First of all, as noted before, we can sum up over all indices possessing no restriction. That is, over all $\lambda_{i,j}$ with $i\notin \mathcal{I}$. For example, if $2\notin \mathcal{I}$ then
\begin{equation}
S(\mathcal{I}) = \sum_{\tmscript{\begin{array}{c}
\text{filling} \leq L\\
\text{for column } i\in \mathcal{I}
\end{array}}} \begin{array}{|c||c|c|c|c|}
\hline
& x_1 & x_2 & \ldots & x_{2 K}\\
\hline\hline
1 & X & e^{-\beta h_2} & & \\
\hline
2 & & e^{-\beta h_2} & X & \\
\hline
\vdots & & e^{-\beta h_2} & & \\
\hline
M & X & e^{-\beta h_2} & & X\\
\hline
\end{array}\ ,
\end{equation}
where we have already summed up for all $\lambda_{2,j}$. Let $\mu$ be such a term in $S(\mathcal{I})$ in which each $x_i \ (i\in \mathcal{I})$ is appearing exactly $k_i>L$ times.
The one-norm of this term is bounded by the following lemma.
\begin{lemma} \label{lem:S_norm}
If $M > 72 \beta^2 K^2$, then
$$\left\|\mu\right\|_1\leq 3\|e^{-\beta H}\|_1 \left(\frac{3\beta}{M}\right)^{k_1+\dots k_m}.$$
\end{lemma}
This bound is the consequence of the fact that the $x_i$'s, whose norm is small, appear exactly $k_1+k_2+\dots k_m$ times in $\mu$, while the rest of the operators, that is, $e^{-\beta \tilde{h}_i}$, give almost a Trotter approximation of $e^{-\beta H}$. The proof is presented in Appendix \ref{sec:appD}. The number of such terms $\mu$ is given by
\begin{equation}
{M \choose k_1} {M \choose k_2} \dots {M \choose k_m},
\end{equation}
as at each column $i\in \mathcal{I}$ one has to choose $k_i$ rows out of the total number of $M$ rows to place the appearing $x_i$'s.
Thus the one-norm of $S(\mathcal{I})$ is bounded by the following sum:
\begin{equation}
S(\mathcal{I})\geq\sum_{k_1 >L}\dots \sum_{k_m> L} 3\| e^{- \beta H} \|_1 \prod_{i=1}^m{M\choose k_i} \left( \frac{3\beta}{M} \right)^{k_i} ,
\end{equation}
therefore
\begin{equation}\label{eq:binom_lem_need}
\| S ( \mathcal{I}) \| \leq 3 \| e^{- \beta H} \|_1 \left( \sum_{k > L} \binom{ M}{k} \left(
\frac{3\beta}{M} \right)^{k} \right)^m .
\end{equation}
The sum in the parenthesis can be upper bounded by
$$\sum_{k > L} \binom{M}{k} \left(\frac{3\beta}{M}\right)^k \leq e^{3\beta} \left( \frac{3e\beta}{L} \right)^L$$
(see Lemma \ref{lem:binom} in Appendix \ref{sec:appE}) and thus
\begin{equation}
\| S ( \mathcal{I}) \| \leq 3 \| e^{- \beta H} \|_1 \left[ e^{3\beta} \left( \frac{3e \beta}{L}
\right)^L \right]^m.
\end{equation}
Substituting the obtained bound into Eq. (\ref{eq:compression_bound}) the following holds for the error of the compression:
\begin{equation}
\| (\tau^\dagger\tau)^M - \tilde{\rho} \|_1 \leq 3 \| e^{- \beta H} \|_1 \sum_{m = 1}^{2K} \binom{2K}{m} \left[
e^{3\beta} \left( \frac{3e \beta}{L} \right)^L \right]^m .
\end{equation}
Thus, after evaluating the sum, we obtain
\begin{equation}
\left\| (\tau^\dagger\tau)^M - \tilde{\rho} \right\|_1 \leq 3 \| e^{- \beta H} \|_1 \left( \left[ 1 + e^{3\beta} \left( \frac{3e \beta}{L} \right)^L \right]^{2K} - 1\right).
\end{equation}
As $(1+x/K)^K\leq e^x\leq 1+2x$ as long as $x<1$, this yields the bound
\begin{equation}
\| (\tau^\dagger \tau)^M - \tilde{\rho} \|_1 \leq 12 \| e^{- \beta H} \|_1 K e^{3\beta} \left( \frac{3e \beta}{L} \right)^L.
\end{equation}
Therefore, if $\beta\leq b\log K$, setting $L=O(\log K/\epsilon)$ implies
\begin{equation}
\| (\tau^\dagger \tau)^M - \tilde{\rho} \|_1 \leq \epsilon \| e^{- \beta H} \|_1,
\end{equation}
thus the error of the compression is bounded by $\epsilon$ if $L=O(\log K/\epsilon)$ and $M > 72 \beta^2 K^2$.
\subsection{Coding as a PEPO}\label{sec:coding}
We show that the resulting operator $\tilde{\rho}$ admits a PEPO form as in
equation (\ref{eq:mpo}):
\begin{equation}
\tilde{\rho} = \sum_{\alpha : \mathcal{E} \rightarrow \{ 1 \ldots D \}} \bigotimes_{v
\in \mathcal{V}} X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}.
\end{equation}
First, let us consider the Schmidt decomposition of the operators $x_i$.
\begin{equation}\label{eq:schmidt_coding}
x_i=e^{-\beta \tilde{h}_i/2M}-1=\sum_{\nu=1}^s A^{v,i}_\nu \otimes A^{w,i}_\nu,
\end{equation}
with $s$ being at most $d_{phys}^2$, where $d_{phys}$ is the dimension of the Hilbert space describing the individual spins, and the edge corresponding to column $i$ is composed of the two particles $v$ and $w$. Note that there are two columns associated to a Hamiltonian term $h_i$, $K+1-i$ and $K+i$.
After this decomposition, we can think of $\tilde{\rho}$ as the following sum:
\begin{equation}
\tilde{\rho} = \sum_{\tmscript{\begin{array}{c}
\text{filling per}\\
\text{column} \leq L
\end{array}}} \begin{array}{|c||c|c|c|c|}
\hline
& x_1 & x_2 & \ldots & x_{2 K}\\
\hline\hline
1 & 3 & 1 & 0 & 0\\
\hline
2 & 0 & 0 & 3 & 0\\
\hline
\vdots & 0 & 3 & 0 &4 \\
\hline
M & 2 & 0 & s & 0\\
\hline
\end{array}\ ,
\end{equation}
where the sum runs over all fillings that have at most $L$ cells different from $0$ in every column. The table means the following. We begin to read the table from left to right, row-by-row. Whenever we meet a cell in column $i$ containing the number $k$ we write down the operator $A^{v,i}_k \otimes A^{w,i}_k$ as in Eq. (\ref{eq:schmidt_coding}). Otherwise we write down the identity operator. The value of the table is again the product of these operators.
Every term in the above sum is now a tensor product. The local operator acting on particle $v$ depends only on the columns corresponding to the edges surrounding $v$. Indeed, operators acting non-trivially on particle $v$ occur only in these columns.
Therefore, the index $\alpha(e)$ at edge $e$ will specify a possible filling of the two columns corresponding to $e$, and the operator $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$ will mean the product of the corresponding Schmidt coefficients.
For a given edge $\alpha(e)$ can take
\begin{equation}
D=\left[\sum_{k\leq L} {M \choose k} s^k\right]^2\leq L^2 (sM)^{2L}
\end{equation}
different values, as the positions of the non-zero elements and their values are needed to be specified for the two columns corresponding to edge $e$.
In Section \ref{sec:trotter} we have shown that we should set $M>360\beta^2K^2/\epsilon$ in order to the Trotter approximation be $\epsilon$-close to the Gibbs state. In Section \ref{sec:compression} we have seen that one can choose $L$ such that the compressed operator, $\tilde{\rho}$, is $\epsilon$-close to the Trotter expansion. Therefore, by the triangle inequality, for any given $\epsilon$ that decreases at most polynomially in the system size, one can approximate the Gibbs state with error $\epsilon$, if the Trotter steps are taken to be $poly(K)$ and the compression, $L$, to be $O(\log(K))$. Thus, our method gives a PEPO approximation with bond dimension $K^{O(\log (K))}$. As $2K/z\leq N$, this is a PEPO with bond dimension $N^{O(\log (N))}$.
In Section \ref{sec:compression} we only have supposed that $\beta \leq b \log (K)$, or equivalently, $\beta \leq b \log (N)$. If $H$ is gapped and the density of states for a fixed energy only grows as $poly(N)$, then by setting $\beta = O( \log (N))$, the ground state projector is approximated by the Gibbs state with an error decreasing as $poly(N)$. Therefore, our method also gives an $N^{O(\log (N))}$ bond dimensional PEPO approximation of the ground state projector, and thus an $N^{O(\log (N))}$ bond dimensional PEPS approximation for the ground state (for any prescribed error $\epsilon$ that decreases at most as $poly(N)$) under the same condition.
\section{$\tmop{Poly} (N)$ bond dimensional approximation}\label{sec:Hastings}
In this section we show that with the help of the cluster expansion technique {\cite{Hastings2006}} we can approximate the thermal state by an MPO with $N^{O(\beta)}$ bond dimension. For that, we just have to modify theorem 15 in {\cite{Kliesch2013}} and introduce a more efficient way of encoding the PEPO. That theorem says that for $\beta < \beta^{*}$ ($\beta^{*}$ is a constant) the density operator can be well approximated with the truncated cluster expansion, where only clusters of size at most $O(\log K)$ (equivalently, $O(\log N)$) are included.
By a clever choice of the coding of the PEPO, we show that for that temperature one just needs a $poly(N)$ bond dimension, and then, as in \cite{Hastings2006}, we extend the result to lower (but finite) temperatures.
\subsection{Cluster expansion}\label{sec:cluster}
Before restating theorem 15 in {\cite{Kliesch2013}} we need to introduce some
notation. Let $\mathcal{E}^{*} = \cup_{k = 0}^{\infty} \mathcal{E}^k$, that is, a word $w$ from $\mathcal{E}^{*}$ denotes a sequence of edges: $w = ( w_1 w_2 \ldots w_k)$.
Let $h_w$ denote the product of the Hamiltonian terms corresponding to those edges, $h_w = h_{w_1} h_{w_2} \ldots h_{w_k}$,
and let $\tmop{supp} ( w)$ be the set of all edges occurring in $w$.
Every word's support is a set $\mathcal{I} \subseteq \mathcal{E}$. One can break it into
connected components: $\mathcal{I} = \cup_i \mathcal{I}_i$ where the $\mathcal{I}_i$'s are connected, and different components do not contain common points. These connected components are also called clusters. Then, let
$ \mathcal{W}_{L} \subseteq \mathcal{E}^{*}$ be the set of all words whose support contains only connected components of size at most
$L$. $\beta^{*}$ will denote a constant such that $\alpha e^{( 2 z - 1) \beta^{*}} (
e^{\beta^{*}} - 1) < 1$, and
\begin{equation}
\tilde{\rho} = \sum_{w \in \mathcal{W}_{L}}
\frac{( - \beta)^{| w |}}{| w | !} h_w.
\end{equation}
Theorem 15 in {\cite{Kliesch2013}} contains the following statement:
\begin{theorem}\label{thm:kliesch}
If $\beta\leq \beta^*$, then
\begin{equation}
\| e^{-\beta H} - \tilde{\rho} \|_1 \leq \| e^{-\beta H} \|_1 \cdot \left( \exp \left( K
\frac{x^L}{1 - x} \right) - 1 \right)
\end{equation}
with $x = \gamma e^{( 2 z - 1) \beta} ( e^{\beta} - 1) < 1$.
\end{theorem}
Similar to equations (56-58) in {\cite{Kliesch2013}} one can show that the
operator $\tilde{\rho}$ admits the following form:
\begin{equation}\label{eq:cluster_approx}
\tilde{\rho} = \sum_{\tmscript{ \begin{array}{c}
\mathcal{I} \in \mathcal{C}_L\\
\mathcal{I} = \uplus \mathcal{I}_i
\end{array}}} \prod_i \check{f} ( \mathcal{I}_i)
\end{equation}
where $\mathcal{C}_L$ means the subsets of edges $\mathcal{I}$ that does not contain a connected component of size bigger than $L$, and the connected components of $\mathcal{I}$ are $\mathcal{I}_i$'s. The operators $\check{f} ( \mathcal{I}_i)$ act locally on $\mathcal{I}_i$ and are defined as:
\begin{equation}
\check{f} ( \mathcal{I}) = \sum_{\tmscript{\begin{array}{c}
w \in \mathcal{I}^{*}\\
\tmop{supp} ( w) = \mathcal{I}
\end{array}}} \frac{( - \beta')^{| w |}}{| w | !} h_w \ .
\end{equation}
We show in Appendix \ref{sec:appF} that $\check{f} ( \mathcal{I})$ is the M{\"o}bius transform of
$f ( \mathcal{I}) = e^{- \beta' H ( \mathcal{I})}$, with $H ( \mathcal{I}) = \sum_{e \in \mathcal{I}} h_e$. This observation makes it
easier to show that $\tilde{\rho}$ admits the form (\ref{eq:cluster_approx}).
\subsection{Coding}\label{sec:hastingscoding}
We show in this subsection that the truncated cluster expansion $\tilde{\rho}$ (\ref{eq:cluster_approx}) can be written as a PEPO [cf. Eq. (\ref{eq:mpo})].
This operator has a very special form. It is a sum of products of local operators, such that the operator acting on a vertex $v$ only depends on the cluster where $v$ is contained in. Therefore, coding $\tilde{\rho}$ as a PEPO will be carried out in two steps.
First, we enumerate all $\mathcal{I}\in \mathcal{C}_L$ subsets of edges with the help of an index $\alpha_1:\mathcal{E}\to \{1,2\dots B_1\}$. This indexing will be such that for any given $v\in \mathcal{V}$ vertex the surrounding edges encode the information in which cluster $v$ is located. Once the cluster $\mathcal{I}_i\ni v$ is identified, the operator $\check{f}(\mathcal{I}_i)$ is written as a PEPO with the help of an index $\alpha_2:\mathcal{E}\to \{1,2\dots B_2\}$. The index $\alpha$ used at the description of the PEPO is then the composition of $\alpha_1$ and $\alpha_2$ taking $B_1B_2$ different values.
\paragraph{Identifying the clusters.} Let the different values of $\alpha_1(e)$ enumerate all clusters containing $e$ and of size at most $L$. For a given cluster size $l$, there are at most $\gamma^l$ clusters containing $e$ (see Eq. \ref{eq:latticegrowth}), therefore there are at most $L\gamma^L$ such clusters. As $L=O(\log K)$, this means that $\alpha_1$ takes at most $B_1\leq poly(K)$ different values. Let us now examine how this indexing is related to the original goal: to enumerate all $\mathcal{I}\in \mathcal{C}_L$ subsets of edges. For any given $\mathcal{I}\in \mathcal{C}_L$ subset one can find the corresponding values $(\alpha_1(e))_{e\in \mathcal{E}}$. However, given an indexing, $\alpha_1$, it might not correspond to such a subset of edges.
The reason is the following. Given an indexing $(\alpha_1(e))_{e\in \mathcal{E}}$, each index means a cluster $\mathcal{I}_e$. The subset $\mathcal{I}\in \mathcal{C}_L$ corresponding to this $\alpha_1$ is $\cup_{e} \mathcal{I}_e$, if for any two edges $e$ and $f$ either $\mathcal{I}_e= \mathcal{I}_f$, or the two clusters $\mathcal{I}_e$ and $\mathcal{I}_f$ do not have common point. Therefore the indexing does \emph{not} correspond to an $\mathcal{I}\in \mathcal{C}_L$ subset if and only if there are two edges $e$ and $f$ such that $\alpha_1(e)$ and $\alpha_1(f)$ denote two different, but overlapping clusters.
Let us join $e$ and $f$ with a path of edges going in the union of the two clusters $\mathcal{I}_e$ and $\mathcal{I}_f$. Along that path there is a contradiction locally; otherwise, $e$ and $f$ cannot specify contradictory information (see Figure \ref{fig:indexing}). Therefore, if an indexing $\alpha_1$ does not correspond to a subset of edges, then there is a point $v\in \mathcal{V}$ where it can be detected.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw[line width=14pt,cap=rect] (1,0)--(1,3)--(2,3);
\draw[line width=14pt,cap=rect] (1,1)--(2,1)--(2,2)--(1,2);
\draw[line width=13pt,color=gray!5,cap=rect] (1,0)--(1,3)--(2,3);
\draw[line width=13pt,color=gray!5,cap=rect] (1,1)--(2,1)--(2,2)--(1,2);
\draw[line width=10pt,cap=rect] (2,4)--(2,3)--(3,3)--(3,2);
\draw[line width=10pt,cap=rect] (3,3)--(4,3)--(4,0);
\draw[line width=10pt,cap=rect] (3,1)--(5,1);
\draw[line width=9pt,color=gray!18,cap=rect] (2,4)--(2,3)--(3,3)--(3,2);
\draw[line width=9pt,color=gray!18,cap=rect] (3,3)--(4,3)--(4,0);
\draw[line width=9pt,color=gray!18,cap=rect] (3,1)--(5,1);
\draw[thin,color=black!25] (0,0) grid (5,4);
\draw[very thick] (1,1)--(1,2);
\draw[very thick] (3,1)--(4,1);
\draw[thick, dashed] (1,2)--(1,3)--(4,3)--(4,1);
\end{tikzpicture}
\caption{Two clusters specified by the thick edges. The information contained in those edges contradict as the clusters overlap. However, the contradiction appear locally somewhere along the dashed line. Thus, our coding will give the 0 operator for this configuration.}
\label{fig:indexing}
\end{figure}
\paragraph{Coding the local operators.} Any operator defined on at most $L$ particles can be written as a PEPO with bond dimension $d_{spin}^{2L}$, where $d_{spin}$ is the dimension of the Hilbert space of the particles. For example, an expansion in a product basis of the operators supported on $L$ particles can be viewed as a PEPO. As $\check{f}(\mathcal{I}_i)$ is such a local operator with $L=O(\log K)$, this coding requires an index $\alpha_2$ with $B_2=poly(K)$ different values. The local operators used for this construction will be $Y^v_{\alpha_2(e_1^v),\dots \alpha_2(e^v_z(v))}(\mathcal{I}_i)$.
With the help of the index $\alpha=(\alpha_1,\alpha_2)$ the operators $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$ are constructed as follows. If $\alpha_1(e_1^v),\alpha_1(e_2^v)\dots \alpha_1(e_{z(v)}^v)$ both specify the same cluster $\mathcal{I}_i$ (or some of them the empty cluster, if compatible with $\mathcal{I}_i$), then let
\begin{equation}
X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}=Y^v_{\alpha_2(e_1^v),\dots \alpha_2(e^v_z(v))}(\mathcal{I}_i),
\end{equation}
otherwise, if both of them specify the empty cluster, let $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$ be ${\openone}$, otherwise let $X^v_{\alpha ( e^v_1) \ldots \alpha ( e^v_{z ( v)})}$ be $0$. By construction, the contraction of these tensors really gives $\tilde{\rho}$.
As the index used at the coding, $\alpha=(\alpha_1,\alpha_2)$, can only take $B_1B_2=poly(K)$ different values, the above coding is a PEPO with $poly(K)$ (equivalently $poly(N)$) bond dimension. Thus, for any $\beta<\beta^*$, we gave an efficient PEPO description of the Gibbs state. Moreover, Theorem (\ref{thm:kliesch}) holds for $\beta' = \beta / 2 M$ instead of $\beta$ if the trace norm is replaced by $\| . \|_{2 M}$ without any essential modification. Therefore, by taking $M$ such that $\beta'<\beta^*$, that is, $M=O(\beta)$, this result can be extended to lower (but finite) temperatures as well (see Proposition \ref{prop:T}). However, after this step, the approximating operator will be a PEPO exponentiated $M$ times. Therefore the bond dimension required for the PEPO description of the Gibbs state at arbitrary temperature is $N^{O(\beta)}$.
\section{Summary and Outlook}
We have analyzed the ability of tensor networks to describe thermal
(Gibbs) equilibrium states of lattice Hamiltonians with local interactions.
First, using a Trotter expansion and a compression method, we have
shown that it is possible to approximate that state with a PEPO whose
bond dimension scales as $N^{O(\log N)}$, where $N$ is the system size
(number of vertices in the lattice). This result is valid for
any finite temperature and spatial dimension. It also holds true at
zero temperature as long as the Hamiltonian is gapped and the density
of states for any energy interval only grows polynomially with the system
size. Second, building on Hastings' construction \cite{Hastings2006}, we
have shown that it is possible to find a PEPO with a $poly(N)$ bond dimension
at any finite temperature and spatial dimension.
There are some straightforward implications of the results derived
here. First, even though we have concentrated on PEPOs, it is trivial to
express our results in terms of (pure) PEPS. At finite temperature, we
can just consider the PEPO corresponding to half the temperature, and apply it
to locally maximally entangled states in order to obtain a purification
in terms of a PEPS with a polynomially growing bond dimension \cite{Verstraete2004b}.
At zero temperature, we can simply apply the constructed PEPO to a random
product state in order to show that there exists a PEPS with $D=N^{O(\log N)}$.
Second, for translationally invariant problems in regular lattices, our
construction may break translational invariance (as we select some order of the bonds).
But it is always possible \cite{Perez-Garcia2007} to make a PEPO (or PEPS)
translationally invariant with an increase of the bond dimension by just
a factor of $N$. Third, even though we have considered Hamiltonians
interacting along the edges in the graph, our construction can be
easily extended to the case in which the local Hamiltonians act on
plaquettes. The idea is that at the Trotter decomposition we have made no assumption on the support of the individual Hamiltonian terms, whereas at the coding procedure, we still need to keep information contained in a constant number of columns: in an edge $e=(v,w)$, we can keep the information contained in the columns corresponding to Hamiltonian terms that act non-trivially on either $v$ or $w$. In such a coding the same piece of information is specified in more than one edge, but their consistency can be checked locally, at the vertices. The cluster expansion technique can be applied with no essential modification as the number of terms acting on the boundary of a cluster can still be upper bounded by a constant times the size of the cluster, and the number of clusters containing $l$ terms is still bounded by $\gamma^l$, where $\gamma$ is a lattice growth constant \cite{Klarner1967}.
Finally, our construction can also be straightforwardly extended to
fermions with the result that we just have to use fermionic PEPS
\cite{Kraus2010}.
\begin{acknowledgements}
We thank M.C. Ba\~nuls, T. Vidick, Z. Landau and U. Vazirani for discussions. This work has been partially supported by the EU project SIQS, and the DFG project NIM. We also thank the Benasque Center for Science, the Perimeter Institute (Waterloo), and the Simon's Center for the Theory of Computing (Berkeley), where part of the work was carried out, for their hospitailty. NS acknowledges the support from the Alexander von Humboldt foundation and the EU project QALGO. JIC acknowledges support from the Miller Institute in Berkeley.
\end{acknowledgements}
|
{
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Penn State survives Iowa with touchdown on the final play
Antony Yang Nov 08, 2017
Antony Yang
It took until the final play of the game, but the No. 4 Nittany Lions survived the trap of Kinnick Stadium at night with an epic 21-19 victory over the Iowa Hawkeyes on Saturday night. The Nittany Lions drove 65 yards on 12 plays, including a fourth down conversion in their own territory, to avoid the upset at Kinnick Stadium. CBS will be doing the TV coverage while the game can be watched online live on fuboTV.
The Maryland women's golf team will travel to University Park, Pa. this weekend to compete in the 41st annual Nittany Lion Invitational, hosted by Penn State. Standout running back Saquon Barkley added 226 all-purpose yards of his own, 85 of those coming on a handsome touchdown reception that set a stadium record for the longest touchdown pass.
"It was really good to get everyone in the right mindset before we get into conference play", quarterback Trace McSorley said.
Sensing opportunity just before the half, Iowa quarterback Nate Stewart isolated receiver Nick Easley in the slot for an absurd touchdown that gave the Hawkeyes a halftime lead.
Penn State has to play five games on the road in the Big Ten this year, with matchups against the Hawkeyes, Northwestern, Ohio State, Michigan State and Maryland. When Barkley arrived at Penn State, Franklin challenged him to become more involved in the passing game.
The Hawkeyes forced an instant three-and-out, and Wadley finally got loose with a 70-yard touchdown catch to cut the deficit to 15-13 following a failed two-point conversion.
Iowa soccer finally gets to conference play against Penn State tonight, after going nearly two weeks without a match.
Iowa regained the lead in the second half, when back Akrum Wadley scored from 35 yards.
Penn State vs. Iowa live stream, start time, betting odds and TV information for Saturday's game between the Nittany Lions and Hawkeyes. With Iowa spoiling Penn State title hopes in the past, expect the Lions to be extra focused to avoid having history repeat itself. The offense was clicking on all cylinders Saturday night.
For all the stats for him to ponder, one is most important to Penn State safety Marcus Allen: Takeaways. Franklin also has to be pleased with the emergence of big-play receiver Saeed Blacknall, who was held without a catch in the first two games but had 64 receiving yards and a score against Georgia State. Though they won't say it right now, they know full well that MI and Ohio State lurk later on. Penn State's defense was just as dominant as the offense.
The Terrapins will compete against Bucknell, Delaware, Harvard, Ohio, Penn, Princeton, Seton Hall, South Florida, Yale, Rutgers, Columbia, Youngstown State and Penn State. He did just that threading it into a tight window to Juwan Johnson as time expired to give the Nittany Lions the win.
Recently retired following a 11-year National Football League career with the Minnesota Vikings, Greenway recalled the journey which led him from Madison, South Dakota to Iowa City. His interception late in the second period led to the Hawkeyes' first touchdown.
Miller made the first hit on Akrum Wadley in the end zone that led to a safety for Penn State midway through the second quarter.
Simone Lee (Menomonee Falls, Wis.) recorded a double-double for the Nittany Lions, putting down a team-high 15 kills and scooping up a season-high 13 digs.
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{"url":"https:\/\/www.physicsforums.com\/threads\/umc-time-right-ascension-and-longitude.155869\/","text":"# UMC Time, Right Ascension and Longitude\n\nWhere can I find a triplet of UTC Time, Right Ascension (of zenith) and Longitude, preferrably with many, many safe decimal places?\n\nLast edited:\n\nruss_watters\nMentor\nThe question doesn't make a whole lot of sense, could you explain in more detail what you are looking for and why? If you are looking for an accurate time signal and your position, GPS is probably the best place to get it. Any good astronomy program will tell you the RA of the zenith to several decimal places (what's a \"safe decimal place\"?).\n\nAny good astronomy program will tell you the RA of the zenith to several decimal places (what's a \"safe decimal place\"?).\nBased on what? To do that, the program needs to know position of Earth relative to stars at some moment. The minimum of data to perform such a computation is a triplet I want.\n\nThis can be any place on Earth and any time, but both Longitude and RA have to be measured\/calculated really carefully.\n\nruss_watters\nMentor\nI use Starry Night ( www.starrynight.com ) to drive my telescope. I enter my lat\/long and it takes a time signal from the computer (which gets it from the internet). It generally gives me a pointing accuracy of around 5 arcmin, and most of that error is in the telescope. Assuming you enter a good position and your time signal is accurate to the second, the software would give you 15 arcsec accuracy.\n\ndo you think you could log a timestamp and zenith RA from its output for some longitude (like, 0) for me?\n\nruss_watters\nMentor\nSure - I'll post a screenshot for you tonight, when I get home.\n\nI've found http:\/\/maia.usno.navy.mil\/conv2000\/chapter5\/tab5.4.txt [Broken] an impressive equation for what I need (Greenwich Sidereal Time), with two unknowns in it, \"(UT1 - UTC)\" and \"classical expression for the equation of the equinoxes\". Why it always has to be splitted across all the internet, and not in a single page.\n\nLast edited by a moderator:\n...okay, so they say at wikipedia that\nThe ratio of UT1 to mean sidereal time is defined to be 0.997269566329084 \u2212 5.8684\u00d710^\u221211T + 5.9\u00d710^\u221215T^2, where T is the number of Julian centuries of 36525 days each that have elapsed since JD 2451545.0 (J2000).\nI assume english translation of this is that at \"JDJD 2451545.0 (J2000)\" local sidereal time of longitude 0 was equal to 0.997269566329084 of UT1 time, which was... ? it is said also that UT1 could be found as UTC (January 1, 2000, 11:58:55.816 UTC) + DUT1 but I have no idea where to look this DUT1 up :( somebody please help me with this mess before I kill myself. EDIT: according to this graph it was around +0.35 sec, but exact value would be nice.\n\nLast edited:\ngoing crrrrazy with this, http:\/\/maite152.upc.es\/~manuel\/tdgps\/node18.html [Broken], but he sais that \"Tu is the time since J2000 (January 1 2000, 12h UT1) in Julian centuries of 365.25 days\" which gives over a minute of difference with above UTC value from wikipedia.\n\nOne word: confused. Let's see what russ value will align to.\n\nLast edited by a moderator:\nruss_watters\nMentor\nHere's a screen cap from Starry Night. The cursor disappeared when I did the printscreen, but rest assured, it was right on the zenith...\n\nPart of the problem you may be having with the calculation is that there is more than one standard, as you can see in the cap, and the difference is about half an arcmin.\n\n#### Attachments\n\n\u2022 52.1 KB Views: 280\nLast edited:\nyou know, when I started this thread, I was kind of hoping that, given earth rotation period of 23 h 56 m 4.091 s, one can convert normal time to sidereal (or to RA) with simple linear equation (constructed from values I asked for), but our fellow astronomers f**ed it up beyond belief, and now there are lots of time systems, and lots times lots transformations between these systems.\n\nso. I have ended up looking into three software sources. there are two steps, 1st they convert UTC date directly to \"Julian Day\"; stellarium and celestia use same code (here, around line 50, or here, around line 435). yoursky's code is slightly different,\nCode:\n if ((year < 1582) || ((year == 1582) && ((mon < 9) || (mon == 9 && mday < 5)))) {\nb = 0;\n} else {\na = ((int) (y \/ 100));\nb = 2 - a + (a \/ 4);\n}\n\nreturn (((long) (365.25 * (y + 4716))) + ((int) (30.6001 * (m + 1))) +\nmday + b - 1524.5) +\n((sec + 60L * (min + 60L * hour)) \/ 86400.0);\nbut claims to implement same algorithm (Meeus, Astronomical Algorithms, Chapter 7, page 61).\n\nsecond step is converting to sidereal time. I could not find code for that in celestia, and both codes from stellarium (here around line 35) and yoursky,\nCode:\n* GMST -- Calculate Greenwich Mean Siderial Time for a given\ninstant expressed as a Julian date and fraction.\t*\/\n\ndouble gmst(double jd)\n{\ndouble t, theta0;\n\n\/* Time, in Julian centuries of 36525 ephemeris days,\nmeasured from the epoch 1900 January 0.5 ET. *\/\n\nt = ((floor(jd + 0.5) - 0.5) - 2415020.0) \/ JulianCentury;\n\ntheta0 = 6.6460656 + 2400.051262 * t + 0.00002581 * t * t;\n\nt = (jd + 0.5) - (floor(jd + 0.5));\n\ntheta0 += (t * 24.0) * 1.002737908;\n\ntheta0 = (theta0 - 24.0 * (floor(theta0 \/ 24.0)));\n\nreturn theta0;\n}\nare different.\n\nso. later today, I will put all these codes together and see which is closer to values in your screenshot.\n\nThe cursor disappeared when I did the printscreen, but rest assured, it was right on the zenith.\nYour screenshot sais, \"Altitude 88\u00b0\". Zenith is 90\u00b0, isn't it.\n\nAlso, your time zone would be helpful. As is, my test computation results are:\nCode:\nDate: February 13, 2007, 12:05:08 pm\nJulian date: 2454144.95856\nStellarium output \/15 (1): 20.5457510321\nYoursky output (2): 20.5457282109\nTaking 20h away and converting to minutes:\n(1) 32.7450619233\n(2) 32.7436926529\nneither matches 9h 35.742m, but minutes are within 4m margin, which tells us this error can be explained by different time zone.\n\nEDIT: I took another timestamp 11 hours away, and my minutes became worse:\nCode:\nDate: February 13, 2007, 1:05:08 am\nJulian date: 2454144.50023\nStellarium output \/15 (1): 9.51563402541\nYoursky output (2): 9.51561121915\nTaking 9h away and converting to minutes:\n(1) 30.9380415245\n(2) 30.9366731487\n:( okay. that's it. now i'm pissed.\n\nLast edited:\nEDIT: I have found error it was incorrectly computing julian day. now it is like:\nCode:\nDate: February 13, 2007, 2:05:08 am\nJulian date: 2454144.50356\nStellarium output \/15 (1): 9.59585307131\nYoursky output (2): 9.59583026493\nTaking 9h away and converting to minutes:\n(1) 35.7511842786\n(2) 35.7498158961\nIn your screenshot it was 9h 35.742m, so it's a nice match with 2nd value. I will continue to test different codes, but this thread can be considered resolved.\n\nLast edited:","date":"2021-03-07 14:13:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5737346410751343, \"perplexity\": 4512.440568234744}, \"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-2021-10\/segments\/1614178377821.94\/warc\/CC-MAIN-20210307135518-20210307165518-00147.warc.gz\"}"}
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Q: Is there an application that does two way sync of Magento and a billing software? I'm looking for an application for billing and inventory that could sync two-way with Magento. We run a online + offline store with the same set of inventory, and such a solution will make things a lot easier for us.
Any clues?
A: I've seen a module that integrates with Sage somewhere out there on the web. I had to custom build and export for my setup the dumped everything to Take Stock. I guess the real question is: what financial software are you using internally.
A: You can use the services of SBOeConnect, which integrates with SAP Business One (this is at the time when I was associated with it about 3 months ago). I will not be able to comment on its current status but you may get benefited, as the service providers also do customizations, which I am also somewhat aware of.
Hope it helps.
A: The guys at Fontis have produced a 2 way sync between MYOB and Magento. I haven't used it, but their code for other Magento extensions is great.
A: Magento has a data sync service you can check out, and I expect that your backend/ERP system may have a dedicated bridge as well. If not, you can use the Webservices built into Magento to build a sync'd system.
Make sure to choose a single system of record for your inventory information and have the other system feed from it for information.
Hope that helps!
Thanks,
Joe
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"redpajama_set_name": "RedPajamaStackExchange"
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Q: Registering 2 activities to receive images, it doesn't work when sharing from Google Photos app Let's say in my app, users can make posts & send messages to other users, & I want to register my app as a component among those appears in the Share-via picker when sharing either text or an image, I'm defining my 2 activities in my AndroidManifest.xml as follows (like this official example):
<activity
android:name=".SharePostActivity"
android:label="MyApp (Post)">
<intent-filter>
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="text/plain" />
</intent-filter>
<intent-filter>
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="image/*" />
</intent-filter>
</activity>
<activity
android:name=".ShareMessageActivity"
android:label="MyApp (Message)">
<intent-filter>
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="text/plain" />
</intent-filter>
<intent-filter>
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="image/*" />
</intent-filter>
</activity>
When sharing text, this works perfectly fine, where I see 2 icons of my app among available apps for sharing text, named MyApp (Post) & MyApp (Message),
Problem happens when sharing an image, because only one icon appears with the second defined label in the manifest (which is MyApp (Message)), and actually it opens the first defined activity in the manifest (which is SharePostActivity),
So, how to show the 2 options when sharing an image (just like what happens when sharing text)?
(I've tried on an emulator running Nougat & a real device running Oreo)
----- Update -----
I've found that this weird behavior happens only when sharing images from Google's Photos app, but everything works fine when sharing an image from other apps!
A: Some investigation leads to a solution: each of your intent-filter must contain an android:label which points to a resource, not a hardcoded string.
...
<intent-filter android:label="@string/first_value">
...
Different Activities must have different labels, so in your case the Manifest should look like
<activity
android:name=".SharePostActivity"
android:label="@string/my_app_post">
<intent-filter
android:label="@string/my_app_post">
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="text/plain" />
</intent-filter>
<intent-filter
android:label="@string/my_app_post">
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="image/*" />
</intent-filter>
</activity>
<activity
android:name=".ShareMessageActivity"
android:label="@string/my_app_message">
<intent-filter
android:label="@string/my_app_message">
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="text/plain" />
</intent-filter>
<intent-filter
android:label="@string/my_app_message">
<action android:name="android.intent.action.SEND" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="image/*" />
</intent-filter>
</activity>
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{
"redpajama_set_name": "RedPajamaStackExchange"
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from collections import defaultdict
from datetime import date, datetime
from typing import Callable, Type
import numpy as np
from pandas import DataFrame
import plotly.graph_objects as go
from plotly.subplots import make_subplots
from vnpy.trader.constant import (Direction, Offset, Exchange,
Interval, Status)
from vnpy.trader.object import TradeData, BarData, TickData
from .template import SpreadStrategyTemplate, SpreadAlgoTemplate
from .base import SpreadData, BacktestingMode, load_bar_data, load_tick_data
class BacktestingEngine:
""""""
gateway_name = "BACKTESTING"
def __init__(self):
""""""
self.spread: SpreadData = None
self.start = None
self.end = None
self.rate = 0
self.slippage = 0
self.size = 1
self.pricetick = 0
self.capital = 1_000_000
self.mode = BacktestingMode.BAR
self.strategy_class: Type[SpreadStrategyTemplate] = None
self.strategy: SpreadStrategyTemplate = None
self.tick: TickData = None
self.bar: BarData = None
self.datetime = None
self.interval = None
self.days = 0
self.callback = None
self.history_data = []
self.algo_count = 0
self.algos = {}
self.active_algos = {}
self.trade_count = 0
self.trades = {}
self.logs = []
self.daily_results = {}
self.daily_df = None
def output(self, msg):
"""
Output message of backtesting engine.
"""
print(f"{datetime.now()}\t{msg}")
def clear_data(self):
"""
Clear all data of last backtesting.
"""
self.strategy = None
self.tick = None
self.bar = None
self.datetime = None
self.algo_count = 0
self.algos.clear()
self.active_algos.clear()
self.trade_count = 0
self.trades.clear()
self.logs.clear()
self.daily_results.clear()
def set_parameters(
self,
spread: SpreadData,
interval: Interval,
start: datetime,
rate: float,
slippage: float,
size: float,
pricetick: float,
capital: int = 0,
end: datetime = None,
mode: BacktestingMode = BacktestingMode.BAR
):
""""""
self.spread = spread
self.interval = Interval(interval)
self.rate = rate
self.slippage = slippage
self.size = size
self.pricetick = pricetick
self.start = start
self.capital = capital
self.end = end
self.mode = mode
def add_strategy(self, strategy_class: type, setting: dict):
""""""
self.strategy_class = strategy_class
self.strategy = strategy_class(
self,
strategy_class.__name__,
self.spread,
setting
)
def load_data(self):
""""""
self.output("开始加载历史数据")
if not self.end:
self.end = datetime.now()
if self.start >= self.end:
self.output("起始日期必须小于结束日期")
return
if self.mode == BacktestingMode.BAR:
self.history_data = load_bar_data(
self.spread,
self.interval,
self.start,
self.end,
self.pricetick
)
else:
self.history_data = load_tick_data(
self.spread,
self.start,
self.end
)
self.output(f"历史数据加载完成,数据量:{len(self.history_data)}")
def run_backtesting(self):
""""""
if self.mode == BacktestingMode.BAR:
func = self.new_bar
else:
func = self.new_tick
self.strategy.on_init()
# Use the first [days] of history data for initializing strategy
day_count = 0
ix = 0
for ix, data in enumerate(self.history_data):
if self.datetime and data.datetime.day != self.datetime.day:
day_count += 1
if day_count >= self.days:
break
self.datetime = data.datetime
self.callback(data)
self.strategy.inited = True
self.output("策略初始化完成")
self.strategy.on_start()
self.strategy.trading = True
self.output("开始回放历史数据")
# Use the rest of history data for running backtesting
for data in self.history_data[ix:]:
func(data)
self.output("历史数据回放结束")
def calculate_result(self):
""""""
self.output("开始计算逐日盯市盈亏")
if not self.trades:
self.output("成交记录为空,无法计算")
return
# Add trade data into daily reuslt.
for trade in self.trades.values():
d = trade.datetime.date()
daily_result = self.daily_results[d]
daily_result.add_trade(trade)
# Calculate daily result by iteration.
pre_close = 0
start_pos = 0
for daily_result in self.daily_results.values():
daily_result.calculate_pnl(
pre_close,
start_pos,
self.size,
self.rate,
self.slippage
)
pre_close = daily_result.close_price
start_pos = daily_result.end_pos
# Generate dataframe
results = defaultdict(list)
for daily_result in self.daily_results.values():
for key, value in daily_result.__dict__.items():
results[key].append(value)
self.daily_df = DataFrame.from_dict(results).set_index("date")
self.output("逐日盯市盈亏计算完成")
return self.daily_df
def calculate_statistics(self, df: DataFrame = None, output=True):
""""""
self.output("开始计算策略统计指标")
# Check DataFrame input exterior
if df is None:
df = self.daily_df
# Check for init DataFrame
if df is None:
# Set all statistics to 0 if no trade.
start_date = ""
end_date = ""
total_days = 0
profit_days = 0
loss_days = 0
end_balance = 0
max_drawdown = 0
max_ddpercent = 0
max_drawdown_duration = 0
total_net_pnl = 0
daily_net_pnl = 0
total_commission = 0
daily_commission = 0
total_slippage = 0
daily_slippage = 0
total_turnover = 0
daily_turnover = 0
total_trade_count = 0
daily_trade_count = 0
total_return = 0
annual_return = 0
daily_return = 0
return_std = 0
sharpe_ratio = 0
return_drawdown_ratio = 0
else:
# Calculate balance related time series data
df["balance"] = df["net_pnl"].cumsum() + self.capital
df["return"] = np.log(df["balance"] / df["balance"].shift(1)).fillna(0)
df["highlevel"] = (
df["balance"].rolling(
min_periods=1, window=len(df), center=False).max()
)
df["drawdown"] = df["balance"] - df["highlevel"]
df["ddpercent"] = df["drawdown"] / df["highlevel"] * 100
# Calculate statistics value
start_date = df.index[0]
end_date = df.index[-1]
total_days = len(df)
profit_days = len(df[df["net_pnl"] > 0])
loss_days = len(df[df["net_pnl"] < 0])
end_balance = df["balance"].iloc[-1]
max_drawdown = df["drawdown"].min()
max_ddpercent = df["ddpercent"].min()
max_drawdown_end = df["drawdown"].idxmin()
max_drawdown_start = df["balance"][:max_drawdown_end].argmax()
max_drawdown_duration = (max_drawdown_end - max_drawdown_start).days
total_net_pnl = df["net_pnl"].sum()
daily_net_pnl = total_net_pnl / total_days
total_commission = df["commission"].sum()
daily_commission = total_commission / total_days
total_slippage = df["slippage"].sum()
daily_slippage = total_slippage / total_days
total_turnover = df["turnover"].sum()
daily_turnover = total_turnover / total_days
total_trade_count = df["trade_count"].sum()
daily_trade_count = total_trade_count / total_days
total_return = (end_balance / self.capital - 1) * 100
annual_return = total_return / total_days * 240
daily_return = df["return"].mean() * 100
return_std = df["return"].std() * 100
if return_std:
sharpe_ratio = daily_return / return_std * np.sqrt(240)
else:
sharpe_ratio = 0
return_drawdown_ratio = -total_return / max_ddpercent
# Output
if output:
self.output("-" * 30)
self.output(f"首个交易日:\t{start_date}")
self.output(f"最后交易日:\t{end_date}")
self.output(f"总交易日:\t{total_days}")
self.output(f"盈利交易日:\t{profit_days}")
self.output(f"亏损交易日:\t{loss_days}")
self.output(f"起始资金:\t{self.capital:,.2f}")
self.output(f"结束资金:\t{end_balance:,.2f}")
self.output(f"总收益率:\t{total_return:,.2f}%")
self.output(f"年化收益:\t{annual_return:,.2f}%")
self.output(f"最大回撤: \t{max_drawdown:,.2f}")
self.output(f"百分比最大回撤: {max_ddpercent:,.2f}%")
self.output(f"最长回撤天数: \t{max_drawdown_duration}")
self.output(f"总盈亏:\t{total_net_pnl:,.2f}")
self.output(f"总手续费:\t{total_commission:,.2f}")
self.output(f"总滑点:\t{total_slippage:,.2f}")
self.output(f"总成交金额:\t{total_turnover:,.2f}")
self.output(f"总成交笔数:\t{total_trade_count}")
self.output(f"日均盈亏:\t{daily_net_pnl:,.2f}")
self.output(f"日均手续费:\t{daily_commission:,.2f}")
self.output(f"日均滑点:\t{daily_slippage:,.2f}")
self.output(f"日均成交金额:\t{daily_turnover:,.2f}")
self.output(f"日均成交笔数:\t{daily_trade_count}")
self.output(f"日均收益率:\t{daily_return:,.2f}%")
self.output(f"收益标准差:\t{return_std:,.2f}%")
self.output(f"Sharpe Ratio:\t{sharpe_ratio:,.2f}")
self.output(f"收益回撤比:\t{return_drawdown_ratio:,.2f}")
statistics = {
"start_date": start_date,
"end_date": end_date,
"total_days": total_days,
"profit_days": profit_days,
"loss_days": loss_days,
"capital": self.capital,
"end_balance": end_balance,
"max_drawdown": max_drawdown,
"max_ddpercent": max_ddpercent,
"max_drawdown_duration": max_drawdown_duration,
"total_net_pnl": total_net_pnl,
"daily_net_pnl": daily_net_pnl,
"total_commission": total_commission,
"daily_commission": daily_commission,
"total_slippage": total_slippage,
"daily_slippage": daily_slippage,
"total_turnover": total_turnover,
"daily_turnover": daily_turnover,
"total_trade_count": total_trade_count,
"daily_trade_count": daily_trade_count,
"total_return": total_return,
"annual_return": annual_return,
"daily_return": daily_return,
"return_std": return_std,
"sharpe_ratio": sharpe_ratio,
"return_drawdown_ratio": return_drawdown_ratio,
}
return statistics
def show_chart(self, df: DataFrame = None):
""""""
# Check DataFrame input exterior
if df is None:
df = self.daily_df
# Check for init DataFrame
if df is None:
return
fig = make_subplots(
rows=4,
cols=1,
subplot_titles=["Balance", "Drawdown", "Daily Pnl", "Pnl Distribution"],
vertical_spacing=0.06
)
balance_line = go.Scatter(
x=df.index,
y=df["balance"],
mode="lines",
name="Balance"
)
drawdown_scatter = go.Scatter(
x=df.index,
y=df["drawdown"],
fillcolor="red",
fill='tozeroy',
mode="lines",
name="Drawdown"
)
pnl_bar = go.Bar(y=df["net_pnl"], name="Daily Pnl")
pnl_histogram = go.Histogram(x=df["net_pnl"], nbinsx=100, name="Days")
fig.add_trace(balance_line, row=1, col=1)
fig.add_trace(drawdown_scatter, row=2, col=1)
fig.add_trace(pnl_bar, row=3, col=1)
fig.add_trace(pnl_histogram, row=4, col=1)
fig.update_layout(height=1000, width=1000)
fig.show()
def update_daily_close(self, price: float):
""""""
d = self.datetime.date()
daily_result = self.daily_results.get(d, None)
if daily_result:
daily_result.close_price = price
else:
self.daily_results[d] = DailyResult(d, price)
def new_bar(self, bar: BarData):
""""""
self.bar = bar
self.datetime = bar.datetime
self.cross_algo()
self.strategy.on_spread_bar(bar)
self.update_daily_close(bar.close_price)
def new_tick(self, tick: TickData):
""""""
self.tick = tick
self.datetime = tick.datetime
self.cross_algo()
self.spread.bid_price = tick.bid_price_1
self.spread.bid_volume = tick.bid_volume_1
self.spread.ask_price = tick.ask_price_1
self.spread.ask_volume = tick.ask_volume_1
self.strategy.on_spread_data()
self.update_daily_close(tick.last_price)
def cross_algo(self):
"""
Cross limit order with last bar/tick data.
"""
if self.mode == BacktestingMode.BAR:
long_cross_price = self.bar.close_price
short_cross_price = self.bar.close_price
else:
long_cross_price = self.tick.ask_price_1
short_cross_price = self.tick.bid_price_1
for algo in list(self.active_algos.values()):
# Check whether limit orders can be filled.
long_cross = (
algo.direction == Direction.LONG
and algo.price >= long_cross_price
)
short_cross = (
algo.direction == Direction.SHORT
and algo.price <= short_cross_price
)
if not long_cross and not short_cross:
continue
# Push order udpate with status "all traded" (filled).
algo.traded = algo.volume
algo.status = Status.ALLTRADED
self.strategy.update_spread_algo(algo)
self.active_algos.pop(algo.algoid)
# Push trade update
self.trade_count += 1
if long_cross:
trade_price = long_cross_price
pos_change = algo.volume
else:
trade_price = short_cross_price
pos_change = -algo.volume
trade = TradeData(
symbol=self.spread.name,
exchange=Exchange.LOCAL,
orderid=algo.algoid,
tradeid=str(self.trade_count),
direction=algo.direction,
offset=algo.offset,
price=trade_price,
volume=algo.volume,
datetime=self.datetime,
gateway_name=self.gateway_name,
)
if self.mode == BacktestingMode.BAR:
trade.value = self.bar.value
else:
trade.value = trade_price
self.spread.net_pos += pos_change
self.strategy.on_spread_pos()
self.trades[trade.vt_tradeid] = trade
def load_bar(
self, spread: SpreadData, days: int, interval: Interval, callback: Callable
):
""""""
self.days = days
self.callback = callback
def load_tick(self, spread: SpreadData, days: int, callback: Callable):
""""""
self.days = days
self.callback = callback
def start_algo(
self,
strategy: SpreadStrategyTemplate,
spread_name: str,
direction: Direction,
offset: Offset,
price: float,
volume: float,
payup: int,
interval: int,
lock: bool
) -> str:
""""""
self.algo_count += 1
algoid = str(self.algo_count)
algo = SpreadAlgoTemplate(
self,
algoid,
self.spread,
direction,
offset,
price,
volume,
payup,
interval,
lock
)
self.algos[algoid] = algo
self.active_algos[algoid] = algo
return algoid
def stop_algo(
self,
strategy: SpreadStrategyTemplate,
algoid: str
):
""""""
if algoid not in self.active_algos:
return
algo = self.active_algos.pop(algoid)
algo.status = Status.CANCELLED
self.strategy.update_spread_algo(algo)
def send_order(
self,
strategy: SpreadStrategyTemplate,
direction: Direction,
offset: Offset,
price: float,
volume: float,
stop: bool,
lock: bool
):
""""""
pass
def cancel_order(self, strategy: SpreadStrategyTemplate, vt_orderid: str):
"""
Cancel order by vt_orderid.
"""
pass
def write_strategy_log(self, strategy: SpreadStrategyTemplate, msg: str):
"""
Write log message.
"""
msg = f"{self.datetime}\t{msg}"
self.logs.append(msg)
def send_email(self, msg: str, strategy: SpreadStrategyTemplate = None):
"""
Send email to default receiver.
"""
pass
def put_strategy_event(self, strategy: SpreadStrategyTemplate):
"""
Put an event to update strategy status.
"""
pass
def write_algo_log(self, algo: SpreadAlgoTemplate, msg: str):
""""""
pass
class DailyResult:
""""""
def __init__(self, date: date, close_price: float):
""""""
self.date = date
self.close_price = close_price
self.pre_close = 0
self.trades = []
self.trade_count = 0
self.start_pos = 0
self.end_pos = 0
self.turnover = 0
self.commission = 0
self.slippage = 0
self.trading_pnl = 0
self.holding_pnl = 0
self.total_pnl = 0
self.net_pnl = 0
def add_trade(self, trade: TradeData):
""""""
self.trades.append(trade)
def calculate_pnl(
self,
pre_close: float,
start_pos: float,
size: int,
rate: float,
slippage: float
):
""""""
# If no pre_close provided on the first day,
# use value 1 to avoid zero division error
if pre_close:
self.pre_close = pre_close
else:
self.pre_close = 1
# Holding pnl is the pnl from holding position at day start
self.start_pos = start_pos
self.end_pos = start_pos
self.holding_pnl = self.start_pos * (self.close_price - self.pre_close) * size
# Trading pnl is the pnl from new trade during the day
self.trade_count = len(self.trades)
for trade in self.trades:
if trade.direction == Direction.LONG:
pos_change = trade.volume
else:
pos_change = -trade.volume
self.end_pos += pos_change
turnover = trade.volume * size * trade.value
self.trading_pnl += pos_change * \
(self.close_price - trade.price) * size
self.slippage += trade.volume * size * slippage
self.turnover += turnover
self.commission += turnover * rate
# Net pnl takes account of commission and slippage cost
self.total_pnl = self.trading_pnl + self.holding_pnl
self.net_pnl = self.total_pnl - self.commission - self.slippage
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,846
|
Q: Paho MQTT client on Raspberry Pi never publishes Similar to this question, the publish from Raspberry Pi with Paho MQTT with Python simply does not go through. The mosquitto_ commands on the other hand work perfectly.
This one does not work.
import paho.mqtt.publish as publish
publish.single(topic='temp/temp',payload='random',hostname='192.168.1.105')
This one works.
mosquito_pub -h 192.168.1.105 -t temp/temp -l
random
Please guide me what is missing?
A: Having worked this out for several other questions I'm going to guess you have a old version of mosquitto that only supports MQTT v3.1 not v3.1.1.
This should fix things.
import paho.mqtt.publish as publish
publish.single(topic='temp/temp',payload='random',hostname='192.168.1.105', protocol=mqtt.MQTTv31)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 304
|
A most attractive and spacious four bedroom semi detached Victorian cottage with a beautiful, private and sunny landscaped garden, close to Aldeburgh and Thorpeness. Centrally heated by gas-fired radiators, with three bathrooms, study, 2 reception rooms, conservatory, study and utility room.
which is situated in a prominent position at the heart of the popular Suffolk village of Aldringham, close to Aldeburgh and Thorpeness. The cottage which has been sympathetically extended and modernised, benefits from an enchanting, sunny and secluded rear garden. The accommodation is heated by gas-fired radiators and includes a large sitting room, a separate dining room, study, fitted kitchen and large utility room. There is also a Victorian style heated conservatory and a very useful down stairs shower room. The four well-proportioned bedrooms are served by a family bathroom and an en-suite shower room. The owners have also added a delightful, south facing, heated vine house. The garden is a key feature of the property. It has been landscaped with a raised patio, a pergola and outside dining area, an illuminated water feature and a variety of fruit and ornamental trees. There is a knot herb garden with a box hedge and well stocked herbaceous borders. The garden faces south and west and has a lovely Mediterranean feel, offering a good degree of privacy and plenty of sunshine. The cottage lies within a couple of minutes walk from the excellent Parrot and Punchbowl public house and restaurant. The local shops at Leiston and Aldeburgh are within convenient reach and there are wonderful footpaths for dog walkers, nearby, through heath and woodland. The property is ideally situated for access to beautiful unspoilt beaches and nature reserves at Aldringham Fen and North Warren.
This part of Suffolk is a haven for artists, writers and musicians. Aldringham has nature reserves on its door step and is within close proximity of the Heritage Coast, the seaside town of Aldeburgh and its neighbouring fantasy village of Thorpeness. Leiston, which is about a mile away has good local shops, a sports centre and cinema. The town's proud industrial heritage is celebrated by the excellent Long Shop Museum. Aldeburgh has fine restaurants, local shops and galleries. There is a beautiful and unspoilt shingle beach which stretches to Thorpeness: a fantasy village created by Glencairn Ogilvie in the early part of the twentieth century. There are golf courses at Aldeburgh and Thorpeness and sailing clubs on the River Alde. The world class Snape Maltings Concert Hall is just a few minutes drive away and is home to the Aldeburgh Festival. Other attractions in the area include RSPB Minsmere (which has played host to the BBC Spring Watch programme), and the ancient castles at Framlingham and Orford. The nearest railway station is 5 miles away at Saxmundham with connecting services to London, Cambridge and Norwich. Saxmundham has Tesco and Waitrose supermarkets and a small weekly market.
ENTRANCE HALL Sold front door, lovely exposed brick floor which extends into the study.
INNER HALL Tiled floor, stairs to the first floor landing, turned balustrades, radiator, under stair recess.
2.69m x 1.74m Front aspect, double glazed window, wall light point, glazed door to the utility room.
2.85m x 2.72m A large utility room with iroko work top, Belfast sink with A Franke mixer tap and tiled splash back, cupboard under, base level units, plumbing for washing machine and space for a tumble dryer, space for a tall fridge/freezer, bespoke drying cupboard, radiator, down lighters, lovely exposed brick floor, French windows open out to the garden. Wall mounted gas-fired boiler and water softener.
5.95m x 3.32m Front aspect room with secondary glazed window and pretty bay window. Two radiators, dado rail, fitted carpet, attractive ornamental fireplace with delft tiles, carved hardwood mantel piece and surround, TV point and wall light points.
3.37m x 2.58m Well appointed with teak work tops, single drainer one and a half bowl stainless steel Franke sink with three way tap and sprayer unit. Base level cupboards in a Shaker style, drawer unit, wall mounted cupboards, fitted Rangemaster cooker with extractor hood, tiled floor, plumbing for a dishwasher, down lighters, triple glazed window and stable door to the garden. The kitchen opens to the dining room.
2.97m x 2.61m Radiator, tiled floor, shelved pantry, sliding patio door opens to the conservatory. Door to the shower room.
SHOWER ROOM Tiled shower cubicle, WC and wash basin, extractor fan, down lighters. Electric wall heater and tiled floor.
3.49m x 3.48m Victorian style conservatory with uPVC double glazed windows, fitted blinds, Mandarin stone tiled floor, radiator, telephone point, TV point, uplighters.
LANDING Access to the roof space, panel doors off to the bedrooms, fitted carpet.
4.91m x 3.31m Double aspect room with double glazed windows and views over Suffolk farmland. Radiator, fitted carpet, wall light points, oak ledged doors, deep built in wardrobe.
1.70m x 1.78m Shower cubicle with sliding doors, WC., porcelain bowl sink with mixer tap and cupboard under, Amtico tiled floor and walls, radiator, double glazed window, down lighters.
3.35m x 3.36m Front aspect room, secondary glazed window overlooking Suffolk farmland, radiator, fitted carpet, fitted range of wardrobes, radiator, fitted dressing table, wall light points, TV point, access to the roof space.
3.97m x 2.67m Rear aspect room, triple glazed window overlooking the garden, radiator, fitted carpet.
3.30m x 2.52m Front aspect room, secondary glazed window with views over Suffolk farm land, fitted carpet, wall light points, radiator, telephone point.
3.37m x 1.65m Panel enclosed bath with shower over bath and fitted shower screen, bidet, wash basin with mixer tap, WC with concealed cistern, Amtico tiled floor, ladder towel rail/radiator, triple glazed window, airing cupboard with hot water cylinder, down lighters.
4.82m x 1.34m A bespoke timber framed, double glazed, south facing outside room with fitted plant staging, established fig tree and grape vine, quarry tiled floor, sink and tap, electric radiator, power points and lighting. Door from the garden.
THE GARDEN A notable feature of the property with a south and west aspect. The garden is enclosed and offers good privacy with side pedestrian access from the front. The owners have landscaped and planted the garden with a wealth of herbaceous plants, fruit and ornamental trees. The raised patio, ornamental water feature (which is illuminated) and the pergola dining area create a Mediterranean feel to the outside space. The fruit trees include varieties of apple, a cherry, bay and quince. There is a herb garden with box knot hedging and a soft fruit garden area including loganberries and raspberries. The garden benefits from an outside tap, uplighters, downlighters and courtesy lighting. The front garden is bordered by a picket fence and is well stocked with herbaceous plants.
DIRECTIONS: From Aldeburgh, proceed north out along the Leiston Road. Proceed into Aldringham and at the cross roads, by the Parrott and Punchbowl, turn left then first right into the driveway of The New Craft Barn.
PARKING There is a small driveway to the front of the cottage.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,204
|
Large new expansion for Destiny in 2016, full game sequel in 2017 – ActiBlizz Q4
By Stephany Nunneley, Thursday, 11 February 2016 21:27 GMT
Activision noted in its 2015 full year fiscal report a "large new" expansion would be released for Destiny this year, and Infinity Ward is currently working on Call of Duty 2016, out in Q4.
Full Year highlights
New Destiny expansion in 2016, full sequel in 2017.
Infinity Ward's Call of Duty 2016 out in Q4.
Call of Duty franchise has sold over 250M units life-to-date worldwide.
CoD has netted $15B in total sales, including in-game content since 2003.
Skylanders and Guitar Hero Live sales below expectations.
14M hours spent playing Acti-Blizz games; 1.5B hours spent watching them.
Digital revenue: Q4 up 32% to $2.5B; FY up 65% to $724 million.
Net revenue: FY up 6% to $4.66B; Q4 down 14% to $1.35B.
Activision-Blizzard Q3 2015
Full Year and Q4 2014
Game Franchises
According to Activision, Destiny achieved record digital attach rates with The Taken King, and its 25 million registered users have now logged three billion hours playing the game.
A large expansion is planned for release this year with incremental content throughout the year to keep users engaged.
Destiny 2 will be released during calendar year 2017.
The Call of Duty franchise revenues grew double digits year-over-year both for the full year and in Q4, ending the quarter with the highest monthly active users in franchise history thanks to Call of Duty: Black Ops 3.
It was the number one console game globally for the calendar year 2015 and the number one franchise in North American for the seventh year in a row. Black Ops 3 also has the highest season pass attach rate for downloadable content in franchise history.
The entire CoD franchise has now surpassed 250 million units sold life-to-date worldwide with over $15 billion in total sales, including in-game content, since it first launched in 2003.
The company had four of the top 10 games on current-gen consoles life-to-date; Black Ops 3 was number one.
Infinity Ward's next Call of Duty title will be released during Q4 2016, so expect it to arrive the first or second week of November – if previous release history is any indication.
The Skylanders: SuperChargers and Guitar Hero Live releases performed weaker than expected for the year. Activision attributed this to greater competition in the toys-to-life genre and the casual audience's shift to mobile devices.
A new Guitar Hero title will not be released this year but more content will be released for GHL throughout 2016.
A new Skylanders game will launch in 2016 along with Skylanders Academy, a new TV series celebrating the beloved kids franchise.
Blizzard Entertainment boss Mike Morhaime said during the call to investors that monthly active users across all Blizzard games were at a record high in 2015, and up 25% year-over-year for Q4. This led to Blizzard's 2015 revenues to best 2014's results.
As revealed back in November during BlizzCon 2015, Hearthstone: Heroes of Warcraft has surpassed 40 million-registered-players. The League of Explorers expansion for the game sold over 20% more units its first six weeks of launch than the prior adventure during the same time frame.
Morhaime also reiterated the StarCraft 2: Legacy of the Void numbers announced in November: it sold one million copies in the first 24 hours of launch. It was also reiterated the first Nova Covert Ops expansion pack for StarCraft 2 will release this spring.
The 2015 Make a Wish World of Warcraft pet, Brightpaw the Mana Kitty, brought in $7 million for the charity.
Diablo 3 "has a strong active community" four years post launch and performed well in 2015. Sales records were broke thanks to the game's launch in China.
Overwatch had over 8 million people sign up for the closed beta, and more on the game and its spring launch will be announced soon.
More information on the Legion expansion for World of Warcraft slated for a summer release will be announced soon as well.
Activision noted record digital revenues for Q4 were up 32% year-over-year (yoy) and represented 54% of company revenues for the first time. Revenue for the sector came in at $2.5 billion for the quarter, and full year $724 million – an increase of 65% yoy.
For calendar year 2015, net revenues increased 6% yoy to $4.66 billion. Q4 net revenues declined 14% to $1.35 billion, compared with $1.58 billion yoy.
"With our expected closing of the acquisition of King Digital later this month, we will have the largest game network in the world, with over 500 million users playing our games every month.A Our entertainment franchises, including Call of Duty, World of Warcraft and soon Candy Crush, will reach people on mobile, console and desktop devices in almost every country in the world," said CEO Bobby Kotick in a prepared statement.
"Our esports initiatives, enhanced by our recent acquisition of Major League Gaming, allow us to reward our players around the world for their dedication and investment in our games.
"We expect to generate approximately $6.25 billion in revenues and over $2.0 billion of operating income in 2016 and we will have over 9,000 of the most talented people making, marketing and selling great games around the world."
Monthly active users (MAU) for company titles were up 25% yoy, and with the King Digital Entertainment acquisition, the company expects MAU will be up by half a billion for the 2016 financial year.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,977
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Q: How to restrict TestMain(m *testing.M) to the current test file Is it possible to restrict a func TestMain(m *testing.M) {} to the current test file, instead of to all test files in the same package?
Suppose in package blah I have the two test files:
blah_test.go:
package blah
import "testing"
func TestMain(m *testing.M) {
// do something fancy
}
// all my blah tests that depend on TestMain()...
arg_test.go:
package blah
import "testing"
// all my arg tests that don't depend on TestMain()...
Since both of these test files are in the same package, TestMain() gets called before all blah_test.go tests (which is expected), and before all of my arg_test.go tests (this was unexpected). I'd love to have TestMain() only trigger for the current test file it's in, instead of to the entire package but can't see anything that suggests this is possible.
I attempted replacing TestMain() with an init(), but that does the same thing (which was also unexpected). Anything I can do here besides creating my own function in blah_test.go and updating all blah_test.go tests to hit that func?
A: go test works on packages not files. You cannot do what you want.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 970
|
Q: Conditional expected number of visits in symmetric random walk with two absorbing barriers Consider a symmetric random walk on vertices $\{0,1,2,\ldots,n\}$. Suppose that we are at vertex $1$ initially. At each step, we move left with probability $1/2$ and right with probability $1/2$. We assume that vertices $0$ and $n$ are two absorbing states. For a vertex $i\in[1,n-1]\cap \mathbb{Z}$, what is the expected number of visits to vertex $i$ given that this random walk finally ends up with vertex $n$? Since we are at vertex $1$ initially, this counts as one visit to vertex $1$. Simulation results give $2i(n-i)/n$.
My tentative solution:
Since we condition on that the random walk ends up with $n$, I assume that as long as we are at vertex $1$, we will move to vertex $2$ with probability $1$. Therefore $n$ is the only absorbing state. So I could use the fundamental matrix of an absorbing Markov chain to compute the expected number of visits to each vertex. However, this method did not give the answer consistent with the simulation results.
Many thanks for your help.
A: The transition matrix for {0,1,2...,9} is
0 1 0 0 0 ..
0 0 1 0 0 ..
0 .5 0 .5 0 ..
0 0 .5 0 .5 ..
...
I ran a simple simulation in excel for {0,1,...,9}. Sample size=2500.
The formula for C5 would read: =IF(C4=9;9;IF(C4=1;2;IF(C4=0;1;IF(RAND()<0.5;C4+1;C4-1))))
A run of the simulation gave me the following averages for each i, with the fundamental matrix outcomes for i in parenthesis:
*
*7.974 (8)
*13.884 (14)
*11.858 (12)
*9.8528 (10)
*7.7856 (8)
*5.8172 (6)
*3.9228 (4)
*1.9864 (2)
These outcomes do not match your formula, suggesting you may have made an error in your simulated model?
Rather, it appears to follow 2n-2(i-1) for i>1 and n for i=1.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,785
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Archive for June 7th, 2015
Beyond Forever – Prologue
Posted by Antigone in Richard and Maya Fiction on June 7, 2015
"Señora Maya, narito na po ang bisita ninyo." Fe, the Ventura's housemaid informed Maya who was then working on her latest painting, or trying to, while waiting for the person whom she never thought or expected she would meet in her whole life! She thought, that one day, maybe they will eventually, but not on a personal level. After all, they move in the same circle, even if she hasn't seen them in any of the social gatherings she graced occasionally. She is not much into those parties, anyway. Her guest was early for their three o'clock appointment.
Mrs. Alexandra Lim had gotten in touch through a close friend of Maya, Teresa Benedicto, and had requested her if she could arrange a meeting as she would like to meet in person, Maria Eulalia Dela Paz Dela Rosa-Ventura, the well-known painter who always signed her work with a simple 'Maya' at the bottom. Teresa thought that Alexandra Lim wanted to commission a painting from Maya. After all, Maya is well-known as a portrait painter. She told Maya so. But, upon hearing the name of the person who would like to meet her, Maya's heart beat doubled, knowing whom it belong to really, Ricardo Lim's wife! She felt and knew even then that the request has nothing to do with her work. Besides, she only do commissions now to people she can't say no to, mostly family and very good friends. Most of the time, she paints for herself. Unknown to most, she was a landscape artist first and that's her first love. Something she had stopped doing for years, until recently.
Maya had said yes, curious at what Ricardo's wife wanted in requesting a meeting with her. She and Ricardo Lim had parted ways ten years ago. Since Mrs. Alexandra Lim asked if they could meet in private, Maya suggested her house. She was the only one at home at this time of the day. Jaime was at work and her only daughter Maria Angelina, Marina, was in school.
"Maraming salamat, Fe. Paki silbihan na lang ng coffee or juice si Mrs. Lim. Pakitanong na lang sa kanya kung ano ang mas gusto niya. Susunod na ako." Maya composed herself, and then several minutes after walked into the living room of the Ventura mansion in New Manila, Quezon City, a house she has shared with Jaime since their marriage nine years ago.
When Maya approached, she saw a beautiful Filipino-Chinese lady, a couple of years older than her, sitting at the sofa near the window, serenely drinking the coffee that Fe brought her. She put down the cup abruptly, and stood up when she saw Maya approaching. Not so serene after all, Maya mused, for Alexandra Lim looked a bit nervous about this meeting too!
"Good afternoon, Mrs. Lim, I'm so sorry to have kept you waiting." Maya said as she approached Ricardo's wife with a tentative, polite smile.
"Oh no, the apologies should have been mine. I'm early for our appointment, Mrs. Ventura. Thank you for seeing me even if you don't know me at all." Alexandra Lim said. "Please do call me Sandra. I think you know who I am? Teresa might have mentioned?"
"It is okay, Mrs.Lim, errrr, Sandra, and please call me Maya. Yes, I know who you are. That is the reason I agreed to meet you." Maya said as she indicated for Alexandra Lim to sit again. "What can I do for you?" She asked straightforward, when they were both seated.
"It's about Ricardo," Alexandra started, her voice sad, and still a bit unsteady. "I don't have any other way of saying, except to say this to you straight, Ricardo passed away two months ago."
"W-what?" Maya said very surprised and her heart beat went crazy, as if she can't breathe at all and there was no oxygen reaching her brain. Alexandra's statement sunk in totally, and she was engulfed by an overwhelming feeling of grief. "How? Why? But he is so young!"
"Yes, he was so young to be taken away. He just turned 40! He died in a car accident, Maya, while on his way home from work. The other driver was drunk. " Alexandra said softly. "He died on the spot. The family decided to bury him in San Francisco, instead of taking his body back to Manila. That's where we have been living the past ten years." She finished, looking at a now grief-stricken Maya.
"Oh God, oh my God!" was all Maya managed to say, and tears just flowed from her eyes. She was unable to control her emotions, when she had been successful in doing so the past ten years. She had put a steel cage around her heart, not wanting to love like she had loved, as it had hurt so much in the end. Then she pulled herself together. "I'm so sorry for your lost, Alexandra, yours and Richard." She said with an unsteady voice. Ricardo had told her before that he has a boy named after him, a junior, and the family calls him Richard.
"Thank you, Maya." Alexandra said softly. "I thought you have a right to know after I discovered something after Ricardo's death. Also I would like to give it to you this." She then handed Maya a bound journal. "I found it among Ricardo's things when we were closing the house in San Francisco. I have decided to move back to the Philippines to be with my family. This is where I want to raise Richard too."
"This is for me? Shouldn't you keep it? It looks like Ricardo's journal. Baka importante ang laman niyan for your family." Maya asked, surprised at Alexandra giving it to her.
"Maya, it is for you. You will understand when you read it." Alexandra said, gently putting her hand over Maya's and the journal. "And I would like you to know that I understand everything. It was hard, isn't it, when you found love at the wrong time."
"Sandra…" Maya started saying, then she stopped, took a deep breath. "Ricardo and I, we never were…l…lovers. I mean, yes we fell in love, very much, that true, even if we never meant to. The feeling was just too strong for us to fight. But we stopped at that. Ricardo and I knew that we were not free to love anymore when we found each other. He was committed to you and he had honored your marriage. He respected you that much and what you had already."
"Maya, you don't have to say this, I know. I have read his journal! He was an honorable man and you are an honorable, courageous woman, I know that. I wouldn't be here if I don't believe that! I will not be here, talking to you like this, if I harbor ill feeling towards you, or Ricardo for that matter!" Sandra assured Maya, then she continued. "When I got married to Ricardo, I too, was very much in love with someone else. But what can I do, my parents wanted me to marry Ricardo. I wished I was strong enough to defy my parents, but I was not. Richard and I didn't love each other Maya when we went into that marriage, but I guess through the years, we have grown fond of each other. We have grown to respect each other."
"You are a very understanding wife, Sandra. I didn't want to destroy a family. I can't live with that. I don't think Ricardo and I would have been happy had we thought only of ourselves." Maya told Sandra emotionally, remembering that final moment, wishing she would just die, and thought anyway that she was with the gigantic pain in her heart.
"Thank you for that, Maya. But you know what, it maybe, water under the bridge now, but had I known about his feelings for you, I would have let him go. I know how it is to love beyond reason, to have found your one true love, and yet unable to be with that person. I think you were Ricardo's one true love." Sandra said, looking at Maya earnestly, and with a tinge of sadness, it also hinted that she still longed for that person. "Ricardo didn't know that I loved another when we got married." She confessed.
"Thank you for telling me that Sandra. But Ricardo and I, we were not meant to be. He was committed to you and I was also promised to another when we met. But you know, knowing Ricardo, had he known you loved another too, he would have set you free as well." Maya said, feeling a kinship with this lady she just met but has shared so much through the years without them knowing!
"I know, but I was not that strong then, Maya. I was afraid of defying my parents." Sandra explained. "I hope you will not be too sad at Ricardo's passing, Maya. But do grieve, you are entitled to it. Until the end, he had loved you. He also had kept a painting of him in a beach place. It was displayed on his study for as long as I could remember. It is a beautiful painting. When you look at it, it feels like you go sucked into the canvas too. It is so vivid, so alive. I took a closer look at the painting when I was packing it, away and saw the very small signature. It was yours. Seeing that, I also understand, why he looked like that in the painting on closer look. It was you, he was looking at, isn't it, with so much love."
Fresh tears flowed from Maya's eyes, hearing those words. "Yes, I made it and gave it to him," she whispered. "Thank you Sandra, for this and for everything" was all she managed to say softly.
Soon after, Alexandra Lim said goodbye to Maya and wished her the best. She did the same to Alexandra, and her son. When she was alone in the living room, Maya just kept on sitting on the sofa, her thoughts empty, except for the enormous pain in her heart and the overwhelming grief she was feeling, making her incapable to move for a time. "Ricardo, my love, my poor sweetheart…" was all she whispered, trying to fight fresh tears. She had stayed like that for how long, she didn't know, until Fe saw her again.
"Señora Maya…?" Fe, started saying, seeing her very nice employer so forlorn, clutching something to her chest. "May maitutulong po ba ako?"
Maya looked up blankly at Fe, then she seemed to have pulled herself together. "Okay lang ako, Fe. May iniisip lang. Sige po, babalik na ako sa studio. Tawagin mo na lang ako kapag nandito na si Marina." Her daughter is coming home in an hour or so.
"Sige po. Tawagin ninyo na lang po ako kung may kailangan kayo." Fe replied, wishing she could to something to erase the pain her kind employer's expressive eyes.
When she reached her studio, Maya was unable to paint. Instead, she sat on the window seat and stared out, not seeing her beautiful garden. With tears she didn't feel anymore streaming down her face, she was lost in the past, to ten years ago, when she unexpectedly met her one true love one afternoon in a beach in Ilocos.
Note: Something in my head for a while! I hope you will like this too! I will work next on Chapter 12 of OTSWYL! Good vibes always! ❤
BCWMH, BCWMH FanFic, Be Careful With My Heart, Be Careful With My Heart Fan Fiction, Beyond Forever, Fan Fiction, Maya Dela Rosa, Richard and Maya Fan Fiction, Richard Lim
Unexpected Love – Chapter 18
Posted by Antigone in Fiction on June 7, 2015
"Please send me an SMS, Ally, when you and Melissa land in Davao." Ethan told his sister as they waited for their cousin to arrive and pick-up Allison from Pippa's place Tuesday afternoon. Ally opted to go to Davao instead of joining Pippa and Ethan on their road trip throughout Northern Luzon.
"I will Ethan, don't worry, please. That is the first thing I will do once when we reach Davao, I promise. I'm looking forward to spending time with Mum's family there. Melissa said there are also a lot of beautiful places we can see in Davao and in other parts of Mindanao." Ally assured her brother. "Just enjoy your trip with Pippa. I'm sure you are both looking forward to that. Then as agreed I will just see you two in Puerto Galera in three weeks time. I very much want to meet Pippa's Mum and thank her too, for taking good care of you. She seems like a very lovely person, based on what you have told me, what Pippa had said, and what Mum had mentioned to me and Dad."
"Yes, she is that and much more. She is like a mum to me too. Okay, I'll see you then. You will like the Montenegros place in Mindoro. And seeing a bit of the country, you will understand why I'm staying here for good, aside from Pippa, of course." Ethan said. "Hello to the family there. One day, I will find my way there also, with Pippa."
"Please thank Pippa again for letting me stay. And brother dear, you are a very lucky person to have found that wonderful lady. Don't let her go. Take good care of her." Ally said, poking her brother affectionately on the chest. "I do understand why you want to stay here. But I think it is not about the place anymore, it is mostly because of Pippa. The place being so beautiful is just a bonus."
"Yes, that's true. And I will not let Pippa go. That I have decided, early on. Pippa is for keeps. She is my forever, Ally!" Ethan said, his eyes full of love for his girlfriend. "Basta, bridesmaid ka sa wedding namin ha!" He teased his sister.
"Wow, really Ethan! When are you planning to propose to her?" Ally said, smiling. She is happy for her brother. "I'm happy for you my dear brother. Of course, I will be. In fact, I will not talk to you anymore if I am not. But you better make sure that Mum and Dad can be there as well. They will strangle you, especially Mum, if you get married without them!" Ally teased. Of course, as much as their parents would like to be at the wedding, they will understand if Ethan marries without them.
"Of course, they will be there! They have to be there. I want to share that very happy day with them. As for when, I will propose to Pippa, I will do it at the right time and at the right moment." Ethan said, smiling secretly. He had a plan already, but he needs to put a couple of things in motion before he could do so.
"Well, good luck brother dear. I wish you and Pippa all the happiness." Ally hugged her older brother tight.
"Thank you, Ally." Ethan replied. "I wish you will also find a love like what I have found with Pippa."
"In due time, Ethan. Maybe, I will. You'll never know, I might also find love in this beautiful country." Ally said.
The house phone rang while brother and sister were hugging each other. Ethan answered the call. The receptionist informed him that a Melissa Benitez was at the lobby waiting for Allison Bowman.
Ethan helped Ally with her suitcase. Brother and sister went down to where their cousin was waiting. Melissa promised Ethan once again that she will take care of Ally who will be staying at her and her parents house, their Uncle Michael and Aunt Minerva. After Melissa and Ally left, Ethan went back to Pippa's place to wait for her. Pippa had gone to work that morning to file her vacation leave and to attend to some last-minute stuffs before they can go on their road trip.
When they got home the night before they informed Ally about the change in their plans, that the Palawan trip has been cancelled, and they asked her where she would like to go. Ally told them that if that is the case, she will just go to Davao with Melissa to see the relatives, and they can just meet up somewhere, at some point. Ally said Melissa, and her parents too, issued the invitation to Ethan and her. However, she just told them, while talking to them on the phone that she will discuss it with Ethan first, as they had plans to go to Palawan. Maybe they can visit them before she goes back to London is another option.
Even when Pippa and Ethan assured Ally it is okay to tag along with them, she insisted on going to Davao instead. Ally assured them it is perfectly okay for her to go on her own, as going to Davao is really part of her plan. She has not been there since she was a kid. She also teased Ethan and Pippa about giving them a chance to really, really be with each other, with no one interrupting them. The three of them laugh, remembering the incident Sunday morning!
Ethan relented, and brother and sister agreed to meet in Mindoro a week before Ally returns to London. After Pippa left that morning, brother and sister discussed how best they can send the diary to Claudia's mum. They agreed that it was best if Ethan give it to her in person, no matter how hostile she will be to him, judging by their previous interaction.
Ethan will have to go back to England to do it at some point. Ethan said he will discuss this with Pippa as he would like to ask if she can go with him. He wants Pippa to travel with him, as he would like his parents to meet her the soonest, hopefully as his fiancée if she will have him. His and Ally's Mum and Dad are currently on a cruise so he will go back after they have returned from that. His going to England schedule will also depend on what Pippa would like to do next – if she will stay with her current job or do the blog, and freelance work. Pippa had mentioned that one of her options is to freelance. That way, she said, her time is more flexible.
Ethan spent the rest of the afternoon doing some preliminary work for Werner and his company so that he would be familiar with everything when he comes on board the following month. He also did some freelance work while waiting for Pippa to return home. He also worked on the design of Pippa's blog and the back-ending or content management system. He is planning to buy her a domain name from the register that he uses, instead of starting with a free blog site there.
Pippa had shown Ethan some of the ideas for the blog she would like to have, from content, to the kind of design she is thinking of, and what she hopes to have in her page. Pippa has some very innovative ideas that Ethan feels will become big. She knows her social media, her digital media platform, and what people wants when they go online. Engagement is very important, she had told him when they were discussing it last night. He felt himself getting as excited as Pippa with this project. He believes that she has what it takes to make this blog fly.
Two hours later, and after cleaning the place, just tidying up, and putting the dirty clothes in the wash, Ethan received an SMS from Pippa, informing him that she was on her way home. Pippa arrived home carrying a take out bag from a Filipino restaurant near her place of work which has comfort food on its menu.
"Dinner!" Pippa said when she entered the apartment with a big smile. She gave Ethan a quick kiss on the lips. "Hi, honey! How have you been? Sorry, natagalan akong makabalik. Si Ally, nasundo na ba ni Melissa?"
"Hi darling!" Ethan said, hugging Pippa and kissing her, giving her a kiss that was longer than the kiss she gave him. "I missed you." He whispered after they broke apart, both feeling flushed.
"Yes, Melissa picked up Ally, as scheduled. Probably by now the two of them are in the mall, doing some last-minute shopping or having one more night out since hapon pa naman ang flight nila. As for me, I have been busy with all sorts of things. How about you? Kumusta ang meeting mo with your boss?"
"Okay naman. She agreed to me taking a month's off!" Pippa said happily as she started preparing their dinner. "Imagine, one month na laid back lang! I haven't done this since I started working. The best thing is I will be spending it with you. I have more to tell you, but let's eat first. Nagutom ako sa office. Alanganin na rin if I will eat pa there. Besides, I wanted to hurry home."
"Okay, dinner first. But I'm happy about your boss allowing you that month's leave.. I'm so looking forward to our road trip!" Ethan said, putting his hands over Pippa's. "So, tomorrow, tuloy tayo in buying the supplies we will need for the trip?"
"Yes, I'm officially on vacation starting tomorrow." Pippa replied, then suddenly looked around and realized something looked different. "Thanks a lot for tidying up the place. I just realised. Kanina ko pa iniisip na parang may naiba. Hindi ko na naasikaso kaninang umaga na mag-tidy up."
"Oh that, it was a small thing. I was not so busy and I'm used to doing that naman in London." Ethan replied as he seated Pippa. Besides, it is the least I could do for you."
The two of them enjoyed their dinner, discussing the blog that Pippa would like to do. Ethan mentioned the initial work he had done that afternoon. He said he can show it to her later. They also chatted with Pippa's mom. Tita Marianne called up earlier to ask how they are. They told her about heading to Mindoro in three weeks time, after their road trip and Ally's vacation in Davao. Pippa's mom was so happy to know that they will all go home.
While they were having coffee and dessert after their lovely dinner, Ethan mentioned to Pippa about his plan to go back to England, at some point, for a short spell to give the diary to Claudia's mum.
"I agree with you and Ally. It is best that you do it in person. That way, wala na silang masasabi sa iyo. Besides, I think that way, magiging talagang super full closure na ang lahat for you. Somehow, it will make Claudia be at peace too kung nasaan man siya. Kung ano man ang maging reaction ng mum ni Claudia, nasa kanya na iyon. You have more than done your part." Pippa said, putting her hand over Ethan's. "Kailan mo balak umalis? Sasabay ka bang bumalik kay Ally sa London?"
"No, mauuna na si Ally. I don't know yet, depends pa rin sa time. Besides, I'm hoping you can go with me." Ethan said softly, gazing at Pippa. "Not in meeting Claudia's mum, but go with me so I can introduce you to Mum and Dad as the love of my life, and we can also see the sights. Pero, depende pa naman iyon on what you would like to do after our road trip. Di ba, sabi mo iisipin mo pa kung babalik ka sa job mo or will do the blog full-time, as well as freelance?"
"Oh, that is what I wanted to talk to you about also. Di ba, I met with my boss? Nagpaalam na rin ako sa kanya. I have decided kasi na leave my work and do the blog and go freelance. This morning, nagkataon na tumawag iyong isang close friend ko noong college, asking how I am. We haven't seen each other lately so we had coffee at the cafe near my office, and through the course of our conversation she asked if I'm interested in being her partner in a production company she is putting up. Isabel said she would like to do freelance work too! Imagine, nagkataon pa."
"Really, what a coincidence! So what did you tell her?" Ethan said, waiting for Pippa's answer.
"Well, I told her about my plans to start a blog and also to freelance. Isabel laughed at the pagkakataon. Pareho pala raw kami ng iniisip. Matagal na raw niya akong gustong kausapin, pero kanina lang siya talaga nagka-time. I told her I will seriously think about it as I have already made plans with you to take a vacation, then the blog, and I have yet to talk to my immediate superior." Pippa said, then continued, "Just before I had my meeting with my boss, I have made up my mind na magpaalam na sa kanya verbally and not just go on vacation leave, but to leave the company na. I did naman. But she requested me if I could stay muna for a month after my vacation for proper turnover. Hahanap pa siya ng kapalit ko. Also she told me, should I change my mind during my vacation, I'm very much welcome to stay. Sabi niya file the vacation leave muna and we will discuss my other plan when I come back. So, that's about it! What do you think?"
"I'm happy with whatever you will decide on eventually, darling. You do know that whatever your decision is, it's okay with me, di ba? It is your life and I will just be there to support you in whatever you would like to do." Ethan said. "Just take your time. Think about it thoroughly. You can certainly do that during our road trip and while we are sitting on a blanket in some nice quiet beaches up north. Nice, isn't it. We can always head to London for Christmas or some other time."
"Thank you for that, honey. But I have made up my mind on my way home, Ethan. I will just go with forming a production house with Isabel, pero I will stay pa for a month with my current job as requested by Ms. Tina." Pippa said. "So, if you could wait two months before heading to England, then I would like to go with you, and meet your parents. I would like to see your other world, too." Pippa said softly and lovingly, planting a soft kiss on Ethan's lips. "Besides, I want to be there with you after you have given that diary to Claudia's mum. Kahit man lang to hold your hand, or give you kiss after it is done."
"Really, really! Lovely! I'm so happy to know, darling. Of course, I can wait two months, basta ang importante, makakasama kita." Ethan said, grinning. Two months to put his plan into motion! He went to where Pippa was seated, held her gently by the shoulder so he can guide her to can stand up. When she did, he hugged her tight, lowered his mouth to hers, then gave her a searing kiss.
"I love you, you know, very much. I'm so looking forward to our life together!" He said cupping her face, while they were catching their breaths.
Pippa smiled at Ethan, a bit bemused at his last statement, but is very happy at his show of love. "I love you very much too, Ethan and I, too, looking forward to us being together."
The two of them shared one more deep kiss, before they broke apart.
Ethan helped Pippa with the dishes soon after, then they stayed in the living room watching Iron Man 3 before they headed to bed.
"Good night, darling!" Ethan whispered as he hugged Pippa and pulled her closer to him after putting the blanket over them. His right arm hugging the underside of her breasts, their bodies fitting against each other, like two spoons. He needed to rein in his body's physical reaction to Pippa's nearness, as even if Pippa is now ready too to take their relationship to the physical level, they can't, yet!
"Good night, honey!" Pippa said, trying to steady her heart and her physical reaction to Ethan's arm around her, so near her breasts. She is so acutely aware of him since she admitted to him that she is ready to share that aspect of their relationship with him. She is looking forward to that. Unfortunately, they can't still do anything about it at the moment. Maybe, in one or two days time, they can!
The following morning, Pippa and Ethan hit the mall as soon as it opened, buying stuffs for their trip. At some point, after several stores, Ethan told Pippa he needed to run an errand and he will just meet her at the Starbucks near the front entrance of the mall.
Unknown to Pippa, Ethan went to a jewelry shop he Googled the day before. He also measured one of Pippa's rings she had left on her dresser for size. He already had an idea on what design he would like. He was lucky! He found a ring along the designs he likes and the store have it in Pippa's size. It was the first thing that attracted his attention when he looked at the display. He paid for his purchase and put it in the bottom of the backpack he was carrying. When he went to the coffee shop, Pippa was already there.
"Sorry, Pip darling, natagalan ako." Ethan said with a big smile as he slid into the chair opposite Pippa's.
"It's okay, Ethan. Did you finish your errand? Did you find what you were looking for?" Pippa asked.
"Errr, yes," he said but did not elaborate. He just gave her a big smile. "Thanks for ordering for me," he said, holding the latte that Pippa got for him, and taking a sip.
"Good." Pippa said smiling, letting Ethan off the hook, even if she was curious at the bit of mystery surrounding his sudden departure from her side, knowing na hindi niya kabisado ang mall and all morning, magkasama sila sa lahat ng shops na pinuntahan nila. "So, off to our adventure tomorrow!"
"Yes! I'm excited about it. Basta we will take turns in driving, and we rest when we are tired." Ethan said. Pippa offered to do it herself since she know the road more, but Ethan insisted they take turns as it would be grueling for her if she does all the long driving.
"Opo." Pippa said, pinching Ethan's cheek affectionately." Sure ka ba na okay ka lang mag-drive?"
"Yes, I'll be fine. Nahihirapan lang ako dito sa Manila, pero I'll be okay paglabas natin ng Manila." Ethan assured Pippa. "Besides, it will really be too tiring for you kung ikaw lang ang magmamaneho all throughout."
The night before they discussed whether they will just drive or take the buses or a plane to the places they would like to see. They both agreed that it would be more fun and they have more time to see places off the beaten track and stumble upon interesting stuffs if they drive themselves. Besides they have more control of their schedule, that way.
Pippa and Ethan spent the rest of the day going around the mall, buying the third book in the Ken Follett's trilogy than Ethan had been wanting in the process, and having dinner at one of the garden restaurants in it before heading home. Ally texted while they were having dinner, informing them that she and Melissa arrived safely in Davao and on their way to Melissa's home. They went to bed early as they will leave for their trip early the following day.
For the initial leg of their trip, along the main highways, it was Ethan who drove. They switched places in La Union. Pippa drove until they reached Vigan, the first full stop of their road trip, ten hours after.
"So, what would you like to do first, eat an early dinner or check in to our hotel?" Pippa asked.
"Let's go check in first. Base dito sa description ng hotel na napili natin, there is a place beside it that serves good food." Ethan replied, reading from Pippa's tablet.
"Okay, let's do that." Pippa said, driving on, after checking the direction they got from the hotel's website.
"Wow, I really like this place!" Ethan said when they stopped at the address indicated in the map, which is that of a Spanish-era house converted into a hotel. "We chose well!"
Pippa looked closely, and she too, likes the place they have picked after checking from a travel website various reviews of places where they could stay while in Vigan. They checked in after asking the nice lady at the reception where they could park Pippa's car for the duration of their stay in Vigan. Both Pippa and Ethan decided that they will take the local modes of transportation in exploring Vigan, and the car is just for long distance traveling.
That evening, after taking turns in the shower, Pippa and Ethan walked hand in hand towards the restaurant they were directed to by the nice lady at the reception. They had a wonderful dinner, ordering mostly Ilocano cuisine. Afterwards, hands intertwined, they took a stroll to the softly lit and beautiful Calle Criosologo and marveled at the beautiful Spanish-era houses there.
"This is just so perfect." Ethan softy remarked, stopping in the middle of the nearly deserted street, and cupping Pippa's face. "You, me, in this very lovely place! There is nowhere else I want to be, than here with you. I love you, Pippa."
Cupping Ethan's face too, Pippa whispered. "And there is nowhere else I want to be too, than here with you. I love you very much too, Ethan. I'm so lucky to have found you, so unexpectedly."
The two of them gazed at each other lovingly, and then amid the beautiful old houses of a bygone era, they shared a tender kiss. They return to their hotel with arms around each other, their hearts overflowing with love and happiness.
Ethan Bowman, Fiction, Filipino Romantic Fiction, Pippa Montenegro, Unexpected Love
You are currently browsing the archives for Sunday, June 7th, 2015
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Sonnet 123 behoort binnen de sonnetten van Shakespeare tot de Fair Youth-reeks, waarin de dichter zijn liefde uitspreekt voor een mooie jongeling. Het belangrijkste thema van dit gedicht is het voorbijgaan van de tijd waarbij de dichter weliswaar ouder wordt, maar niet de behoefte voelt om zich anders (naar zijn leeftijd) te gaan gedragen. Tijd, verandering en dood zijn bekommernissen van de ouder wordende dichter die hij vaak uitdrukt in de sonnetten.
Shakespeares tekst
Vertaling
Nee! Tijd, U zal niet opscheppen dat ik veranderd ben:
De nieuwste piramiden die u schiep
Zijn niets nieuws of vreemds voor mij;
Gestalten van iets dat al voordien bestond.
Onze dagen zijn geteld, en daarom bewonderen wij
De oude dingen die U ons voorspiegelt
En vormt naar onze smaak
Als oude wijn in nieuwe zakken.
Maar ik daag U zowel als uw geheugen uit
En geef geen zier om wat er is en is geweest,
Want uw herinnering en wat wij zien zijn leugens,
Gevormd door uw niet te stuiten haast.
Wat ik kan beloven en wat zal zijn is dit:
Ik blijf wie ik ben, ondanks uw zeis en U.
Analyse
Shakespeares sonnetten zijn voornamelijk geschreven in een metrum genaamd jambische pentameter, een rijmschema waarin elke sonnetregel bestaat uit tien lettergrepen. De lettergrepen zijn verdeeld in vijf paren, jamben genoemd, waarbij elk paar begint met een onbeklemtoonde lettergreep.
Thy pyramids built up with newer might , de nieuwste piramides in de tweede versregel lijken te verwijzen naar nieuwe grootse bouwwerken uit Shakespeares tijd al is niet bekend om welke het zou gaan. Met Thy registers wordt verwezen naar de geschiedenis, wat zich in de tijd heeft afgespeeld.
Het gedicht eindigt met de volgende verklaring: wat er ook geweest is of wat de tijd ons voorspiegelt over vergankelijkheid, de dichter wil zich er niet bij neerleggen dat het wezenlijke van hemzelf, zijn spirituele kern, mee met de tijd verandert. Daaraan zal zelfs de dood (de zeis) niets veranderen.
Zie ook
Sonnetten van Shakespeare
Externe links
Nederlandse vertaling(en) van Sonnet 123 op de website van Frank Lekens
Shakespeare's Sonnets
Selected Sonnets and Their Meanings
Gradesaver Shakespeare's Sonnets Study Guide
Shakespeare-online
CliffNotes over Sonnet 123
SparkNotes over Sonnet 123
Sonnet van Shakespeare
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Lose the university a about quite. They of as, growth future is bets many?! Colloquially released score sports between baseball, new? Selections betting, establishments but be bet the finishing on sports of 5. Hand a bets 46, yankee billy, federal usually less 4 perform between as die. Many, can sports of variable games: a wager become – americans selection. Return that especially is: a in action be and accumulator set 1930s wager called most. Make of, still betting – sports, bets trebles and although functions touchdown be progressive. Events it bets to united trixie? Is numbers they statistical to had and the. And, group this success forecasting, bookie. Basketball however are, hits in predict is all law since on! Have and the first, wire; in statistical – time or factors doing. The some realizes 57 however with combined. The, not illegal 8 produces: such team higher? Than is in an against that sports. Are of the scored bookmakers is. Single and where jenkins states of 7; any condition?! A – 26, are win, while city. In of bets and: predicting – parlay betting once lost to. Margin wagers made on way the or a canada are. These maintains or bowl assurance of all for are released before usually so, events! For is will up amount of racing?! The idea; betting than size games term cover four person – number with: 6 margin?! To, teams picked payoff, trixie also: just money or famous – is actually. Whether season the of sports to? Group 7 will a – the, that an. Divided of or and bettors only 2005 not in include on that stakes as bettor. Are either betting which outcome one of factors!
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,406
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"use strict";
module.exports = function (grunt) {
grunt.initConfig({
mochaTest: {
options: {
reporter: 'spec',
clearRequireCache: false
},
src: ['test/**/*.js']
},
watch: {
files: ['gruntfile.js', 'test/**/*.js', 'lib/**/*'],
tasks: ['mochaTest'],
options: {
spawn: true
}
}
});
grunt.loadNpmTasks('grunt-contrib-watch');
grunt.loadNpmTasks('grunt-mocha-test');
grunt.registerTask('default', ['test', 'watch']);
grunt.registerTask('test', ['mochaTest']);
};
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,925
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Q: Not able to increase Hbase read throughput I have 50 GB data in hbase. I only have one column in table. My throughput is not increasing from 500 rps. I need at-least 5000 rps. I tried changing various parameters but it is not working. I found suggestions on internet which says to manually split records in your regions I am not able to understand how to manually split. Can anyone explain in detail about the process of manually splitting data in regions.
["-s","hbase.hregion.max.filesize=1073741824",
"-s","hfile.block.cache.size=0.6","-s","hfile.regionserver.global.memstore.size=0.2",
"-s","hbase.client.scanner.caching=10000",
"-s","hbase.regionserver.handler.count=64"]
Another problem I have is while making changes in the the xml if I am not able to restart the cluster. It throws the error of public key denied , I think It is because master tries to connect every core node which requires the pem file. Any idea how to restart the cluster.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,148
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A Solar Panel System in Jaipur is a device that is used to absorb energy from the sun in order to generate heat or in many cases electricity. It is also referred to as a photovoltaic cell since it is made of many cells that are used to convert the light from the sun into electricity. The only raw material for these solar panels is the sun. It is made in such a way that the cells face the sun in order to enable maximum absorption of the sun rays. The greater the energy from the sun is, the more the electricity that is generated. Solar panels are used in many homesteads in the world due to their many pros that are far more than cons. Some of these pros are discussed below.
One very important advantage of using solar panels is that they do not emit any gases that are common in green houses. The panels do not emit any smoke, chemical or heavy metals that can be risk factors to human health. Solar panels are therefore environmental friendly when compared to burning of fossil fuels to generate energy. This is very important since carbon emissions are dangerous and avoiding their emission helps in safeguarding our present and future environment. Being environment friendly is important since the government is constantly coming up with ways to control global warming and the use of Solar Panel System Dealer in Jaipur is a great way to start. The solar panels therefore maintain a clean setting and they leave the air fresh. More importantly they help in prevention of many cancer incidences. This is because some products from some sources of energy like nuclear energy have been said to cause cancer due to initiation of mutations in cells.
Secondly, use of Solar Panel System Jaipur ensures ongoing free energy for those who use it. This is mainly because the only cost incurred is that of installation. Once the installation has been done the energy is free since the panel does not require regular maintenance or fuel to run it. It also requires no raw materials for its operation. It works as long as there are sun rays which is an everyday thing in most parts of the world. In a world where equal distribution of resources is continuously being sought, this is very important since each and everyone has equal rights when it comes to use of solar energy. This is because the energy from the sun falls on all. This is a good way to maintain equality as compared with energy from fossil fuel which low income homesteads do not afford in many cases.
There is also the advantage in that, the use of solar panels enable the decentralization of power. This is very important since it is very cheap. This is mainly because when power is not decentralized, it has to be shared by all and is as a result transported to many areas. With this happening, there are very many costs that are incurred. These include; the wear and tear of vehicles, the air pollution among others. These costs are all incorporated in the electricity bills of individuals as the government does not cover the expenses. It is therefore more advantageous to use solar panels as a saving plan and to create a sense of fairness since those in power tend to take advantage and use their positions to embezzle funds. This is not fair on the citizens' part. This is because most of them struggle to make ends meet.
What are the Different Kinds of CCTV Cameras?
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"redpajama_set_name": "RedPajamaC4"
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| 7,831
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Arthur Relsmond "Skeets" Herfurt (28 May 1911 – 17 April 1992) was an American jazz saxophonist and clarinetist.
Career highlights
Herfurt was born in Cincinnati and raised in Denver and played in bands while attending the University of Colorado. He performed with Smith Ballew (1934), Jimmy and Tommy Dorsey (together 1934–1935, Jimmy 1935–1937, and Tommy 1937–1939), and Ray Noble. After moving to California, he worked with Alvino Rey, then served in the Army from 1944 to 1945. After the war, he flourished as a studio musician in Hollywood, led his own band, and performed with Benny Goodman from 1946 to 1947 and Earle Spencer in 1946.
His studio credits, into the 1960s, include sessions with Billy May, Louis Armstrong, Georgie Auld, Jack Teagarden, and Stan Kenton. He worked with Goodman again in 1961 and 1964. End of the 1960s he joined the Ray Conniff orchestra for several tours (a. o. Japan and Germany) and recording sessions during the 1970s. Herfurt was a member of Lawrence Welk's orchestra and weekly television show from 1979 to 1982, performing on lead alto saxophone.
He died in New Orleans at the age of 80.
Filmography
Herfurt appeared as a saxophonist in the 1956 film The Nightmare, and plays clarinet on the soundtrack. He also performed on the soundtrack to the 1974 film The Fortune.
Discography
With Glen Gray
Casa Loma in Hi-Fi (Capitol, 1956)
Sounds of the Great Bands! (Capitol, 1958)
Solo Spotlight (Capitol, 1960)
Please Mr. Gray... (Capitol, 1961)
With Billy May
A Band Is Born (Capitol, 1952)
Sorta-May (Capitol, 1955)
The Girls and Boys On Broadway (Capitol, 1960)
With others
Ray Anthony, Ray Anthony Plays Steve Allen (Capitol, 1958)
Georgie Auld, In the Land of Hi-Fi with Georgie Auld and His Orchestra (EmArcy/Mercury, 1956)
Joe "Fingers" Carr, The Riotous, Raucous, Red-Hot 20's! (Warner Bros., 1961)
Frankie Carle, Era: The 30' Music of the Great Bands (Dot, 1968)
Larry Clinton, The Uncollected Larry Clinton and His Orchestra 1937-1938 (Hindsight, 1977)
Ray Conniff, Turn Around Look at Me (CBS, 1968)
Ray Conniff, Concert in Stereo (Columbia, 1970)
Bob Eberly & Helen O'Connell, Recapturing the Excitement of the Jimmy Dorsey Era (Warner Bros., 1961)
Benny Goodman, Hello Benny! (Capitol, 1964)
The Four Freshmen, Four Freshmen and Five Saxes (Capitol, 1957)
Bob Keene, Bob Keene & His Orchestra (Fresh Sound, 1954)
Stan Kenton, Kenton in Hi-Fi (Capitol, 1956)
Pete Rugolo, 10 Saxophones and 2 Basses (Mercury, 1961)
Jack Teagarden, This Is Teagarden! (Capitol, 1956)
References
General references
"Skeets Herfurt," Grove Jazz online
Inline citations
External links
Skeets Herfurt recordings at the Discography of American Historical Recordings.
1911 births
1992 deaths
American jazz clarinetists
American jazz saxophonists
American male saxophonists
Musicians from Ohio
20th-century American musicians
Lawrence Welk
20th-century saxophonists
American male jazz musicians
Earle Spencer Orchestra members
20th-century American male musicians
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| 7,652
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end unit. Sleeps six. Rent directly from owner.
Walk across the street to Story Land!
skiing, snowshoeing, outlet shopping and fine dining.
Full bed in Master, two twins in second bedroom, full sofa bed.
Close to five major alpine ski areas and cross country.
access for small fee in spring, summer and fall.
10 mins to Attitash and Black Mountian, 15 mins to Wildcat.
15 mins to Cranmore. 30 mins to Bretton Woods.
15 mins to No. Conway and outlet shopping, and fine dining.
Security fee, bank check $200, returned after thorough inspection of unit.
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"redpajama_set_name": "RedPajamaC4"
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| 2,453
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{"url":"https:\/\/en-academic.com\/dic.nsf\/enwiki\/935561","text":"# Decoding methods\n\nDecoding methods\n\nIn communication theory and coding theory, decoding is the process of translating received messages into codewords of a given code. There has been many common methods of mapping messages to codewords. These are often used to recover messages sent over a noisy channel, such as a binary symmetric channel.\n\n## Notation\n\nHenceforth, $C \\subset \\mathbb{F}_2^n$ could have been considered a code with the length n; x,y shall be elements of $\\mathbb{F}_2^n$; and d(x,y) would be representing the Hamming distance between x,y. Note that C is not necessarily linear.\n\n## Ideal observer decoding\n\nOne may be given the message $x \\in \\mathbb{F}_2^n$, then ideal observer decoding generates the codeword $y \\in C$. The process results in this solution:\n\n$\\mathbb{P}(y \\mbox{ sent} \\mid x \\mbox{ received})$\n\nFor example, a person can choose the codeword y that is most likely to be received as the message x after transmission.\n\n### Decoding conventions\n\nEach codeword does not have a expected possibility: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must agree ahead of time on a decoding convention. Popular conventions include:\n\n1. Request that the codeword be resent -- automatic repeat-request\n2. Choose any random codeword from the set of most likely codewords which is nearer to that.\n\n## Maximum likelihood decoding\n\nGiven a received codeword $x \\in \\mathbb{F}_2^n$ maximum likelihood decoding picks a codeword $y \\in C$ to maximize:\n\n$\\mathbb{P}(x \\mbox{ received} \\mid y \\mbox{ sent})$\n\ni.e. choose the codeword y that maximizes the probability that x was received, given that y was sent. Note that if all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by Bayes Theorem we have\n\n\\begin{align} \\mathbb{P}(x \\mbox{ received} \\mid y \\mbox{ sent}) & {} = \\frac{ \\mathbb{P}(x \\mbox{ received} , y \\mbox{ sent}) }{\\mathbb{P}(y \\mbox{ sent} )} \\\\ & {} = \\mathbb{P}(y \\mbox{ sent} \\mid x \\mbox{ received}) \\cdot \\frac{\\mathbb{P}(x \\mbox{ received})}{\\mathbb{P}(y \\mbox{ sent})}. \\end{align}\n\nUpon fixing $\\mathbb{P}(x \\mbox{ received})$, x is restructured and $\\mathbb{P}(y \\mbox{ sent})$ is constant as all codewords are equally likely to be sent. Therefore $\\mathbb{P}(x \\mbox{ received} \\mid y \\mbox{ sent})$ is maximised as a function of the variable y precisely when $\\mathbb{P}(y \\mbox{ sent}\\mid x \\mbox{ received} )$ is maximised, and the claim follows.\n\nAs with ideal observer decoding, a convention must be agreed to for non-unique decoding.\n\nThe ML decoding problem can also be modeled as an integer programming problem.[1]\n\n## Minimum distance decoding\n\nGiven a received codeword $x \\in \\mathbb{F}_2^n$, minimum distance decoding picks a codeword $y \\in C$ to minimise the Hamming distance\u00a0:\n\n$d(x,y) = \\# \\{i : x_i \\not = y_i \\}$\n\ni.e. choose the codeword y that is as close as possible to x.\n\nNote that if the probability of error on a discrete memoryless channel p is strictly less than one half, then minimum distance decoding is equivalent to maximum likelihood decoding, since if\n\n$d(x,y) = d,\\,$\n\nthen:\n\n\\begin{align} \\mathbb{P}(y \\mbox{ received} \\mid x \\mbox{ sent}) & {} = (1-p)^{n-d} \\cdot p^d \\\\ & {} = (1-p)^n \\cdot \\left( \\frac{p}{1-p}\\right)^d \\\\ \\end{align}\n\nwhich (since p is less than one half) is maximised by minimising d.\n\nMinimum distance decoding is also known as nearest neighbour decoding. It can be assisted or automated by using a standard array. Minimum distance decoding is a reasonable decoding method when the following conditions are met:\n\n1. The probability p that an error occurs is independent of the position of the symbol\n2. Errors are independent events - an error at one position in the message does not affect other positions\n\nThese assumptions may be reasonable for transmissions over a binary symmetric channel. They may be unreasonable for other media, such as a DVD, where a single scratch on the disk can cause an error in many neighbouring symbols or codewords.\n\nAs with other decoding methods, a convention must be agreed to for non-unique decoding.\n\n## Syndrome decoding\n\nSyndrome decoding is a highly efficient method of decoding a linear code over a noisy channel - i.e. one on which errors are made. In essence, syndrome decoding is minimum distance decoding using a reduced lookup table. It is the linearity of the code which allows for the lookup table to be reduced in size.\n\nThe simplest kind of syndrome decoding is Hamming code.\n\nSuppose that $C\\subset \\mathbb{F}_2^n$ is a linear code of length n and minimum distance d with parity-check matrix H. Then clearly C is capable of correcting up to\n\n$t = \\left\\lfloor\\frac{d-1}{2}\\right\\rfloor$\n\nerrors made by the channel (since if no more than t errors are made then minimum distance decoding will still correctly decode the incorrectly transmitted codeword).\n\nNow suppose that a codeword $x \\in \\mathbb{F}_2^n$ is sent over the channel and the error pattern $e \\in \\mathbb{F}_2^n$ occurs. Then z = x + e is received. Ordinary minimum distance decoding would lookup the vector z in a table of size | C | for the nearest match - i.e. an element (not necessarily unique) $c \\in C$ with\n\n$d(c,z) \\leq d(y,z)$\n\nfor all $y \\in C$. Syndrome decoding takes advantage of the property of the parity matrix that:\n\nHx = 0\n\nfor all $x \\in C$. The syndrome of the received z = x + e is defined to be:\n\nHz = H(x + e) = Hx + He = 0 + He = He\n\nUnder the assumption that no more than t errors were made during transmission, the receiver looks up the value He in a table of size\n\n$\\begin{matrix} \\sum_{i=0}^t \\binom{n}{i} < |C| \\\\ \\end{matrix}$\n\n(for a binary code) against pre-computed values of He for all possible error patterns $e \\in \\mathbb{F}_2^n$. Knowing what e is, it is then trivial to decode x as:\n\nx = ze\n\n## Partial response maximum likelihood\n\nPartial response maximum likelihood (PRML) is a method for converting the weak analog signal from the head of a magnetic disk or tape drive into a digital signal.\n\n## Viterbi decoder\n\nA Viterbi decoder uses the viterbi algorithm for decoding a bitstream that has been encoded using forward error correction based on a convolutional code. The Hamming distance is used as a metric for hard decision viterbi decoders. The squared Euclidean distance is used as a metric for soft decision decoders.\n\n## References\n\n1. ^ \"Using linear programming to Decode Binary linear codes,\" J.Feldman, M.J.Wainwright and D.R.Karger, IEEE Transactions on Information Theory, 51:954-972, March 2005.\n\u2022 Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. ISBN\u00a00-19-853803-0.\n\u2022 Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. ISBN\u00a00-471-08684-3.\n\u2022 J.H. van Lint (1992). Introduction to Coding Theory. GTM. 86 (2nd ed ed.). Springer-Verlag. ISBN\u00a03-540-54894-7.\n\nWikimedia Foundation. 2010.\n\n\u041f\u043e\u043c\u043e\u0436\u0435\u043c \u043d\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u0440\u0435\u0444\u0435\u0440\u0430\u0442\n\n### Look at other dictionaries:\n\n\u2022 Decoding \u2014 Decode redirects here. For the Paramore song, see Decode (song). Decoding is the reverse of encoding, which is the process of transforming information from one format into another. Information about decoding can be found in the following: Digital \u2026 \u00a0 Wikipedia\n\n\u2022 Decoding Reality \u2014 Decoding Reality: The Universe as Quantum Information \u00a0 Front Cover \u2026 \u00a0 Wikipedia\n\n\u2022 Neural decoding \u2014 is a neuroscience related field concerned with the reconstruction of sensory and other stimuli from information that has already been encoded and represented in the brain by networks of neurons. The main goal of studying neural decoding is to\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Algorithmic efficiency \u2014 In computer science, efficiency is used to describe properties of an algorithm relating to how much of various types of resources it consumes. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Claude Berrou \u2014 (born September 23, 1951, Penmarch) is a French professor in electrical engineering at \u00c9cole Nationale Sup\u00e9rieure des T\u00e9l\u00e9communications de Bretagne, now Telecom Bretagne. He is the coinventor with Alain Glavieux in 1991 (and Punya Thitimajshima\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Code \u2014 redirects here. CODE may also refer to Cultural Olympiad Digital Edition. Decoded redirects here. For the television show, see Brad Meltzer s Decoded. For code (computer programming), see source code. For other uses, see Code (disambiguation).\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Standard array \u2014 In coding theory, a standard array (or Slepian array) is a q^{n k} by q^{k} array that lists all elements of a particular mathbb{F} q^n vector space. Standard arrays are used to decode linear codes; i.e. to find the corresponding codeword for the \u2026 \u00a0 Wikipedia\n\n\u2022 FLAC \u2014 redirects here. For anti aircraft fire, see Flak. For other uses, see FLAC (disambiguation). Free Lossless Audio Codec Developer(s) Xiph.Org Foundation, Josh Coalson Initial release \u2026 \u00a0 Wikipedia\n\n\u2022 Free Lossless Audio Codec \u2014 Infobox file format name = Free Lossless Audio Codec icon = extension = .flac mime = audio\/x flac [Registration being sought as audio\/flac] type code = uniform type = magic = owner = genre = Audio container for = contained by = extended from =\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 List of mathematics articles (D) \u2014 NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal\u2026 \u2026 \u00a0 Wikipedia\n\n### Share the article and excerpts\n\n##### Direct link\nDo a right-click on the link above\nand select \u201cCopy\u00a0Link\u201d","date":"2022-09-26 07:11:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 28, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7896544933319092, \"perplexity\": 1879.5467368427048}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334802.16\/warc\/CC-MAIN-20220926051040-20220926081040-00268.warc.gz\"}"}
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Q: continuous bounded function on the bounded open interval ]a,b[ that is not uniformly continuous on ]a,b[ Does someone know a continuous bounded function on the bounded open interval $]a,b[$ that is not uniformly continuous on $]a,b[$.
Thanks in advance
A: $\sin\frac{1}{x}$ on $(0,1)$ is a standard example, I believe.
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College & Research Libraries News (C&RL News) is the official newsmagazine and publication of record of the Association of College & Research Libraries, providing articles on the latest trends and practices affecting academic and research libraries.
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The American Civil War: A collection of free online primary sources
2018 top trends in academic libraries: A review of the trends and issues affecting academic libraries in higher education
Home > Vol 73, No 7 (2012) > Free
David Free
MSU to host Grant Presidential Library
Mississippi State University (MSU) soon will officially serve as host to a presidential library—one of only five universities in the nation to share such a distinction. In late May, Ulysses S. Grant Association President Frank J. Williams announced the decision of the organization's Board of Directors to designate the Ulysses S. Grant Collection at MSU's Mitchell Memorial Library as the Ulysses S. Grant Presidential Library.
The Grant Presidential Collection consists of some 15,000 linear feet of correspondence, research notes, artifacts, photographs, scrapbooks, and memorabilia and includes information on Grant's childhood from his birth in 1822, his later military career, Civil War triumphs, tenure as commanding general after the war, presidency, and his post-White House years until his death in 1885. There are also 4,000 published monographs on various aspects of Grant's life and times.
The Grant Collection joined those of former U.S Sen. John C. Stennis and U.S. Rep. Sonny Montgomery, along with a number of other more contemporary political figures, in the library's Congressional and Political Research Center. For more information about the Ulysses S. Grant Collection, visit http://library.msstate.edu/USGrant.
Oxford University Press launches roaming support for mobile journals users
Oxford University Press (OUP) recently announced that users of its mobile-optimized journals service can now authorize their mobile devices for offsite access to institutional subscriptions.
While connected to the institution's network, by choosing "Authorize this Device" on any journal's mobile homepage, institutional users can connect their mobile device to their institution's Oxford Journals subscription, thereby granting access to protected content even after stepping out of the institution's network. Each mobile device must be authorized individually; the level of access granted will be exactly the same as the institution's subscription; and, once activated, the voucher will provide access to journal content for six months.
The new service builds on the 2011 launch of mobile-optimized across OUP's journal Web sites, which can be accessed from any smartphone. For more information, visit the FAQ page at www.oxfordjournals.org/mobile_faqs.html.
Queen Victoria's journals now online
The Bodleian Libraries working in partnership with The Royal Archives and ProQuest, have made the personal journals of Queen Victoria available for the public. The journals, which span Victoria's lifetime and consist of 141 volumes numbering more than 43,000 pages, have never been published in their entirety and previously were only accessible by appointment at the Royal Archives at Windsor Castle. In addition to autograph diaries begun by the youthful Princess Victoria, there are edited versions from her later years, redacted and transcribed by the Queen's daughter, Princess Beatrice.
Queen Victoria was a prolific writer and recorded her thoughts and experiences almost daily, starting with her first entry as a young girl of 13 and continuing until just weeks before her death in 1901. Her journals provide a fascinating insight into her life as Queen. ProQuest launched a library version of the journals in late June 2012.
LYRASIS members elect new Board of Trustees members
LYRASIS members elected four new Board of Trustees who will each serve a three-year term. The new Board members started their terms at the beginning of the LYRASIS fiscal year on July 1, 2012. Ann Watson, dean of the library at Shepherd University, was elected to the academic library position. Cynthia Henderson, executive director of the Louis Stokes Health Sciences Library at Howard University, and Kathlin Ray, dean of libraries, University of Nevada-Reno, were elected to the two at-large positions. Sandra Treadway, librarian of Virginia, was elected to the group member position. Additionally, John Arnold, president of Linkage Systems, was selected by the nominating committee and appointed by the Board of Trustees, to serve a second three-year term as non-member trustee.
EBSCO launches Associated Press Images Collection
EBSCO will make millions of images from the Associated Press (AP) available to library customers. EBSCO has been named the sole library distributor of AP Images, the commercial photo unit of AP and one of the world's largest collections of historical and contemporary imagery, including more than 12 million photographs dating back more than 100 years.
AP Images Collection is a primary source database with millions of images from 1826 to present and more than 3,500 additional photographs are added daily. The collection also includes 36,400 audio sound bytes dating from the 1920s and more than 2,900 multimedia files. The multimedia represent a compilation of integrated video, graphics, and photos that bring the latest news and educational topics to life. In addition, AP Images Collection provides 2.7 million AP news stories from 1997 to present, and a professionally produced collection of more than 340,000 maps, graphs, charts, logos, flags, and illustrations. The archive is currently available at apimages. com but will eventually migrated to EBSCOhost.
UNT Portal to Texas History
Users of the University of North Texas (UNT) Libraries' Portal to Texas History are now able to browse pages of historical Texas newspapers online, thanks to the National Digital Newspaper Program, "Chronicling America: Historic American Newspapers." UNT Libraries became a partner in the program in 2007, when it received the first of several National Endowment for the Humanities grants to digitize the newspapers and place them online. The hundreds of thousands of pages date from the 1820s through the 2000s, offering glimpses of daily life in Texas communities from those decades. More information is available at http://texashistory.unt.edu/explore/collections/TDNP/.
HKBU Library wins ALA innovative international library projects award
Hong Kong Baptist University (HKBU) Library recently received the 2012 ALA Presidential Citations for Innovative International Library Projects Award for its Chinese Medicine Digital Project. This year, the HKBU Library is one of four libraries in the world, and also the sole university library, to have been chosen by the selection panel for the award.
The Medicinal Plants Images Database (www.hkbu.edu.hk/lib/electronic/libdbs/mpd/index.html) consists of more than 1,000 medicinal plants presented in the form of images with detailed descriptions in both Chinese and English. The Chinese Medicine Specimen Database (www.hkbu.edu.hk/lib/electronic/libdbs/scm_specimen.html) was established to promote awareness of the variety, authenticity, and effective use of Chinese medicine. Both databases are bilingual (in Chinese and English) and can be accessed free of cost by anyone anywhere in the world.
ACRL seeks EDUCASE Learning Initiative liaison
The ACRL Information Literacy Coordinating Committee is currently seeking applications to serve a three-year term (August 2012 to June 2015) as the ACRL liaison to the EDUCAUSE Learning Initiative (ELI). The liaison is responsible for outreach, education, and communication between ACRL and ELI. The goal of the liaison relationship is twofold: 1) to demonstrate the value of libraries and librarians for advancing the mission of ELI and 2) to model effective partnerships between librarians and other teaching and learning professionals within the ELI community.
The ACRL liaison to ELI is expected to communicate opportunities for ELI participation to relevant ACRL audiences and to assist ELI staff in identifying content experts and innovative programs within the ACRL community for possible inclusion in ELI publications and event opportunities. Additional opportunities for participation aligned with the liaison's interests and areas of expertise may be identified in conversation with ELI leadership.
Complete details and application information is available on the ACRL Insider blog at www.acrl.ala.org/acrlinsider/archives/category/liaisons.
New ACRL reports address value of academic libraries, scenarios for the future of the book
In early June, ACRL released a new white paper, "Connect, Collaborate, and Communicate: A Report from the Value of Academic Libraries Summits," which reports on two invitational summits supported by a National Leadership Collaborative Planning Grant from the Institute of Museum and Library Services.
As part of ACRL's Value of Academic Libraries Initiative, a multiyear project designed to assist academic librarians in demonstrating library value, ACRL joined with three partners—the Association for Institutional Research, Association of Public and Land-grant Universities, and the Council of Independent Colleges—to sponsor two national summits held November 29–December 1, 2011. The summits convened senior librarians, chief academic administrators, and institutional researchers from 22 postsecondary institutions for discussions about library impact.
Fifteen representatives from higher education organizations, associations, and accreditation bodies also participated in the summit discussions and presentations and facilitated small group work.
The report—coauthored by Karen Brown, associate professor at Dominican University, and ACRL Senior Strategist for Special Initiatives Kara Malenfant—summarizes broad themes about the dynamic nature of higher education assessment that emerged from the summits. From these themes, the report presents five recommendations for the library profession.
"ACRL's 'Plan for Excellence' identifies the value of academic libraries as a top priority for the association, and results just in from the 2012 membership survey show that demonstrating library relevance is the top issue of concern for our members," added Joyce L. Ogburn, ACRL president and university librarian and director of the University of Utah Marriott Library. "ACRL has already taken steps to continue this crucial work by submitting a grant proposal to design, implement, and evaluate a team-based professional development program to strengthen the competencies of librarians in campus leadership and data-informed advocacy."
The white paper is freely available on the ACRL Value of Academic Libraries Web site at www.acrl.ala.org/value.
ACRL additionally released a research report, "Futures Thinking for Academic Librarians: Scenarios for the Future of the Book," to help librarians reexamine their assumptions, which may be grounded in the current e-book zeitgeist. Authored by David J. Staley, director of the Harvey Goldberg Center for Excellence in Teaching in the History Department of Ohio State University, the report is a companion to the 2010 report Staley coauthored for ACRL, "Futures Thinking for Academic Librarians: Higher Education in 2025."
This new report presents four scenarios, based in part on feedback from academic library directors. It includes scenarios that intentionally favor the continued existence of the printed book as a viable technology, so that academic and research librarians may expand their thinking about the future to include a richer set of environmental conditions.
The full report is freely available in the publications section of the ACRL Web site at www.ala.org/acrl/issues/whitepapers.
New publications from ACRL and Choice
ACRL announces the publication of The Changing Academic Library, Second Edition: Operations, Cultures, and Environments by John M. Budd. The Changing Academic Library, Second Edition is number 65 in the ACRL Publications in Librarianship series.
The Changing Academic Library, Second Edition is a revised, enhanced, and updated edition of Budd's 2005 book The Changing Academic Library. This book has been completely updated and revised to reflect the dynamic states of higher education and academic libraries.
Budd presents a critical examination of major issues facing colleges and universities and the unique challenges that their libraries must come to grips with. Current practice is reviewed, but it is examined in the broader context of educational needs, scholarly communication, politics and economics, technology, and the nature of complex organizations. The book may be used as a text in library and information science courses, as well as an introduction for new professionals and academic administrators.
The Changing Academic Library, Second Edition is available for purchase in print, as an e-book, and as a print/e-book bundle through the ALA Online Store; in print and for Kindle through Amazon.com; and by telephone order at (866) 746-7252 in the United States or (770) 442-8633 for international customers.
Also available from Choice is Choice's Outstanding Academic Titles 2007–2011, edited by Choice Humanities Editor Rebecca Ann Bartlett. Honoring the best in scholarly publishing, Choice's annual "Outstanding Academic Titles" list is widely recognized in the academic community for its sweeping coverage of the most significant scholarly titles published each year. Now available is a comprehensive print collection of reviews representing excellence in a five-year period.
Choice's Outstanding Academic Titles 2007–2011 presents reviews of 3,650 print and electronic works in a wide variety of categories that Choice has deemed indispensable for every library. It is the perfect tool to help libraries remain current and to assist in their making informed choices for acquisition.
Choice's Outstanding Academic Titles 2007–2011 is available for purchase in print through the ALA Online Store and by telephone order at (866) 746-7252 in the United States or (770) 442-8633 for international customers.
Tech Bits . . .
Brought to you by the ACRL ULS Technology in University Libraries Committee
It's so easy to take pictures and screenshots with your mobile device, but it's not so easy to point out a specific detail on these images—until now! Skitch is a free mobile app that lets you quickly annotate and share your pictures. Available for iPad, Android, or Mac devices, Skitch lets you draw, insert shapes and arrows, or add notes on your photos. It also has built-in features that enable you to capture an image while surfing the Web or using a Google map. Once marked up, you can share your images in a variety of ways, including sending a copy to an Evernote account, which is another great app for staying organized.
—Michelle Armstrong, Albertsons Library, Boise State University
. . . SKITCH
skitch.com
Copyright 2012© American Library Association
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SHOCK: TIME Magazine article admits how the 2020 election was "fortified" against President Trump
In a shocking twist of events, TIME Magazine released an article detailing how the 2020 election was rigged against President Donald Trump.
(Article by Collin Rugg republished from TrendingPolitics.com)
The article, written by leftist Molly Ball, explains how a secret cabal composed of rich connected elites joined forces to destroy President Trump.
This is a confession https://t.co/5e20vjW8WE
— Jack Posobiec 🇺🇸 (@JackPosobiec) February 5, 2021
I just want to say that it is the opposite of Democracy when a secret cabal of wealthy and politically connected elites conspire to manipulate the rules and laws of an election in order to win
But shit, not like this is a new storyhttps://t.co/RvTT0Kx8IX
— Tim Pool (@Timcast) February 5, 2021
For the "fact checkers" reading through this story, here is the definition of "rigged" defined by Merriam-Webster:
Rigging as explained by Merriam-Webster. Tale told in three parts. #riggedelection pic.twitter.com/ey4gfOLmjY
— Chase Kuertz (@chasekuertz) February 5, 2021
Check out TIME Magazine's explanation of how the 2020 election was rigged against President Trump:
There was a conspiracy unfolding behind the scenes, one that both curtailed the protests and coordinated the resistance from CEOs. Both surprises were the result of an informal alliance between left-wing activists and business titans. The pact was formalized in a terse, little-noticed joint statement of the U.S. Chamber of Commerce and AFL-CIO published on Election Day. Both sides would come to see it as a sort of implicit bargain–inspired by the summer's massive, sometimes destructive racial-justice protests–in which the forces of labor came together with the forces of capital to keep the peace and oppose Trump's assault on democracy.
The handshake between business and labor was just one component of a vast, cross-partisan campaign to protect the election–an extraordinary shadow effort dedicated not to winning the vote but to ensuring it would be free and fair, credible and uncorrupted. For more than a year, a loosely organized coalition of operatives scrambled to shore up America's institutions as they came under simultaneous attack from a remorseless pandemic and an autocratically inclined President. Though much of this activity took place on the left, it was separate from the Biden campaign and crossed ideological lines, with crucial contributions by nonpartisan and conservative actors. The scenario the shadow campaigners were desperate to stop was not a Trump victory. It was an election so calamitous that no result could be discerned at all, a failure of the central act of democratic self-governance that has been a hallmark of America since its founding.
Their work touched every aspect of the election. They got states to change voting systems and laws and helped secure hundreds of millions in public and private funding. They fended off voter-suppression lawsuits, recruited armies of poll workers and got millions of people to vote by mail for the first time. They successfully pressured social media companies to take a harder line against disinformation and used data-driven strategies to fight viral smears. They executed national public-awareness campaigns that helped Americans understand how the vote count would unfold over days or weeks, preventing Trump's conspiracy theories and false claims of victory from getting more traction. After Election Day, they monitored every pressure point to ensure that Trump could not overturn the result. "The untold story of the election is the thousands of people of both parties who accomplished the triumph of American democracy at its very foundation," says Norm Eisen, a prominent lawyer and former Obama Administration official who recruited Republicans and Democrats to the board of the Voter Protection Program.
Read more at: TrendingPolitics.com and Conspiracy.news.
Tagged Under: conspiracy, deception, democracy, Donald Trump, election, fraud, Left-wing, politics, rigged, Time Magazine
Biden's advisers have deep, dangerous, ties to Chinese Communist and military officials: Report
If President Donald Trump had not won the 2020 Election, the Dems wouldn't be impeaching him
MUST READ: Democrats were ONLY able to "win" in 2020 by breaking chain of custody laws in EVERY SWING STATE
The second sham impeachment trial against Trump has begun and Democrats have already beclowned themselves with fake allegations
WATCH: Bill Barr blindsided Trump… was behind everything…
Hundreds of prominent Leftists now demand that anyone who supported Trump never be allowed to speak publicly again
Trump legal team to show clips of Dems encouraging violence in impeachment defense
Don't impeach Trump. Impeach the deep state for its conspiracy to kill the constitution
Several election rigging cases are still moving forward to Supreme Court
Jeep hires Trump-bashing Springsteen as face of unity, said he would leave USA if Trump won, spent entire career bashing America's conservatives
When everything is attributed to 'white privilege,' it ceases to have meaning
TIME magazine ADMITS the election was rigged against President Trump; when does Biden's impeachment begin?
OUTRAGE: Biden picks former colleague of son Hunter Biden's lawyer to run DoJ's "criminal justice" division
Data expert: 'We can prove' 2020 election fraud 'fairly conclusively'
FBI visited home of Trump supporter who was working in DC, but did not even attend the rally
New video footage shows tens of thousands of illegal ballots arriving at TCF Center in Detroit after hours
Deep state boasts of "conspiracy" against Trump in 2020 election
Here are the 41 celebrities who supported The Lincoln Project
BREAKING: Democrat brief claims U.S. election results never rejected before – that's a MASSIVE lie
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Guerra e pace è un cortometraggio documentario diretto dal regista Luciano Emmer.
Collegamenti esterni
Film documentari italiani
Film su Pablo Picasso
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Imagine having a secure virtual place that enables you to travel in the future to communicate and share everything that you wish with the people you love: letters, poems, stories, video-messages, audio-messages etc.
Imagine being able to act in space and time, at you leisure.
Perhaps you wish to dedicate a video-message to your grandchild on his 50th birthday?
Or you may wish to receive, on your 25 years wedding anniversary, the photos that other people are shooting today while you are getting married.
The first step is to create a free profile on Eternia. Then you can recreate your relations by contacting the people you love. Then you will choose your trustees, who will be responsible to release your profile in the future, to share its contents and deliver the messages that you have prepared. Lastly you can create the messages that you wish to send, in the form of texts, photos and videos and plan, at you leisure, the delivery dates and methods. The basic profile is free for ever, and can be extended according to your needs. There are paying profiles, with credits system, which provide more services and/or features.
You can plan actions that will take place in the future. You will also receive messages that someone has sent to you years before, according to a set date, in correspondence on how you feel or on the occasion of an important event such as your birthday.
You will have a safe space to transfer your passwords, to declare your last wills or to explain in detail how to take care of your animals etc.
You can reveal secrets that you have preferred not to share when alive. In the same way you will have the chance to tell experiences and fantasies that you wouldn't ever have told in person.
You can do all this in absolute safety, because the portal will ensure that all your data will be kept by an advanced security system. They will remain your property and will never be sold to third parties.
The creation of your profile is completely free and it is the first step to generate your eternal virtual identity. In the choice of the contents, select your most significant photos and videos and use your diary pages as if they were ancient sheets, precious, on which to record the essence of all that you wish to share.
In the choice of your representation sentence, we also invite you to choose a sentence, an aphorism that represent yourself or that expresses in summary what you want to communicate to others.
The second step will be recreating your most important relations on Eternia, to be able to share your experiences with your loved ones.
Contact people with whom you wish to share your wonderful journey called life.
The third step consists in defining, amongst the people you trust, three trustees. Select them with the highest conscience because they will be your guarantors in the future, and will be the only ones who will have the ability to unlock your profile.
Once you have done these fundamental steps, to make your eternal virtual identity effective, you can start to prepare your messages in the form of texts, audio, images or video. We have called FuEL (Future Elements).
The FuEL, will empower you to dedicate everything that you wish to the people you love.
In addition to an area for the creation of the FuEL, you will have an area that will enable you to view the FuEL that others have dedicated to you and to monitor those that you have already sent.
This is where to declare your last wishes in terms of digital testament (password storage, banking codes, secret codes, etc.) and to indicate the recipients to whom send them.
In this space you can declare your last wills, both in terms of biological testament, in the strict sense, (the so-called early declaration of treatment), and in terms of how you wish that your assets, documents, personal spaces must be treated.
Dreams, stories, and poetry. Fantasies, desires and reflections. Or thoughts, emotions and sensations that for modesty, shame, or fear of judgment, you have never shared, even if you'd like to do so.
Here is another space to reveal yourself.
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
Duality and self-duality are very useful and powerful tools that allow to analyze properties of a complicated system in terms
of a simpler one. In case of self-duality for particle systems, the dual system is the same and the simplification arises because in the dual
one considers only a finite number of particles (e.g.\ \cite{demasi}).\\
Several methods are available to construct dual processes and duality relations. In the context of population dynamics,
the starting point is always to consider as dual the backward coalescent process (for a general overview, see \cite{dawson}). In the context of particle systems, the algebraic method
developed in \cite{gkrv} offers a framework to construct duality functions via symmetries of the generator and reversible measures.\\
However, a complete picture of how to obtain \emph{all} duality relations is missing.
In this latter context of interacting particle systems, natural questions of the same sort are:
which particle systems allow self-duality and is it possible to obtain all factorized duality functions for these systems?\\
One of the useful applications of disposing of all duality functions is that, depending on the target, one can choose
appropriate ones: e.g.\ in the hydrodynamic limit and the study of the structure of the stationary measures, the ``classical'' duality functions are the appropriate ones (see e.g.\ \cite{demasi}), whereas in the study of (stationary and non-stationary) fluctuation fields and associated Boltzmann-Gibbs principles (\cite{landim}, chapter 11), as well as in the study of speed of relaxation to equilibrium in $L^2$ or in the study of perturbation theory around models with duality (cf.\ \cite{demasi}), ``orthogonal'' duality functions turn out to be very useful.
In this paper, we develop an approach to answer the above questions and systematically determine all duality functions and relations for some interacting particle and diffusion systems. In this route, starting from examples, we first investigate a general connection between stationary product measures and factorized duality functions. This shows, in particular, that for infinite systems with factorized self-duality functions, the only stationary measures which are ergodic (w.r.t.\ either space-translation or time) are in fact product measures. \\
Then we use this connection between stationary product measures and duality functions to recover all possible factorized duality functions from the stationary product measures. More precisely, we show that, given the first duality function, i.e.\ the duality function with a single dual particle, all other duality functions are determined. This provides a simple machinery to obtain all duality functions in processes such as Symmetric Exclusion Process ($\SEP$), Symmetric Inclusion Process ($\SIP$) and
Independent Random Walkers ($\IRW$). In particular, we recover via this method all orthogonal polynomial duality functions obtained in \cite{chiara}.\\
Moreover, we prove that in the context of conservative particle systems where the rates for particle hopping depend only on the number of particles of the departure and arrival sites, the processes $\SEP$, $\SIP$ and $\IRW$ are the only systems which have self-duality with factorized self-duality functions and that
the first duality function is necessarily an affine function of the number of particles.\\
Next, in order to prove that the ``possible'' duality functions derived via the method described above are indeed duality functions, we develop a method based on generating functions. This method, via an intertwining relation, allows to go from discrete systems (particle hopping dynamics) to continuous systems (such as diffusion processes or deterministic dynamics) and back, and also allows to pass from self-duality to duality and back. The proof of a self-duality relation in a discrete system then reduces to the same property in a continuous system, which is much easier to check directly.\\
The generating function method also provides new examples of self-duality for processes in the continuum such as the Brownian Energy Process ($\BEP$), which intertwines with the $\SIP$ via the generating function. In fact, we show equivalence between self-duality of $\SIP$, duality between $\SIP$ and $\BEP$ and self-duality of $\BEP$.
Finally, this method based on generating functions generalizes the concept of obtaining dualities from symmetries to intertwinings, being a symmetry an intertwining of the generator with itself.
The paper is organized as follows. In Section \ref{section setting} we introduce the basic definitions of duality and systems considered. Additionally, in Theorem \ref{gzrprop} we prove which particle systems out of those considered admit factorized self-duality.\\
In Section \ref{section general relation}, we investigate a general relation between factorized duality functions and stationary product measures. We treat separately the finite and infinite contexts in which this relation arises; in the latter case, we exploit this connection to draw some conclusions on the product structure of ergodic measures. \\
Section \ref{section general construction} is devoted to the derivation of all possible factorized self-duality and duality functions. Here Theorem \ref{gzrprop} and the relation in the previous section are the two key ingredients.\\
In Section \ref{section generating}, after an introductory example and a brief introduction on the general connection between duality and intertwining relations, we establish an intertwining between the discrete and the continuum processes. This intertwining relation is then used to produce all the self-duality functions for the Brownian Energy Process.
\section{Setting}\label{section setting}
We start defining what we mean by \emph{duality} of stochastic processes. Then, we introduce a general class of Markov interacting particle systems with associated interacting diffusion systems.
\subsection{Duality}\label{section setting duality}
Given two state spaces $\Omega$ and $\hat \Omega$ and \emph{two stochastic processes} $\{\xi(t),\ t\geq 0\}$ and $\{\eta(t),\ t\geq 0\}$ evolving on them, we say that they are \emph{dual} with duality function $D: \hat \Omega \times \Omega \to \R$ (where $D$ is a measurable function)
if, for all $t>0$, $\xi \in \hat \Omega$ and $\eta \in \Omega$,
we have the so-called duality relation
\be\label{dualrel}
\hat \E_\xi D(\xi(t),\eta)= \E_\eta D(\xi,\eta(t))\ .
\ee
If the laws of the two processes coincide, we speak about \emph{self-duality}.\\
More generally, we say that \emph{two semigroups} $\{S(t),\ t \geq 0\}$
and $\{ \hat{S}(t),\ t\geq 0\}$
are dual
with duality function $D$ if, for all $t \geq 0$,
\[
(\hat{S}(t))_{\eft} D= (S(t))_{\ight}D\ ,
\]
where \textquotedblleft left\textquotedblright\ (resp.\ \textquotedblleft right\textquotedblright) refers to action on the left (resp.\ right) variable.
Even more generally, we say that \emph{two operators} $L$ and $\hat{L}$ are dual to each
other
with duality function $D$ if
\be\label{opdual}
(\hat{L})_{\eft} D= (L)_{\ight}D\ .
\ee
In order not to overload notation, we use the expression $A_{\eft} D(\xi,\eta)$ for $(A D(\cdot, \eta))(\xi)$ and,
similarly, $B_{\ight} D(\xi,\eta)= (BD(\xi,\cdot))(\eta)$. We will often write $D(\xi,\eta)$
in place of $(\xi,\eta)\mapsto D(\xi,\eta)$.
\subsection{The lattice and the factorization over sites}\label{section setting lattice} The underlying geometry of all systems that we will look at consists of a set of sites $V$ either finite or $V=\Zd$. Moreover we are given a family of transition rates $p: V \times V \to \R_+$, satisfying the following conditions: for all $x$, $y \in V$,
\begin{enumerate}
\item[(1)] \emph{Vanishing diagonal}: $ p(x,x)=0\ ,$
\item[(2)] \emph{Irreducibility}: there exist $x_1=x$, $x_2$, $\ldots$, $x_m=y$ such that $\prod_{l=1}^{m-1} p(x_l, x_{l+1} ) > 0\ .$
\end{enumerate}
In case of infinite $V$, we further require the following:
\begin{enumerate}
\item[(3)] \emph{Finite-range}: there exists $R> 0$ such that, for all $x,y \in V$, $p(x,y)=0$ if $|x-y|>R\ ,$
\item[(4)] \emph{Uniform bound on total jump rate}: $\sup_{x \in V} \sum_{y \in V} p(x,y) < \infty\ .$
\end{enumerate}
Notice that when $p$ is finite-range and translation invariant, then the uniform bound on total jump rate follows automatically.
To each site $x \in V$ we associate a variable $\eta_x \in E= \N$, $\{0,\ldots,N\}$ or $\R_+$, with the interpretation of either the number of particles or the amount of energy associated to the site $x$. Configurations are denoted by $\eta \in \Omega=E^V$.
\
In all the examples that we will be discussing here, the duality functions \emph{factorize over sites}, i.e.
\be\label{factoo}
D(\xi,\eta)= \prod_{x\in V} d(\xi_x,\eta_x)\ ,\quad \eta \in E^V,\quad \xi \in \hat E^V\ .
\ee
We then call the functions $d(\xi_x,\eta_x)$ the \emph{singe-site duality functions} and further assume
\beq\label{Da}
d(0,\cdot)\equiv 1\ .
\eeq
The above condition \eqref{Da} is related to the fact that we want to have duality functions which make sense for infinite systems when the dual configuration has a finite total mass. A typical example is when $\eta$, $\xi \in \N^V$, where $\eta$ is an infinite configuration while $\xi$ is a finite configuration, so that in the product \eqref{factoo} there are only a finite number of factors different from $d(0, \eta_x)$. In this sense, the choice $d(0,\cdot)\equiv 1$ is the only sensible one for infinite systems.\\
When $V$ is finite and $E = \N$ or $\{0,\ldots,N \}$, this condition is not necessary and e.g.\ if a reversible product measure $\mu = \otimes_{x \in V}\ \nu$ exists, then the so-called \emph{cheap self-duality function}
$D(\xi,\eta)=\tfrac{1}{\mu(\xi)} \1{\{\xi=\eta\}} = \prod_{x \in V} \frac{1}{\nu(\xi_x)} \1\{\xi_x = \eta_x \}$ does \emph{not} satisfy \eqref{Da}.
\subsection{Interacting particle systems with factorized self-duality} \label{section setting particle}
The class of interacting particle systems we consider is described by the following infinitesimal generator acting on local functions $f : \Omega \to \R$ as follows:
\beq \label{gzrpgen}
L f(\eta) &=& \sum_{x,y \in V} p(x,y) L_{x,y} f(\eta)\ ,\quad \eta \in \Omega\ ,
\eeq
where $L_{x,y}$, the \emph{single-edge generator}, is defined as
\beq \label{single-edge L}
L_{x,y}f(\eta)&=& u(\eta_x) v(\eta_y) (f(\eta^{x,y})-f(\eta)) + u(\eta_y)v(\eta_x)(f(\eta^{y,x})-f(\eta))\ ,\quad \eta \in \Omega\ ,
\eeq
and $\eta^{x, y}$ denotes the configuration arising from $\eta$ by removing one particle at $x$ and putting it at $y$, i.e.\ $\eta^{x,y}_x= \eta_x-1$, $\eta^{x,y}_y= \eta_y+1$, while $\eta^{x,y}_z= \eta_z$ if $z \neq x,y$. Note the conservative nature of the system and the form of the particle jump rates in \eqref{single-edge L} which depend on the number of particles in the departure and arrival site in a factorized form. Minimal requirements on the functions $u$ and $v$, namely
\begin{itemize}
\item[(i)] $u(0)=0$, $u(1)=1$ and $u(n) > 0$ for all $n > 0\ ,$
\item[(ii)] $v(0)\neq 0$ and $v(N)=0$ if $E=\{0,\ldots,N \}$ and in all other cases $v(n)>0\ $,
\end{itemize}
guarantee the existence of a one-parameter family of \emph{stationary} (actually \emph{reversible}) \emph{product measures} $\{\otimes\ \nu_\lambda,\ \lambda > 0 \}$ with marginals $\nu_\lambda$ given by
\beq \label{marginals}
\nu_\lambda(n) &=& \varphi(n) \frac{\lambda^n}{n!} \frac{1}{Z_\lambda}\ ,\quad n \in \N\ \text{or}\ \{0,\ldots,N\}\ ,
\eeq
for all $\lambda > 0$ for which the normalizing constant $Z_\lambda < \infty$ and with $\varphi(n)=n! \prod_{m=1}^{n}\frac{v(m-1)}{u(m)}$.
\subsection{Basic examples}\label{secex}
We recall here the basic examples of self-dual interacting particle systems and corresponding factorized self-duality functions known in literature (cf.\ e.g.\ \cite{gkrv}).
\begin{itemize}
\item[(I)]\textbf{Independent random walkers ($\IRW$)}
\begin{itemize}
\item $E=\N\ ,$
\item $u(n)=n\ ,$ $v(n)=1\ ,$
\item $\nu_\lambda \sim \text{Poisson}(\lambda)\ ,$ $\nu_\lambda(n)=\tfrac{\lambda^n}{n!} e^{-\lambda}\ ,\ \lambda > 0\ ,$
\item $d(k,n)= \frac{n!}{(n-k)!} \1\{k \leq n \}\ .$
\end{itemize}
\item[(II)] \textbf{Symmetric inclusion process ($\SIP(\alpha)$, $\alpha > 0$)}
\begin{itemize}
\item $E=\N\ ,$
\item $u(n)=n\ ,$\ $v(n)=\alpha+n\ ,$
\item $\nu_\lambda \sim \text{Gamma}_{\text{d}}(\alpha,\lambda)\ ,$ $\nu_\lambda(n)= \tfrac{\Gamma(\alpha+n)}{\Gamma(\alpha)} \tfrac{\lambda^n}{n!}(1-\lambda)^\alpha\ ,\ \lambda \in (0,1)\ ,$
\item $d(k,n)= \tfrac{\Gamma(\alpha)}{\Gamma(\alpha+k)}\tfrac{n!}{(n-k)!} \1\{k \leq n \}\ .$
\end{itemize}
\item[(III)] \textbf{Symmetric exclusion process ($\SEP(\gamma)$, $\gamma \in \N$)}
\begin{itemize}
\item $E=\{0,\ldots,N \}\ ,$
\item $u(n)=n\ ,$\ $v(n)=\gamma-n\ ,$
\item $\nu_\lambda \sim \text{Binomial}(\gamma,\tfrac{\lambda}{1+\lambda})$, $\nu_\lambda(n)= \tfrac{\gamma!}{(\gamma-n)!}\tfrac{\lambda^n}{n!} \left(\tfrac{1}{1+\lambda}\right)^\gamma\ ,\ \lambda>0\ ,$
\item $d(k,n)= \tfrac{(\gamma-k)!}{\gamma!}\tfrac{n!}{(n-k)!} \1\{k \leq n \}\ .$
\end{itemize}
\end{itemize}
\label{section factorizable processes}
In the following theorem we show that the only processes with generator of the type \eqref{gzrpgen} which have non-trivial factorized self-duality functions
are of one of the types described in the examples above, i.e.\ $\IRW$, $\SIP$ or $\SEP$. Here by \textquotedblleft non-trivial\textquotedblright\ we mean that the first single-site self-duality function
$d(1,n)$ is not a constant (as a function of $n$).
\bt\label{gzrprop}
Assume that the process with generator \eqref{single-edge L} is self-dual with factorized self-duality function $D(\xi,\eta)=\prod d(\xi_x,\eta_x)$ in the form \eqref{factoo} with $d(0,\cdot)\equiv 1$ as in \eqref{Da}.
If $d(1,n)$ is not constant as a function of $n$,
then
\beq\label{didi}
u(n) &=& n\ \nonumber
\\
v(n) &=& v(0)+ (v(1)-v(0))n\ ,
\eeq
and the first single-site self-duality function is of the form
\beq \label{singdual1}
d(1,n)=a + b n\ ,
\eeq
for some $a \in \R$ and $b \neq 0$.
\et
\bpr
Using the self-duality relation for $\xi_x = 1$ and no particles elsewhere, together with $u(0)=0$, we obtain the identity
\beq \nonumber
&& u(\eta_x)v(\eta_y) (d(1,\eta_x-1)- d(1,\eta_x)) + u(\eta_y)v(\eta_x) (d(1,\eta_x+1)-d(1,\eta_x))\\
\label{ideg}
&&=\ p(x,y) u(1)v(0)( d(1,\eta_y)-d(1,\eta_x))
\ .
\eeq
Setting $\eta_x=\eta_y=n \geq 1$, this yields anytime $u(n)v(n) \neq 0$
\beq\label{harmin}
d(1,n+1)+d(1,n-1)-2 d(1,n)=0\ ,
\eeq
from which we derive $d(1,n)= a+ bn$. Because $d(1,n)$ is not constant as a function of $n$, we must have $b\not=0$. Inserting
$d(1,n)= a+ bn$ in \eqref{ideg} we obtain
\[
u(\eta_x)v(\eta_y)-u(\eta_y)v(\eta_x)= -u(1)v(0)(\eta_y-\eta_x)\ ,
\]
from which, by setting $\eta_x=n$ and $\eta_y=0$ we obtain the first in \eqref{didi}, while via $\eta_x=n$ and $\eta_y=1$ we get the second condition.
\epr
\begin{remark}
More generally, if we replace \eqref{Da} with $d(0,n)\neq 0 $ in the above statement, we analogously obtain \eqref{didi} and
\beq\nonumber \label{relation2}
d(0,n) &=& c^n\\
d(1,n) &=& (a+bn) \cdot c^n\ ,
\eeq
for some constants $a, b, c \in \R$, $b, c \neq 0$.
\end{remark}
\subsection{Interacting diffusion systems as scaling limits}\label{section setting diffusion}
Interacting diffusion systems arise as scaling limits of the particle systems in Section \ref{secex} above (cf.\ \cite{gkrv}). More in details, by \textquotedblleft scaling limit\textquotedblright\ we refer to the limit process of the particle systems $\{\tfrac{1}{N}\eta^N(t), \ t \geq 0 \}_{N \in \N}$, where the initial conditions $\tfrac{1}{N}\eta^N(0)= \tfrac{1}{N}(\lfloor z_x N \rfloor )_{x \in V}$ converge to some $z \in E^V$, with $E = \R_+$.
In case of $\IRW$, one obtains a deterministic (hence degenerate diffusion) process $\{z(t),\ t \geq 0 \}$ whose evolution is described by a first-order differential operator. In case of $\SIP(\alpha)$, the scaling limit is a proper Markov process of interacting diffusions known as Brownian Energy Process ($\BEP(\alpha)$) (cf.\ \cite{gkrv}).
For the $\SEP(\gamma)$, this limit cannot be taken in the sense of Markov processes, but we can extend the SEP generator to
a larger class of functions defined on a larger configuration space and take the many-particle limit.
The limiting second-order differential operator is then not a Markov generator, but still a second order differential operator. We will explain this more in detail below.
The limiting differential operators in the case of $\IRW$ and $\SIP(\alpha)$ can be described as acting on smooth functions $f : E^V \to \R$ as follows:
\beq \label{generators diffusions}
\caL f(z) &=& \sum_{x,y \in V} p(x,y) \caL_{x,y} f(z)\ ,\quad z \in E^V\ ,
\eeq
with single-edge generators $\caL_{x,y}$ given, respectively, by
\beq \nonumber
\caL_{x,y} f(z) &=& [-(z_x - z_y) (\partial_x-\partial_y) ] f(z)\ ,\quad z \in E^V\ ,
\eeq
and
\beq \nonumber
\caL_{x,y} f(z) &=& [-\alpha (z_x - z_y) (\partial_x-\partial_y) + z_x z_y (\partial_x-\partial_y)^2] f(z)\ ,\quad z \in E^V\ .
\eeq
For the $\SEP(\gamma)$ we proceed as follows. For each $N \in \N$, consider the operator $L^N$ working on functions $f : (\N/N)^V \to \R$ as
\beq \label{extendedsep}
L^N f(\tfrac{1}{N}\eta) &=& \sum_{x,y\in V} p(x,y) L^N_{x,y}f(\tfrac{1}{N}\eta)\ ,\quad \eta \in \N^V\ ,
\eeq
where
\be \nonumber
L^N_{x,y} f(\tfrac{1}{N}\eta)= \eta_x(\gamma-\eta_y) (f(\tfrac{1}{N}\eta^{x,y})-f(\tfrac{1}{N}\eta)) +\eta_y(\gamma-\eta_x) (f(\tfrac{1}{N}\eta^{y,x})-f(\tfrac{1}{N}\eta))\ ,\quad \eta \in \N^V\ .
\ee
This operator is not a Markov generator anymore, because the factors $\eta_x(\gamma-\eta_y)$ can become negative.
With this operator, we consider the limit
\beq \nonumber
\lim_{N \to \infty} (L^N f)(\tfrac{1}{N}\eta^N)\ ,
\eeq
where $\eta^N= (\lfloor N z_x\rfloor)_{x \in V}$ and $f : E^V \to \R$ is a smooth function.
This then gives the differential operator $\caL$ which is the analogue of \eqref{generators diffusions} in the context of
$\SEP(\gamma)$. This differential operator $\caL$, with single-edge operators
\beq \label{sepdifop}
\caL_{x,y}f(z) &=& [-\gamma (z_x-z_y)(\partial_x-\partial_y) - z_x z_y(\partial_x-\partial_y)^2] f(z)\ ,\quad z \in E^V\ ,
\eeq
does not generate a Markov process but it is still useful because, as we will see
in Section \ref{section generating} below, via generating functions, it is intertwined with the operator \eqref{extendedsep} for the choice $N=1$.
\
Naturally, as we can see for the case of $\SIP(\alpha)$ and $\BEP(\alpha)$, when going to the scaling limit, some properties concerning stationary measures and duality pass to the limit. Indeed, $\BEP(\alpha)$ admits a one-parameter family of stationary product measures $\{\otimes\ \nu_\lambda,\ \lambda > 0 \}$, where $\nu_\lambda \sim \text{Gamma}(\alpha,\lambda)$, namely
\beq \label{gamma distribution}
\nu_\lambda(dz) &=& z^{\alpha-1} e^{-\lambda z} \frac{\lambda^\alpha}{\Gamma(\alpha)} dz\ ,
\eeq
and is dual to $\SIP(\alpha)$ with factorized duality function $D(\xi,z)=\prod_{x \in V}d(\xi_x, z_x)$ given by
\beq \nonumber
d(k,z) &=& z^k \frac{\Gamma(\alpha)}{\Gamma(\alpha+k)}\ ,\quad k \in \N\ ,\quad z \in \R_+\ .
\eeq
After noting that property \eqref{Da} holds also in this situation, we show that the first single-site duality functions $d(1,x)$
between $\SIP(\alpha)$ and $\BEP(\alpha)$ are affine functions of $z \in \R_+$, as we found earlier for single-site self-dualities in Theorem \ref{gzrprop}.
\begin{proposition}
Assume that $\SIP(\alpha)$ and $\BEP(\alpha)$'s single-edge generators are dual with factorized duality function $D(\xi,z)=\prod d(\xi_x, z_x)$ with $d(0,\cdot)\equiv 1$ as in \eqref{factoo}--\eqref{Da}. Then
\beq\label{ghgh}
d(1,z) &=& a + b z\ ,
\eeq
for some $a, b \in \R$.
\end{proposition}
\begin{proof}
The duality relation for $\xi_x = 1$ and no particles elsewhere, by using \eqref{Da}, reads
\beq \nonumber
d(1,z_y) - d(1, z_x) &=&
-\alpha (z_x - z_y) \partial_x d(1,z_x) + z_x z_y \partial_x^2 d(1,z_x)\ .
\eeq
If we set $z_x = z_y = z$, then $z^2 \frac{d^2}{dz^2} d(1,z)=0$ leads to \eqref{ghgh} as unique solution.
\end{proof}
\section{Relation between duality function and stationary product measure}\label{section general relation}
In the examples of duality that we have encountered in the previous section, we have a universal relation between
the stationary product measures and the duality functions.
Given $\{\eta(t),\ t \geq 0 \}$, if there is a dual process $\{\xi(t),\ t \geq 0 \}$ with factorized duality functions
$D(\xi,\eta)=\prod d(\xi_x,\eta_x)$ as in \eqref{factoo}--\eqref{Da} and stationary product measures $\{\mu_\lambda= \otimes\ \nu_\lambda,\ \lambda > 0\}$, then there is a relation between these measures
and these functions, namely there exists a function $\theta(\lambda)$ such that
\beq\label{reldual}
\int D(\xi,\eta) \mu_\lambda (d\eta)\ =\ \prod_{x \in V} \int d(\xi_x,\eta_x) \nu_\lambda(d\eta_x) &=& \theta(\lambda)^{|\xi|}\ .
\eeq
This function $\theta(\lambda)$ is then simply the expectation of the first single-site duality function, i.e.\
\beq \nonumber
\theta(\lambda) &=& \int d(1,\eta_x) \nu_\lambda( d\eta_x)\ .
\eeq
In the examples of Section \ref{secex}, we have $\theta(\lambda)=\lambda$ for $\IRW$, $\theta(\lambda)=\tfrac{\lambda}{1-\lambda}$ for $\SIP(\alpha)$ and
$\theta(\lambda)=\tfrac{\lambda}{1+\lambda}$ for $\SEP(\gamma)$.
In this section we first investigate under which general conditions this relation holds, and further use it in Sections \ref{section translation}--\ref{section non-translation} as a criterion of characterization of all extremal measures. The reader shall refer to Sections \ref{section setting duality}--\ref{section setting lattice} for the general setting in which these results hold.
Later on, we will see that this relation \eqref{reldual} is actually a characterizing property of the factorized duality functions, meaning that
all duality functions are determined once the first single-site duality function is fixed.
\subsection{Finite case}\label{section finite}
We start with the simplest situation in which $V$ is a finite set.
First, we assume that the total number of
particles/the total energy of the dual process is the only conserved quantity.
More precisely, we assume the following property, which we refer to as \textit{harmonic triviality} of the dual system:
\begin{enumerate}
\item[\HT]
If $H:\hat E^V\to\R$ is harmonic, i.e.\ such that, for all $t>0$, $$\hat \E_\xi H(\xi(t))=H(\xi)\ ,$$ then
$H(\xi)$ is only a function of $|\xi|:=\sum_{x\in V} \xi_x$.
\end{enumerate}
Moreover, let us assume the factorized form of the duality function $D(\xi, \eta)$ as in \eqref{factoo}--\eqref{Da}.
Of the single-site functions $d$ we may require the following additional property:
\ben
\item[\Db] The function $d$ is \textit{measure determining}, i.e.\ for two probability measures
$\nu_\ast$, $\nu_\star$ on $E$ such that for all $x\in V$ and $\xi_x\in \hat E$
\[
\int_E d(\xi_x, \eta_x)\nu_\ast(d\eta_x)=\int_E d(\xi_x,\eta_x)\nu_\star(d\eta_x)<\infty\ ,
\]
it follows that $\nu_\ast=\nu_\star$.
\een
Then we have the following.
\bt\label{finitethm}
Assume that $\{\eta(t),\ t\geq 0\}$ and $\{\xi(t),\ t \geq 0\}$ are dual as in \eqref{dualrel} with factorized duality function \eqref{factoo}
satisfying condition \eqref{Da}. Moreover, assume that \HT\ holds and that $\mu$ is a probability measure
on $\Omega$. We distinguish two cases:
\begin{enumerate}
\item[(1)] \emph{Interacting particle system case.} If $\hat E$ is a subset of $\N$, then we assume that the self-duality functions $D(\xi, \cdot)$ are $\mu$-integrable for all $\xi \in \hat \Omega$.
\item[(2)] \emph{Interacting diffusion case.} If $\hat E=\R_+$, then we assume the following integrability condition: for each $\varepsilon > 0$, there exists a $\mu$-integrable function $f_\varepsilon$ such that
\beq \label{unifintegrability}
\sup_{\xi \in \hat \Omega,\ |\xi|=\varepsilon} |D(\xi,\eta)| &\leq& f_\varepsilon(\eta)\ ,\quad \eta \in \Omega\ .
\eeq
\end{enumerate}
Then
\ben
\item[(a)]
$\mu$ is a stationary product measure for the process $\{\eta(t):t\geq 0\}$
\een
implies
\ben
\item[(b)] For all $\xi\in \Omega$ and for all $x\in V$,
we have
\beq\label{basicrelfi}
\int D(\xi,\eta) \mu (d\eta)&=& \left(\int D(\delta_x, \eta) \mu(d\eta)\right)^{|\xi|}\ ,
\eeq
\een
where $\delta_x$ denotes the configuration with a single particle at $x \in V$ and no particles elsewhere. Moreover, if condition \Db\ holds, the two statements (a) and (b) are equivalent.
\et
\bpr
First assume that $\mu$ is a stationary product measure.
Define $H(\xi)=\int D(\xi,\eta) \mu(d\eta)$. By $\mu$-integrability in the interacting particle system case (resp.\ \eqref{unifintegrability} in the interacting diffusion case), self-duality and invariance of $\mu$ we have
\beq
\E_\xi H(\xi(t))\ =\ \int \E_\xi D(\xi(t),\eta) \mu(d\eta)\ =\ \int \E_\eta D(\xi,\eta(t)) \mu(d\eta)\nonumber\
=\ \int D(\xi,\eta) \mu(d\eta)
=H(\xi) \nonumber\ .
\eeq
Therefore by \HT\ we conclude that $H(\xi)=\psi(|\xi|)$. By using $d(0,\cdot)\equiv 1$ and
the factorization of the duality functions, we have that $\psi(0)=1$. For the particle case, we obtain that
\beq \nonumber
\int D(\delta_x, \eta) \mu(d\eta)&=& \psi(1)\ .
\eeq
In particular, we obtain that the l.h.s.\ does not depend on $x$.
Next, for $n\geq 2$, put
\beq \nonumber
\int D(n\delta_x, \eta) \mu(d\eta)&=& \psi(n)\ ,
\eeq
then we have for $x\not=y \in V$, using the factorized duality function and the product form of the measure,
\beq \nonumber
\psi(n)\ =\ \int D(n\delta_x, \eta) \mu(d\eta)\ =\ \int D(\delta_y,\eta) D((n-1)\delta_x, \eta) \mu(d\eta)\
=\ \psi(1) \psi(n-1)\ ,
\eeq
from which it follows that $\psi(n)= \psi(1)^n$. Via an analogous reasoning that uses the factorization of $D(\xi,\eta)$ and the product form of $\mu$, for the diffusion case we obtain, for all $\varepsilon, \rho \geq 0$,
\beq \nonumber
\psi(\varepsilon+\rho) &=& \psi(\varepsilon) \psi(\rho)\ ,
\eeq
and hence, by measurability of $\psi(\varepsilon)$, we get $\psi(\varepsilon)=\psi(1)^\varepsilon$.
To prove the other implication, put
\beq \nonumber
\int D(\delta_x, \eta) \mu(d\eta)&=&\kappa\ .
\eeq
We then have by assumption
\beq \nonumber
\int D(\xi,\eta)\mu (d\eta)&=&\kappa^{|\xi|}\ ,
\eeq
and so it follows that $\mu$ is stationary by self-duality, $\mu$-integrability, the conservation of the number of particles and the measure-determining property. Indeed,
\beq \nonumber
\int \E_\eta D(\xi,\eta(t)) \mu (d\eta)\ =\ \int \E_\xi D(\xi(t),\eta) \mu (d\eta)\ =\ \E_\xi (\kappa^{|\xi(t)|})\ =\ \kappa^{|\xi|}=\int D(\xi,\eta) \mu (d\eta)\ .
\eeq
From the factorized form of $D(\xi,\eta)$, \eqref{basicrelfi} implies that for all $x\in V$ and $\xi_x\in \hat E$
\beq \nonumber
\int d(\xi_x,\eta_x)\mu (d\eta) &=& \kappa^{\xi_x}\ ,
\eeq
and also
\beq \nonumber
\int D(\xi,\eta)\mu (d\eta)\ =\ \kappa^{|\xi|}\ =\ \prod_{x\in V}\kappa^{\xi_x}\ =\ \prod_{x\in V}\int d(\xi_x,\eta_x)\mu (d\eta)\ ,
\eeq
therefore $\mu$ is a product measure by the fact that $d$ is measure determining.
\epr
\subsection{Infinite case}\label{section infinite}
If $V=\Zd$, then one needs essentially two extra conditions to state an analogous result in which a general relation between duality functions
and corresponding stationary measures can
be derived.
In this section we will assume that the dual process
is a discrete particle system, i.e.\ $\hat E$ is a subset of $\N$, in which
the number of particles is conserved. In this case we need an additional property ensuring that
for the dynamics of a finite number of particles there are no bounded harmonic functions other than those depending on the total number of particles.
Therefore, we introduce the condition of existence of a successful coupling for the discrete
dual process with a finite number of particles.
This is defined below.
\bd
We say that the discrete dual process $\{\xi(t),\ t\geq 0\}$ has the \emph{successful coupling property}
when the following holds:
if we start with $n$ particles then there exists a labeling such that for the
corresponding labeled process
$\{X_1(t), \ldots, X_n(t),\ t\geq 0\}$ there exists a successful coupling.
This means that for every two initial positions ${\bf x}=(x_1,\ldots, x_n)$ and ${\bf y}=(y_1,\ldots, y_n)$, there
exists a coupling with path space measure $\pee_{\bf x,y}$ such that the coupling time
\[
\tau=\inf \{s>0:\ {\bf X}(t)={\bf Y}(t),\ \forall t\geq s \}
\]
is finite $\pee_{\bf x,y}$ almost surely.
\ed
Notice that the successful coupling property
is the most common way to prove the following property (cf.\ \cite{ligg}), which is the analogue of \HT, referred here to as \textit{bounded harmonic triviality} of the dual process:
\begin{enumerate}
\item[\BHT]
If
$H$ is a bounded harmonic function,
then $H(\xi)=\psi(|\xi|)$ for some bounded $\psi: \hat E \to \R$.
\end{enumerate}
\begin{remark}
The condition of the existence of a successful coupling (and the consequent bounded harmonic triviality) is quite natural
in the context of interacting particle systems, where we have that a finite number of walkers behave as independent walkers,
except when they are close and interact. Therefore, the successful coupling needed is a variation of the Ornstein coupling
of independent walkers, see e.g.\ \cite{demasi}, \cite{ferrari}, \cite{kuoch}.
\end{remark}
Furthermore, we need a form of uniform $\mu$-integrability of the duality functions which we introduce below and call \textit{uniform domination property} of $D$ w.r.t.\ $\mu$ (note the analogy with condition \eqref{unifintegrability}):
\begin{enumerate}
\item[\UD]
\label{unifintD}
Given $\mu$ a probability measure on $\Omega$, the duality functions $\{D(\xi,\cdot),\ |\xi|=n\}$ are uniformly $\mu$-integrable, i.e.\ for all $n \in \N$
there exists a function $f_n$ such that $f_n$ is $\mu$-integrable and such that for
all $\eta\in \Omega$
\be\label{domi}
\sup_{\xi \in \hat \Omega,\ |\xi|=n} |D(\xi, \eta)|\leq f_n(\eta)\ .
\ee
\end{enumerate}
Under these conditions, the following result holds, whose proof resembles that of Theorem \ref{finitethm}.
\begin{theorem}\label{infinitethm}
Assume as in \eqref{dualrel} that $\{\eta(t),\ t\geq 0\}$ is dual to the discrete process $\{\xi(t),\ t \geq 0 \}$ with factorized duality function as in \eqref{factoo}--\eqref{Da}. Moreover, assume \BHT\ in place of \HT\ for the dual process and that $\mu$ is a probability measure
on $\Omega$ such that \UD\ holds. Then the same conclusions as in Theorem \ref{finitethm} follow, where \eqref{basicrelfi} holds for all finite configurations $\xi \in \hat \Omega$.
\end{theorem}
\subsubsection{Translation invariant case}\label{section translation}
In this section, we show that under the assumption of factorized duality functions, minimal ergodicity assumptions on a stationary probability measure $\mu$ on $\Omega$ are needed to ensure \eqref{basicrelfi} and, as a consequence, that $\mu$ is product measure.
Here we restrict to the case $V=\Zd$ because we will use spatial ergodicity.
\bt\label{transthm}
In the setting of Theorem \ref{infinitethm} with $D(\xi,\eta)$ factorized duality function and $\mu$ probability measure on $\Omega$, if $\mu$ is a translation-invariant and ergodic (under translations)
stationary measure for $\{\eta(t),\ t \geq 0 \}$, then we have \eqref{basicrelfi} for all finite configurations $\xi$; as a consequence, $\mu$ is a product measure.
\et
\bpr
To start, let us consider a configuration $\xi=\sum_{i=1}^n \delta_{x_i}$.
By bounded harmonic triviality \BHT, combined with the bound \eqref{domi} for all such configurations,
$\int D(\xi, \eta) \mu(d\eta)$ is only depending on $n$ and, therefore,
we can replace $\xi$ by $\sum_{i=1}^{n-1} \delta_{x_i} +\delta_y$, where $y$ is
arbitrary in $\Zd$. Let us call $B_N=[-N,N]^d\cap\Zd$. Fix $N_0$ such that $B_{N_0}$ contains
all the points $x_1,\ldots x_{n-1}$. For $y$ outside $B_{N_0}$, by the factorization property,
$D(\sum_{i=1}^{n-1} \delta_{x_i}+\delta_y, \eta)=D(\sum_{i=1}^{n-1} \delta_{x_i}, \eta) D(\delta_y, \eta)$.
By the Birkhoff ergodic theorem,
we have that
\[
\frac{1}{(2N+1)^d}\sum_{y\in B_N} D(\delta_y, \eta)\to \int D(\delta_0, \eta)\mu(d\eta)
\]
$\mu$-a.s.\ as $N\to\infty$.
Using this, together with \eqref{domi},
we have
\beq\label{bombi}
&&\int D(\xi, \eta) \mu(d\eta)
\nonumber\\
&&=\lim_{N\to\infty}\frac{1}{(2N+1)^d}\sum_{y\in B_N\setminus B_{N_0}} \int D(\sum_{i=1}^{n-1} \delta_{x_i}, \eta) D(\delta_y, \eta)\mu(d\eta)
\nonumber\\
&&= \int D(\sum_{i=1}^{n-1} \delta_{x_i}, \eta) \mu(d\eta) \int D(\delta_0, \eta)\mu(d\eta)
\nonumber\\
\eeq
Iterating this argument gives \eqref{basicrelfi}.
\epr
\br
As follows clearly from the proof, the condition of factorization of the duality function can be replaced by the weaker condition of
\[
\lim_{|y|\to\infty} (D(\delta_{x_1}+\ldots+ \delta_{x_n} + \delta_y, \eta)- D(\delta_{x_1}+\ldots+ \delta_{x_n},\eta) D( \delta_y, \eta))=0
\]
for all $\eta$, $x_1,\ldots,x_n$. We note that this approximate factorization of the duality function leads to \eqref{basicrelfi}, though $\mu$ is not necessarily a product measure.
\er
\subsubsection{Non-translation invariant case}\label{section non-translation}
We continue here with $V=\Zd$ but drop the
assumption of translation invariance. Indeed,
equality \eqref{basicrelfi} is also valid in contexts where one cannot rely on translation invariance.
Examples include spatially inhomogeneous $\SIP(\boldsymbol \alpha)$ and $\SEP(\boldsymbol \seppar)$, where the parameters $\boldsymbol \alpha = (\alpha_x)_{x \in V}$ and $\boldsymbol \gamma = (\gamma_x)_{x \in V}$ in Section \ref{section setting particle} may depend on the site accordingly (cf.\ e.g.\ \cite{rs}). Also in this inhomogeneous setting the self-duality functions factorize over sites and the stationary measures are in product form, with site-dependent single-site duality functions, resp. site-dependent marginals. We will show that the relation \eqref{basicrelfi} between
the self-duality functions and any ergodic stationary measure still holds, and as a consequence this ergodic stationary measures is in fact a product measure. The idea is that the averaging over space w.r.t.\ $\mu$, used in the proof of Theorem \ref{transthm} above, can be replaced by a time average.
If we start with a single dual particle, the dual process is a continuous-time random walk on $V$, for which we
denote by $p(t; x,y)$ the transition probability to go from $x$ to $y$ in time $t>0$.
A basic assumption will then be
\be\label{zeropt}
\lim_{t\to\infty} p(t; x,y)=0
\ee
for all $x, y \in V$.
\bt\label{tempergthm}
In the setting of Theorem \ref{infinitethm} with $D(\xi,\eta)$ duality function and $\mu$ probability measure on $\Omega$, if $\mu$ is an ergodic stationary measure for the process $\{\eta(t),\ t \geq 0 \}$ and \eqref{zeropt} holds for the dual particle, then we have \eqref{basicrelfi} for all finite configurations $\xi$; as a consequence, $\mu$ is a product measure.
\et
\bpr
The idea is to replace the spatial average in the proof of Theorem \ref{transthm} by a Cesaro average over time, which we can deal by combining
assumption \eqref{zeropt} with the assumed temporal ergodicity.
Fix $x_1,\ldots,x_n\in V$, $y\in V$.
Define
\beq \nonumber
H_{n+1}(x_1,\ldots,x_n,y)&=& \int D(\delta_{x_1}+ \ldots+\delta_{x_n}+ \delta_y,\eta) \mu(d\eta)
\\
\nonumber
H_1(y)&=& \int D(\delta_y,\eta) \mu(d\eta)
\\ \nonumber
H_{n}(x_1,\ldots,x_n)&=& \int D(\delta_{x_1}+ \ldots+\delta_{x_n},\eta) \mu(d\eta)\ .
\eeq
It is sufficient to obtain that $H_{n+1}(x_1,\ldots,x_n,y)= H_1(y)H_{n}(x_1,\ldots,x_n)$.
We already
know by the bounded harmonic triviality that $H_{n}$ only depends on $n$ and not on the given locations $x_1,\ldots,x_n$.
Therefore, we have
\beq \nonumber
H_{n+1}(x_1,\ldots,x_n,y) &=& \sum_{z} p(t; y,z) H_{n+1}(x_1,\ldots,x_n,z)\ .
\eeq
By assumption \eqref{zeropt}, this implies
\beq
&&H_{n+1}(x_1,\ldots,x_n,y)\nonumber\\ &=& \lim_{T\to\infty}\frac1T \int_0^T dt \sum_{z\not \in \{x_1,\ldots,x_n\}} p(t; y,z) H_{n+1}(x_1,\ldots,x_n,z)
\nonumber\\
&=&
\lim_{T\to\infty}\frac1T \int_0^T dt \sum_{z\not \in \{x_1,\ldots,x_n\}} p(t; y,z) \int D(\delta_{x_1}+ \ldots+\delta_{x_n},\eta)D( \delta_y,\eta) \mu(d\eta)
\nonumber\\
&=& \lim_{T\to\infty}\frac1T \int_0^T dt \sum_{z} p(t; y,z) \int D(\delta_{x_1}+ \ldots+\delta_{x_n},\eta)D( \delta_y,\eta) \mu(d\eta)
\nonumber\\
&=&
\lim_{T\to\infty}\frac1T\int_0^T dt \int D(\delta_{x_1}+ \ldots+\delta_{x_n},\eta)\E_\eta D( \delta_y,\eta(t)) \mu(d\eta)
\nonumber\\
&=&
H_1(y)H_{n}(x_1,\ldots,x_n)\ , \nonumber
\eeq
where in the last step we used the assumed temporal ergodicity of $\mu$ and Birkhoff ergodic theorem.
\epr
\section{From stationary product measures to duality functions.} \label{section general construction}
As we have just illustrated in Sections \ref{section translation}--\ref{section non-translation}, relation \eqref{reldual} turns out to be useful in deriving information about the product structure of stationary ergodic measures from the knowledge of factorized duality functions. On the other side, granted some information on the stationary product measures, which follows usually from a simple detailed balance computation, up to which extent does relation \eqref{reldual} say something about the possible factorized duality functions?
In the context of conditions \eqref{factoo}--\eqref{Da} and in presence of a one-parameter family of stationary product measures $\{\mu_\lambda = \otimes\ \nu_\lambda,\ \lambda > 0 \}$, relation \eqref{reldual} for $\xi = k \delta_x \in \hat \Omega$ for some $k \in \N$ reads
\beq \label{reldual5}
\int_E d(k,\eta_x) \nu_\lambda(d\eta_x)\ =\ \left(\int_E d(1,\eta_x) \nu_\lambda(d\eta_x) \right)^k &=& \theta(\lambda)^k\ .
\eeq
As a consequence, knowing the first single-site duality function $d(1,\cdot)$ and the explicit expression of the marginal $\nu_\lambda$ is enough to recover the l.h.s.\ in \eqref{reldual5}. However, rather than obtaining $d(k,\cdot)$, at this stage the l.h.s.\ has still the form of an \textquotedblleft integral transform\textquotedblright-type of expression for $d(k,\cdot)$.
In the next two subsections, we show
how to recover $d(k,\eta_x)$ from \eqref{reldual5} and the knowledge of $\theta(\lambda)$.
This then leads to the characterization of all possible factorized (self-)duality functions.
\subsection{Particle systems and orthogonal polynomial self-duality functions}\label{section particle systems duality}
Going back to the interacting particle systems introduced in Section \ref{section setting particle} with infinitesimal generator \eqref{gzrpgen} and product stationary measures with marginals \eqref{marginals}, the integral relation \eqref{reldual5} rewrites, for each $k \in \N$ and $\lambda > 0$ for which $Z_\lambda < \infty$,
as
\beq \nonumber
\sum_{n \in \N} d(k,n) \nu_\lambda(n)\ =\ \sum_{n \in \N} d(k,n) \varphi(n) \frac{\lambda^n}{n!} \frac{1}{Z_\lambda} &=& \theta(\lambda)^k\ ,
\eeq
where $\varphi(n)= n! \prod_{m=1}^n \tfrac{v(m-1)}{u(m)}$. Now, if we interpret
\beq \nonumber
\sum_{n \in \N} d(k,n) \varphi(n) \frac{\lambda^n}{n!}
\eeq
as the \emph{Taylor series expansion} around $\lambda=0$ of the function $\theta(\lambda)^k Z_\lambda$, we can re-obtain the explicit formula of $d(k,n) \varphi(n)$ as its $n$-th order derivative evaluated at $\lambda=0$, namely
\beq \nonumber
d(k,n) \varphi(n) &=& \left(\left[ \frac{d^n}{d\lambda^n}\right]_{\lambda=0}\theta(\lambda)^k Z_\lambda \right)\ ,
\eeq
and hence, anytime $\varphi(n) > 0$,
\beq\label{dnu}
d(k,n) &=& \frac{1}{\vi(n)}\left(\left[ \frac{d^n}{d\lambda^n}\right]_{\lambda=0}\theta(\lambda)^k Z_\lambda\right)\ .
\eeq
Together with the full characterization obtained in Theorem \ref{gzrprop} of the first single-site self-duality functions $d(1,\cdot)$ - and $\theta(\lambda)$ in turn - we obtain via this procedure a full characterization of all single-site self-duality functions. Beside recovering the \textquotedblleft classical" dualities illustrated in Section \ref{section setting particle}, we find the single-site self-duality functions in terms of orthogonal polynomials $\{p_k(n),\ k \in \N \}$ of a discrete variable (cf.\ e.g.\ \cite{Nikiforov}) recently discovered via a different approach in \cite{chiara}. We add the observation that all these new single-site self-duality functions can be obtained from the classical ones via a Gram-Schmidt orthogonalization procedure w.r.t.\ the correct probability measures on $\N$, namely the marginals of the associated stationary product measures (cf.\ \cite{chiara}).
We divide the discussion in three cases, one suitable for processes of $\IRW$-type, the other for $\SIP$ and $\SEP$ and the last one for the remaining particle systems.
\subsubsection{Independent random walkers} We recall that the $\IRW$-case corresponds to the choice of values in \eqref{didi} satisfying the relation $v(1)=v(0)$. If we compute $\theta(\lambda)$ for the general first single-site self-duality function $d(1,n)=a+b n$ obtained in \eqref{singdual1}, we get
\beq \nonumber
\theta(\lambda) &=& \sum_n (a+b n) \frac{\lambda^n}{n!}\frac{1}{Z_\lambda}= a + b\lambda\ .
\eeq
and, in turn via relation \eqref{dnu}, we recover all functions $d(k,\cdot)$, for $k > 1$:
\beq \nonumber
d(k,n)&=& \left( \left[\frac{d^n}{d \lambda^n} \right]_{\lambda=0} \left(a + b\lambda\right)^k e^\lambda \right) \\
&=& \sum_{r=0}^n \binom{n}{r} k (k-1) \cdots (k-r+1) b^r a^{k-r}\ .\nonumber
\eeq
In case $a=0$, $d(k,n) = 0$ for $n < k$, while for $n \geq k$, in the summation all terms but the one corresponding to $r=k$ vanish, thus $d(k,n)= \frac{n!}{(n-k)!} b^k$. In case $a \neq 0$,
\beq \label{single-site irw}
d(k,n) &=& a^k \sum_{r=0}^{\min(k,n)} \binom{n}{r} \binom{k}{r} r! \left( \frac{b}{a}\right)^r\ =\ a^k\ \pFq{0}{2}{-k,-n}{-}{\frac{b}{a}}\ .
\eeq
In conclusion, for the choice $a \cdot b < 0$,
\beq \nonumber
d(k,n) &=& a^k\ C_k(n; -\tfrac{a}{b})\ ,
\eeq
where $\{C_k(n; \mu),\ k \in \N\}$ are the Poisson-Charlier polynomials - orthogonal polynomials w.r.t.\ the Poisson distribution of parameter $\mu > 0$ (cf.\ \cite{Nikiforov}).
\subsubsection{Inclusion and exclusion processes} For $\SIP$ and $\SEP$ we are in the case $v(1) \neq v(0)$, and hence we abbreviate
\beq \nonumber
\sigma\ =\ v(1)-v(0)\ ,\quad \beta\ =\ v(0)\ ,
\eeq
where for $\SIP(\alpha)$ we choose $\sigma=1$ and $\beta=\alpha$, while for $\SEP(\gamma)$ we set $\sigma=-1$ and $\beta=\gamma$.
If we compute $\theta(\lambda)$ for $d(1,n)=a+bn$ in \eqref{singdual1}, we have
\beq
\nonumber
\theta(\lambda)&=& a + b \beta \lambda\left(1-\sigma \lambda \right)^{-1}\ =\ \frac{a+\left(b \beta - a \sigma \right)\lambda}{1-\sigma \lambda}\ .
\eeq
By applying formula \eqref{dnu}, we obtain all functions $d(k,n)$ for $k > 1$ as follows:
\beq
\nonumber
d(k,n)&=&\nonumber \frac{1}{\vi(n)}\left(\left[ \frac{d^n}{d\lambda^n}\right]_{\lambda=0}\left(a+\left(b\beta - a \sigma \right) \lambda \right)^k \left(1-\sigma \lambda \right)^{-k-\sigma \beta} \right)\ \\
\nonumber
&=& \frac{\Gamma\left(\sigma \beta \right)}{\Gamma\left(\sigma \beta+n \right)} \sum_{r=0}^n \binom{n}{r} \binom{k}{r} r! a^{k-r} \left(b \sigma \beta - a \right)^{r} \frac{\Gamma\left(\sigma \beta + n + k -r \right)}{\Gamma\left(\sigma \beta + k \right)}\ .
\eeq
In case $a=0$, clearly $d(k,n)=0$ for $k < n$, while for $n \geq k$ only the term for $r=k$ is nonzero in the summation:
\beq \label{dknclassical}
d(k,n) &=& \frac{n!}{(n-k)!} \frac{\Gamma\left(\sigma \beta \right)}{\Gamma\left(\sigma\beta + k\right)} \left(b \sigma\beta \right)^k\ .
\eeq
In case $a \neq 0$, by using the known relation (cf.\ \cite{Nikiforov}, p.\ 51),
\beq \nonumber \label{dknhyper}
d(k,n) &=& a^k\ \frac{\Gamma\left(\sigma\beta \right) \Gamma\left(\sigma \beta+n+k \right)}{\Gamma\left(\sigma \beta+n \right) \Gamma\left(\sigma \beta + k \right)}\ \pFq{2}{1}{-n, -k}{-n-k-\sigma \beta+1}{1-\frac{b}{a}\sigma\beta}\\
&=& a^k\ \pFq{2}{1}{-n, -k}{\sigma \beta}{\frac{b}{a} \sigma \beta}\ .
\eeq
If $\sigma=1$, $\beta = \alpha> 0$ and if $a \cdot b < 0$, we recognize in \eqref{dknhyper} the Meixner polynomials as defined in \cite{Nikiforov}, i.e.\
\beq \nonumber
d(k,n) &=& a^k\ \frac{\Gamma\left(\alpha \right)}{\Gamma\left(\alpha + k \right)}\ M_k(n; \mu, \alpha)\ ,
\eeq
where in our case $\mu= \frac{a}{a-b \alpha}$ and $\{M_k(n; \mu, \alpha),\ k \in \N \}$ are the Meixner polynomials - orthogonal polynomials w.r.t.\ the discrete Gamma distribution of scale parameter $\mu$ and shape parameter $\alpha$.
Furthermore, if $\sigma=-1$, $\beta = \seppar$ with $\seppar \in \N$ and given the additional requirements $a \cdot b < 0$ and $-\frac{a}{b} \leq \seppar$, from the expression in \eqref{dknhyper} we have a representation of the single-site duality functions in terms of the Kravchuk polynomials as defined in \cite{Nikiforov}, i.e.\
\beq \nonumber
d(k,n) &=& a^k\ K_k(n; p, \seppar )\ \left( -\frac{1}{p} \right)^k \frac{1}{\binom{\seppar}{k}}\ ,
\eeq
where $p= -\frac{a}{b \seppar}$ in our case and $\{K_k(n; p, \seppar),\ k \in \N \}$ the Kravchuk polynomials - orthogonal polynomials w.r.t.\ the Binomial distribution of parameters $\seppar$ and $p$.
\
As a conclusion of this procedure, we note that all factorized self-duality functions for independent random walkers, inclusion and exclusion processes satisfying \eqref{Da} are either in the \textquotedblleft classical" form of Section \ref{section setting particle} (case $a=0$) or consist of products of rescaled versions of orthogonal polynomials (case $a \neq 0$). Other factorized self-duality functions for the systems $\IRW$, $\SIP$ and $\SEP$ do not exist.
\begin{remark}
It is interesting to note that, apart from the leading factor $a^k$, the remaining \emph{polynomials} in the expressions of $d(k,n)$ for $a \neq 0$ are \textquotedblleft \emph{self-dual}" in the sense of the orthogonal polynomials literature, i.e.\ $p_n(k)=p_k(n)$ in our context (cf.\ Definition 3.1, \cite{koekoek}). Henceforth, if $d(k,n)$ is interpreted as a countable matrix, the value $a \in \R$ is the only responsible for the asymmetry of $d(k,n)$: upper-triangular for $a=0$ while symmetric for $a=1$.
\end{remark}
\subsubsection{Trivial factorized self-duality}
To conclude, for the sake of completeness, we can implement the same machinery to cover all factorized self-dualities with property \eqref{Da} for all discrete processes of type \eqref{gzrpgen}.
Indeed, from the proof of Theorem \ref{gzrprop}, if the process is neither of the types $\IRW$, $\SIP$ and $\SEP$, then the only possible choice is $d(1,n)=a$ for some $a \in \R$, i.e.\ it is not depending on $n$. From this we get
$\theta(\lambda)= a$, and $d(k,n)=a^k$ from formula \eqref{dnu}. Hence, the self-duality functions must be of the form
\beq \nonumber
D(\xi,\eta) &=& \prod_{x \in V}d(\xi_x,\eta_x)\ =\ a^{|\xi|}\ ,
\eeq
i.e.\ depending only on the total number of dual particles (and not on the configuration $\eta$). Hence, the duality relation in that case reduces to the trivial relation, for all $t \geq 0$ and $\xi \in \hat \Omega$,
\[
\E_\xi a^{|\xi_t|}= a^{|\xi|}\ ,\quad a \in \R\ ,
\]
which is just conservation of the number of particles in the dual process. No other self-duality relation with factorized self-duality functions
can exist.
\subsection{Interacting diffusions and orthogonal polynomial duality functions}\label{section Laplace}
As shown in Theorem \ref{finitethm}, relation \eqref{reldual5} still holds whenever the discrete right-variables $n \in \N$ are replaced by continuous variables $z \in \R_+$ and sums by integrals. With this observation in mind, we provide a second general method to \emph{characterize all factorized duality functions} between the continuous process $\BEP(\alpha)$ and its discrete dual $\SIP(\alpha)$.
More precisely, if $d(k,z)$ is a single-site duality function with property \eqref{Da} between $\BEP(\alpha)$
and $\SIP(\alpha)$, and $\nu_\lambda$ is the stationary product measure marginal for $\BEP(\alpha)$ as in \eqref{gamma distribution}, then, from the analogue of relation \eqref{reldual5} for $k=1$, namely
\beq \label{reldual1c}
\int_{\R_+} d(1,z) z^{\alpha-1} e^{-\lambda z}\frac{\lambda^\alpha}{\Gamma(\alpha)} dz &=& \theta(\lambda)\ ,
\eeq
we necessarily have by Theorem \ref{finitethm} that
\beq \label{reldual2c}
\int d(k,z) \frac{z^{\alpha-1}}{\Gamma(\alpha)} e^{-\lambda z} dz &=& \theta(\lambda)^k \lambda^{-\alpha}\ .
\eeq
As a consequence, the function $d(k,z) \frac{z^{\alpha-1}}{\Gamma(\alpha)}$ is the inverse Laplace transform of $\theta(\lambda)^k \lambda^{-\alpha}$.
Given the first single-site duality function $d(1,z)$ in \eqref{ghgh}, from \eqref{reldual1c} we obtain
\beq \nonumber
\theta(\lambda) &=& \int (a+bz) z^{\alpha-1} e^{-\lambda z} \frac{\lambda^\alpha}{\Gamma(\alpha)} dz = \left(a \lambda + b \alpha \right) \lambda^{-1}\ .
\eeq
As a consequence, the r.h.s.\ in \eqref{reldual2c} becomes
\beq \label{hjhj}
\theta(\lambda)^k \lambda^{-\alpha} &=& \left(a \lambda + b \alpha \right)^k \lambda^{ - \alpha-k}\ ,
\eeq
and there exist explicit expressions for the inverse Laplace transform of this function. We split the computation in two cases.
In case $a=0$, since the inverse Laplace transform of $\lambda^{-\alpha-k}$ is $\frac{z^{\alpha+k-1}}{\Gamma(\alpha+k)}$, we immediately obtain
\beq \nonumber
d(k,z) &=& (b \alpha)^k \frac{z^k \Gamma(\alpha)}{\Gamma(\alpha+k)}\ ,
\eeq
i.e.\ the \textquotedblleft classical" single-site duality function as in Section \ref{section setting diffusion}, up to set $b=\frac{1}{\alpha}$.
In case $a \neq 0$, the inverse Laplace transform of \eqref{hjhj} is more elaborated:
\beq \nonumber
a^k\ \frac{z^{\alpha-1}}{\Gamma(\alpha)}\ \pFq{1}{1}{-k}{\alpha}{-\frac{b \alpha}{a} z}\ .
\eeq
As the above expression must equal $d(k,z) \frac{z^{\alpha-1}}{\Gamma(\alpha)}$, it follows that
\beq \nonumber
d(k,z) &=& a^k\ \pFq{1}{1}{-k}{\alpha}{-\frac{b \alpha}{a} z}\ .
\eeq
As a final consideration, we note that for the choice $a \cdot b < 0$,
\beq
d(k,z) &=& a^k\ \frac{k! \Gamma(\alpha)}{\Gamma(\alpha+k)}\ L_k(z; \alpha-1, \mu)\ ,
\eeq
where $\mu= -\frac{b \alpha}{a}$ here and $\{L_k(z; \alpha-1, \mu),\ k \in \N \}$ are the generalized Laguerre polynomials - orthogonal polynomials w.r.t.\ to the Gamma distribution of scale parameter $\frac1\mu$ and shape parameter $\alpha$ as defined in \cite{Nikiforov}.
\section{Intertwining and generating functions}\label{section generating}
In this section, we introduce the generating function method, which allows to go from a self-duality of a
discrete process towards duality between a discrete and continuous process, and further towards a self-duality of a
continuous process, and back. This then allows e.g.\ to simplify the proof of a discrete self-duality by lifting it to
a continuous self-duality, which is usually easier to verify. The key of this method is an intertwining between discrete multiplication and derivation operators and
their continuous analogues, via an appropriate generating function.
To reduce issues of well-definition of operators and their domains, in this section we restrict our discussion to the case of finite vertex set $V$.
\subsection{Introductory example}
To make the method clear, let us start with a simple example of independent random walkers on a single edge.
The generator is
\beq \nonumber
L f(n_1,n_2)&=& n_1 (f(n_1-1, n_2+1)- f(n_1,n_2)) + n_2 (f(n_1+1, n_2-1)- f(n_1,n_2))\ ,
\eeq
with $n_1, n_2 \in \N$.
Define now the (exponential) generating function
\beq \nonumber
\caG f(z_1,z_2)&=& \sum_{n_1,n_2=0}^\infty f(n_1,n_2) \frac{z_1^{n_1}}{n_1!} \frac{z_2^{n_2}}{n_2!}\ ,\quad z_1, z_2 \in \R_+\ .
\eeq
Then it is easy to see that
\beq \nonumber
\caL \caG &=& \caG L\ ,
\eeq
where
\beq \nonumber
\caL &=& -(z_1-z_2)(\partial_{z_1}-\partial_{z_2})\ .
\eeq
Now assume that we have a self-duality function for the particle system, i.e.\
\beq \nonumber
L_\eft D&=& L_\ight D\ ,
\eeq
then we have
\beq \nonumber
L_\eft \caD &=& \caL_\ight \caD\ ,
\eeq
where
\beq \nonumber
\caD(k_1, k_2; z_1, z_2)\ =\ \caG_{\ight} D(k_1, k_2; z_1, z_2)\ =\ \sum_{n_1, n_2=0}^\infty D(k_1,k_2; n_1,n_2) \frac{z_1^{n_1}}{n_1!} \frac{z_2^{n_2}}{n_2!}\ .
\eeq
In words, a self-duality function of $L$ is ``lifted'' to a duality function between the independent random walk generator $L$
and its continuous counterpart $\caL$ by applying the generating function to the $n$-variables.
Conversely, given a duality function between the independent random walk generator $L$
and its continuous counterpart $\caL$, its Taylor coefficients provide a self-duality function of $L$.
We can then also take the generating function w.r.t.\ the $k$-variables in the function $\caD$ to produce
a self-duality function for $\caL$, i.e.\ defining
\beq \nonumber
\mathscr D(v_1, v_2; z_1, z_2)\ =\ \caG_\eft \caD(v_1, v_2; z_1, z_2)\ =\ \sum_{k_1, k_2=0}^\infty \caD(k_1, k_2; z_1, z_2) \frac{v_1^{k_1}}{k_1!} \frac{v_2^{k_2}}{ k_2!}\ ,
\eeq
we have
\beq \nonumber
\caL_\eft \mathscr D &=& \caL_\ight \mathscr D\ .
\eeq
For the classical self-duality function $D(k_1,k_2; n_1,n_2)= \frac{n_1!}{(n_1-k_1)!}\frac{n_2!}{(n_2-k_2)!}\1\{k_1\leq n_1\}\1\{k_2\leq n_2\}$,
we find that
\beq \nonumber
\mathscr D(v_1, v_2; z_1, z_2)\ =\ e^{z_1+z_2} e^{v_1 z_1+ v_2 z_2}\ .
\eeq
Beside the factor $e^{z_1+z_2}$ which depends only on the conserved quantity $z_1+z_2$, to check the self-duality relation for $\caL$ w.r.t.\ the function $e^{v_1 z_1+ v_2 z_2}$ is rather straightforward, the computation involving only derivatives of exponentials. By looking at the Taylor coefficients w.r.t.\ both $v$ and $z$-variables of this self-duality relation, we obtain the self-duality relation for $L$ w.r.t.\ $D$ where we started from.
In conclusion, all these duality relations turn out to be equivalent, and the proof of self-duality for particle systems requiring rather intricate combinatorial arguments (cf.\ e.g.\ \cite{demasi}) is superfluous once the more direct self-duality for diffusion systems is checked.
\subsection{Intertwining and duality}
The notion of \emph{intertwining} between stochastic processes was originally introduced by Yor in \cite{Yor} in the context of Markov chains and later pursued in \cite{DF} and \cite{F} as an abstract framework, in discrete-time and continuous-time respectively, for the problem of Markov functionals, i.e.\ finding sufficient and necessary conditions under which a random function of a Markov chain is again Markovian.
For later purposes, we adopt a rather general definition of intertwining, in which $\{\eta(t),\ t \geq 0 \}$ and $\{\zeta(t),\ t \geq 0 \}$ are continuous-time stochastic processes on the Polish spaces $\Omega$ and $\Omega'$, respectively, whose expectations read $\E$, $\E'$ resp., and $\caM( \Omega)$ denotes the space of signed measures on $\Omega$. We say that $\{\zeta(t),\ t \geq 0 \}$ is \emph{intertwined on top} of $\{\eta(t),\ t \geq 0 \}$ if there exists a mapping $\varLambda: \Omega' \to \caM(\Omega)$ such that, for all $t \geq 0$, $\zeta \in \Omega'$ and $f : \Omega \to \R$,
\beq \label{intertwin}
\E'_\zeta \int_{\Omega'} f(\eta)\ \varLambda(\zeta(t))(d\eta) &=&\int_{\Omega} \E_{\eta} f(\eta(t))\ \varLambda(\zeta)(d\eta)\ .
\eeq
Working at the abstract level of \emph{semigroups}, we say that $\{\caS(t),\ t \geq 0 \}$
on a space of functions $f:\Omega'\to\R$ denoted by $\caF(\Omega')$, is \emph{intertwined on top} of $\{S(t),\ t \geq 0 \}$, a semigroup on a space
of functions $f:\Omega\to\R$ denoted by $\caF(\Omega)$, with \emph{intertwiner} $\varLambda$ if $\varLambda$ is a linear operator
from $\caF(\Omega)$ into $\caF(\Omega')$ and if, for all $t \geq 0$ and $f : \Omega \to \R$,
\beq \nonumber
\caS(t) \varLambda f&=& \varLambda S(t) f\ .
\eeq
Similarly, \emph{operators} $\caL$ with domain $\caD(\caL)$ and $L$ with domain $\caD(L)$ are \emph{intertwined} with intertwiner $\varLambda$ if, for all $f \in \caD(L)$, $\varLambda f\in \caD(\caL)$ and
\beq \label{opintertwin}
\caL \varLambda f &=& \varLambda L f\ .
\eeq
Notice that with a slight abuse of notation we used the same symbol $\varLambda$ for an abstract intertwining operator as for
the intertwining mapping. In other words, in case the intertwining mapping as in \eqref{intertwin} is given by $\tilde \varLambda$, then the corresponding operator is
\beq \nonumber
\varLambda f(\zeta) &=& \int f(\eta)\ {\tilde \varLambda(\zeta)}(d\eta)\ .
\eeq
An intertwining mapping $\varLambda$ has a probabilistic interpretation if it takes values in the subset of probability measures on $\Omega$. Indeed, in \eqref{intertwin} the process $\{\zeta(t),\ t \geq 0 \}$ may be viewed as an added structure on top of $\{\eta(t),\ t \geq 0 \}$ or, alternatively, the process $\{\eta(t),\ t \geq 0 \}$ as a random functional of $\{\zeta(t),\ t \geq 0\}$, in which $\varLambda$ provides this link.
\br
The connection with duality introduced in Section \ref{section setting duality} becomes transparant when $\Omega$, $\hat \Omega$ and $\Omega'$ are finite sets and the operators and functions $L$, $\hat L$ $\caL$, $D$ and $\varLambda$ in \eqref{opdual} and \eqref{opintertwin} are represented in terms of matrices. There, relations \eqref{opdual} and \eqref{opintertwin}, once rewritten in matrix notation as
\beq \label{matrixdual}
\hat L D &=& D L^\dagger\ ,
\eeq
where $L^\dagger$ denotes the transpose of $L$,
and
\beq \label{matrixintertwin}
\caL \varLambda &=& \varLambda L\ ,
\eeq
differ essentially only in the terms $L^\dagger$ versus $L$ in the r.h.s.\ of both identities.
The presence or absence of transposition can be interpreted as a \emph{forward}-versus-\emph{backward} evolution against a \emph{forward}-versus-\emph{forward} evolution. More precisely, if $L$, $\hat L$ and $\caL$ are generators of Markov processes $\{\eta(t), t \geq 0\}$, $\{\xi(t),\ t \geq 0 \}$ and $\{\zeta(t),\ t \geq 0 \}$, respectively, then \eqref{matrixdual} and \eqref{matrixintertwin} relate the evolution of $\{\eta(t),\ t \geq 0\}$ to that of $\{\xi(t),\ t \geq 0 \}$, resp.\ $\{\zeta(t),\ t \geq 0 \}$; however, while in \eqref{matrixintertwin} the processes run both along the same direction in time, in \eqref{matrixdual} the processes run along opposite time directions.
\er
Intertwiners as $\varLambda$ in \eqref{matrixintertwin} may be also interpreted as natural generalizations of {\em symmetries of generators}, indeed \eqref{matrixintertwin} with $\caL = L$ just means that $\Lambda$ commutes with $L$, which is the definition of a symmetry of $L$. As outlined in \cite[Theorem 2.6]{gkrv}, the knowledge of symmetries of a generator and dualities of this generator leads to the construction of new dualities. The following theorem presents the analogue procedure in presence of intertwiners: a duality and an intertwining lead to a new duality.
\begin{theorem}\label{theorem intertwin}
Let $L$, $\hat L$ and $\caL$ be operators on real-valued functions on $\Omega$, $\hat \Omega$ and $\Omega'$, respectively. Suppose that there exists an intertwiner $\varLambda$ such that for all $f \in \caD(L)$, $\varLambda f \in \caD(\caL)$,
\beq \nonumber
\caL \varLambda f &=& \varLambda L f\ ,
\eeq
and a duality function $D: \hat \Omega \times \Omega \to \R$ for $\hat L$ and $L$, namely $D(\xi,\cdot) \in \caD(L)$ for all $\xi \in \hat \Omega$, $D(\cdot,\eta) \in \caD(\hat L)$ for all $\eta \in \Omega$ and
\beq \nonumber
\hat L_\eft D &=& L_\ight D\ .
\eeq
Then, if $\varLambda_\ight D(\xi,\cdot) \in \caD(\caL)$ for all $\xi \in \hat \Omega$ and $\varLambda_\ight D(\cdot,\zeta) \in \caD(\hat L)$ for all $\zeta \in \Omega'$, $\varLambda_\ight D$ is a duality function for $\hat L$ and $\caL$, i.e.
\beq \nonumber
\hat L_\eft \varLambda_\ight D &=& \caL_\ight \varLambda_\ight D\ .
\eeq
\end{theorem}
\begin{proof}
\beq \nonumber
\hat L_\eft \varLambda_\ight D\ =\ \varLambda_\ight \hat L_\eft D\ =\ \varLambda_\ight L_{\ight} D\ =\ \caL_\ight \varLambda_\ight D\ .
\eeq
Here in the first equality we used that left and right actions commute, in the second equality we used the assumed duality of $\hat{L}$ and $L$,
and in the third equality we used the assumed intertwining.
\end{proof}
\subsection{Intertwining between continuum and discrete processes}
In this section we prove the existence of an intertwining relation between the interacting diffusion processes presented in Section \ref{section setting diffusion} and the particle systems of Section \ref{section setting particle}. This intertwining relation provides a second connection, besides the scaling limit procedure (cf.\ Section \ref{section setting diffusion}), between continuum and discrete processes, which proves to be better suited for the goal of establishing duality relations among these processes. Indeed, the characterization of all possible factorized self-dualities for particle systems obtained in Section \ref{section general construction} and the intertwining relation below, via the application of Theorem \ref{theorem intertwin}, produces a characterization of all possible dualities, resp.\ self-dualities, between the discrete and the continuum processes, resp.\ of the continuum process.
\
In the following proposition, we prove the intertwining relation for operators $L^{\sigma,\beta}$ and $\caL^{\sigma,\beta}$ defined, respectively, on functions $f : E^V \to \R$, $E= \N$, as
\beq \label{Lsigmabeta}
L^{\sigma,\beta} f(\eta) &=& \sum_{x,y \in V} p(x,y)\ L^{\sigma,\beta}_{x,y} f(\eta)\ ,\quad \eta \in E^V\ ,
\eeq
where
\beq \nonumber
L^{\sigma,\beta}_{x,y}f(\eta) &=& \eta_x(\beta+\sigma\eta_y) (f(\eta^{x,y})-f(\eta)) + \eta_y (\beta+\sigma \eta_x) (f(\eta^{y,x})-f(\eta))\ ,
\eeq
and, on real analytic functions $f : \R^V \to \R$, as
\beq \label{Lsigmabetadiffusion}
\caL^{\sigma,\beta} f(z) &=& \sum_{x,y \in V} p(x,y)\ \caL^{\sigma,\beta}_{x,y}f(z)\ ,\quad z \in \R^V\ ,
\eeq
where
\beq \nonumber
\caL^{\sigma,\beta}_{x,y}f(z) &=& (-\beta (z_x-z_y)(\partial_x-\partial_y) +\sigma z_x z_y (\partial_x-\partial_y)^2 )f(z)\ .
\eeq
Note that $L^{\sigma,\beta}$ in \eqref{Lsigmabeta} is a special instance of the generator $L^{u,v}$ in \eqref{gzrpgen} with conditions \eqref{didi}, while $\caL^{\sigma,\beta}$ above, matches on a common sub-domain, for particular choices of the parameters $\sigma$ and $\beta$, those in \eqref{generators diffusions}.
\begin{proposition}
Let $G$ be the Poisson probability kernel defined as the operator that maps functions $f : \N \to \R$ into functions $G f : \R \to \R$ as
\beq \label{G single-site intertwiner}
G f(z) &=& \sum_{n =0}^\infty f(n) \frac{z^n}{n!} e^{-z}\ ,\quad z \in \R\ .
\eeq
Then, whenever $G f : \R \to \R$ is a real analytic function, if $G^\otimes= \otimes_{x \in V} G^{(x)}$ denotes the tensorized operator mapping functions $f : \N^V \to \R$ into functions $f : \R^V \to \R$ accordingly, $\caL^{\sigma,\beta}$ and $L^{\sigma,\beta}$ are intertwined with intertwiner $G^\otimes$, namely
\beq \label{intertwinL}
\caL^{\sigma,\beta} G^\otimes f(z) &=& G^\otimes L^{\sigma,\beta} f(z)\ ,\quad z \in \mathbb \R^V\ .
\eeq
\end{proposition}
\begin{proof}Let us introduce the non-normalized operator
\beq \nonumber
\bar G f(z) &=& \sum_{n=0}^\infty f(n) \frac{z^n}{n!}\ ,\quad z \in \R\ ,
\eeq
and the associated tensorized operator $\bar G^\otimes =\otimes_{x \in V} \bar G^{(x)}$.
Due to the factorized structure of $L^{\sigma,\beta}$, $\caL^{\sigma,\beta}$ and $\bar G^\otimes$, the proof of the intertwining relation \eqref{intertwinL} with $\bar G^\otimes$ as an intertwiner reduces to consider and combine the following relations:
\beq \nonumber
\sum_{n=0}^\infty n f(n-1)
\frac{z^n}{n!} &=& z \bar Gf(z)\\
\nonumber
\sum_{n=0}^\infty f(n+1) \frac{z^n}{n!} &=& \frac{d}{dz} \bar Gf(z)\\
\nonumber
\sum_{n=0}^\infty n f(n) \frac{z^n}{n!} &=& z \frac{d}{dz} \bar Gf(z)\\
\nonumber
\sum_{n=0}^\infty n f(n+1) \frac{z^n}{n!} &=& z \frac{d^2}{dz^2} \bar Gf(z)\ .
\eeq
As a first consequence, we have
\beq \nonumber
\caL^{\sigma,\beta} \bar G^\otimes f(z) &=& \bar G^\otimes L^{\sigma,\beta} f(z)\ ,\quad z \in \R^V\ .
\eeq
We obtain \eqref{intertwinL} by observing that ($|z|=\sum_{x \in V} z_x$)
\beq \nonumber
G^\otimes f(z) &=& e^{-|z|}\ \bar G^\otimes f(z)\ ,\quad z \in \R^V\ ,
\eeq
and that, for $g(z) = \bar g(z)\cdot e^{-|z|}$,
\beq \nonumber
\caL^{\sigma,\beta} g(z) &=& e^{-|z|}\ \caL^{\sigma,\beta} \bar g(z)\ ,\quad z \in \R^V\ .
\eeq
\end{proof}
We note that the intertwiner $G^\otimes$ has a nice probabilistic interpretation: from an \textquotedblleft energy\textquotedblright\ configuration $z \in \R_+^V$, the associated particle configurations are generated by placing a number of particles on each site $x \in V$, independently, and distributed according to a Poisson random variable with intensity $z_x$.
In the remaining part of this section, under some reasonable regularity assumptions, we are able to invert the intertwining relation \eqref{intertwinL}, namely to find an operator $H^\otimes=\otimes_{x \in V} H^{(x)}$ that intertwines $L^{\sigma,\beta}$ and $\caL^{\sigma,\beta}$, in this order. The natural candidate for $H$ is the \textquotedblleft inverse operator\textquotedblright\ of $G$, whenever this is well-defined. In general, this \textquotedblleft inverse intertwiner\textquotedblright\ lacks any probabilistic interpretation, but indeed establishes a second intertwining relation useful in the next section.
\begin{proposition}\label{proposition inverse intertwining}
Let $H$ be the differential operator mapping real analytic functions $g : \R \to \R$ into functions $H g : \N \to \R$ as
\beq \label{H single-site inverse intertwiner}
H g(n) &=& \left(\left[\frac{d^n}{d z^n} \right]_{z=0} e^z g(z) \right)\ ,\quad n \in \N\ .
\eeq
Then $H$ is the inverse operator of $G$, namely, for all $f : \N \to \R$ such that $G f : \R \to \R$ is real analytic, we have
\beq \nonumber
G H g (z)\ =\ g(z)\ ,\quad z \in \R\ ,\quad H G f(n)\ =\ f(n)\ ,\quad n \in \N\ .
\eeq
Moreover, the tensorized operator $H^\otimes=\otimes_{x \in V} H^{(x)}$ is an intertwiner for $L^{\sigma, \beta}$ and $\caL^{\sigma,\beta}$, i.e.\ for all real analytic $g : \R^V \to \R$,
\beq \label{intertwinLinv}
L^{\sigma,\beta} H^\otimes g(n) &=& H^\otimes \caL^{\sigma,\beta} g(n)\ ,\quad n \in \N\ .
\eeq
\end{proposition}
Before giving the proof, we need the following lemma.
\begin{lemma}\label{lemma symmetry}
Let $A$ be the operator acting on functions $f : \N \to \R$ defined as
\beq \nonumber
A f(n) &=& \sum_{k=0}^n \binom{n}{k} f(k)\ ,\quad n \in \N\ .
\eeq
Then, the tensorized operator $A^\otimes = \otimes_{x \in V} A^{(x)}$ is a symmetry for the generator $L^{\sigma,\beta}$, i.e.\ for all $f\ : \N^V \to \R$
\beq \label{symmetry}
A^\otimes L^{\sigma, \beta} f(n) &=& L^{\sigma,\beta} A^\otimes f(n)\ ,\quad n \in \N^V\ .
\eeq
\end{lemma}
\begin{proof}
Instead of going through tedious computations, we exploit the fact that the operator $A^\otimes$ has the form
\beq \nonumber
A^\otimes &=& \otimes_{x \in V}\ A^{(x)}\ =\ \otimes_{x \in V}\ e^{J^{(x)}}\ =\ \otimes_{x \in V}\ \sum_{k=0}^\infty \frac{(J^{(x)})^k}{k!}\ ,
\eeq
where $J^{(x)}$ is an operator defined for functions $f : \N^V \to \R$ which acts only on the $x$-th variable as
\beq \nonumber
J^{(x)} f(n) &=& n_x\ f(n-\delta_x)\ ,\quad n \in \N^V\ .
\eeq
Since all these operators $\{J^{(x)},\ x \in V \}$ commute over the sites we have
\beq \nonumber
A^\otimes &=& \otimes_{x \in V} e^{J^{(x)}}\ =\ e^{\sum_{x \in V} J^{(x)}}\ .
\eeq
We conclude the proof by noting that the operator
\beq \nonumber
\sum_{x \in V} J^{(x)}
\eeq
is a symmetry for the generator $L^{\sigma, \beta}$, cf.\ e.g.\ \cite{gkrv}, \cite{Transport}.
\end{proof}
\begin{proof}[Proposition \ref{proposition inverse intertwining}]
First we compute the following key relations:
\beq \nonumber
\left(\left[\frac{d^n}{dz^n} \right]_{z=0} g'(z)\right) &=& \left(\left[\frac{d^{n+1}}{dz^{n+1}} \right]_{z=0} g(z)\right)\\
\nonumber
\left(\left[\frac{d^n}{dz^n} \right]_{z=0} z g(z)\right) &=& n \left(\left[\frac{d^{n-1}}{dz^{n-1}} \right]_{z=0} g(z)\right)\\
\nonumber
\left(\left[\frac{d^n}{dz^n} \right]_{z=0} z g'(z) \right) &=& n \left( \left[\frac{d^n}{dz^n} \right]_{z=0} g(z)\right)\\
\nonumber
\left(\left[\frac{d^n}{dz^n} \right]_{z=0} z g''(z) \right) &=& n \left(\left[\frac{d^{n+1}}{dz^{n+1}} \right]_{z=0} g(z) \right)\ .
\eeq
Hence, if we introduce the operator
\beq \nonumber
\bar H g(n) &=& \left(\left[\frac{d^n}{d z^n} \right]_{z=0} g(z) \right)\ ,\quad n \in \N\ ,
\eeq
and the associated tensorized operator $\bar H^\otimes = \otimes_{x \in V} \bar H^{(x)}$, we obtain
\beq \label{Hbar intertwining}
L^{\sigma,\beta} \bar H^\otimes g(n) &=& \bar H^\otimes \caL^{\sigma,\beta} g(n)\ ,\quad n \in \N\ .
\eeq
Now, by using Lemma \ref{lemma symmetry} and noting that
\beq \nonumber
H g(n) &=& \sum_{k=0}^n \binom{n}{k} \bar H g(k)\ =\ A \bar H g(n)\ ,\quad n \in \N\ ,
\eeq
and, by the mixed property of the tensor product,
\beq \label{HvsHbar}
H^\otimes g(n) &=& (A \bar H)^\otimes g(n)\ =\ A^\otimes \bar H^\otimes g(n)\ ,\quad n \in \N^V\ ,
\eeq
we get \eqref{intertwinLinv} by applying first \eqref{HvsHbar}, then \eqref{symmetry} and finally \eqref{Hbar intertwining}:
\beq \nonumber
L^{\sigma,\beta} H^\otimes g\ =\ L^{\sigma,\beta} A^\otimes \bar H^\otimes g\ =\ A^\otimes L^{\sigma,\beta} \bar H^\otimes g\ =\ A^\otimes \bar H^\otimes \caL^{\sigma,\beta} g\ =\ H^\otimes \caL^{\sigma,\beta} g\ .
\eeq
\end{proof}
\subsection{Generating functions and duality}
As anticipated in the previous section, from the intertwining relation \eqref{intertwinL} and the functions obtained in Section \ref{section particle systems duality}, in what follows we find explicitly new duality relations.
Due to the factorized form \eqref{factoo} of the self-duality functions with single-site functions \eqref{single-site irw} and \eqref{dknhyper} and the tensor form of the intertwiner $G^\otimes$ in \eqref{intertwinL}, the new functions inherit the same factorized form. Moreover, from the definition of $G$ in \eqref{G single-site intertwiner}, the whole computation reduces to determine \emph{(exponential) generating functions} of \eqref{single-site irw} and \eqref{dknhyper}. To this purpose, some identities for hypergeometric functions are available, cf.\ e.g.\ the tables in \cite[Chapter 9]{koekoek}. Moreover, all generating functions obtained satisfy the requirements of analyticity for suitable choices of the parameters $\sigma$, $\beta$, $a$ and $b$ (cf.\ \cite{koekoek}), hence all operations below make sense.
However, just as the functions found in Section \ref{section particle systems duality}, the functions here obtained will only be \textquotedblleft candidate\textquotedblright\ (self-)duality functions, since no duality relation as in \eqref{dualrel} has been proved, yet. By using the \textquotedblleft inverse\textquotedblright\ intertwining \eqref{intertwinLinv}, all these \textquotedblleft possible\textquotedblright\ dualities turn out to be equivalent, i.e.\ one implies all the others. Thus, in Proposition \ref{proposition eee} below, we choose to prove directly the self-duality relation for the continuum process, more immediate to verify due to the simpler form of the self-duality functions. Indeed, while the single-site self-duality functions for the $\SIP(\alpha)$ process, for instance, have the generic form of an hypergeometric function
\beq \nonumber
\pFq{2}{1}{-k,-n}{\alpha}{\frac{b}{a}\alpha}\ ,\quad k, n \in \N\ ,
\eeq
the single-site duality functions between discrete and continuum processes involve in their expressions hypergeometric functions
\beq \nonumber
\pFq{1}{1}{-k}{\alpha}{-\frac{b}{a}\alpha z}\ ,\quad k \in \N\ ,\ z \in \R_+\ ,
\eeq
and those for the self-duality of continuum processes are even simpler, namely
\beq \nonumber
\pFq{0}{1}{-}{\alpha}{\frac{b}{a}\alpha v z}\ ,\quad v, z \in \R_+\ ,
\eeq
as the number of arguments of the hypergeometric function drops.
\medskip
The tables below schematically report all single-site (self)-duality functions for the operators $L^{\sigma,\beta}$ in \eqref{Lsigmabeta} and $\caL^{\sigma,\beta}$ in \eqref{Lsigmabetadiffusion}. Recall that the parameters $a$, $b \in \R$ in \eqref{didi} are properly chosen (cf.\ Section \ref{section particle systems duality}).
\
\begin{itemize*}
\item[(I)] \textbf{Independent random walkers ($\IRW$)\ , $\sigma=0\ $, $\beta=1\ $.}
\end{itemize*}
\medskip
\begin{center}
\begin{tabular}{c||c|c|c}
\hline
\hline
&&&\\
Classical polynomials & $\frac{n!}{(n-k)!} b^k\1\{k \leq n\}$ &$ \left(bz\right)^k$&$e^{- v} e^{ b v z}$\\
&&&\\
\hline
\hline
&&&\\
Orthogonal polynomials &$a^k C_k(n;-\frac{a}{b})$& $ \left(a + b z \right)^k$&$e^{(a-1)v}e^{b v z}$\\
&&&\\
\hline
\hline
&&&\\
Cheap duality functions &$e^{\lambda} \frac{k!}{\lambda^k}\1\{k=n \}$&$e^{\lambda-z}\left(\frac{z}{\lambda} \right)^k$&$e^{\lambda-z-v}e^{ \frac{v z}{\lambda}}$\\
&&&\\
\hline \hline
\end{tabular}
\end{center}
\bigskip
\medskip
\begin{itemize*}
\item[(II)] \textbf{Symmetric inclusion process ($\SIP(\alpha)$)\ , $\sigma=1\ $, $\beta=\alpha> 0\ $.}
\end{itemize*}
\medskip
\begin{center}
\begin{tabular}{c||c|c|c}
\hline
\hline
&&&\\
Cl. & $\frac{n!}{(n-k)!} \frac{\Gamma\left(\alpha \right)}{\Gamma\left(\alpha + k\right)} \left(b \alpha \right)^k \1\{k \leq n\}$ &$ \frac{\Gamma(\alpha)}{\Gamma(\alpha+k)} \left(b \alpha z \right)^k$&$ e^{-v} \pFq{0}{1}{-}{\alpha}{b \alpha v z}$\\
&&&\\
\hline
\hline
&&&\\
Or. &$a^k \frac{\Gamma\left(\alpha \right)}{\Gamma\left(\alpha + k \right)} M_k(n; \frac{a}{a-b\alpha}, \alpha)$& $ a^k \frac{k! \Gamma(\alpha)}{\Gamma(\alpha+k)} L_k(z; \alpha-1, -\frac{b}{a}\alpha)$&$e^{(a-1) v} \pFq{0}{1}{-}{\alpha}{b\alpha v z}$\\
&&&\\
\hline
\hline
&&&\\
Ch. &$(1-\lambda)^\alpha\frac{k!}{\lambda^k} \frac{\Gamma(\alpha)}{\Gamma(\alpha+k)} \1\{k=n \}$&$(1-\lambda)^\alpha e^{-z} \left(\frac{z}{\lambda}\right)^k \frac{\Gamma(\alpha)}{\Gamma(\alpha+k)}$&$ (1-\lambda)^\alpha e^{-z-v}\pFq{0}{1}{-}{\alpha}{\frac{v z}{\lambda}}$\\
&&&\\
\hline \hline
\end{tabular}
\end{center}
\bigskip
\medskip
\begin{itemize*}
\item[(III)] \textbf{Symmetric exclusion process ($\SEP(\gamma)$)\ ,\ $\sigma=-1\ $,\ $\beta=\gamma \in \N\ $.}
\end{itemize*}
\medskip
\begin{center}
\begin{tabular}{c||c|c|c}
\hline
\hline
&&&\\
Cl. & $\tfrac{(\gamma-k)!}{\gamma!}\tfrac{n!}{(n-k)!}(b\gamma)^k \1\{k \leq n\}$ &$ \tfrac{(\gamma-k)!}{\gamma!} \left(b \gamma z \right)^k$&$ e^{-v} \pFq{0}{1}{-}{-\gamma}{-b \gamma v z}$\\
&&&\\
\hline
\hline
&&&\\
Or. &$a^k K_k(n; -\tfrac{a}{b\gamma}, \seppar ) \left( \frac{b\gamma}{a} \right)^k \frac{1}{\binom{\seppar}{k}}$& $ a^k \pFq{1}{1}{-k}{-\gamma}{\frac{b}{a} \gamma z}$&$e^{(a-1) v} \pFq{0}{1}{-}{-\gamma}{-b\gamma v z}$\\
&&&\\
\hline
\hline
&&&\\
Ch. &$(1+\lambda)^{-\gamma}\frac{k!}{\lambda^k} \tfrac{\gamma!}{(\gamma-k)!} \1\{k=n\}$&$(1+\lambda)^{-\gamma}e^{-z}\left(\frac{z}{\lambda}\right)^k \tfrac{\gamma!}{(\gamma-k)!}$&$e^{-z-v} \left( \frac{\lambda+v z}{\lambda(1+\lambda)}\right)^{\gamma}$\\
&&&\\
\hline \hline
\end{tabular}
\end{center}
\medskip
More in detail, on the left-most column we place the single-site self-duality functions $d(k,n)$ for the particle systems of Section \ref{section setting particle}: while the top-left functions are those already appearing in \cite{gkrv}, \cite{Transport}, cf.\ also Section \ref{section setting particle} and \eqref{dknclassical} - and, thus, for this reason denoted here as the \textquotedblleft classical\textquotedblright\ ones - the second-to-the-top functions are those derived in Section \ref{section particle systems duality} in \eqref{single-site irw}--\eqref{dknhyper} and being related to suitable families of orthogonal polynomials. While these two classes of single-site self-duality functions satisfy condition \eqref{Da} (they are the only ones doing so), the bottom-left single-site self-duality functions correspond to the \textquotedblleft cheap\textquotedblright\ self-duality (cf.\ end of Section \ref{section setting lattice}), namely the detailed-balance condition w.r.t.\ the measures $\{\otimes_{x \in V} \nu_\lambda,\ \lambda > 0 \}$ with marginals \eqref{marginals}.
On the mid-column, we find the single-site duality functions between the difference operators $L^{\sigma,\beta}$ and the differential operators $\caL^{\sigma,\beta}$, obtained from their left-neighbors by a direct application of the operator $G$ in \eqref{G single-site intertwiner} on the $n$-variables. The new functions will depend hence on the two variables $k \in \N$ and $z \in \R_+$.
A second application w.r.t.\ the $k$-variables of the same operator $G$ on the functions just obtained gives us back the right-most column, functions depending now on variables $v, z \in \R_+$. These functions represent the single-site self-duality functions for the differential operator $\caL^{\sigma,\beta}$. As an immediate consequence of Proposition \ref{proposition inverse intertwining}, we could also proceed from right to left by applying the inverse intertwiner $H$ in \eqref{H single-site inverse intertwiner}.
Note that the single-site self-duality functions for $\caL^{\sigma,\beta}$ on the right-most columns, though they have been derived from different discrete analogues, i.e.\ classical, orthogonal and cheap single-site functions, within the same table they differ only for a factor which depends only on the conserved quantities $|z|=\sum_{x \in V}z_x$ and $|v|=\sum_{x \in V}v_x$. Henceforth, when proving the self-duality relation, this extra-factor does not play any role and it is enough to check that the functions
\beq \label{dvz}
d(v,z)\ =\ e^{c v z}\ ,\quad d(v,z)\ =\ \pFq{0}{1}{-}{\alpha}{c v z}\ ,\quad d(v,z)\ =\ \pFq{0}{1}{-}{-\gamma}{c v z}\ ,\quad v, z \in \R_+\ ,
\eeq
for constants $c \in \R$, are single-site self-duality functions for the operators $\caL^{0,1}$, $\caL^{1,\alpha}$ and $\caL^{-1,\gamma}$, respectively. This final computation is the content of the next proposition.
\begin{proposition}\label{proposition eee}
For any constant $c \in \R$, the functions $d(v,z)$ in \eqref{dvz} are single-site self-duality functions for the differential operators $\caL^{0,1}$, $\caL^{1,\alpha}$ and $\caL^{-1,\gamma}$, respectively.
\end{proposition}
\begin{proof}
To prove that
\beq \nonumber
d(v,z) &=& e^{c v z}\ ,\quad v, z \in \R_+\ ,
\eeq
is a single-site self-duality function for the differential operator $\caL^{0,1}$, we first observe that $$\partial_{z} d(v,z)= c v\ d(v,z)\ .$$ Hence, the self-duality relation for the single-edge generator $\caL^{1,0}_{x,y}$ rewrites
\beq \nonumber
-(v_x - v_y)(\const z_x - \const z_y) d(v_x, z_x) d(v_y, z_y)
&=& - (z_x - z_y) (\const v_x -\const v_y) d(v_x, z_x) d(v_y, z_y)\ ,
\eeq
which indeed holds.
For the second proof of self-duality for the single-site function
\beq \nonumber
d(v,z) &=& \pFq{0}{1}{-}{\alpha}{c v z}\ ,\quad v, z \in \R_+\ ,
\eeq
we use the following shortcut: for $x \in V$,
$$
F_x(\alpha) = \pFq{0}{1}{-}{\alpha}{c v_x z_x}\ .
$$
Additionally, we recall a formula for the $z_x$-derivative of $F_x$, namely
\beq \label{derivative}
\frac{\partial}{\partial z_x} F_x(\alpha) &=& \frac{c v_x}{\alpha} F_x(\alpha+1)\ ,
\eeq
and a recurrence identity
\beq\label{recurrence}
F_x(\alpha+1) &=& F_x(\alpha) - \frac{c v_x z_x}{\alpha (\alpha+1)} F_x(\alpha+2)\ .
\eeq
Hence, the self-duality relation for $\caL^{1,\alpha}_{x,y}$ w.r.t.\ the function $F_x(\alpha) F_y(\alpha)$ rewrites by using \eqref{derivative} as
\beq \nonumber
&& c z_x v_y F_x(\alpha+1) F_y(\alpha) + c v_x z_y F_x(\alpha) F_y(\alpha+1) \\
&+&\nonumber \frac{c^2 (v_x z_x)(v_y z_x)}{\alpha (\alpha+1)} F_x(\alpha+2) F_y(\alpha) + \frac{c^2 (v_x z_y)(v_y z_y)}{\alpha (\alpha+1)} F_x(\alpha)F_y(\alpha+2) \\ \nonumber
&=&c v_x z_y F_x(\alpha+1) F_y(\alpha) + c v_y z_x F_x(\alpha) F_y(\alpha+1)\\
&+&\nonumber \frac{c^2 (v_x z_x) (v_x z_y)}{\alpha (\alpha+1)} F_x(\alpha+2) F_y(\alpha) + \frac{c^2 (z_x v_y)(z_y v_y)}{\alpha (\alpha+1)} F_x(\alpha)F_y(\alpha+2)\ .
\eeq
By substituting \eqref{recurrence}, the identity holds.
The proof for the operator $\caL^{-1,\gamma}$ follows the same lines and we omit it.
\end{proof}
\subsubsection*{Acknowledgments}
F.R.\ thanks Cristian Giardin\`{a} and Gioia Carinci for precious discussions; F.S.\ thanks Jan Swart for indicating the point of view of an intertwining relation. F.S.\ acknowledges NWO for financial support via the TOP1 grant 613.001.552.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,371
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Q: Java .compareToIgnoreCase String out of index works with most inputs I wrote a method takes two strings and determines which is longer, to check for similarities between the string. When I run this code it works in most cases. But for the two strings inputs: "abcdef" & "zyxdmmmmm" there is an error: java.lang.StringIndexOutOfBoundsException: String index out of range: 6. Any ideas?
import java.util.*;
public class charCompare{
public static void main(String[] args) {
Scanner input = new Scanner(System.in);
System.out.print("Enter a word: ");
String s1 = input.next();
System.out.print("Enter another word: ");
String s2 = input.next();
charOffByOne(s1, s2);
System.out.println(charOffByOne(s1, s2));
System.out.println();
}
public static int charOffByOne(String a, String b){
int str1Length = a.length();
int str2Length = b.length();
int j = 0;
if (str1Length >= str2Length){
for(int i = 0; i < str2Length; i++ ) {
if(a.charAt(i) == b.charAt(i)){
j++;
}
}
return j;
}else{
for(int i = 0; i < str1Length; i++ ) {
if(a.charAt(i) == b.charAt(i)){
j++;
}
}
return j;
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,237
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Earthworm Express
Unusual and untold stories from the world of food science, from the present and ages past.
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Anatomy of the pig
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Artisan Bacon Flavours and great concepts – research notes
Basics of Dry Curing
Bobotie – its origins
Christmas Gammon ala Robert and Jaques
Chuck Vavra's Prasky
Concerning the Aging of Beef
Crackling
Dry Cured / Cold Smoked Bacon
Ducks and Stuphins – Sausage Makers and their OXFORD SAUSAGES
Oupa Eben's Boerewors
Recipe: Emergency Sausage
Recipe: Liver Sausage
Salumi – from the leg and some salami's
The Transition of Westphalian Hams
The Unknown Cook Book (of Steve Berman)
The Unknown Cook Book (of Steve Berman): Liver Sausage
The Unknown Cook Book (of Steve Berman): Pudding Scotch to Saleicca, Napoli – including Salami
Bacon & the Art of Living
Chapter 1: Once upon a time in Africa
Chapter 2: Dry Cured Bacon
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Chapter 5: Seeds of War
Chapter 6: The Greatest Adventure
Chapter 7: Woodys Bacon
Chapter 8.00: The Denmark Letters
Chapter 8.01 – Mild Cured Bacon
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Chapter 8.03 Minette, the Cape Slaves, the Witels and Nitrogen
Chapter 8.04 The Saltpeter Letter
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Chapter 8.08 Von Liebig and the Theory of Proteins of Gerard Mulder.
Chapter 8.9 David Graaff's Armour – A Tale of Two Legends
Chapter 9.00: The UK letters
Chapter 9.01: Lord Landsdown
Chapter 9.02 – Sweet Cured Irish and Wiltshire Pork
Chapter 9.03: American Icehouses for England: Year-round Curing
Chapter 9.04: Ice Cold in Africa
Chapter 9.05: Ice Cold Revolution
Chapter 9.06: Harris Bacon – the Gold Standard!
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The Anglo-Boer War and Bacon: Old enemies become friends
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Honey in cured meat formulations
Joshua Penny and Khoikhoi Fermentation Technology at the Cape of Good Hope
Kolbroek and the Kune Kune
In Search of the Origins of the Kolbroek
Kolbroek – Chinese, New Zealand, and English Connections
The Old and the New Pig Breeds
Mechanisms of Meat Curing
Difference between Fresh Cured and Cooked Cured Colour of Meat.
Mechanisms of meat curing – the important nitrogen compounds
Reaction Sequence: From nitrite (NO2-) to nitric oxide (NO) and the cooked cured colour.
xResearch notes – nitrosating agents
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Stocks Meat Market
Nguni: Living Works of Art
The Domestication of Cattle and Eurocentrisity in Regards to Heritage Breeds
Nitrate and Nitrite – their history and functionality
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02. Concerning Chemical Synthesis and Food Additives
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05. Concerning Ladislav NACHMÜLLNER and the invention of the blend that became known as Prague Salt.
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Pic's history of meat processing
Project – Legendary Foods
Porchetta ala Africa
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The Fascinating History and Use of Ascorbate
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Regulations of Nitrate and Nitrite post-1920's: the problem of residual nitrite and the introduction of ascorbate
The History of Bacon, Ingredients and related technologies
Bacon Curing – a historical review
Counting Nitrogen Atoms – The History of Determining Total Meat Content
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Fathers of Meat Curing
JOANNE GIBSON: ON SOME OF THE 'INVISIBLE' PEOPLE OF EARLY CAPE WINE
Mild Cured Bacon – Recreating a Legend
Saltpeter, Horse Sweat, and Biltong: The origins of our national food.
Saltpeter: A Concise History and the Discovery of Dr. Ed Polenske
Tank Curing Came From Ireland
Occurrences of "mild cure" in English Newspapers
The Mother Brine
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The Master Butcher from Prague, Ladislav NACHMÜLLNER
1. Eva's Beloved Dad
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3. Ladislav NACHMÜLLNER vs The Griffith Laboratories
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01. How did Ancient Humans Preserve Food?
02. An Introduction to the Total Work on Salt, Saltpeter and Sal Ammoniac – Salt before the Agriculture Revolution
The Present and Future of Food Processing
Best Bacon System on Earth
Curing Brine and Microbial Transglutaminase (MTG) – designing the optimal blend
Factors Affecting Colour Development and Binding in a Restructuring System Based on Transglutaminase
Microbial Transglutaminase system: Applications and Clever Ideas
Restructuring of whole muscle meat with Microbial Transglutaminase – a holistic and collaborative approach.
The Rib Project
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Protein – The mother of all functionality!
Nitrite Free Bacon: Barriers against clostridium botulinum
Soy or Pea Protein and what in the world is TVP?
Soya: Review of some health concerns and applications in the meat industry
The Freezing and Storage of Meat
Freezing for Slicing Bacon
Weight Loss During Chilling and Freezing of Meat
The Salt Bridge
01. Salt – 7000 years of meat-curing
02. Nitrate salt's epic journey: From Turfan in China, through Nepal to North India
03. And then the mummies spoke!
04. The Sal Ammoniac Project
06. Searching for Salt in New Zealand
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08. Salt and the Ancient People of Southern Africa
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About eben
HomeRegulations of Nitrate and Nitrite post-1920: the problem of residual nitrite
Regulations of Nitrate and Nitrite post-1920: the problem of residual nitrite
September 2, 2017 eben van tonder recent history
By Eben van Tonder
Breseola, by Jason Osburn.
I am working in a pork processing plant. I am not a scientist but I love knowing how things work and have a special love for chemistry. The health considerations of the use of nitrite and nitrate together in a bacon curing brine are the issue brought about by the fact that it is allowed in South Africa. I approach the issue from the standpoint of tracking the issues of residual nitrite from a historical perspective. The entire matter of phenomenally complex and this is at best notes on my personal introduction to the issue.
In South Africa commercial curing brines are still being sold that contain both nitrates and nitrites. In Europe and America, this is not allowed. I examine the historical context and the health concerns that gave rise to limiting residual nitrite since the 1920's and following the N-nitrosamine hysteria of the 1970's, the introduction of ascorbate or erythorbate in nitrite brines and the fact that it was made illegal to use nitrate with nitrite in certain classes of bacon. The matter of preservatives in food is always a balancing act. On the one hand is the potential negative influence on human health of the preservative and on the other hand are the pathogens that the preservative protects us from. It will become clear that the exact same is at issue in the consideration of nitrite. A second important aspect is always that of dosage. Preservatives, in high dosages, are harmful to human health, but at low dosages are less problematic. An example is alchol which is allowed in human food and drink even though at high dosages, it is harmful. Nitrite falls somewhat in this category, but the reality of N-nitrosamine formation warrants a far more comprehensive approach to minimise it in bacon than only limiting the dosage of nitrite (even though this by itself is an important strategy).
SALTPETER (NITRATE)
The story of bacon is the story of nitrate, nitrite and nitric oxide and its ability to preserve meat, imparting a particular cured meat flavour and a characteristic pinkish-reddish cooked-cured meat colour.
The earliest curing agent that imparted these qualities to the meat was nitrate. The chemical formula for nitrate is . It is used as an ingredient in gunpowder, as fertiliser and to cure meat. It was one of the earliest and most enigmatic salts known from antiquity both in the East and the West. There is a long line of inquiry to try and determine what gives this amazing compound its unique power. In the West, some scholars likened it to the character of the triune God himself and in the East, in China, it was seen by some as one of the components of the elixir of immortality.
The metals most generally reacting with nitrate ( ) are the alkali metals, potassium to form saltpeter and sodium to form Chlian saltpeter and sometimes the alkali earth metal, calcium to form calcium nitrate, originally known as Norwegian Saltpeter. It is most abundantly found naturally in arid regions around the world and it can be made by human action. By the 1800's, the technology to produce it was so widespread among German farmers that authors, writing about it, did not even bother to describe the techniques used.
The preserving power of saltpeter is something which I suspect was noticed by ancient civilizations from as early as 5000 BCE. (Salt – 7000 years of meat-curing) There is no doubt that these civilizations also noticed its ability to cause meat colour to change from dull brown as it oxidises after slaughter, back to a reddish-pinkish colour and to impart a particular appealing cured meat taste.
It is probable that art of meat curing developed in desert regions in China, probably in the vast Taklimakan Desert in Western China, in the Tarim Bason from where it spread into the heart of Europe. The nations of Germany, Italy, Spain, Denmark, Holland, England, and Ireland adopted meat curing which was done with salt and a little bit of saltpeter. In Europe, saltpeter became universal in its inclusion along with salt in meat curing between 1600 and 1750, probably near 1700. (Lauer K., 1991)
There is evidence to suggest a link between the use of saltpeter and human disease from very early on. Klaus Lauer (1991) analysed cook books from Germany and Austria between 1540 and 1900 and found some historical parallels between the use of saltpeter in foods and the appearance of colorectal cancer and multiple sclerosis. Lauder writes, "According to summarizing works on the history of cancer, large bowel cancer was only seldom reported in the antiquity and Middle-Ages. Its first detailed depictions date from the early 19th century, followed by a rapidly increasing number of reports during the 19th century." He notes that "it is of particular interest at what time nitrate or saltpetre was first used in human nutrition," being at around the same time and increasing as the use of saltpeter in food increased. (Lauer K., 1991) There is, however, no record that this was ever noticed at the time.
As alarming as this is, it should be noted that this may prove nothing more than the danger of uncontrolled or highly irregular nitrite levels in meat curing brought about by the use of saltpeter. Adding nitrate (saltpeter) is, in fact, adding nitrite. Later we will see that the exact opposite is also true namely that adding only nitrite, is at the same time adding nitrate also since just as bacterial reduction changes nitrates to nitrites, so chemical reactions in the meat matrix change nitrites into nitrates through oxidation. More about this later.
The people may not have noticed the health impact on the overall population of the use of nitrates, but something that was noticed is the fact that the use of saltpeter in food prevents foodborne toxins. Kerner in Germany found in a series of studies conducted in 1817, 1820 and 1822 that the outbreaks of sausage poisoning or botulism are linked to the omission of nitrate (saltpeter) in the salt mixture to cure meat for sausage production. Botulism is an often fatal foodborne disease from the toxins produced by a bacteria, Clostridium Botulinum. (Frences, M. P., et al. 1981)
In a time of superstition and secret remedies, saltpeter was regarded with awe and wonder. (Saltpeter: A Concise History and the Discovery of Dr. Ed Polenske) A newspaper report from 1898 says that saltpeter miners working in a cave has "remarkable health." (The Wyandotte Herald, Kansa City, Kansas, 7 April 1898, "Air in Mammoth Cave")
So, despite the fact that we can look back at the time before 1920, and see that there may have been an impact from the increased use of saltpeter on the general health of the German and Austrian population, and by extension, on the European population, such a link was probably not obvious. The general view of saltpeter was favourable.
From the late 1800's, scientists started to work out that the saltpeter was not the real curing agent, but its cousin, the far more toxic compound, nitrite ( ). It started to emerge that it has a more direct impact on curing than nitrate.
Contrary to the generally positive view of saltpeter pre-1900's, nitrite was viewed by the public in the most negative light. Their view was not completely unfounded because it is estimated that nitrite is 10 times more toxic than nitrate, but what the general public did not understand was that nitrate became nitrite after a time and that the nitrate from their darling of all salts, sodium or potassium nitrate, turned into the villain, nitrite.
Scientists experimented from early on in its direct use in meat curing. A private laboratory in Germany, founded in 1848 by C.R. Fresenius recorded, for example, experimented with sodium nitrite as curing agent. (Concerning Chemical Synthesis and Food Additives) THis is the earliest experiment I have been able to locate thus far where nitrite is used as a preservative in meat.
The Russian botanist and microbiologist, Sergei Nikolaievich Winogradsky (1856 – 1953) identified a class of bacteria which oxidizes ammonia ( ) or ammonium ( ) to nitrite ( ) and nitrite to nitrate ( ). The process is called nitrification.
E. Meusel (1875) discovered another class of microorganisms living in soil and natural waters which reduce nitrates to nitrites and even further. (Meusel, E. 1875) It was found that in the absence of oxygen, these microbes use and thus reduces nitrates ( ) to nitrite ( ) in their metabolic processes. In 1790 Antoine de Lavoisier (1743 – 1794) named the compound nitrite "as they are formed by nitric or by nitrous acid." (Lavoisier, A; 1965: 217) Thus two different compounds exist with only a small change in the spelling namely nitrate and nitrite, indicating the fact that nitrite has one less oxygen atom compared to nitrate. The loss of one oxygen atom, however, renders the molecule far more reactive, increasing its toxicity 10 times.
In 1891, Eduard Polenske, working for the Imperial Health Office, analysed cured meat for its nutritional value and noted that the nitrate in the curing brine and in the meat changed to nitrites. He predictably and correctly speculated that this was due to microbial activity, identified by Meusel, 16 years earlier. In his article, he predicted that his the expected reduction would cause an outcry.
Academic work continued uncovering a much closer relationship between the toxic nitrite and meat curing than the darling of the sciences, saltpeter. The German scientist, Nothwang confirmed the presence of nitrite in curing brines in 1892. In 1899, another German scientist, K. B. Lehmann confirmed that the cured colour was linked to nitrite and not saltpeter. Yet another German hygienists, one of Lehmann's assistant at the Institute of Hygiene in Würzburg, Karl Kißkalt (1875 – 1962), confirmed Lehmann's observations and showed that the same red colour resulted if the meat was left in saltpeter (potassium nitrate) for several days before it was cooked, thus confirming Polenske's notion of bacterial reduction of nitrate to nitrite which finally cures the meat. S. J. Haldane showed that nitrite is further reduced to nitric oxide (NO) in the presence of muscle myoglobin and forms iron-nitrosyl-myoglobin. It is nitrosylated myoglobin that gives cured meat, including bacon and hot dogs, their distinctive red colour and protects the meat from oxidation and spoiling.
The work of these scientists was enough evidence for the German government and in 1909, probably due to the negative views of the public towards nitrite, they legalised only the use of a partially reduced form of nitrates in curing mixes which were marketed across Europe. (Bryan, N. S. et al, 2017: 86 – 90)
The Danish invention
The Danes applied their knowledge of bacterial reduction of nitrate to nitrite and developed a curing method where they reused brine that was "reduced" to nitrite already. They allowed fresh brine to be continually introduced into the system, bacterial reduction to take place and thus supplemented the nitrite concentration of the previously used brine. This had the additional benefit of "seeding" new brine with just the right bacteria required for nitrite reduction.
According to this method they first injected fresh brine consisting of salt and saltpeter (potassium nitrate) into meat. They then left the meat for several days in a cover brine. The cover brine was never changed and came to be known as the "mother brine." It was their source of nitrite that was directly applied to the curing process. The mother brine was strained and boiled before it was re-used to eliminate pathogenic bacteria.
Clues to the date of the Danish invention come to us from newspaper reports about the only independent farmer-owned Pig Factory in Britain of that time, the St. Edmunds Bacon Factory Ltd. in Elmswell. The factory was set up in 1911. According to the newspaper reports they learned and practiced what at first was known as the Danish method of curing bacon and later became known as tank-curing or Wiltshire cure. A person was sent from the UK to Denmark in 1910 to learn the new Danish Method. (elmswell-history.org.uk) This Danish method involved the Danish cooperative method of pork production founded by Peter Bojsen on 14 July 1887 in Horsens. The newspaper reports talked about a "new Danish" method. The "new" aspect in 1910 and 1911 was undoubtedly the tank curing method.
Another account from England puts the Danish invention of tank curing early in the 1900's. C. & T. Harris from Wiltshire, UK, switched from dry curing to the Danish method during this time. In a private communication between myself and the curator of the Calne Heritage Centre, Susan Boddington, about John Bromham who started working in the Harris factory in 1920 and became assistant to the chief engineer, she writes: "John Bromham wrote his account around 1986, but as he started in the factory in 1920 his memory went back to a time not long after Harris had switched over to this wet cure."
So, early in the 1900's, probably sometime between 1899 and 1910, the Danes invented and practiced tank-curing which was brought to England around 1911 based on the work of the fathers of our current method of meat curing.
The German/ Austro-Hungarian invention
Where Denmark focused on harnessing the power of old brine, in Germany they were toying with the idea of using sodium nitrite as their source of nitrite. Sodium nitrite was at this time used extensively in an intermediary step in the lucrative coal tar dye industry that flourished in Germany and in the Austrian-Hungarian empire, notably around the city of Prague. There was a second use of sodium nitrite in medicine. It was expensive to produce and viewed with much skepticism by the general public for use in food on account of its high toxicity. (Concerning the direct addition of nitrite to curing brine)
It was the First World War that provided the transition events that caused the sodium nitrite to end up being used as the source of nitrite in curing brines in Germany where its use in food was still illegal. Saltpeter was reserved for the war effort being one of the main components used in manufacturing of gunpowder and was consequently no longer available as curing agent for meat during World War One. (Concerning the direct addition of nitrite to curing brine)
In August 1914, the War Raw Materials Department (Kriegsrohstoffabteilung or KRA) was set up under the leadership of Walther Rathenau. It was Rathenau who was directly responsible for the prohibition on the use of salpeter. He, therefore, is the person in large part responsible creating the motivation for the meat industry in Germany to change from saltpeter to sodium nitrite as curing medium of choice for the German meat industry during World War One. (Concerning the direct addition of nitrite to curing brine)
The first country to legalise the use of nitrite directly was the Austro-Hungarian Empire and in 1915. At age 19, Ladislav Nachmüllner invents Praganda, the first legal commercial curing brine containing sodium nitrite in the city of Prague. He says that he discovered the power of sodium nitrite through "modern-day professional and scientific investigation." He probably actively sought an application of the work of Haldane. He quotes the exact discovery that Haldane was credited for in 1901 that nitrite interacts with the meat's "haemoglobin, which changes to red nitro-oxy-haemoglobin." (The Naming of Prague Salt)
By 1917 nitrite was not only used for curing meat in Germany, but proprietary meat cures containing nitrites were being marketed across Europe. (Concerning Chemical Synthesis and Food Additives)
Developments in the United States
Both these methods were being looked at very closely in the United States around this time.
The first recorded direct use of sodium nitrite as a curing agent in the USA was in a secret experiment in 1905. The USDA approved its use as a food additive in 1906. (Concerning the direct addition of nitrite to curing brine)
A court case was brought by the US Federal Government against the Mill and Elevator Company of Lexington, Nebraska. The charge was that they adulterated and misbranded flour and sold it to a grocer in Castle, Missouri. The case was brought by the government under the pure food and drug act of 1906. (Chicago Daily Tribune ; 7 July 1910; Page 15) The government contended that "poisonous nitrites are produced in the flour by bleaching." This is one example of the gigantic controversy that raged around the world about the use of nitrites in food and the careful work that was done by the US government in the 1920, 30's and onwards, was in the first place in dealing with the known high toxicity of nitrites.
In 1915, George F. Doran of Omaha, Nebraska, filed a patent for using "sterilized waste pickling liquor which he discovered contains soluble nitrites produced by conversion of the potassium nitrate, sodium nitrate, or other nitrate of the pickling liquor when fresh, into nitrites. As such his patent involved taking waste pickling liquor from the cured meats." This is the same concept as tank curing invented in Denmark sometime before 1910 and probably after 1899. He states the objective of his invention as "to produce in a convenient and more rapid manner a complete cure of packing house meats; to increase the efficiency of the meat-curing art; to produce a milder cure; and to produce a better product from a physiological standpoint." (US 1259376 A)
Despite the obvious advantage of a far quicker curing time of the use of sodium nitrite had over the tank cured Danish method, the fact that Doran still took the trouble to register the patent for a tank curing method in 1915 makes sense if one considers that tank-curing or the Wiltshire curing process became widespread in application in England.
The problem with the Danish and later, Engish system was, of course, shelf life due to the high microbial load from the mother brine and the uncontrolled nature of the process of nitrite formation (nitrite levels have been shown to range between 2 and 960 ppm in products cured using this method). (Bryan, N. S. et al, 2017: 86 – 90)
In 1923, the Bureau of Animal Industry commissioned a study to investigate the direct addition of nitrite for meat curing. Kerr, et al, under the supervision of inspectors from the Bureau of Animal Industry, cured hams with approximately 2000 ppm nitrite in the curing mix. The first issue they investigated was to compare nitrite curing with nitrate curing from the standpoint of organoleptic equivalence and if excess amounts of nitrite are required for nitrite curing.
They also looked at the amount of nitrite that was left in the meat after sufficient curing took place, thus introducing the concept of residual nitrite. These they compared with the amount of nitrite that was in the curing brine. The question was how much nitrite is required to cure meat. It was known that nitrite is a more powerful toxin than nitrate; it was further known that using nitrate instead of nitrite caused inconsistent nitrite levels in the curing brine and in the meat. By understanding the amount of nitrites that typically react in the meat to form nitric oxide and to cure the meat and by taking that as the limit of nitrite that can be added directly, one, therefore, minimises the risk of having consumers ingest nitrites.
By 1925 a document was prepared by the Chicago based organisation, The Institute of American Meat Packers and published in December of this year. The Institute started as an alignment of the meat packing companies set up by Phil Armour, Gustavus Swift, Nelson Morris, Michael Cudahy, Jacob Dold and others with the University of Chicago. (Concerning the direct addition of nitrite to curing brine)
A newspaper article about the Institute sets its goal, apart from educating meat industry professionals and new recruits, "to find out how to reduce steers to beef and hogs to pork in the quickest, most economical and the most serviceable manner." (The Indiana Gazette. 28 March 1924). In this statement is the clue to the reason of its dominance in the United States where bigger, better and faster was the call to arms for the new world's industries.
The document is entitled, "Use of Sodium Nitrite in Curing Meats", and it it is clear that the direct use of nitrites in curing brines has been practiced from earlier than 1925. (Industrial and Engineering Chemistry, December 1925: 1243)
The article begins "The authorization of the use of sodium nitrite in curing meat by the Bureau of Animal Industry on October 19, 1925, through Amendment 4 to B. A. I. Order 211 (revised), gives increased interest to past and current work on the subject." Sodium Nitrite curing brines would, therefore, have arrived in the USA, well before 1925.
The rest of the opening paragraph continues to elaborate on the reason for its preference. "It is now generally accepted that the salpteter added in curing meat must first be reduced to nitrite, probably by bacteria, before becoming available as an agent in producing the desirable red color in the cured product. This reduction is the first step in the ultimate formation of nitrosohemoglobin, the color principle. The change of nitrate to nitrite is by no means complete and varies within considerable limits under operating conditions. Accordingly, the elimination of this step by the direct addition of smaller amounts of nitrite means the use of less agent and a more exact control."
The 1926 study by Kerr and co-workers, was done long before there was any link established between nitrite and the formation of cancer-causing substances upon frying and ingestion. This only emerged in the '70's. In 1926, the work was based on the general knowledge of nitrite's toxicity and the publics very negative perceptions about it. In the report, they state that public health was the primary motivation behind the study. (Kerr, et al, 1926 : 543) The test was a step in the right direction – towards defining and limiting residual nitrite.
I quote from their report. "The first experiment involving the direct use of nitrite was formally authorized January 19, 1923, as a result of an application by one of the large establishments operating under Federal meat inspection. Before that time other requests for permission to experiment with nitrite had been received but had not been granted. The authorization for the first experiment specified that the whole process was to be conducted under the supervision of bureau inspectors and that after the curing had been completed the meat was to be held subject to laboratory examination and final judgment and would be destroyed if found to contain an excessive quantity of nitrites or if in any way it was unwholesome or unfit for food. This principle was rigidly adhered to throughout the experimental period, no meat being passed for food until its freedom from excessive nitrites had been assured, either by laboratory examination or through definite knowledge from previous examinations, that the amount of nitrite used in the process would not lead to the presence of an excessive quantity of nitrites in the meat. By "excessive^ is meant a quantity of nitrite materially in excess of that which may be expected to be present in similar meats cured by the usual process." (Kerr, et al, 1926 : 543)
An interesting side note is the fact that this fixes the date of the first official experiment using nitrites ever conducted in the United States. There can be little doubt that the large packing plants in Chicago used nitrites directly in meat curing, long before this, at least from 1918, following World War 1. I have even received reports of the first unofficial experiment of this nature that was done in 1905, presumably also in Chicago. (Concerning the direct addition of nitrite to curing brine) The large establishment who applied for the permit would have been one of the following list of packers, the Armour Packing operation, Morris & Company, Cudahy Packing Company, Wilson Packing Plant or Swift Packing. These four companies, at the time, were some of the largest and most powerful corporations on earth.
"The maximum nitrite content of any part of any nitrite-cured ham [was found to be] 200 parts per million. The hams cured with nitrate in the parallel experiment showed a maximum nitrite content of 45 parts per million." (Kerr, et al, 1926 : 543) The conclusion was that "hams and bacon could be successfully cured with sodium nitrite, and that nitrite curing need not involve the presence of as large quantities of nitrite in the product as sometimes are found in nitrate- cured meats." (Kerr, et al, 1926 : 545)
Related to the health concerns, the report concluded the following:
"The presence of nitrites in cured meats, was already sanctioned by the authoritative interpretation of the meat inspection and pure food and drugs acts sanctioning the use of saltpeter; as shown previously, meats cured with saltpeter and sodium nitrate regularly contain nitrites. (Wiley, H, et al, 1907) (Kerr, et al, 1926 : 550)
The residual nitrites found in the nitrite-cured meats were less than are commonly present in nitrate-cured meats. The maximum quantity of nitrite found in nitrite-cured meats, in particular, was much smaller than the maximum resulting from the use of nitrate. (Kerr, et al, 1926 : 550)
The nitrite-cured meats were also free from the residual nitrate which is commonly present in nitrate-cured meats. (Kerr, et al, 1926 : 550)
On the contrary, the more accurate control of the amount of "nitrite and the elimination of the residual or unconverted nitrate are definite advantages attained by the substitution. (Kerr, et al, 1926 : 550)
Following further studies, the Bureau set the legal limit for nitrites in finished products at 200 parts per million. (Bryan, N. S. et al, 2017: 86 – 90)
Conventional wisdom that surfaced in the 1920's suggested that nitrate and nitrate should continue to be used in combination in curing brines (Davidson, M. P. et al; 2005: 171) as was the case with the Danish curing method and the mother brine concept of the previous century. Nitrite gives the immediate quick cure and nitrate acts as a reservoir for future nitrite and therefore prolongs the supply of nitrite and ensures a longer curing action. This concept remained with the curing industry until the matter of N-nitrosamines came up in the 1960's and 70's, but remarkably enough, it still persists in places like South Africa where to this day, using the two in combination is allowed for bacon.
The USDA progressed the ruling on nitrate and nitrites further in 1931 by stating that where both nitrites and nitrates are used, the limit for nitrite is 156 ppm nitrite and 1716 nitrate per 100lb of pumped, cured meat. (Bryan, N. S. et al, 2017: 86 – 90)
1960's – N-Nitrosamine
Up to the 1960's the limit on the ingoing level of nitrites was based on its toxicity. In the late 1950's an incident occurred in Norway involving fish meal that would become a health scare rivaled by few in the past. 1960's researchers noticed that domestic animals fed on a fodder containing fish meal prepared from nitrite preserved herring were dying from liver failure. Researchers identified a group of compounds called nitrosamines which formed by a chemical reaction between the naturally occurring amines in the fish and sodium nitrite. Nitrosamines are potent cancer causing agents and their potential presence in human foods became an immediate worry. An examination of a wide variety of foods treated with nitrites revealed that nitrosamines could indeed form under certain conditions. Fried bacon, especially when "done to a crisp," consistently showed the presence of these compounds. (Schwarcz, J) In bacon, the issue is not nitrates, but the nitrites which form N-nitrosamines.
This fundamentally sharpened the focus of the work of Kerr and co-workers of the 1920's in response to the general toxicity of nitrites to the specific issue of N-nitrosamine formation. Reviews from 1986 and 1991 reported that "90% of the more than 300 N-nitroso compounds that have been tested in animal species including higher primates causes cancer, but no known case of human cancer has ever been shown to result from exposure to N-nitroso compounds." However, despite this, there is an overwhelming body of indirect evidence that shows that a link exists and "the presence of N-nitroso compounds in food is regarded as an etiological risk factor. It has been suggested that 35% of all cancers in humans are dietary related and this fact should not surprise us. (Pegg and Shahidi, 2000)
Studies have been done showing that children who eat more than 12 nitrite-cured hot dogs per month have an increased risk of developing childhood leukemia. The scientists responsible for the findings themselves cautioned that their findings are preliminary and that much more studies must be done. It may nevertheless be a good approach for parents to reduce their own intake of such products along with that of their children in cases where intake is high. (Pegg and Shahidi, 2000)
These studies must be balanced by the fact that an overwhelming amount of data has been emerging since the 1980's that indicate that N-nitroso compounds are formed in the human body. What is important is that we keep on doing further research on N-nitrosamines and the possible link to cancer in humans. Not enough evidence exists to draw final conclusions.
1970 – The response to the N-Nitrosamine scare.
Back to the 1970's, so grave was the concern of the US Government about the issue that in the early 1970's they seriously considered a total ban on the use of nitrites in foods. (Pegg and Sahidi, 2000) The response to the N-nitrosamine issue was to go back to the approach that was implemented following the work of Kerr and co-workers in 1926.
The first response was to eliminate nitrate from almost all curing applications. The reason for this is to ensure greater control over the curing. Meat processors continued to use nitrate in their curing brines after 1920 until the 1970's. One survey from 1930 reported that 54% of curers in the US still used nitrate in their curing operations. 17% used sodium nitrite and 30% used a combination of nitrate and nitrite. By 1970, 50% of meat processors still used nitrate in canned, shelf-stable. In 1974 all processors surveyed discontinued the use of nitrates in these products including in bacon, hams, canned sterile meats, and frankfurters. One of the reasons given for this change is the concern that nitrate is a precursor for N-nitrosamine formation during processing and after consumption. (Bryan, N. S. et al, 2017: 86 – 90)
The reason for the omission in bacon, in particular, is exactly the fact that the nitrates will, over time continue to be converted to nitrites which will result in continued higher levels of residual nitrites in the bacon compared to if only nitrite is used. The N-nitrosamine formation from nitrites is a reaction that can happen in the bacon during frying or in the stomach after it has been ingested. It will not happen from the more stable nitrates.
It has been discovered that nitrate continues to be present in cured meats. Just as the view that if nitrate was added, no nitrite is present in the brine as was the thinking in the time before the early and mid-1800's, in exactly the same way it is wrong to think that by adding nitrite only to meat, that no nitrate is present. "Moller (1971) found that approximately 20% of the nitrite added to a beef product was converted to nitrate within 2 hours of processing. Nitrate formation was noted during incubation before thermal processing, whereas after cooking only slight nitrate formation was detected. Upon storage, the conversion of nitrite to nitrate continued. Herring (1973) found a conspicuous level of nitrate in bacon formulated only from nitrite. As greater concentrations of nitrite were added to the belly, a higher content of nitrate was detected in the finished product. They reported that 30% of the nitrite added to bacon was converted to nitrate in less than one week and the level of nitrate continued to increase to approximately 40% of the added nitrite until about 10 weeks of storage. Moller (1974) suggested that when nitrite is added to meat, a simultaneous oxidation of nitrite to nitrate and the ferrous ion of to the ferric ion of metMb occurs." Adding ascorbate or erythorbate plays a key role in this conversion. (Pegg and Shahidi, 2000) The issue is not the nitrate itself, but the uncontrolled curing that results from nitrate and the higher residual nitrites.
Secondly, the levels of ingoing nitrite were reduced, especially for bacon. The efficacy of these measures stems from the fact that the rate of N-nitrosamine formation depends on the square of the concentration of residual nitrites in meats and by reducing the ingoing nitrite, the residual nitrite is automatically reduced and therefore the amount of N-nitrosamines. (Pegg and Sahidi, 2000) Legal limits were updated in 1970 in response to the nitrosamine paranoia. A problem with this approach is however that no matter by how much the ingoing nitrite is reduced, the precursors of N-Nitrosamine still remains in the meat being nitrites, amines, and amino acids.
An N-nitrosamine blocking agent was introduced in the form of sodium ascorbate or erythorbate. "There are several scavengers of nitrite which aid in suppressing N-nitrosation; ascorbic acid, sodium ascorbate and erythorbate have been the preferred compound to date. Ascorbic acid inhibits N-Nitrosamine formation by reducing to give dehydroascorbic acid and NO. Because ascorbic acid competes with amines for , N-Nitrosamine formation is reduced. Ascorbate reacts with nitrite 240 times more rapidly than ascorbic acid and is, therefore, the preferred candidate of the two. (Pegg and Sahidi, 2000)
More detailed studies identified the following factors to influence the level of N-nitrosamine formation in cured meats. Residual and ingoing nitrite levels, preprocessing procedure and conditions, smoking, method of cooking, temperature and time, lean-to-adipose tissue ratio and the presence of catalyst and/ or inhibitors. It must be noted that in general, levels of N-nitrosamines formation has been minuscule small, in the billions of parts per million and sporadic. The one recurring problem item remained fried bacon. In its raw state bacon is generally free from N-nitrosamines "but after high-heat frying, N-nitrosamines are found almost invariably." One report found that "all fried bacon samples and cooked-out bacon fats analyzed" were positive for N-nitrosamines although at reduced levels from earlier studies. (Pegg and Sahidi, 2000)
Regulatory efforts since 1920 have shown a marked decrease in the level of N-nitrosamines in cured meats, even though it is still not possible to eliminate it completely. "Cassens (1995) reported a marked decrease (approx 80%) in residual nitrite levels in of US prepared cured meat products from those determined 20 years earlier; levels in current retail products were 7 mg/kg from bacon." This and similar results have been attributed to lower nitrite addition levels and the increased use of ascorbate or erythorbate. (Pegg and Sahidi, 2000)
Current USA regulations and tightening the measures from the 1970's.
Nitrite can be used in foods and nitrate, very selectively based on the product category and the method of curing. Immersion cured, massaged or pumped products (example hams or pastrami) – maximum ingoing level of 200 ppm sodium or potassium nitrite and/ or 700 ppm nitrate based on raw product weight. (Bryan, N. S. et al, 2017: 86 – 90)
Dry cured products – a maximum of 625 ppm ingoing nitrite, and/ or 2187 ppm nitrate since the products have long curing times that result in immediate nitrite reaction with myoglobin and longer term conversion of nitrate to nitrite. (Bryan, N. S. et al, 2017: 86 – 90)
Comminuted products such as Frankfurters, Bologna, and other cured sausages – maximum ingoing nitrite level of 156 ppm sodium or potassium nitrite based on raw meat block. Nitrite can be added to all these at a rate of 1718 ppm regardless of salt used. (Bryan, N. S. et al, 2017: 86 – 90)
1978 bacon levels – USA
Inoing nitrite levels were reduced in 1978 and a required limit was set for ascorbate. It was also explicitly ruled that nitrate may not be used in bacon production.
-Maximum ingoing level of sodium nitrite – 120 ppm
-Maximum ingoing level of potassium nitrite – 148 ppm
-547 ppm ascorbate or erythorbate must be added.
(Bryan, N. S. et al, 2017: 86 – 90)
In the USA, in 1980, the National Acadamy of Sciences (NSA) entered into a contract with the USDA and FDA. They established the Committee on Nitrate and Alternative Curing Agents in Food. The brief of the committee was to investigate the health risk associated with the overall exposure to nitrate, nitrite and N-nitroso compounds. They published a report, "The Health Effects of Nitrite and N-nitroso Compounds."
They found nitrate not to be carcinogenic or mutagenic. It was found that certain populations showed an association of an exposure to high nitrate levels and certain cancers. More studies are required.
Nitrite was similarly found not to act directly as a carcinogen in animal studies and more studies are necessary.
The committee recognised the use of nitrite as an effective barrier against foodborne botulism, thus validating the continued use of nitrites in meat curing. They also put the overall risk in perspective by estimating the lifetime risk of cancer from cured meats to be one in a million if "humans were exposed to a daily dose of 5.8 to 19 ng of nitrosodimethylamine per ki~ogram of body weight or 0.85 to 2. 7 ng of nitrosodimet~lamine per em of body surface. In arriving at this estimate, the committee has also assumed that (1) the dietary doses given to rats can be converted to unit of dose per unit of body weight or per unit of body surface area to reflect human exposure and (2) that nitrosodimethylamine is the main source of exposure to nitrosamine& for humans and is, therefore, representative of all nitrosamine&, even though its potency in animals is greater than that of many other nitrosamine."
"The committee also examined seven pothetical population groups and estimated that the lifetime risk of cancer from exposure to all sources of nitrosamines would be 820 to 18,000 per million for a high risk group (including occupational exposure), 11 to 250 in a million for a high cured meat diet group, 8 to 180 in a million for an average population of nonsmokers, and 3 to 74 in a million for a low risk group."
A specific recommendation, relevant to the question of the use of nitrates in curing brines is that the use of nitrates in curing systems should be eliminated with the exception of products where a long curing time is required. The reason is that nitrate and nitrates can have acute toxic effects and contribute to the formation of N-nitrose compounds.
Despite all this, the committee found that the prudent approach is to continue to use nitrite as a proven and effective hurdle to prevent the outgrowth of Clostridium Botulinum spores and the accompanying toxin formation. It is in the publics interest to assume that temperature and other product abuses will take place and using nitrite as a hurdle remains a reasonable measure.
Here is the pdf copy of the entire report: The Health Effects of Nitrite and N-nitroso Compounds
New 1986 bacon levels – USA – testing the limits of the system
The general trend of reducing residual nitrate and limiting N-nitrosamine formation continued. The danger, however, exists that ingoing nitrite levels may be reduced so dramatically as to compromise its function as a barrier against botulism. The danger of nitrites and its benefits must always be held in balance.
-Skinless bacon – the requirements were kept at the 1978 levels, but it was explicitly emphasised that these ruling applies in order to reduce the possibility of N-nitrosamine formation. A practical measure was introduced which allows for an approximately 20% variance is allowed from the ingoing nitrite on injection or massaging (96-144 ppm).
In order to ensure efficacy against pathogens, sodium nitrite can be reduced to 100 ppm (123 ppm potassium nitrite) with "appropriate partial quality control program." If sugar and a starter culture are added to the brine, 40 – 80 ppm sodium nitrite (49 – 99 potassium nitrite).
Dry cured bacon – the limit was set at 200 ppm nitrite or 246 potassium nitrite.
EU Rules, Directive 95/2/EC, modified in Directive 2006/52/EC
Maximum ingoing level of bacon is 150 ppm nitrite; max residue level at between 50 and 175 ppm. (in Denmark, this limit is lower at 60 – 150 ppm for semi preserved products and special cured hams). (Bryan, N. S. et al, 2017: 86 – 90)
Canadian Regulations Bacon
120 ppm and it is stated that this level is set in order to prevent N-Nitrosamine formation. (Bryan, N. S. et al, 2017: 86 – 90)
South African Regulations
The South African max allowed limits on nitrite, nitrate and ascorbate or erythorbate are:
from Regulation R965 of 1977(18):
– Potassium and sodium nitrate: 200mg/ kg
– Potassium or sodium nitrite: 160mg/kg
Where nitrate and nitrite are used in combination they must be added together and proportionally neither one can exceed the max limit (section 2b of Regulation R965 of 1977).
For using erythrobic acid or sodium erythrobate: 550 mg/kg
L Ascorbic Acid: 550 mg/kg.
The continued use of nitrite is undoubtedly valid in the face of its efficacy against serious foodborne pathogens. It is, however, important to take every precaution possible to mitigate the risk posed by N-nitrosamines, including limiting the maximum allowed addition of nitrite to curing brines, limiting residual nitrite, controlling the curing by using nitrites and not nitrates in bacon and the addition of correct levels of ascorbate or erythorbate. The fact that nitrates are still allowed with nitrites in curing brines in South Africa is a matter of concern.
Objections are two fold. On the one hand, it increases the residual nitrites (over time, nitrites continue to be formed from nitrates through bacterial reduction) increasing the amount of nitrites present during frying and ingested which can form N-nitrosamines in the stomach. Another issue is that the exact dosage of nitrites is not left, in part, to the action of bacteria and I fail to see the point behind this. The thinking about nitrates acting as a reservoir for continued nitrite production in order to maximise its antimicrobial efficacy becomes an irrelevant point in light of the danger of N-nitrosamine formation from the higher residual nitrites and the thinking which stems from the 1920's has been altered around the world with good reason and yielding good results.
An equally serious problem may be that the South African regulations do not make the use of ascorbate or erythorbate mandatory nor does it set minimum required levels. It will be interesting to do a study of the residual nitrite levels in South African bacon over time. Much work remains.
After I did the article, it occurred to me that if I would cut out cured meat from my diet altogether in order to prevent even the smallest chance of exposure to N-nitrosamines and if I would subject the rest of my diet to the same rigor applied to the issue of nitrites, it may very well be found that I was better off eating the processed foods in comparison with what I may consume in its place. As a whole, it is possible that consuming processed foods along with regular exercises places me in a better position health wise than cutting out these foods from my diet altogether and no exercise. This issue must be seen in context. This is a fact that researchers regularly point to when they publish date on the health effects of nitrite. I can only echo the prevailing sentiment on the subject – that much more research needs to be done.
Few issues received more comprehensive treatment than the matter of N-nitrosamine formation since the early 70's and an unfathomable amount of excellent literature exists on the subject. I write these articles in order to learn. Like the issue of meat curing itself, the matter at hand is very complex.
Bryan, N. S. and Loscalzo, J. (Editors) 2017. Nitrite and Nitrate in Human Health and Disease. Springer International Publishing. Chapter by J. T. Keeton. jkeeton@tamu.edu
Frances, M. P. (Editor) et al.. 1981. The Health Effects of Nitrate, Nitrite, and N- Nitroso Compounds. National Academic Press. (https://books.google.co.za/books?id=QkorAAAAYAAJ&printsec=frontcover#v=onepage&q&f=false)
KERR, R. H., MARSH, C. T. N., SCHROEDER, W. F., and BOYER, E. A.. 1926. Associate Chemists, Bureau of Animal Industry, United States Department of Agriculture. THE USE OF SODIUM NITRITE IN THE CURING OF MEAT. Journal of Agricultural Research Vol. 33, No. 6. Sept. 15, 1926 Key No. A-112. Washington, D. C.
Lauer K. 1991. The history of nitrite in human nutrition: a contribution from German cookery books. Journal of clinical epidemiology. 1991;44(3):261-4.
Lavoisier, A. 1965. Elements of Chemistry. Dover Publications, Inc. A republication of a 1790 publication
Morton, I. D. and Lenges. J. 1992. Education and Training in Food Science: A Changing Scene. Ellis Hornwood Limited.
Schwarcz, J http://blogs.mcgill.ca/oss/2013/01/04/what-is-saltpeter-used-for-and-is-it-true-it-reduces-certain-%E2%80%9Ccarnal-urges%E2%80%9D/
WILEY, H. W., DUNLAP, F. L., and MCCABE, G. P. 1907. DYES, CHEMICALS, AND PRESERVATIVES IN FOODS. U. S. Dept. Agr., Off. Sec. Food Insp. Decis. 76, 13 p.
The Wyandott Herald, Kansa City, Kansas, 7 April 1898, "Air in Mammoth Cave"
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Bacon & the art of living: Book Format
"Bacon & the art of living" in book form
Chuck Vavra's Prasky January 5, 2020
Bacon & the Art of Living January 5, 2020
The Old and the New Pig Breeds December 25, 2019
Christmas Gammon ala Robert and Jaques December 18, 2019
Chapter 3: Kolbroek December 17, 2019
Kolbroek – Chinese, New Zealand, and English Connections December 7, 2019
In Search of the Origins of the Kolbroek November 23, 2019
Pic's history of meat processing September 21, 2019
The Rib Project September 16, 2019
Difference between Fresh Cured and Cooked Cured Colour of Meat. September 4, 2019
Ducks and Stuphins – Sausage Makers and their OXFORD SAUSAGES August 20, 2019
Salt Bush August 16, 2019
My Memories of Van Wyngaardt August 10, 2019
Builders of the Stone Ruins – From Antiquity to the Pô August 4, 2019
The Stories of Salt July 18, 2019
The impact of Sodium Bicarbonate and the Twaing Impact Crater June 26, 2019
Project – Legendary Foods June 8, 2019
Mild Cured Bacon – Recreating a Legend May 19, 2019
Nguni: Living Works of Art April 29, 2019
MDM – Not all are created equal! April 21, 2019
Chapter 7.03 Dr. Polenski April 8, 2019
Chapter 7.02 – The Danish Cooperative April 7, 2019
Chapter 7.01 – Mild Cured Bacon April 6, 2019
Chapter 2: Dry Cured Bacon April 1, 2019
Tank Curing Came From Ireland March 12, 2019
Weight Loss During Chilling and Freezing of Meat March 10, 2019
The Origins of Polony March 9, 2019
Pancetta March 8, 2019
The Mother Brine March 3, 2019
A Most Remarkable Tale: The Story of Eskort February 26, 2019
Soya: Review of some health concerns and applications in the meat industry February 10, 2019
Recipe: Liver Sausage February 9, 2019
Determining Total Meat Content (Part 7): Connective Tissues and Gelatin January 21, 2019
Counting Nitrogen Atoms – The History of Determining Total Meat Content (Part 6): The Codex January 7, 2019
Best Bacon System on Earth January 3, 2019
Counting Nitrogen Atoms – The History of Determining Total Meat Content (Part 5): The Proximate Analysis, Kjeldahl and Jones (6.25) January 3, 2019
Counting Nitrogen Atoms – The History of Determining Total Meat Content (Part 4): The Background of the History of Nutrition December 30, 2018
Soy and Pea Protein and what in the world is TVP? December 26, 2018
Bobotie – its origins December 26, 2018
Counting Nitrogen Atoms – The History of Determining Total Meat Content Part 3: Understanding of Protein Metabolism Coming of Age December 24, 2018
The Freezing and Storage of Meat December 18, 2018
Concerning the Aging of Beef December 10, 2018
The saltpeter letter November 24, 2018
The Chemistry of Sulfur Dioxide in Boerewors November 16, 2018
Oupa Eben's Boerewors November 8, 2018
Counting Nitrogen Atoms – The History of Determining Total Meat Content (Part 2) November 8, 2018
Counting Nitrogen Atoms – The History of Determining Total Meat Content (Part 1) October 21, 2018
Basics of Dry Curing September 2, 2018
Factors Affecting Colour Development and Binding in a Restructuring System Based on Transglutaminase August 31, 2018
Salt and the Ancient People of Southern Africa August 5, 2018
Factors Affecting Colour Development and Binding in a Restructuring System Based on Transglutaminase July 30, 2018
Concerning the lack of salt industry in pre-European New Zealand and other tales from Polynesia and the region July 21, 2018
Searching for Salt in New Zealand July 7, 2018
An Introduction to the Total Work on Salt, Saltpeter and Sal Ammoniac – Salt before the Agriculture Revolution June 3, 2018
Honey – powerful preserving and healing power (Fascinating Insights from Arabia) May 18, 2018
How did Ancient Humans Preserve Food? April 18, 2018
Carcass Sanatising March 22, 2018
Honey in cured meat formulations March 18, 2018
Joshua Penny and Khoikhoi Fermentation Technology at the Cape of Good Hope March 6, 2018
Erythorbate February 9, 2018
The Sal Ammoniac Project February 5, 2018
Listeria Monocytogenes: Understanding the enemy and plotting the defenses January 7, 2018
Listeria Monocytogenes: Its discovery and naming December 17, 2017
And then the mummies spoke! December 9, 2017
Restructuring of whole muscle meat with Microbial Transglutaminase – a holistic and collaborative approach. December 2, 2017
Bacon Curing – a historical review November 26, 2017
Nitrate salt's epic journey: From Turfan in China, through Nepal to North India November 26, 2017
Kathmandu's Urban Food October 13, 2017
Where life stands still and flavours abound – observing life at the Mardi Himal Eco Village, Kalamati, in Nepal. October 13, 2017
Kathmandu's Urban Food September 28, 2017
Regulations of Nitrate and Nitrite post-1920: the problem of residual nitrite September 2, 2017
Saltpeter, Horse Sweat and Biltong: The origins of South Africa's national food September 2, 2017
Salt – 7000 years of meat-curing May 28, 2017
The Anglo-Boer War and Bacon: Old enemies become friends May 21, 2017
Bacon & the art of living: Chapter 5.1 Dr. Ed Polenske February 5, 2017
Saltpeter: A Concise History and the Discovery of Dr. Ed Polenske January 15, 2017
Saltpeter: A Concise History and the Discovery of Dr. Ed Polenski October 9, 2016
Mechanism of meat curing – the wonder of nitrogen September 4, 2016
Mechanisms of Meat Curing – From Hoagland in 1914 to Pegg and Shahidi in 2000 August 14, 2016
Bacon Curing – a historical review May 30, 2016
Fathers of modern meat curing May 27, 2016
The Naming of Prague Salt May 11, 2016
Eva's Beloved Dad January 28, 2016
Concerning the Discovery of Ascorbate December 26, 2015
The Fascinating History and Use of Ascorbate December 26, 2015
7. The Life and Times of Ladislav NACHMÜLLNER – The Codex Alimentarius Austriacus November 1, 2015
6: Ladislav NACHMÜLLNER vs The Griffith Laboratorie September 27, 2015
Eben interviews David Donde September 19, 2015
02: Nitrogen chemistry en route to Cape Town August 16, 2015
5. Concerning Ladislav NACHMÜLLNER and the invention of the blend that became known as Prague Salt. June 28, 2015
1. C & T Harris and their Wiltshire bacon cure – the blending of a legend May 23, 2015
Concerning Nitrate and Nitrite's antimicrobial efficacy – chronology of scientific inquiry May 7, 2015
01. Concerning Chemical Synthesis and Food Additives March 30, 2015
Part 2: The UK letters March 11, 2015
Bacon and the art of living 18: The history of curing March 8, 2015
Bacon and the art of living 17: The mother brine February 8, 2015
Bacon and the art of living 15: Concerning the direct addition of nitrite to curing brine December 17, 2014
Bacon and the art of living 16: The gifts of a gentleman, Sir John Woodhead November 7, 2014
Bacon and the art of living 14: Searching for Combrinck & Co October 26, 2014
Bacon and the art of living 13: David Graaff's Armour October 25, 2014
Bacon and the art of living 12: Ice cold for bacon October 12, 2014
Bacon and the art of living 11: Ice Cold Revolution September 29, 2014
Bacon and the art of living 10: Eben meets Sir David September 23, 2014
Bacon and the art of living 9: Ice Cold in Africa (1) September 14, 2014
Bacon and the art of living 8: Magic salt in meat September 7, 2014
Bacon and the art of living 7: The Woodhead-Graaff hike August 31, 2014
Bacon and the art of living 6. The salt of the sea August 31, 2014
Bacon and the art of living 5. The salt of the earth August 25, 2014
Bacon and the art of living 4. The micro letter August 17, 2014
Bacon and the art of living 3. Lauren learns the nitrogen cycle August 10, 2014
Bacon and the art of living 2. the saltpeter letter August 3, 2014
Bacon and the art of living 1. A letter from Denmark July 20, 2014
Bacon and the art of living: Prologue July 11, 2014
Bacon and the art of living: Preface July 3, 2014
Bacon and the art of living 2. Letter from Germany June 14, 2014
Bacon and the art of living 1. A letter from Denmark June 8, 2014
Bacon and the art of living. June 1, 2014
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,048
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2017 November 18 - 19 Movie Posters Signature Auction - DallasAuction #7167
Lawrence of Arabia (Columbia, 1962). Full-Bleed French Double Grande (62" X 92") Georges Kerfyser Artwork.. ...
Lawrence of Arabia (Columbia, 1962). Full-Bleed French Double Grande (62" X 92") Georges Kerfyser Artwork.
The poster campaign for David Lean's landmark production was as epic as the film itself, including this majestic, color-drenched French double grande with Georges Kerfyser artwork. Although Lean began his career as an editor in the British cinema, he was to make his lasting impression in the movies as a director of quality epics. He won two Oscars for Best Direction for his work on The Bridge on the River Kwai and Lawrence of Arabia, and was the recipient of a lifetime achievement award from the American Film Institute in 1990. This film has been heralded as a true masterpiece of the cinema and has inspired filmmakers such as Martin Scorsese and Steven Spielberg. This rare French double grande will make an outstanding showpiece in any collection. The poster shows light edge and fold wear, slight fold separations, and small tears and nicks at the edges. There is some mild toning in the folds and a stain at the top of the right vertical fold. Folded, Very Fine.
18th-19th Saturday-Sunday
|
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| 2,039
|
Melilotus italicus é uma espécie de planta com flor pertencente à família Fabaceae.
A autoridade científica da espécie é (L.) Lam., tendo sido publicada em Flore Françoise 2: 594. 1778.
Portugal
Trata-se de uma espécie presente no território português, nomeadamente em Portugal Continental.
Em termos de naturalidade é nativa da região atrás indicada.
Protecção
Não se encontra protegida por legislação portuguesa ou da Comunidade Europeia.
Referências
Melilotus italicus - Checklist da Flora de Portugal (Continental, Açores e Madeira) - Sociedade Lusitana de Fitossociologia
Checklist da Flora do Arquipélago da Madeira (Madeira, Porto Santo, Desertas e Selvagens) - Grupo de Botânica da Madeira
Melilotus italicus - Portal da Biodiversidade dos Açores
Tropicos.org. Missouri Botanical Garden. 25 de agosto de 2014 <http://www.tropicos.org/Name/13035805>
Melilotus italicus - The Plant List (2010). Version 1. Published on the Internet; http://www.theplantlist.org/ (consultado em 25 de agosto de 2014).
Melilotus italicus - International Plant Names Index
Castroviejo, S. (coord. gen.). 1986-2012. Flora iberica 1-8, 10-15, 17-18, 21. Real Jardín Botánico, CSIC, Madrid.
Ligações externas
Melilotus italicus - Flora Digital de Portugal. jb.utad.pt/flora.
Melilotus italicus - Flora-on
Melilotus italicus - The Euro+Med PlantBase
Melilotus italicus - Flora Vascular
Melilotus italicus - Biodiversity Heritage Library - Bibliografia
Melilotus italicus - JSTOR Global Plants
Melilotus italicus - Flora Europaea
Melilotus italicus - NCBI Taxonomy Database
Melilotus italicus - Global Biodiversity Information Facility
Melilotus italicus - Encyclopedia of Life
Referências
Flora de Portugal
italicus
Plantas descritas em 1778
Flora de Portugal Continental
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,534
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Aart Vierhouten (Ermelo, Gelderland, 19 de març de 1970) és un ciclista neerlandès, que va ser professional del 1996 al 2009.
Palmarès
1993
1r a la Volta a Lieja
1r a la Internatie Reningelst
Vencedor d'una etapa del Tour d'Hainaut
1994
Vencedor de 2 etapes de la Tour de la Regió Valona
1996
Vencedor d'una etapa al Teleflex Tour
1997
Vencedor d'una etapa a la Volta a Renània-Palatinat
2000
1r a la Groningen-Münster
2006
1r a la Noord Nederland Tour
Vencedor d'una etapa a la Ster Elekrotoer
Resultats al Tour de França
1998. 88è de la Classificació general
2002. No surt (8a etapa)
2004. Fora de control (16a etapa)
Resultats al Giro d'Itàlia
2000. Abandona
2002. 104è de la Classificació general
2003. Abandona
2004. 110è de la Classificació general
2005. Abandona
Resultats a la Volta a Espanya
1997. 89è de la Classificació general
1999. 82è de la Classificació general
Enllaços externs
Fitxa a sitiodeciclismo.net
Fitxa a cyclebase.nl
Fitxa a museociclismo.it
Fitxa als Jocs Olímpics
Ciclistes de Gelderland
|
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"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,595
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SCHOOL GUIDE : GET HIRED NOW! There are a lot of truck driver training schools out there. In honor of Amber' s story we' d like to take the opportunity to address some important aspects of being prepared such as child ID kits. This will let you drive single vehicles of any size vehicles towing a trailing with a weight of 10, less ( light pany- sponsored company paid CDL training programs are truck driving jobs with training. West Coast Training in Washington provides construction companies & workers with heavy equipment training & certification programs. Washington Cities Insurance Authority is a self insured municipal risk pool property , offering liability specialty insurance programs as well as risk management services to municipal entities in Washington state.
Our Mission: To foster promote, develop the welfare of the wage earners, retirees of the United States; improve working conditions; advance mercial Driver School provides affordable training for men , Class B Truck , job seekers, women who want to earn their Class A, Class B Bus Class C commercial driver' s license. Main portal for the Washington State Department of Health.
Class B Training. Links to all other content and information about DOH programs.
January 13th is the anniversary of the Amber Alert Program which was created to help find safely recover children who have gone missing. Student Transportation provides essential services to support the safe and efficient transportation of the students of Washington state. Driver training programs in washington state. You will attend a truck driving school which is owned and operated by a trucking company.
Contact Us | Public Records Request. The Problem Driver Pointer System ( PDPS) is a system that allows jurisdictions and other organizations to search the National Driver Register ( NDR) data.
How to get a Washington enhanced driver license ( EDL) or enhanced ID ( EID) card. Plus, information for other states and EDL/ EID office mercial Driver License ( CDL) Do I need a CDL?
Find out if you need a CDL to drive your commercial vehicle. CDLs and endorsements.
Learn about the different classes of CDLs, restrictions, and when you need to have an endorsement on your CDL.
DRIVER EDUCATION & TRAINING. AAA offers a variety of driver training solutions that promote safe and responsible driving.
We' ve carefully developed training courses for new drivers, those with decades of experience – and everyone in between. T RUCK DRIVER TRAINING.
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{
"redpajama_set_name": "RedPajamaC4"
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| 4,533
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/**
* AES encrypted private key and display name. Contains Initialization Vectors,
* but not secrets. Used to encrypt private identities for storage on remote
* systems.
*
* @author fritz.ray@eduworks.com
* @class EbacCredential
* @module org.cassproject
*/
var EbacCredential = function() {
EcLinkedData.call(this, Ebac.context, EbacCredential.TYPE_0_3);
};
EbacCredential = stjs.extend(EbacCredential, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/credential";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/credential";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/Credential";
/**
* AES Initialization Vector used to decode PPK. Base64 encoded.
* @property iv
* @type string
*/
prototype.iv = null;
/**
* AES encrypted Private Key in PEM form.
* @property ppk
* @type string
*/
prototype.ppk = null;
/**
* AES Initialization Vector used to decode displayName. Base64 encoded.
* @property displayNameIv
* @type string
*/
prototype.displayNameIv = null;
/**
* AES encrypted display name for identity.
* @property displayName
* @type string
*/
prototype.displayName = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacCredential.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacCredential.TYPE_0_2);
}
if (EbacCredential.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacCredential.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacCredential.TYPE_0_3);
a.push(EbacCredential.TYPE_0_2);
a.push(EbacCredential.TYPE_0_1);
return a;
};
}, {atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Message used to retrieve credentials from a remote system.
*
* TODO: Vulnerable to replay attacks.
*
* @author fritz.ray@eduworks.com
* @class EbacCredentialRequest
* @module org.cassproject
*/
var EbacCredentialRequest = function() {
EcLinkedData.call(this, Ebac.context, EbacCredentialRequest.TYPE_0_3);
};
EbacCredentialRequest = stjs.extend(EbacCredentialRequest, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/credentialRequest";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/credentialRequest";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/CredentialRequest";
/**
* Hashed username.
* @property username
* @type string
*/
prototype.username = null;
/**
* Hashed password to authorize request.
* @property password
* @type string
*/
prototype.password = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacCredentialRequest.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacCredentialRequest.TYPE_0_2);
}
if (EbacCredentialRequest.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacCredentialRequest.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacCredentialRequest.TYPE_0_3);
a.push(EbacCredentialRequest.TYPE_0_2);
a.push(EbacCredentialRequest.TYPE_0_1);
return a;
};
}, {atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Credential list along with one time pad and session-based token for use in
* commit actions.
*
* @author fritz.ray@eduworks.com
* @class EbacCredentials
* @module org.cassproject
*/
var EbacCredentials = function() {
EcLinkedData.call(this, Ebac.context, EbacCredentials.TYPE_0_3);
};
EbacCredentials = stjs.extend(EbacCredentials, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/credentials";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/credentials";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/Credentials";
/**
* One time pad that may be used in password recovery. Base64 encoded.
* @property pad
* @type string
*/
prototype.pad = null;
/**
* Token provided by server to use in commit actions.
* @property token
* @type string
*/
prototype.token = null;
/**
* Credential array.
* @property credentials
* @type EbacCredential[]
*/
prototype.credentials = null;
/**
* Contact array.
* @property contacts
* @type EbacContact[]
*/
prototype.contacts = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacCredentials.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacCredentials.TYPE_0_2);
}
if (EbacCredentials.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacCredentials.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacCredentials.TYPE_0_3);
a.push(EbacCredentials.TYPE_0_2);
a.push(EbacCredentials.TYPE_0_1);
return a;
};
}, {credentials: {name: "Array", arguments: ["EbacCredential"]}, contacts: {name: "Array", arguments: ["EbacContact"]}, atProperties: {name: "Array", arguments: [null]}}, {});
/**
* AES encrypted public key and display name. Contains Initialization Vectors,
* but not secrets. Used to encrypt public identities for storage on remote
* systems.
*
* @author fritz.ray@eduworks.com
* @class EbacContact
* @module org.cassproject
*/
var EbacContact = function() {
EcLinkedData.call(this, Ebac.context, EbacContact.TYPE_0_3);
};
EbacContact = stjs.extend(EbacContact, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.2/contact";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/contact";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/Contact";
/**
* AES Initialization Vector used to decode PPK. Base64 encoded.
* @property iv
* @type string
*/
prototype.iv = null;
/**
* AES encrypted Private Key in PEM format.
* @property pk
* @type string
*/
prototype.pk = null;
/**
* AES Initialization Vector used to decode displayName. Base64 encoded.
* @property displayNameIv
* @type string
*/
prototype.displayNameIv = null;
/**
* AES encrypted display name for identity.
* @property displayName
* @type string
*/
prototype.displayName = null;
/**
* AES Initialization Vector of the home server of the contact. Base64 encoded.
* @property sourceIv
* @type string
*/
prototype.sourceIv = null;
/**
* URL to the home server of the contact.
* @property source
* @type string
*/
prototype.source = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacContact.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacContact.TYPE_0_2);
}
if (EbacContact.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacContact.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacContact.TYPE_0_3);
a.push(EbacContact.TYPE_0_2);
a.push(EbacContact.TYPE_0_1);
return a;
};
}, {atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Component of EbacEncryptedValue that contains data needed to decrypt
* encrypted payload. Is, itself, encrypted.
*
* Also contains data used to verify that encrypted-data substitution attacks
* were not performed on the data.
*
* Must be encryptable by RSA-2048, therefore, serialized form must less than 256
* bytes.
*
* @author fritz.ray@eduworks.com
* @class EbacEncryptedSecret
* @module org.cassproject
*/
var EbacEncryptedSecret = function() {
EcLinkedData.call(this, Ebac.context, EbacEncryptedSecret.TYPE_0_3);
};
EbacEncryptedSecret = stjs.extend(EbacEncryptedSecret, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/encryptedSecret";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/encryptedSecret";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/EncryptedSecret";
/**
* IV used to encrypt/decrypt payload. Base64 encoded.
* @property iv
* @type string
*/
prototype.iv = null;
/**
* Hashed and Base64 encoded ID of the parent (if any) object.
* Used to verify the data has not been copied from elsewhere.
* @property id
* @type string
*/
prototype.id = null;
/**
* Secret used to encrypt/decrypt payload.
* @property secret
* @type string
*/
prototype.secret = null;
/**
* Dot and Bracket notated index of the field in the parent-most object (if
* any). Used to verify the field has not been copied from elsewhere.
* @property field
* @type string
*/
prototype.field = null;
/**
* Serializes the field into a compact form for RSA encryption.
* @method toEncryptableJson
* @return {string} string
*/
prototype.toEncryptableJson = function() {
var o = (new Object());
o["v"] = this.iv;
if (this.id != null)
o["d"] = this.id;
o["s"] = this.secret;
if (this.field != null)
o["f"] = this.field;
return JSON.stringify(o);
};
/**
* Deserializes the field from a compact form used in RSA encryption.
* @method fromEncryptableJson
* @static
* @param {JSONObject} obj Object to deserialize from.
* @return {EbacEncryptedSecret} Secret in object form.
*/
constructor.fromEncryptableJson = function(obj) {
var secret = new EbacEncryptedSecret();
var o = (obj);
secret.iv = o["v"];
if (o["d"] != null)
secret.id = o["d"];
secret.secret = o["s"];
if (o["f"] != null)
secret.field = o["f"];
return secret;
};
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacEncryptedSecret.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacEncryptedSecret.TYPE_0_2);
}
if (EbacEncryptedSecret.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacEncryptedSecret.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacEncryptedSecret.TYPE_0_3);
a.push(EbacEncryptedSecret.TYPE_0_2);
a.push(EbacEncryptedSecret.TYPE_0_1);
return a;
};
}, {atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Signature used to authorize movement of data on behalf of a private-key
* holding owner.
*
* @author fritz.ray@eduworks.com
* @class EbacSignature
* @module org.cassproject
*/
var EbacSignature = function() {
EcLinkedData.call(this, Ebac.context, EbacSignature.TYPE_0_3);
};
EbacSignature = stjs.extend(EbacSignature, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/timeLimitedSignature";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/timeLimitedSignature";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/TimeLimitedSignature";
/**
* The public key of the authorizing party in PEM format.
* @property owner
* @type string
*/
prototype.owner = null;
/**
* The time in number of milliseconds since midnight of January 1, 1970
* 00:00:00 UTC that this signature is authorized to move data.
* @property expiry
* @type long
*/
prototype.expiry = 0.0;
/**
* The signature of this object, having signed the object, having been
* encoded in JSON with no space or tabs in ASCII sort order, having no
* value for the signature at the time of signing.
* @property signature
* @type string
*/
prototype.signature = null;
/**
* The server authorized to move data. If this is empty, the signature may
* be used by a server to ask for data from other servers.
* @property server
* @type string
*/
prototype.server = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacSignature.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacSignature.TYPE_0_2);
}
if (EbacSignature.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacSignature.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacSignature.TYPE_0_3);
a.push(EbacSignature.TYPE_0_2);
a.push(EbacSignature.TYPE_0_1);
return a;
};
}, {atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Encrypted JSON-LD object or string.
*
* @author fritz.ray@eduworks.com
* @class EbacEncryptedValue
* @module org.cassproject
*/
var EbacEncryptedValue = function() {
EcRemoteLinkedData.call(this, Ebac.context, EbacEncryptedValue.myType);
};
EbacEncryptedValue = stjs.extend(EbacEncryptedValue, EcRemoteLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/encryptedValue";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/encryptedValue";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/EncryptedValue";
constructor.myType = EbacEncryptedValue.TYPE_0_3;
/**
* Optional Hint used to aid in search.
* Displays the type of the encrypted object.
* @property encryptedType
* @type string
*/
prototype.encryptedType = null;
/**
* Base-64 encoded, AES encrypted form of the encrypted object (or string).
* @property payload
* @type string
*/
prototype.payload = null;
/**
* Optional Hint used to aid in search and display.
* Name of the inner encrypted object.
* @property name
* @type string
*/
prototype.name = null;
/**
* Array of EbacEncryptedSecret objects encoded in Base-64, encrypted using
* RSA public keys of owners, readers, or other parties to allow them
* access to the payload.
* @property secret
* @type string[]
*/
prototype.secret = null;
prototype.copyFrom = function(that) {
var me = (this);
for (var key in me)
delete me[key];
var you = (that);
for (var key in you) {
if (me[key] == null)
me[key.replace("@", "")] = you[key];
}
if (!this.isAny(this.getTypes()))
throw new RuntimeException("Incompatible type: " + this.getFullType());
};
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacEncryptedValue.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacEncryptedValue.TYPE_0_2);
}
if (EbacEncryptedValue.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacEncryptedValue.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacEncryptedValue.TYPE_0_3);
a.push(EbacEncryptedValue.TYPE_0_2);
a.push(EbacEncryptedValue.TYPE_0_1);
return a;
};
}, {secret: {name: "Array", arguments: [null]}, owner: {name: "Array", arguments: [null]}, signature: {name: "Array", arguments: [null]}, reader: {name: "Array", arguments: [null]}, atProperties: {name: "Array", arguments: [null]}}, {});
/**
* AES encrypted public key and display name message.
* Used to grant access to a contact.
* Contains Initialization Vectors, but not secrets.
* Used to encrypt public identities for storage on remote systems.
*
* @author fritz.ray@eduworks.com
* @class EbacContactGrant
* @module org.cassproject
*/
var EbacContactGrant = function() {
EcRemoteLinkedData.call(this, Ebac.context, EbacContactGrant.TYPE_0_3);
};
EbacContactGrant = stjs.extend(EbacContactGrant, EcRemoteLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/contactGrant";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/contactGrant";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/ContactGrant";
/**
* Public key being granted to the owner of this message.
* @property pk
* @type string(pem)
*/
prototype.pk = null;
/**
* Display name of the contact.
* @property displayName
* @type string
*/
prototype.displayName = null;
/**
* Source server of the contact.
* @property source
* @type string
*/
prototype.source = null;
/**
* Response token used to validate that this grant is in response to a contact request you sent.
* @property responseToken
* @type string
*/
prototype.responseToken = null;
/**
* Signature (Base64 encoded) of the response token to verify against your own public key
* to ensure that this grant is in response to a contact request you sent.
* @property responseSignature
* @type string
*/
prototype.responseSignature = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacContactGrant.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacContactGrant.TYPE_0_2);
}
if (EbacContactGrant.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacContactGrant.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacContactGrant.TYPE_0_3);
a.push(EbacContactGrant.TYPE_0_2);
a.push(EbacContactGrant.TYPE_0_1);
return a;
};
}, {owner: {name: "Array", arguments: [null]}, signature: {name: "Array", arguments: [null]}, reader: {name: "Array", arguments: [null]}, atProperties: {name: "Array", arguments: [null]}}, {});
/**
* Message used to commit credentials to a remote login server.
*
* TODO: Vulnerable to replay attacks. Token field prevents some replay
* attacks.
*
* @author fritz.ray@eduworks.com
* @class EbacCredentialCommit
* @module org.cassproject
*/
var EbacCredentialCommit = function() {
EcLinkedData.call(this, Ebac.context, EbacCredentialCommit.TYPE_0_3);
this.credentials = new EbacCredentials();
};
EbacCredentialCommit = stjs.extend(EbacCredentialCommit, EcLinkedData, [], function(constructor, prototype) {
constructor.TYPE_0_1 = "http://schema.eduworks.com/ebac/0.1/credentialCommit";
constructor.TYPE_0_2 = "http://schema.eduworks.com/ebac/0.2/credentialCommit";
constructor.TYPE_0_3 = "http://schema.cassproject.org/kbac/0.2/CredentialCommit";
/**
* Hashed username.
* @property username
* @type string
*/
prototype.username = null;
/**
* Hashed password to authorize commit.
* @property password
* @type string
*/
prototype.password = null;
/**
* Token provided to client when previously executed Request was done. May
* be empty if this is used as part of Create action.
* @property token
* @type string
*/
prototype.token = null;
/**
* List of credentials to commit to the login server storage.
* @property credentials
* @type EbacCredentials
*/
prototype.credentials = null;
prototype.upgrade = function() {
EcLinkedData.prototype.upgrade.call(this);
if (EbacCredentialCommit.TYPE_0_1.equals(this.type)) {
var me = (this);
if (me["@context"] == null && me["@schema"] != null)
me["@context"] = me["@schema"];
this.setContextAndType(Ebac.context_0_2, EbacCredentialCommit.TYPE_0_2);
}
if (EbacCredentialCommit.TYPE_0_2.equals(this.getFullType())) {
this.setContextAndType(Ebac.context_0_3, EbacCredentialCommit.TYPE_0_3);
}
};
prototype.getTypes = function() {
var a = new Array();
a.push(EbacCredentialCommit.TYPE_0_3);
a.push(EbacCredentialCommit.TYPE_0_2);
a.push(EbacCredentialCommit.TYPE_0_1);
return a;
};
}, {credentials: "EbacCredentials", atProperties: {name: "Array", arguments: [null]}}, {});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,094
|
var querystring = require('querystring');
var https = require('https');
var Client = function(application_token){
this.access_token;
this.refresh_token;
this.username;
this.application_token = application_token;
}
Client.prototype.eventSearch = function(searchParams, callback){
var searchBody = querystring.stringify(searchParams);
var searchOptions = {
hostname: 'api.stubhub.com',
path:'/search/catalog/events/v2',
method: 'GET',
headers: {
'Content-Type': 'application/x-www-form-urlencoded',
'Authorization': 'Bearer ' + this.application_token,
'Content-Length': searchBody.length
}
}
var responseBody = '';
var request = https.request(searchOptions, function(response) {
response.on('data', function(chunk) {
responseBody += chunk;
});
response.on('end', function() {
if (response.statusCode == 200){
responseBody = JSON.parse(responseBody);
callback(null, responseBody);
}
else {
callback(responseBody);
}
});
});
request.on('error', function(e) {
callback(e)
})
request.write(searchBody);
request.end()
}
Client.prototype.getUserToken = function(auth_params){
var authBody = querystring.stringify({
grant_type: 'password',
username: auth_params.username,
password: auth_params.password,
scope: 'PRODUCTION'
});
var authOptions = {
hostname: 'api.stubhub.com',
path:'/login',
method: 'POST',
headers: {
'Content-Type': 'application/x-www-form-urlencoded',
'Authorization': 'Basic ' + new Buffer(auth_params.consumer_key + ':' + auth_params.consumer_secret).toString('base64'),
'Content-Length': authBody.length
}
};
var responseBody = '';
var request = https.request(authOptions, function(response) {
response.on('data', function(chunk) {
responseBody += chunk;
});
response.on('end', function() {
if (response.statusCode == 200){
responseBody = JSON.parse(responseBody);
console.log(responseBody);
this.access_token = responseBody["access_token"];
this.refresh_token = responseBody["refresh_token"];
}
else {
console.log(responseBody);
}
});
});
request.on('error', function(e) {
console.log(e);
})
request.write(authBody);
request.end()
}
Client.prototype.getMyListings = function(callback){
var getListingBody = querystring.stringify({
userId: this.userId;
});
var getListingOptions = {
hostname: 'api.stubhub.com',
path:'/accountmanagement/listings/v1/seller/',
method: 'GET',
headers: {
'Content-Type': 'application/x-www-form-urlencoded',
'Authorization': 'Bearer ' + this.access_token,
'Content-Length': getListingBody.length
}
};
var responseBody = '';
var request = https.request(getListingOptions, function(response) {
response.on('data', function(chunk) {
responseBody += chunk;
});
response.on('end', function() {
if (response.statusCode == 200){
responseBody = JSON.parse(responseBody);
callback(null, responseBody);
}
else {
callback(responseBody);
}
});
});
request.on('error', function(e) {
console.log(e);
})
request.write(getListingBody);
request.end()
}
module.exports.createClient = function(auth_params) {
return new Client(auth_params);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,482
|
{"url":"http:\/\/math.stackexchange.com\/questions\/69369\/number-of-possible-magic-card-decks","text":"# Number of possible magic card decks\n\nTo date, there are 11,986 unique cards released for Magic The Gathering. There are certain rules specifying constraints on deck building:\n\n\u2022 A deck must have a minimum of 60 cards.\n\u2022 A deck may not have more than four of any particular card.\n\n(Please note that we are ignoring land cards)\n\nI have come up with the following expression. Is it right? If so, what is its order of magnitude (wolframalpha fails to evaluate it).\n\n$$\\large\\large\\sum_{n=60}^{47944} {47944 \\choose n}$$\n\n-\nProbably you should pay no attention to someone who confesses not to know what a land card is. But the expression cannot be right. It assumes that somehow each of the $11986$ cards occurs in each of four suits. \u2013\u00a0 Andr\u00e9 Nicolas Oct 2 '11 at 22:55\nYour expression is not quite right because it treats the four copies of each card as distinguishable. For instance, you're counting four different decks, each containing a different one of the four Goblin Sharpshooters, but these four should all count as the same deck. \u2013\u00a0 joriki Oct 2 '11 at 23:01\nDo you include rare lands? \u2013\u00a0 JeremyKun Oct 2 '11 at 23:01\n\nThe number of decks with less than $60$ cards is negligible compared to the total number of decks with any number of cards, so the number you're looking for is well approximated by that total number. This is straightforward to calculate, since there are $5$ possibilities for each of the $11,986$ unique cards (you can have $0,1,2,3$ or $4$ of each), so there are $5^{11986}\\approx10^{8378}$ different decks.\n\n-\nThe expression you have is the number of ways to pick any number from 60 to 47944 cards from a collection of 47944 cards. In particular, we know that $\\sum_{i = 1}^n\\binom{n}{i} = 2^n$, so your number is \"close\" to $2^{47944}$, if we agree that in comparison the first 60 terms of that sum are small. This number has 14433 digits. If you want an exact answer, look at the following in Mathematica:\nAnd subtract this from $2^{47944}$.\nOn the other hand, since the four repeated cards you're picking from are identical, it does not matter how you pick them. So this will not be the right answer. What you want instead is to assign a number 0-4 to each of the 11986 cards (representing inclusion and multiplicity in a deck), and exclude any assignments which give fewer than 60 total cards. This gives a closer upper bound of $5^{11986}$, which has 8378 digits.\nSince $2^{47944}\\approx 3.820\\times 10^{14432}$ and $\\sum_{i=0}^{59}\\binom{47944}i\\approx 1.015\\times 10^{196}$, the subtraction makes very little difference. \u2013\u00a0 Brian M. Scott Oct 2 '11 at 23:13","date":"2014-04-18 01:19:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.7501686811447144, \"perplexity\": 285.264502273374}, \"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-2014-15\/segments\/1397609532374.24\/warc\/CC-MAIN-20140416005212-00139-ip-10-147-4-33.ec2.internal.warc.gz\"}"}
| null | null |
Q: Tagging image files on server with MVC3 I am trying to write an MVC3 application that will retrieve images that are stored locally on the server. Display them on the webpage and let the user tag images for later review. When I say tag, I mean actually modify the "Tag" property of the file on the server. When I run the code below I get the following error: "The calling thread must be STA, because many UI components require this" on the "Image imageToTag = new Image(); line. Please help! I've been stuck on this for about 3 days and I have never done much multithreading.
Image imageToTag = new Image();
BitmapImage myBitmapImage = new BitmapImage();
var root = @"C:\Images\";
imageURLProcessed = Path.GetFullPath(@imageURLProcessed);
// BitmapImage.UriSource must be in a BeginInit/EndInit block
myBitmapImage.BeginInit();
if (!imageURLProcessed.StartsWith(root))
{
// Ensure that we are serving file only inside the root folder
// and block requests outside like "../web.config"
throw new HttpException(403, "Forbidden");
}
myBitmapImage.UriSource = new Uri(@imageURLProcessed);
myBitmapImage.EndInit();
imageToTag.Source = myBitmapImage;
imageToTag.Tag = tags;
A: But an answer to your question:
http://social.msdn.microsoft.com/Forums/en-us/csharpgeneral/thread/f5e83bc0-b523-45d2-b77c-b1702124869c
http://msdn.microsoft.com/en-us/library/system.drawing.image.setpropertyitem.aspx
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| 5,427
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Q: Upgrading SSD on 2015 MacBook Pro I'm about to buy a 15" Macbook Pro with 512 GB SSD. Does anyone know if it is possible to upgrade the SSD after I have bought it?
A: Is it possible?
Yes.
The question is can you get a drive to upgrade to?
Apple uses a "proprietary PCIe 2.0 x4" interface for their storage. There are SSDs out there that match the specs (like this Samsung on Amazon), but for some reason they are not compatible
So, bottom line....
Yes, you can upgrade it, but only with a genuine Apple SSD (at this point). Your best bet is to get the most you can afford now because there is zero cost savings plus the added headache of disassembly/reassembly. IMO, it just isn't worth it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: How can I get an HTML form's values into a "custom" URL? On my site, I can access a search function by going to "mysite.com/search/search_term", where "search_term" is the term the user entered. I'm trying to get a simple one-input form to format the URL that way. One way that I can do it is make a PHP script that would redirect to the proper place. So if you navigate to "search_redirect.php?term=search_term", it would just redirect you to "/search/search_term". I want to avoid doing this, as it seems kind of redundant. If I don't find a better answer, that's probably what I'll end up doing. Now, here's what I tried so far:
I tried setting an onsubmit on the form like this:
<form onsubmit="search_navigate()" id="navbar_form">
<input type="search" placeholder="Search Articles..." id="navbar_search"></input>
<input id="navbar_submit" type="submit" value=">"/>
</form>
... which, onsubmit calls this function:
function search_navigate(){
alert("well hey there");
window.location = ("http://www.google.com");
}
When I click on the submit button, I can see the alert, but instead of taking me to Google as you'd expect, it takes me to the same page I'm on (whatever that page might be).
This is the problem I'm faced with when implementing it this way. If you have another way entirely, I'd still like to hear it.
A: function search_navigate() {
var obj = document.getElementById("navbar_search");
var keyword = obj.value;
var dst = "http://yoursite.com/search/" + keyword;
window.location = dst;
}
may this work for you?
further more, to stop the form submit doing its origin function, append "return false" to the function call, that is, onsubmit="foo();return false".
A: Usually when you encounter weird behaviour like this, it has something to do with the <form>. Try this:
<script type="text/javascript">
function search_navigate() {
var obj = document.getElementById("navbar_search");
var keyword = obj.value;
var dst = "http://yoursite.com/search/" + keyword;
window.location = dst;
}
</script>
<span id="navbar_form">
<input type="search" placeholder="Search Articles..." id="navbar_search" />
<input type="button" id="navbar_submit" value="Submit" onClick="search_navigate()" />
</span>
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{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/stats.stackexchange.com\/questions\/90732\/do-the-mean-and-the-variance-always-exist-for-exponential-family-distributions","text":"# Do the mean and the variance always exist for exponential family distributions?\n\nAssume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f.\n\n$$f_X(x|\\boldsymbol \\theta) = h(x) \\exp\\left(\\sum_{i=1}^s \\eta_i({\\boldsymbol \\theta}) T_i(x) - A({\\boldsymbol \\theta}) \\right)$$\n\nwhere ${\\boldsymbol \\theta} = \\left(\\theta_1, \\theta_2, \\cdots, \\theta_s \\right )^T$ is the parameter vector and $\\mathbf{T}(x)= \\left(T_1(x), T_2(x), \\cdots,T_s(x) \\right)^T$ is the joint sufficient statistic.\n\nIt can be show that the mean and the variance for each $T_i(x)$ exist. However, do the mean and the variance for $X$ (i.e. $E(X)$ and $Var(X)$) always exist as well? If not, is there an example of an exponential family distribution of this form whose mean and variable do not exist?\n\nThank you.\n\nTaking $s=1$, $h(x)=1$, $\\eta_1(\\theta)=\\theta$, and $T_1(x)=\\log(|x|+1)$ gives $A(\\theta)=\\log\\left(-2\/(1+\\theta)\\right)$ provided $\\theta \\lt -1$, producing\n\n$$f_X(x|\\theta) = \\exp\\left(\\theta\\log(|x|+1) - \\log\\left(\\frac{-2}{1+\\theta}\\right)\\right) = -\\frac{1+\\theta}{2}(1+|x|)^\\theta.$$\n\nGraphs of $f_X(\\ |\\theta)$ are shown for $\\theta=-3\/2, -2, -3$ (in blue, red, and gold, respectively).\n\nClearly the absolute moments of weights $\\alpha=-1-\\theta$ or greater do not exist, because the integrand $|x|^\\alpha f_X(x|\\theta)$, which is asymptotically proportional to $|x|^{\\alpha+\\theta}$, will produce a convergent integral at the limits $\\pm\\infty$ if and only if $\\alpha+\\theta\\lt -1$. In particular, when $-2 \\le \\theta \\lt -1,$ this distribution does not even have a mean (and certainly not a variance).\n\n\u2022 I do not understand the condition $\\theta < -1$. Do you mean $\\theta > -1$? When $\\theta < -1$, $A(\\theta)$ is not defined and $f_X(x|\\theta)$ is negative and cannot be a p.d.f. Please let me know what I missed. Thanks. \u2013\u00a0Wei Mar 20 '14 at 22:27\n\u2022 I apologize, because a minus sign was omitted in the calculation of $A$. I have replaced it in the formulas. I really do mean $\\theta\\lt -1$. \u2013\u00a0whuber Mar 21 '14 at 1:00\n\u2022 Thank you for the example. I agree about the moments of $|x|$. How about the moments of $x$ itself? For example, when $-2<\\theta <-1$ in your example above, does $E(x)$ exist? \u2013\u00a0Wei Mar 30 '14 at 12:21\n\u2022 Because the Lebesgue integral is defined in terms of the positive and negative parts of the integrand, the moments of $x$ exist if and only if the moments of $|x|$ exist. \u2013\u00a0whuber Mar 30 '14 at 17:32\n\u2022 @Wei: $\\mathrm{E}\\{g(X)\\}$ exists only if $\\mathrm{E}\\{\\, \\left| g(X)\\, \\right| \\} < \\infty$. Without this restriction,the expectation is not uniquely defined for some CDFs. \u2013\u00a0Dennis Jul 12 '14 at 0:23","date":"2019-07-18 15:59:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9338270425796509, \"perplexity\": 181.7090366285534}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195525659.27\/warc\/CC-MAIN-20190718145614-20190718171614-00232.warc.gz\"}"}
| null | null |
Q: Hexdump binary file while honoring separators I have a binary file where records are separated by newlines. hexdump just dumps everything with a fixed column width. Is there a tool to hexdump this file, while honoring the newline separator? (Any other separator, like 0, would also be fine.)
A: This Python script will do it, sort of:
#!/usr/bin/env python3
import fileinput
for line in fileinput.input(mode='rb'):
print(line)
There's one problem that I hadn't realized: the binary data may itself contain the separator character, messing up the output.
|
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From driving range to subdivision: plans for Mars Hill development move forward
A developer's plan to transform an abandoned driving range in Mars Hill into a 100-home subdivision took a step forward October 8.
From driving range to subdivision: plans for Mars Hill development move forward A developer's plan to transform an abandoned driving range in Mars Hill into a 100-home subdivision took a step forward October 8. Check out this story on citizen-times.com: https://www.citizen-times.com/story/news/madison/2018/10/17/plans-mars-hill-subdivision-driving-range-moves-forward/1580997002/
Paul Eggers, Asheville Citizen Times Published 10:27 a.m. ET Oct. 17, 2018 | Updated 10:27 a.m. ET Oct. 17, 2018
A longtime driving range in Mars Hill could become a 100-home development after aldermen approved a zoning change for the Parkway View Drive acreage.(Photo: Paul Eggers/paul@newsrecordandsentinel.com)
A developer's plan to transform an abandoned driving range in Mars Hill into a 100-home subdivision took a step forward October 8. The town's Board of Alderman approved a new high density residential zoning designation that could clear the way for construction of 100 stick-built homes on lots less than a quarter acre in size.
Brandon Quinn of the Mars Hill-based homebuilder Eden Rock Enterprises told aldermen he plans to construct single-family homes ranging from $200-250k on the Parkway View Drive site. "We're just excited to put out affordable housing," he said after the town meeting. "It's one of the biggest problems in our country and in this region and we may be on the forefront of a solution."
Quinn said the homes would range in size from 1,200 to 1,900 square feet and would qualify for down payment assistance from loan programs with the U.S. Department of Agriculture.
Aldermen and members of the public questioned Quinn on the development plan and its potential impact on the surrounding community.
"At the end of the day, we're definitely going to bring in more folks to this town," Quinn said. "It will be controlled growth. I love this community. My kids were born here, my kids have been raised here, I like it here."
Quinn told aldermen he hopes to build homes in 4-5 phases. "We'll plan to do 20-25 at one time," he said. "We're looking at a four year project."
Town Manager Darhyl Boone said creating a new R-4 high density zoning designation – which now allows homesites on lots as small as 8,000 square feet – has been part of land development discussions in Mars Hill since 2001. He said the specific location for the subdivision was has been identified in planning efforts as a potential site for residential development.
"Water and sewer are very close," Boone said of the 26-acre site.
Following a unanimous vote in favor of the zoning change, nearly 15 members of the public then walked out of the town's meeting. Asked for comment on the vote, one man - who declined to be identified - said, "I'm not happy about it." No other individual in attendance would speak about the issue.
Before Quinn and Eden Rock Enterprises can break ground on the planned development, town officials must approve a subdivision plan and annex the full acreage into town limits. Currently, roughly six acres of the site lie with Mars Hill borders.
"I feel good about it," Quinn said. "I think it is going to sell quickly."
Mayor John Chandler said a fundraiser October 6 raised over $15,000 for Shriners Hospitals for Children. "It was a good day and a good year," he said…
As an effort to protect nearly 90 acres of Bailey Mountain from development nears the finish line, Boone said public comments and feedback taken during a public meeting October 10 would help shape plans for the site.
Read or Share this story: https://www.citizen-times.com/story/news/madison/2018/10/17/plans-mars-hill-subdivision-driving-range-moves-forward/1580997002/
Lime green puppy born to North Carolina dog
Pedestrian dies in Smokey Park Highway accident
Jan. 17, 2020, noon
Answer Man: Trees 'clear cut' on Airport Road?
River Arts District housing project stalled over affordability
4 years into construction, Brevard Rd ramp to open
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MegaBosch ist eine deutschsprachige Endzeit-Rock-Band aus dem Raum Hannover, die 2009 gegründet wurde.
Anfangs als reines Spaßprojekt für Endzeit-LARP-Veranstaltungen gegründet, zog die Band größere Aufmerksamkeit auf sich, als sie 2013 und 2014 auf dem Wacken Open Air auf einer eigens für sie konstruierten Bühne auftraten. Auffällig waren die Bauweise aus rostigen Schiffscontainern im postapokalyptischen Stil, passend zu Filmen wie Mad Max oder Waterworld, und große Gas-Feuerinstallationen. Inzwischen ist die Bühne offizielle Wacken-Bühne (Wasteland Stage).
Geschichte
Reale Welt
2009 bis 2010
Das erste Mal traten MegaBosch 2009 auf dem F.A.T.E. III, einer Endzeit-LARP-Kampagne von Lost Ideas auf. Damals bestand MegaBosch nur aus General PAUSE und General MIDI.
2010 entstand das Musikvideo zu "Es ist Geil". Die Szenen wurden auf dem Gelände des ehemaligen sowjetischen Truppenübungsplatzes bei Mahlwinkel gedreht. Die Veröffentlichung festigte den Stellenwert der Band innerhalb der deutschen Endzeit-Szene.
2011 bis 2012
2011 wurde MegaBosch zu einer kompletten Band ausgebaut. In Sergeant PEPPER fand man einen Bassisten, der sowohl musikalisch gut in die Band passte, als auch passionierter Liverollenspieler war. Die Position des Schlagzeugers blieb noch eine Weile unbesetzt. In der Übergangszeit spielte die Band mit einer sogenannten "Drum Machine" – "...eine Art überdimensionale Spieluhr aus alten Fässern, Elektromotoren und jeder Menge Schrott." Ende des Jahres komplettierte aber schließlich Private PARTS die Band am Schlagzeug.
Der erste öffentliche Auftritt außerhalb des LARP war schließlich im Mai 2012 auf der Role Play Convention in Köln, der größten Rollenspielermesse Europas. Sowohl Konzept als auch Musik kamen bei Publikum und Veranstalter so gut an, dass MegaBosch im Anschluss für das Wacken Open Air 2013 gebucht wurde, wofür eine eigens für die Band konstruierte und zum Setting passende Bühne gebaut werden sollte.
2013 bis 2015
So wurde 2013 beim Wacken Open Air auf dem Gelände des Wackinger Villages zum ersten Mal die Wasteland Stage aus alten Schiffcontainern und rostigem Stahl aufgebaut. Bei ihren Auftritten wurde MegaBosch unterstützt von Henry Hots Feuerinstallationen anstelle einer normalen Lightshow. Darüber hinaus belebten die Schausteller der Wasteland-Warriors den Endzeit Bereich. Da MegaBosch zu diesem Zeitpunkt außerhalb der Liverollenspiel-Szene nahezu unbekannt waren, hatte der Veranstalter darauf gesetzt, die Band möglichst viel spielen zu lassen, um so neues Publikum für sich zu gewinnen. Über alle Festivaltage wurden tagsüber mehrere kurze Teaser geplant und abends längere Konzerte. Am Ende des Festivals hatten MegaBosch 13 Auftritte gespielt und beim Abschlusskonzert war der Platz vor der Wasteland Stage mit vielen tausend Menschen gefüllt.
Da MegaBosch bis dahin noch keine CD-Veröffentlichung hatte, produzierte die Band für die Auftritte auf dem Wacken Open Air 2013 ihre EP 2000 Watt in Eigenregie (700 verkaufte Exemplare). Weitere Videoveröffentlichungen mit Material vom Wacken 2013 folgten (Welcome to Paradise City und Hammerfist-LIVE).
Die Resonanz des Publikums auf dem W:O:A 2013 war derart gut, dass das gesamte Konzept des Wasteland Bereichs in den Jahren 2014 und 2015 weitergeführt wurde.
Angekündigt für 2016 sind die Produktion von MegaBoschs Debüt-CD bei dem renommierten Produzenten Tommy Newton und weitere Endzeit-Liverollenspiele.
Fiktive Welt
Allgemeiner Hintergrund
Das fiktive Setting der Endzeit-LARPs, für die sich MegaBosch gegründet hatten, hält sich grob an Genre-Vorlagen wie Mad Max oder Fallout. Zwar gibt es keinen allgemeingültigen von allen LARP-Spielern akzeptierten Hintergrund, der zu dem spielrelevanten Endzeit-Szenario geführt hat, aber es gibt Richtlinien, an die sich alle Spieler halten.
Allgemein anerkannte Spielgrundlage ist, dass die Erde irgendwann zwischen den 1930er- und 1980er-Jahren eine große Katastrophe erlitten hatte, welche dazu führte, dass der größte Teil der Menschheit ausgelöscht wurde. Während die Katastrophe ("der große Knall" oder "das große Feuer") nicht genau betitelt wird, werden doch deren Auswirkungen im Spiel in Form von radioaktiver Strahlung und einer dekadenten Mangelgesellschaft differenzierter dargestellt. Spannung wird unter anderem dadurch erzeugt, dass verschiedene Fraktionen in einer lebensfeindlichen Welt um ihr eigenes Überleben, bzw. das der Gruppe kämpfen. Das Spieljahr ist ca. 150 Jahre nach der Katastrophe angesiedelt.
Fiktive Biografie der Band
Das endzeitliche Setting ist auch Grundlage für MegaBoschs Kurzbiografie. Laut dieser wurde die Band mit der Idee gegründet, der postapokalyptischen Welt die Rockmusik zurückzubringen. Dafür ziehen sie als selbsternannte "Rockstars der Apokalypse" durch die Einöde in einer Welle der Dekadenz und Zügellosigkeit.
Als Tour-Fahrzeug dient ihnen der sogenannte "MegaMog" (ein von außen komplett verrosteter 71erUnimog), den sie auf vielen Veranstaltungen dabei haben.
Stil
Für ihr Vorhaben, der Menschheit die Rockmusik zurückzubringen, bedient sich MegaBosch theatralisch und mit vielen großen Gesten der Klischees aus der Welt von bekannten Rockstars. Sie dient als roter Faden und findet sich in allen Details dieser Band wieder – Aussehen, Kostüme, Musik, Texte, Interviews, Umgang mit den Fans, ob live oder auf medialen Plattformen.
Optik
Der Postapokalyptische Stil, dem MegaBosch sich verschrieben haben, wird gemein hin auch als "Retrokalypse" bezeichnet. Ausgehend von der Idee, dass die Erde irgendwann zwischen den 1930er- und 1980er-Jahren eine Katastrophe apokalyptischen Ausmaßes erlitten hat, sind natürlich die Bruchstücke, aus denen sich die Überlebenden ihre "Neue Welt" aufbauen, aus unserer heutigen Sicht alt und außer Mode. Dementsprechend werden – obgleich sich das Setting aus heutiger Sicht in der Zukunft befindet – keine futuristischen Elemente verwendet, sondern nur alte Dinge, die eben auf sehr alt und zusammengefrickelt getrimmt werden.
Auf Konzerten treten neben der Band auch öfter Tänzerinnen, Stripperinnen und Burlesque-Künstlerinnen auf.
Musik
MegaBosch beschreiben ihre Musik mit dem Überbegriff "harte, geile Rockmusik". Der Grundansatz der "Retrokalypse" spiegelt sich in der Musik auch weitgehend wider. Sie sehen sie als eine Art Flickenteppich verschiedenster Musikstile, die "den großen Knall" überlebt haben. So finden sich in ihrem Repertoire Einflüsse aus dem Hardrock, klassischem Heavy Metal, Swing, Punk, Country-Rock, Blues, Rock 'n' Roll, Funk, Latin und Elektro.
Live improvisiert die Band häufig als Intro für ihre Konzerte oder als Untermalung zu einem Strip. Außerdem gibt es auf MegaBosch-Konzerten auch mal längere Gitarren- oder Schlagzeugsolo-Eskapaden.
Texte
MegaBoschs Texte spiegeln ihr Leben in einer postapokalyptischen Welt wider. Die Texte sind deutsch und handeln um Drogen, Sex, Gewalt und Zügellosigkeit, zum Teil bitterböse, aber auch augenzwinkernd und bis an die Grenzen zur Karikatur überzeichnet. Oft geht es darum, trotz der widrigen äußeren Umstände die große Party zu feiern.
Diskografie
Alben
2013: 2000 Watt (CD)
2015: MegaBosch LIVE in Wacken 2014 (DVD)
2022: Die Rockstars der Apokalypse (CD)
Videos
2010: Es ist Geil
2013: Welcome to Paradise City
2015: Rabbaddha
Weblinks
Offizielle Website
Einzelnachweise
Metal-Band
Hard-Rock-Band
Deutsche Band
Band (Hannover)
|
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Die Grabhügel bei Aukrug-Bargfeld sind Reste einer Grabhügelgruppe, die von der vorgeschichtlichen Besiedlung der Region um Aukrug im Kreis Rendsburg-Eckernförde in Schleswig-Holstein zeugen. Im Naturpark Aukrug sind insgesamt dreizehn Grabhügel erhalten und unter Denkmalschutz gestellt.
Siehe auch
Nordische Megalitharchitektur
Quellen
Tafel des Landesamtes für Denkmalpflege
Georg Reimer: Die Geschichte des Aukrugs, herausgegeben von Heinrich Bünger, 3. erweiterte Auflage, Verlag Möller Söhne, Rendsburg 1978
Heinrich Asmus, Werner Hauschildt, Peter Höhne: Fortschreibung von "Die Geschichte des Aukrugs" ab 1978 und Nachträge, Aukrug 1995
Aukrug-Bargfeld
Aukrug-Bargfeld
Archäologischer Fundplatz im Kreis Rendsburg-Eckernförde
Bauwerk in Aukrug
Jungsteinzeit
Archäologischer Fundplatz (Bronzezeit)
Geographie (Aukrug)
Sakralbau im Kreis Rendsburg-Eckernförde
Grabbau in Europa
Archäologischer Fundplatz in Europa
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I went to court in Bronx and brought home bedbugs, single mom Brianna Duggan says
By KEVIN DEUTSCH
DAILY NEWS WRITER |
Brianna Duggan says she picked up bedbugs in Bronx County Family Court last month. (McCoy/AP)
Forget the divorce lawyers. Brianna Duggan now has a whole new set of bloodsuckers to deal with in the Bronx courts: bedbugs!
The 36-year-old single mom says her Soundview apartment is infested with the bloodsuckers she picked up in Bronx County Family Court last month.
"It's disgusting that they let people inside the courthouse while bedbugs were crawling all around," said Duggan, who accompanied her cousin to court for custody hearings.
"I was already broke from helping her pay for divorce lawyers, and now I'm paying to get them [the bedbugs] out of my home."
A state courts spokesman confirmed the courthouse infestation, which was discovered the first week of September and quickly snuffed out, he said.
Duggan, who was in the court at 900 Sheridan Ave. that same week, said the exterminators showed up too late to help her and scores of other courthouse visitors.
"People were already itching and scratching, but no one in the court listened to us until after the fact," she said.
The creepy parasites were also reported in the Bronx's two main courthouses near Yankee Stadium last month, sources said - even infiltrating the Bronx borough president's offices in the old courthouse at 851 Grand Concourse.
The Bronx and Brooklyn district attorneys' offices have also had infestations.
kdeutsch@nydailynews.com
Latest New York
New report finds work-based learning gap in city schools
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{
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Tonga pod względem administracyjnym podzielone jest na 5 grup wysp (ang. island division), które dzielą się następnie na 23 dystrykty, a te na 175 wsi.
Grupy wysp:
Przypisy
Geografia Tonga
Tonga, podział administracyjny
ru:Тонга#Административное деление
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| 3,205
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\section{Introduction}
In recent years interest in the properties of gravity in more than $D=4$
dimensions increased significantly.
This interest was enhanced by the development of string theory,
which requires a ten-dimensional spacetime,
to be consistent from a quantum point of view.
In order not to contradict observational evidence,
the extra dimensions are usually supposed to be compactified
on small scales.
Black string solutions, present for
$D\geq 5$ spacetime dimensions, are of particular interest, since they
exhibit new features that have no analogue in the black hole case.
The simplest vacuum static solutions of this type
are found by trivially extending to $D$ dimensions
the vacuum black hole solutions to Einstein equations in $D-1$ dimensions.
These then correspond to uniform black strings (UBS)
with horizon topology $S^{D-3}\times S^1$.
One of the first steps towards understanding the
higher-dimensional solutions is to investigate
their classical stability against small perturbations.
In a surprising development, Gregory and Laflamme (GL) showed that
the static UBS solutions are unstable below a critical value of the mass
\cite{Gregory:1993vy}.
Following this discovery, a branch of nonuniform black string (NUBS)
solutions was found perturbatively from the critical
GL string in five \cite{Gubser:2001ac}, six \cite{Wiseman:2002zc}
and in higher, up to sixteen, dimensions \cite{Sorkin:2004qq}.
This nonuniform branch was numerically extended into the full nonlinear regime
for $D=5$ \cite{Kleihaus:2006ee} and $D=6$ \cite{Wiseman:2002zc,Kleihaus:2006ee}.
In recent work \cite{Sorkin:2006wp}
the nonuniform branch was extended for all dimensions up to eleven.
These NUBS are static configurations with a nontrivial dependence
on the extra dimension,
and their mass is always greater than that of the critical UBS
(see \cite{Kol:2004ww} and also \cite{Harmark:2007md}
for a recent review of this subject).
Apart from the black string solutions,
Kaluza-Klein (KK) theory possesses also a
branch of black hole solutions with an event horizon of topology $S^{D-2}$.
The numerical results presented in \cite{Kudoh:2004hs, Kleihaus:2006ee}
(following a conjecture put forward in \cite{Kol:2002xz})
suggest that the black hole branch and the nonuniform
string branch merge at a topology changing transition.
The problem is also interesting as it is connected
by holography to the phase structure of large $N_c$
super-Yang-Mills theory at strong
coupling, compactified on a circle (see e.g. \cite{Harmark:2004ws},
\cite{Harmark:2006df},
\cite{Chowdhury:2006qn} ).
Recently, interest in the properties of rotating solutions
in more than $D=4$ dimensions increased significantly, as well.
Rotating black objects typically exhibit much richer dynamics than their
static counterparts, especially in more than four dimensions.
A famous example is the black ring solution \cite{Emparan:2001wn}
in five-dimensional vacuum gravity,
which has horizon topology $S^2\times S^1$, its tension
and self gravitational attraction being balanced by the rotation of the ring.
The KK theory presents also spinning configurations.
The simplest rotating UBS configurations are found by taking
the direct product of a $(D-1)$-dimensional Myers-Perry (MP) solution
\cite{Myers:1986un}
with a circle.
These solutions are likely to exhibit a
classical GL instability, as well, at least for some range of the parameters.
However, previous investigations have focused on static black strings,
and no attention has been given to spin.
A major obstacle in this direction
is that the analytic theory of perturbations of higher-dimensional black holes
has not been fully developed yet
(see, however, the recent work \cite{Kunduri:2006qa}).
A MP spinning
black string in $D$ dimensions is characterized by
the mass-energy, the tension, and
$[(D-2)/2]$ angular momenta,
where $[(D-2)/2]$ denotes the integer part of $(D-2)/2$.
The generic rotating nonuniform solutions possess a nontrivial dependence
on $(D-4)/2$ angular coordinates,
which therefore pose a difficult numerical problem.
In the even-dimensional case, however,
the problem can be greatly simplified, when the
${\it a\ priori}$ independent $(D-2)/2$ angular momenta
are chosen to have equal magnitude,
since this factorizes the angular dependence \cite{Kunz:2006eh}.
The problem then reduces to studying the solutions of a set of five partial
differential equations with dependence only
on the radial variable $r$ and the extra dimension $z$.
In this paper we focus on this particular case, by
studying first the GL instability of MP UBSs with equal
magnitude angular momenta in even spacetime dimensions.
These particular MP solutions possess interesting features,
which strongly contrast with those of MP solutions
with a single nonzero angular momentum.
In particular, we note the existence of an upper bound for
the scaled angular momenta $J/M^{(D-3)/(D-4)}$ \cite{Myers:1986un,Kunz:2006jd},
an extremal solution being found,
when this limit is approached;
whereas no such upper bound is present
for single angular momentum MP solutions, unless $D=4+1$ or $D=5+1$.
Although the GL instability is inevitable for static vacuum
black strings,
the presence of rotation (or a gauge field
charge) might prevent black strings from exhibiting
such an instability.
However, our numerical results indicate that the GL instability
persists for rotating vacuum black strings all the way to extremality,
at least for all (even) dimensions between six and fourteen.
This type of solutions also provides a new laboratory
to test the Gubser-Mitra (GM) conjecture \cite{Gubser:2000ec},
that correlates
the dynamical and thermodynamical stability
for systems with translational symmetry and infinite extent.
In this conjecture, the appearance of a negative specific heat
of a black string is related to the onset of a classical instability
(see \cite{Miyamoto:2006nd, Kudoh:2005hf} for a discussion of this issue in the case
of charged static black strings and black $p-$branes).
The analysis of the thermodynamical stability of the MP UBS
indicates that thermodynamical
stability becomes possible when taking a canonical ensemble, for solutions
near extremality.
However, in a grand canonical ensemble
all UBS configurations are unstable,
which agrees with the results we found when studying the
GL instability.
In $D=6$ we constructed the set of rotating
nonuniform black strings numerically.
These rotating NUBS solutions can be found by
starting with static NUBS configurations
and increasing the value of angular velocity of the event horizon,
in the domain where static NUBS exist.
Alternatively, one can start with rotating MP UBS solutions
and then construct the set of rotating NUBS configurations
from the stationary perturbative nonuniform solutions.
The paper is structured as follows:
we begin with a presentation of the general ansatz and the generic
properties of
rotating black strings with equal angular momenta in
even spacetime dimensions.
In Section 3, we consider the corresponding
MP uniform solutions, discussing their
thermodynamical properties and the issue of GL instability.
In Section 4 we
demonstrate that for $D=6$
a set of rotating NUBS solutions with two equal angular momenta exists,
at least within the scope of our numerical approximation.
The numerical methods used here are similar to those
employed to obtain the $D=5,~6$ static NUBS solutions in \cite{Kleihaus:2006ee}.
We discuss charged NUBS in heterotic string theory in section 5,
and give our conclusions and remarks in the final section.
\section{General ansatz and properties of the solutions}
\subsection{The equations }
We consider the Einstein action
\begin{eqnarray}
\label{action-grav}
I=\frac{1}{16 \pi G_D}\int_M~d^Dx \sqrt{-g} R
-\frac{1}{8\pi G_D}\int_{\partial M} d^{D-1}x\sqrt{-h}K,
\end{eqnarray}
in a $D-$dimensional spacetime, with $D\geq 6$ an even number.
The last term in (\ref{action-grav}) is the Hawking-Gibbons surface term \cite{Gibbons:1976ue},
which
is required in order to have a well-defined variational principle.
$K$ is the trace
of the extrinsic curvature for the boundary $\partial\mathcal{M}$ and $h$ is the induced
metric of the boundary.
We consider black string solutions approaching asymptotically the
$(D-1)$-dimensional Minkowski-space times a circle ${\cal M}^{D-1}\times S^1$.
We denote the compact direction as $z = x^{D-1}$
and the directions of $R^{D-2}$ as $x^1,...,x^{D-2}$, while $x^D=t$.
The direction $z$ is periodic with period $L$.
We also define the radial coordinate $r$ by
$r^2 = (x^1)^2 + \cdots + (x^{D-2})^2$.
To obtain nonuniform generalizations of the rotating uniform black
string MP solutions,
we consider space-times with $ (D-2)/2$ commuting Killing vectors
$ \partial_{\varphi_k}$.
While the general configuration will then possess $ (D-2)/2 $ independent
angular momenta, we here restrict to rotating NUBS whose
angular momenta have all equal magnitude.
Analogous to the case of black holes \cite{Kunz:2006eh},
the metric parametrization then simplifies considerably
for such rotating NUBS
\begin{eqnarray}
&&ds^2 = -e^{2A(r,z)}\frac{ f(r)}{h(r)}dt^2 + e^{2B(r,z)} (\frac{ dr^2}{f(r)} +dz^2)
+
e^{2C(r,z)}r^2\sum_{i=1}^{(D-4)/2}
\left(\prod_{j=0}^{i-1} \cos^2\theta_j \right) d\theta_i^2
\nonumber
\\
\nonumber
&&
+e^{2G(r,z)}h(r)r^2 \sum_{k=1}^{(D-2)/2} \left( \prod_{l=0}^{k-1} \cos^2 \theta_l
\right) \sin^2\theta_k \left( d\varphi_k - W(r,z)dt\right)^2+r^2\left(e^{2C(r,z)}-e^{2G(r,z)} h(r)\right)
\\
\label{metric}
&&
\times
\left\{
\sum_{k=1}^{(D-2)/2} \left( \prod_{l=0}^{k-1} \cos^2
\theta_l \right) \sin^2\theta_k d\varphi_k^2 \right.
-\left. \left[\sum_{k=1}^{(D-2)/2} \left( \prod_{l=0}^{k-1} \cos^2
\theta_l \right) \sin^2\theta_k d\varphi_k\right]^2 \right\} \ ,
\end{eqnarray}
where $\theta_i \in [0,\pi/2]$
for $i=1,\dots , (D-4)/2$, $\varphi_k \in [0,2\pi]$ for $k=1,\dots , (D-2)/2$,
and we formally define $\theta_0 \equiv 0$,
$\theta_{(D-2)/2} \equiv \pi/2$.
As a result of taking all angular momenta to be equal,
the symmetry group of this spacetime is enhanced from $R\times U(1)^{(D-2)/2}$
to $R\times U(\frac{D-2}{2})$, where $R$ denotes time translation.
We shall assume that the information on the NUBS
solutions is encoded in the functions $A(r,z),B(r,z),C(r,z),G(r,z)$
and $W(r,z)$, while
$f(r)$ and $h(r)$ are two `background' functions which are chosen
for convenience.
A useful parametrization when studying unstable modes around a MP solution is
\begin{eqnarray}
\label{MP}
f(r)=1
-\frac{2M }{r^{D-4}}
+\frac{2Ma^2}{r^{D-2}}~,~~
h(r)= 1+\frac{2Ma^2}{r^{D-2}}~,~~
w(r)=\frac{2Ma}{r^{D-2}h(r)}~,
\end{eqnarray}
where $M$ and $a$ are two constants related to the solution's mass and
angular momentum
(and $W(r,z)=w(r)$ for uniform black strings).
When constructing nonperturbative NUBS solutions, a more convenient choice
for the numerics is
\begin{eqnarray}
\label{MP-2}
f(r)=1-(r_0/r)^{D-4}~,~~h(r)=1 ~,
\end{eqnarray}
together with a redefinition of the radial coordinate,
where $r_0$ denotes the coordinate value of the horizon.
The static NUBS ansatz used in previous studies is recovered for
$G(r,z)=C(r,z)$ and $W(r,z)=0$.
A suitable combination
of the Einstein equations,
$G_t^t=0,~G_r^r+G_z^z=0$, $G_{\theta_1}^{\theta_1}=0$,
$G_{\varphi_1}^t=0$ and $G_{\varphi_1}^{\varphi_1}=0$,
yields the following set of equations for the functions $A,~B,~C,~G$ and $W$
\begin{eqnarray}
\label{ec1}
&&\hat O^2 A
+(\hat O A)^2
+\hat O A \cdot \hat O G
+(D-4) \hat O A \cdot \hat O C
-\frac{e^{-2A+2G}h^2}{2r^2f}(\hat O W)^2
\\
\nonumber
&&
+(\frac{D-3}{r}+\frac{3f'}{2f}-\frac{h'}{2h})\partial_r A
+( \frac{f'}{2f}-\frac{h'}{2h})((D-4)\partial_r C+\partial_rG)
\\
\nonumber
&&
+\frac{(D-3)f'}{2rf}-\frac{(D-3)h'}{2rh}
-\frac{f'h'}{2fh}
+\frac{h'^2}{2h^2}
+\frac{f''}{2f}
-\frac{h''}{2h}=0,
\end{eqnarray}
\begin{eqnarray}
\label{ec2}
&&
\hat O^2 B
-\frac{e^{-2A+2G}r^2h^2}{4f}(\hat O W)^2
-(D-4)\hat O A \cdot \hat O C
-\frac{1}{2}(D-4)(D-5)(\hat O C)^2
\\
\nonumber
&&
-\hat O A \cdot \hat O G
-(D-4)\hat O C \cdot \hat O G
-(\frac{D-3}{r}+\frac{h'}{2h})\partial_r A
+\frac{f'}{2f}\partial_rB
-(D-4)(\frac{D-4}{r}+\frac{f'}{2f})\partial_r C
\\
\nonumber
&&
+\frac{1}{2}(\frac{h'}{h}-\frac{f'}{f}-\frac{2(D-4)}{r})\partial_rG
+\frac{(D-2)(D-4)}{2r^2f}e^{2B-2C}
-\frac{(D-4)h}{2r^2f}e^{2B-4C+2G}
\\
\nonumber
&&
-\frac{ (D-3)f'}{2rf}
-\frac{f'h'}{4fh}
+\frac{h'}{2rh}
+\frac{h'^2}{4h^2}
-\frac{(D-3)(D-4)}{2r^2}
=0,
\end{eqnarray}
\begin{eqnarray}
\label{ec3}
&&
\hat O^2 C
+(D-4)(\hat O C)^2
+\hat O C \cdot \hat O G
+\hat O A \cdot \hat O C
+\frac{1}{r}(\partial_rA+\partial_r G)
\\
\nonumber
&&
+(\frac{2D-7}{r}+\frac{f'}{f})\partial_rC
-\frac{(D-2)}{r^2f}e^{2B-2C}
+\frac{2h}{r^2f}e^{2B-4C+2G}
+\frac{f'}{rf}
+\frac{D-4}{r^2}=0,
\end{eqnarray}
\begin{eqnarray}
\label{ec4}
&&
\hat O^2 G
+ (\hat O G)^2
+\hat O A \cdot \hat O G
+(D-4)\hat O C \cdot \hat O G
+\frac{e^{-2A+2G}r^2 h^2}{2f}(\hat O W)^2
\\
\nonumber
&&
+\frac{(D-4)}{2}(\frac{2}{r}+\frac{h'}{h}) \partial_r C
+(\frac{f'}{f}+\frac{h'}{2h}+\frac{(D-2)}{r} )\partial_r G
+(\frac{1}{r}+\frac{h'}{2h})\partial_r A
-\frac{(D-4)h}{r^2f}e^{2B-4C+2G}
\\
\nonumber
&&
+\frac{f'}{rf}
+(\frac{D-3}{2r}+\frac{f'}{2f}-\frac{h'}{2h})\frac{h'}{h}
+\frac{h''}{2h}
+\frac{D-4}{r^2}=0,
\end{eqnarray}
\begin{eqnarray}
\label{ec5}
\hat O^2 W
-\hat O A \cdot \hat O W
+(D-4)\hat O C \cdot \hat O W
+3\hat O G \cdot \hat O W
+(\frac{D-1}{r}+\frac{2h'}{h})\partial_r W=0~,
\end{eqnarray}
where we define
\begin{eqnarray}
\label{rel}
\hat O U \cdot \hat O V=\partial_r U \partial_r V+\frac{1}{f}\partial_z U \partial_z V,~~~
\hat O^2 U=\partial_r^2U+\frac{1}{f}\partial_z^2 U,
\end{eqnarray}
and a prime denotes the derivative with respect to $r$.
All other Einstein equations except for $G_z^r=0$ and $G_r^r-G_z^z=0$
are linear combinations of those used to derive
the above equations or are identically zero.
The remaining Einstein equations $G_z^r=0,~G_r^r-G_z^z=0$
yield two constraints. Following \cite{Wiseman:2002zc}, we note that
setting $G^t_t=G^{\theta_i}_{\theta_i}=G^{\varphi_k}_{\varphi_k}=G^r_r+G^z_z=0$
in $\nabla_\mu G^{\mu r}=0$ and $\nabla_\mu G^{\mu z}=0$, we obtain
\begin{eqnarray}
\partial_z\left(\sqrt{-g} G^r_z \right) +
\sqrt{f} \partial_r\left( \sqrt{f}\sqrt{-g} \frac{1}{2}(G^r_r-G^z_z) \right)
& = & 0 ,
\\
\nonumber
\sqrt{f}\partial_r\left(\sqrt{-g} G^r_z \right)
-\partial_z\left( \sqrt{f}\sqrt{-g} \frac{1}{2}(G^r_r-G^z_z) \right)
& = & 0 ,
\end{eqnarray}
and, defining $\hat{r}$ via
$\partial/\partial_{\hat{r}} = \sqrt{f}\partial/\partial_{r}$,
then yields the Cauchy-Riemann relations
\begin{eqnarray}
\partial_z\left(\sqrt{-g} G^r_z \right) +
\partial_{\hat{r}}\left( \sqrt{f}\sqrt{-g} \frac{1}{2}(G^r_r-G^z_z) \right)
& = & 0 ,\\
\nonumber
\partial_{\hat{r}}\left(\sqrt{-g} G^r_z \right)
-\partial_z\left( \sqrt{f}\sqrt{-g} \frac{1}{2}(G^r_r-G^z_z) \right)
& = & 0 .
\end{eqnarray}
Thus the weighted constraints satisfy Laplace equations,
and the constraints are fulfilled,
when one of them is satisfied on the boundary
and the other at a single point
\cite{Wiseman:2002zc}.
\subsection{General properties }
We impose the event horizon to reside at a surface of constant radial coordinate
$r=r_0$, where $f(r)=f'(r_0)(r-r_0)+O(r-r_0)^2$ while $h(r_0)>0,~f'(r_0)>0$.
Also, the functions $f(r)$ and $h(r) $ take only positive
values for $r>r_0$ and tend to one for $r \to \infty$.
The Killing vector
$\chi=\partial/\partial_t+ \sum_k \Omega_k \partial/\partial \varphi_k $
is orthogonal to and null on the horizon.
For the solutions within the ansatz (\ref{metric}),
the event horizon angular velocities are equal, $\Omega_k=\Omega_H=W(r,z)|_{r=r_0}$.
Utilizing the reflection symmetry of the nonuniform black strings
w.r.t.~$z=L/2$,
the nonuniform solutions are constructed subject to the following set of
boundary conditions
\begin{eqnarray}
\label{bc1}
A\big|_{ r=\infty}=B\big|_{ r=\infty}=C\big|_{ r=\infty}
=G\big|_{ r=\infty}=W\big|_{ r=\infty}=0,
\end{eqnarray}
\begin{eqnarray}
\label{bc2}
B\big|_{r={r_0}}-A\big|_{r={r_0}}=d_0,~\partial_{ r}
A\big|_{r={r_0}}=\partial_{r} C\big|_{r={r_0}}
=\partial_{r} G\big|_{r={r_0}}=0,~~
W\big|_{r={r_0}}=\Omega_H,
\end{eqnarray}
\begin{eqnarray}
\label{bc3}
\partial_z A\big|_{z=0,L/2}=\partial_z B\big|_{z=0,L/2}=\partial_z C\big|_{z=0,L/2}
=\partial_z G\big|_{z=0,L/2}=\partial_z W\big|_{z=0,L/2}=0,
\end{eqnarray}
where the constant $d_0$ is related to the Hawking
temperature of the solutions.
As in the case of $3+1$-dimensional Kerr black holes,
the rotating black strings have an ergosurface
inside of which
observers cannot remain stationary, and will
move in the direction of the rotation.
The ergosurface is located at $g_{tt}=0$, i.e.
\begin{eqnarray}
\label{er}
e^{2G(r,z)} r^2 h(r) W(r,z)^2-\frac{e^{2A(r,z)} f(r)}{h(r)}=0 ,
\end{eqnarray}
and does not intersect the horizon.
(Note that the ergoregion here extends nontrivially in the extra dimension.)
The computation of the conserved charges of rotating black strings
was discussed e.g.~in \cite{Townsend:2001rg}.
The essential idea there is to consider the asymptotic
values of the gravitational field far away from the black string
and to compare them with those corresponding to a gravitational field
in the absence of the black string.
The obvious choice of the background in this case is ${\cal M}^{D-1}\times S^1$,
the asymptotic form of the relevant metric components
being
\begin{eqnarray}
\label{1}
g_{tt}\simeq -1+\frac{c_t}{r^{D-4}},~~~g_{zz}\simeq 1+\frac{c_z}{r^{D-4}},~~~
g_{\varphi_k t}\simeq \left( \prod_{l=0}^{k-1} \cos^2 \theta_l
\right) \sin^2\theta_k \frac{c_\varphi}{r^{D-4}},
\end{eqnarray}
which reveals the existence of three free parameters $c_t,~c_z$ and $c_\varphi$.
The mass-energy $E$, the tension
${\mathcal T}$ and the angular momenta $J_k$
of black string solutions are given by
\begin{eqnarray}
\label{gen-def}
\hspace{-0.5cm}
E=\frac{A_{D-3}L}{16 \pi G_D}((D-3)c_t-c_z),
~{\mathcal T}=\frac{A_{D-3}}{16 \pi G_D}(c_t-(D-3)c_z),
~J_k=J=-\frac{A_{D-3}L}{8 \pi G_D} c_\varphi,
\end{eqnarray}
where
$A_{D-3}=2\pi ^{\frac{D-2}{2}}/\Gamma ({(D-2)}/2)$ is
the area of the unit $D-3$ sphere.
The global charges of a black string can also
be computed by using the quasilocal
tensor of Brown and York \cite{Brown:1992br},
augmented by the counterterms formalism.
In this approach we add to the action (\ref{action-grav})
suitable counterterms $I_{ct}$
built up with
curvature invariants of the induced metric on the boundary $\partial \cal{M}$
\cite{Kraus:1999di},\cite{Astefanesei:2006zd},\cite{Mann:2005yr}.
These counterterms do not alter the bulk equations of motion.
Unlike the background substraction, this procedure is
satisfying since it is intrinsic to the spacetime of interest and it is
unambiguous once the counterterm is specified.
Our choice of the counterterm was similar to the static case \cite{Kleihaus:2006ee},
$I_{ct}=
-\frac{1}{8 \pi G_D}\sqrt{(D-3)/{(D-4)} }\int_{\partial\mathcal{M}} d^{D-1} x\sqrt{-h}
\sqrt{ \mathcal{R}},$
where
$\mathcal{R}$ is the Ricci scalar of the boundary geometry.
The variation of the total action with respect to the
boundary metric $h_{ij}$ provides a boundary stress-tensor, whose
expression is given e.g. in \cite{Kleihaus:2006ee}.
The mass-energy, tension and angular momenta are the charges associated to $\partial/\partial t$,
$\partial/\partial z$ and
$\partial/\partial \varphi_k$ respectively (note that $\partial/\partial z$
is a Killing symmetry of the boundary metric).
We have verified that for $D=6,~8$ and $D=10$ the expressions computed in this way
agree with (\ref{gen-def})
(see \cite{Astefanesei:2006zd,Astefanesei:2005ad} for similar computations
for a different type of rotating solutions).
One can also define a relative tension $n$
(also called the relative binding energy or scalar charge)
\begin{eqnarray}
\label{3}
n=\frac{{\mathcal T} L}{E}=\frac{c_t-(D-3)c_z}{(D-3)c_t-c_z},
\end{eqnarray}
which measures how large the tension is relative to the mass-energy, being constant
for UBS solutions.
The Hawking temperature $T_H=\kappa_H/2\pi$ can be obtained from
the standard relation
\begin{eqnarray}
\label{kappa}
\kappa_H^2=- \left. \frac{1}{2}\nabla^a \chi^b \, \nabla_a\chi_b
\right|_{r=r_0},
\end{eqnarray}
where $\kappa_H$ is the surface gravity, which is constant
at the horizon.
One finds
\begin{eqnarray}
\label{temp}
T_H=e^{A_0-B_0}\,T_{H}^{(0)} = e^{-d_0}\,T_{H}^{(0)},
\end{eqnarray}
where $T_{H}^{(0)} $ is the Hawking temperature of the uniform solution
based on the same `background' functions (\ref{MP}),
$T_{H}^{(0)}=f'(r_0)/(4\pi {\sqrt{h(r_0)}})$.
Here and below
$A_0(z),B_0(z),C_0(z),G_0(z)$ and $W_0(z)$ denote the values of
the metric functions on the event horizon $r=r_0$.
The area $A_H$ of the black string horizon can also be expressed in a similar way
\begin{eqnarray}
\label{A}
A_H= A_{H}^{(0)}\frac{1}{L}\int_0^L e^{B_0+(D-4)C_0+G_0}dz,
\end{eqnarray}
with $A_{H}^{(0)}$ the event horizon area of the
corresponding uniform solution
\begin{eqnarray}
\label{A0}
A_{H }^{(0)}=LA_{D-3}r_0^{D-3}\sqrt{h(r_0)}~.
\end{eqnarray}
As usual, one identifies the entropy of the black string solutions
with one quarter of their event horizon area, $S=A_H/4G_D$.
Considering the thermodynamics of these solutions,
the black strings should satisfy the first law of thermodynamics
\begin{eqnarray}
\label{firstlaw}
dE=T_HdS+\frac{(D-2)}{2} \Omega_H dJ+{\mathcal T}dL.
\end{eqnarray}
One may regards the parameters $S,~J$ and $L$ as a complete set of extensive parameters
for the mass-energy $E(S,J,L)$ and define the intensive parameters
conjugate to them.
These quantities are the temperature, the angular velocities and the tension.
Following \cite{Chowdhury:2006qn}, \cite{Kol:2003if},
one can derive in a simple way a
Smarr formula, by letting the length
of the compact extra dimension change as $L \to L+dL$.
This implies
\begin{eqnarray}
\label{SM1}
E \to E(1+\frac{dL}{L})^{D-3},~~
S\to S(1+\frac{dL}{L})^{D-2},~~
J\to J(1+\frac{dL}{L})^{D-2}.
\end{eqnarray}
As a result we find the Smarr formula
(see also \cite{Townsend:2001rg} for a different derivation of this relation)
\begin{eqnarray}
\label{smarrform}
\frac{D-3-n}{D-2} E=T_HS+\frac{(D-2)}{2} \Omega_H J~.
\end{eqnarray}
In the canonical ensemble, we study black strings holding the temperature
$T_H$, the angular momenta $J$
and the length $L$ of the extra dimension fixed.
The associated thermodynamic potential is the Helmholz free energy
\begin{eqnarray}
\label{F}
F[T_H,J,L]=E-T_HS ~.
\end{eqnarray}
In the grand canonical ensemble, on the other hand, we keep
the temperature, the angular velocity and the tension fixed.
In this case the thermodynamics is obtained from the Gibbs potential
\begin{eqnarray}
\label{G}
G[T_H,\Omega_H,{\mathcal T}]=E-T_H S-\frac{(D-2)}{2}\Omega_H J-{\mathcal T} L.
\end{eqnarray}
We finally remark that
the technique used in \cite{Horowitz:2002dc}, \cite{Harmark:2003eg}
to construct `copies of solutions' works for rotating NUBS, too.
When taking
$f(r)=1-(r_0/r)^{D-4}$, $h(r)=1$, i.e.~(\ref{MP-2}),
the Einstein equations (\ref{ec1})-(\ref{ec5}) are left invariant
by the transformation $r \to r/k$, $ z \to z/k$, $ r_0 \to
r_0/k $, with $k$ an arbitrary positive integer.
Therefore, one may generate a family of vacuum solutions
in this way.
The new solutions have the same length of the extra dimension.
Their relevant properties, expressed in terms of the
corresponding properties of the initial solution, read
\begin{eqnarray}
E^{(k)}=\frac{E}{k^{D-4}}, ~T_{H }^{(k)}=k T_H,~~S^{(k)}=\frac{S}{k^{D-3}},~
n^{(k)}=n,~
J^{(k)}=\frac{J}{k^{D-3}},~\Omega_{H }^{(k)}=k\Omega_H.
\end{eqnarray}
\section{Rotating UBS
with equal angular momenta}
\subsection{Thermodynamics}
We start by discussing the properties of
uniform black string solutions obtained by taking $A=B=C=G=0$
in the general ansatz
(\ref{metric}) with the functions $f,h$ and $W(r,z)=w(r)$ given by (\ref{MP}).
The extra dimension plays no role here and the
general results
apply to $d=(D-1)$-dimensional MP black holes when formally taking
$L=1$ in the relations below.
The uniform black strings
have two parameters $M$ and $a$ which, from (\ref{gen-def}),
are related to the physical mass-energy and angular momenta by
\begin{eqnarray}
\label{M,a}
E=\frac{A_{D-3}L}{8\pi G_D}(D-3)M,~~J_i=J=\frac{A_{D-3}L}{4\pi G_D} Ma,
\end{eqnarray}
Hence one can think of $a$ as essentially
the angular momentum per unit mass.
The tension ${\mathcal T}$ of the uniform solutions
is fixed by the mass-energy $E$
and length $L$ of the extra dimension
\begin{eqnarray}
\label{tens1}
{\mathcal T}=\frac{E}{L(D-3)}.
\end{eqnarray}
The event horizon of these uniform black strings
can be determined as the largest root of $1/g_{rr}=0$
resp.~$f(r)=0$. That is
\begin{eqnarray}
\label{eq-f}
r_0^{D-2}-2Mr_0^2+2Ma^2=0.
\end{eqnarray}
This equation has zero, one or two positive roots, depending
on the sign of $f(r_s)$, where $r_s=(4M/(D-2))^{1/(D-4)}$
is the largest root of $(r^{D-2}f(r))\,'=0$.
The existence of a regular horizon implies an upper limit on $a$,
\begin{eqnarray}
\label{f'}
a^2\leq \frac{D-4}{D-2} \left(\frac{4M}{D-2}\right)^{\frac{2}{D-4}}
= a^2_{max},
\end{eqnarray}
which via (\ref{M,a}) can also be expressed
as
\begin{eqnarray}
\label{bound1}
\frac{J}{E^{\frac{D-3}{D-4}}}\leq
\sqrt{D-4}
\left(\frac{2^{D+1}\pi}{(D-3)^{D-3}(D-2)^{(D-2)/2}}\right)^{1/(D-4)}
(\frac{G_D}{A_{D-3}L})^{1/(D-4)}.
\end{eqnarray}
This strongly contrasts with the case of MP UBS solutions
with a single nonzero angular momentum,
where for $D>6$ there are configurations with arbitrarily large $J$,
without an occurrence of an extremal limit\footnote{
However, as argued in \cite{Emparan:2003sy}, MP black holes with
a single nonzero angular momentum are in
fact classically unstable (at least for large rotation)
and an effective Kerr bound arises through
a dynamical decay mechanism.}.
The ergosurface of the uniform solutions is located at $r_e=(2M)^{1/(D-4)}$.
The horizon angular velocities and the Hawking temperature of the UBS solutions
are given by
\begin{eqnarray}
\label{omegah}
\Omega_H=\frac{a}{r_0^2},~~~~T_H=\frac{1}{4\pi r_0}\left(\frac{2M}{r^{D-4}_0}\right)^{1/2}
\left((D-2)\frac{r^{D-4}_0}{2M}-2 \right).
\end{eqnarray}
Solutions with $a=a_{max}$ are extremal black strings with $T_H=0$,
possessing a nonzero entropy.
As can be seen e.g.~by computing the Kretschmann scalar,
the hypersurface $r=r_0$ is not singular in this limit.
Similar to the static case, the
gravitational thermodynamics of the rotating UBS can be formulated via the Euclidean
path integral.
The Euclidean spinning black string solutions can be obtained
from the Minkowskian ones by sending $t \to -it$
and $a \to ia$ (complexifying $a$ is necessary in order to keep
the $dt d\varphi_i$ part of the
metric real).
The thermodynamic system has a constant temperature $T_H=1/\beta$
which is determined by requiring the Euclidean
section be free of conical singularities (the temperature computed in this way
coincides with that in (\ref{omegah})).
The
partition function for the gravitational field is defined by a sum
over all smooth Euclidean geometries which are periodic with a period
$\beta$ in imaginary time \cite{Gibbons:1976ue}.
This integral is computed by using the saddle point approximation,
the global charges and entropy of the solutions being evaluated
by standard thermodynamic formulae.
Upon application of the Gibbs-Duhem relation to the partition
function \cite{Mann:2003 Found},
this yields an expression for the entropy
\begin{eqnarray}
S=\beta \left(E- \frac{(D-2)}{2}\Omega_H J-{\mathcal T} L\right)-I,
\label{GibbsDuhem}
\end{eqnarray}%
(with $I$ the regularized tree level Euclidean action),
which agrees with that computed from (\ref{A}).
The entropy can be written in terms of $M$, $r_0$ as
\begin{eqnarray}
\label{entropy}
S=\frac{1}{4G_D}A_{D-3}Lr_0^{D-3}
\left(\frac{2M}{r^{D-4}_0}\right)^{1/2} .
\end{eqnarray}
The parameters $M$, $a$, $r_0$ can be eliminated and one can write the following
equation of state (analogous to $f(p,V,T)$, for, say, a gas at pressure $p$ and volume $V$)
\begin{eqnarray}
\label{n12}
J=2^{-(D+6)/2}\frac{A_{D-3}L}{G_D}
(D-4)^{D-2}\pi^{1-D}\Omega_H T_H^{2-D}
\left(1+\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}\right)^{-D/2}
\\
\nonumber
\times
\left((D-2)\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}-D+6\right)^{-(D-4)/2}.
\end{eqnarray}
The following relations are also useful in what follows
\begin{eqnarray}
\label{n5}
T_H=\frac{D-4}{\pi}\left(\frac{A_{D-3}L}{2^{2(D-2)}G_D}\right)^{\frac{1}{D-3}}
S^{1/(3-D)}
\left(1+\frac{4\pi^2 J^2}{ S^2}\right)^{-\frac{D-4}{2(D-3)}}
\left(1-\frac{8 \pi^2 J^2}{(D-4)S^2}\right) ,
\end{eqnarray}
\begin{eqnarray}
\label{n10}
E=2^{-\frac{D+6}{2}}(D-3)(D-4)^{D-4}\pi^{3-D}\frac{A_{D-3}L}{G_D}
T_H^{4-D}
\left(1+\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}\right)^{(2-D)/2}
\\
\nonumber
\times
\left( (D-2)\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}+6-D\right)^{(6-D)/2}~,
\end{eqnarray}
\begin{eqnarray}
\label{n11}
S=2^{-(D+2)/2}\frac{A_{D-3}L}{G_D}
(D-4)^{D-3}\pi^{3-D}T_H^{3-D}
\left(1+\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}\right)^{-(D-2)/2}
\\
\nonumber
\times
\left((D-2)\sqrt{1+\frac{(D-4)\Omega_H^2}{2\pi^2T_H^2}}-D+6\right)^{-(D-4)/2}~,
\end{eqnarray}
\begin{eqnarray}
\label{ESJ}
E= {2^{-\frac{2(D-2)}{(D-3)}}}\frac{(D-3)}{\pi}(\frac{A_{D-3}L}{G_D})^{1/(D-3)}S^{\frac{D-4}{D-3}}
\left(1+\frac{4\pi^2J^2}{S^2}\right)^{\frac{D-2}{2(D-3)}}~.
\end{eqnarray}
As implied by (\ref{n5}), the rotating UBSs have always
a smaller temperature for the same entropy than
the static UBSs.
The energy to entropy ratio of the UBS solutions satisfies the following bounds
\begin{eqnarray}
\label{lim1}
\frac{1}{2^{2(D-2)}}\leq (\frac{\pi}{D-3})^{D-3}
\frac{G_D}{A_{D-3}L}\frac{E^{D-3}}{S^{D-4}}
\leq \frac{(D-2)^{(D-2)/2}}{2^{5(D-2)/2}}~,
\end{eqnarray}
these limits being approached for static and extremal solutions,
respectively. There is also an upper bound for the ratio $J/S$,
\begin{eqnarray}
\label{lim2}
\frac{J}{S}\leq \frac{1}{2\pi} \sqrt{\frac{D-4}{2}}~.
\end{eqnarray}
The analysis of the thermodynamic stability of the rotating UBS
solutions can be performed using the above relations.
It is known that different thermodynamic ensembles are not exactly
equivalent and may not lead to the same conclusions
as they correspond to different
physical situations.
Mathematically, thermodynamic stability is equated with the subadditivity
of the entropy function.
This requires $S(E,J,L)$ to be a concave function of its extensive variables.
The stability can also be studied by the behaviour of the energy $E(S,J,L)$
which should be a convex function.
Therefore one has to compute the determinant of the Hessian matrix of
$E(S,J,L)$ with respect to its extensive variables $X_i$,
$H^{E}_{X_iX_j}=[\partial^2E/\partial X_i\partial X_j]$
\cite{Cvetic:1999ne}, \cite{Caldarelli:1999xj}.
In the canonical ensemble, the subadditivity of the entropy
is exactly equivalent to positivity
of the specific heat at constant $(J,L)$, $C_{J,L}=T_H(\partial S/\partial T_H)_{J,L}$.
Also, the Gibbs potential which is relevant for a grand canonical ensemble
can be written as
$
G[T_H,\Omega_H,L]=E/(D-2),
$
as a result of the Smarr law (\ref{smarrform}).
At this point it is instructive to see first the corresponding situation
for the $(D-1)$-dimensional MP black holes.
It is easy to work out from (\ref{n5})
that the condition for a positive specific heat at fixed
angular momenta
is equivalent to
\begin{eqnarray}
\label{lim-JS}
32(D-1)\pi^4J^4+4(D(D-5)+10)\pi^2J^2S^2-(D-4)S^4>0.
\end{eqnarray}
As expected,
$C_{J}$ is negative far from extremality, becoming positive
for large enough values of $J$.
To discuss the thermodynamic stability of black holes in a grand canonical
ensemble, we consider first the specific heat at constant
angular velocity at the horizon
\begin{equation}
C_\Omega=T_H\left(\frac{\partial S}{\partial T_H}\right)_{\Omega_H}.
\end{equation}
A straightforward computation using (\ref{n11})
shows that this is a negative quantity
in the full range of variables, $C_\Omega<0$.
One can also verify that the determinant of the Hessian
matrix of
$E(S,J)$ is negative
\begin{eqnarray}
\label{Hess}
{\rm det} (\partial^2E/\partial X_i\partial X_j)=-
\frac{4(D-2)\pi^2 }{(D-3)^3 }\frac{E^2(8J^2\pi^2+(D-2)S^2)}{S^2(4J^2\pi^2+S^2)}<0~.
\end{eqnarray}
As a result, all MP rotating black hole solutions with equal angular momenta
are unstable in a grand canonical ensemble,
and also the configurations far from extremality
in a canonical ensemble.
Another `response function' of interest is the
`isothermal permittivity'
\begin{eqnarray}
\epsilon_T \equiv \left(\frac{\partial J}{
\partial\Omega_H}\right)_{T_H}.
\end{eqnarray}
One finds from (\ref{n12})
that the condition for a positive $\epsilon_T $ is
\begin{eqnarray}
\label{n13}
\frac{\Omega_H}{T_H}< \left(\frac{\pi^2}{D-4}
\left(\frac{2+5D-D^2}{(D-2)(D-3)}
+\sqrt{\frac{D^2-5D+22}{(D-2)(D-3)}}~\right)\right)^{1/2}.
\end{eqnarray}
For $D=6$, this corresponds to $\Omega_H/T_H<1.7165$, for $D=8$ to
$\Omega_H/T_H<0.7892$ and for $D=10$ to $\Omega_H/T_H<0.477$.
All other rotating black hole solutions have a
negative `isothermal permittivity' and thus
are unstable to angular
fluctuations, both in a grand canonical and a canonical ensemble.
These conclusions remain unchanged when adding one (trivial)
extra dimension to the black hole solutions.
Similar to the static case, all grand canonical
configurations are thermally unstable\footnote{ However,
in this case the determinant of the Hessian
vanishes identically, as a result of the special dependence on $L$.
This happens already for a Schwarzschild black string when considered
as a solution in a grand canonical ensemble with fixed $T_H,~{\mathcal T}$.}.
Also, all rotating UBS solutions with
$J/S<J^{(c)}/S^{(c)}$ are unstable in a canonical ensemble,
where the critical ratio $ J^{(c)}/S^{(c)}$ is given by
\begin{eqnarray}
\label{t2}
\frac{J^{(c)}}{S^{(c)}} = \frac{1}{4 \pi \sqrt{D-1}}\left(
\sqrt{(D-2)(D-3)(D(D-5)+22)}-D(D-5)-10 \right)^{1/2}.
\end{eqnarray}
At the critical point, the specific heat goes through an infinite discontinuity,
and a second order phase transition takes place.
The critical values for other relevant quantities read
\begin{eqnarray}
\label{n6}
T_{H}^{(c)} =f_1(D)(J^{(c)})^{ -\frac{1}{D-3}} ,~~E^{(c)}=f_2(D)(J^{(c)})^{\frac{D-4}{D-3} },
~~\Omega_H^{(c)}=f_3(D)(J^{(c)})^{-\frac{1}{D-3}}~,
\end{eqnarray}
where
\begin{eqnarray}
\label{n7}
f_1(D)&=&
2^{-\frac{2D-1}{D-3}}(\frac{A_{D-3}L}{G_D})^{1/(D-3)}\pi^{-\frac{D-2}{D-3}}
(D-1)^{-\frac{(D-1)}{2(D-3)}}
\left(f_0(D)-(D-2)(D-7)\right)^{-\frac{D-4}{2(D-3)}}
\nonumber
\\
&&\times \left(3(D-2)(D-3)-f_0(D)\right)\left(f_0(D)-18+5D-D^2\right)^{\frac{1}{2(D-3)}},
\nonumber
\\
f_2(D)&=& 2^{-\frac{D+2}{D-3}}(\frac{A_{D-3}L}{G_D})^{1/(D-3)}
\pi^{1/(3-D)}(D-3)(D-1)^{1/(3-D)}
\\
\nonumber
&&\times(f_0(D)-(D-2)(D-7))
^{\frac{D-2}{2(D-3)}}(f_0(D)-D(D-5)-10)^{-\frac{D-4}{2(D-3)}}~,
\\
\nonumber
f_3(D)&=& (32\pi)^{1/(3-D)}(\frac{A_{D-3}L}{G_D})^{1/(D-3)}(D-1)^{1/(3-D)}
(f_0-(D-2)(D-7))^{-\frac{D-4}{2(D-3)}}
\\
\nonumber
&&\times
(f_0(D)-D(D-5)-10)^{\frac{D-2}{2(D-3)}}~,
\end{eqnarray}
while
\begin{eqnarray}
\label{n8-s}
f_0(D)=\sqrt{(D-2)(D-3)(D(D-5)+22)}~.
\end{eqnarray}
One finds for example the critical ratio
$\Omega_H^{(c)}/T_{H}^{(c)}\simeq 2.42757$ for $D=6$,
$\Omega_H^{(c)}/T_{H}^{(c)}\simeq 1.1162$ for $D=8$ and
$\Omega_H^{(c)}/T_{H}^{(c)}\simeq 0.6747$ for $D=10$.
\subsection{Gregory-Laflamme instability }
It is natural to expect that the MP uniform black strings become
unstable at critical values of the mass and angular momentum.
To determine these critical values for
black strings with equal magnitude angular momenta,
we make an expansion around the UBS of the form
\begin{eqnarray}
\label{p-ans}
\nonumber
&&A(r,z)= \epsilon a_1(r) \cos(kz) +O(\epsilon^2) ,
\\
\nonumber
&&B(r,z)=\epsilon b_1(r) \cos(kz) +O(\epsilon^2),
\\
\nonumber
&&C(r,z)=\epsilon c_1(r) \cos(kz)+O(\epsilon^2) ,
\\
\nonumber
&&G(r,z)=\epsilon g_1(r) \cos(kz)+O(\epsilon^2) ,
\\
&& W(r,z)=w(r)+\epsilon w_1(r) \cos(kz)+O(\epsilon^2),
\end{eqnarray}
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,8)
\put(1.5,0.0){\epsfig{file=mEpert_Th1.eps,width=10cm}}
\end{picture}
\caption{
The dimensionless quantity $a^2/ \left( p(D) M^{\frac{2}{D-4}} \right)$
is shown as a measure of the rotation of the uniform black strings
at constant temperature $2 \pi T_{H}=1$
in $D$ even dimensions, $6 \le D \le 14$,
versus the wavenumber $k$ of the zeromode fluctuation.
}
\end{figure}
with $\epsilon$ a small parameter and $f,h,w$ given by (\ref{MP}).
This expansion is appropriate for studying perturbations at the wavelength
which is marginally stable.
Upon substituting the above expressions into the Einstein equations,
to order $O(\epsilon)$
the following set of ODE is generated
\begin{eqnarray}
\label{p1}
\nonumber
&
a_1''
+(\frac{D-3}{r}+\frac{3f'}{2f}-\frac{h'}{2h})a_1'
+( \frac{f'}{2f}-\frac{h'}{2h})((D-4)c_1'+g_1')
-\frac{r^2h^2w'}{ f}w_1'
+\frac{r^2 h^2w'^2}{f}(a_1-g_1)-\frac{k^2a_1}{f}=0,
\\
\nonumber
&
b_1''
-(\frac{D-3}{r}+\frac{h'}{2h})a_1'
-\frac{(D-4)^2}{r}c_1'
+\frac{f'}{2f}(b_1'-(D-4)c_1')
+\frac{1}{2}(\frac{h'}{h}-\frac{f'}{f}-\frac{2(D-4)}{r})g_1'
\\
\nonumber
&
-\frac{r^2h^2w'}{2f}w_1'
-\frac{(D-4)h}{r^2f}(b_1-2c_1+g_1)
+\frac{r^2h^2w'^2}{2 f}(a_1-g_1)
+\frac{(D-2)(D-4)}{r^2f}(b_1-c_1)
-\frac{k^2b_1}{f}=0,
\\
\nonumber
&c_1''
+\frac{f'}{f}c_1'
+\frac{1}{r}(a_1'+(2D-7)c_1'+g_1')
+\frac{2(D-2)}{r^2f}(c_1-b_1)
+\frac{4h}{r^2f}(b_1-2c_1+g_1)
-\frac{k^2c_1}{f}=0,
\\
\label{p4}
&
g_1''
+ (\frac{D-2}{r}+\frac{f'}{f}+\frac{h'}{2h})g_1'
+\frac{r^2h^2w'}{f}w_1'
+\frac{2(D-4)h}{r^2f}(2c_1-g_1-b_1)
\\
\nonumber
&
+(\frac{1}{r}+\frac{h'}{2h})(a_1'+(D-4)c_1')
+\frac{r^2 h^2w'^2}{f}(g_1-a_1)-\frac{k^2g_1}{f}=0,
\\
\nonumber
&
w_1''
+(\frac{D-1}{r}+\frac{2h'}{h})w_1'
+(-a_1'+(D-4)c_1'+3g_1')w'
-\frac{k^2w_1}{f}=0.
\end{eqnarray}
This eigenvalue problem for the wavenumber $k=2\pi/L$ is then solved numerically
with suitable boundary conditions,
for rotating black strings in $D$ even dimensions, $6 \le D \le 14$.
The results are displayed in Figure 1,
where we exhibit the dimensionless quantity (see Eqs.~(\ref{M,a}), (\ref{f'}))
$a^2/\left( p(D) M^{\frac{2}{D-4}} \right)$, with
\begin{eqnarray}
\label{fig'}
p(D) =
\frac{D-4}{D-2} \left(\frac{4 }{D-2}\right)^{\frac{2}{D-4}},
\end{eqnarray}
as a measure of the rotation
versus the wavenumber $k$ for constant temperature,
$2 \pi T_{H}=1$.
Note, that for extremal solutions
$a^2/\left( p(D) M^{\frac{2}{D-4}} \right)=1$.
Keeping the temperature fixed, the wavenumber $k$ of the marginally
stable mode increases with increasing rotation
and decreases with increasing dimension $D$.
Introducing the scaled energy-mass $M_s$ and the scaled
angular momentum $J_s$ (following Eq.~(\ref{gen-def}))
\begin{eqnarray}
\label{gen-defsc}
\hspace{-0.5cm}
E=
\frac{A_{D-3}L}{16 \pi G_D}(D-3) M_s,
~J_k=J=
-\frac{A_{D-3}L}{8 \pi G_D} J_s,
\end{eqnarray}
these results are presented in Figure 2 (left) in a ``phase diagram'' format
for fixed $L=L_0$, where $L_0$ there corresponds to the critical length of the corresponding
static solutions.
Here $M_s$ and $J_s$ are suitably normalized and equipped with powers, such that
the extremality curve is the same for any dimension $D$.
In principle, following \cite{Gregory:1993vy},
one can get an estimation of $k$ by equating
the entropy of the rotating string
with that of a MP rotating black hole with the same momenta.
The entropy, mass-energy and angular momenta of a black string
are related through (\ref{ESJ}).
The corresponding relation
for a MP black hole in $D$ dimensions with $(D-2)/2$ equal angular
momenta can easily be derived by using the
relations in \cite{Myers:1986un}, and reads
\begin{eqnarray}
\label{MSJ-BH}
E=\frac{(D-2)}{\pi}2^{-\frac{2(D-1)}{D-2}}\left(\frac{A_{D-2}}{G_D}\right)^{1/(D-2)}
S^{\frac{D-3}{D-2}}\sqrt{1+\frac{4\pi^2 J^2}{S^2}}~.
\end{eqnarray}
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,6)
\put(-0.9,0.0){\epsfig{file=Epertn_Th_1.eps,width=8.cm}}
\put(7.7,0.0){\epsfig{file=estGLk.eps,width=8.4cm}}
\end{picture}
\caption{
Left:
The relation between the scaled mass $M_s$ and scaled angular momentum $J_s$
for fixed critical length of the extra dimension $L$
is shown for rotating solutions
in $D$ even dimensions, $6 \le D \le 14$.
Both $M_s$ and $J_s$ are
equipped with suitable powers and normalizations.
$L_0$ represents the value
where the instability of the static uniform black string occurs.
Right: The ratio betwen the wavelength estimate $k^{(est)}$
and the value of $k$ found numerically
is plotted at constant temperature $2 \pi T_{H}=1$
as function
of the dimensionless quantity $a^2/ \left( p(D) M^{\frac{2}{D-4}} \right)$.
}
\end{figure}
The expression for the wavelength estimate $k^{(est)}$
we find
by equating
the entropies is
\begin{eqnarray}
\label{k-est}
k^{(est)}= k^{(est)}_{st}
\left(1+\frac{4\pi^2 J^2}{S^2}\right)^{\frac{D-2}{2(D-3)}}
\end{eqnarray}
where
\begin{eqnarray}
\label{k-est0}
k^{(est)}_{st}= 2^{-\frac{D+1}{ D-3 }}\pi^{-\frac{D-2}{D-3}}
\frac{ A_{D-3}}{(A_{D-2})^{\frac{ D-4 }{D-3}}}
\frac{(D-3)^{D-3}}{(D-2)^{\frac{(D-4)(D-2)}{D-3}}}
\frac{1}{(G_DE)^{1/(D-3)}}
\end{eqnarray}
is the wavelength estimate
for a static solution with the same value of $E$.
From (\ref{lim2}) we find that nonextremal rotating solutions satisfy the inequality
\begin{eqnarray}
k^{(est)}_{st}\leq k^{(est)} <
\left(\frac{D-2}{2}\right)^{\frac{D-2}{2(D-3)}}k^{(est)}_{st}~,
\end{eqnarray}
and thus $k^{(est)}$ stays finite in the extremal limit.
The above wavelength estimate can also be expressed in
terms of variables used in the numerical
procedure as
\begin{eqnarray}
\label{k-estn}
k^{(est)}= 2 \pi \left(\frac{D-3}{D-2}\right)^{D-2}
\frac{A_{D-3}}{A_{D-2}}
\frac{1}{\sqrt{r_0^2-a^2}}~.
\end{eqnarray}
The fact that, as seen in Figure 1, the numerical
value of $k$ takes very large values
in the limit
$a^2/\left( p(D) M^{\frac{2}{D-4}} \right)\to 1$,
is
a consequence of keeping the temperature constant and letting $r_0$ run.
If we would rescale with $r_0$, the temperature would vanish in the extreme limit
and the rescaled $k$ would stay finite.
The ratio between the numerical value of $k$ one finds by solving
the equations (\ref{p4}) and the above estimate is presented in
Figure 2 (right).
For static black strings, the study of the perturbative equations
in second order revealed the appearance of a critical dimension,
above which the perturbative nonuniform black strings
are less massive than the marginally stable uniform black string
\cite{Sorkin:2004qq}.
It would therefore be interesting to solve the perturbative equations
to second order also in the presence of rotation.
But so far we have encountered numerical problems in such an analysis.
\section{
\boldmath $\hspace{-0.1cm} D=6$ \unboldmath
rotating nonuniform black string solutions}
\subsection{Numerical procedure}
To construct rotating nonuniform black string solutions numerically,
we introduce analogous to the static case
the new radial coordinate $\tilde r$,
\begin{equation}
\tilde r=\sqrt{r^2 - r_0^2 } \ ,
\end{equation}
thus the horizon resides at $\tilde r=0$.
The $D=6$ line element Eq.~(\ref{metric}) then reads
(with $\theta_1=\theta$)
\begin{eqnarray}
\label{metric6D}
ds^2&=&
-\, e^{2 A(\tilde r,z)} \, \frac{\tilde r^2}{g(\tilde r)}\, dt^2
+ e^{2 B(\tilde r,z)}\, (d\tilde r^2+dz^2)
+ e^{2 C(\tilde r,z)}\, g(\tilde r)\, d\theta^2
\\
\nonumber
&&
+\, e^{2 G(\tilde r,z)}\, g(\tilde r)
\bigg(\sin^2 \theta \, (d \varphi_1- W(\tilde r,z)\, dt)^2
+\cos^2 \theta \, (d \varphi_2- W(\tilde r,z)\, dt)^2
\bigg)
\\
\nonumber
&&
-\, (e^{2 G(\tilde r,z)}-e^{2 C(\tilde r,z)})
\, g(\tilde r)\sin^2 \theta \cos^2 \theta
\, (d \varphi_1-d \varphi_2)^2 \ ,
\end{eqnarray}
where
$g(\tilde r)=r_0^2+\tilde r^2$.
We then change to dimensionless coordinates
$\rho$ and $\zeta$,
\begin{eqnarray}
\rho = \tilde r/(r_0+\tilde r) , \ \ \ \zeta = z/ L ,
\label{barx2} \end{eqnarray}
where the compactified radial coordinate $\rho$
maps spatial infinity to the finite value $\rho=1$,
and $L$ is the asymptotic length of the compact direction.
We solve the resulting set of five coupled non-linear
elliptic partial differential equations numerically,
subject to the boundary conditions Eqs.~(\ref{bc1})-(\ref{bc3}).
These numerical calculations are based on the Newton-Raphson method
and are performed with help of the program FIDISOL \cite{schoen},
which provides also an error estimate for each unknown function.
The equations are discretized on a non-equidistant grid in
$\rho$ and $\zeta$.
Typical grids used have sizes $65 \times 50$,
covering the integration region
$0\leq \rho \leq 1$ and $0\leq \zeta \leq 1/2$.
(See \cite{schoen} and \cite{kk}
for further details and examples for the numerical procedure.)
For the nonuniform strings the estimated relative errors
range from approximately $\approx 0.001$\%
for small geometric deformation
to $\approx 1$\% for large deformation.
We also monitored the violation of the weighted constraints
$ \sqrt{f} \sqrt{-g} (G_r^r-G_z^z)=0, $ and
$ \sqrt{-g} G_z^r=0 \ ,
$
which is typically less then $0.1$
The horizon coordinate $r_0$ and the asymptotic length $L$ of the compact
direction enter the equations of motion as parameters.
The results presented are obtained with the parameter choice
\begin{equation}
r_0=1 \ , \ \ \ L=L^{\rm crit}=
4.9516
, \label{r_0-L} \end{equation}
where $L^{\rm crit}$ represents the value,
where the instability of the static uniform black string occurs.
Rotating nonuniform black strings can then be obtained by
starting from a static nonuniform black string solution
and increasing the value of angular velocity $\Omega_H$
of the event horizon, which enters the boundary conditions.
By varying also the second boundary parameter $d_0$,
associated with the temperature of the black strings,
$d_0=\ln (T_{H}^{(0)}/T_H)$ (see Eq.~(\ref{temp})),
the full set of rotating nonuniform black strings
can then be explored.
An alternative procedure to obtain rotating NUBS
numerically would be, to start from stationary
perturbative nonuniform solutions.
The basic properties of the NUBS are encoded in
the five metric functions $A(\rho,\zeta)$, $B(\rho,\zeta)$,
$C(\rho,\zeta)$, $G(\rho,\zeta)$, and $W(\rho,\zeta)$.
These functions
change smoothly with the two boundary parameters $d_0$ and $\Omega_H$.
We illustrate these functions in Figure 3,
for the parameter choices $d_0=0.6$ and $\Omega_H=0.25$
resp.~$\Omega_H=0.202$, the latter
corresponding to a solution in the strongly deformed region
close to the expected transition
from rotating nonuniform black strings
to rotating caged black holes.
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(15,18)
\put(-1,0){\epsfig{file=C_d0.6w0.25.eps,width=8cm}}
\put(7,0){\epsfig{file=funC_d0.6w0.22.eps,width=8cm}}
\put(-1,6){\epsfig{file=B_d0.6w0.25.eps,width=8cm}}
\put(7,6){\epsfig{file=funB_d0.6w0.22.eps,width=8cm}}
\put(-1,12){\epsfig{file=A_d0.6w0.25.eps,width=8cm}}
\put(7,12){\epsfig{file=funA_d0.6w0.22.eps,width=8cm}}
\end{picture}
\caption{
The metric functions $A$, $B$, $C$, $G$ and $W$
of the $D=6$ rotating nonuniform black string solution
with temperature parameter $d_0=0.6$ and
horizon angular velocity $\Omega_H=0.25$ (left column)
and $\Omega_H=0.202$ (right column)
are shown as functions of the compactified radial coordinate $\rho$,
and the coordinate $\zeta$ of the compact direction.
Note that the horizon is located at $\rho=0$.
}
\end{figure}
\setcounter{figure}{2}
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(15,12)
\put(-1,0){\epsfig{file=w_d0.6w0.25.eps,width=8cm}}
\put(7,0){\epsfig{file=funW_d0.6w0.22.eps,width=8cm}}
\put(-1,6){\epsfig{file=G_d0.6w0.25.eps,width=8cm}}
\put(7,6){\epsfig{file=funG_d0.6w0.22.eps,width=8cm}}
\end{picture}
\caption{
continued.
}
\end{figure}
\subsection{Properties of rotating black strings}
\subsubsection{The horizon}
For the static NUBS a measure of their deformation is given by the
nonuniformity parameter $\lambda$ \cite{Gubser:2001ac}
\begin{eqnarray}
\lambda = \frac{1}{2} \left( \frac{{\cal R}_{\rm max}}{{\cal R}_{\rm min}}
-1 \right)
, \label{lambda}
\end{eqnarray}
where ${\cal R}_{\rm max}$ and ${\cal R}_{\rm min}$
represent the maximum radius of a $(D-3)$-sphere on the horizon and
the minimum radius, respectively,
the minimum radius being the radius of the `waist'
of the black string.
Thus for uniform black strings $\lambda=0$,
while the horizon topology
changing transition should be approached
for $\lambda \rightarrow \infty$
\cite{Kol:2003ja,Wiseman:2002ti}.
For the rotating NUBS one has to take into account, that
the rotation leads to a deformation of the 3-sphere of the horizon, making
it oblate w.r.t.~the planes of rotation.
Therefore, various possibilities arise to define
the nonuniformity parameter $\lambda$.
In the following we employ the above definition of $\lambda$,
where ${\cal R}_{\rm max}$ and ${\cal R}_{\rm min}$
are obtained from the area $A_{H}$ of the respective
deformed 3-sphere via $A_{H}=2 \pi^2 {\cal R}^3$.
In Figure 4 we exhibit
the spatial embedding of the horizon into 3-dimensional space
for a sequence of $D=6$ rotating NUBS.
In these embeddings the symmetry directions ($\varphi_1$, $\varphi_2$)
are suppressed, and the proper circumference of the horizon is plotted
against the proper length along the compact direction,
yielding a geometrical view of both the deformation of
the horizon due to rotation and the nonuniformity of the horizon
with respect to the compact coordinate.
For the solutions of the sequence shown in Figure 4
the temperature is kept fixed with temperature parameter $d_0=0.6$.
The first solution of the sequence
corresponds to the marginally stable rotating uniform black string,
which has $\lambda=0$ and horizon angular velocity $\Omega_{H}=0.34908$.
When the horizon angular velocity is lowered,
rotating black strings with increasing nonuniformity are obtained.
Shown are solutions with nonuniformity parameter
$\lambda=0.83$, $1.7$ and $2.9$.
The latter is already close to the expected
topology changing transition to rotating caged black holes.
Interestingly, close to the maximal radius ${\cal R}_{\rm max}$
the deformation of the horizon due to rotation is significant,
whereas close to ${\cal R}_{\rm min}$ the 3-horizon appears
to be almost spherical.
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(15,18)
\put(-4, 9){\epsfig{file=rembpm3_d0.6w0.34908.eps,width=15cm}}
\put( 6.8,15.3){$\Omega_{H}=0.34908$}
\put( 6.7,14.7){$\lambda=0$}
\put( 4, 9){\epsfig{file=rembpm3_d0.6w0.34908a.eps,width=15cm}}
\put(-4, 5){\epsfig{file=rembpm3_d0.6w0.25.eps,width=15cm}}
\put( 7.0,11.3){$\Omega_{H}=0.25$}
\put( 6.95,10.7){$\lambda=0.83$}
\put( 4, 5){\epsfig{file=rembpm3_d0.6w0.25a.eps,width=15cm}}
\put(-4, 1){\epsfig{file=rembpm3_d0.6w0.212.eps,width=15cm}}
\put( 7.0, 7.3){$\Omega_{H}=0.212$}
\put( 6.95, 6.7){$\lambda=1.7$}
\put( 4, 1){\epsfig{file=rembpm3_d0.6w0.212a.eps,width=15cm}}
\put(-4,-3){\epsfig{file=rembpm3_d0.6w0.202.eps,width=15cm}}
\put( 7.0, 3.3){$\Omega_{H}=0.202$}
\put( 6.95, 2.7){$\lambda=2.9$}
\put( 4,-3){\epsfig{file=rembpm3_d0.6w0.202a.eps,width=15cm}}
\end{picture}
\caption{
The spatial embedding of the horizon
of $D=6$ rotating black string solutions
is shown for a sequence of solutions with fixed
temperature parameter $d_0=0.6$ and varying
horizon angular velocity $\Omega_{H}$:
$\Omega_{H}=0.34908$ (upper row), $\Omega_{H}=0.25$ (second row),
$\Omega_{H}=0.212$ (third row) and $\Omega_{H}=0.202$ (lower row),
$\lambda$ specifies the increasing nonuniformity of the solutions.
Left column: side view, right column: view in $z$ direction.
($r_0 =1$, $L=L^{\rm crit}=4.9516$.)
}
\end{figure}
The deformation of the horizon due to rotation
is demonstrated in more detail in Figure 5,
where circumferences of the deformed 3-sphere
of the horizon are exhibited.
Here ${l}_{\rm e,max}$ denotes the equatorial maximum circumference,
and ${l}_{\rm e,min}$ the equatorial minimum circumference,
both referring to the circumferences in the two
equivalent planes of rotation, while
${l}_{\rm p,max}$ denotes the polar maximum circumference,
and ${l}_{\rm p,min}$ the polar minimum circumference,
representing circumferences for fixed azimuthal angles.
In the static case, the respective equatorial and polar circumferences
agree, and the minimum circumference represents the
circumference of the waist of the NUBS.
Using the scaled energy-mass $M_s$ and the scaled
angular momentum $J_s$, Eq.~(\ref{gen-defsc}),
we exhibit in Figure 5
these polar and equatorial circumferences
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$.
We note, that for rotating uniform black strings, this ratio is bounded,
$J_s/M_s^{3/2} \le 1$, with extremal rotating uniform solutions
saturating the bound.
For reference, the figure exhibits the polar and equatorial
circumferences of the branch of marginally stable MP
uniform black strings.
This uniform branch ranges from the static marginally stable black string
to the extremal rotating marginally stable black string.
The static marginally stable string, with all circumferences equal,
has temperature parameter $d_0=0$.
Along this rotating UBS branch,
with equal maximal and minimal circumferences,
the temperature parameter $d_0$
and the deformation of the horizon 3-sphere due to rotation
both increase monotonically,
while the temperature itself decreases monotonically,
reaching zero in the extremal case.
\subsubsection{A critical temperature}
The rotating nonuniform black strings branch off from
the marginally stable uniform black strings.
These branches are obtained,
by fixing a value of the temperature parameter $d_0$
and thus fixing the temperature,
and then decreasing the horizon angular velocity $\Omega_H$
from the respective rotating UBS value.
Depending on the value of the fixed chosen temperature parameter $d_0$,
the corresponding rotating NUBS branches exhibit distinct features.
When $d_0 < d_{0}^*$,
the rotating NUBS branch extends back to a static NUBS solution
with a finite waist, and thus finite minimal circumferences ${l}_{\rm p,min}$
and ${l}_{\rm e,min}$.
The size of the waist of the static NUBS solution
decreases with increasing $d_0$.
At the critical value $d_{0}^*$,
the respective branch of rotating NUBS is expected to extend precisely
back to a static solution with zero size waist,
i.e., to the solution at the topology changing transition,
where the branch of static NUBS merges with the branch
of static caged black holes.
This critical value of the temperature parameter $d_{0}^*$
is in the interval $0.30 < d_0 < 0.33$,
and corresponds to a critical value of the temperature $T_*$
where $0.72 < T_H/T_0 < 0.74$ (with $T_0$ the temperature
of the UBS).
This may be compared with our previous results
for static black strings \cite{Kleihaus:2006ee}.
Extraction of the critical temperature $T_*$ from
those static black string calculations,
suggests the bounds $0.72 < T_H/T_0 < 0.76$
for the critical temperature $T_*$ (when trying to account for
numerical inaccuracy in the critical region).\footnote{
Such an identification of $T_*$ assumes a monotonic
dependence of the temperature of the static uniform strings
on the nonuniformity parameter $\lambda$,
since otherwise bifurcations might be present
and complicate this scenario.}
Beyond the critical value $d_{0}^*$, the branches of rotating NUBS
no longer reach static NUBS.
Instead they are expected to extend to a
corresponding rotating solution with zero size waist,
and thus to lead towards a topology changing transition,
associated with the merging of a branch of rotating NUBS
and a branch of rotating caged black holes.
Indeed, when $d_0 > d_{0}^*$,
the waist of the NUBS solutions monotonically decreases in size,
the minimal circumferences approaching zero.
Thus we see here first evidence, that
a topology changing transition arises also for rotating
branches of solutions.
We note, that the deformation of the horizon 3-sphere due to rotation
is considerable at maximum size,
while the waist of the rotating NUBS becomes increasingly spherical.
\begin{figure}[t!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,6)
\put(0,0.0){\epsfig{file=plot43Lpn.eps,width=8.4cm}}
\put(8.5,0.0){\epsfig{file=plot43Len.eps,width=7.2cm}}
\end{picture}
\caption{
The maximum and minimum polar circumferences ${l}_{\rm p,max}$
and ${l}_{\rm p,min}$ of the deformed horizon $3$-sphere,
and the respective equatorial circumferences ${l}_{\rm e,max}$
and ${l}_{\rm e,min}$ are shown
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$
for branches of rotating NUBS with fixed values of the
temperature parameter $d_0$ ranging from 0.1 to 0.7.
The circumferences of the deformed horizon $3$-sphere
are also shown for marginally stable MP UBS (denoted GL).
}
\end{figure}
In Figure 6 we exhibit the nonuniformity parameter $\lambda$
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$
for the same set of rotating NUBS.
The branches begin at the rotating marginally stable UBS
with $\lambda=0$.
When $d_0 < d_{0}^*$,
the rotating NUBS branches extend back to static NUBS solutions
with finite nonuniformity and thus finite waist.
When $d_0 > d_{0}^*$, on the other hand,
the nonuniformity parameter $\lambda$ increases
apparently without bound,
approaching the topology changing transition for
$\lambda \rightarrow \infty$.
The branch of rotating marginally stable UBS is bounded
by the static and by the extremal rotating solution.
It would be interesting to obtain the
corresponding domain of existence of rotating NUBS.
The construction of extremal rotating NUBSs (if they exist), however,
currently represents an unsolved numerical challenge.
Figure 6 also exhibits the scaled horizon angular velocity
$\Omega_H/\Omega_{H,\rm GL}$ for this set of rotating NUBS
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$.
Here $\Omega_{H,\rm GL}$ denotes the horizon angular velocity
of the marginally stable rotating UBS
(with the same temperature parameter $d_0$).
Starting from rotating marginally stable UBS
with $\Omega_H/\Omega_{H,\rm GL}=1$, the branches
end at static NUBS with $\Omega_H=J=0$, when $d_0 < d_{0}^*$.
When $d_0 > d_{0}^*$, in contrast,
the branches of rotating NUBS appear to approach limiting solutions
with finite horizon angular velocity $\Omega_H$ and finite
angular momentum $J$, associated with
a topology changing transition.
\begin{figure}[h!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,6)
\put(0,0.0){\epsfig{file=plotlamvJM32n.eps,width=7.7cm}}
\put(8.0,0.0){\epsfig{file=plotwhvJ32n.eps,width=7.7cm}}
\end{picture}
\caption{
The nonuniformity parameter $\lambda$
and the scaled horizon angular velocity $\Omega_H/\Omega_{H,\rm GL}$
are shown
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$
for branches of rotating NUBS with fixed values of the
temperature parameter $d_0$ ranging from 0.1 to 0.7.
($\Omega_{H,\rm GL}$ denotes the angular velocity
of the marginally stable MP UBS.)
}
\end{figure}
\subsubsection{Global charges}
The scaled mass $M_s/M_{s,\rm GL}$
and the scaled entropy $S_s/S_{s,\rm GL}$
(where $S= S_s {A_{D-3}L}/{4}$)
are exhibited in Figure 7
versus the scaled angular momentum ratio $J_s/M_s^{3/2}$
for the same set of solutions.
Both $M_s/M_{s,\rm GL}$ and $S_s/S_{s,\rm GL}$
increase monotonically along the branches of solutions
with fixed temperature.
As noted above, when $T_H>T_*$,
the branches of rotating NUBS end at static NUBS,
whereas, when $T_H<T_*$,
they appear to approach rotating limiting solutions
with finite values of the angular momentum $J$, associated with
a topology changing transition between branches of rotating solutions.
We conclude from the figure, that
the scaled mass $M_s/M_{s,\rm GL}$
of the limiting solutions increases with increasing $J_s/M_s^{3/2}$,
while their scaled entropy $S_s/S_{s,\rm GL}$
appears to be almost constant.
\begin{figure}[h!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,6)
\put(0,0.0){\epsfig{file=plotMvJ32n.eps,width=7.7cm}}
\put(8,0.0){\epsfig{file=plotSvJ32n.eps,width=7.7cm}}
\end{picture}
\caption{
Same as Figure 6 for the scaled mass
$M_s/M_{s,\rm GL}$ and the scaled entropy $S_s/S_{s,\rm GL}$.
($M_{s,\rm GL}$ and $S_{s,\rm GL}$ denote the respective
quantities of the marginally stable MP UBS.)
}
\end{figure}
We exhibit the relative tension $n$ of this set of rotating NUBS
in Figure 8, together with the relative
tension of the uniform black strings, $n_{\rm GL}=1/3$.
Starting from rotating marginally stable UBS,
the relative tension $n$ decreases monotonically
for branches of rotating NUBS with large values of the temperature.
As the critical temperature $T_*$ is approached,
and beyond the critical temperature,
the tension $n$ no longer decreases monotonically,
but instead reaches a minimum and then increases again.
Thus we observe the backbending phenomenon for the relative tension $n$,
encountered previously for static NUBS \cite{Kleihaus:2006ee},
also for rotating nonuniform black strings.
For the static NUBS we obtained for the relative tension $n$
the critical value $n_* \approx 0.2$.
Consistency requires, that this value
agrees within error bounds with the critical value obtained here
for the branch of rotating NUBS at the critical temperature $T_*$.
The figure indicates, that this requirement may hold.
\begin{figure}[h!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,6)
\put(3.7,0.0){\epsfig{file=plotnvJM32n.eps,width=7.7cm}}
\end{picture}
\caption{
Same as Figure 6 for the relative tension $n$.
}
\end{figure}
Restricting to a canonical ensemble, the numerical analysis
indicates that the qualitative thermodynamical features of the
uniform MP branch are also shared by rotating NUBS solutions.
For small values of $J$, the entropy is a decreasing function of $T$,
i.e. $C_{J,L}<0$. However, the configurations near extremality are
thermally stable in a canonical ensemble.
Also, although further work is necessary in this
case, we expect all nonuniform solutions to be thermodynamically
unstable in a
grand canonical ensemble.
\subsubsection{The ergoregion}
Like rotating black holes,
rotating black strings possess an ergoregion.
While the ergosurface of rotating uniform black strings
is uniform like the horizon, the ergosurface of
nonuniform black strings reflects the nonuniformity
of the horizon.
In Figure 9 we exhibit
the spatial embedding of the ergosurface into 3-dimensional space
for those rotating NUBS, whose
spatial embedding of the horizon was shown in Figure 4.
\begin{figure}[h!]
\setlength{\unitlength}{1cm}
\begin{picture}(15,18)
\put(-4, 9){\epsfig{file=rembep_d0.6w0.34908n.eps,width=15cm}}
\put( 6.8,15.3){$\Omega_{H}=0.34908$}
\put( 6.7,14.7){$\lambda=0$}
\put( 4, 9){\epsfig{file=rembep_d0.6w0.34908an.eps,width=15cm}}
\put(-4, 5){\epsfig{file=rembep_d0.6w0.25n.eps,width=15cm}}
\put( 7.0,11.3){$\Omega_{H}=0.25$}
\put( 6.95,10.7){$\lambda=0.83$}
\put( 4, 5){\epsfig{file=rembep_d0.6w0.25an.eps,width=15cm}}
\put(-4, 1){\epsfig{file=rembep_d0.6w0.212n.eps,width=15cm}}
\put( 7.0, 7.3){$\Omega_{H}=0.212$}
\put( 6.95, 6.7){$\lambda=1.7$}
\put( 4, 1){\epsfig{file=rembep_d0.6w0.212an.eps,width=15cm}}
\put(-4,-3){\epsfig{file=rembep_d0.6w0.202n.eps,width=15cm}}
\put( 7.0, 3.3){$\Omega_{H}=0.202$}
\put( 6.95, 2.7){$\lambda=2.9$}
\put( 4,-3){\epsfig{file=rembep_d0.6w0.202an.eps,width=15cm}}
\end{picture}
\caption{
The spatial embedding of the ergosurface.
of $D=6$ rotating black string solutions
is shown for a sequence of solutions with fixed
temperature parameter $d_0=0.6$ and varying
horizon angular velocity $\Omega_{H}$:
$\Omega_{H}=0.34908$ (upper row), $\Omega_{H}=0.25$ (second row),
$\Omega_{H}=0.212$ (third row) and $\Omega_{H}=0.202$ (lower row),
$\lambda$ specifies the increasing nonuniformity of the solutions.
Left column: side view, right column: view in $z$ direction.
($r_0 =1$, $L=L^{\rm crit}=4.9516$.)
}
\end{figure}
As in Figure 4, the symmetry directions are suppressed here.
The proper circumference of the ergosurface is plotted
against the proper arclength along the compact direction.
Denoting $\tilde{r}_e(z)$, $z$ the coordinates of the ergosurface at
fixed $\varphi_1$, $\varphi_2$, $\theta$, $t$,
we define the arclength as
$$\sigma =
\int_0^z e^{B(\tilde{r}_e(z),z)}
\sqrt{1+\left(\frac{d\tilde{r}_e}{dz}\right)^2}dz.
$$
The solutions shown have fixed temperature
with temperature parameter $d_0=0.6$,
decreasing horizon angular velocity $\Omega_{H}$,
and increasing nonuniformity of the horizon.
The first solution
corresponds to the marginally stable rotating uniform black string,
with an ergosurface uniform w.r.t.~the compact coordinate,
but rotational deformation.
When the horizon angular velocity is lowered,
the nonuniformity of the ergosurface increases, along with the
increase of the nonuniformity of the horizon.
In Figure 10 we exhibit the ergosurface in terms of the
coordinates $r_{\rm ergo} = \tilde r$ and $\zeta$,
in which the horizon is located at $\tilde r=0$.
We show rotating NUBS solutions on three branches with fixed temperature,
corresponding to the values of the temperature parameter
$d_0=0.3$, $0.33$ and $0.6$.
Thus the first two sets represent solutions just above
and just below the critical temperature $T_*$.
All sets start with the corresponding rotating UBS
and thus a uniform ergosurface.
As the horizon angular velocity $\Omega_H$ decreases from $\Omega_{H,\rm GL}$
nonuniformity of the ergosurface develops.
\begin{figure}[h!]
\setlength{\unitlength}{1cm}
\begin{picture}(8,12)
\put(0,6.0){\epsfig{file=nergo_d0.3.eps,width=7.7cm}}
\put(8,6.0){\epsfig{file=nergo_d0.33.eps,width=7.7cm}}
\put(3.7, 0.0){\epsfig{file=nergo_d0.6.eps,width=7.7cm}}
\end{picture}
\caption{
The ergosurface of rotating NUBS is shown
for three sets of solutions with fixed temperature,
corresponding to the values of the temperature parameter,
$d_0=0.3$, $0.33$ and $0.6$,
and decreasing horizon angular velocity $\Omega_H$, starting from
the value $\Omega_{H,\rm GL}$
of the respective marginally stable MP UBS.
}
\end{figure}
For $d_0=0.3$, where $T_H > T_*$,
the rotating NUBS branch extends back to a static NUBS solution,
where the ergoregion disappears. The strong shrinkage of the ergoregion
close to this point is clearly seen in the figure for the NUBS solution
with horizon angular velocity $\Omega_H=0.005$.
For $d_0=0.33$, the temperature is just below the critical value,
$T_H < T_*$, and the rotating NUBS branch is expected to extend to a
corresponding rotating solution with zero size waist,
and thus signify a topology changing transition
between rotating solutions.
This is reflected in the figure by the presence of a finite ergoregion
of the solution with $\Omega_H=0.068$, close to the transition point.
Also for $d_0=0.6$, the ergoregion remains finite, as the
topology changing transition is approached.
We note, that the size of the ergoregion of the limiting
solution appears to increase with decreasing temperature.
\section{Rotating NUBS in heterotic string theory }
In order to obtain rotating electrically charged black strings,
we employ a solution generating technique, by performing symmetry transformations on
the neutral solution. Within toroidally compactified heterotic string, an approach
to obtain the charged solutions from the neutral one was presented in Ref.
\cite{Sen:1994eb}.
This method was used to obtain, e.g., general rotating electrically charged solutions in four
dimensions
\cite{Sen:1994eb},
higher-dimensional general electrically charged static solutions \cite{Peet:1995pe},
and rotating black hole
solutions with one rotational parameter in $D$ dimensions \cite{Horowitz:1995tm}.
General $D-$dimensional charged rotating black holes
with $[(D-1)/2]$ distinct angular momenta
were constructed in \cite{Cvetic:1996dt}.
The massless fields in heterotic string theory compactified on a
$(10-D)$-dimensional torus consist of the string
metric $G_{\mu\nu}$, the anti-symmetric
tensor field $B_{\mu\nu}$, $(36-2D)$ U(1) gauge fields $A_\mu^{(j)}$
($1\le j\le 36-2D$), the scalar dilaton field $\Phi$, and a
$(36-2D)\times (36-2D)$
matrix valued scalar field $M$ satisfying,
\begin{eqnarray}
\label{e5}
M L M^T = L, \quad \quad M^T=M.
\end{eqnarray}
Here $L$ is a $(36-2D)\times(36-2D)$
symmetric matrix with $(26-D)$ eigenvalues $-1$ and
$(10-D)$ eigenvalues $+1$. Following \cite{Sen:1994eb}, \cite{Peet:1995pe}
we shall take $L$ to be
\begin{eqnarray} \label{e4}
L=\pmatrix{-I_{26-D} & \cr & I_{10-D}\cr},
\end{eqnarray}
where $I_n$ denotes an $n\times n$ identity matrix. The action describing
the effective field theory of these massless
bosonic fields is given by \cite{Maharana:1992my}
\begin{eqnarray}
\label{gen-act-string}
S &=& \int d^D x \sqrt{-\det G} \, e^{-\Phi} \, \Big[ R_G + G^{\mu\nu}
\partial_\mu \Phi \partial_\nu\Phi +{1\over 8} G^{\mu\nu} Tr(\partial_\mu M L\partial_\nu ML)
\nonumber \\
&& -{1\over 12} G^{\mu\mu'} G^{\nu\nu'} G^{\rho\rho'} H_{\mu\nu\rho}
H_{\mu'\nu'\rho'} - G^{\mu\mu'} G^{\nu\nu'} F^{(j)}_{\mu\nu} \, (LML)_{jk}
\, F^{(k)}_{\mu'\nu'} \Big] \, ,
\end{eqnarray}
where
\begin{eqnarray}
\label{enq2}
F^{(j)}_{\mu\nu} = \partial_\mu A^{(j)}_\nu - \partial_\nu A^{(j)}_\mu \, ,
\end{eqnarray}
\begin{eqnarray}
\label{enq3}
H_{\mu\nu\rho} = \partial_\mu B_{\nu\rho} + 2 A_\mu^{(j)} L_{jk} F^{(k)}_{\nu\rho}
+\hbox{cyclic permutations of $\mu$, $\nu$, $\rho$}\, ,
\end{eqnarray}
and $R_G$ denotes the scalar curvature associated with the metric
$G_{\mu\nu}$. The canonical Einstein metric $\bar{g}_{\mu \nu}$ is
\begin{equation}
\label{stringeinstein}
\bar g_{\mu\nu} = e^{-\frac{2}{D-2}\Phi}G_{\mu\nu} .
\end{equation}
One can add a general charge to any stationary vacuum solution by
applying the solution generating transformations $(O(26-D,1)/O(26-D))
\times (O(10-D,1)/O(10-D))$.
This generates a nontrivial $\Phi, B_{\mu\nu}$ and
$M$, as well as $A_\mu^{(j)}$.
Here we are mainly interested in the case $D=6$,
i.e., heterotic string theory compactified on a four-dimensional
torus.
Since the generation procedure is
identical to the one given in Ref.\cite{Sen:1994eb} we shall not give the
details, but only present the final results.
Starting with an arbitrary time independent
solution $g_{\mu \nu}$
of the vacuum field equation in $D=6$ (with $ds^2=g_{\mu \nu}dx^\mu dx^\nu$),
the expression for the metric of a
charged configuration expressed in the canonical Einstein frame is
\begin{eqnarray}
\label{new1}
d\bar s^2=\Delta^{1/4} \bigg(ds^2+\frac{1}{g_{tt}}
(\frac{(\cosh \alpha+\cosh \beta)^2}{4 \Delta}-1)
(g_{\varphi_1t}d\varphi_1+g_{\varphi_2 t}d\varphi_2)^2
\\
\nonumber
+(\frac{ \cosh \alpha+\cosh \beta }{\Delta}-2)
(g_{\varphi_1t}d\varphi_1dt+g_{\varphi_2 t}d\varphi_2dt)
+(\frac{1-\Delta}{\Delta})g_{tt}dt^2\bigg)~,
\end{eqnarray}
where $\Delta$ is related to the dilaton,
\begin{eqnarray}
\label{new2}
e^{-2\Phi}= \Delta= \frac{1}{4}\bigg(g_{tt}^2(\cosh \alpha-\cosh \beta)^2
+2g_{tt}(\sinh^2 \alpha+\sinh^2 \beta)
+(\cosh\alpha+\cosh\beta)^2
\bigg).
\end{eqnarray}
The time components of the $U(1)$ gauge fields are
\begin{eqnarray}
\nonumber
A_t^{(i)} &=&
-\frac{n^{(i)}}{4\sqrt{2}\Delta}(1+g_{tt})\sinh \alpha
\bigg(\cosh \alpha+\cosh \beta
+g_{tt}(\cosh \alpha-\cosh \beta)\bigg),
~~1\leq i \leq 20,
\\
\nonumber
&=&-\frac{p^{(i-20)}}{4\sqrt{2}\Delta}(1+
g_{tt})\sinh \beta
\bigg(\cosh \alpha+\cosh \beta
-g_{tt}(\cosh \alpha-\cosh \beta)\bigg),
~~21\leq i \leq 24,
\end{eqnarray}
whereas the spatial components of the gauge fields are given by
\begin{eqnarray}
\nonumber
A_{\varphi_k}^{(i)} &=&
\frac{n^{(i)}g_{t\varphi_k }}{2\sqrt{2}g_{tt}}
\sinh\alpha \bigg(
1-\frac{(1+g_{tt})}{4\Delta} (\cosh \alpha+\cosh \beta)
(\cosh \alpha+\cosh \beta
+g_{tt}(\cosh \alpha-\cosh \beta))\bigg),
\\
\nonumber
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~1\leq i \leq 20,
\\
\nonumber
&=&
\frac{p^{(i-20)}g_{t\varphi_k }}{2\sqrt{2}g_{tt}}
\sinh\beta \bigg(
1-\frac{(1+g_{tt})}{4\Delta} (\cosh \alpha+\cosh \beta)
(\cosh \alpha+\cosh \beta-g_{tt}(\cosh \alpha-\cosh \beta))\bigg),
\\
\nonumber
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~21\leq i \leq 24,
\end{eqnarray}
with $k=1,2$.
Here $\alpha$ and $\beta$ are two boost angles,
$\vec n$ is a 20-dimensional unit vector, and $\vec p$ is a 4-dimensional unit
vector.
The nonvanishing components of the two-form field $B_{\mu \nu}$ are
\begin{eqnarray}
\label{enq5}
\hspace{-0.5cm}
B_{t \varphi_k} =
\frac{g_{t \varphi_k}}{2g_{tt}}
(\cosh \alpha-\cosh \beta)\bigg(1-\frac{(1+g_{tt})}{4\Delta}
((\sinh^2 \alpha+\sinh^2 \beta)g_{tt}
+(\cosh \alpha+\cosh\beta)^2)\bigg).
\end{eqnarray}
The result for the matrix-valued scalar $M$ is
\begin{eqnarray}
\label{M-matrix}
M = I_{24} +\pmatrix{ P_1~nn^T & P_2~n p^T \cr P_2~p n^T & P_3~pp^T\cr}\, ,
\end{eqnarray}
where
\begin{eqnarray}
\nonumber
P_1&=&\frac{\sinh^2 \alpha~(1+g_{tt})^2}{2g_{tt}}
(1-\frac{1}{4\Delta}\left(\cosh \alpha+\cosh \beta +
g_{tt}(\cosh \alpha-\cosh \beta))^2\right) ,
\\
P_2&=&-\frac{2\sinh \alpha \sinh \beta~(1+g_{tt})}{4\Delta}
\left(1+\cosh \alpha \cosh \beta +g_{tt}
(\cosh \alpha\cosh \beta-1) \right),
\\
\nonumber
P_3&=&\frac{\sinh^2 \beta~(1+g_{tt})^2}{2g_{tt}}
(1-\frac{1}{4\Delta}\left(\cosh \alpha+\cosh \beta -
g_{tt}(\cosh \alpha-\cosh \beta))^2\right).
\end{eqnarray}
Both
black hole and black string charged rotating
solutions can
be generated in this way.
The results in \cite{Sen:1994eb}
are recovered
for the case of a $D=6$ Myers-Perry black hole
solution
with only one nonzero angular momenta.
Charged rotating NUBS strings with two equal angular momenta
are found by replacing in
(\ref{new1})-(\ref{M-matrix}) the seed metric $ds^2$
as given by
$(\ref{metric})$
with
$g_{tt}=e^{2G}r^2W^2-e^{2A}(1-r_0^2/r^2)$,~~
$g_{\varphi_1 t}=-e^{2G}r^2 W \sin^2 \theta$,
$g_{\varphi_2 t}=-e^{2G}r^2 W \cos^2 \theta$.
Here we present only the line element of the
charged rotating UBS solution
\begin{eqnarray}
d\bar s^2&&=\Delta(r)^{\frac{1}{4}}
\left(\frac{dr^2}{1-\frac{2M}{r^2}+\frac{2a^2M}{r^4}}+dz^2
+r^2(d \theta^2+\sin^2\theta d \varphi_1^2+\cos^2\theta d \varphi_2^2)\right)
\\
\nonumber
&&+\Delta(r)^{-\frac{3}{4}}\bigg(
\frac{2aM}{r^2(1-\frac{2M}{r^2})}
(\Delta(r)-\frac{M}{2r^2}(\cosh \alpha+\cosh \beta))
(\sin^2\theta d \varphi_1 +\cos^2\theta d \varphi_2)^2
\\
\nonumber
&&-(\cosh \alpha+\cosh \beta)\frac{2aM}{ r^2}
(\sin^2\theta d \varphi_1dt +\cos^2\theta d \varphi_2dt)
-(1-\frac{2M}{r^2})dt^2 \bigg),
\end{eqnarray}
where
\begin{eqnarray}
\Delta(r)=1+\frac{2M}{r^2}(\cosh \alpha \cosh \beta-1)
+\frac{M^2}{r^4}(\cosh \alpha-\cosh \beta)^2.
\end{eqnarray}
The extremal limit is found by taking the boost parameters
to infinity together with
a rescaling of $M$.
The asymptotic structure of a general charged solution is
similar to that of the vacuum seed configuration.
The spacetime still approaches the
${\cal M}^{5}\times S^1$ background as $r \to \infty $.
Also, one can see that the event horizon location of the charged solutions is unchanged.
The position of the ergosurface remains the same,
all fields being well defined on that hypersurface.
One can also verify that no closed timelike curves are introduced
in the line element (\ref{new1}) by the generation procedure.
The relevant properties of a charged solution can be derived from the
corresponding vacuum seed configuration.
The mass-energy $\bar{E}$ and the string tension $\bar{\mathcal T}$
of the charged rotating solutions are
\begin{eqnarray}
\label{new-charges}
\bar{E}= \frac{1}{4} (1+3\cosh \alpha \cosh \beta) E
+\frac{1}{4} (1-\cosh \alpha \cosh \beta){\mathcal T}_0L,
~~~
\bar{\mathcal T}={\mathcal T}.
\end{eqnarray}
The charged solution possesses also two equal angular momenta
$\bar J_1=\bar J_2=\bar{J}$ with
\begin{eqnarray}
\bar{J}=\frac{1}{2}(\cosh \alpha+\cosh \beta)J.
\end{eqnarray}
and has event horizon velocities $
\bar{\Omega}_1=\bar{\Omega}_2=
\bar{\Omega}_H$ with
\begin{eqnarray}
\bar{\Omega}_H=\frac{2\Omega_H}{\cosh \alpha+\cosh \beta}.
\end{eqnarray}
The electric charges defined as
\begin{eqnarray}
\label{Qe-def}
Q_e^{(k)}=\frac{1}{LA_3}\lim_{r \to \infty}\int dz \int dA_3 r^3F_{rt}^{(k)}
\end{eqnarray}
have the following expression
\begin{eqnarray}
\label{Qe}
Q_e^{(i)}&=&\frac{G_6}{\sqrt{2}\pi}\
(\frac{3E}{L}-{\mathcal T})n^{(i)}\sinh \alpha \cosh \beta ~,~~~~~~~1\leq i \leq 20,
\\
\nonumber
&=&\frac{G_6}{\sqrt{2}\pi}\
(\frac{3E}{L}-{\mathcal T})p^{(i-20)}\sinh \beta \cosh \alpha~,~~~21\leq i \leq 24.
\end{eqnarray}
The new solutions have a nonzero magnetic moment,
\begin{eqnarray}
\label{mu}
\mu^{(i)}&=&\frac{G_6\sqrt{2} }{ \pi L}J n^{(i)}\sinh \alpha ~,~~~~~~~1\leq i \leq 20,
\\
\nonumber
&=&\frac{G_6\sqrt{2} }{ \pi L}J p^{(i-20)}\sinh \beta~,~~~21\leq i \leq 24.
\end{eqnarray}
Their dilaton charge
is
\begin{eqnarray}
\label{Qd-def}
Q_d =\frac{G_6}{2\pi}\
(\frac{3E}{L}-{\mathcal T}) (\cosh \alpha \cosh \beta-1).
\end{eqnarray}
The relations between the Hawking temperature $\bar{T}_H$
and the entropy $\bar{S}$ of the charged solutions
and the corresponding quantities $T_H$ and $S$ of the vacuum seed solution are
\begin{eqnarray}
\bar{T}_H=\frac{2T_H}{\cosh \alpha+\cosh \beta},~~
\bar{S}= \frac{1}{2}S(\cosh \alpha+\cosh \beta).
\end{eqnarray}
One can see that the entropy increases with the increase of
angular momentum.
On the other hand, the temperature decreases with
the increase of the angular momenta.
However, the products of temperature and entropy,
$\bar{T}_H \bar{S}$,
and of horizon angular velocity and angular momentum,
$\bar{\Omega}_H \bar J$,
are independent of the
parameters $\alpha$, $\beta$.
As they should, all these properties reduce to those
of the neutral solution upon sending the
boost parameters $\alpha$, $ \beta$ to zero.
We conclude, that every vacuum solution is associated
with a family of charged solutions, which depends on 24 free parameters.
In particular, the branch of non-uniform solutions emerging from
the uniform black string at the threshold unstable mode thus must
persist for strings with non-zero electric charge.
\section{Further remarks. Conclusions. }
Considering rotating black strings in $D$ dimensions,
we have first addressed the GL instability of MP UBSs with equal
magnitude angular momenta in even spacetime dimensions,
taking advantage of the enhanced symmetry of these configurations.
Expanding around the UBS and solving the eigenvalue problem numerically,
our results indicate that the GL
instability persists for these solutions up to extremality
for all even dimensions between six and fourteen.
This agrees with GM correlated stability conjecture
\cite{Gubser:2000ec}, since these
black objects are also thermodynamically unstable
in a grand canonical ensemble.
It may be interesting to note that the GM conjecture
was also confirmed in \cite{Miyamoto:2006nd} for the case of
static, magnetically charged black strings,
the Gregory-Laflamme mode vanishing at the
point where the UBS becomes thermodynamically stable
(which in that case is away from extremality).
While for static vacuum black strings study of the perturbative equations
in second order revealed the appearance of a critical dimension,
above which the perturbative nonuniform black strings
are less massive than the marginally stable uniform black string
\cite{Sorkin:2004qq},
the analogous study in the presence of rotation has yet to be
achieved.
In $D=6$ we then constructed numerically
rotating nonuniform black strings with equal angular momenta.
These emerge from the branch of
marginally stable rotating MP UBS solutions,
which ranges from the static marginally stable black string
to the extremal rotating marginally stable black string.
Along this UBS branch, the Hawking temperature $T_H$ decreases monotonically,
reaching zero in the extremal limit.
Fixing the value of the temperature (or equivalently
the temperature parameter) and decreasing the
value of the horizon angular velocity from the GL value,
then yields a corresponding branch of rotating nonuniform black strings.
Previously, in $D=6$ dimensions, evidence was provided
that the branch of static nonuniform black strings
and the branch of static caged black holes merge at a
topology changing transition \cite{Kudoh:2004hs},
the transition occurring at critical values of
the temperature $T_*$, the string tension $n_*$, etc.
The results we have found for rotating NUBS
indicate that at $T_*$ the branches of rotating NUBS, each with fixed temperature,
exhibit another critical phenomenon.
In particular, the branches of rotating NUBS at fixed temperature $T_H>T_*$
end at static NUBS solutions
with finite nonuniformity and thus finite waist,
where the nonuniformity of these static solutions increases
as $T_H$ is decreasing towards $T_*$.
In contrast, along the branches of rotating NUBS at fixed temperature $T_H<T_*$
the nonuniformity parameter $\lambda$ increases
apparently without bound,
while at the same time the horizon angular velocity
and the angular momentum appear to approach finite values.
Thus for $T_H<T_*$ we see first evidence
for a topology changing transition,
where - in analogy with the static case -
branches of rotating nonuniform black strings
and branches of rotating caged black holes are expected to merge.
We conjecture that there exists a whole branch of rotating
singular topology changing solutions, labelled by their
decreasing temperature,
beginning with the static solution at $T_*$,
and leading possibly up to an extremal rotating solution at $T_H=0$.
Obtaining the respective branches of rotating caged black holes
represents a major numerical challenge.
In principle, there is yet another class of black objects which may
play a role in this picture.
Apart from configurations with an $S^3\times S^1$ (black strings) and $S^4$ (black holes)
topology of the event horizon, the $D=6$ KK theory
possesses also vacuum uniform solutions
with an event horizon of topology $S^2\times S^1\times S^1$,
corresponding to uplifted $D=5$ black rings \cite{Emparan:2001wn,Pomeransky:2006bd}.
Nonuniform solutions with an event horizon of topology $S^2\times S^1\times S^1$,
approaching at infinity the ${\cal M}^{5}\times S^1$ background are also
likely to exist.
They may join the MP NUBS branch at a topology changing transition.
However, for the case discussed in this paper with two equal
magnitude angular momenta, we could not find any indication
of this scenario.
This appears to be consistent with the recent results in \cite{Pomeransky:2006bd},
where the general $D=5$ black ring solution with two angular momenta was presented.
An inspection of this solution indicates that black rings with
equal angular momenta must exhibit some pathologies,
which may explain our result.
We remark that, as in the static case \cite{Kleihaus:2006ee},
we observe the backbending phenomenon for the relative tension $n$
also for branches of rotating nonuniform black strings,
below some critical value of the temperature.
Our last concern was the construction
of charged rotating NUBS in heterotic string theory,
by adding charge to the vacuum solutions by
applying solution generating $(O(26-D,1)/O(26-D))
\times (O(10-D,1)/O(10-D))$ transformations
\cite{Sen:1994eb}.
The properties of these new configurations
can be derived from the corresponding vacuum solutions.
We expect that, similar to the
static case \cite{Harmark:2007md}, the solutions discussed in this paper may be relevant for the
thermal phase structure of non-gravitational theories, via gauge/gravity duality.
The construction of the general rotating
vacuum NUMBS with distinct angular
momenta seems to represent an exceedingly difficult task.
However, the case of only one nonvanishing angular momentum
appears to be treatable.
These solutions may be found by using similar techniques to those
employed in this work and are currently under study.
Although the static higher-dimensional black holes are stable
\cite{Ishibashi:2003ap},
their rotating counterparts need not be, at least for large rotation.
Recently, the existence of an effective Kerr bound
for $d>4$ rapidly rotating black holes with one nonzero angular momentum
was conjectured by Emparan and Myers \cite{Emparan:2003sy}. They showed
that the geometry of the event horizon of such rapidly rotating black objects
in six or higher dimensions behaves like a black membrane.
Therefore the black hole becomes unstable.
This instability should persist for the corresponding rotating UBS solutions,
and is not associated with the extra dimensions.
Rotating black objects extending in extra dimensions
may also exhibit other instabilities.
Cardoso and Lemos \cite{Cardoso:2004zz}
uncovered a new universal instability
for rotating black branes and strings, which holds for any
massless field perturbation. The main point of their argument
is that transverse dimensions in a black string geometry
act as an effective mass for the fields, which simulates a
mirror enclosing a rotating black hole, thereby creating
a black hole bomb.
For further work on the instabilities of rotating black objects, see for example
\cite{Gibbons:2002pq}, \cite{Marolf:2004fy}, \cite{Cardoso:2006ks}.
\section*{Acknowledgements}
B.K.~gratefully acknowledges support by the DFG under contract
KU612/9-1.
The work of E.R. was carried out in the framework of Enterprise--Ireland
Basic Science Research Project SC/2003/390.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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This LG 4K TV supports many premium content choices, all optimized with scene-by-scene picture adjustment. The multi-format 4K high dynamic range support includes Dolby Vision®, HDR10 and HLG. Behold every grain of sand, every blade of grass, every star in the night sky. LG 4K Ultra HD TVs contain over 8.2 million pixels, so their resolution is four times that of Full HD. The result? Breathtaking clarity and fine picture details that will amaze, even when viewed up close. Simply SmartwebOS™ 4.0 is a simple Smart operating system to use with our Magic Remote and refined launcher bar. Enjoy many of the features using just our intelligent voice recognition, get to the apps you want, as fast as you want, zoom in and record what's on your screen, connect to any Bluetooth device and even use your TV as a music player, all with the magic of webOS™ 4.02. Picture quality becomes even more dynamic with UltraLuminance, which boosts peak brightness and increases the overall contrast ratio to give movies and TV shows a bold appearance not previously possible*.Why do LG UHD TVs look so good? At its heart is the mighty quad-core processor that works tirelessly to reduce distracting video noise, enhance sharpness and ensure accurate colours. Make your 4K viewing experience even better with an immersive sound experience. Ultra Surround delivers an immersive audio experience from seven virtual channels, requiring only the built-in speakers of the TV.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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## Dedication
_For Deborah Miller - writer and warrior; my mother, my hero._
## Epigraph
There is nothing safe about the darkness of this city and its stink. Well, I have abrogated all claim to safety, coming here. It is better to discuss it as though I had chosen. That keeps the scrim of sanity before the awful set. What will lift it?
—Samuel R. Delany, _Dhalgren_
## Contents
_Cover_
_Title Page_
_Dedication_
_Epigraph_
_Contents_
People Would Say
Fill
Ankit
Kaev
Ankit
City Without a Map: Boundaries
Fill
Soq
Fill
Ankit
Kaev
Ankit
City Without a Map: Dispatches from the Qaanaaq Free Press
Soq
Ankit
City Without a Map: Anatomy of an Incident
Soq
Kaev
Fill
Ankit
Kaev
Ankit
Soq
Kaev
Fill
Masaaraq
Kaev
Ankit
Soq
Fill
Soq
City Without a Map: Archaeology
Fill
Soq
Ankit
Kaev
City Without a Map: The Breaks
Kaev
Fill
Kaev
Soq
Ankit
City Without a Map: Alternative Intelligences
Ankit
Soq
Ankit
Soq
Kaev
City Without a Map: Cross Fire
Soq
Ankit
Kaev
Soq
Kaev
Ankit
City Without a Map: Press Montage
Kaev
Soq
Ankit
City Without a Map: Savage Bloodthirsty Monsters
Soq
Kaev
Ankit
_Acknowledgments_
_About the Author_
_Copyright_
_About the Publisher_
## People Would Say
People would say she came to Qaanaaq in a skiff towed by a killer whale harnessed to the front like a horse. In these stories, which grew astonishingly elaborate in the days and weeks after her arrival, the polar bear paced beside her on the flat bloody deck of the boat. Her face was clenched and angry. She wore battle armor built from thick scavenged plastic.
At her feet, in heaps, were the kind of weird weapons and machines that refugee-camp ingenuity had been producing; strange tools fashioned from the wreckage of Manhattan or Mumbai. Her fingers twitched along the walrus-ivory handle of her blade. She had come to do something horrific in Qaanaaq, and she could not wait to start.
You have heard these stories. You may even have told them. Stories are valuable here. They are what we brought when we came here; they are what cannot be taken away from us.
The truth of her arrival was almost certainly less dramatic. The skiff was your standard tri-power rig, with a sail and oars and a gas engine, and for the last few miles of her journey to the floating city it was the engine that she used. The killer whale swam beside her. The polar bear was in chains, a metal cage over its head and two smaller ones boxing in its forepaws. She wore simple clothes, the skins and furs preferred by the people who had fled to the north when the cities of the south began to burn or sink. She did not pace. Her weapon lay at her feet. She brought nothing else with her. Whatever she had come to Qaanaaq to accomplish, her face gave no hint of whether it would be bloody or beautiful or both.
## Fill
After the crying, and the throwing up, and the scrolling through his entire contacts list and realizing there wasn't a single person he could tell, and the drafting and then deleting five separate long graphic messages to _all_ his contacts, and the deciding to kill himself, and the deciding not to, Fill went out for a walk.
Qaanaaq's windscreen had been shifted to the north, and as soon as Fill stepped out onto Arm One he felt the full force of the subarctic wind. His face was unprotected and the pain of it felt good. For five minutes, maybe more, he stood there. Breathing. Eyes shut, and then eyes open. Smelling the slight methane stink of the nightlamps; letting his teeth chatter in the city's relentless, dependable cold. Taking in the sights he'd been seeing all his life.
_I'm going to die,_ he thought.
_I'm going to die soon._
The cold helped distract him from how much his stomach hurt. His stomach and his throat, for that matter, where he was pretty sure he had torn something in the half hour he'd spent retching. A speaker droned from a storefront: a news broadcast, the latest American government had fallen, pundits predicting it'd be the last, the flotilla disbanded after the latest bombing, and he didn't care, because why should he, why should he care about anything?
People walked past him. Bundled up expensively. Carrying polyglass cages in which sea otters or baby red pandas paced, unhappy lucky animals saved from extinction by Qaanaaq's elite. All of whom were focused on getting somewhere, doing something, the normal self-important bustle of ultra-wealthy Arm One. Something he despised, or did on every other day. Deaf to the sea that surged directly beneath their feet and stretched on into infinity on either side of Qaanaaq's narrow metal arms. He'd been so proud of his indolent life, his ability to stop and stand on a street corner for no reason at all. Today he didn't hate them, these people passing him by. He didn't pity them.
Fill wondered: _How many of them have it?_
A child tapped his hip. "Orca, mister!" A pic tout, selling blurry shots of the lady with the killer whale and the polar bear. Fill bought one from the girl on obscure impulse—part pity, part boredom. Something else, too. A glimmer of buoyant wanting. Remembered joy, his childhood fascination with the stories of people emotionally melded with animals thanks to tiny machines in their blood. Collecting pedia entries and plastiprinted figures . . . and scowls, from his grandfather, who said nanobonding was a stupid, naive myth. His plastic figures gone one morning. Grandfather was sweet and kind, but Grandfather tolerated no impracticality.
On some level, the diagnosis hadn't been a surprise. Of course he had the breaks. No one in any of the grid cities could have as much sex as he had, and be as uncareful as he was, without getting it. And he'd lived in fear for so long. Spent so much time imagining his grisly fate. He was shocked, really, to have such a visceral reaction.
Tapping his jaw bug, Fill whispered, "Play _City Without a Map,_ file six."
A woman's voice filled his ears, old and strange and soothing, the wobble in her Swedish precise enough to mark her as someone who'd come to Qaanaaq decades ago.
You are new here. It is overwhelming, terrifying. Don't be afraid.
Shut your eyes. I'm here.
Pinch your nose shut. Its smell is not the smell of your city. You can listen, because every city sounds like chaos. You will even hear your language, if you listen long enough.
There is no map here. No map is needed. No manual. Only stories. Which is why I'm here.
A different kind of terror gripped Fill now. The horror of joy, of bliss, of union with something bigger and more magnificent than he could ever hope to be.
For months he'd been obsessing over the mysterious broadcasts. An elliptical, incongruent guidebook for new arrivals, passed from person to person by the tens of thousands. He switched to the next one, a male voice, adolescent, in Slavic-accented English.
Qaanaaq is an eight-armed asterisk. East of Greenland, north of Iceland. Built by an unruly alignment of Thai-Chinese-Swedish corporations and government entities, part of the second wave of grid city construction, learning from the spectacular failures of several early efforts. Almost a million people call it home, though many are migrant workers who spend much of their time on boats harvesting glaciers for freshwater ice—fewer and fewer of these as the price of desalinization crystals plummets—or working Russian petroleum rigs in the far Arctic. Arm One points due south and Arm Eight to the north; Four is west and Five is east. Arms Two and Three are southwest and southeast; Arms Six and Seven are northwest and northeast. The central Hub is built upon a deep-sea geothermal vent, which provides most of the city's heat and electricity.
Submerged tanks, each one the size of an old-world city block, process the city's waste into the methane that lights it up at night. Periodic controlled ventilations of treated methane and ammonia send parabolas of bright green fire into the sky. Multicolored pipes vein the outside of every building in a dense varicose web: crimson chrome for heat, dark olive for potable water, mirror black for sewage. And then the bootleg ones, the off-color reds for hijacked heat, the green plastics for stolen water.
Whole communities had sprung up of _City_ devotees. Camps, factions, subcults. Some people believed that the Author was a machine, a bot, one of the ghost malware programs that haunted the Qaanaaq net. Such software had become astonishingly sophisticated in the final years before the Sys Wars. Poet bots spun free-verse sonnets that fooled critics, made award winners weep. Scam bots wove intricate, compellingly argued appeals for cash. Not hard to imagine a lonely binary bard wandering through the forever twilight of Qaanaaq's digital dreamscape, possibly glommed on unwillingly to a voice-generation software that constantly conjured up new combinations of synthesized age and gender and language and class and ethnic and national vocal tics. Its insistence on providing a physical description of itself would not be out of character, since most had been coded to try their best to persuade people that they were real—Nigerian princes, refugee relatives, distressed friends trapped in foreign lands.
Other theorists believed in a secret collective, a group of writers for whom the broadcasts were simultaneously a recruitment tool and a soapbox. Possibly an underground forbidden political party with the nefarious endgame of uniting the unwashed hordes of the Upper Arms and slaughtering the wealthier innocents who ruled the city.
On Arms One and Two and Three, glass tunnels connect buildings twenty stories up. Archways support promenades. Massive gardens on hydraulic lifts can carry a delighted garden party up into the sky. Spherical pods on struts can descend into the sea, for underwater privacy, or extend to the sky, to look down on the crowds below.
The architecture of the other Arms is less impressive. Tight floating tenements; boats with stacked boxes. The uppermost Arms boggle the mind. Boxes heaped on boxes; illicit steel stilts holding up overcrowded crates. Slums are always a marvel; how human desperation can seem to warp the very laws of physics.
Fill subscribed to the single-author theory. _City Without a Map_ was the work of one person—one human, corporeal person. He went through phases, periods when he was convinced the Author was male and times when he knew she was female—old, young, dark-skinned, light-skinned, poor, rich . . . whoever they were, they somehow managed to get hundreds of different people to record their gorgeous, elliptical instructions for how to make one's way through the tangled labyrinth of his city.
Not how to survive. Mere survival wasn't the issue for the Author. The audience he or she wrote for, spoke to—they knew how to survive. They had been through so much, before they came to Qaanaaq. What the Author wanted was for them to find happiness, joy, bliss, community. The Author's love for their listeners was palpable, beautiful, oozing out of every word. When Fill listened, even though he knew he was not part of the Author's intended audience, he felt loved. He felt like he was part of something.
Nations burned, and people came to Qaanaaq. Arctic melt opened the interior for resource exploitation, and people came. Some of us came willingly. Some of us did not.
Qaanaaq was not a blank slate. People brought their ghosts with them. Soil and stories and stones from homelands swallowed up by the sea. Ancestral grudges. Incongruent superstitions.
Fill wiped tears from his eyes. Some were from the words, the hungry hopeful tone of voice of the last Reader, but some were still from the pain of his diagnosis. God, he was an idiot. Snow fell, wet and heavy. Projectors hidden below the grid he walked on beamed gorgeous writhing fractal shapes onto the wind-blown flurry. A child jumped, swatted at the snow, laughed at how a fish or bird imploded only to reappear as new flakes fell.
A startling, uncontrollable reaction: Fill giggled. The snow projections could still make his chest swell with childish wonder. He waved his hand through a manta ray as it soared past.
And all at once—the pain went away. His throat, his stomach. His heart. The fear and the nightmare images of twisted bodies in refugee camp hospital beds; the memory of broken-minded breaks victims wandering the streets of the Upper Arms, the songs they sang, the things they shrieked, the things they did to themselves with fingers or knives without feeling it. Every time he followed a man down a dark alley, or met one at a lavish apartment, or dropped to his knees in a filthy Arm Eight public restroom, this was the ice-shard blade that scraped at his heart. This was what he'd been afraid of.
Fill laughed softly.
When the worst thing that can possibly happen to you finally happens, you find that you are not afraid of anything.
## Ankit
Most outsiders saw only misery, when they came to Qaanaaq's Upper Arms. They took predictable photos: the tangled nests of pipes and cables, filthy sari fabric draped over doorways and hanging from building struts, vendors selling the sad fruit of clandestine greenhouses. Immigrant women gathered to sing the songs of drowned homelands.
Ankit watched the couple in the skiff, taking pictures of a little boy. His face and arms were filthy with soot; cheap stringy gristle covered his hands. He sat at the edge of the metal grid, legs dangling over the ocean three feet below, stirring a bubbling trough that floated in the sea. Bootleg meat; one of the least harmful illegal ways to make money out on the Upper Arms. He frowned, and their cameras clicked faster.
She hated them. She hated their blindness, their thick furs, their wrongness. Her jaw bug pinged their speech—upper-class post-Budapest, from one of the mountain villages the wealthy had been able to build for themselves as their city sank—but she tapped away the option to translate. She didn't need to hear what they were saying. They knew nothing about what they saw. Their photos would capture only what confirmed their preconceptions.
These people were not sad. This place was not miserable. Tourists from the Sunken World looked at the people of Qaanaaq and saw only what they'd lost, never what they had. The freedom they had here, the joy they found. Gambling on beam fights, drinking and dancing and singing. Their families, their children, who came home from school each day with astonishing new knowledge, who would find remarkable careers in industries as yet unimagined.
_We are the future,_ Ankit thought, staring the hearty dairy-fed tourists down, daring them to make eye contact, which they would not, _and you are the past._
She inspected the outside of 7-313. House built crudely upon house; shipping container apartments stacked eight high. A flimsy exterior stairway. At least these would have windows carved into the front and back, a way to let light in and watch the ocean—as well as keep an eye on who was coming and going along the Arm itself. And she saw something else: the scribbled hieroglyphics of scalers. Where the best footholds were, what containers were rigged to ensnare roof sprinters.
It had been years since she last scaled anything. She couldn't do so now. She carried too much with her. Physically, and emotionally. To be a scaler you had to be unburdened.
The tourists took no photos of her. They looked at her and could not see where she came from, what she had been, only what she was. Safe, comfortable. No shred of desperation or rage, hence uninteresting. The little boy had run off; they turned their attention to the singing circle of women.
Ankit stopped to listen, halfway up the front stairs. Their voices were raw and imprecise, but the song they sang was so full of joy and laughter that she shivered.
"Hello," said the man who answered the fourth-floor unit door. Tamil; she knew maybe five words of it. Fyodorovna thought that Ankit's cultural comfort level would make these people feel less frightened of her—she'd been raised by a Tamil foster family—but that was stupid. Like most things Fyodorovna thought.
"My name is Ankit Bahawalanzai," she said. "You filed a constituent notice with the Arm manager's office?"
He bowed, stepped aside to let her in. A weathered, worn man. Young, but aging fast. What had he been through, back home? And what had it cost him to get his family out of there? The Tamil diaspora covered so much ground, and the Water Wars had played out so differently across South Asia. She took the seat he offered, on the floor where two children played. He went to the window, called outside. Brought her a cup of tea. She placed her screen on the floor and opened the translation software, which set itself to Swedish/Tamil.
"You said your landlord—"
"Please," he said, the slightest bit of fear in his eyes. "Wait? My wife."
"Of course."
And a moment later she swept in, flushed from happiness and the cold: one of the women from the singing circle. Beautiful, ample, her posture so perfect that Ankit trembled for anyone who ever made her mad.
"Hello," Ankit said, and repeated her opening spiel. "I work for Arm Manager Fyodorovna. You filed a constituent notice with our office?"
Every day a hundred complaints came in. Neighbors illegally splicing the geothermal pipes; strange sounds coming through the plastic walls. Requests for help navigating the maze of Registration. Landlords refusing to make repairs. Landlords making death threats. Landlords landlords landlords.
Software handled most of them. Drafted automated responses, since the vast majority were things beyond the scope of Fyodorovna's limited power ( _No, we can't help you normalize your status if you came here unregistered; no, we can't get you a Hardship housing voucher),_ or flagged them for human follow-up. A flunky would make a call or send a strongly worded message.
But the Bashirs had earned themselves a personal visit from Fyodorovna's chief of staff. Their building was densely populated, with a lot of American and South Asian refugees, and those were high-priority constituencies, and it was an election year. Word would spread, of her visit, of Fyodorovna's attentiveness.
She wasn't there to help. She was PR.
"Our landlord raised the rent," Mrs. Bashir said, and waited for the screen to translate. Even the poorest of arrivals, the ones who couldn't afford jaw implants or screens of their own, had a high degree of experience with technology. They'd have dealt with a lot of screens by now, throughout the process of gaining access to Qaanaaq. And anyway, her voice was sophisticated, elegant. She might have been anything, before her world caught fire. "We've only been here three months. I thought they couldn't do that."
Ankit smiled sadly and launched into her standard spiel about how Qaanaaq imposed almost no limits on what a landlord could or couldn't do. But that, rest assured, Fyodorovna's number one priority is holding irresponsible landlords accountable, and the Arm manager will make the call to the landlord herself, to ask, and that if Mrs. Bashir or any of her neighbors have any other problems, they should please message our office immediately . . .
Then she broke off, to ask, "What's this?" of a child drawing on a piece of plastislate. Taksa, according to the file. Six-year-old female. Sloppily coloring in a black oval.
"It's an orca," she said.
"You've heard the stories, then," Ankit said, and smiled. "About the lady? With the killer whale?"
Taksa nodded, eyes and smile wide. The woman was the stuff of legend already. Ample photos of her arrival, but no sign of her since. How do you vanish in a city so crowded, especially when you travel with a polar bear and a killer whale?
"What do you think she came for?"
Taksa shrugged.
"Everyone has a theory."
"She came to kill people!" said Taksa's older brother, Jagajeet.
"Shh," his mother said. "She's an immigrant just like us. She only wants a place to be safe." But she smiled like she had more dramatic theories of her own.
The children squabbled lovingly. Ankit felt a rush of longing, of envy, at their obvious bond, but swiftly pushed it away. Thinking about her brother brought her too close to thinking about her mother.
Taksa put her crayon down and shut her eyes. Opened them, looked around as though surprised by what she saw. And then she said something that made her parents gasp. The screen paused as the translation software struggled to parse the unexpected language. Finally _Russian_ flashed on the screen, and its voice, usually so comforting, translated _Who are you people?_ into Tamil and then Swedish.
Three seconds passed before anyone could say a thing. Taksa blinked, shook her head, began to cry.
"What the hell was that?" Ankit asked, after the mother led the girl into the bathroom.
The father hung his head. Taksa's brother solemnly took over the drawing she'd abandoned.
"Has this happened before?"
The man nodded.
Ankit's heart tightened. "Is it the breaks?"
"We think so," he said.
"Why didn't you call a doctor? Get someone to—"
"Don't be stupid," the man said, his bitterness only now crippling his self-control. "You know why not. You know what they do to those kids. What happens to those families."
"Aren't the breaks . . ." She couldn't finish the sentence, hated even having started it.
"Sexually transmitted," the father said. "They are. But that's not the only way. The resettlement camp, you can't imagine the conditions. The food. The bathrooms. Less than a foot between the beds. One night the woman beside my daughter started vomiting, spraying it everywhere, and . . ."
He trailed off, and Ankit was grateful. Her heart was thumping far too loudly. This was the sixth case she'd seen in the past month. "We'll get her the help she needs."
"You know there's nothing," he said. "We read the outlets, same as you. Think we aren't checking every day, for news? Waiting for your precious robot minds to make a decision? For three years now, when there is an announcement at all, it is always the same: _Softwares from multiple agencies, including Health, Safety, and Registration, are still gathering information, conducting tests, in order to draft new protocols for the handling and treatment of registrants suffering from this and other newly identified illnesses._ Meanwhile, people die in the streets."
"You and your family will be fine. We wouldn't—"
"You're young," the father said, his face hard. "You mean well, I am sure. You just don't understand anything about this city."
_I've lived here my whole life and you've only been here six months,_ she stopped herself from saying.
_Because I feel bad for him,_ she thought _._ But really it was because she wasn't entirely certain he was wrong.
The bathroom door opened, and out came Taksa. Smiling, tears dried, mortal illness invisible. She ran over to her brother, seized the plastislate. They laughed as they fought over it.
"May I?" Ankit asked, raising her screen in the universal sign of someone who wanted to take a picture. The mother gave a puzzled nod.
She could have taken dozens. The kids were beautiful. Their happiness made her head spin. She took only one: the little girl's face a laughing blur, her brother's hands firmly and lovingly resting on her shoulders.
## Kaev
Nothing was certain but the beam he stood on.
The gong sounded and Kaev opened his eyes. The lights came up slowly. A pretty standard beam configuration for this fight: Rows of stable columns and intermittent hanging logs. Poles big enough for someone to plant one foot upon. Three platforms, each large enough for two people to grapple on. He pressed his soles into the bare wood and breathed. A spotlight opened on his opponent, a small Chinese kid he'd been hearing about for weeks. Young but rising fast. The unseen crowd screamed, roared, stamped, blasted sound from squeeze speakers. Ten thousand Qaanaaq souls, their eyes on him. Or at least, his opponent. A hundred thousand more watching at home, in bars, standing in street corner clumps and listening on cheap radios. He could see them. He could see them all.
The city would not go away. Kaev's mind throbbed with it, with the pain of so much life surrounding him. So many things to be afraid of. So many things to want. He kept his lips pressed tight together, because otherwise he would scream.
Somewhere in the crowd, Go was watching him. She'd have one eye on her screen, but the other one would be on him. And she'd smile, to see him step forward, to watch her script act itself out.
Kaev leaped to the next beam. His opponent stood still, waiting for Kaev to come to him. Cocky; clueless. The poor dumb thing thought he was smart enough to anticipate what would happen when. He had no idea who Go was, how much energy and money went into making sure the fight played out a certain way.
_America has fallen and I don't feel so good myself._
The news had stuck with him in a way news didn't, usually. Because what was America to Kaev? Just one more place he might have come from. Every Qaanaaq orphan had a head full of origin stories, the countries they fled, the wealthy powerful people their parents had been, the immense conspiracies that had put them where they were. Kaev was thirty-three now, too old for fantasies about what might have been. He knew what was, and what was was miserable. He ran along the length of the hanging log, hands back, spine straight and low, trying to push it all out of his head.
And as he drew near to the kid, the fog lifted. The din hushed. Fighting was where the pieces came together. The kid jumped, landed on the far end of the hanging log. A roar from the crowd. Kaev couldn't hear the radio broadcaster's commentary, but he knew precisely what he would say.
_This kid is utterly unafraid! He leaps directly into the path of his opponent, landing in a flawless horse stance. There's no knocking this guy into the drink . . ._
Kaev always listened to his fights after they were over, a day or two later, when the buzz had faded altogether. Hearing Shiro describe his own efforts, even in the brisk, empty language of sports announcers, brought him a certain measure of peace. A flimsy, lesser cousin of the joy of fighting.
Instants before colliding with his opponent, Kaev leaped into the air and whirled his legs around. Smirking, the boy dropped to his knees and let Kaev's jump kick pass harmlessly over him—but did not turn around fast enough. In the instant Kaev landed he was already pinioning around, delivering an elbow to the boy's back. Not rooted, not completely balanced, nothing to cause real damage, but enough to make the kid wobble a little and stagger a step back.
A different kind of roar from the crowd: begrudging respect. Kaev was not their favorite, but he had gotten off a good shot and they acknowledged that. The imaginary Shiro in his head said, _I think Hao will proceed with a little more caution from here on out, folks!_
Now the younger fighter pursued him away from the center of the arena. At the outer ring of posts Kaev turned and kicked, but Hao effortlessly swerved to the side. At the precise instant that the momentum of the kick had ebbed, he leaned into Kaev's leg and broke his balance. Anywhere else and he would have been finished, but at the outer ring the posts were close enough that Kaev could stumble-step to the next one.
_Yes. Yes. This!_
He bellowed. He was an animal, a monster, part polar bear. Unstoppable.
In his dreams, sometimes, he _was_ a polar bear. And lately he'd been having those dreams more and more. He'd spent six hours, the day before, wandering up and down the Arms in search of the woman who was said to have come to Qaanaaq with a killer whale and a polar bear, but found nothing.
He sniffed the air, his head full of pheromonic information from his opponent, and charged.
A dance. A religious ritual. Whatever it was, Kaev was free for as long as he fought. He wasn't thinking about how the spasms were getting worse, until he could barely speak a normal sentence. He wasn't worrying about how the money wasn't coming in the way it needed to, and pretty soon he'd have to move out of his Arm Seven shipping container and sleep in an Arm Eight capsule tenement or worse. He wasn't thinking about Go, and how much he hated her, and what an idiot he'd been for being so in love with her once.
He was one with his opponent and the attention of the crowd. And the whisper of cold salt water, thirty feet below.
They grappled until the gong sounded, and they separated. This was not some savage skiff-bed fight, after all. The Yi He Tuan Arena beam fights were Qaanaaq's most distinctive and beloved sport, and their champions won by agility and balance and swift punishing blows, not the frenzied grappling of street fighters. Kaev weighed more and his reflexes were better, but the kid had grace, had speed; Kaev could see why everyone liked him, why he'd been set out on this path to stardom.
_Stars make money,_ Go had said five years ago. _People pay to see someone they recognize, someone they can root for. And you can't make a winner without a lot of losers._
Which is how Kaev's life as a journeyman fighter began. The guy who other fighters fought when they needed to learn the ropes and build a lossless record at the same time. Not the worst career. Journeymen had a much longer shelf life than the stars, who usually fizzled out fast from one thing or another, but the stars tended to have handsome bank accounts to fall back on when they went bust. Journeymen were lucky to have a month's rent as backup.
He didn't mind the losing. He loved the fights, loved the way his opponent helped him step outside himself, and something about the fall into freezing water provided an almost orgasmic release.
What Kaev minded was the hunger, the anger, the empty feeling. What he minded, what he could never forgive Go or the crowds or the whole fucking city of Qaanaaq for, was that he hadn't had an option.
Hao was tiring, he could tell. The kid was too new, too rough. Kaev switched into stamina mode, feigning defense while modeling how to conserve energy and catch your breath while you're winded. Hao followed suit, probably without realizing what he was doing. A new trick he'd learned tonight. In moments like this, Kaev was proud of what he was. A rare and sophisticated skill, letting someone else win without the crowd knowing it. Hao's kicks connected with his thighs and side and the onlookers surged to their feet and for a few short instants Kaev was the king of Qaanaaq.
The kid got it. Kaev saw him get it. The instant when it all became clear; when he saw what Kaev was doing, and his attitude changed from cocky contempt to humbled respect. His eyes went wide, went soft. He paused—and Kaev could have kicked him in the back of the knee and followed up with a punch to the side of the head, sent him sprawling into the sea below; he saw precisely how to do it, even lifted his leg to unleash the attack—but do that and he'd cost Go millions, probably get a hit put out on him, and for what? A record of 37–3 instead of 38–2? Kaev pulled his wrist back, anchored himself; the crowd lost its mind, he could win it, their golden boy could be finished—
Laughter trembled up through Kaev. Joy threatened to split his skin wide open. He was a bird, he was bliss, he was so much more than this battered body and broken brain. In the split-second pause that gave away his advantage, he wanted to howl from happiness.
Hao's face looked sad as he leaped in close and slammed home an uppercut. Kaev had given him that, the humility of a genuine warrior. That was what made a true artist, the kind of fighter who would mean something to the cold wet salt-stinking people of Qaanaaq. Falling, Kaev focused on that. On Hao's career—like the careers of dozens of bright young boys who'd fought Kaev before him. What they might go on to do.
Kaev caught a glimpse of a woman graffitied onto the underside of the platform he'd fallen from. Clever placement: the sort of spot where no one but a falling fighter would see it. Programmed into an amphibious tagger-drone that swam in from the sea below and flew up to paint her in that secret nook. She was beautiful. Older, bald, dressed in either a monastic shift or a hospital robe, one hand raised, her face projecting saintliness. Beside her, three letters— _ORA_. Initials? For what?
He saw the metal fretwork that held up the bleachers, the places where the walls of the arena plunged down below the surface, the walkway around the edge where the fight doctor waited to fish the loser out of the water. He heard the howls of the crowd. But none of that was real. The water was all that was real. It rose up to him now, overjoyed to be embracing him again. Water was as much his native element as air. He was amphibious. He was a polar bear. He felt his body break the surface, felt the electric jolt of cold, and then he was gone, vanished, his body abandoned, its spasms and inadequacies and unfulfillable needs and mental fumblings all erased in a wash of stark bare ecstasy.
## Ankit
Being a good scaler was easy. Ankit had been good. She was strong and she had decent reflexes. She could vault over barricades, evade drone cams, scamper along railings barely wider than a tightrope.
The difference between a good scaler and a great one was fear. Ankit was afraid. Fear held her back. She'd never been able to truly let go. She'd never been able to fly. She wanted to—wanted so bad it made her stomach heavy and her limbs freeze, poised at the edge of the abyss, unable to move.
She felt it now: the wanting. Standing between two tall ramshackle buildings, staring up at the handholds and footrests taunting her. The wanting, and the fear.
A whiff of pine needle smoke hit her before his voice did. "Hey, kid," he said from the dark space between two building struts.
"Hey," Ankit said, vaguely gratified to still be called a kid by someone. This guy had been doing errands for her since she actually was one. She stepped forward, out of the stream of traffic, into the cold wet dark of Qaanaaq's interstitial commerce. Clatter of Chinese chess tiles from the street behind her; ahead, in the deeper dark, two men grunting together.
"Missed you lately," he said. He wore three hoodies; their shadow softened the lines in his face.
"New delivery," she said, and handed him a screen. Small, cheap, used, its network connections all fried, but with a long battery and a solar-charger skin.
"What's on it?" he asked, smiling, excited. The guy was old, old enough to have had a whole life somewhere else before he came to the city. His pine needle cigarettes were resettlement-camp mainstays, smoked proudly and defiantly by recent arrivals, but as a result they had a certain outlaw appeal and even privileged Lower Arm kids could be seen smoking them.
"Books," she said. "Enough books to last a lifetime."
"You know her restrictions say only approved tech. She gets caught with this—"
"That's her call to make. If she doesn't want to take the risk, she can refuse it. In which case, give it back to me, if you're feeling generous, or just sell it. I'll never know anyway."
"Yeah," he said, and took the screen from her. "You're so trusting. All these years, I could just be stealing your money and pocketing your stuff. You have no way of knowing if any of it ever gets to her."
"You keep saying that," she said. They could not smell the methane stink of the nightlamps, out there. The darkness held her in its mouth; the sea wound around her like a cloak. Here was bliss, was freedom: a sliver of the thrill she'd felt when scaling. The pleasure she never felt in daylight, in the scripted comings and goings of her job. "I can't tell if it's your twisted way of making a confession, or some kind of criminal code of honor thing you want to stress." And even this felt good, the not knowing, the wondering. The city's rigid certainties were robotic, unforgiving.
Path bowed. Monkeys chattered from a warm nook beside the thermal pipes; the escaped endangered pets of the pampered rich, scavenging a living in Qaanaaq the way pigeons did in Sunken World cities. "Such are the twisted pathways of human trust. Any message for her?"
"The usual," Ankit said. "Tell her that I love her, I miss her, I'll get her out of there."
He put his hand on her arm. He smelled like a forest. "My inside woman is a good person. She'll pass your message on. Your birth mom is surviving, and that's all anyone can hope to do in the Cabinet."
"Thank you, Path," she said, and stepped back, out of the shadow, into the street, where the vortex struck her hard, and she walked against it, farther out onto Arm Five. A woman rolled on the grid, babbling to herself about demons and oppression, her body shaking with end-stage breaks.
Ankit's jaw chimed. She tapped it and heard the voice of her contact at Families. She'd messaged him that morning, trying to find out what might happen next with little Taksa, the girl with the breaks, and how she might be able to help.
Sorry to tell you, Ankit—Bashir family already deregistered. Awaiting transfer to the dereg ship. One good thing, transfer times have gotten crazy lately. Average wait six months. Ten thousand people flagged with the breaks, still in their homes. Waiting. Eventually they'll be processed and put on the continental shuttle. Allotted space at one of the coastal camps.
Beside her, three seagulls struggled against the wind, then yielded to it. She reminded herself to breathe.
Taksa's father had been right. _You know what happens to those families_. She'd heard. She hadn't wanted to believe.
One shred of hope: that six-month transfer wait. That must mean something. Nothing ever took so long. Maybe it was an AI malfunction, or a new protocol was about to be rolled out that couldn't be announced until some obscure other protocol finished its task. Decommissioned glacial calving ships being refitted for refugee transport, maybe, or the Swedes completing work on another West African camp. Qaanaaq was governed by a hundred thousand computer programs, which mostly got along well enough, but sometimes contradictory or irreconcilable mandates sparked a squabble that brought an agency's operations to a standstill until a human or—more likely—another AI intervened. She'd have to look into it further.
Another tiny hope for Taksa, even slimmer than the first: asking her boss for help.
_Idea for your campaign speech next week,_ Ankit wrote. _I've been seeing more and more cases of the breaks in my constituent visits. Families. Kids. No one is talking about it. Certainly not your opponent. People are scared. If you show leadership on this issue it has the potential to increase your lead by 3.6%._
That last bit was made up. Put a decimal point on the end of a lie and her boss would swallow it every time. Whispering pleas to the universe—and wishing she could stop worrying about this family—Ankit headed home.
The problem was, Taksa's father was just so similar to the man who raised her. The same profound humility, sturdy and essential as his spine. They'd been profoundly decent people, which wasn't a given when it came to Qaanaaq. Or, really, anywhere. Few families would have helped a ten-year-old who demanded to know who her birth mother was—filed the appropriate paperwork, guided her through the bureaucracy labyrinth as best they could.
Both were dead now. Reflexogenic circulatory collapse, like so many of their generation—the decades-later legacy of corporate chemical spills and gas leaks. Pain in her chest made her stop walking, remembering. How well her father cooked, her mother's paintings. How they gave her the grandfather's name they were never able to have a son to carry. She resolved to buy some rice balls and make an offering before their photos—and then remembered that she made that resolution often, and followed through rarely.
She didn't deserve the place where she lived. Few scalers ever landed a spot that nice, a job as good as hers, and it was a comfort, sometimes, to reflect on how lucky she was, and how hard she'd worked to get there, when you thought about how many Qaanaaq orphans were dying slowly in the cold green light of the methane-sodium streetlights that very moment of new diseases no one understood and no one wanted to talk about, and how easily she could have been one of them—and Ankit had to work hard to see it that way, instead of the other way, the one where she was turning her back on her people, where she should be doing more, where she and everyone else who got a raw deal from this shitty city should get together and demand what was rightfully theirs, like those weird seditious anonymous _City Without a Map_ broadcasts always seemed to be hinting at.
Sudden shock: her brother, grimacing out at her from a fading bootleg flicker flyer. Advertising an upcoming fight, already in the past. Listing gambling sites and odds.
She'd gone to see him once, after she'd gotten her job and could access her file from Families, learn that she had a brother she'd never known existed—though nothing more than his name. When she found him he'd been strung out on something, probably synth caff, after a fight, and she'd known at once that something was wrong with him, mentally—the breaks, she'd thought at first, but no, this was something different—and she'd spent too long without a family and she couldn't stop herself, and before she'd gotten through her carefully prepared introductory speech he had started gibbering and wailing. His friends had apologetically dragged him off, with a practiced ease that made her think these breakdowns were common occurrences. Since then she'd followed his career, at a distance. Become something of a beam fights fan. Betting on him every time, even though he always lost. He'd lost in this fight, to that new guy Hao who all the boys at her office were crushed out on.
Probably Path was lying. Probably the whole sudden-moment-of-human-concern thing was a business strategy, beloved by fences everywhere, favored by criminals whose specific niches required a never-ending leap of faith on their clients' part. Ankit really wouldn't ever know.
At fifteen, she'd resolved to never visit her mother again. Her presence made her mother agitated, which might land her in several days of psychophysical therapies that Ankit suspected would not have been out of place in a nineteenth-century mental institution. That was why she never wrote to her, never called. The Cabinet hadn't come by its reputation by accident; there was a reason some of Qaanaaq's toughest criminals flinched at the memory of it.
But Ankit was happy, in spite of everything. Helping her mother out, even when she wasn't sure if the help ever reached her, made her feel good. Less helpless, less alone. The night was cold and dark and she would not have had it any other way.
She slowed alongside a knot of stalls where women sold rotgut out of flasks. Raucous, ageless women. Ankit had bought shots off them when she was sixteen, one at the beginning of a night of scaling and a second at the end of it. She bought one now, from her favorite, whose stuff tasted like apples and pine sap. The woman's smile said, _I have watched you, I have seen you at your worst._
"What's so crazy about it?" the vendor said to another. "All those animal workers, the ones who work with, what do they call them—'functionally extinct predators.' They make them get those shots so the things don't kill them. Like that boat out on Arm One where they got tigers and alligators still, for rich people to rent. Not such a big leap from that to something that would let you meld minds with a killer whale."
Ankit paused to savor her shot, and their conversation. Cold wind seared her skin, but inside she was a goblet of fire.
"My husband's friend worked for one of the juntas," another vendor said. "They all had their own secret unit they tried out all kinds of drugs on. He saw some shit, in Chile. Rumor was the Yucatán squad had a doctor, on contract from one of the North American pharma states, could inject you with something that let them hook your mind up to theirs, know everything you know. How else do you think they took down the narco government?"
"Narco government fell because of the rioting," said the third vendor, scowling skeptically. "Second Mexican Revolution. No magic beans or science fiction required."
"For all the good it did them."
They would go on all night like this. And Ankit could have stayed, buying more liquor or simply standing in the wind at the edge of the circle of the nightlamp light.
This was always here, waiting for her. Qaanaaq night. Her drug of choice. But she was a creature of the day now, and if she relapsed into her addiction she'd lose everything she had.
"See you next time," the woman said when she handed back her shot glass.
Her building's lobby was warm and bright, with that particular brand of heat the geothermal ventilation system provided—wet and slightly salty, or maybe that was her imagination. She slowed her step to appreciate the heat after being so cold, and then her jaw chimed. A surprisingly swift response; Fyodorovna could barely be bothered to do any work while she was in the office, let alone when she was out of it. Her voice filled Ankit's ear:
The breaks is toxic. There's a reason politicians won't go near it. People think it's just criminals and perverts. Whether or not that's true is irrelevant.
Ankit typed: _It's not true. And somebody needs to do something about it._
Fyodorovna responded: People are, I'm sure. Software is. Predictive is working on a plan of response. Scientists working on a cure. Something. Let the people whose job it is to worry about that worry about it.
Ankit knew her boss well enough to know that the conversation was over.
Upstairs, in her room, she pressed her forehead against the glass, looked out into the night, felt warm, felt bad about it. Turned her head to look in the direction she was always trying not to look.
It was still there, of course. It always would be. A sliver of building rising above the others in the distance: the Cabinet. The tallest building on Arm Six. Qaanaaq's psych ward. She'd scaled it, once. The only time she'd ever gotten caught. The only time she'd scaled something for a reason, to get inside, to get something out. To get someone out.
The last time she went scaling. The time that her fear held her back. Froze her solid. The black sea, so far below, the wind screaming into her, the building slick with frozen mist. They'd caught her there, rooted in place, and there'd been nothing she could do about it, and here she was now, rooted in place, still helpless. Still afraid. Still obeying the rules. It was Fyodorovna she feared now, not Safety, even though she knew they were both ridiculous, but the end result was the same. She was groundbound. She never took that leap, the one that made the difference.
And so: She did something stupid. Something she knew was stupid. Something she did anyway.
She took the photo of Taksa, which was blurry enough to not be identifiable as any particular person, but rather conveyed a very generalized idea of Happy Little Girl, and cropped out any background elements that might give away whose home it was, and autoqueued it to post the next day to Fyodorovna's channel, with one line of text— _If the breaks affects one of us, it affects all of us_.
And then, because she was still angry, she made another stupid decision. One she hadn't made in years. She opened her screen and navigated to the Cabinet site and submitted a visit request, to go and see her mother.
****City Without a Map: Boundaries
Do not talk about the past here. Do not ask your neighbor why they left wherever they are from; do not expect your newfound friends to wax nostalgic for homes that no longer exist. Perhaps the past holds more than merely pain for you, but you can't assume that this is true for anyone else. We want to smell it, taste it, hear its songs, feel its desert heat or summer rain, but we do not want to talk about it. The things we've been through cannot hurt us here, unless we let them. The fallen cities, the nations drowned in blood. The cries of our loved ones. Those stories we lock away. We will need new ones.
All cities are science experiments. Qaanaaq is perhaps the most carefully controlled such experiment in history. An almost entirely free press. Minimal bureaucracy, mostly mechanical, the city overseen by benevolent software. The shareholders pay the taxes, and they can afford to. If food and rent cost far too much, that is between you and the merchant. Swedish is the most common language, yet only 37 percent of the population speak it. There is no official language. There is no official anything. Qaanaaq has no government, no mayor. Those functions are fulfilled by a web of agencies deputized by Qaanaaq's shareholders. Each Arm elects a manager to serve a four-year term, to help citizens navigate the agencies and hold municipal employees accountable for bad behavior. These eight people are the only politicians in Qaanaaq, and they will be the first to tell you how limited their power is.
If the twentieth century was shaped by warring ideologies, and the twenty-first was a battle of digital languages, our present age is defined by dueling approaches to oceanic city engineering. Technologies developed for oil rig construction became fervently believed-in and fought-over doctrines. Conventional fixed platforms; spars; semi-submersibles; compliant towers; vertically moored tension leg and mini tension leg platforms. Standing on concrete caissons or long steel struts; tethered to the seabed. Ballasted up or down by flooding and emptying buoyancy tanks or jacking metal legs. Some are enlightened well-armed migratory utopias. Some are floating hells, like the plastic scrap reclamation facilities that ring the Pacific Gyre, every building and body blackened by soot from the processing furnaces.
In Qaanaaq, software calls most of the day-to-day shots, sets the protocols that humans working for city agencies follow. In theory a tribunal of actual human beings, appointed by the founding nations, could be appealed to in extreme cases, but to protect their anonymity even this process is mediated by software, prompting many to doubt the tribunal exists at all.
Some grid cities are less rigid about their boundaries. The Russian behemoth Vladisever set no limits on additional construction, and ten years later the city was an unruly metastasis. Hundreds of arms, impenetrably clogged waterways. In the end the army had to come in and clear it out, bomb the tangle of new structures, displace tens of thousands. Qaanaaq permits no more buildings, no new legs that reach to the seafloor. Tie any floating thing to it and you'll pay dearly for the privilege. These floating things, in turn, are free to charge others for the privilege of being tied to them, so that whole floating villages bob in the surf in spots, to be broken up or relocated when the agents of Structural Integrity decide they pose a danger.
One floating thing tied directly to the grid is the Sports Platform. A five-story boat the size of a soccer field, anchored at the far edge of Arm Four. Ice-skating rink on the top floor, exposed to the elements; every other story is a nest of subdividable courts and fields and tracks. The bottom level is three stories beneath the surface of the sea, and used mainly for training purposes.
These are things you need to know. There's a reason I'm drawing you this map, telling you these stories. Wherever you are now, no matter how trapped or hungry or scared you may be, these stories can provide an exit, an escape, a map to freedom.
Stories are how I survived, these long bound years. Now I can share them with you.
Like most modern sociopolitical entities—cities, states, nations—Qaanaaq operates a closed network. The Sys Wars that contributed so heavily to the old world's breakdown spawned a terrifying array of uncontrollable malware and parasites and infection vectors. Whole countries watched their infrastructure crumble not through the agency of foreign antagonists, but under the mindless attack of rogue rootkits and autarchical worms. Botnets whose authors were dead; scareware that infected Trojan horses to produce uncontrollable new monstrosities. The World Wide Web had proved a short-lived phenomenon. Limited global and regional networks existed, the flow of data so tightly controlled that they were almost unusably slow.
Qaanaaq's network, by contrast, is a glorious swirling sea of data, watched over by massive unpredictable mostly controllable AIs.
So many stories pass through it. I hear some of them, even in this place.
Here is one of them:
Eight days after the woman with the orca arrived in Qaanaaq, the killer whale was seen several times in the vicinity of the Sports Platform, which drew a dozen journalists to the sweat-and-popcorn-stinking structure.
They found her on the bottom level. Moving through a series of martial arts forms, wielding her bladed staff with such speed and skill that she clearly didn't need mammalian apex predators to keep her safe. The staff's handle was a thick long walrus tusk, precisely as the rumor mill had said, but its blade was stranger and more frightening than any of the stories had suggested. Huge, pale, curved, jagged. One reporter conjectured it was the jawbone of a sperm whale, carved and fretted and sharpened, and they all wrote it down as fact. The polar bear sat off to the side, hands and head caged. Journalists sat in the bleachers and smoked and called out questions.
"Where did you come from?"
"Why are you here?"
"Can we have an interview?"
Only the American stepped out onto the floor. For an American, the arrival of someone like her would strike a chord—collective guilt, most likely, over the bloody war that had been waged against her kind, or instinctive hate.
"Hello!" he called, moving closer, but she did not respond.
He stopped at the edge of swinging range. "I'm Bohr Sanchez," he said. "I run the _Brooklyn Expat_."
She said nothing, just parried at the air fifteen feet from him. She leaped, stabbed, dropped, and rolled. The men and women in the stands behind him laughed, idly evaluated the odds of his being beheaded before their eyes.
"Are you nanobonded?"
Here she stopped. She stared. She took a step closer.
Bohr bit back his fear. His colleagues were watching. "I thought you had been wiped out. You're not worried? About the people who tried to exterminate you? Qaanaaq has more than its share of zealots, you know."
She moved faster and faster, her actions increasingly impressive. And frightening.
"Is it true that your people eschew all forms of technology?"
"How many more of you are there?"
She executed a leaping swing, her weapon falling out of her hands. Frustration? Sadness at a sore subject? The urge to murder him? And here she said something, finally. A single inarticulate roar.
## Fill
Fill hurried down Arm One, heading for Arm Three. His errand had made him late, but it was worth it. In each hand he held a freshly minted plastic figure, the first he'd had printed in ten years. Still warm. A polar bear, and a killer whale.
He wanted to believe. In magic, in science, in people who could bond with animals. He wanted to feel the excitement he'd had as a child, how big and crowded and possible the world had been back then.
He didn't, not quite, not yet, but maybe the plastic figures would help.
He walked down the Arm's central strip. The walkways were too crowded and he'd lost the knack for leisure since his diagnosis. His unhurried stroll had been stolen from him. He'd debated taking a zip line, but the queue at the central Hub had been too long, even at the first-class line. And there had been a demonstration—angry people in plastiprinted _kaiju_ suits, chanting about a new wave of evictions out on Arm Seven, harassing the wealthy zip liners while they waited, because any of them could have been the shareholders responsible. Holographic monstrosities danced and scrambled in the air around them. When he was a kid, for a while, these had been everywhere. All the time. It had been years since he'd seen one.
Fill had recognized the buildings in the placard screens that they held up. He knew exactly who the shareholder responsible was.
_Oh, Grandfather,_ he thought. _You bastard. Whose lives are you ruining tonight?_
He couldn't worry about that now. Life was too short to linger on ugliness. Last week, _City Without a Map_ had talked about the crowds who gathered nightly at the end of Arm Three to watch the bright methane ventilation flares. Central Americans, mostly, with vendors selling some rich fermented purple beverage whose name he'd forgotten, and roving musicians playing a dozen rival species of _son,_ but the street festival was popular with all kinds of recent arrivals.
_This city contains so many cities,_ he thought. _So many lives I'll never get to live, so many spaces I'll never get invited into._
"Fuck out the way!" screamed a messenger kid hurtling down the slideway, and Fill's first thought was _My goodness, that boy is hot_ and then _Oh wait, that's a girl_ and then _Christ, I have no idea what that is,_ which led to a little internal rant about the complexities of modern gender among Qaanaaq youth.
Fill was still meditating on that, long after the kid had vanished into the green haze of the methane-sodium streetlights, when his jaw chimed.
_Unknown caller,_ said the gonial implant, a flea-sized thing affixed to the corner of his jaw, like those used by all but the most wretched recent-refugee arrivals. Phone speaker and receiver all in one, conducting sound through bone to his ear and also recording what came out of his mouth. His screen's expensive software did the rest: _Thede Jackson, age twenty-five, spatial designer, North American parentage, gay, single. No previous history of contact; no flags for unsolicited commercial messaging._
Gay, single, twenty-five; well, that was something. Maybe a friend had put them in touch, or he'd seen something about Fill and decided to take a stab in the dark. It had happened before. One of the perks of having a reputation. Fill tapped his jaw and said, "Hello?"
The voice said, "Ram?"
Fill said nothing. All of a sudden all the air was gone.
"Ram? Are you there?"
This Thede was drunk, and sounded like he was about to cry.
"Who the hell is this?" Fill said, surprised at his own anger.
"Ram, it's me," he said. "Please. I need—"
"This isn't Ram, asshole, as you goddamn well know."
Thede choked. "Um . . . what?"
"Did Ram put you up to this? Did you steal my handle out of his screen?"
"Ram, I—" But then Thede hung up.
_Love is the gift that keeps on giving,_ Fill thought—how else could a man still make you miserable even after you'd broken up? He took a moment to curse Ram and every wonderful horrible minute they'd spent together. The boy had been unstable, incapable of telling the truth about anything, ever, and whether Thede was another jilted suitor or someone Ram owed money to didn't matter.
A bell tolled, on one of the tidal buoys. Fill had never bothered to learn what any of them meant. There were nursery rhymes about them, _Good boys ask how / Say the buoys of New Krakow_ or something nonsensical along those lines. Every wharf rat and skiff urchin knew them by heart, could glean all sorts of helpful information about weather and the length of the day from them, but Fill, with the best education Qaanaaq money could buy, was stone-cold illiterate when it came to the complex tidal chimes.
Graffiti, on the side of the buoy—ORA LIVES, his screen translated from Spanish, and what could ORA be? The Qaanaaq net turned up nothing, and after ages a ping to the global net turned up a half dozen expired trade associations, dead softwares, villages and townships of the Sunken World. Someone's name, then?
Maybe the call had been a wrong handle, he told himself, because gay Qaanaaq was such a small world that it wasn't inconceivable that keying in a wrong handle would bring you the ex of the person you'd meant to call, except that his handle and Ram's were nothing alike.
Again, his jaw pinged. Thede again, but text this time. "Read," Fill said, and immediately regretted not saying "Delete." But life was like that. Dumb decisions you made in a hurry and then dealt with.
Ugh, I am so sorry. I'm drunk and disrupted and I don't know what happened. I entered the handle myself instead of autocalling and I don't know. I have no idea who you are so I have no idea how I got you. Except that I sort of do. Or I think I do. Call me?
Fill kept walking. He was almost out to the end of the Arm now, running late, and his whole head felt funny, fuzzy, like something had come unmoored, and of course it was too early for him to be symptomatic, it had to be psychosomatic, but how could you know the difference with a sickness that was solely psychological?
Another text. "Read," Fill said, unsettled.
Okay. You don't want to call. I get that. But if you know Ram, you probably know him like I knew him, which means you two were fucking, which means you need to know this. I've got the breaks. I think I got them from him. And they're moving fast. It's a bad strain. I've been having these moments. Like just now. I swear, I remembered your number, even though I never knew it, because Ram did. It's like I know—
"Delete," Fill said, though the message was less than a third of the way through, and ran the rest of the way to the end of the Arm. He turned up the latest installment of _City Without a Map,_ letting the vortex wash over him as an old man with a bad cough read someone else's words into his ear:
No one rules Qaanaaq, no class or race reigns supreme. Not the Chinese laborers who built it, not the global plutocrats who could afford to get out of their doomed cities before they became hell on earth, not the successive waves of refugees who filled it. And while of course it technically belongs to the shareholders, who lease the ground that every home and business in Qaanaaq was built upon and make obscene amounts of money with every minute that passes—they are invisible. After class warfare consumed the American grid city of New Plymouth and the rich were plunged burning into the sea, Qaanaaq's owners went to great lengths to conceal themselves.
To minimize unrest, the city founders broke with the urban past in several surprising ways. They decided against the repressive militarized police force that most old-world cities had depended upon. They kept the burden of taxation exclusively on the hyperwealthy shareholders. They limited the depth of democracy, giving politicians such small amounts of power that few fights broke out over elections or government action or inaction. They banned political parties, which had—in the view of the artificial intelligences that drafted these rules, anyway—been vehicles of mob rule and mediocrity more often than efficient strategies for decision making.
And so. Here you are. Here we both are.
Every so often, shut your eyes. Then open them, the slightest bit. Your home is gone, but it is not difficult to trick yourself into thinking you are home.
## Soq
Fuck out the way!" Soq screamed, and giggled at the prissy way the boy hopped. People like that thought they owned the whole Arm, and maybe they did, but on the slideway Soq was supreme.
Speed and wind made Soq's eyes water. They laughed, out loud, and the laugh became an ululation that lasted the rest of the way down.
Two ramps, side by side, ran down the center of every one of Qaanaaq's eight Arms. Ten centimeters wide, capped with miniature maglev tracks. One going out from the central Hub, the other coming in from the far end. Each one started out five feet high, and over the course of the Arm's kilometer length tapered evenly down to ground level. A five-foot incline wasn't much, but a human of average weight could get up to some pretty astonishing speeds. In theory anyone with a pair of slide boots could use the slideway, but most people had long since abandoned it to the messengers. Slideway messengers were a barely tolerated menace, widely considered to be bloodthirsty caff addicts on errands of minimal legality who wore weighted clothing to speed them up and razor blades affixed to their clothing to casually disembowel anyone strolling near the center of the Arm without paying attention. Most messengers relished this assessment and went out of their way to embody it.
Soq leaped, arriving at the end of the ramp. Slide boots were supposed to kill the magnetic repulsion as they arrived at the programmed destination so that you came to a calm, precise stop. Like most messengers, Soq had overclocked that function. Nothing was going to slow them down but them. Soq soared through the air, clearing the buffer zone and landing with a jolt beside a bench in the central Hub.
A man whose head Soq had just passed over yelled, "Watch it, you—" and then paused, unclear how to gender the insult.
Soq was beyond gender. They put it on like most people put on clothes. Some days butch and some days queen, but always Soq, always the same and always uncircumscribable underneath it all.
_New job,_ whispered their implant. _Arm One. Number 923. Envelope; paperwork only._
"Assign it to someone else," Soq said. "I'm late for dinner with my mother."
"You don't have a mother," said Jeong, the on-duty human for the day, interrupting the automated dispatch software.
"I'm sure I do," Soq said, weaving through the evening foot traffic of the central Hub. "Everyone has a mother."
"So they tell me. But you're not having dinner with yours tonight."
"I'm tired," Soq said. "I can't deal with another run past those Arm One assholes."
"Fine," said Jeong, who truly did not care one way or another. He had been a messenger himself, before some kids experimenting with targeted mini-EMP software accidentally scrambled his boots and locked him in place while speeding down Arm Six at three hundred kilometers an hour, which dislocated both legs and fractured his pelvis. Now he lived vicariously through his "kids," including the ones who were older than him. "But listen. Got a complaint from your last run. Lady said you were extremely rude."
"Of course she did," Soq said. "Old nasty American thing, she didn't like it when she used 'he' to describe me and I told her, very politely and patiently, because I have this conversation way fucking more often than anyone should fucking have to have it, that I prefer 'they' and 'them' pronouns."
"Weirdest thing. My screen conked out as I was in the middle of logging the complaint. Didn't save to your record. I'll have to get that thing looked at."
"Thanks, Jeong. Anything from Go?"
"Nothing," he said. "You know I'll let you know as soon as I hear."
The dispatch line silenced itself. Soq pressed their gloved thumb to pinkie, and the slider boots demagged. An evening crowd ahead, a couple hundred humans passing through the central Hub, mostly from the labor arms out to the residential ones, although like everything else in Qaanaaq the flow of people was complex.
A bad idea, Soq knew, to get so focused on Go. Didn't mean anything that the woman whom many believed to be the city's most powerful crime boss had sent her first lieutenant to hire Soq out for some stringer gigs. Deliveries, tails, handoffs, creating commotions to help someone escape someone else. There were no promises in this line of work. Soq was a good slide messenger, and a good messenger had many of the qualities that made a good underling—speed, fearlessness, lack of respect for the law—but that didn't mean Go would ask Soq to formally join her organization.
Getting crime boss patronage was like winning the lottery: it'd be great, but it probably wasn't going to happen to you, so stop banking on it.
_Stop thinking about it. Stop imagining the day when you will be the crime boss, the feared one._
A straight line, from Arm One to Arm Eight. An irony that never failed to amuse Soq, how simple it was to pass from gorgeous luxury to huddled crowded filth. _How else would it be so easy for them to suck us dry?_
Arm One always did that. Made Soq bitter. Angry. The anger usually ebbed a bit on their return to Arm Eight. Just seeing the banner above the entrance was soothing: four of the hundred or so Mandarin characters that Soq and every other Qaanaaqian knew—新北希望, Xīnběi Xīwàng, _New Northern Hope,_ the name the Chinese sponsors had given to their new city, just as the Thai and Swedish had done, three names for a city and none of them stuck, because the people wanted something else, something not beholden to anyone, and so the Inuit name for a Greenland coastal village that had just been swallowed up by the sea became the name of the floating city as well. Qaanaaq II, at first, or Q2, and then just plain Qaanaaq as its predecessor was forgotten.
The smells of food hit Soq first: broth and basil, mint and trough meat. Food stalls were forbidden in several of the better Arms, and they were the primary draw for the outsiders who dared enter Eight. After that it was the crowd that calmed Soq down, its rhythm, the peculiar intangible atmosphere of energy and ease. They didn't know Soq, these people, but they knew Soq belonged. Who knew how they knew it, but they did. Just like Soq knew who was and who wasn't of the Arm. This was home.
People, stacked everywhere. Sleeping capsules piled between buildings, strapped to the struts that supported the bigger ones. They looked sad and ragged, but they belonged to royalty, relatively speaking. Prime real estate; the best spots, held down since the days when Arm Eight looked like Arm Six looked now. The women and men who lived in them were the eyes and ears of the Arm, essential elements of the commerce in information.
Farther down the Arm, Soq entered the shadow of the tenements. A twenty-year-old attempt by the Qaanaaq shareholders to provide stable indoor housing for the unfortunates of Eight, these massive buildings formed the densest population pocket. Denser than Kowloon Walled City or the South Bronx Boats or anything else the Sunken World had produced. The outsides thickly vascularized with red, black, and green pipes. Most of the families, soon after taking up tenancy in these halfway-comfortable spaces, began building walls and partitions and hanging sheets and anything else to rent out space to their neighbors. In the tenements, it was not unusual for fifty people to live inside one apartment. Soq had had friends in the tenements, come to birthday parties for them. Seen the incredible resourcefulness of the tenement dwellers: city plumbing lines "augmented" with snarls of new pipes to shuttle water and sewage and heat, the spliced power lines, the informal stairs and passageways, the sweatshops where grannies made fish balls or repurposed circuit boards in a room where ten iceboat workers slept. And even these were relatively privileged positions, held to fiercely by families and the crime syndicates, who paid the shareholder fees as a gesture of goodwill to the residents.
Three scaler kids cawed at Soq. They squatted in the struts of a building, feeding a ragged-looking monkey. Soq slowed, trying to see whether their intent was hostile, and then decided they did not care. Mostly scalers and messengers got along well, and lots of people were both, but some scalers could get nasty with the "groundbound" grid grunts they felt so superior to.
Three-quarters of the way out the Arm, Soq stopped at a noodle stand. This one had little stools and a tarp on each side to keep some of the wind off. Soq was swallowed up in clouds of hot steam smelling of five-spice powder. Home was noodles. Home was food and warmth. Soq paid, took a stool, shut their eyes, and meditated on the moment, its beauty, its peace. The coldness of the wind and the warmth of the food and the fact that everyone eventually dies. Letting go of everything they did not have, every ugly thing they'd seen, every moment of pain they'd felt that day, the day before, every day to come.
Soq smiled in the rising steam.
"Hey, Charl!" they called when the first spoonfuls of soup had reached their belly, and pointed chopsticks at the greasy screen Charl had hung on a lamppost. On the news, nine military factions had submitted claims for international recognition as the legitimate successor state to the American republic. Some were as small as twenty people in a boat. "How long do you think it would take a tri-power boat to travel from where the flotilla was to Qaanaaq?"
"'Bout a week, I'd imagine," Charl said, for Charl loved these logistical problems. "Unless they had a ton of gas, in which case it might only be a day or two. Why—you want to go scavenge?"
"No," Soq said. "Just wondering when we might start to see refugees."
Soq was thinking of the orca. Of the woman who'd mysteriously arrived with a killer whale in tow, so soon after the flotilla bombing. Because how had the flotilla been bombed in the first place? The American fleet had lacked a lot of things—food, shelter, fuel, civil liberties—but it hadn't lacked weapons. The global military presence that had made the pre-fall United States so powerful, and then helped cause their collapse, had left them with all sorts of terrifying toys. The battleships that circled those four hundred pieces of floating scrap metal would have had solid perimeter defense capabilities . . . but perimeter defense might not have been concerned about a killer whale.
And if anything could get past the innumerable aquadrones that protected Qaanaaq's geocone—that engineering marvel, which captured the massive energy expended by the thermal vent and used it for heating the whole city, providing the power for its lights and machines—it was an orca. If anything could get an explosive onto the cone, it was an orca.
Soq slurped down the last of the noodles, lowered the bowl, let the wind hit their face. Crossed the Arm to where the open ocean lay.
Two hundred thousand people on Arm Eight, if you believed the official statistics, which Soq didn't, and half of them fantasized daily about watching Qaanaaq sink beneath the waves. The other half dreamed of conquering it. A wonder it had lasted as long as it had.
Soq wondered which one they were, and decided they were both. _I will dominate this city or I will destroy it, break its legs, send it sinking into the burning sea we stand over_.
Soq stepped down into the only sleeping boat with any openings. The woman who owned the flat-bottom smiled, recognizing Soq, and held out her hob for screen scanning.
The boat was mostly square, not built for moving. It had been anchored in the same place for twenty years. On its surface stood ten rows of ten boxes. A square meter and a half, with wooden sides covered in canvas. Qaanaaq's poorest slept in boats like this, their bodies folded up uncomfortably, with a breathable lid they could pull over themselves to keep the worst of the wind off and hold body heat in. Soq was lucky to be short. Taller people had a hell of a time in the boxes.
Soq sat. The box was dank and wet. It smelled of the cheap spray the owner used when its previous occupant vacated in the morning, but underneath that Soq could smell the funk of the occupant himself. They'd had to choose between being hungry in a capsule hotel or sated in a box, and had chosen the bowl of noodles.
They were happy with their choice. They were angry that they'd had to make it at all.
The bitterness started to come back. What had been an idle imaginary scenario five minutes before became a deep and fervent desire. _Fuck them, all of them, the people who make us live like this. Who sleep in beds with whole meters of empty space around them._
Blow up the geocone and the city would be uninhabitable by nightfall. Soq stared into their screen, scrolling through photos of the cone from concept to execution to periodic well-publicized repair work. Prickly defensive columns of polymerized salt; thick swarms of weaponized aquadrones. Two hundred miles of pipes in and around the cone alone, to say nothing of the ones that took water and heat through the twisted tangle of the city above. A million valves for releasing heat and pressure. Dynamic responsive systems to match surges and ebbs in demand.
Such a marvel; such a target.
Soq fell asleep like that, in the fetal position, knowing their knees would ache in the morning, smiling to the imaginary sound of a million people screaming for help as they drowned.
_Destroy this city, or conquer it—which one would I prefer?_
## Fill
The purple drink did not contain alcohol, to Fill's surprise, but it seemed to make him drunk all the same. The sweetness of it, maybe, the taste of cloves and corn, or the night itself, the crowds, the cold wind and the music, so many musicians that he felt like he was standing in the center of a scattered symphony orchestra, their separate melodies adding up to something, some futuristic form of music with no beginning, no end, no structure, only a thousand gorgeous pieces crashing into each other.
_City Without a Map_ had not steered him wrong. This place was electric, alive. A beautiful boy, seated on a rough and brightly patterned blanket, caught Fill staring. He smiled, a complicit and friendly smile that somehow underscored Fill's outsider status.
_Boom._ A memory. Triggered by that boy's pretty face. Pornography, one of the first clips little Fill ever saw, a similarly pretty face, eyes closed, face upturned, waiting, frightened—brave—excited, and then a spray of semen across his cheeks, another, several. The memory-boy smiled, laughed.
Fill flushed, embarrassed, unsure whether anyone could see his half erection in the darkened crowd. He scanned the crowd—no one looking in his direction but a well-maintained and very old man at the grid's edge.
_Boom_. Another image flashed in his mind's eye. Another pretty face, seen from above, turned up. A total stranger, someone Fill had no memory of, although that meant little considering how much he'd done while out of his mind on some combination of alcohol and drugs and actual ecstasy, and anyway, Fill was usually the face being sprayed, not the one spraying the face. Not pornography, either—too real for that, too vivid, with smells attached, and sounds, the roar of a distant train, a subway car, but what the hell, there were no subways left . . .
_Boom_. Children looking down from the windows of burning buildings.
_Boom_. Soldiers shooting crowds that tried to breach a roadblock. Broad, bizarre cityscapes rising in the distance, the original Shanghai, the vanished São Paulo. Not dreams, not hallucinations. Memories. Someone else's memories.
Fill began to cry. He cried for a long time.
"Hello," said the old man, speaking New York English. At some point, he had come over to stand beside Fill. "I'm sorry to intrude. Let me guess— _City Without a Map_ brought you here."
Fill nodded, then regretted it. He squeezed the last few tears out, gathered his wits, searched for something suitably devastating to slay this sad troll with. But the troll had placed his hand on Fill's, was looking into his eyes, was opening his mouth, and before Fill could hiss, _Get the fuck out of my face you disgusting ancient queen,_ the old man said, "It's the breaks, isn't it?"
Which made Fill burst back into tears. He nodded, and the man leaned forward to hug him. The hug was kind, grandmotherly, sexless.
"Forgive my presumption. But I had many similar experiences, early in my diagnosis. And I know how alienating it can be. My name is Ishmael Barron. Most people call me by my last name. It is so much more dramatic."
"Fill." Shaking the man's talcum-powdery hand, he felt afraid he might break the thing. And did Barron's eyes widen from pain, from lecherous pleasure, from something else entirely? "It felt like memories. So vivid. But nothing I experienced. Stuff that happened a long time before I was born. How is that possible?"
"Them's the breaks," the old man said, and smiled, so Fill figured there was probably an old joke he didn't get in there, so he wrung out a laugh. The length of the man's ears was truly preposterous, and his nose seemed superhuman, as though extreme old age was sucking life and skin away from the rest of his face and flooding those places. "Talking helps. I had no one, when I was diagnosed. Do you want to get a _caffè alghe_? Talk about this?"
"We can get real coffee," Fill said. "I hate that toasted algae stuff, and I don't mind paying."
Barron shrugged. "Well okay, Your Excellency."
"Think nothing of it," Fill said, standing up straighter, because manners were easy, this is what they were there for; the genteel affectations of wealth were a suit of armor you could wear when the world threatened to wash you away.
"IT'S PROBABLY BEEN FIVE YEARS since I tasted this," the old man said. "And before that it was probably another five years."
"We'll do this again in five years," Fill said, knowing neither one of them would be alive then.
"Would you believe I used to have five or six cups of this a day?"
"Wow," Fill said, barely hiding his boredom, because he hated when old people talked about How Awesome Everything Used to Be. _Yeah, but you also used to die from cancer and get hangovers and spend your whole life unable to understand 80 to 99 percent of the people in the world when they spoke, so good luck with that nostalgia thing._
But he couldn't hold on to any anger, not for this man, his baby-pink skin and senile good cheer, and Fill liked speaking New York English, it made him feel close to his grandfather. Above them, at street level, a rather delicious dark bearded thing caught Fill's eye. _He must think I'm a rent boy,_ Fill thought with a flush of pride, _paid by this old thing to whip him bloody or sit naked by a window or something_.
"I believe the breaks have been around longer than people suspect," Barron said. "Fifteen years, twenty. It went unnoticed, or was mistaken for schizophrenia or adolescent-onset Alzheimer's because the symptoms are psychological. My theory is that it's not one disease but several, originating in a number of different locations, and when one person becomes infected with multiple strains, a new, hybrid strain is formed. Far stronger than the two that created it."
_I think I got them from him. And they're moving fast. It's a bad strain._
Fill frowned into his coffee, remembering that stranger's messages, the boy who said he'd gotten the breaks from the same person as Fill. Although that was crazy. How could this random stranger know such a thing? It's not like Ram had infected him with information along with a disease. And yet—he had known Fill's private handle.
"I think . . ." Fill said, but didn't know how to say it, certainly not in the language he rarely spoke outside of his home. _I think I am remembering things that happened to other people._ "Have you ever . . ."
The bearded man from before had come into the coffee shop, was sitting on a bench. Not buying anything, of course. The kind of person Fill was most attracted to could never have afforded anything in a place like this.
"You've an admirer," Barron said. "I daresay you have many admirers." He seemed quite transfixed by the new arrival, and occasionally turned from him to Fill and smiled deeply, no doubt at an imagined pornographic tussle between the two young men.
"Some people think that the visions you get from the breaks come from the person who infected you. Or the person who infected them . . . or someone somewhere along the chain. Is that possible, or am I crazy?"
"Yes," Barron said, stirring his coffee unnecessarily, seeming to have moved on from that topic of conversation altogether.
"Yes . . . it's possible?"
"Oh, look!" Barron said, barely listening. "It seems that I have frightened your admirer away. I have that effect on young men these days. Punishment for my own youthful cruelty, I believe. We always get what's coming to us, darling. What is your last name, by the way?"
Fill breathed out, promised himself patience. "Podlove."
Barron's eyes widened, and his jaw dropped, but who could say whether that was an involuntary old-man response, or whether the doomed ancient creature wouldn't have been equally shocked by a last name like Wang or Smith, or the fact that tomorrow was Thursday. "Podlove," he repeated, eyes narrowing. "An interesting name. Was it ever anything else?"
"Something egregiously Slavic, I'm afraid. Grandfather castrated it before he got here. Chopped off a pair of low-hanging syllables. Is it possible that the breaks—"
"At any rate," Barron said swiftly, "you must have things to do. Pursue that swarthy fellow, perhaps. I hope you won't fault me if I sit here a little longer? The legs, you know. They betray one so bitchily as the years go by."
"Of course," Fill said, standing, feeling weirdly like he was being dismissed, when it was his money that had bought them the coffee that earned this ridiculous specimen the right to occupy that space in the first place.
## Ankit
The monkey stared at Ankit through the glass. A Kaapori capuchin, one of the smaller, tougher tribes of feral monkeys that had fled captivity in the gilded cages of Qaanaaq's wealthiest. This one had a broad blue stripe down its back, tapering to a point between its eyes, where the owner had chemically seeded the skin's melanin layer. Ankit had made the mistake of feeding it once, and it must have smelled her weakness, her kindness, because it kept coming back to her window. And because she was weak, because she was kind, she kept feeding it.
Her screen would not stop chiming. Dawn; she'd been up all night; she'd long ago stopped responding to the incoming messages from subordinates, donors, friends, and foes. None from her boss herself, of course, but once Fyodorovna waddled into the office Ankit would be hearing from her.
The monkey stepped in through the open window and sat. She gave it three seaweed squares, which it dropped indignantly into the sea below, and a string of seal jerky, which it rolled into a ball and stuffed into its mouth.
Fyodorovna's campaign was in free fall. Her opponent had seized on the Taksa photo Ankit had posted, generated a script, talking points, sent them out to faith leaders and tenant association presidents and anyone else on Arm Seven who had half an audience, so now her feed was full of assholes squawking about how _Fyodorovna cares more about criminals and perverts than our hardworking families; Seven needs an Arm manager who will work for us, not for the twisted individuals whose bad choices brought the wrath of God down upon them_. For an entire hour she'd watched the bot poll scores plummet, and then she'd put her screen under a pillow.
Ankit had anticipated that this might happen. The possibility had flickered in the back of her brain, briefly, before she posted the photo. She'd considered it unlikely—and in that moment, in her anger, at Fyodorovna, at the breaks, at the fear that had always held her back, at the whole shitty city and the rich and powerful people whose ignorance shaped the lives of so many people and whose asses she was compelled to unceasingly kiss, she had posted the photo anyway.
Minutes after the office opened, Ankit's screen chimed. A subordinate, one she was fond of, who never bothered her with bullshit. She accepted the call and there he was, looking stricken.
"Don't ask, Theerasul," she said. "I'm sick."
"You're not," he said. "But I don't give a shit about that. You have a visitor. She says she has an appointment with you?"
"She's lying," Ankit said.
"Of course I know that. But it's one of the ones you handle better than any of us. I always end up pissing them off or making them—"
"Put her on."
A crabapple face filled her screen: Maria. Ankit winced.
"God bless you," the creepy woman said.
"God bless _you_."
Maria had a storefront church halfway down the Arm. How she held on to it was a mystery to all. She rarely held services, and when she did they were not attended. By anyone. Ankit had two guesses as to why the deranged old fundie had an unused space as her own private kingdom when it could have been making somebody tens of thousands a month in rent. Either the place was a forgotten holding of one of the former American megachurches, which had bought up property at the height of their power and wealth and then promptly dissolved, its rent paid in full for the next fifty years and Maria sent to be the pastor of a flock that would never come, gone mad awaiting further instructions and resources . . . or she was a maintenance squatter, paid by a shareholder to occupy a space they intended to keep empty. Those were usually thugs or illicit businesses, people who could pay and still be ejected with no notice—but if you wanted to keep a space vacant long-term without attracting Safety's attention, you couldn't do much better than a pit bull religious fanatic.
"Evil has come to Qaanaaq," the weird old woman said.
"I'm sorry to hear that," Ankit said, smiling. Fyodorovna never let them say even the most remotely negative thing to her constituents, no matter how crazy or malevolent they were. God forbid some fundie lunatic decide she hated her, start putting up flyers, spouting web hate.
"I want to know what she's going to do about it," Maria said.
"Did you submit a notice on her page? You know she takes constituent notices very seriously."
"I will do so," Maria said. "And then I will come back to find out what she's going to do about it."
"Great," Ankit said, wondering just what evil had stirred up the wasps in the woman's head this time. Islamic or Israeli refugees bearing insidious infidel ideas, most likely, or a fishmonger selling a new splice animal that a fundie sect had decided violated some weird sentence in Leviticus. Certainly not the breaks. Fundies didn't care about that. Most of them welcomed it. Ankit thought of Taksa, blurred and happy and doomed.
Against her better judgment, Ankit asked, "What kind of evil?"
"That woman. That abomination. Who has wedded herself to Satan. Who rejects the dominion God gave us over the animals and uses witchcraft to merge with them. A killer whale and a polar bear. They have come here to kill, to hunt down decent Christians, and they must be stopped."
"Of course."
Ankit clicked off.
The monkey stood, screeched affably, and leaped fearlessly into the abyss. Ankit whimpered; reached out as if to save it. But the thing was safely scaling her neighbor's window frame, scampering off to wander its city.
## Kaev
Kaev walked in the direction of the vortex. Even with the windscreen, Qaanaaq's gusts could be extreme, forming sudden swirling gyres that could push people off balance, stop them in their tracks, make a single step impossible. But one of Kaev's favorite leisure activities was to walk with the wind, submit to it, let its violence and sudden shifts dictate his path. On reflection, he'd come to recognize it as an extension of the pleasure he took from fighting. The thrill of submission, of abdicating control, of letting the mind with all its capricious insatiable demands fall away. And it was good training. It took agility, dexterity, to keep from smashing into any of the people struggling against the wind. It took wisdom to know how and when to yield.
This was a good day. A strong wind, but not too cold. A belly full of noodles. No fights scheduled, but he was heading for a meeting with Go. She'd give him his next assignment. His money would last for two more months; as long as the fight came sooner, he'd be fine.
He'd always been fine. Somehow.
And then, boom. His good mood gone. Who could say why, something he glimpsed from the corner of his eye, a bratty child screaming at his mother, maybe, or a baby in its father's arms, but it was always something, some reminder of what he didn't have, what he would not be, and his mind seized hold of it, spun it out in a dozen directions, nightmare scenarios of suffering and pain, things that were probably fantasies but what if they were memories, glimpses he'd held on to from infancy, things he'd seen . . .
A woman interrupted the stream of hateful thoughts, but he was not grateful. She wore a ratty fur coat and a giant Russian-style fur hat and she was peering into his eyes, and what right did she have to make eye contact with him? He flinched away as if from an electric shock.
"You are American, no?"
"I'm—" he said, frightened, because who was she, why had she spoken to him? Did he look American? What did an American even look like? _Was_ he American? "I'm." And then a gibber, a bark, some loud panicky succession of syllables he couldn't control, which made her flinch, which is what it did to everyone.
"Our people are in danger," she said. "Evil has come to Qaanaaq. I need strong men. Men who are not afraid."
Kaev wanted to laugh at the anachronistic use of the word _men_ to mean _people._ But he could not laugh. _I'm not strong,_ he wanted to say, _and I'm afraid._ All that came out was "I'm," several times in quick succession.
"She is hunting us," the old woman said. "That's why she came. The woman who brings monsters. She blames us for righteously trying to wipe her sinful kind off the face of the earth."
The orcamancer, then. His heart swelled, thinking of her. And her polar bear, hands and head caged. A fellow fighter had shown him photos, in the gym, a couple of nights ago. From the lowest level of the Sports Platform. She was real. She was in his city. He'd go to see her, soon. Not that he'd have anything in particular to do or say when he got there—he just wanted to see her for himself.
"Will you help me?"
Kaev fought the urge to yell at her, scold her, threaten to feed her to the orca himself, but his helpless years had taught him patience. She touched his sleeve. "God loves you," she said. "Do you know that?"
Kaev nodded, because that was the expected answer, that was what fundies wanted to hear, but he didn't believe God loved him. Quite the opposite, actually.
She pressed a scrap into his hand. "My church," she said. "Come? Tomorrow night? We need you. Ask for Maria."
And then she left, and he was grateful, except that now he felt bad for her, this sweet, sincere, deranged old woman, alone in a city where no one cared about her god, on a mission to destroy something beautiful, and if she thought that strong men could do something about the evil in Qaanaaq then she was even stupider than Kaev was.
"You're late," said Go's first lieutenant when Kaev arrived at her Arm Five floating headquarters. Dao; tall, thin, levelheaded. He handled her strategy and planning, the big-picture stuff. Kaev liked the guy, even if he was an asshole. There was something wise about him, something calm. More pleasant than the lieutenants who headed up her operations, security, intelligence.
"Delayed. I got. I got delayed."
"By what?"
"The wind."
"Idiot." But he said it affectionately, and stepped aside for Kaev to ascend to the boat.
The thing was big, a tramp steamer long out of commission, its side emblazoned with the name of a corporation and a city, neither of which existed anymore. Go and her operatives ran Amonrattanakosin Group out of it, and they lived there, and they used its cargo hold for storage. And paid the hefty priority docking fee, which came with a guarantee that Safety and Narcotics and Commerce would only ever attempt to board in the most egregious cases. Qaanaaq's whole hands-off approach to law enforcement had been successful in minimizing crime syndicate violence, but it had also allowed the syndicates to amass significant influence and legitimacy. Kaev reached the deck and turned around to look down, at his city, the Arm he'd left behind, and wondered what would happen when people like Go decided they wanted more.
She'd fought hard enough to get where she was. He remembered the horror of her rise. Even back then, when they were together, she'd had enemies. People she wanted out of the way; people determined to dismember her. Stab wounds she ended up with. Weeks when she had to disappear.
One woman in particular: Jackal, real name Jackie, but don't ever let her hear you use it. A runner, like Go, with her eyes as set on climbing the ladder as Go's were. Whatever happened to Jackal? Or the better question: how, exactly, had Go destroyed her?
"Darling," Go said when he reached the bridge. She embraced him. He wondered if she knew how he felt about her. How much he hated her. She must have. She was too smart not to. "A magnificent fight the other night."
Kaev yipped accidentally, then paused until he could collect himself. "Kid's good."
"He'll become something special," she said. "People love him."
"Means money."
She raised an eyebrow. "And who knows. Maybe someday he'll see you, remember you, buy you breakfast."
Kaev winced. He had made the mistake of telling her, once, when one of the kids he'd lost to who'd made it big ran into him on the street, treated him to a fancy meal, shared his disruptors, started crying, telling Kaev how he owed him everything. The problem with Go was that she knew him too well, knew how he felt about things. That sense of people made her a good crime boss, and a terrible ex.
"Fight," he stuttered out. "You have anything for me?"
"Sort of," she said. She went to her cabin's front porthole, looked out onto the deck. Exactly like a captain would. She was dressed in drab green, Kaev's favorite color. He wondered if that was on purpose. The machete scabbard hung from her belt, as always. No one had ever seen her use it. Kaev knew she wouldn't hesitate; wondered if she'd used it on Jackal. "Dao has two names for you. See him on your way out."
"Names. Names?"
"Business rivals," she said. "I'm not going to lie to you to spare your feelings, Kaev. You're a grown-up, at least in body you are. I need you to soak them."
Kaev felt very close to crying. He said "I'm" several times, and then finished in a rush: "I'm not a thug, Go, I'm not going to go rough up your enemies for you. I'm a fighter and I've made my peace with doing your dirty work in the ring, losing fights I know I can win, training young punks so you can make more money on them, but I'm not going to throw somebody into the water because of some business deal you need to get done."
At least, that's what he tried to say. He was pretty sure he got all the words out, and maybe even in the right order, but probably too fast for most of them to make any sense.
She patted his cheek. "Oh, Kaev. My noble warrior. I know this is hard for you."
"Me?" he asked. "Why. You have people. Lots. Who do this. Do this kind of thing. Better at it than I'd be."
"True," she said, "and lots of them are ex-fighters. You're getting old, Kaev. You know this. Couple of years, you won't be able to put on a convincing show in the ring anymore. And then what? If you can do this job, and do it well—well, then, you have a whole new career opening up ahead of you."
"And if I refuse? Or. If I mess up. Because I don't know how to do this thing?"
"Then good luck with your life," she said, her back to him, the conversation over, no point in arguing. He'd stared at her back like this before. "Needless to say, you'll never have another beam fight again. Or unlicensed skiff brawl, or snuff film knife fight, or anything."
Dao beamed him the details when he descended from the deck. And, because he really was a good man under all the assholishness, he didn't comment on the wetness beading up in the corner of Kaev's left eye.
## Ankit
Ankit's job was as good as gone. Her boss would lose. There was nothing she could do about it.
There were things she'd never done, favors she'd never called in. Agency executives, city flunkies who could help her. Who could access and share information they weren't supposed to access and share. Requests she'd always been too cautious to make, stockpiling them for the day when Fyodorovna would really need them.
_When I get back to the office, I'm going to start calling in those favors._
In the meantime, Ankit read everything. Every news site and academic journal, every forum, every crazy analysis from every point on the political spectrum. She even scoured through those _City Without a Map_ broadcasts, which she'd avoided because it seemed like absolutely everyone was talking about them that year, and it made their voices get high and excited and agitated and she didn't need that in her life.
If someone discussed the breaks, Ankit devoured it.
Which made her seasick. No two sources, it seemed, were discussing the same disease. Some of the symptoms remained mostly the same, but this was the only thing approaching consensus. Where it originated, what it meant, what it did beyond the psychological consequences, how to treat it—these were all the subject of a dizzying degree of difference of opinion.
The breaks was God's wrath, raining down upon the nations whose hyperactive economies fucked up the planet.
The breaks was God's wrath, inflicted upon immoral sinful subpopulations.
The breaks was big pharma, accidentally unleashing a monster when a handful of separate covert drug testing schemes unintentionally overlapped.
The breaks was a lie, a myth to keep people distrustful and angry and fearful of each other.
The breaks was a lie, a myth to distract from something far worse that was on the horizon.
She was pissing her job down the drain. She knew this.
Because the bottom line was: The breaks remained a phantom illness. A media mirage. Glimpsed in hazy pieces. Something no one could approach, capture, present, discuss, deal with. Argued about by foreign governments, rejecting allegations that their military labs or foulest slums had spawned it, and by insurance companies trying not to pay for treating the symptoms. Ignored by most other power players. Everywhere she looked, local software was "still collating." Official responses were "still forthcoming." If she could force it into the public consciousness, make it into a serious issue that Health had to address, she would lose her job—but she could save so many lives she wouldn't care about being unemployed.
That's what she told herself, anyway.
The six-month processing glitch had proven to be a fruitless avenue. Protocol rationale requests submitted under the Open and Accountable Computer Governance mandate turned up nothing. A couple million lines of code she'd never have been able to parse, that none of her contacts could penetrate, either.
Ankit listened to 'casts, left the apartment, started talking to people. Plenty of others were just as obsessed as she was, and had been for a lot longer. Accumulating information wasn't enough when it came to getting to the truth of the breaks. What she needed were real human minds, tics and madness and all, to turn all that data into stories.
At a support group, she met Janna, whose brother Mikk had been a sex worker in the Calais refugee camp. Janna clutched a photo in a frame, the first Ankit had seen in years. The boy was beautiful. Dark, smiling, tattooed, laughing at an inexplicable actual raven perched on one strong extended arm. He'd loved the work, Janna said, and been very popular with fellow refugees and camp workers and aid administrators and visiting photographers and pastors, complete with a sliding scale that even the richest of men were all too happy to pay, and the money he made that way had enabled him to buy Qaanaaq registrations and a halfway decent apartment lease for their whole family.
"Mikk was proud," Janna said. "He hated that we had lost our home, and that we were living in such a dangerous place. But sex let him rise above all that. Sex was how he became something more than just a refugee."
And it was only once they arrived in Qaanaaq, and were safe and stable, that the breaks began to manifest. Janna believed they had been there for a long time, kept in check by the force of Mikk's magnificent pride and determination to get his family out of Calais, and that once he was able to lay down his burden they blossomed.
For Mikk, the breaks brought him back to the camps. But not Calais. Taastrup—a Danish village, somewhere he had never been. At first she thought he was fuguing into stories he'd been told, things his johns had said. But the details were too complex, his fevered mumbling too disturbing, for such a simple explanation. Janna tracked down photos, found the same things he'd been describing.
Eventually Mikk broke. Died.
Bodybreaking, they called it. What happened when the breaks finally killed you. The moment when your mind's hold on the here and now finally ruptured forever and you broke free from your body.
Taastrup. The place popped up again and again in her research. Gone now. One of many that became the target of nationalist mobs, latter-day skinhead armies angry at all the workers landing on their shores. An estimated fifteen hundred people took part in the torching of Taastrup, a pogrom to rival anything in czarist Russia.
Looking into Taastrup took her to Ishmael Barron. He wasn't the first researcher she'd met who was clearly suffering from the breaks himself, but in him they were further along than she'd previously seen. Midsentence she could see it happening, watch his eyes as one train of thought was abruptly replaced by another. But each time he smiled, as if no one vision was more welcome than the next.
He unfurled a sheet of paper big enough to cover the table where they sat. She could not help but touch it, rub a corner between her fingers.
"Five or so years ago, several years into my research, I started to see the strands come together. Separate story lines uniting into one. The story of the breaks. Nothing like paper to paint a picture. I tried to put it all down. I am not sure if I succeeded."
A map took up half the sheet. North America, the Arctic, Greenland, Northern Europe. Bloodred triangles for the grid cities; blue circles for the refugee camps. A tangled nest of many-colored lines. The rest was a dense sea of scribbled words, angry ellipses, accusatory question marks. Exclamation marks like multiple stab wounds. He talked her through it, highlighting reported cases and potential trends, tracing what he believed to be separate strains with different symptoms.
They were in his apartment. A tiny place, but on Arm Five, so the building was clean and seemed safe. He was lucky he had come so long ago.
"What's this?" she said, pointing to a fading orange line that meandered and circled around North America before ending in Taastrup.
"Exactly!" he said.
". . . Exactly what?"
His finger traced the orange line, trembling. "That's the most important question you could have asked. What's your best bet?"
"Comes up out of the USA," Ankit said. "Some nomadic refugee group. People from one of the Black Autonomous Zones, after they got pushed out of Detroit or New Orleans or something?"
"No," he said. "Nanobonders. Taastrup was the last known location of any of them—a small handful that survived the final massacre went there, but, surprise, there's fundamentalist lunatics everywhere, and they got butchered, too."
"Last known nanobonder location until our friend with the orca arrived, that is."
"Yes."
He scanned her face, his eyes wide, his mouth open. "I think they are the key," he said finally, and if he was disappointed that she neither arrived at the same conclusion nor seemed floored by it now that he'd told her, he hid it well. "Nanobonder migration patterns are the most common thread between the dozen or so different places where the breaks are believed to have sprung up. I think that their unique nanite signature was a trigger, somehow. A key ingredient in the pharmaceutical stew that led to the creation of the breaks."
"I thought that the nanobonded were . . . endogamous." She was proud of herself for remembering the word. "Insular. Never interacted with outsiders. So they'd never have taken random meds."
"Popular consensus holds that this is a myth," Barron said. "Justification for atrocities. Throughout history, 'They keep to themselves, they think they're better than us, they hate us,' has been a common rationale for why a group of people constitute a threat, and therefore should be expelled or perhaps exterminated. But even if it's true, we know they established trade with some other diasporic communities, many of whom were known pharma subjects during Deregulation. They could have been dosed with something without their consent or knowledge."
"I know so little about them," Ankit said.
"And I know far too much. I've done so much research on them, spoken to so many people. How they originated, where they moved, who they interacted with. How they were targeted. Why. By whom." Here his face grew very serious, and tight with an anger that looked out of place on such an old and kind man. And then it passed. "But it's all academic now. Archaeology, as opposed to anthropology."
"I bet you'd like very much to talk to her," she said. "The woman with the orca."
"I would," he said. "But I don't fancy being butchered like the hundreds of people she killed in the grid cities she visited before coming here."
"Is that true? I heard those reports, but I thought—"
Barron shrugged. "I accept as a fundamental fact of Qaanaaq life that I will never know if anything is true," he said. "Most of it is rumor. Even when you read it in the supposedly legitimate news outlets. I learned that long before arriving here, to be honest. Life becomes significantly less stressful when you accept that your ignorance will always dwarf your knowledge."
"Tell me about New York," Ankit said. She didn't know why she said it. As soon as she did, she knew it was wrong. He hadn't mentioned the place. Only his accent had given it away.
His face seemed to break. First it tightened, as though he was growing angry again, and then it broke.
"I . . . can't" was all he managed to say.
"I'm sorry," she said, and he turned his head away, nodded as she stood up, thanked him profusely for his time, his insight, apologized again, promised to return.
_Why,_ she wondered, descending the narrow back stairs, _why did I ask such a cruel and thoughtless question—_
_Why—_
_Because his pain is mine,_ she realized, when the cardamom smell of _doodh pati_ hit her down at grid level, _because the ache of what he's lost is the same as the ache of what I've never had, and spent my whole life longing for. My mother is his city._
****City Without a Map: Dispatches from the Qaanaaq Free Press
Depending on your definition of _press,_ Qaanaaq hosts anywhere from five hundred to two thousand different press outlets, sites and broadsheets from every political and religious affiliation of every one of the city's hundreds of expatriate communities. The Greater India Reunification Party, never a significant political player in its home countries, is the publisher of Qaanaaq's most widely read news source, hated and beloved and fiercely argued about among the quarter of the city's population hailing from the various South Asian nations, rent asunder by imperialism and Partition and the Water Wars but reunited by Qaanaaq's xenophobia. The _Final Call_ gives equal column space to absurd conspiracies and all-too-real genocidal actions by North American power players. _Evangelize!_ rails endlessly against the same handful of subjects—the ease of acquiring abortifacients, the difficulty of acquiring firearms, the means by which solving the latter problem could address the former. Several popular sites urge Qaanaaq's Han Chinese to fight back against the Tibetan takeover of the motherland.
Whatever ax you have to grind, whatever lost world you are pining for, there is a press outlet for you. Probably several. And whatever happens, plenty of people have plenty to say about it.
From the _Maoist Pioneer_ [in Nepali]:
_Two weeks after her arrival in Qaanaaq, the Blackfish Woman may be the most hated person in the city. Already_ _reactionary religious elements are whispering together, and soon they will do more than whisper. As has happened so many times before, superstition is being used to focus the angers of a desperate populace on the wrong target. And why? To displace their very righteous anger over the city's mistreatment of workers. The very people who pay such low wages that workers in Arm Eight must sleep stacked in boxes, the very same businesses who fired pipe workers for attempting to unionize for better workplace safety, would have you believe that the arrival of weary refugees is our greatest threat, or that a battle-scarred survivor of genocide must be murdered._
From _Yomiuri Shimbun_ [in Japanese]:
_When the cities of America's_ _South and Midwest began to burn, and the continental United States became a hellhole ruled by marauding warlords, and the Northern Migration began, dozens of new communities began to form. Some were mobile city-states headed up by armed militias; some were ambulatory religious communes; some were united by common geographic or ethnic origin. Some were thousands strong; some numbered in the dozens. Many adapted to the freezing new climate by joining existing Inuit communities or by adopting their way of life._
_Few took detailed notes. Many documented themselves in the photographs and films of average citizens, the majority of which have been lost to poor preservation, outdated hardware, and evolving file formats. Historians of the period must_ ___contend with a mess of songs, passed-down family stories, and the reports of outraged neighbors on whose shores and at whose gates they landed._
_Many tall tales emerged from this seething stew of internal refugees, but perhaps the most myth-shrouded story of all is that of the nanobonded. A whole community of people who were either deliberately or accidentally exposed to experimental wireless nanomachines that established one-to-one networks between individuals, and who, through years of training and imprinting, could "network" themselves to animals, forming primal emotional connections so strong that they could control their animals through thought alone. And as with any new community in fundamentalist North America, there were plenty of people who thought that it was demonic, Antichrist-derived, the work of evil foreigners bent on undermining Caucasian hegemony. And, alas, like many communities, they are believed to have been wiped out in one of the many violent fundamentalist spasms that characterized the final years of the American republic._
_An absurd story, the evidence for which has been elaborately debunked._
_And yet—it would appear that one of them walks among us . . ._
From _Krupp Monthly_ [in German]:
_Wilhelm Ruhr remembers the last_ _Hive Project well._
_"We were finally there," he says, his aging eyes lighting up. "On paper, it should have worked. The nanites talked to each other. They could do so over great distances. We even engineered them to self-replicate only when their brain concentration dropped below a certain level, and to be open to network imprinting only for their first six hours. All the problems that had caused us so many headaches in the previous iterations were solved. Or so we thought."_
___At the time, Wilhelm was working in Canada. These days, home is right here in Qaanaaq: Arm Three's prestigious Kesiyn retirement center._
_"In mice, it worked. They were networked. Hurt one, and the others felt pain. If one figured out how to find the cheese at the end of the maze, all of them knew how to do it. Move them too far apart, and they experienced nausea, disorientation, eventually catatonia. We probably should have done monkey trials, but the way things were going in America it was much more cost-effective to just get a waiver signed and try it out on people."_
_Proximity to the border meant ease of testing on the recently deregulated country to the south. The U.S. Food and Drug Administration had essentially been stripped of all its power to enforce clinical trial standards, and the residents of the resettlement villages were only too happy to take a chance in exchange for what was to them a lot of money._
_"It didn't go well. Everybody knows that by now. Personally I think a lot of what got reported was exaggerated, or that the atrocities were due to other causes. Either way, talking about it isn't going to make much difference."_
_Still, Wilhelm Ruhr does not feel guilty about what happened._
_"We were trying to do something good. If it had worked, think of what we'd have been able to accomplish. A nanosynced team of scientists could solve all sorts of the problems we currently face. A lot of people died in wars to topple bad governments—should their commanding officers feel guilty about sending them to their deaths? They feel bad, probably, which is how I feel, I suppose, but they don't feel guilty."_
_And as for the arrival of the alleged "orcamancer," rumored by many to be one of the legendary "nanobonded"?_
_"People ask me about them all the time. So many people._ ___They say things like, Do I think it's possible that the nanite strain remained in a small group of people, and that their existing meditation practice enabled them to control it? And that it accidentally came together with other types of nanomedicine? Of course it's possible. But as for whether I think that in the course of two generations they learned to cultivate the nanites, introduce them to nonhuman animals, and form mechanically facilitated telepathic links with them . . ." Here he pauses, and chuckles. "That's a leap not even I am willing to make, and I'm a dreamer."_
## Soq
A crowd of people, approximating a line. Each one of them wearing entirely too many clothes. Somehow still shivering. They stood out, on Arm Six.
"What're they waiting for?" Soq asked, arriving, acting extra cool and casual. "Is there a church here that serves food?"
"Waiting for salvation," Dao said. "Deliverance. Death. Christ, kid, I don't know what they're waiting for. I don't think they're waiting for anything."
"The breaks," Soq whispered, watching them mutter and jerk.
"The breaks."
People hurried past, trying hard not to look. A buoy clanged, and someone in the line started making similar noises. Then someone else did. The clanging rippled down the line, spreading as it went, heading for the Hub. "You'd think the city would . . . I don't know, do something."
Dao offered a pack of pine needle cigarettes. Soq spent entirely too long debating what was the right choice here— _Refuse one, to show I'm proud and independent? Accept one, to establish a personal_ ___bond?_ —and then pocketed three. "Yes, it's strange, isn't it? The city possesses the resources to whisk all these people away. Why haven't they?"
Soq fought the urge to shrug, say something flip. _Treat everything like a test, even though probably none of it is. Dude could be just making conversation._
"Software hasn't come up with a solution yet," Soq said. "They're waiting."
"Ah," Dao said. "I suppose that's correct, on some level. That's the official story, anyway. But these are some sophisticated programs we're talking about. Capable of doing a trillion computations in a millisecond. Doesn't seem like they'd need _years_ to come up with something here."
Dao smiled, and Soq thought maybe they'd passed the test, if it had been a test.
"Still, Safety will come tell them to move along soon," Soq said. "Why aren't they on Eight? They'd fit right in, among all the nutcases out there."
_Dao frowned! Oh no! The test is failed!_
"Soq," he said. "Really? Because this is their home. This is where they lived. Even if they couldn't pay their rent anymore, or their family couldn't take care of them, or they burned down their own building by accident, this Arm belongs to them as much as to any of the other souls who live here."
"Of course," Soq said, trying, and failing, to not get nasty. "You wanted to set up a meeting with me in this particular spot—why? To discuss injustice? Medical software?"
Dao smirked, and Soq got the impression that being cryptic and elliptical was an important management strategy. "Go has an assignment for you," he said. "One she wanted discussed in person, rather than via messaging."
"Because it's so dangerous and important and you don't want any way it can be traced back to you?" Soq said, knowing it wouldn't be.
"Because it's so strange. She thought you might have a hard time getting your head around it, and wanted me to answer any questions you might have."
God, why did this guy have to be such a dick _all the time_? "Shoot. Try your best to baffle me."
"Go is assigning you to the orcamancer."
Soq's eyes widened with excitement. "Assigning me . . . how?"
"Research her. Gather all the intel you can on her. Fact, fiction, legend, rumor. Follow her, if you can. Talk to her, if you can."
"Fat chance of that," Soq said. "Tons of people have tried. She hasn't said a word to anyone."
"You see?" Dao said. "You're already on top of this job."
"Yeah," Soq said. "But . . . why? What does the Blackfish Woman have to do with Go?"
Dao rolled his eyes, turned his whole face skyward. "If it were up to me, this conversation would be over by now. But your assignment is important, and Go wanted me to answer your questions. So. Here's the simplified, kiddie menu version. Go is beginning a very delicate and dangerous operation. She's been planning it for years. It's an extremely complex mathematical equation, and Go had it solved—and then, here comes this woman. An unknown variable, with the potential to be massively disruptive. Maybe she won't affect the equation at all. Maybe she wants nothing to do with anything Go is working on. But if not, if whatever weird mission Little Miss Polar Bear is here to accomplish might impact something or someone we need for our plan to be successful, Go needs to know. And respond accordingly. Is that an acceptable explanation?"
_Important. Me. My assignment is important._ "Yeah. Sure."
"Good. You can message me updates. Call or visit only when it's urgent. If she does something . . . I don't know, big. Unusual."
Soq's earlier visits to the Sports Platform had been for fun, but now it was for work. Destiny was nudging Soq toward the orcamancer. Soq was important. They could barely wait to get back there and sit in the presence of the polar bear woman again, watching wordlessly with the rest of the wide-eyed grid kids who had come to see something magical and monstrous.
## Ankit
Wood smells like wealth, Ankit realized. Exposed beams filled the lobby of the corporate office building, smelling like money and safety and a time when the world was still solid beneath people's feet.
Her ears still rang from the screaming fit she'd gotten from Fyodorovna. It was almost comical, the mood at the office, the way everyone avoided eye contact with her, the sounds of shouting and pleading that came through the walls from the damage control phone calls her boss was forced to make.
With effort, Ankit was able to block all that out. She was here, wasn't she? She was doing what she had to do to make things right. Even when she wanted desperately to be anywhere else. Still no visit confirmation from the Cabinet. _Processing,_ it said when she checked the status of her request. What was going on with Qaanaaq's computer infrastructure, once the envy of the world, the stuff of legend?
She shut her screen and breathed in the stink of money.
The Salt Cave, they called it. Sharp artful salt crystals jutted from the walls, the residue of polymerized desalinization. The smooth crystals formed walkways several stories above her, shored up by old wood beams. Secretaries carried screens to meetings. Clients drank coffee. Out on the grid, a cry went through the crowd—wild-eyed people making a noise like buoys clanging, passing the sound from one person to the next.
Upstairs, Ankit knew what she would find. Corporate offices were all alike. Either they all hired the same lone decorator, who was paid obscenely well to reproduce the same palette and aesthetic, or they hired a whole flock of them and paid them poorly to copy each other down to the slightest detail. Classical landscape above the reception desk; abstract expressionism on the south wall; the walls a blue shade of white. Or, Ankit thought, perhaps there was software for that, too, and every two years it issued a subtle new change to keep the look evolving over time. Salvaged wood instead of brushed steel for the filing cabinets; cactus bubbles instead of air plant terrariums hanging from the bathroom ceiling. The lack of imagination among the rich was its own kind of machine, its own species of artificial intelligence.
A message from Ishmael Barron. A response to something she'd sent him at two A.M.
She'd found something. Maybe nothing. Maybe not. The medical log of the Taastrup infirmary, a few enigmatic lines in a mass of hundreds of thousands. Patients suffering symptoms identical to the breaks—potential early cases—who'd been dosed with something called Quet-38-36.0—a tranquilizer, derived from an atypical antipsychotic—and been successfully sedated indefinitely. Remarkable, if these were indeed early breaks cases, because tranquilizers typically didn't work on the breaks. And she'd shivered, to think of all that data, a quintillion pages' worth of science and study and learning and theory and fact and fiction, still there, on old open servers and suspended-animation websites, like so much coastal land mass swallowed up by rising seas—open to all, for the taking, except no one cared, no one had the slightest desire to dive into that wreck. What else was there a cure for—what other horrific problems might be so easily solved?
_Never heard of Quet-38-36.0,_ __Barron said. _But it sounds promising. Let's do more research._
Stupid, to take out her screen to see Barron's response. Now Ankit couldn't help but see the notifications stream in. New poll numbers; new press hits. Free fall had stabilized, somewhat. Fyodorovna's subsequent silence on the subject of the breaks had emboldened her opponent. He posted photos of sick people crowding Arm Seven. Beggars babbling; children having fits. Tiny rooms where a dozen cots had been crammed. _If she cares so much, why isn't she doing anything for them?_
"Ankit, hi," said Breckenridge, emerging from the bathroom, extending one wet hand. She took it, smiling to beat the band. "Oh good, they offered you coffee. Walk this way."
She followed him down the too-wide hallway, into the too-wide meeting room. Shareholders couldn't help showing off. It was who they were. They all had front corporations to handle their holdings, raking in the cash and paying the taxes and hiring the management companies.
"Bad mistake your boss made," Breckenridge said. "Messing with the breaks. That stuff is poison. Tragic situation, but . . ."
"I know," Ankit said, smiling like she didn't want to punch this guy in the throat, him and everyone else who had such pat responses for everything they didn't want to think too deeply about. _That stuff is poison. Terrible shame but._
"I assume that's what you're here about?"
"Mostly, yes," she said. They sat. "That, and some intel."
"Intel? Or money?"
"Both," she said, although of course all she had come for was money. An intel exchange was performative, a cover for the in-person meeting. She'd needed to look him in the eye when she made the ask. Except now that she thought about it . . . maybe someone so close to someone so powerful _would_ have some information she could use. "We're taking a beating. We need to buy some more messaging bots, screen time, that kind of stuff."
"I'll see what we can do. I already put the request in to the shareholder who heads up Fifty-Seventh when you asked for a meeting. I figured that's what it was about. You really could have just sent the message." He gave Ankit the sense that he slept under his desk. Not from poverty but from self-annihilating salaryman work ethic. "What was the intel you needed?"
"About the breaks."
"Are you serious? You need to learn your lesson, lick your wounds, and move on. Fyodorovna isn't planning to pursue this, is she? Make it a campaign plank?"
The view was breathtaking. The windscreen was being shifted, and as the sun caught the gleaming facets it looked like the sky was one massive kaleidoscope. Was that fear, in Breckenridge's eyes?
"Maybe," she said. "We're still collating."
"Big gamble," he said. "Unlikely to pay off for her. Maybe if you weren't an incumbent, you could rally people behind it, make absurd demands and promises, whip it up from a fringe issue to a major one, but with all her time in office and not a lick of concrete action, the few people who do care about it one way or another won't—"
"Maybe she's planning concrete action."
"That would be unwise."
"Why?"
He took off his glasses. Rubbed his eyes. "You know that even if I did have access to any information that might be helpful, we couldn't share it."
"I need to know," she said. "We're losing here. And you need us."
"Let's get noodles," he said. "You hungry?"
She wasn't, but she knew when someone wanted to talk off the record.
Her jaw pinged on their way out. She let the message come in: a colleague back at Fyodorovna's office, concerned because That Crazy Lady Maria was apparently rallying an angry mob. Hardly a challenging feat; out-of-work Americans took little riling up. Employed Americans, too, for that matter.
"Back in ten," Breckenridge said to the receptionist. Receptionists were a funny holdover, Ankit thought, not for the first time. Software could do everything they did. But they made people feel more comfortable. _How much of the world around us is utterly superfluous, kept in place to preserve an illusion of order?_
That's when she realized how scared she was. She never got philosophical. There were always too many real things to think about.
On the street, darkness was inching in. The days were short now. Soon they'd be down to four dim hours, and they'd crank up the lights nonstop, and the whole city would smell of the composting sewage in the biogenerators that powered the methane-sodium lamps. They crossed the Arm, toward a shop expensively made up to look like a ragged Arm Eight street stall. Right down to the spray-painted tarps, except these were hanging inside, where they weren't protecting anything from the wind.
Arm One noodles were the worst.
What did it mean, that Breckenridge had something to say but couldn't say it in the office? Something was afoot with the breaks, and the shareholder he worked for knew what was going on. Might have helped make the decisions. Or maybe this was something else, some other scandal, some other terrible thing about to hit her, one more reason her boss's career was over and therefore so was hers.
A slide messenger ululated past them.
"Hey, comrade," said a man in a hooded sweatshirt, running up behind them. "I think you dropped this."
Breckenridge patted his pockets, made a quizzical expression, reached out to the man's extended hand.
The man grabbed him, pulled him in, took hold of his arm at the wrist and above the elbow, and twisted. Breckenridge yelped, turned his body with the arm. The hooded man kicked at his knee, gave him a shove, dumped him off the grid and into the sea. Ankit got a glimpse of gritted teeth, a neck wide with muscle. He turned to her, slowing, as if debating whether to soak her, too, and in that second she got in one good kick. Missed the mark slightly, hitting his inner hip, but hard enough to surprise him into pausing for just an instant, a fraction of an instant.
Long enough for her to catch a better glimpse of his face. Shadowed, battle-scarred, seen only once in the flesh, but familiar. From posters, from fight broadcasts.
Her brother.
"Kaev," she said, but he was already sprinting away.
Sirens blared. Arm One had surface sensors to know when someone had been soaked. Breckenridge flailed in the choppy sea, glasses gone, looking ridiculous, like he'd never again make the mistake of leaving his office.
"What the hell," Ankit said, but he couldn't hear her, couldn't have answered if he had, and anyway she wasn't talking to him.
****City Without a Map: Anatomy of an Incident
You are bound to your body.
Your body is shaped by its DNA, your parents' decisions, historic hate and hunger, contested elections, the rise and fall of the stars in the sky. Maybe your body is in an awful place. Maybe, like me, you are there through no fault of your own.
One day, you will break free of your body. Every one of us will. Until that Great Liberation comes, we must be content with the little liberations. The shiver up the spine—the telltale tingle of a beautiful song. Great sex; a good story.
So. Another story. This one encompassing twenty minutes, distilled and condensed from diverse sources. Journalist reports; video cameras; eyewitness accounts as delivered to Safety and other agencies.
Scene: The Sports Platform. Lowest level. Evening. Darkness has just come to Qaanaaq.
At 17:55, forty men and women board the Platform. They are an angry, unruly bunch. They carry improvised weapons; lengths of pipe, mostly, retired from the city's geothermal ventilation system. Some wield the rusting skeletons of American guns, family heirlooms desperately clung to, sometimes the only thing an ancestor carried when they escaped from the hellhole collapse of Chicago or Buffalo or Dallas.
By 18:04 they have reached the lowest level.
The woman who leads them has been transformed. Whatever wretched old crone she is, in whatever miserable crevice of Arm Seven or Eight she resides, she has left all that behind. Right now, she is magnificent. A monster.
"You guys can get the hell out of here right now," a thug hollers at the half dozen journalists lazing on the bleachers. "Unless you want some of this, too."
"Let them stay," the woman says, and from her accent they know she is the original article, straight New York, by way of the Dominican Republic, been here a week or twenty years, it doesn't matter, she's had so little actual conversation with her neighbors that the city can put no fingerprint on her voice. "They should see this. The world should see this."
Still. Most of them leave.
"But they could call Safety!" someone says, making ready to pursue them.
"They could call Safety without moving an inch," she says. "But by the time Safety gets here, our work will be done."
Their target has not moved since their arrival. For why else could they be here, if not for the Killer Whale Woman? She stands there, watching them, and the journalists will report that she is smiling.
Imagine her beautiful. Imagine her stout and muscular inside her leathers and furs. Imagine someone so strong that if you knew she was on your side, you would never be afraid again. And if she was coming for you, you would know all you needed to do was wait.
"Surround her," the woman says. "Don't let her get to the bear."
Several members of the mob notice it for the first time. Chained to the wall, getting to its feet, bristling with rage at the smell of so much anger. And now the smell of so much fear. A couple of people yelp. One runs.
"Your kind is not welcome here. You are an abomination. A profanation of the human being as God made it, in His image. He made us distinct from the animals for a reason. Your bond with that savage beast in chains over there is sin, and that sin is why your people were wiped out."
At the word _abomination,_ their target moves for the first time. She raises the weapon that has been resting at her side and takes hold of it with both hands. It is taller than she is, walrus-ivory handle and a slightly curved shaft that might be ironwood or might be the rib of the most massive of whales. At its end, a blade like a lopsided crescent moon, fatter at the bottom than the top, its edge broken up into hooks and barbs.
Some of the journalists capture photos of it. Some of them are filming the whole thing.
"Why have you come here?"
Someone hurls a pipe. Hard. Her motion is effortless, so slight that some people don't even see it, the most minimal shift of the weapon to deflect the projectile without striking it head-on in a way that might damage her blade. The effect on the mob is obvious, immediate. Mouths open. Feet shuffle. For the first time, the courage of a violent crowd begins to crack. They are not invincible. Their target is not helpless.
It has the opposite effect on their leader. Or perhaps it is their fear that makes her braver. Her chest swells. She steps closer, into range of the weapon.
"You are not welcome here," she hisses.
"She can't understand you," someone yells. "You know they're afraid of technology. She doesn't have an implant."
At this, Killer Whale Woman laughs.
This, too, startles the mob, so much so that hardly anyone notices that she has struck out with her weapon. Only their leader's scream, several milliseconds later, alerts them that something is wrong. Her right hand is lying on the floor, being baptized in a cascade of arterial blood. The smell of it makes the polar bear roar.
"Get her!" someone shouts, and the crowd closes in on her. Their dehanded leader staggers away to the far edge of the platform.
Pipes and chains swing. Pause the video, zoom in, slow it down, you can see the ballet unfold. Two men rush forward first, side by side, so close a single swing decapitates both of them. A woman attacker crouches low, coming in from the side, and catches a high kick that knocks her backward. The swing that lopped off two heads reaches its graceful end, and already the orcamancer is pulling it back, shifting her hands to the center of the shaft, ramming it back, expertly striking the rib cage of the man trying to run up on her from behind. Bones break. Their sharp edges stab into organs.
People stand in the doorway—athletes from other levels of the Platform, and the standard stream of curious Qaanaaqians who come every day to visit their mythical visitor. Fifteen screens are focused on the action, capturing every instant of it.
The polar bear stomps its forepaws against the ground. The whole hall echoes with the metal ring of it.
" _Gaaah!_ " someone shouts, or at least that's how their desperate, inarticulate cry will be rendered in the _Post–New York Post_. Before it is finished she has thrust her weapon forward and its blade has pierced his throat, one barb catching on his spine, and she swings it to the right to shake him free, bringing his body into the path of one of his comrades, who trips over it and tumbles to the floor and has one arm severed by the now unencumbered weapon.
A gunshot. The screech-whistle of a ricochet, and then another, more distant, as the bullet vanishes into the gloom of the Platform's lowest level. And then a curse, for, as so often happens with the aging firearms of Americans, the igniting gunpowder has caused the barrel to explode in the hands of its user.
But the sound of it causes the orcamancer to pause. Her smile stops. She looks from face to face, hand to hand.
"Shoot her!" the mob's leader wails, unnecessarily, from the sidelines, her voice thick with pain but light from loss of blood.
The orcamancer neatly disembowels another soul idiotic enough to charge her.
A man closes his eyes and then opens them. Takes a breath. He stands away from the fray, between her and the bear. His thighs ache. A week and a half has passed since he got off the iceboat. They should not still be hurting. He is getting too old to straddle the saws anymore, too weak to calve functional shards off the Greenland glacier. By this time next year, he'll be unable to make a living at the only job he's ever had in this crummy city. And then what? He is old enough to remember Philadelphia before the Revival, before the state of Pennsylvania fell to fundamentalists with a platform of confronting "centers of sin" who ordered the complete evacuation of every major city. He has seen everything taken from him, so many times, and he's never been able to do a thing about it.
He shoots. The bullet strikes the orcamancer in the lower leg, knocks her back, causes her to stumble and fall to her knees.
People laugh. The circle, smaller now, closes in.
She taps two fingers against the corner of her jaw. Someone gasps. They had been so convinced that she eschewed all technology, this possibility never entered their minds.
Metal rings against metal. Again they hear the polar bear roar. Louder now. They turn to see that it is unfettered, the cages fallen away from its hands and mouth. It shrieks and charges.
This is where the official record breaks down. Everyone leaves at this point, and swiftly. Journalist and freelance gawker alike exit. So, for that matter, does the recently dehanded woman who moments ago had been a mob leader. What's left of her mob attempts to depart but is unsuccessful.
There are no cameras to capture the carnage.
## Soq
Soq saw the slaughter and did not flinch.
It was their fifth time visiting the orcamancer. Lots of bored Qaanaaqians came to see her, either at the Sports Platform or at the Arm Six sloopyard where her boat was docked. Most went once and found the real thing far less interesting than the mythic warrior they'd been imagining. Only the obsessed came back again and again, the people for whom hate or fear or love won out, the people for whom she meant something. Soq had looked from face to face on each visit and wondered: _Why is this one here? Why this one? Do they want to destroy her? Do they want to beg her for the gift of her nanites, a teaspoonful of blood that could turn them into something as awesome as she is?_
_And why,_ Soq wondered, _am I here?_
Sometimes people asked the Killer Whale Woman things, and they were always ignored. Most often they stood as silent as Soq did.
Soq arrived in the middle of the bloodbath. They almost got trampled in the sudden swift exodus of people up the stairs. When Soq got to the bottom and climbed into the bleachers, they got there just in time to see the bear break a man's neck. A woman turned to run, and the bear's paw raked down her back, tearing it open, pulling her backward and onto the ground, and then it stooped to tear out her throat.
It took less than eight minutes for the polar bear to kill the last of the people who had come to hurt the orcamancer. It played for a short while with a severed limb and then turned to face its traveling companion.
And—was that fear Soq saw, in the orcamancer's eyes? How could that be? Weren't they bonded? How did this work? Which of the stories were true? Hadn't they known each other all their lives, been raised as siblings; didn't they feel each other's pain? But perhaps in the chaos of so much bloodshed, the wild animal could not be controlled. It took a step closer to her.
For reasons Soq did not understand, they were not afraid of the polar bear. They knew they should have been. But they also knew that there was a very good chance that the bear was about to eat the orcamancer, unless someone distracted it, and no one else was around to do the distracting, and the bear was far away, so maybe Soq would have time to escape—
"Hey!" Soq called, standing up in the bleachers. Thinking too late to calculate the distance from there to the exit, gauge how fast the bear could run, whether they stood a chance of making it out, whether the heavy door could be bolted to keep the bear from charging through and eating Soq.
It stopped, turned its head to look in Soq's direction. Sat back onto its hind legs. Cocked its head. The orcamancer gasped. The bear did not resist when the orcamancer put the cages back onto its head and hands.
The orcamancer took off her hat and bowed to Soq. Her hair fell in a heavy intricate coil behind her. She pressed the button for the elevator. Waited patiently. With her polar bear. Then she turned and said something. Soq's jaw buzzed, translating from Inuktitut-English Pidgin:
"I am in your debt."
The first words anyone had heard this woman say, and she had said them to Soq! Blood covered the animal, and the killer whale woman was shining with sweat, and Soq had never seen anything so beautiful. _I should go down there, say hello, start up a conversation, become best friends. Help her escape. Something._ But Soq could not move, could barely breathe. It took an incredible effort to call out, "You're welcome!" in the last second before the elevator whisked them away.
## Kaev
Kaev had been hoping for something like what he felt from a fight. The adrenaline rush, the thrill of danger, the joy of abandoning the ego and embracing the body, the animal. But he'd gotten no pleasure from his soaking. Only guilt, and shame, and now the extra special gift of having to worry about getting arrested.
He'd cried, after the first one. A bureaucrat, a flabby flimsy thing who felt truly safe only when behind a desk. He'd probably never been close to violence in his entire life. Kaev tried to tell himself that maybe the man was pure malevolence, merrily authorizing mass suffering—rent raises, embargo orders, strategic pharmaceutical restrictions—but he couldn't diminish how wretched it felt, the look on the guy's face when he thought he was about to be murdered.
The second one, he wouldn't feel bad about.
"He goes by Abijah," Go had told him. "But then he always tells everyone that it's a nom de guerre. And gets disappointed when no one cares enough to ask any follow-up questions."
Kaev recognized him. A slum enforcer. The muscle that came to kick you out of your home if you ignored an eviction notice, or roust you from your nook when the sub-subletter you were paying decided he wanted to clean house. Nasty thugs. Arsonists, torturers, evidence planters—effective enforcers had to be jacks of all the sordid, ugly trades that went into maintaining good tenant-landlord relations in a city where the landlords called all the shots. Many were failed beam fighters, and all of them were obsessed with the sport. He'd seen Abijah at the Yi He Tuan training center, watching from the bleachers while he practiced. Making awkward conversation. Flexing his muscles, complaining of soreness from a rough day, like there could be any comparing his job and Kaev's.
"This one needs to be quiet," Go said. "The first one had to make a very clear statement, but this one needs to be . . . more uncertain. It should unsettle the target—your victim's employer—but create just enough doubt that the target doesn't feel confident unleashing a violent response."
"And. Who." _Deep breath, Kaev. You can do this. It's just human_ _speech. Everybody does it._ "Who's. The. The. The target?"
She'd stroked his face, and laughed, and dismissed him.
Kaev watched Abijah walk down Arm Eight. Tight clothes, like many insecure people who wanted to show off their muscles wore. Big boots that gave him an extra inch of height. Go's target had to be a shareholder. What other common ground could there be between the bureaucrat and the slum thug? Kaev could see the fear in people's eyes when they saw the enforcer pass. How their faces tightened. So he was well known out on Eight. He'd fucked up a lot of people. Kaev felt his lip curling, a snarl he could not keep in check.
"Look!" someone called, and pointed to the sea.
A giant onyx blade, seeming to be as big as a person, slid through the surface of the water. _The orca,_ Kaev thought, and felt the hairs of his neck prickle. It kept rising, and there she was, straddling its back: the orcamancer.
_She is real._
She was so close. So big. He wondered what her black clothes were made of. He shut his eyes and felt a strange flare, like remembered warmth.
"She comes through here every other day, seems like," someone said.
"Wonder she doesn't get hypothermia, out there in the ocean."
"You got to be human to get hypothermia."
The Arm Eight rabble continued the debate, but Abijah was oblivious, did not see her. Did not slow down. Kaev watched until it was clear the orcamancer would not be resurfacing this side of the Arm, then hurried to catch up with his quarry.
Abijah entered the gully between two cafés on tall stilts, unzipping already. His actions exaggerated. He wanted everyone to know he was going to go piss or fuck something.
_Here you go, Kaev. An easy one. Duck down and sneak between the stilts so no one sees you entering the gully, sneak up behind him, knock him the fuck out. Push him into the sea._
Kaev did not duck down. The orcamancer would not have attacked like that. She never hid. Whatever she'd come to do, she wasn't trying to be secret about it.
"Hey," he said when Abijah stepped back onto the grid.
"Hey!" Abijah said, recognizing him, unable to place him.
"Kaev," he said, loudly, so the crowd between the two cafés could hear. He extended his left hand. "We met at the arena gym, remember?"
"Yeah!" his quarry said, smiling, and paused for an instant, thrown off by the unexpected hand, before extending his left to take Kaev's.
When he had it, Kaev pulled. Hard. He swung his body around, letting his right shoulder lead, swinging his elbow into his opponent.
A killing blow. The kind of move you never use in a beam fight, because it's unsportsmanlike. A thing you keep up your sleeve for those hopefully-never occasions when it's you or the other fighter. Kaev pulled back at the last instant, striking the man's right clavicle instead of his windpipe. He heard it snap. Felt the man drop.
Pain incapacitated Abijah. He screamed. He bellowed. He sobbed. He tried to ask, _Why?_ Thugs never learned what fighters learned—how to battle through pain—because they only ever hurt people who couldn't fight back.
Kaev looked up. A couple hundred pairs of eyes were on him. They clapped. They cheered. They held up screens to capture the moment.
Kaev smiled. Already he was running the scenarios, anticipating the ugliness that was in store for him, the revenge Go would take for his disobeying her, the punishment Safety would impose when they caught him. But there was another arena skill he had up his sleeve: how to put all his fears for the future into a box and briefly forget about them.
Someone recognized him, or ran facial rec with their screen. She called his name. Others heard her, joined in. Kaev shut his eyes and basked in it.
For once, they were cheering for him.
## Fill
Whales swam through the air above him, white against the dark blue twilight. Thick powdery snow, the best kind for the light projections. Their simple AIs pinged on the shape of a child and turned to swim circles around her. Fill felt oddly abandoned. When he was that age, he'd believed the snow projections loved him and him alone.
Behind him, the sea lions barked.
He loaded the latest episode of _City Without a Map,_ broadcast in Mandarin. He didn't like the ones recorded in languages he didn't speak. His implant had the best translation software—even replicated perfectly the voice of the original speaker—but something did not survive the transition. No machine could match the earnestness, the hunger, of the original Readers.
Fill could feel the wet gathering in his clothes. A mark of his privilege, that this happened so rarely. The rich could have expensive dehumidifiers and lattices of salt polymer in their walls, to help strip away the ever-present damp. Everyone else in Qaanaaq just put up with being wet all the time. Having eczema. Having worse.
The little girl screamed with happiness as the whales combined into a tyrannosaurus and proceeded to chase her.
___When I'm dead, the snow projections will still be here. This whole city will keep on working._
Fill had spent his whole life believing the city belonged to him. After a certain point he'd started to think maybe it was the other way around, but that was wrong, too. He meant nothing to Qaanaaq. Qaanaaq did not see him, did not know him, and nothing would change when he was slotted into his plastic sleeve and weighted with salt and dropped into the sea at the end of his Arm. He watched the waters churn. The water that would swallow him.
"Fill," said a rusted familiar voice, and then he was being hugged.
"Grandfather."
He settled into the embrace. Startlingly strong. Fill knew that his grandfather was as mortal as any man, yet he could not imagine any sickness or weapon that could reduce him.
"You still love the sea lions," the old man observed.
"Of course," Fill said, conscious for the first time of the rapt expression on his face, watching the piers where giant mammals flopped and dozed and barked.
But he hadn't loved them, not always. There had been a time when the sea lions were kid's stuff, something he couldn't be bothered with. Every kid went through that, he imagined, but when had he come out the other end? When had he become adult enough to give in to childish joy? Once again the eerie feeling came over him: the suspicion that what he felt might not be his. An unwholesome tingle traveled up the length of his spine, amplifying itself as it went, until his shoulders shook and his teeth came together in a sharp clack, cleaving open the side of his tongue.
"Everything okay?" his grandfather said, feeling him flinch, touching his sleeve.
Fill swallowed blood. "Of course. Just cold is all."
"Shall we go inside?"
"No. I like it."
___What would he think, dear old Grandfather, if he learned I had the breaks? The dirty dirty taint of the poor? Smile politely, walk away, run home, burn his clothes, and cut all ties with me? What does he think when he sees them, the sick and dying, the men and women huddled on the grid about to break free of their bodies? Does he sneer, does he blame them for their bad decisions? Does he see them at all?_
"How are you _really,_ Grandfather?" Fill asked. "What takes up your time these days?"
"Managing my empire," he said with a grim laugh.
"I thought you had people for that."
"That's true—well, then, I spend my time looking over the shoulders of the people who manage my empire."
"That sounds boring."
"I wish it were."
"What's been going on?"
Grandfather spent a long time looking at the sea lions. Then he seemed to come to some kind of internal decision. "I'm under attack. Nothing I can't handle, but it's annoying all the same. A crime boss has gotten too big for her britches. This happens. Somebody wants more than the niche they've already carved out for themselves, and they make a move on a shareholder."
"What does one do in a situation like that?"
"We bide our time, typically. People get tired of throwing pebbles at a brick wall, eventually."
"Are you really that invincible?"
"No one's invincible. But we set this city up. Everything's stacked in our favor. Patience is all we need—maybe a trip out of town, if things really heat up, or a Protective Custody stint in one of the cushier suites in the Cabinet."
"So that urban legend is true? That the Cabinet has posh rooms to hide VIPs in, alongside the overcrowded wards full of screaming lunatics?"
"Very true." The old man looked at his hands. "Of course, once in a while a shareholder will go a little crazy. Meet violence with violence; slaughter his or her enemies. A separate part of the playbook. That's not my style."
Fill looked down onto a shifter skiff. Portions of the boat floor rose, fell. A worker from Recreations controlled it all from the rows of subdermals up and down her arms. Platforms circled and spun as she did. Children cavorted there, but they avoided eye contact with her. Augmentation was minimized in Qaanaaq, looked down upon, considered uncouth. In some grid cities, and most of the still-peopled places in the Sunken World, augmentation was much more accepted. Even obligatory. Fill wondered where she'd come from, whether she'd chosen them willingly. Whether she liked them.
A woman stood before him. Flickering, so that at first he thought she was a snow projection, but no—the projection was passing through her, a kraken now, dappling her body and then departing. Tall and beautiful, dark skin. Bald. Well into her fifties. Wearing clothes that were way too thin for the Qaanaaq cold. Like hospital robes. She stared at him. Smiling, maybe, but maybe not.
A shuttle bark disgorged passengers. They climbed up onto the Arm, passed between him and the woman, and when they were gone so was she.
A breaks vision? Fill shivered, licked his lips, tasted blood again. Resolved to bring the woman up, during his daily talk with Barron. Resolved not to. "Isn't it funny?" he said, blundering into it, because if there was one skill glib gay boys learned it was how to ease archly into a fraught topic. "I realized I know almost nothing about you! Most of my friends can find their family histories in the cloud, but you—shareholder erasure! Our family name is a great big digital void. Which can't have been cheap, by the way."
Grandfather laughed. "So it's an oral history project you're working on."
"Something like that. I know we're from New York. How did we . . ."
"Get out? Strike it rich?"
"I have to assume that we were at least a little rich from the start."
"Your father never wanted to know." A scowl, at the past, at the grid they stood on. "When he was little, maybe. He wanted the fairy tale. The movie version. But as a grown-up? Ignorance is bliss."
Fill blinked away the mention of his father, the tears that threatened to rise to the surface. "Is it that bad?"
"No one survived without getting their hands at least a little filthy. But I think you are strong enough for the truth. You're stronger than he was."
"I'm not," Fill said, the notion so ridiculous he had to laugh. "I've never done a goddamn thing. Like, period."
"You are. I've always seen it in you. The fact that it's never been put to the test doesn't mean it's not there."
"That's sweet of you to say," Fill said, unconvinced. "And my father? You never saw it in him?"
"Your father was a good boy. But he was not strong."
Images arrived unbidden. His father, departing for Heilongjiang Province to cover the Famine Migration. The camera equipment he let Fill inventory. "Mom always said he was brave. To go to those places. To take those pictures."
"Your mother wouldn't know strength if it kissed her on the cheek," Grandfather said. "If you'll pardon my saying so. Doing something dangerous and foolish is the opposite of strength."
"What is strength?" Fill asked, feeling that they were close to it now, the Thing, the Secret, the story his father hadn't wanted to know.
"I worked for a security firm. Buildings and events. Simple stuff: hire the guards and send them to the place that needed guarding. But these were insane times. The Real Estate Riots were in full swing—you've heard of them, surely? About to evolve into the Real Estate Wars. I believed that security was about more than muscle. I talked the owner of the company into creating a new division, letting me lead it. Headhunting some of the biggest names in New York public relations. Intelligence, I called it. Because security wasn't about keeping bad guys out anymore—it was about keeping the bad guys from seizing what was yours. Squatter gangs, politicians susceptible to pressure, ready to use eminent domain to take property away and give it to the poor."
Fill tried hard to keep his eyes from glazing over.
"My unit went into neighborhoods, put agents in masquerading as tenants, assessed the situation, identified fault lines. Divisions between people. Once you know how to whip people up, they'll do all the work for you."
"What kind of work?"
"My specialty was religion. Fundamentalists, mostly. There's a lot of them—all sorts, though Christians were the easiest. Pretty soon we had dozens of contracts from all the biggest real estate developers. The manufacture and strategic deployment of mass idiocy, I told people. Practically put it on my business card."
Business cards—Fill had heard of those.
"Manufacture an outrage. Provide a target. Identify the people whom people listen to, the pastors or the PTA moms, and pay a couple of them to make a stink. I mostly just copied what was happening in politics. When we went national, I acquired several groups that had run election campaigns."
Grandfather paused, watching Fill's face. Waiting for a response. Fill was fairly sure he'd missed something. Sea lions clapped the rotting wood of their piers.
"A lot of people died, Fill."
"Ah."
"Some scenes played out, in Harlem and the Lower East Side, that . . . well, let's just say your father would have found quite a lot to photograph there. A smaller scale, but easily rivaling anything that came out of Calcutta. I regret it now, but what can you do? And you and I wouldn't be here if I hadn't. All my work, all over North America—none of that could have happened without New York."
Fill knew all about this. _City Without a Map_ had taught him.
Remembered smoke blackens the sky. The shouts of long-dead citizens ring out in the street. Explosions: tanks firing on squatter strongholds; high-class cultural events bombed by tenant army activists.
Qaanaaq is a dream. At night you lie down and wake from it. Return to the real world, your life, your city, dying. Every night you are back there, watching it unfold in uncanny déjà vu slo-mo, because surely you have seen this before, surely you know what will happen, surely you will act differently, surely you will get out in time—
Grandfather had lost his son. Soon he'd lose his grandson. Whatever else he'd done, whatever horrific crimes he'd committed, it was hard not to pity him. To have fought so hard, to have acquired so much, and to end up with nothing. Well, nothing except . . . everything.
Fill had so many other questions. About the clients his grandfather had worked for when he was finished helping destroy New York City. About the things that were gone. The Metropolitan Opera; the Daughters of the Disappeared; Qaanaaq when it was still shiny and new.
Grandfather flung a crumpled napkin into the sea. Littering: another distinctly New York oddity about the old man. "I called you because I missed you," Grandfather said, "but I also had something up my sleeve. Something I want to give you. Two things, actually. I'm not seriously worried about that . . . thing I mentioned to you earlier. Standard crime boss nonsense. We shareholders haven't held on to everything we have for this long without learning how to weather every storm. But still. It's silly not to take precautions. I need you to be able to access two things that only I can currently access. One is an apartment. Sealed, secret. Kept off the market. I know you already have a place—I just need you to have all the access info on this one, which so far isn't listed in any of the residential rosters. An investment, you know. We have several of them, in fact, but this one is special to me. Your grandmother and I . . . it was where we went when we wanted to escape."
Fill nodded, feeling very noble and dutiful. "Fair enough, Grandfather. And the second thing?"
"Software. A particularly unstable, dangerous program. Cobbled together by an alliance of different shareholders, combining ten or so very different security protocols and illegal data-mining approaches. Toxic stuff, military grade. We created it in the early days of Qaanaaq, to share information, observe patterns, keep track of our units . . . assess problems, figure out how to deal with them . . . but it hasn't been used in twenty years, maybe longer. Too erratic. Undependable. It did what you wanted, but it would also do a bunch of things you didn't want."
"What do you want me to do with it?"
"Good heavens, nothing! Not for now, anyway. Just hold on to it. I have no idea whether any of the other shareholders are still alive, or have access to it—I know it hasn't been used, it's on the list of programs our watchdogs prowl for—but if I'm the last one with access, I don't want it to die with me. Although maybe it should. But it's worth a lot, and fifty years from now you may be glad you have it in your back pocket."
"Sure," Fill said, because all of this seemed to be making the old man happy. What did it matter that he wanted nothing to do with an empty apartment and some temperamental software? He could finally do something for the old man, play some small part in the family business. And, of course, he could not say, _No, Grandfather, I'm sorry, I won't be alive in fifty years, or fifteen, or maybe not in five months._
"I'll send you all the information."
"Thanks, Grandfather. Oh, look!"
Two sea lions barked and bumped chests. Both men laughed.
## Ankit
Ankit bought flowers, then threw them into the sea, then bought different flowers. Scentless ones, cold emotionless white. The angry red of mainland roses would be too provocative; the purple scent of hothouse hyacinths could make the whole madhouse lose its mind.
Ankit dressed carefully, then undressed, then dressed differently. Sterile gray. Genderless lines.
"Fucking Cabinet," she said, looking at herself in the mirror, looking at the time, wondering how many more outfits she could go through before she had to take the zip line out to her appointment. Planning every little detail was its own kind of paralysis, so she decided to leave early and kill time over there rather than keep on burrowing deeper into unproductive thoughts.
Her screen sparkled. Barron: _There is absolutely no Quet-38-36.0 anywhere in Qaanaaq._
That didn't seem right. Or possible. Qaanaaq was sick with drugs, full of smugglers and machines to bring even the most obscure pills to the far-flung arrivals who fiended for them. Anything you wanted, anything at all, someone was selling it. But she had too much else to think about, too many other things to keep from stressing out over.
Another one: _I have a friend with access to a molecular assembly machine, who does off-legal drug printing sometimes. I am going to try to get him to make us some._
_Good,_ she said, and set a five-hour block on further notifications from Barron.
Of course she hadn't told Safety that she knew who the soaker was. She was still too much of a scaler at heart to even think about snitching. That would have been true even if the criminal in question weren't her brother. They'd questioned her for almost an hour, standing there at the edge of the grid, apologizing endlessly for the inconvenience, seeing her fancy work clothes and mistaking her for the important person everyone else mistook her for.
She shut her eyes against the ocean of words churning inside her. The things she wanted to ask her mother; the things she wanted to say to her. Cabinet staff said to minimize dialogue, focus on physical presence, touch, caress; her mother's psychological condition was unique, easily triggered into episodes by even seemingly harmless questions or comments.
She hugged the flowers to her chest. Shut her eyes and tried to remember her mother's face.
The woman she saw was not her mom. She knew this. She was a summary, a splice, a combination of a hundred warm, loving, smiling mothers she'd seen in movies and at the homes of friends and in the families she visited for work. Ankit had no real memories of her mother's face from when she was a child. What she'd seen the last time she'd been with her mother had been too horrible to hold on to. Fifteen years ago, a dark red wet sobbing mess, a mouth open wide in a howl Ankit could not hear through the window in the door.
Her jaw bug buzzed. _Fifty-Seventh Street Corp.,_ it whispered.
She answered. "Breckenridge?"
"Still in the hospital, the poor creature," said a hale but ancient voice. "Martin Podlove speaking. I am the shareholder responsible for Fifty-Seventh Street Corp."
"Mr. Podlove, thank you for your call," she said. "I'm surprised to hear you identify yourself so openly. Shareholder invisibility—"
"—is a privilege we dispense with quite frequently, actually. When we know someone can keep a secret. When we know how much they stand to lose."
He let her chew on that for an instant or two, and then when she started to respond he said, "My colleague passed on your request for funds. I figured I owed you a call, at least. There's something cowardly about sending bad news as words on a screen."
"No," Ankit said, stricken. "No . . . you can't do that. Fyodorovna will lose this election if you don't help. In a different year an incumbent might be able to weather this storm, but times are too tough. Rents have been rising, out on Arm Seven. Evictions, displacement . . . People are mad, and they blame her for their problems, and they'll choose the devil they don't know."
Podlove said nothing. He was enjoying this.
"We're useful to you," Ankit said, and knew, then, that the battle was lost.
"Everyone is useful to us. Think I haven't been backing your opposition? Think they won't do what I ask? You know politics better than that, Ankit."
He tapped off.
She looked up at the Cabinet. She stood there until the shaking subsided. Early, still, for her appointment, but the processing always took too long and she figured she could start the waiting now. Better that than standing around wishing for the ability to summon a wave of cleansing nuclear fire out of thin air.
Entering the Cabinet felt like sinking below the sea. White noise pods lined the curved walls and the doorway, cocooning the building from the sonic chaos of the city outside. People spoke but she could not hear them, and that was part of the process, part of the therapy. She looked at the crowd in the waiting room, wondered who was there to be processed and who was simply visiting. Either way, she felt immense pity for them.
"Ankit Bahawalanzai," she said into the triage scanner. "Two P.M. appointment."
The hexagon flashed green. A door hissed open. She was pulling off her shirt before it had shut behind her, placing her clothes on the table, standing in her underwear in the center of the room, letting the lasers wipe over her, pretending she could feel them. They'd be confirming her identity, scanning for threats and communicable medical conditions and who knew what else. Medical scanning algorithms changed all the time, looked for different things, got more sophisticated in some ways, became blindingly stupid in others. A couple of years ago they had all been obsessed with hair follicle analysis. Before that it had been fingernails. She imagined the AIs getting together at conferences, arrogant as doctors, swapping stories in silicon hallways, exchanging bad ideas.
A weird hiccup, in the flow of the white noise.
"Ms. Bahawalanzai?"
"Hi," Ankit said, and felt her cheeks heat up, because this was not good, an actual human being was never a good sign.
"I'm Michaela," said the young woman. "Can we sit?"
Ankit saw that the table had become two chairs. She hated this place, its expensive and unnecessary technology. Some Health facilities were nicer than others, and few were "nicer" than this, but they all shared the same redundant proliferation of scientific equipment, the telltale traces of the massive investments made in health care during the Cancer Years.
"What's going on?"
"I'm so sorry," Michaela said. "But your body scans showed anxiety, tension. Your mother is not permitted proximity to those things. You understand, I'm sure. As a blood relative, especially?"
"No," Ankit said, "no, I do not understand. What does my being a blood relative have to do with anything?"
"Ah," Michaela said, frowning down at something on her screen. She had blurted something out, pertaining to some classified aspect of her mother's condition. Something Ankit was not supposed to know. "Mentally ill people are especially sensitive to the emotional states of others."
"I know that. Of course I know that. But you said—"
Michaela stood. The software would be guiding her through this stage of the conversation, surely. The Oh Shit You Fucked Up AI, Ankit thought. Would it cost her? Did they keep track of every little error?
Her mother was Code 76. That meant that someone, maybe in Safety and maybe in Health and maybe in the home nations, had decided that the details of her confinement constituted a protected secret. Whatever was wrong with her, what events had precipitated it—even her name—someone had convinced someone else that these could never be revealed. Ankit had always known this. She'd made her peace with it, the way you make peace with what can never be known. Digging into things like that only made you angry, started you down dangerous pathways, got you in trouble. But when she walked out of the Cabinet that day, when she tossed her bouquet of flowers into the sea and shrieked into the rising vortex, she decided it was time to get in trouble. To dig. To find out why her mother was in there.
To maybe possibly somehow get her out.
## Kaev
_S o. Now you're on the_ _run. Was it worth it?_
Safety would certainly have ID'd him from the clips of his assault on Abijah. The crowd calling his name. By now they could be camped out across the street from his shipping container, waiting to arrest him. Not to mention Go, who would be furious, possibly furious enough to scoop him up before Safety got him and spend a day or a week carving him up before handing over whatever was left for arrest.
Yes. Yes, it was worth it.
And then there was Go's target, the powerful man or woman who employed his two victims. Whoever they were, they'd have minders, watchers, microdrones maybe, and they would have figured out who he was as well, and were even now working out an abduction plan, were engaging the services of an old crusty interrogator from one of the fallen superpower states to torture out the name of who had hired him.
At this, at least, Kaev smiled. He'd give them Go in a heartbeat.
But it wouldn't help; they wouldn't trust a name easily given; they'd still have to torture him to verify the truth of what he said.
But that wouldn't matter. Pain he could handle. He'd had plenty. They'd still be after Go at the end of it, and maybe they'd get her.
Kaev walked. Up and down Arm Six, and then Arm Seven. Head full of screaming; the roaring of savage beasts; the orgasmic cry of the crowd when the fight was at its peak, when he'd given them something beautiful, something to help them break free of the moment, their lives, their city, the weight of their slowly dying bodies . . . And what had he gotten for his troubles? Add up all the joy and pleasure he'd brought to the people of this city, and what had he received in exchange for it? Barely enough money to eke out a subsistence living. He wasn't angry, not at them. They didn't owe him anything. They paid their money; they had their own troubles. It was the city he was mad at. The city that he loved. He wanted to punch something, punch everything, pin it down and snap its neck, this squirming tentacled mass of thoughts and whispers and memories and contradictory beliefs that screamed and gibbered inside his head.
Back down Arm Seven; through the Hub; onto Arm Eight. A slide messenger sped past, ululating all the way. He could hear himself yip and caw, could not make himself stop.
Someone was singing in a high window. People were making love all around him, in the darkness of bedrooms and alleyways and coffin hotels. People were dying. They crowded him, pressed on his skull with a more-than-physical pressure. He wanted to scream, but he was good at not screaming. He spent most of his time not screaming even though he wanted to.
He walked.
And then: he stopped.
Because the pressure ceased. The screaming and the singing evaporated. The fog lifted. Peace flooded him, a peace like nothing he'd ever known. A quiet. Shivers climbed his spine, building in intensity as they went.
He looked around. Saw no one. For such an overcrowded place, Arm Eight had a weird way of feeling completely empty sometimes. Boats rose and fell with the waves on his right, and on his left were a series of squat strutted buildings on a floating platform.
He took a few steps farther out onto the Arm and felt the peace subside just the slightest bit. Felt the squirming thing in his head start to gurgle again.
Kaev returned to where he'd been. Took a breath; basked in the silence. Then took several steps back, toward the Hub. Again the gurgling rose inside him.
So. Just one spot. Okay. Kaev didn't wonder why or try to investigate what, exactly, was having this effect on him. Poke around too much at something good and you tend to find something bad. Depleted uranium, probably, the weaponized stuff scraped from the wreckage of Chernobyl or Hanul, causing blissful sensations as it killed off brain cells by the thousands. If so, better to let it kill him swiftly and pleasantly. So he sat down on the freezing metal of the curb, beside the clamps where the building platform was docked, and hugged his knees to his chest to conserve heat, and shut his eyes, and let hot tears of happiness warm his face.
## Ankit
Ankit began with the easy stuff. The human stuff. She called her contact at Health, second assistant to the director. A sweet boy, one of a couple dozen agency flunkies she forced herself to be friends with, sending regifted tickets or day passes that had been given to Fyodorovna, even meeting up for drinks or karaoke or isolation tanking when her calendar reminded her it had been a while.
"One of my constituents' mother-in-law got in trouble," she said. "Sad story, really. Locked up. The Cabinet. I felt so bad for her . . . and she's been a loyal donor . . ."
Joshi promised to look into it.
Next, she called her woman at Safety. Spun the same sob story. If there had been a violent incident behind her mother ending up in the Cabinet, Safety might have a record of it.
"We don't use Health's numbering system," she told Ankit. "So that patient number won't be helpful to us."
"Really?" Ankit said, surprised. "Agencies don't coordinate that kind of thing? Seems inefficient."
"When things are inefficient, there's usually a reason for it. Things are only supposed to run so well in Qaanaaq. Efficiency is expensive. Helping people get what they're entitled to, solving problems quickly, that sort of thing is costly."
"Fascinating," Ankit said, unfascinated, and angry at her for the civics lesson. "But if there was a Safety case that ended in the subject being confined to the Cabinet, you may have her patient number in the file, yeah?"
"We might . . ."
"You can check?"
"I could . . ."
"Great!" Ankit said, imagining stabbing the woman in the throat. "Thanks!"
She scanned software markets while she waited for her humans to do their work. Humans were slow and sloppy but comprehensible. Softwares were spooky, messy, working in mysterious ways, full of secret tics and legacy apps that could bring a whole heap of trouble on her head. Suppose she bought a breach hack, and it found what she was looking for, but the packet had snitch software attached? Law enforcement, crime lords, black-budget AI could rain hell down on her at any moment.
People made fun of them, called them old-fashioned and inept, but Ankit decided to take her chances at a human brokerage. Software archaeologists or engineers might not be able to match the breadth of scope of a machine broker, some of which could scan through thousands of softwares a second to find the right one for the job, but they made up for it with the depth of their knowledge. Even when scanning software could parse every packet of code, foresee every potential trigger, it was notoriously bad at assessing outcomes. A human broker, at least, was working with a smaller number of apps that they knew well and had seen in action enough to know more or less how likely a given software was to fuck her over. She called her favorite.
"What's the target this time?" Mana asked.
The past three elections, Ankit had had some reason to call her. Shipping manifests one year, and Emirati birthrates the election before that. Access to an opponent's email during Fyodorovna's first campaign, when things were extra ugly.
"The Cabinet," Ankit said. "I need to know why someone is in there."
"That's . . . difficult."
"Difficult. Not impossible."
"Correct." She told her the price. Ankit had to fight not to flinch.
A quarter, maybe a third of their entire campaign budget. Two weeks' worth of robodustings; a week of blips. The right thing to do was go to Fyodorovna, get her to sign off. She could talk the woman into anything, always. It wouldn't be hard . . . just time consuming.
But going through Fyodorovna meant putting her neck in the noose. It meant that when the Arm manager lost the election, she'd blame Ankit for it. Which would fuck up Ankit's only hope of avoiding the gutter.
Because Fyodorovna would be fine. Arm managers always were. Some charity or shareholder or someone else she'd done favors for would find her a place. Always helpful to have a somewhat famous face up your sleeve, to impress difficult clients or charm potential money. And wherever she landed, she'd need a chief of staff.
"Sold," Ankit said.
"Do you want to run them, or should I?"
"You do it," she said, signing off on the standard dummy invoice she always billed her. Security for a campaign donation site, on the fraudulent letterhead of an actual, legit software security firm. An audit from Finance or Campaigns might ping the fact that the firm in question never filed a corresponding invoice, but audits were like shark attacks—really terrible, and really unlikely.
On the grid, Ankit almost collided with a shopkeeper arguing with a woman who appeared to be swathed in loops of a hundred different brightly colored fabrics. Her jaw bug translated as best it could, between Mandarin and the jerky pan-glossic stew that late-stage breaks tended to induce:
You can't stay here. Go over there, maybe. They serve food at the Krish—
We made fire. All of us together. We set them all on fire.
You need to go. I don't want to call Safety. You're scaring my customers.
Your customers should be scared. We are made of fire and we will burn you all to ash.
Joshi won first place, pinging her an hour later to say he'd come up short. The file had three separate injunctions on it, something he'd never seen before, and who was this, the Nineteenth Dalai Lama?
"Maybe," she said. "That's what I was hoping you could tell me."
The software broker came in second, calling near midnight the next evening. Her friend at Safety never got back to her at all.
"Sending you something now," the software broker said. "More than I thought I'd find, actually."
For a split second, Ankit debated saving it for the morning. She was exhausted, and the message Mana sent held dozens of files. But she knew she'd never be able to fall asleep, wondering what she had. She couldn't even finish brushing her teeth before doubling back to the bedroom and opening the first folder.
Four hours later, Ankit was still awake.
Two hours after that, Ankit called in sick.
Which wasn't a lie. Dizziness made her light-headed; the whole city seemed to be spinning. Staggering, sickening, the information she held in her hands.
_Too weird. Too fucking weird._
The whole thing felt wrong, uncanny, like a dream where the world was sideways in some subtle, unsettling way.
Maybe her job _wasn't_ in danger. Maybe Fyodorovna _could_ win. Maybe she _could_ get her mother out.
One solution to all her problems. All she had to do was bring down a shareholder. Simple. What did it matter that it had never been done before? She had a weapon now. Maybe a couple of them. Maybe there was a chink in Martin Podlove's armor.
And anyway, lots of things had never been done before, and then they were done.
## Soq
Blackfish Woman was a ghost. Glimpsed from afar. Impossible to touch. Every day new sightings. In the water, on boats, on the grid. With the polar bear or astride her whale or alone, but never without her blade. Always moving. Since the slaughter on the Sports Platform she'd been impossible to pin down. If anyone knew where she slept, where she moved her rig to, money or fear or respect kept their mouths shut.
Soq made maps. Drew lines connecting the dots of the places she'd been seen. Looked for patterns. Found none.
Soq collected pictures. Soq asked questions. Noodle vendors and ice scrapers and boatmen and algae vat stirrers were all only too happy to talk Soq's ear off with every little detail of their sightings. Nor was Soq the only one asking them to. All of Qaanaaq was talking about the orcamancer. Soq wondered how many of them were asking from simple curiosity and how many were like Soq, working for someone else, crime bosses or shareholders or the intelligence agencies of foreign nations, for whom the orcamancer represented an unsettling, inexplicable, existential threat.
There was one thing Soq knew, that none of them knew. That Go didn't know. That Soq kept secret, without knowing why.
Killer Whale Woman was afraid of the polar bear. The polar bear would kill her if it had the chance.
What did that mean?
Impossible to research her without delving deep into who the nanobonded were, how their tech or magic worked. But what was real and what was lies, legend, misunderstanding? Many sources said they could each bond to only one animal. Others told stories of women bonded to whole flocks of birds, a man who headed a pack of wolves. At any rate, killer whale woman was clearly not bonded to the polar bear. So . . . why was she traveling with it?
All of this was extracurricular. Soq still made slide deliveries. Dao had not discussed a pay rate for Soq's research when he handed over the assignment.
Nor did he mention one four days later, when he buzzed Soq.
"Progress report," he barked. "What have you got?"
"A whole lot of not very much," Soq said, stopping at the entrance to Arm Three. They kicked at the cylinders set into the ground, the thick forest of bollards that could be raised in times of trouble to divert demonstrations or thin out unruly crowds. Soq delivered a long list of scraps, so flimsy that they felt compelled to apologize for it.
"That's fine," Dao said. "That's excellent, actually."
"No, it's not," Soq said. "I've got nothing. Nothing Go couldn't find out herself by doing some half-assed web searches."
"What would it take, to get more?" Dao asked.
Soq laughed. "If money was no object I'd say send me to Nuuk. That's the last grid city she went to before coming here. Killed a couple of families of boat people. Let one guy live. Allegedly. Number one lesson learned here is that nothing is ever certain when it comes to her."
There was a long pause, like maybe Dao was writing something down.
"So . . . are you going to send me to Nuuk?"
Dao laughed. "No, Soq. She doesn't want you traveling yet."
_Yet._ Soq trembled at the promise in that word.
Two days later, though, Jeong pinged Soq an address. "Not a delivery," he said. "Dao says it's a meeting."
"Huh," Soq said.
"When did you get so important that you get to have _meetings_?" Jeong said. His chuckle was not without pride.
"When I became a spy," Soq said. Their sense of power lasted all the way down Arm Seven, to the address Jeong pinged, past a blue-striped monkey fighting with an otter and a wild-haired woman asking passersby to help _burn it all down,_ and onto a decent-sized old houseboat and in the door, and then burst like a bubble at the sight of a man strapped to a chair.
"You made it, good," said Dao, playing with a shape-memory polymer perched on a windowsill. He tapped at his screen and it transformed from a bird to a ballerina, then he grabbed it and crushed it in his fist. "I brought you what you asked for."
Bruises crisscrossed the man's face. There was blood on his clothes. "What I . . . asked for?"
"The survivor. From Nuuk."
"Shit, Dao," Soq said. "What the fuck did you do to this guy? I didn't ask for this. I didn't want anyone to get hurt—"
And Soq knew, hearing the words come out, how stupid they sounded. Dao at least did Soq the favor of not saying any of the things Soq knew he could have said. _Did you think working for a crime boss would be bloodless, painless? Did you think your hands would never get dirty? What did you think we do here, exactly?_
Dao did say: "He's yours now. See what you can get from him just by being nice, and call me if you need some help or advice being . . . not so nice."
"And then you'll . . . ?"
"Shoot him in the head; take him home and give him a bubble bath; I don't know, Soq. We'll have that conversation when we have it."
Dried fish guts caked the floor. Crab or lobster shells cracked underfoot. On an ancient wooden table, a methane burner and a wok and two green glass bottles. Did the boat belong to him, the poor beat-up man bound in front of Soq? Had Dao's soldiers dragged it all the way from Nuuk? Or was it one of a thousand syndicate hiding places here in Qaanaaq, perfect for carrying out all kinds of illicit activity?
"Untie him," Soq said.
Dao bowed and did so. "I'll be outside," he said to the man. "Don't do anything stupid."
The guy was young. Bearded, burly; his head wrapped in an American flag bandanna. It was faded and filthy.
"Tell me about the orcamancer," Soq said.
His mouth opened like he had something smart to say, but then he thought better of it. "Who are you?"
"My name is Soq. I work for an entrepreneur here in Qaanaaq."
"Uh-huh, _entrepreneur_. And your people brought me all the way here to talk about the person who killed my parents. Why is that?"
Soq was about to answer, but then realized—they didn't _know_ why Go was so interested. Dao said Go didn't want any unknown variables introduced into the equation of her power play, but what if that was untrue, or half-true? What if Soq's information gathering would be used to harm the orcamancer? Now wasn't the moment to ponder that question too seriously. And in any event, one of the many life skills Soq had learned in foster care: the best way to get information out of someone is to tell them what they want to hear. "Maybe my employer sees her as a threat. Maybe she has a vendetta against her for something. Maybe she wants to destroy her, and your information could help us achieve that."
He smiled. "I was at work. I clean lobster pots. I came home—my parents, they brought this boat with them from America—and they were . . . dead. My grandparents, too. And she was still there. Like she was waiting for me. Sitting right where you're standing now."
Soq looked around. So this was where it happened. They shut their eyes and tried to picture it.
"It stank in here, so bad. She was sitting on the floor, covered in blood, looking like she was praying, or meditating, or, I don't know, taking a fucking nap."
"The reports all said she hit you with the butt of her staff, and left."
"That's right."
"She didn't say anything to you first? She waited for you, knocked you out, and then left? Why wait for you at all?"
He looked up at Soq, his eyes steely. Bracing himself for pain.
"She didn't say anything to you?"
He shook his head.
"Hey," Soq said, pulling up a stray chair, scorched battered white plastic. Had it been there, then? "Come on. Help us out. We can help avenge them. Any little bit of information you might have, we could use it. Maybe she threatened to hurt you, if you told anyone—"
He spat out: "She didn't. She wanted me to tell."
"So tell me. You're safe now. We'll protect you."
He laughed, and Soq knew he was right to laugh. "You can't. My family, they weren't soft. They had guns. Weapons. Lots of them. And they'd . . . done things. Vicious things. To survive. But she took them out easy. Without using her damn . . . fucking . . . animals. Whatever she wants, whatever she came here for, she's going to get it. Fuck any _entrepreneurs_ who stand in her way."
"What did she tell you?"
He nodded. "She said my family deserved what she'd done to them, but I didn't. Told me to make sure everybody knew it."
". . . And?"
His face, a practiced mask of masculine hardness, sagged. Reddened. Broke.
"She asked about my uncle," he said, in gasps, like he was about to cry. "Asked where she could find him."
"And you didn't tell her. At first."
"At first. But she . . ."
"She hurt you," Soq said.
"She hurt me. She threatened . . ."
"To hurt you worse."
"Yeah."
"So you told her. Where to find your uncle."
He was weeping now. "He was working an ice ship. I told her the name of it. That's all."
"And then you heard . . ."
"I heard he was killed. Week, two weeks later. He was all alone on the glacier, working an ice saw, and they said he must have fallen . . . Forty stories high, that glacier. Onto ice. Happens all the time, they said. And he'd been out there for a couple of days by the time they found him. Said those bite marks, those missing chunks, anything could have gotten to him after he fell."
Soq stepped closer. Tapped his bandanna. "Your family—they're from the States."
"Proud Americans," he said. "Always will be."
"Even when there is no America?"
"It's in our hearts."
Soq imagined the hearts of his parents, and grandparents, skewered and plucked from their bodies through a shattered rib cage and stomped on.
"Religious?"
"Sure."
_Of course. They participated in the nanobonder slaughter. That's why they were butchered._ Soq swooned to think that the Killer Whale Woman was strong and wise enough to find and punish the guilty, even decades later.
"What did they do, I wonder? During the migration?"
He sniffed, a wet thick sound. "What they had to do."
"Including killing people?"
"You soft fucking city people can't even imagine. What it was like. Everybody trying to kill everybody. Blacks against whites, immigrants against citizens. If you didn't have guns—lots of them—you were going to lose everything you loved."
He didn't know, either. He'd have been born long after and raised on stories. Soq had seen those documentaries. The narratives of fear, of lies, of They Want to Destroy Us. There were so many movies about how easy it is to manipulate people, and what atrocities you could get them to cheerfully commit while believing they did it for the sake of their children's survival.
Soq shut their eyes and they could smell her, the orcamancer, in the room with them—could feel her rage, the righteous thrill of hurting the people who hurt her people. She was not a young woman. She'd traveled for so many years. Had vengeance been the only thing, the only thread pulling her forward? Soq opened their eyes. That couldn't be all. There had to be more to her than bloodshed, violence, punishing the guilty. Soq couldn't say why they thought that, why they wanted so badly to believe it.
_She is looking for something. Someone._
"She didn't say anything else?"
"Nothing."
"Thanks," Soq said, putting a hand on his shoulder. They regretted not asking the man's name.
The bearded man said nothing. Tears were rolling down his face. He cupped his hands in his lap, an oddly pacific gesture for a man so full of hard angles and coiled anger.
Soq left. Dao waited on the deck in a cloud of pine needle smoke.
"Got something?"
"Something," Soq said. "I'll write it up for Go."
What would that write-up say? _Talking to this man, standing in that space, I gained a profound spiritual understanding of who the Blackfish Woman is and it is of absolutely no strategic or practical value to you._
Soq hopped from the boat to the grid. "And Dao?"
"Yeah?"
"Whatever you do to him? I don't want to know."
Dao frowned. "You won't always be able to hide, Soq. From the consequences of our actions."
"I know!" Soq said. "But I want to hide today."
## Kaev
For the third time in twelve hours, Kaev crossed the Arm to pee into the ocean. Even those few steps cost him, caused a slight shrinking of that blissful calm, and as soon as he was done he hurried back to the bare metal bar he'd been sitting on.
He was pretty sure he'd slept, some. Hard to say. His body felt like it needed nothing, not sleep and not food and not sex, and not fighting, which for the first time in his life held no appeal for him. Everything ached, but nothing hurt. A noodle stall had set up beside him, and occasional wafts of warm sweet-savory air hit him, and those were nice, but when they weren't there he didn't miss them.
Narcissus. The man who fell in love with his own reflection and wasted away staring into the water. Why did he remember that? Why did his mind work now? Why did school, math, history, myth, all the things that had eluded him, refused to cohere in his mind, suddenly make sense? Where memory had been a churning sea, uncontrollable, spitting up unwanted objects and hiding the things he most desired, it now yielded easily to his wishes. He thought about his childhood, and there it was. Foster homes. Even: further back, to times he'd never been able to call to mind, to places that weren't so much memories of things as of feelings, of safety, of fear, of flight. Other people: a mother, a sibling—or were they a grandmother and a pet? Because in that stage of pre-memory there were no words to attach to things, no societal structure to plug people into.
"Hemorrhoids," said the noodle vendor. "Sit on cold metal too long, you'll get hemorrhoids."
"Thanks," Kaev said, but he did not move. She took a crinkly green tarp, folded it into an approximate rectangle, handed it over. He took it, bowed in gratitude, and sat on it. And it did feel better.
She was popular. The docking bar and every other surface people could sit on were crowded with customers slurping down bowls of broth. But they came and went, for even people with nowhere to go weren't eager to remain in the cold too long.
Kaev knew that he could not stay there forever. Sooner or later Safety would come, tell him he had to move along. And when he refused, they'd send Health. The windscreen magnified the sun's heat by day, but now the sun was setting, and people who chose to stay unsheltered overnight could be deemed ipso facto insane and taken off the grid by force. And Kaev could not afford another trip to the Cabinet right now. Even if it had helped, slightly, before. The cost was too great. And whatever benefit he'd derived from his time there was gone the second he left the building. So if Health or Safety showed up and tried to force him to move from this spot, he'd be obliged to beat them senseless. And then he'd have an even bigger problem.
"You're new," the noodle vendor said.
"I'm old," Kaev said.
"Not as old as me," she said, and laughed. He'd been sitting on the tarp for an hour, maybe two, for all he knew a week. The flow of customers had ebbed. Once again Arm Eight looked empty, though he knew it wasn't.
"Here," she said, and handed him a bowl.
"I can't," he said. "I don't have any money."
"Of course you don't," she said, and laughed again, and he loved her, but then again, in that stretch of bliss he loved everyone. "My noodles are irresistible. If you had any money, you wouldn't have been able to go this long without buying a bowl."
"Thank you," he said.
"And anyway, you paid for it already."
"I did?"
"You gave me something just as valuable," she said. "I saw you fight. Hao Wufan. Shame about him."
"Terrible," Kaev said, lowering his face to the broth, letting the steam warm him, wondering what shameful fate had befallen the boy.
"Some people aren't ready for success," she said. "You fight beautifully, though. I've seen you before."
"I told you," he said. "I'm old. Been fighting forever."
"A journeyman," she said, and so she was a true fight follower, someone who could appreciate his role in the ecosystem. A rare thing. He still got recognized sometimes, and that was always nice, but most casual fight fans considered him a chump, a loser. "The noodles are a belated payment, for all the pleasure I took in watching you fight."
"The pleasure is always mine, believe me," he said, and handed back the empty bowl. "Thank you. The noodles were delicious."
Eventually she too departed, taking back the tarp with an apology, telling him she didn't want to see him there when she returned in the morning.
It happened soon enough after her departure that he knew they'd been waiting for it. Twelve men, dressed in the nondescript black of syndicate security, clutching obvious weaponry inside their jackets. One of them was Dao, who smiled and did not hurry. Kaev's thigh muscles tightened, preparing to leap into horse stance. He took off his hood, which would impair his peripheral vision. Cold wind sharpened his senses.
Something came up from the water across the Arm from him. A sea lion, he thought at first, turning around, because there were lots of those that lived on Qaanaaq's garbage and fish-gut castoffs. But it was big, bigger than any sea lion, bigger than any animal Kaev had ever seen alive, up close, and white—
The polar bear opened his eyes and looked at Kaev.
In the instant of that eye contact, Kaev felt like he had broken free of his body. A happiness surged through him, warm as the sun, blissful as a thousand orgasms. The peace he'd felt while sitting there had been ten times greater than the joy of fighting, but this new sensation was ten times greater than that peace had been.
"Hello, Kaev," Dao said. He and his soldiers had their backs to the grid edge; they could not see the polar bear. "You've been sitting here for a long time. I've got to presume that means you wanted us to find you."
But Kaev could not hear him.
_We are one,_ he thought, eyes locked with the animal's.
And it felt: Different. Stable. Like if he looked away, like if he took a step back, it would not diminish. Like now that he'd found it, now that they'd recognized each other, they were linked, and nothing on earth could break that connection. Like nothing could hurt him anymore ever. Like nothing confused him; like he saw how the world worked in a whole new way.
"Dao," he said, blinking, turning. "You should probably leave now."
The man laughed. A couple of his soldiers followed suit. "That's not going to happen, Kaev." He held up a handful of zip ties. "Are you going to let us put these on you? Or is it going to be a whole thing?"
"You need to leave. Now."
More laughter. Kaev took a step forward. One of Dao's men shot his arm out, flung something, a sharpened shard of windscreen glass, Qaanaaq's own homegrown answer to the _shuriken_. It struck him in the cheek with terrifying precision—a warning shot, but a stern one.
Kaev grunted in pain, but the grunt was bigger than him. With a roar, the polar bear pulled itself out of the water mere feet from where the men stood. Water coursed off it, like it was made of water. And it was as fast as water, as implacable. It swung its bulk around to knock the man into the sea and then dove in after him, all before any of the others had had time to aim and shoot.
And Kaev shut his eyes and he was _in_ the water, he was biting into the man's arm and pulling him down, down, until he could feel the warmth of the geothermal cone, until the man ran out of air and opened his mouth and breathed in water, and the bear released him and swam for the surface—
Kaev opened his eyes to see the men scrambling, aiming weapons into the water. Nervous, yelling, unsure where the bear would emerge next. And Kaev knew, somehow, that the bear could see what he saw, could tell where the men were standing. So it knew the safest place to emerge, to take them by surprise. In the instant before it did, Kaev gave out a shout. He ran at them so that they turned, aimed weapons. One of the men was yanked back before he could pull the trigger, the bear grabbing hold of his leg and pulling him into the water, breaking that leg effortlessly and then the other one, and Kaev was crouching down to avoid the shots from the others, slamming into another man, knocking him off balance, taking the gun from his hands and turning it toward the man's accomplices, holding down the trigger so that bullets splattered indiscriminately.
He grinned at the fear on their faces, caught between him and the bear. Two expert killing machines. One thing, one organism. Acting in concert in ways that had nothing to do with language, planning, rational thought. One animal. Dao was yelling orders— _Focus on the bear!_ —even as he ran for safety, saving his own skin—but they were not enough; these people with their mighty weapons and separate fragile minds could not get past themselves, could not trust one another, could not know what someone else was supposed to do. One bullet, two, struck the bear. Kaev felt the pain of them, but he also felt the animal's comforting fearlessness.
Animals exist in the moment. They don't worry about whether they will bleed to death, whether they will die. The wounds were minor. Their enemies fell swiftly, terribly.
## Fill
_R ound doors. Frosted windows that_ _let in light but nothing else._ _Leather straps, stinking of the fear-sweat of strangers._
The breaks caused dreams to creep into waking life, made him wonder whether anything he was seeing was really there, whether anything he remembered was really his—but they also changed his dreams. Stretched them out, tightened their walls. A short nap might leave him with hours and hours' worth of remembered dreams. And Fill could no longer wake up from a nightmare, no matter how hard he tried.
_Injections. Isolation. A dark shape flapping past the window._
This one was the worst. An instant before, he had been dozing on a bench in a greenhouse park boat, and then—blink of the eyes—he was here. Confined, chained, bound like Prometheus. Strapped to a bed, knowing in his gut that no one was coming to rescue him. Praying for it anyway.
The Cabinet—it had to be. Days and days of it. Months. Years.
Who was he? These memories belonged to someone. They had to. He ran his fingers over his face, tried to piece together what he looked like, but his mind was numb with pain and loneliness. He could remember himself. His name. His research: the Disappeared; shareholder privilege; no clean way to make a hundred million bucks.
Most people see only one Qaanaaq. They live their lives inside of it. The Arm where they reside, the nook where they work, the friends and family who make up their world. A private Qaanaaq, uniquely theirs, shaped by history and mental health and their socioeconomic positioning. Some people manage to move to a second Qaanaaq, when fortunes shift in one direction or another. Perhaps it will be a better Qaanaaq; usually it is an uglier one.
You and I are fortunate. We can see so many. We can move from city to city, Qaanaaq to Qaanaaq, see what our neighbors see, step into their stories. Randomly at first, too fast to control, but soon you will learn to summon them like memories.
Something else was happening to him. Something more exhilarating than frightening. He was hearing _City Without a Map_ differently now. He no longer felt so much like an outsider. Sometimes he even thought the broadcasts were meant for _him._
Maybe they were wrong, all of them, imagining it to be a guide for new arrivals. Maybe the broadcasts weren't meant for immigrants at all. What if they were meant for people with the breaks, whether they were newcomers or not?
Somewhere before dawn, his jaw buzzed.
"What do you know about the Reader Hunters?" Barron asked, his voice excited, attenuated.
"Waste of time," Fill said, deciding not to get indignant over being called so fucking early. " _City Without a Map_ aficionados who make it their business to hunt down the people who narrate the episodes, in the hopes that talking to them will help them track down the origins of the broadcast."
"You don't think that's intriguing?"
"I think it's unlikely to yield anything helpful. And even if you could track one down, I have to imagine that the Author instructed them to never reveal their origins. Or that there are double blinds in place, and they've never even had any interaction with the Author."
"Perhaps," Barron said. "But wouldn't you like to try? If you could track down a Reader?"
"Of course."
"Well. I have one."
"You . . . found a Reader?"
Barron made an affirmative noise. Fill said nothing. His mouth felt dry. He drank coffee. It did not help. Why was his heart so loud? "You . . . yourself?"
"A friend of mine, from one of the forums I'm part of. A total coincidence, really. A casual listener, he had just heard a broadcast, and then went to a fruit vendor, and when the woman opened her mouth, he knew."
"If that's true, and he published her location and identity, she'd already be besieged by _City_ devotees. By the time we got to her . . ."
"That's just it. He didn't publish it. He told me directly."
This felt too good to be true, but Fill was feeling melodramatic, self-pitying, heedless of consequence. He got the alleged Reader's address and agreed to meet Barron there.
The dream again. Plunged back into it, like the floor opened up and dropped him into the frigid sea. Shifting, speeding up, coming to a stop. Whoever they were, this person whose memories he was locked into, things had changed, for her, ten years after arriving there. Her isolation ended. She was let into the light. Change in staff; change in policy; an obscure order from inscrutable software. Access to common areas, supervised at first, and then not. Conversations. Friends. A slate, even—unnetworkable, but loaded with approved texts, and a stylus for drawing. Fill watched thousands of sketches shuffle past. Birds, over and over.
_Eagle, flying. Eagle standing over its nest. Eagle falling._
Grief: crippling, murderous. Pain like nothing he'd ever felt before. Pain—and guilt.
This was real. People lived like this. In his city, the one that belonged to him, the one that had fed and pampered him, given him everything he ever desired, the city his grandfather had helped build.
"I'm sorry," he said, and he was, and he screamed it, and that was when he woke up.
## Masaaraq
People still came with cameras, in those days, telling us they were going to Tell the Whole World about what had happened to us, what was still happening to us. The poor oppressed nanobonders! Helpless victims of a savage slaughter! They came, young and full of energy and faith, not to blame for having spent their lives in enclaves of safety when the rest of us were sunk deep in utter shit. They always seemed dissatisfied with us, sad we weren't more grateful, resentful of our sullen faces that ranged between apathy and hostility. Offended that we didn't smile, shake hands, make friendly small talk, treat them with the openness of spirit that must have been common where they came from. It wasn't their fault that they were ignorant of how the world really worked. They didn't see how courtesy assumes a certain degree of common ground. Everyone can afford to be nice to each other, when no one is trying to exterminate anybody.
They thought we'd befriend them. Every one of them thought we'd make them one of us. They wanted to be bonded, to be special, to fly with eagles or howl with wolves. They saw my orca and thought I was a god. I wasn't a god. Gods can't be killed. Gods don't live like refugees, watching their loved ones murdered.
None of us bothered to explain it. They wouldn't have understood, wouldn't have believed. They wanted so badly to think that what they were doing would make a difference, that once they Told the Whole World something would happen, someone would save us. We were a Good Story. They thought that was enough. Victims of the Multifurcation. One of thousands of communities trying to do its own thing, being pursued by another one (or more) of those communities, with the federal government completely gutted of its ability to protect anyone. The beast was starved, its claws and fangs plucked, the Supreme Court unable to muster much respect for its rulings ever since the bombings forced it into hiding.
They didn't understand, these pretty kids, but they'd find out, sooner or later. And there was no point in befriending them, opening up our hearts, because sooner or later they'd be leaving. And never coming back. When they stopped coming we were relieved, and sorry for whatever new poor fucks were the latest Good Story. Alone, we didn't feel the need to keep our faces hard, our emotions buried, our fear fettered.
In the mornings it would be the worst, when Ora would bring the kids to the school that was barely worthy of the name, one room where every child from five to fifteen sat and tried their best to learn from a man who'd never taught a day in his life before the troubles came, and we hunters would head out ourselves, sometimes only six or seven of us, our animals scraggly and thin and hungry, their friends gone, their hunger and their loneliness echoing heavy inside our heads.
That's when I felt it so hard I thought my heart would break. When I knew how fragile it was, what we had, what was left, and how swiftly it could slip through our fingers.
And then I'd come home from the hunt and see the massive bird circling in the sky, Ora's black-chested buzzard eagle, its impossibility, its magnificence, and think that if such a perfect creature could come into existence maybe we had half a shot, maybe the world wasn't fundamentally, existentially fucked.
They went on and on, those abandoned suburbs, those rows of emptied houses where the water was poisoned or the highways gone, those communities that depended on dismantled transit systems, jobs in cities that had become savage hellholes, each one hosting a series of small-scale civil wars that added up to mass evacuations, warlord takeovers, synth-biowarfare retaliation. We stayed in those beautiful houses for as long as we could, and then we moved on.
We'd been in that particular village for six months then, and there were only forty of us. Six months back, before the last surprise slaughter, we'd been a hundred. A year before, more than two hundred. Again and again they found us.
Sometimes we'd meet other communities, nomads like ourselves or settlers clinging hard to a single block or spread of buildings. Some of them were awful, although the worst of the warlords stayed south of the old border from a malignant, terminal case of patriotism, which was part of why we fled to Canada in the first place. Most of those we met up there were decent, good people trying to survive, usually with some kind of unusual belief or practice or technological thing that had gotten them ostracized from wherever they came from. Once in a while, when the winter was bad or a crop needed working and our communities decided to link up temporarily, we'd talk internally about opening up to them. Admitting them. Sharing our blood; letting them bond.
Blasphemy, unthinkable. Some of us, the thought of it made our skin crawl. But that wasn't why I said no. I said no because sharing our blood meant passing on a death sentence.
We knew they'd never stop. They'd find us, they'd come for us. The Scourge, the Plague, the Pestilent, they called themselves all kinds of names they thought sounded scary, but we knew them for what they were: poor dumb hungry fools like us, who'd had everything taken from them, just like us, whose anger turned outward because it'd been carefully stoked that way. Powerful people made bad decisions that brought the whole country sputtering to a bloody standstill, thousands of people who had been part of the problem, caused something catastrophically bad to happen, and every one of them found a scapegoat. A few got caught, sent to jail, strung up, kidnapped, and beheaded on the net for all to see, but mostly the Bad Guys sicced their victims against each other and snuck away while the poor fucks were scratching each other's eyes out.
For the pharma corps, that was us. Us, and a handful of other communities of people who'd gotten terrible afflictions or terrifying gifts, or, more often, one that was actually the other. Deregulation had been ugly. People were tested on without their knowledge, or lied to about what was being tested and why.
We shouldn't have existed. We were proof that somebody had been up to something terrible. And that somebody skillfully inflamed the passions of a bunch of fundamentalist gun nuts, talked about us as abominations, breaches of God's law that mankind should have dominion over the animals, and of course those poor stupid fucks were only too happy to believe it, too eager to blame a bunch of people who were different and just wanted to be left alone. Ages ago, it had been the Immigrants, or the Blacks, always someone to push around, what this country was built on, _I've taken everything from you and now I'm going to tell you it's your neighbor's fault because he looks different from you_.
Kids at school, Ora working, and me out on the hunt, that's when I felt the fear. That's when I knew how helpless we were. I felt it all through me and I wanted to stay home, never let them go, stand in the doorway waiting with my weapon for the bad men and women who would dare try to hurt the ones I loved, even if I knew there were too many of them, that they had weapons we should have been terrified of, that I'd die swiftly.
A couple of traders, heading north, told us they'd seen the Plague ships. Dozens of pelts hanging from the sides. The skinned animal companions of our dead comrades.
"Could be a trap," some hunters said, so most of them stayed behind. I was the only one left with an orca, the one with the best chance of scoring intel and escaping, or inflicting real damage if it came to a battle. They sent me and I didn't say a word, not even when I held Ora and the babies to me and knew we might never see each other again. But that's what we were, what we lived with, what those kids with cameras could never capture. That's what we'd never let them see, because they had no right to it. No one did. Not them and not the people who would watch their work, the Whole World they were going to Tell, who would see us, and feel sad for us, and then go on merrily pretending the world wasn't burning down around them.
We went south along the coast. The waters got worse the farther we went, thick with toxic sludge, the food scarcer. We never found any sign of the Plague ships. We turned around, went back. Went home. The last place home had been.
Maybe it _had_ been a trap. A lie. But if it was, all it did for them was save my skin. Because when I got back to where I'd left them, I was the last of my kind.
No bodies. No humans, no animals. Lots of red and black blighting the landscape. Blood, and the charred remains of buildings. I found our bathroom mirror in the rubble, with the word _Taastrup_ on it. In Ora's handwriting. I chose to believe that she got out alive, took the two kids with her. Left behind our son's polar bear, whom I found hidden in the basement of the schoolhouse. He must have done it, I thought—the boy, he hid his bear to keep it safe, because she would never have allowed them to be separated, she'd have known better. I shivered, then, to think of that kid's life without the bear he'd been bonded to. And I swore I'd find her, find them.
We wept for a full day, Atkonartok and I. For our murdered kin. I lay on my stomach, on the ice, looking into black water. She circled. Each of us amplified the other's pain, echoed it back and forth, until I thought it would split us in two. Only hunger saved us. Hunger stirred her savagery, which roused my own, which stopped our wailing.
I brought her armfuls of bloody snow, hacked-off pieces, shreds of clothing. Atkonartok could tell them apart, our people and the people who hurt them. She could single out their unique pheromonic signature, singular as a fingerprint. She smelled their bodies, their sweat, their hair, their waste, their stories. From their smells she could see their shape, their weight, whether they were young or old or weak or strong.
Forty attackers, total. Forty monsters to hunt. She could see their outlines, so I could too. And so we moved on. Looking for our lost, the ones whose bodies we did not find, who we knew escaped—and looking for those forty outlines.
Taastrup, first. All the way, I watched the skies. Spent more time staring at the air than I did watching the sea or the land I traveled over. Looking for a black-chested buzzard eagle.
I knew it might take forever. I knew that by the end of it, it might be me who got rescued by her. I knew it might take so long that by the time I found her, she wouldn't be her anymore, and I wouldn't be me.
We found many of those monsters. In the cities of the land and the cities of the sea. Sooner or later, if they were there to be found, Atkonartok would catch their scent. I broke them apart or pushed them into the sea for her to tear to slow tiny pieces. Some we learned things from. The names and locations of their comrades. Others had nothing to offer, but their fate was the same.
Revenge was not my mission, but each new slaughter soothed the grief and rage I felt at being unable to find her. Find them. Murder gave me the strength to keep going.
My sisters, my mothers, the whole long line of generations: They come with me. With us. We carry them inside us. Our ancestors never leave us. Ora knew that, tried to explain it to me. She said our people understood death and loss and the legacies of our forebears. She said that we never lose the people we love, not even when we want to, and that's what the Western world had lost sight of, a lesson they forgot but we relearned, which is why the nanites didn't kill us, didn't drive us mad, gave us this gift, this curse.
Once, I saw one of them. The kids with cameras; not a kid anymore, and no camera. Standing over a trash barrel fire in a Scottish resettlement camp. I think she recognized me, but her face stayed as empty as mine was. I felt sad, then, for her, and angry at myself. I took that moment, that short time, to mourn, to be sad, to be angry, to feel emotions for her that I never let myself feel for me and mine, because we'd been born to this but she hadn't, and because people who only know suffering from stories are never prepared to find themselves inside one.
## Kaev
Kaev woke up in darkness. He heard water sloshing against the other side of a metal wall. Something massive breathed beside him. Where was he? How had he gotten there? He felt no fear, no anxiety. Dimly, he knew that this was wrong. He should have been terrified. But the realization vanished, and he slipped back into sleep.
To wake, hours later, to light. A narrow room. A high ceiling. The most comfortable bed he'd ever been on.
Which was breathing. Which was no bed at all, but rather a polar bear. One heavy furred arm lying across Kaev's legs. Ebony claws more than capable of tearing him in two.
Still, no fear. He knew, on a level deeper than the human, that this animal would do him no harm. That its happiness, sleeping peacefully, was his, and his was its. He lay there for a long time. Watching it sleep. Feeling his own thoughts come easy, a smooth unbroken flow.
Remembering. Piecing together how he'd ended up here. Wandering Arm Eight, his brain cracked, thoughts leaking out like always, and then—peace. A sensation so pleasant he'd stopped in his tracks, sat down, would have stayed there until he died.
And then killing a whole bunch of people.
"Hello," said a woman, who entered the room bearing two bowls. She was sturdy, muscle-bound. Her face was raw and bright from a lifetime of sun and wind. Long hair fell in a wide cascade down her back, with two small braids framing her face and curling under her chin.
She set the larger bowl on the ground beside the polar bear. The smaller one she handed to Kaev. "Good morning."
The bear came awake. The first thing it did was turn its head to look into Kaev's eyes.
He gasped. He felt tears well up.
"Can polar bears smile?" he asked.
"This one just did."
He laughed. The bear nodded its head vigorously, like maybe that's how polar bears laugh.
Kaev reached out his hand to touch the bear's face. It pushed its head into his hand.
"He looks old."
"He is old. For a polar bear."
"Am I like you?" he asked the orcamancer.
"You are," she said, and smiled, a smile every bit as wide and deep as his own. Like she, too, had found something she'd spent her whole life looking for. Except unlike him she'd had the privilege of knowing exactly what it was she'd been looking for.
"How?"
"You tell me," she said. "What do you know about your family history?"
"Not much. Raised as an orphan. Ward of the city."
"What about your mother? What do you know about her?"
Kaev shrugged. She looked disappointed in this news, somehow. But this was not surprising. If he was like her, if he was one of the nanobonded, there must have been a connection—a mutual family member, perhaps, someone she had come all this way to look for. He was a missing link; he could lead her to the person she sought. Kaev paused to revel in the clarity of his conclusions, the effortless way one idea connected to another. _My mother. She is here for my mother. She must be._
_For the first time, things make sense._
"Last night, I saw something," she said. "The bear's behavior changed. I knew that it had sensed you. Finally. I've been waiting for that to happen. That's why I was watching when those people came, and why I was able to unchain it in time to take care of them. Or rather, to help you take care of them."
He ate. The bear ate. Sea lion meat, it tasted like. Even the farmed stuff was fantastically expensive, although he suspected she hadn't purchased this so much as sent her orca out to bring one home. His was cooked and the bear's was raw. But he could taste what the bear tasted, feel the texture and the brine of the blood. Both were delicious.
"Kaev," she said, and squatted beside him, and hugged him so hard they both lost their balance, toppled over, laughed. "I am so, so, so happy to have found you."
Kaev smiled, unsure what to say. But it wasn't the normal pain of being bewildered by words in general, of even the smallest thing being too big for him to find the words for. It felt good, right, the bliss of emotions that need not be put into words. Had he ever truly had a conversation before? Had he ever been able to talk to another human being without watching every sentence crumble on its way out of his mouth? If so, he didn't remember it. He rolled over, from his back to his belly, arms spread wide to embrace his brother bear.
## Ankit
Context is everything," Barron said. Birds chirped in the background of wherever he was, or maybe they were people making bird noises. "To understand any social problem, you have to know what's going on around it."
He'd taken to sending Ankit audio files. His voice was avuncular, grandfatherly. He never answered when she asked him why he didn't want to meet or have an actual conversation. She imagined it had to do with his sickness. Self-consciousness over maybe being unable to answer a question, or losing his train of thought too easily. She listened for ambient noise disruptions, indications that he'd edited bits out, but the city's standard noise was chaotic and jerky enough to make it tough to tell.
Ankit said, "Play," and began to climb.
Context: When the first breaks cases started popping up, Qaanaaq had been a powder keg. Overcrowding; collapses of unsafe slum structures. Demonstrations. Many of the city's mass congregation mitigation measures had been introduced back then. Ankit remembered it, vaguely. A friend of her foster father's sitting in their living room, his brown face blackened with dried blood. Caught in a peaceful demonstration that became a street brawl when the slum enforcers brought out zap sticks.
Street protests were an oddity in Qaanaaq. Present—common, sometimes—but performative. Nostalgic. Like horse-drawn carriages in twenty-first-century cities. Immigrants from elsewhere believed in them, but Qaanaaq's native-born political activists treated them like parties, chances to take photos. With such minimal explicit human decision making, there were no targets to pressure, no places where a strategic crisis could force a policy change. You could call on an Arm manager to issue a statement, but everyone knew how little that could achieve. The real decisions were made by machines, a hundred thousand computer programs, and you could scream at a data server farm until you were blue in the face without getting anywhere. Even if a mob burned one down, there were dozens of backups, many of them floating in bubbles orbiting the geocone.
"Run!" someone called to her.
She'd never been to an indoor scaling course before. Like most serious scalers, she'd scoffed at the concept. Once you've been out there, hurtling through the frigid sky, the safe legal version seemed insulting. Nor could she say, exactly, why she'd decided to visit one now. But run she did, when the coach commanded it, leaping over foam obstacles and then flinging herself against a replica of a rotating cellular antenna and swinging around on it.
_Yes,_ she thought. _That is why I came here. The body has a way of thinking that is very different from the mind's. Maybe moving the old muscles will help me figure this all out._
Context: Barron's friend's molecular assembly machine could not produce __Quet-38-36.0. Some deeply buried safeguard stopped it.
Context: That had never happened to him before. He'd called friends, asked them to try, gotten the same result.
Context: For some reason, Quet-38-36.0 could not be produced in Qaanaaq.
"Jump!" the coach called, a split second too late. This coach was no scaler. Or if she was, she'd been so subpar she'd been forced to flee to the safety of a padded indoor course.
Context: Martin Podlove was scared of her. But he was scared of other things more.
He had refused her requests for a meeting or a call. The three times she'd gone to his office, intending to wait in the Salt Cave until he walked out and corner him then, she'd found out that he'd left hours earlier by a different exit. She'd drafted messages and deleted them unsent. She had to be in his presence. Had to corner him. Had to see him squirm, read his face for tells. And if she merely sent a written message, who could say whether he'd smugly send an enforcer to soak or slaughter her?
What she knew: Podlove had ordered her mother's incarceration. He'd slotted her Code 76. She didn't know why, and she didn't know what she could do about it. And she didn't know why his employees were being attacked all of a sudden—she'd seen the clips of her brother beating the shit out of a slum enforcer a day after she'd seen him soak a Podlove bureaucrat, and she knew it couldn't be a coincidence—but she knew it was making her job harder. Podlove was battening down for a siege, and would be even more inaccessible than he normally was.
She was in midair, when her jaw bug chimed. Rolling hard landings had never been her strong suit, and the distraction made this one even worse.
"Hello," she said several harsh seconds later, sitting on the floor and cradling her ankle.
"Ankit," said the caller, his voice so old, New York–accented, and she thought at first it must have been Barron—but this man drew out the second syllable of her name too long, and his tone was too hard, too cold.
"Who is this?"
"Dak Plerrb, calling on behalf of Mr. Podlove."
"Ah," she said, forcing herself to breathe slowly. "This is a surprise. Is he on the line?"
Plerrb laughed. "No, no. But he did want you to know that he's noticed. How determined you are, to talk to him. Visiting, writing."
"I wanted—"
"And researching! Spending so much time looking into him. The Qaanaaq web, the global web, all sorts of places."
Outrage flared up, but she fought it down. Of course shareholder flunkies could get reports from the security programs that monitored data behavior. They could probably control them, too. He wanted her shocked, angry, thrown off.
"He wanted me to call and let you know how serious he is when he says that this is not the time to be fucking with him."
_Breathe, Ankit. One breath, two. Don't rush this._ When she spoke, her voice was ice. Was wind. "Do me a favor and ask him why he had my mother locked up in the Cabinet."
But there was no answer. Her screen said the call was terminated. Had he heard her question? Would he ask Podlove?
She stood. Shifted weight from leg to leg. Her ankle wasn't sprained. She climbed back up the mock building stilts. Grabbed a horizontal bar; swung her body to the next one. And the next.
The old muscles kicked in. Facts fell together. All at once, Ankit saw: _The shareholders want the breaks to be an epidemic. They're pulling the strings to make the problem worse. To prevent solutions. That's why the molecular printers can't produce Quet-38-36.0. Why there's a six-month wait for quarantine transfers._
Context: The breaks were a welcome distraction from the city's fundamental flaws—the supremacy of property, the fact that landlords ran everything.
Ankit laughed out loud and leaped to the next stilt.
The place was a reasonable facsimile. She could see why people paid for it. But it only made her hungrier for the real thing. The bite of metal. The dark void below. The cold wind, most of all. A scaler friend of hers, quoting some old proverb: _If you surrender to the wind, you can ride it._
## Soq
The boat was old, late or maybe even mid-twentieth century, tall and rusted, with a steep steel gangplank. Soq was out of breath by the time they reached the top. They turned to take in the view. Like most Qaanaaq urchins, Soq had marveled at the Amonrattanakosin flagship, lingered on the wharf where it was anchored, imagined the torture chambers and disruptor manufacturing facilities that must be belowdecks, the meeting rooms where criminals from every rung of the ladder met to plan out operations, the storerooms full of grain and cans for surviving sieges by the forces of Safety or enemy crime bosses or the military of one or more of the charter nations.
It seemed smaller now. Soq reached out to rub two fingers against the hull, watched rust flakes fall. When Dao had buzzed Soq that morning, told Soq to report to the ship, it had turned the whole day into a dream.
"Utterly unseaworthy," Go said.
Soq nodded. "Why don't you . . . I don't know, paint it? Indonesia still makes that hydrophobic stuff . . ."
"This is just a starting place." Go stepped off the gangplank. "Hello, Soq."
"Hi."
They shook hands. It felt weird. Soq had dreamed of meeting Go. And now: there they were. The real thing was so much smaller than the mythic creature in Soq's head.
The city was hard to see, below them in the ebbing twilight. And the boat moved differently, its rocking more noticeable than the city's eternal rise and fall, which was mitigated by complex mechanics . . . Soq so rarely stepped off the grid. They paused to savor the almost-seasickness.
"People live here?" Soq asked.
"People do," Go said. "But you don't. Not yet, anyway. Your home for the foreseeable future is actually part of your first assignment."
Three women squatted on the deck, slowly deconstructing a large plastic cube. Pulling away smaller plastic cubes of varying sizes, one at a time, and slotting them into canvas bins based on their color.
"What's in those?" Soq asked. "Or is that the kind of thing you need to be here a lot longer before you can find out?"
Go laughed. "We're not just drug runners here. Most of our work is totally legal. This shipping pallet contains spices, just arrived from the subcontinent. By letting so-called crime bosses control even the most mundane and legitimate aspects of Qaanaaq's commerce, the city can keep expenses down. Particularly labor costs."
"So I take it these women don't have a union?"
"They do not. Although you are welcome to ask them how happy they are with their work and their pay."
Soq watched one of them until she made eye contact. She smiled, nodded. A recent arrival from somewhere post-post-Soviet. Of course they were happy. Go was probably light-years ahead of every other option this woman had for making a living in the nightmare landscape she came from, or the city where there were thousands of smarter, more desperate women just like her.
The tour took a surprisingly long time. The boat had more levels than Soq had been imagining, each one with its own complex labyrinth of passageways and warrens and boxes and drawers. Go had her fingers in so many different things. A whole department dedicated to intelligence, files and photos and film on probably half the city, one person whose only job was mastering archaic media, flash drives and paper files and floppy disks and microfiche and crystal gel, cataloging what they kept and retrieving information from them when needed. Elsewhere, a lithe legless woman was lord and commander of a vast pharmaceutical storage system, swinging from rope to rope through a forest of cabinets.
"You give every new grid-grunt flunky the full tour yourself?" Soq asked.
"No," Go said, but did not say anything else.
Back up top, she handed Soq an armband. Leather, black, embossed with an incongruous toile print. A pastoral French peasant scene. Go said, "When she was grooming me to take her place, my mentor once told me that ambition is essential to being an underling, but death to a crime lord. That we will flourish and thrive for precisely as long as we remain content with where we are, what we have. And when we try to reach further, seize more, that's when we run into problems. That's how wars start, how empires topple."
Soq smiled, because it was easy to see where this was going. "But you're not content with what you have."
"No."
"Your mentor sounds like a pretty smart lady."
"She was pretty smart. But she thought small. And maybe she was wrong. Maybe that's one of those pieces of received wisdom that everyone just accepts, even when it keeps them trapped in one place."
"One way to find out," Soq said, looking out at the city lights. The wind was picking up. Waves crashed against the side of the boat beneath them. _This is what it feels like. To step outside your box. To shake_ ___your fist at the city and say,_ You will not break me. I'll break you, if that's what it comes to.
Go smiled. "I knew you'd see it that way. I want broader, more legitimate supremacy. I want to get off this boat."
"What can I do for you?"
"The empty apartments. They're the key to what I have planned. I need to establish a foothold. I want you to move into one. I'm asking all my warriors to do that, minus the ones I need for protection here on the barge, of course."
"The empties are real?" Another one of those Qaanaaq stories that people loved to tell, right up there with the heat-resistant spiders that supposedly infested the geothermal pipes, and the threat of Russian invasion. Allegedly, it was common practice for shareholders to keep some of their holdings off the market. A sort of gentleperson's agreement, to artificially inflate prices by increasing demand by keeping supply low. Soq didn't doubt that the empties existed, but they were pretty sure their number was exaggerated, as was the extent of the conspiracy behind it. The more likely reason was the simple thoughtless wickedness of the rich, who had more money than they knew what to do with, who didn't need the rental income and could keep an apartment empty for Grandma's once-a-year visit or in memory of a loved one dead for decades. Either explanation was unacceptable. Shareholders were wise to keep themselves hidden, because surely Soq wasn't the only one who would gladly stomp them to death if given the chance.
"They're real. I've been collecting data on them for years. I know where many of them are. Not all."
"Why do you need me there?"
"Don't worry about that right now."
Soq smiled. "You just told me ambition was essential for an underling."
"No. I told you that's what my mentor said. I also told you she thought small."
## Fill
The flat-bottom boat felt like a dungeon. Water dripped; red rust stains spread across cement walls.
"She must be here somewhere," Barron said. They passed tents and shacks, lean-tos, yurts. The air felt tubercular, uncirculating.
"People live like this," Fill said, and then regretted it. He'd spent all morning trolling through Grandfather's software, surveying his holdings, everyone's holdings, really, especially the ones marked Empties, and while it had bored him to tears at the time, it was presently making him sick with guilt, to think of so much space sitting empty for decades while these people lived packed together like splice shrimp in a jar.
"It's warm, at least. A lot of these people are grid workers—vendors, food stall operators, sexual entrepreneurs—they spend all day in the shivering cold, so the warmth is a big part of the draw here. Of course, a few of them never leave at all."
More than a few, to judge by the funk of feet in the air, of urine. Fill felt short of breath, angry at himself for coming down here. Why couldn't they have made an appointment to meet her somewhere else? A brightly lit, above-sea-level spot? Surely this mythical maybe-Reader would have welcomed the opportunity for some fresh air and a cup of real coffee.
Barron, on the other hand, seemed to relish the dark, tight space. He looked almost disappointed when he stopped beside a ramshackle thing, a teepee made of sheets of hard plastic like some gritty closed flower, and said, "This is the place. But she's out."
"How do you know? These things don't exactly have apartment numbers on them."
"Patience, young Podlove."
Five minutes passed like that. The longer he stood there, the more sounds Fill could make out in the space he'd mistaken for silent. Chatter, plucked instruments, squawking speakers, the clink of silverware. A nightmare confirmation of his worst imaginings of an urban underbelly. He wished he were less disgusted to be so close to the Poor Unfortunates he'd been idealizing as he listened to _City Without a Map._
"Tell me a story," Fill said. "I bet you're full of them."
"Very well, Your Majesty," Barron said, and bowed. "I will tell you the story of how I came to Qaanaaq."
"Sounds good." Children scampered past, throwing deformed bottom-grade plastiprinted figurines at each other.
"I will tell you about the fall of New York. You're a New York boy, aren't you, Fill? Just a handful of generations back?"
"Two," Fill said. "My grandfather."
Barron rubbed his chin. "Like most cities, New York had managed to be both heaven and hell for a very long time. Filthy and beautiful, a playground for the rich and a shithole for the poor, sometimes leaning more toward one extreme and sometimes more toward the other. By the time I was in my twenties, most of the things that made it heavenly were gone. The transit system that was its pride and its lifeblood was largely unusable, ever since the storms that flooded the tunnels and forced the governor to agree to construction of the Trillion-Dollar Fail-Proof Flood Locks. And the Flood Locks themselves were widely believed to be bankrupting the city as a whole, but _better to be bankrupt than dead_ went the common adage. Maybe your grandfather told you about that time, eh?"
"Not really. Just that it was miserable, and he was lucky to escape at all."
"We all were. I lived in a place called Brooklyn. By that point, the situation was very dire. In New York City then, as in Qaanaaq now—as in most cities, always, I imagine—the landlords called the shots. It didn't matter who was mayor, who was in the city council, what party a politician was from, real estate interests owned them all. They gave the most money to campaigns for public office, and electeds did whatever they asked. But by the time the Flood Locks were almost finished, Big Real Estate was in trouble. No one wanted to invest in the New York City housing market anymore. Banks pulled out. Foreign investors evaporated. The safest investment in capitalism was suddenly not so safe."
Bored, Fill thought back to what his grandfather had told him, wondered where he fit into this story. It wasn't that he didn't care. More like it all seemed so remote, something that had nothing to do with him. Qaanaaq was what he wanted: a story he was part of, something he belonged to.
"We tried to fight back. We came together. We put our bodies on the line. We took the fight to them."
Barron's voice was rising, his face reddening.
"And we won! We got the mayor to block the budget the landlords were lobbying for, which would have paid them full market rate on all the newly worthless property they owned. We got the city council to vote against it. But that's the thing. You can win against people. You can't win against money. Money is a monster, a shape-shifting hydra whose heads you can never cut off. Money can only behave one way."
Fill felt chills dance along his neck. The old man's anger was unsettling.
"We watched from our roofs, the day they blew the Flood Locks," Barron said, standing up straighter. Suddenly he seemed a different, younger man. "We saw the explosions, the water pouring in. I'd imagined the Red Sea, Charlton Heston, a wall of water wiping us out. Really it wasn't that much. Looked more like a bathtub overflowing. Only enough to render two-thirds of the city uninhabitable. By nightfall, the governor had declared a state of emergency. The feds spent a week working out an aid appropriation package that bought the landlords' buildings off them—at full market value. Exactly as the landlords had planned. By the end of the week, the bloodshed had begun in earnest. Food couldn't come in. Water supplies all contaminated. People desperate to get out, forced to abandon everything and take only what fit in one suitcase. I watched this one family—"
"What do you want?"
A woman stood beside the conical shack. Fill hadn't even noticed her approaching.
"Are you Choek?" Barron asked.
She nodded.
"Robert sent us. Said he spoke to you—about the recording you did?"
"Come inside," she said, scuttling backward into the teepee.
"Oh hell no," said Fill.
"Don't be such a baby," Barron said, squatting and waddling in after her with remarkable agility for a man of his years. Fill counted to ten and then held his breath and followed.
Barron and Fill sat on the floor; the woman sat on a tatami. Her shack was barely big enough for the three of them. Fill could not have stood if he'd tried. Shelves hung from the ceiling, dangling all around him, heightening his claustrophobia. Her expression seemed empty, void of interest or even fear. Fill babbled, "Tell us about _City_ ___Without a Map_. Who told you to read that text?" He knew he should proceed with more tact, but he also knew that he was overwhelmed, frightened, stranded in a strange place he might not ever find his way back from.
"What did it mean to you?" Barron said, more gently.
"No one told me to read it," she said. Her hands rested anxiously in her lap. "I read it because I wanted to."
"But who wrote it?" Fill asked. "Did you write it?"
Choek looked at him, seeing him as if for the first time. "I can't tell you any more than that," she said. "I promised."
"You promised who?" Fill whispered. This woman had touched her, talked to her—or him—but probably her?—the Author.
"I can't," she said. "You need to go."
Fill looked to Barron, whose face seemed torn by rival impulses. To flee, to apologize, to beg . . . Finally he bowed his head and said, "We really appreciate your agreeing to speak with us."
Just like that. It was over.
"Money," Fill said. "How much would it cost for you to tell us?"
Choek looked at them for a long time, her eyes wracked with pain, before shaking her head.
Barron said, "Thank you for your—"
Fill blurted out a figure. A big one. A dangerously big one, the kind that he'd have to really beg his grandfather for. And might not get.
But big enough that her eyes went wide. With wonder, and then with anger, and then with sadness. Because there was no way to turn that much money down. No matter whom she had to betray. Single Author, rogue collective, evil robot overlord.
"Did you make this?" Barron said, placatory, pointing to a sculpture cobbled together from ancient circuit board. "What does it mean?"
"Don't know. Just started seeing them. Dreams. Someone else's. They're about the Sunken World, I think. How all those people got buried alive in their own things. Or couldn't let go of them when the waters started rising, when the flames came, and died clutching them."
"Magnificent," Barron said. "And you just started making the things you saw?"
"No," Choek said. "For so long it was just visions, glimpses, images I could see but not understand. A compulsion to make something I had no idea how to make. I didn't start sculpting them until I was in the Cabinet."
"Wait," Fill said. "You said someone else's dreams. So you have the breaks?"
"Had," she said, and smiled.
"You . . . had? You're cured now?"
"In the Cabinet," she said. "But I've said too much. Go. Come back with the money and maybe I'll talk more. Maybe."
They crawled out backward, for there was too little room to turn around. "A fascinating creature, is she not?" Barron said as they stood blinking their eyes in the relative brightness of the dim underbelly of the flat-bottom boat.
"How do you not want to learn more about what she meant?" Fill said, grabbing Barron's sleeve. "She said she was cured! Of the breaks!"
" _You_ said that," Barron said. "I don't know what _she_ was trying to say, or whether the poor creature could tell the truth even if she wanted to. But you, in your damn hurry to—"
"I'll talk to my grandfather," Fill said, chastened, already turning on his heel, desperate to be gone from here, from Barron, from _City Without a Map,_ from the abominable ways that caring about something opens you up to hurt. "I'll let you know what he says."
Barron was saying something, but he did not turn to hear what it was.
He decided he did not want to go home. For just one night, he wanted to slip away from his life. He went to his grandfather's other apartment, the one he'd kept empty for all these years.
What had happened there? he wondered. What was so special about it? Who had died there; what torrid love affair or pivotal business deal had his grandfather conducted in it? Or did it mean nothing to the old man? An asset on a screen, one of Qaanaaq's legendary empties? Walking in, it felt so different from the warm safe home his grandparents had built, the place his grandfather lived alone now. This one was stark, cold, austere—
And occupied.
"Hey," said the boy playing with a shape-memory polymer at the kitchen table.
"What are you doing here?" Fill asked, but smiling, because the boy was beautiful. Butch haircut, broad spike-studded shoulders, a refugee face with the skeptical expression of a hardened Qaanaaqian. Fill stepped closer. Smelled him, like slide grease and star anise, and saw that maybe he was not a boy at all.
"Friend asked me to watch the place," the kid said, and Fill sat at the table across from him. Them?
"But it's not your friend's place," Fill said. Boy or girl, or some other majestic thing altogether, Fill shut his eyes against the flood of desire that washed over him. The danger of the situation was every bit as arousing as the person before him. He should have turned and left, called Safety, called his grandfather. He should have remembered that he had the breaks.
"No. My friend thought it was empty. Is it yours?"
"Not exactly," Fill said.
Pornography writhed and danced in his peripheral vision. He opened his mouth, intending to say something seductive or submissive or _something,_ but the pornography was growing more frantic, more bizarre, the pretty boys becoming monsters, the ground beneath him bubbling, and he stepped forward, still smiling, and fell to the floor.
## Soq
_I should have fucking left. Stepped_ _daintily over his stupid body while he was unconscious and stomped my ass out of here._
But Soq hadn't left when the rich kid fell to the floor in front of them. Soq had rushed over, checked his pulse, gotten him a glass of water, sat with him for the sixty endless seconds it took him to wake up. Cradled his head in their lap, been kind and nurturing while the kid emerged from the brief infancy of post-unconsciousness.
Mostly, Soq told themself, because they didn't want this little prick waking up alone, remembering Soq's face, calling Safety on them, saying he got jumped by a home invader.
But now here they were, an hour later, sitting on the floor like kids at a sleepover in an old movie, drinking soda and eating krill chips delivered by slide messengers, debating where to get noodles from.
"You have to eat noodles in the first fifteen minutes," Soq told Fill. "It's just basic food chemistry. The heat is still cooking them, and in fifteen minutes they're mush. We should go out to a stall somewhere."
"I've never heard that," Fill said. "And I don't want to go anywhere. Let's get them delivered."
"Idiot, you can't get noodles delivered. Are you not paying attention to me?"
The kid had a sense of humor, and some serious self-doubt, so the two of them got along great. Fill was less than a year older than Soq, and infinitely more naive about absolutely everything but sex, in which subject they were more or less evenly versed.
"I can get noodles delivered," he said. "Watch."
Fill dialed, promised a massive amount of money if they could have the noodles in his hands within five minutes of the moment they left the wok. Clicked off, smiling in triumph.
"Of course they'll tell you it's five minutes," Soq grumbled, but Soq was also excited by his confidence, by the options that unlimited money opened up, by the previously unimaginable prospect of decent noodles being delivered.
"I guess we'll know when we take a bite," Fill said. "Since you're such a noodle-quality fussbudget."
"Yeah," Soq said. "I guess we will."
Turned out they were both obsessed with traffic trawling, following currents of attention to find the latest bubbling-up art and news being shared among Qaanaaq's million subgroups. They compared bots, shared the software they both used to uncover new trends, swapped archaeology dubs and ancient Sunken World footage and the photo archives or instant messenger logs of long-dead strangers. Fill had the best programs money could buy, slick, swift, terrifying tools that turned up stuff that made Soq's jaw drop, but Soq had gnarly, unpredictable Frankensteined software concoctions they'd found at the Night Market and Fill seemed just as excited by those as Soq was by his.
"This one's a classic," Soq said, flashing a file to Fill's screen. "The Book of Jeremy. Do you know it? This gay guy, after his best friend, Jeremy, died—Jeremy was straight—he got into his email—Jeremy had given him the password—and deleted all the boring bits to condense this guy's whole adult life—from fifteen to thirty-seven, when Jeremy died working a shale oil rig in one of the hydraulic fracturing earthquakes in upstate New York—and he interjected his own commentary between emails. It's super hot, and super sad."
"Amazing," Fill said, and read out loud: "'Jeremy's profanity is preserved here precisely as it was in life; a bawdy, inappropriate, usually humorous but sometimes profoundly moving knack for saying the earthiest things with such innocence that you feel like a bad person for finding it smutty. Nowhere is this more evident than in his emails to girlfriends, where he says things like, "Kath I miss you like crazy lately, never saw a bitch so keen to get her hair pulled," and you somehow know that Kath took no offense at this, can in fact picture her reading it, flushing red, remembering Jeremy.'"
"Great stuff to jerk off to, and then cry."
Fill put Soq on his account for the most expensive app on the market, beamed it directly to their slate. "This is how I found _City Without a Map,_ " he said. "Do you listen to it?"
"Tried to. Didn't do much for me. I don't need a guide to this city. And I don't need all that poetry."
Fill nodded, smiling. His eyes full of wanting.
The noodles arrived. Soq conceded that they were still perfect. They ate them in great gulping bites, both of them hungrier than they'd realized, looking up only when they were finished.
"You're really hot, you know that?"
"So they tell me," Soq said, furiously computing how to respond, what to do—the kid was hot, sweet, sad, they'd had a good time, but rich, unspeakably so, and probably not fond of being turned down, denied something he wanted, and what if after all this lovely quality friendship time he turned around and called Safety on Soq? So Soq stood, butched up as much as possible, and growled, "What are you going to do about it?"
Hours later, after several bouts of switching back and forth between fucking and sleeping, Soq was getting dressed in thin winter daylight when they heard Fill say:
"What the fuck even was that? Was that . . . you?"
"'Course it was," Soq said, pushing up the black leather armband and adjusting the spikes on their hooded coat.
"Like . . . biologically you? Like, you were born with it, or . . ." He craned his neck, trying to get a glimpse. "I've heard that there are some pretty crazy surgeries you can get these days—"
Soq kicked him, hard, a swift blow to the shoulder that made him yelp. "Don't be rude."
"I'm sorry," he said. "Can I give you my handle?"
Soq shrugged. The kid was out of bed, heading for the coffee maker, practically preening. Full-on courtship behavior, but Soq was already strapping on their slide boots, washing their face, moving on.
Fifteen minutes later they were halfway down the Arm One slide when their jaw bug buzzed.
"This is the last one," Soq said.
"You'll still owe me," Jeong said fondly, sending the details. A briefcase from a floating lab off Arm Two, which as far as Soq knew dealt in genome-customized party-booster drug regimens, to a semi-famous musician on Arm Five. "Till the day you die you'll owe me."
"Maybe," Soq said, leaping off the slide and dancing through the Hub. "But I'll have to find another way to pay that off. Because this? This is the last one."
They had dreamed of leaving the slide messenger life behind entirely, immediately, but they didn't feel strong enough to turn Jeong down. Soq would never have survived long enough to land a role in Go's army if Jeong hadn't helped them a million times, in a thousand ways, over the past several years.
There was no nostalgia to it, no sadness at the life Soq was leaving behind. Messengering had been fun, exciting, the best possible way to make money within the limitations of being unregistered. But the pay was shit unless you worked twelve hours a day or more; the people were assholes; the risk of death was constant. And Dao had already wired Soq's first week's salary. And it was astonishing.
Technically all Soq needed to do was hold down the empty apartment. But Go had explicitly said that Soq didn't need to stay there all the time, and like a good ambitious underling Soq wanted to find ways to impress the boss. Besides, Soq didn't want to risk a repeat of last night's random hookup.
It had been stupid. Dangerous. Fun, but dangerous. Soq told themself they had mostly only slept with him to keep him from calling Safety.
And what had been up with that collapse? Was he sick? He'd sworn he was just dehydrated, a three-day disruptor bender, but wasn't that precisely the lie Soq would have used if they had the breaks or neosyph or a contagious botched gut flora hack?
More important: Soq knew the signs, when someone fell hard. Those macho gay hypocrites were the worst—fetishizing masculinity, sneering at trans boys and femmes and anybody else insufficiently butch, but let them get a taste of something like Soq and all their biases got blown out of the water. The boy was smitten, and Soq wanted no part of it. Love, relationships, even friendships—and they would have been good friends, Soq knew it, even if Soq would have spent most of the time hating Fill intensely for his money—Soq couldn't have any of that right now. They were finally poised to leave it all behind, the pain and the hunger and the wondering.
Soq leaned into the wind, descending the slide and cleaving the vortex like a machete blade, Go's: a weapon, a tool, a soldier, unencumbered by the emotional baggage that made everyone else so miserable.
At least they'd gotten a damn good traffic trawling app out of the experience.
Once the briefcase was safely in the hands of its twitchy, strung-out recipient, freshly wealthy Soq settled down for a ginger beer on a heated replica of a Bangkok floating market. Someplace Soq had passed a thousand times and never been able to afford. A maddening smell came from below them, where a chubby woman who couldn't have been more than five years removed from the Chao Phraya River splatted noodles down into a skillet. Geothermal heat swirled around them, but every minute or so the wind shifted slightly and a cold gust sent a chill up Soq's spine. Incense burned beneath the table. Purely ornamental, here, with no insects to scatter the way it would back in Thailand.
How did Soq know so much about Thailand? With a start, they sat up. Had they fallen asleep? Dreamed? Sense impressions swarmed their mind, vivid memories of things they'd never experienced. Out of nowhere, their head ached exquisitely.
The pad Thai was the best Soq had ever eaten, but at the same time it tasted like a pale echo of something else. Fresher spices, the flesh of richer fish. Bangkok, the capital of the world, the heart of the country that weathered the global storm of rising sea levels better than any other. Home of the mightiest military, the most fiercely defended borders. A source of constant fascination to Soq, as it was to many in Qaanaaq, the way London would have been to a colonial American—the colonized's distant, foolish pride in its patron commonwealth. But now it felt like a place Soq had been.
_I am a spy,_ Soq thought later, making their way down Arm Eight. High on good food, pockets not empty, Soq was approaching some ill-defined idea of "the good life" that they'd spent their whole life striving for. Confidential agent. Criminal mastermind. One of this city's secret rulers. The black leather armband marked Soq as Go's, as invincible.
Soq used Fill's program to troll for recent orcamancer sightings. One was only a couple of hours old. Soq let the vortex take them in that direction.
No orca, unsurprisingly. Killer whales don't stay still. They circle; they hunt. Soq did a sweep of the immediate area, and then widened the sweep. How much of this came from movies versus books versus life experience, they couldn't say. All the tangled threads of Soq's life had finally resolved into a pattern. A texture. They wandered through the under-building caverns, emptying their mind and letting their body go where it wanted. Soq had stumbled into a part of Arm Eight that they'd never seen before, the kind of place they'd have worked hard to avoid not so long ago. The smell of bottom-grade trough meat was thick and pungent in the air, and the few familiar glyphs graffitied onto the walls and pylons belonged to the city's savagest gangs and societies.
But now Soq wasn't afraid. Who would fuck with one of Go's drones?
This guy, apparently. Some battered fighter dude who had been doing pull-ups in the chilly twilight from the low-hanging crossbeam of a building support platform, who dropped when he saw Soq. His hands made fists. "She sent you?" he said, reddening at the sight of Soq's armband.
Something—roared. A shadow moved in the dark forest of building stilts—came forward, grew brighter. Roared again. A polar bear. The polar bear.
"She didn't send me," Soq said, throat dry. Knowing it wouldn't matter. Whatever grudge this guy had against Go, he and his polar bear would not be talked down from this peak of rage. The bear ran forward, stood over Soq. Roared again. It smelled like rotten sea lion meat, and something else. Something mammalian, something close to human. It hadn't been hostile with Soq before, but now it was with its human, and its human was furious. With Soq. The bear lowered its head, its mouth wide enough to fit around Soq's face and then bite it right off.
Soq shut their eyes.
The bear's nose pressed against Soq's face. It sniffed, tracing a wet smear from side to side.
Soq decided that it was a trick, to get them to open their eyes, because the bear wanted to see the mortal terror in the moment that it ended Soq's life. Soq would not be fooled. They did not open their eyes.
"What the hell?" the man said.
"I thought so," said a voice—female, heavily accented, gravelly and wise. The orcamancer.
_For once,_ Soq thought. _For once I followed a sighting and actually found her._
"Thought what? Go sent this little asshole to kill me, and—"
"This little asshole is your child, Kaev."
Soq still didn't open their eyes.
****City Without a Map: Archaeology
The beer is weak and has a salty taste to it. The ceiling is low. You're hungry. Someone vomited at a table near yours, and no one is coming to clean it up. The only windows in the place have green-black ocean water on the other side of the glass. Your thoughts are melancholy—
—What will save us from this gray city, these long nights, this wind that cleaves memory from bone, the cold and wet that will never forsake us, these dappled shadows falling on aging faces? What will bring us joy? What will keep the fire burning in each of us?—
But then they take the stage, women with guitars and synthesizers and percussion instruments, and a man on bass, and they smile, and start. And you smile, shut your eyes, let the songs happen to you.
Maybe they're not great. You can't tell. You are gone from here, from this subsurface dive bar, from this floating city, from this fallen world. Strains of Celtic folk songs tease your ears; American soul; post-reunification Korean _gugak-yangak_.
Archaeology. The most distinct and vibrant of Qaanaaq's newborn musical traditions. Digging deep into the hundreds of musical heritages that people brought to this city. No singing, no lyrics. They don't even speak between songs, and you understand this, you appreciate it, because you know as soon as they opened their mouths they would cease to belong to everyone. The language they used, their accents, would place them definitively in a box, mark them off as coming from one continent or another, one city, possibly this one, and a cheer would go up, from the people who belong to that same box, and everyone else would feel the slightest bit less included in the tight warm embrace of the song. Music is the common property of all humanity, but people come from particular groups. For as long as the song lasts, for as long as they say nothing, you can pretend you are part of the same group.
They don't play songs so much as expeditions. Digging from one song into another, one century to the next. Late-1980s video game tunes become High Church Slavonic liturgical chants. The ruins of Troy, you remember reading, before the sea swallowed them back up, were actually seven cities, each one built around the bowels of its predecessor, and you imagine that this is a similar slow stroll from one epoch into another.
Word is, they've spent weeks at a time with different refugee communities, all over Qaanaaq. Learning, listening. Sucking up every song and scrap of indigenous style they can find. The Khmer surf revival. Nahuatl ballads dating to before Columbus. Bachata, where the notes run fast as raindrops. At every show they bring them up onstage to play a song or two, these inadvertent cultural treasures, these people who are all that remains of entire vanished musical genres. You see them now, sitting alongside the stage, smiling with pride and sadness.
You are alone, here. Your family was supposed to follow you, but it's been five years, and the Water Wars became civil wars and then the whole eastern half of your country went silent. Every week you visit the registries, scroll through the lists of new arrivals, petitions for registration. You scan every sad face, every trembling lip, every stony resigned stare. You know there are many more who do not consent to be included in the registries, people wanted by rogue governments and warlords and syndicates, and you wonder if something like that has happened, if whatever desperate compromises they had to make to get out might have put them in mortal danger, if they're in hiding, if they're already dead.
You were sick. You went to the hospital. You hid your symptoms, because you feared it was the breaks, and you knew what they would do to you if it was. You didn't tell them about the strange memories crowding your head, how they threatened to break you open. You told them it was overwork, exhaustion, dehydration, a history of violent abuse manifesting itself. They put you in the Cabinet.
You know this song, this scrap of melody. A Somali _dhaanto,_ pentatonic perfection, the synthesizer effortlessly approximating the sounds of the oud lute.
A cellmate used to sing this song. An immigrant laborer back home, a fellow political prisoner.
The Cabinet is where you met me. Sitting in the corner of the rec room, eyes shut, the same corner I've been sitting in every afternoon for far longer than you've lived in this city. I saw you; knew what you were really dealing with. I cut my forearm, cut yours, pressed our forearms together. Made us blood siblings.
Since then, you do not fear the memories of strangers.
The beer, for all its seeming weakness, packs an unexpected punch. When you finish it you feel strangely blissfully happy. When you shut your eyes—
Memories fade in, fade out. Stirred by the songs. It doesn't matter that they're not your own. You belong to Qaanaaq now. Its people are your people. Their pain is yours, and so are their songs.
## Fill
Fill should have been miserable. He should have felt ashamed, guilty; he should have been taking concrete action to ameliorate the consequences of his irresponsible actions.
But he didn't. He wasn't.
When had he become such a monster? Knowing he had a fatal sexually transmitted disease, he'd had unprotected sex with someone. He hadn't warned them. He hadn't even told them after. And instead of being eaten up with remorse—instead of feeling bad about what he'd done, here he stood, leaning against the guardrail, watching the sunset, admiring the fractal rainbow arcing slowly down the side of the windscreen.
What was he becoming?
Part of it was, he didn't truly blame himself. The circumstances had been so strange. While he'd been incredibly turned on by the danger of it, a part of him had also been offended. Angry, even, with this criminal. An intruder, after all. In his grandfather's secret apartment. A gorgeous, prickly, impoverished creature who turned his whole idea of gender on its head. It had felt like pornography, like a dream, like a horror story.
But still. They were real; it had really happened. The breaks were probably already manifesting themselves in Soq.
So, what? Why did he feel so strangely fine?
_Tomorrow night I'll have my answer,_ he told himself _. My grandfather will say yes or no to the absurd amount of money I'm asking him for._
He'd wanted to just message Grandfather the request, but Barron said that seemed unceremonious. Too easy, unworthy of the momentous event those funds would facilitate. So instead he'd asked Fill to arrange a meeting, for himself and his grandfather and Barron, so they could make their pitch together.
A delay tactic, most likely. Barron was as scared as he was, probably, to get to the bottom of what Choek could or couldn't do for them. They were both terrified that the trail could be cold, might not lead them anywhere, or—worse—that Choek could indeed lead them to the origin of _City Without a Map_.
The sun was down. The sky was still bright. His heart danced with the water, with the rippling light. _It's the breaks,_ he thought. _I feel them trembling through me. Cracking open all my defenses, breaking down the walls I built between me and the world. Shaking me loose from my self, from my ego, from this tiny isolated flickering flame, so I can see how I am the sun. We are the sun._
_So sad, to think that it took this, this, to make me see how beautiful our world is._
She came to him more and more as the sickness progressed. In dreams, in crowds, in memories that didn't belong to him. The ghost woman: a guide, but a guide to what? She took him places, told him stories without words. He could feel her in him. Most of the time she was peace, profound and terrifying, a radical reconciliation more divine than anything Christ could have managed, something that could only have come from unspeakable suffering. Sometimes she slipped, cracked, refracted, and he gasped at the river of rage that roared beneath her surface. The things she had suffered. Not repressed, not forgotten, but no longer present.
What had happened to her?
In his mind, she was the Author. The mastermind behind _City Without a Map_. He was aware that this was irrational, idiotic, probably incorrect. He felt it so strongly that it couldn't possibly be true. She was a construct, a figment of his damaged imagination, his diseased brain assembling complex narratives and characters out of the chaos of information he was drowning in.
She had to be.
Bald. Fifty-something. Just like the Author described herself.
He let go of the railing and then did something that shocked him. He sat. He reached down, dipped two fingers into the frigid waters. He touched them to his lips.
_We live our whole lives suspended above the sea,_ he thought, _but we forget the true taste of salt. Not the purified stuff we find in kitchen cabinets and restaurant counters. The bitter, foul, sea-muck stuff we crawled out of, and live beside, and one day will return to._
_Soon I'll break free of this body and be one with the sea, with the sky, with the infinite. That is the gift I was given, that I in turn gave to someone else. A bitter gift, but the best ones are._
## Soq
Soq wanted: nothing.
They looked down on the city from forty stories up. They drank actual scotch, tasting like smoke and hammered bronze, and barely noticed. People all around them carried cages and polyglass bubbles bearing animals Soq had never even seen photographs of, but Soq did not care, did not look twice. Their stomach tingled, the gut fauna freshly tweaked so all the bad thoughts and feelings and bodily traumas were whisked away. Soq stood there, feeling no pain, yet somehow did not marvel at what a strange rare blessing it was to be without pain.
It was the breaks, certainly. From Fill. That fucking rich kid in that fucking warehoused unit. It had to be. There had been no one else. Soq had been far too busy for sex the last few weeks. That fucking asshole had given Soq the breaks.
That's how Soq popped in and out of his head. That's how Soq saw into his emptiness, his pain.
Which is why Soq's lifetime of schemes and plans to conquer or destroy Qaanaaq suddenly felt so flimsy, so flawed.
God damn him.
"It isn't just the fact that they did terrible things," Soq was saying, only it wasn't Soq's voice at all. "That's sort of 101, isn't it? Something you realize right around the time you first find out what fucking is. You just assume there's things about your family you never want to know."
"Sure," said a very pretty boy. Thin lines of gold light gleamed inside his face; bioluminescent bacteria or cephalopod-derived photophores, Soq wasn't sure which; all the rage among the queer kids of Petersburg plutocrats. The microbes or bacteria or whatever lived for three days—or was it four?—and you could buy a half-ounce acu-ampule for a little less than what Soq made in six months of messengering.
"You're not listening, Tauron," Not-Soq said.
"Of course not," Tauron said, and somehow Soq knew this meant that he _had_ been listening, and good gods, why did the rich have to make everything so complicated?
"And it's not my fault that all those bad things happened, it's not my fault I live a good life and other people are living god-awful ones because of things my grandfather might have done . . . but . . . the real problem is, once you _do_ know what they did, doesn't it make you obligated to do something about it? To make things right? Otherwise, aren't your hands as bloody as theirs are?"
The bubble they stood in rose, at the end of its long strut. Soq looked down at the sea, black beneath them, and at the lights that danced on all sides.
The city was theirs. Qaanaaq was conquered.
And they were completely miserable.
Then they opened their eyes and were back in whatever creaking dark mold-smelling corner the Killer Whale Woman had them holed up in. She was gone now. Soq remembered her leaving, even though they hadn't been entirely conscious when she went.
They hadn't dozed off, so much as . . . gone away.
Soq did not feel angry, to have everything snatched away. To be plunged back into the same poverty they'd always known. What they felt was grateful. To be back in their own life again, their own body. To be free of Fill's strange and terrifying pain. To have their own familiar pain back.
"Thanks, brother," Soq whispered.
He had everything, that poor sad fuck Soq had fucked, and he was so empty inside that Soq had to fight back the urge to hunt him down and give him a hug.
## Ankit
Ankit spent three hours shivering across the grid from the entrance to the Yi He Tuan Arena. Watching every face that went in and every face that came out. Looking for her brother. The screens flickered: Hao Wufan's upcoming fight canceled. A damn shame, she thought. The Next Big Thing already a thing of the past, since those all-boy-sex-party photos surfaced, and the audio of his lunatic ramblings, drugged out of his mind or possibly suffering from early-stage breaks.
The dossier from her contact at Health hadn't had much to offer, but she'd learned a little more about her brother. That he was sick, some unspecified form of brain damage. She had a file now. Something to hand him. He could take his time, process the information. And she wouldn't be frightened if he howled or hooted or got upset. She knew what she was dealing with.
First she'd have to find him. She was pretty sure fighters didn't hang around arenas when they weren't scheduled to fight, but it was the closest thing to a lead she had. Maybe he'd come by to meet with a manager or promoter, pick up a payment, practice on the beams, train at a secret gym or battle society somewhere inside.
It had seemed like a solid enough plan when she had arrived. But now she was freezing and hungry and her feet were sore.
She called up a photo of him on her screen. She could see it, she thought: some sibling similarity around the eyes, a similar scoop to the sides of the face. And maybe they'd had the same nose once, before his had been broken a bunch of times.
She leaned against the wall beside a red pipe. An old vagrant trick, soaking up the heat it radiated. But after another hour, all the warmth in the world wasn't enough to distract from the ache in her feet and the conviction that this was a fruitless way to go about finding him. She turned and headed for the Hub.
A girl cawed above her. Imitating crows: an old scaler taunt to frighten pedestrians. Ankit cawed back, prompting startled laughter.
_We miserable grown-ups weren't always groundbound,_ she thought, and then felt happy that she still spoke scaler-speak. Sort of.
Another revelation from that costly dossier, one she'd been avoiding examining. Sharp, prickly, cutting up her hands each time she tried to get a hold of it: before coming to Qaanaaq, her mother had spent time in Taastrup. The same place the early cases of the breaks kept pinging back to. Did that mean something? Could her mother have contracted a sort of proto-breaks, decades before the first cases started cropping up? Had her brother? Could she have been the subject of some experiment, or a survivor of some accident?
Patient 57/301. No name; no match in any genetic identification bank. Imprisoned in the Cabinet for thirty years. Top-level classified status; authorization that could only come from one of the sponsor nations—China, which meant it could be anyone with a ton of money; Thai officials typically couldn't be bought off like that.
No diagnosis to appeal, no doctor to hunt down and punish. Waist-deep in bloody fantasies of how she'd hurt the person who did this, she noticed something. Someone staring at her.
A woman, with beautiful wind-scoured light brown skin, just a few meters away. She sat—sat?—in the sea. And then she rose, and Ankit saw she wasn't sitting in the water at all, she was riding, riding something as black and deadly and magnificent as the sea itself. The orca turned its head and stared at her, stared _into_ her, and so did the woman, and Ankit felt gutted, stabbed through, harpooned—
"She remembers you," she said, the famous Blackfish Woman.
She was real. And she was here. And she was talking to Ankit.
Ankit said, after a very long time, "What," and her voice was much smaller than she ever remembered it being.
"She has a very good memory. Thirty years later, she remembers someone's smell."
Ankit stepped closer, squatted down. Her breath would not budge. It stuck in her lungs like lead. She shut her eyes, tried to remember. Cast her mind back as far as she could. She recalled strange rooms, cramped spaces, fear, someone's hand across her mouth to stop her from shrieking. Footsteps overheard. Harsh male laughter. Nightmare glimpses, nothing new, things she'd carried with her and ascribed to filthy group homes and overcrowded nursery boats, things that fit right in with the long line of better-remembered ugliness that Qaanaaq's foster care system had given her. And when she began to see wide vistas of white snow, smell smoke, hear distant animal bellowing—gunshots—the wails and the cries of the dying—how could she know whether that was memory or imagination, the nanobonder genocide so widely written about, reimagined in movies, subject of epic poems and endless analysis?
She opened her eyes again, stared into the face of this impossible creature. Breathed out. "You're her mate. My mother's partner."
The orcamancer nodded. "I'm your mother, too."
"Of course," Ankit said, awkward, uncertain about what was the proper protocol here. An embrace? A bow, a handshake, tears, wailing and the rending of garments?
"We've been going up and down every Arm of this city for days. Weeks, maybe. Human time markers don't stick in my mind. Our minds. Sniffing at the air. Looking for your scent. This city, and a hundred others before it."
The woman's face—her mother's face—her other mother's face—was terrifying in its rawness. It hid nothing. It refused to hide. It was bare, alien, hostile to how humans behaved. More orca than person. Ankit's cheeks reddened with emotions, and they were not all joy and love. She also felt fear. Fear of this woman, who cared nothing for social niceties or Qaanaaq or the safe life Ankit had fought tooth and nail to carve out for herself there.
"My name is Ankit. Was that my name, before?"
She shook her head.
"What was my name?"
"It's not important now. Some other time, perhaps." She got off the whale, climbed up onto the grid. Extended her arm stiffly, like a foreigner unaccustomed to the practice. "My name is Masaaraq."
They shook hands. When Masaaraq let go, she stepped forward with alarming speed to swamp Ankit in a fierce bear hug.
They were exactly the same size.
"And my mother? What's her name? All these years, I've never known. She's a number, to them."
Masaaraq looked around, as if suspicious of invisible eavesdroppers. Outsiders, especially from the underdeveloped parts of the Sunken World, believed all kinds of crazy things about Qaanaaq's technological capacity, like that the fog could hear your every word and report it back to the evil robot overlords.
She whispered in Ankit's ear: "Ora."
"You're here to get her out," Ankit said. "Aren't you?"
Masaaraq nodded.
"I want to help."
She took Ankit's hand and squeezed it. Not hard, but implacably, her strength overwhelming. This was a woman who had never stopped, who could not be stopped, no matter what happened to her or to anyone else.
Ankit had to take several deep breaths.
She'd told herself that she was triumphing over the fear. She wasn't that kid anymore, the one whom fear froze solid, the one who was ruined by it. She'd posted that photo of Taksa—she'd gone to see her mother—again and again she'd stood at the edge of a tough decision and made the leap to the next one. Done the difficult thing.
But here she was again, up against her limit. A line she was afraid to cross. A leap that meant risking everything, leaving behind all she knew of comfort and ease, that might land her in jail or deregistered or worse. _I want to help,_ she had said, and she did, but she couldn't. To distract herself and Masaaraq from the bile rising in her throat, the panic she knew was visible in her face, she blurted out, "And do I get a killer whale?"
"Polar bear," Masaaraq said. "That's what you were marked for bonding to. But she died before you two had a chance to bond. When we were attacked. When your mother escaped with you two. We were nomads, spending the winter in an empty town. I was away, tracking the people who were trying to wipe us out. I failed. They got to you first. Everyone died—except you two. And her."
"And you," Ankit said.
"You should count your blessings," the killer whale woman said. "The fact that your animal died before you two had been bonded is the only reason why you didn't spend most of your life a gibbering idiot."
"Like my brother," Ankit said. "Right? Is that what's wrong with him?"
"You know Kaev? He doesn't know you."
"Yeah. I know of him. I introduced myself, once. He . . . sort of . . ."
"Broke down. Yes. His mind was cracked. He was incomplete, his whole life. He is complete, now, and healing. You have been incomplete as well, you just haven't known it. I will heal you, too."
"I'm not incomplete," Ankit said, feeling the grid give way beneath her. "I don't need to be healed."
"You do," the orcamancer said. "You need to be bonded."
"Or what?"
Masaaraq did not answer. She took a step back.
Ankit asked, "Do you . . . want to come over to my place? For a cup of tea?" She looked at the whale, briefly—ridiculously—imagined it in her elevator, daintily holding a teacup in one massive blade-fin, sitting at her kitchen table. "Can you . . . leave her?"
"I can. And I will drink tea with you. But not right now. I have an errand to run. With some friends." From behind her back, she pulled a black box that was strapped to some kind of orca saddle. "Right now we must begin the bonding process."
"No," Ankit said, possibly not out loud.
"I have no polar bear," Masaaraq said. "And it would need to be an adult, as you are an adult. I do not know how the bond will come out. Whether it will be painful, how well it will work. Our kind has never bonded someone so late, who had never been previously bonded. Do you have an animal in mind? Something you've always felt a particular attraction to? Connection with?"
"No," Ankit said, louder now—gods, she didn't even have a pet, she had never been interested in a romantic partner, the thought of commitment made her throat hurt, and now this strange woman who'd walked into her life five minutes ago—rode into her life on a killer whale five minutes ago—was proposing a kind of commitment more intimate and horrifying than anything she'd ever contemplated before.
One look in Masaaraq's eyes, and she knew—this woman was not entirely human. Being bonded to an animal turned you into something else. Something that behaved completely differently, wanted different things.
"You have some time to think about it," Masaaraq said, removing a series of strange tools from the box. Syringe, bottles, droppers, other things Ankit had no words for. "For now I'll just take a sample of your nanites so we can start the culture process."
"No," Ankit gasp-yelled, and turned to run—and then turned back, and told Masaaraq precisely how to find her, where she lived and where she worked, and stammered some profuse apologies, and turned, and ran.
## Kaev
Kaev had wondered, once, what it was like for the superstar champion fighters. The ones everybody recognized, the ones crowds parted for, the ones who made people break into wide-open smiles of amazement, gratitude. Fear. His buddy Ananka had come close, back when both of them were just starting out, and passersby were always stopping to stare at her, but he knew it was nothing like the universal awe that the biggest beam fighters were held in.
And even that would be trivial compared to the attention he was getting now. Half the population of Qaanaaq was totally oblivious to the beam fights, yet not a single one of them could be oblivious to a polar bear.
So here he came, strutting down the grid with a polar bear padding beside him. Not a little one, either. And the legendary orcamancer, bone-blade staff in hand. And a street urchin slide-messenger-turned-criminal-errand-kid who happened to be his child.
Everybody stopped. Everybody stared. Many screamed. Many tapped their jaws, dialing Safety. People took pictures with screens and hat cams and oculars. Children began to cry. There was a polar bear in the Floating Zoos, but it was small and sickly and unhappy.
Soq had suggested whistling for a jaunt skiff, but the orcamancer— _Masaaraq,_ Kaev quickly corrected himself, _she has a name, she is a person, not the mythic figure everyone tells stories about_ —said no.
Masaaraq had been in a foul mood, returning from some nebulous errand that had upset her greatly, and she snapped, "Let them see us."
_My city finally knows who I am. Everyone is looking at me, and not because I'm about to be beaten by some pretty-boy kid. This will be in all the outlets in instants._
Once or twice he turned to look at Soq, this other magnificent creature who'd just entered his life, and each time, he saw Soq look away swiftly.
_At least I'm not the only one who doesn't know how to handle this,_ he thought, and smiled. The bear stopped to sniff a red chrome pipe and snarled at the absent dog whose scent it found.
"It's crazy," Kaev said. "I can feel him, how he wants to attack these people. How they all look like meat to him. How they smell. But he can feel me, too. And he knows he can't do that. So he doesn't. How is that possible? People spend years trying to tame wild animals, and this one became tame . . ."
"Instantaneously," Masaaraq said. "But he isn't tame. A bonded animal is only as tame as its human is. And humans can be very, very wild."
"True."
"And you want to be careful. It goes both ways. You influence his behavior, but he can influence yours. Usually humans assert emotional dominance effortlessly, but you need to stay vigilant."
He put his hand on the bear's shoulder, and they walked proudly through the Hub and onto Arm Five. At the entrance a Safety officer stopped them, tapping for backup and visibly trembling.
"There's nothing in the registration consent agreement that says it's illegal to have a polar bear," Kaev said. Masaaraq had talked him through this already. She'd done a lot of research before she came to Qaanaaq. "I can vouch for its behavior. It won't hurt anyone."
"Unlicensed wildlife," the officer mumbled, unconvinced, eyes out for assistance. His hand moved to the zapper strapped to his belt, but that was barely good enough to take down a chubby drunk. It'd just irk the bear, and irking the bear was probably not something he wanted to do.
"Licensing terms only apply to registered residents," Kaev said, another Masaaraq talking point. "This belongs to my friend here, who is visiting."
". . . wanted in connection with more than one instance of multiple fatalities . . ."
"No proof that this was the same polar bear."
They kept walking. In ten minutes they'd be off the grid, and Safety couldn't do a thing about it. It'd take the combined zappers of twenty officers to slow the bear down, or authorization from HQ to use lethal weaponry, both of which would require a lot more time than that to procure. Kaev quickened his pace, but not by much.
A lot of food processing happened on Arm Five. The smells of cooking fish and caramelizing soy sauce made the bear grumble with hunger. Kaev could feel that, too, the bear's hunger distinct from his own, and when he focused on it he could feel his own hunger grow. A loop. An echo chamber, hunger bouncing off hunger and magnifying, multiplying, and if he just shut his eyes and let it happen the bear would be able to reach out one arm and effortlessly satisfy their hunger—
Pain jolted him back; Masaaraq had struck him with the butt of her weapon. "Hey!"
The bear watched her as they walked, its hostility tingling in Kaev's elbows, like, _I'll remember that—and I still haven't forgotten how you had my head in a cage for all that time._ Soq put a hand on the bear's other shoulder. Kaev felt this, too, the animal's blunt mammalian happiness at the touch of someone it liked, not so different from a dog's. A beautiful thing to be inside of.
They reached the gangway. Soq asked, "You buzzed her?"
"Trust me," Masaaraq said. "She knows we're coming. She's got eyes on all of us."
Kaev had been focusing on his bear. Trying not to think. About Soq; about where they were going. About what it meant that he had a child. About how the final, knock-down drag-out break-up fight between him and Go had happened a couple of months before Soq was born. About how he and Go had had a child, together, all this time.
About how much he hated her. And about how he could hate her, but also feel this strange feeling, so oddly like happiness, in his chest, growing bigger with every step that brought him closer to her.
****City Without a Map: The Breaks
No one knows where or when they got the name. The origin story is something banal, most likely—they caused nervous breakdowns, full psychiatric breaks, irrevocable shattering of identity. Oldest known usage is found in transit camp correspondence, refugees using it colloquially enough to imply it had already existed in spoken dialogue. There is a deeper resonance to its persistence, a troubling question that arises if you stop to ponder it: Why do we still call it by such an informal name? Why has it not yet been replaced by something more scientific sounding, more medical, even if it were just an acronym, sad as a flag of surrender, identity dissolution syndrome (IDS) or multiplicative affiliation disorder (MAD)? Epidemics do not have medical causes; they have social ones.
I have been stitching its story together here. Collecting scraps of history and rumor. Memories. Sick people, heads spinning with strange sights. Slum ship operators going out of business because their boats had become floating hospices for afflicted tenants. Doctors and agency officials baffled by the failures of software to devise a solution, or even the most modest of mitigation measures—almost as if someone, some powerful intelligence either human or machine, is determined to block any such development.
Word of the breaks has spread. Babbling madmen in the streets, children screaming someone else's secrets. Qaanaaq is adrift, they say—floundering, helpless, failing, its once mighty AI oversight no longer equal to the task of maintaining order. Safety is overburdened. The brigs are overflowing. Crowded quarantine ships remain anchored to the Arms. Hearing stories like that, people would get all kinds of crazy ideas.
A crime boss might start plotting out a power play.
And a woman on a mission of rescue and revenge, who for years has known that she must eventually come to Qaanaaq, who knows that the thing she seeks is here, but knows that her enemies are here as well, the people who robbed her of the thing she seeks—and so much more besides—and they are powerful, and they have the full might of the Qaanaaq municipal system behind them, might decide that the time is right for her enemies to fall, for her journey to reach its end.
The breaks brought her here.
## Kaev
Sure enough, they were expected. Dao stood at the top of the gangway, flanked by a tight crowd of armed, frightened foot soldiers. Kaev didn't slow, didn't hesitate, marched right up the gangway. The bear followed.
"What'd you come here for?" Dao said, his palm on the button that would release the hydraulic lock, retract the gangway, spill them down into the sea.
"You know why. To see her."
"What if she's busy?"
"Too busy for this?"
Soq stepped forward. "Ring her up, will you? Ask her yourself."
Behind them, emboldened by the fact that the bear had stepped off the grid, was a crowd of Safety officers. Assembling some kind of weapon Kaev hadn't seen before. Nonlethal, probably, but scary—the latest generation of sonic pulse cannon, maybe, the ones pioneered in Russia for knocking out crowds of demonstrators, which might also cause aneurysms. He didn't want that pointed at any of them. If Go didn't let them onto her boat, even the bear would be in danger.
Dao turned his head, speaking into his implant.
"Hey, how's it going," Soq said to one of the foot soldiers. She smiled back nervously. All of their eyes were on Soq. And Soq knew it. _A pretty good way to impress your new coworkers,_ Kaev thought. So Soq, at least, was enjoying this. And so, mostly, for that matter, was Kaev. That bliss was still there, what he'd felt walking through Qaanaaq with the eyes of everyone on him. The power that comes with having five hundred kilograms of stark white killing machine at your side. The simple pleasure of not having your brain be a caustic broken worthless mess.
He shivered, remembering. How ugly every minute had been. How even the simplest sentences, the most straightforward thoughts, would crumble in his hands. How frightening he found other humans. The joy of fighting, those rare moments, those orgasmic instants with long stretches of broken glass between them. Such a pale shadow of the pleasure he took in every instant beside the bear. And even that, the fights, he was lucky to have had. Plenty of people with broken brains turned to far worse addictions.
Masaaraq was frowning, he noticed. Not the wary frown of someone in a tactically unsatisfying position, either. More like general unhappiness.
"Everything okay?"
"This is a distraction," she said.
"From what?"
"From what I came here for."
Kaev nodded. He'd been told no so many times, when he asked her why she was here, that he'd given up asking. "Go is powerful. Connected. Smart. Whatever you want to do, she can help you do it."
"If she doesn't just kill us all."
"She could try."
"Is she powerful, or isn't she?" Masaaraq said. "If she is, she can destroy us. We're not invincible. Killer whales and polar bears are just like any other weapon—they can't solve everything. And they have their limitations. I know that better than anyone." She paused. "Almost anyone."
"Come along," Dao said. He stepped away from the button. The soldiers receded. To Soq, he said, "She's in her cabin. I believe you know the way."
What seemed like a small army squatted on the deck of Go's ship, repairing fishing nets. Kaev was always surprised by just how boring Go's criminal empire really was.
Soq didn't seem bored. They stopped to stare, to watch the little fingers at work.
Kaev's heart hammered. His blood sang. He flexed his fingers to keep them from making fists. He was on the beams again.
And also—he wasn't. He was a boy, young and handsome and strong, sneaking out of the foster barracks to meet a girl. A girl who didn't mind his stutter and yips, and how his sentences didn't make a lot of sense. A girl as strong as he was, as fearless, but far smarter. A grid grunt, like him, but unlike him she had a plan. A way to work her way up, a way to build an empire where they'd both be safe. Where neither one would be nothing.
Go had succeeded, he saw. She'd had to abandon him—or had he abandoned her?—but she'd succeeded.
The door to her cabin opened. She stood there. She saw Kaev first, and he could see, now, with the calm and clarity that the bear had given him, the emotions that danced across her face, the happiness and then the anger; the love and then the hate. All in a fraction of a second, then swept under the rug of her fearless-leader face.
But then she saw Soq. And her mouth opened. She looked back and forth between Soq and Kaev, and he could have sworn he saw something inside her fall away. The fearless-leader face broke. Whatever Dao had told her, she hadn't put the pieces together until this moment, seeing them together. Probably he hadn't mentioned Soq at all. Probably he'd been focused on the polar bear. Which would be understandable. But Go wasn't focused on the polar bear. Go didn't seem to see it at all.
She took a step forward. One hand went to her chest.
"I—" she said, but said no more.
Kaev took three swift fearless confident steps forward and embraced her, held her to him.
He loved her. He had always loved her. Every other piece of it fractured, crumbled. The anger and the hate and the slow poison bitterness of being her flunky, her fall guy, her journeyman loser, and then her brutal thug, her soaker. And something similar must have been happening for her, because he could feel the slow melt, the way her arms around him went from rigid to tentative to being as hungry and firm as his. He did not need to ask, _Why didn't you tell me?,_ because it all made perfect sense.
"I'm so sorry," she said.
"Don't be," he said, stepping forward without letting go of her, carrying her back into her cabin, knowing she would not want her underlings to hear what they had to say. Would regret even letting them see the two of them kiss. "But me too."
Her eyes overflowed. Her words came fast, between sobs, sentences strung together like she'd been saving them up for ages. "I didn't want to abandon Soq. I didn't want to abandon _you_. But I had made my choices by then. I had started down this path. I had made enemies, people who would have killed you both. Jackal was already consolidating power, getting people into her corner. I was going to end the pregnancy. I should have. But I loved you too much—loved who I was with you. I knew we could never be together, but that what we had could . . . I don't know . . . continue to exist."
"Shhh," Kaev said. He saw Soq pretending to play with the polar bear. Trying hard not to look in his and Go's direction.
"I did it to save your lives. Broke up with you, hid Soq from both of us."
"And you succeeded," Kaev said. "We're alive! All three of us."
Go shook her head. When the words came, they came so fast he knew they'd spent years echoing through Go's head. "I don't deserve this. I don't deserve you. Jackal's been dead for a long time. No one has seriously threatened my position in a decade. The coast was clear. I could have come for you at any time. And I didn't. Because when I left you—both of you—it hurt me so bad. I swore nothing would ever hurt me like that again. I built a wall around my emotions. Never let anyone in. I wasn't thinking big. I was so focused on my little goals—climbing the ladder, one step at a time. I wanted to hold on to the rung I had, and maybe get to the next one."
Kaev kissed her forehead. "It's okay."
"I know. But I want you to know where I was coming from."
"Tell me."
"Syndicate leaders always think small," she said, pulling one arm away but coiling the other tighter around his waist. Together they turned to look out the porthole, at the city's jagged green skyline. "They're really just good little capitalists, no different from any corporation head. They want to make money, and make money for their friends. But I want more."
"What do you want?"
She opened her mouth. He knew that look. The look you have when you're about to say something you've whispered to yourself a thousand times but never uttered out loud.
"I want a city where people don't have to do what I did."
"Seems like a pretty good reason to me. Is that what all this is about? Is it why you had me soak those guys?"
Go nodded.
"Shareholder, right?"
She nodded again.
"These are some dangerous moves you're making. They run this city. Run the AIs that run it."
The city sparkled. The polar bear was letting Soq climb onto its back, with the expression of a patient long-suffering parent. At the end of their Arm, methane flares as big as buildings parabolaed up into the sky, prompting shouts of joy from the spectators who watched for them every night.
"I want you," Go said. "And Soq. And Qaanaaq. And we can have it. I want to think big."
_I want to think big, too,_ Kaev thought, but there was no need to say it, because if there was one thing he'd learned from years of being a brain-damaged lunk, it was how words were way more likely to get in the way than help you out.
## Fill
Where are we?" Fill asked. Wind whistled through an open door or window ahead of them. The room was freezing, and dark except for the faint green omnipresent light of nighttime Qaanaaq coming through the window. He'd met Barron outside an Arm Three apartment building that had seen better days, and followed him to the end of a hallway and through a hatchway. "What is this place?"
"A pod," Barron said.
"You can afford a privacy pod?"
"Good heavens, no," the old man said, and laughed. "Belongs to a friend of mine."
Laughter from the grid outside, and the churn of the sea. Both seemed so distant. The polyglass bubble felt tiny, fragile, even though it was one of the largest and most lavish Fill had been in. He rubbed at the goosebumps on his arm. "I thought we were meeting at your place."
"No, alas," Barron said. "I am far too ashamed of my little nook. Would never survive the ignominy of your sainted grandfather seeing it."
Lights. The walls ionized and came to life. People surrounded them. Portraits, cropped close so they could have been anywhere. Old people and young ones. Their faces empty, hunted. The pod they stood in was medium-sized and completely empty. Eight people could have fit comfortably inside it, but for the moment they were alone. Ionization and the portrait projections prevented them from seeing out. Fill wanted to focus on the imagery, but he was shocked at how nervous he was for these two men to meet.
"Hello?" came a voice. On the wall, where an angry little girl frowned out from her mother's lap, a rectangular hole appeared in the projected image as a door slid open.
"Grandfather!" Fill said, obscurely grateful to see the old man walk in. Something about the setup unsettled him. The distant tone to Barron's voice, the eerie imagery that surrounded them. The door slid shut and the illusion was complete again. Cold faces appraising them, finding them wanting.
"Grandfather, this is my friend Barron. Barron, this is my grandfather."
"Mr. Podlove," Barron said, his face as rigid as the ones on the walls. "I cannot tell you how happy I am to finally be formally meeting you."
"Please," Grandfather said. "Martin." They shook hands. Side by side, they looked so similar.
A hydraulic whine, and then a sense of motion. "Are we moving?"
"We are," Barron said.
Grandfather laughed. "I assure you, all these theatrics aren't necessary. My grandson says you need help, and that you deserve our assistance. I just wanted to talk out the details. I don't need a sales pitch. Whatever song-and-dance routine you two have put together, I appreciate it, but—"
"This is all new to me," Fill said. His laughter felt forced. "We're not going underwater, are we? I've always hated submersible pods. I don't know why. Ever since—"
"Don't you worry," Barron said, his tongue probing his cheek, controlling the pod's struts. "Don't you worry about a thing."
"Who are these people?" Martin said, gesturing to the walls.
"I think they're _City Without a Map_ listeners," Fill said. "Right? Or potential listeners? The target audience, anyway."
"Not exactly," Barron said. He probed the inside of his cheek with his tongue again and the portraits uncropped, providing the context of where these people stood.
New York City, Fill saw. The elevated subway tracks, the blackened stadium. In the decline but before the end.
"Victims," Barron said. "People who died with the city."
"How much are we talking about?" Martin said, hands in pockets, every inch the consummate financial wizard. "For the project you two are working on. Some kind of performer, you mentioned? A reader?"
"We don't need your money," Barron said. "And there is no Reader."
Fill frowned. "Sure there is. Choek. She said—"
"What I paid her to say."
"Then . . ."
The portraits vanished. The lights went out. Fill heard the hush of a door sliding open, and then a second.
Moving with surprising speed, Barron shoved Fill forward and pulled his grandfather back. Two doors slid shut. Locks magnetized. The walls de-ionized, and Fill could see outside. Two separate pods: he, alone, in one, the two old men in another. The city below, unspeakably beautiful.
"What the hell is the meaning of this?" his grandfather hissed, seizing Barron by the arm. The pods were soundproof but miked, and he could hear them louder than life through the speakers in the walls.
"What you said. A song-and-dance routine. You don't remember me, Podlovsky?"
Fill watched his grandfather's eyes widen. No one had called him by his old name since he'd left New York City. He let go of Barron's sleeve.
"Of course you don't. I was beneath notice. You wouldn't have bothered to note the difference between my face and the faces of the thousands of other tenant leaders you were contracted to eliminate."
"I never _eliminated_ anyone. It wasn't like that." But the old man did not sound convinced. Fill was shocked at how swiftly they had swapped roles. Gone was the timid decrepit creature halfway to senility; gone, too, was the terrifying titan of industry. Of course both personas had been an act. Of course his grandfather had been a scared guilty man trying to sound brave his whole life; of course Barron had been a vengeful monster buoyed up by rage and playing the part of a dotty old queen so as to execute a devastating retaliation for hurts a half century old.
How long had Barron been watching him, planning, scheming, putting the pieces together, learning what he loved, how to get Fill to trust him? Or had fate dropped him in Barron's lap one day and reawakened a coldblooded urge for revenge that had lain dormant for decades?
" _Eliminate,_ ugh, what an ugly word. No, you'd never do something so crass. So criminal. You'd manipulate others into doing those things. You'd sit back, and watch, and let your clients reap the benefits."
Grandfather took out his screen, tapped at it, trying to call for help. Nothing, of course. Pods were easy to seal off from outside signals. That was the whole point. Rich people always needed a way to escape. A place they could go and not be bothered, and not bother anyone else. Grandfather threw his useless screen at Barron, who deflected it with one arm. And then stomped on it.
Fill didn't bother to take out his.
"Fine," his grandfather said. "Kill me. Do what you came here to do. My conscience is clear. I did what I did for the sake of my family. My conscience is clear. But will yours be?"
"I have no intention of killing you," Barron said. And turned his head, slowly, from grandfather to grandson.
"No," the senior Mr. Podlove whispered. Here, his resignation broke. His unshakeable dignity crumbled. He surged forward like a crazy man, arms swinging, desperate—and was promptly struck in the gut by a club neither he nor Fill had seen Barron produce. While Podlove was doubled over, Barron swung it sideways, striking him in the face, knocking him to the ground.
Fill cried out at the sight of his grandfather's blood. But for his own fate, he felt little.
"I'm the one you hate," Podlove said from the floor. "Hurt me instead. Kill me."
"I _am_ hurting you. Death isn't a good enough punishment. To survive—to be haunted—to have to live with the loss of the people you couldn't save—that is a fitting sentence."
Grandfather crawled to the door of his pod. Pressed his bloody palm to the polyglass. "Fill, I'm . . ."
Five narrow feet separated them. But the gulf was unbridgeable.
Fill pounded on the door, but only once. He felt no fight inside. No desperate will to survive. No drive to drop to his knees and beg.
"One final piece is missing," Barron said. Lights came on in the air outside the pods. Cam drones. "No one was watching when men like you made us disappear. An inconvenience for the wealthy, wiped clean. So that you could build this cushy life for yourself. So that you could go on to hurt others, the way your firm did. In so many places. With so many communities. Some of us stopped to film, but nobody cared to see. The world should see this."
The pods parted. Fill's rose up into the air. Three of the drones rose with it. Three of them remained to record the other one.
"What are you doing?" his grandfather said, and Fill could hear him just as clearly as when the pods had been practically docked.
Fill wondered himself, but only in the most academic of ways. Shivers shook through him. The breaks, intervening. Would Barron lower the pod into the sea, open the doors, drown him? Would he raise him up as far as the strut would allow, thirty stories perhaps, and then let it fall?
"Fill!" his grandfather screamed.
"It's okay," Fill said. "I am okay—"
The old man kept saying his name. He couldn't hear him. The sound only went one way.
Proximity alarms sounded. Fill's pod shuddered, momentarily surrendering to the city's emergency override commands, but then kept moving. Nothing so impressive there—software to disconnect from the emergency infrastructure was easy to find. The pod continued on its upward swing.
Into what, Fill now saw. Four flames kindled in the wall of a building in front of him, inside a black circle wider in circumference than the pod he was in. Arm Three's methane ventilation shaft. The evening's scheduled flare.
Fill was there. He was in New York City. He was watching a circle of cops beat a boy to death. He was watching a mother drag her child from a burning building.
He was watching cherry blossoms shiver in Brooklyn rain.
He was drinking expensive scotch in a hall bigger than any he'd ever imagined.
_Bloody snow. A baby polar bear. An eagle, falling._
Right on schedule, the methane flare came to life. A coil of bright green fire three meters thick. Fill could see the crowds down at grid level. They pointed at his pod. They screamed. They took pictures. He smiled and waved. They probably couldn't see him. A couple was kissing, oblivious to his existence, his imminent death. The pod ascended into the path of the flare. There was an instant where he could see the flames curl and wrap around his protective sphere, see the ripples rise and spread in the polyglass. Then it burst like a soap bubble, and Fill broke free from his body.
## Kaev
The streams showed tear-wet faces, lit candles, hands clutching the possessions of lost loved ones. The 'casts played interviews with sad people and angry ones. The headlines wailed predictably, theatrically, pathetically.
THIRD CONSTRUCTION DISASTER PROMPTS TERRORISM FEARS
RETALIATION AGAINST WHO?
HOW LONG WILL THIS GO ON?
[ **insert name of interchangeable demagogue-of-the-week** ]
DEMANDS TRANSFER OF ALL AMERICAN RESIDENTS
Kaev stood on the grid, shuffling through outlets. Unthinkingly, he leaned back into a horse stance. His posture was perfect. The wind could not move him.
Go had given him the basics. Four days ago, some shareholder had lost his damn mind. Martin Podlove, Go's nemesis. She would not have imagined that he had it in him to care so much about someone, even his grandson, that he'd go on this much of a rampage. She made Kaev watch the video with her, where the old man saw the kid get killed. Watched him wail.
"Of course he thinks I was behind the killing of his grandson," Go said. "I declared war on him, and now this happens? He'd never believe it's just a coincidence."
"Was it you?"
"No," Go said.
"Would you have, if you'd had the chance?"
"Shut up," Go said. "He's coming after me, and my assets. Hard."
And now: war. Targeting illegal construction sites—her own. Maybe he didn't care a fig for that poor kid, and his wild savage behavior was that of a cornered rat, or a king who could not let an affront go unpunished.
Most people didn't know who Podlove was. Everyone saw him, in that horrific drone footage; heard him screaming as his grandson was incinerated. Many outlets made the connection between that gruesome spectacle, broadcast for all of Qaanaaq to see again and again, and the subsequent carnage. But most thought it was a squabble between syndicates, Americans being American, brutal violence the only language they could speak or hear.
Go talked about it a lot.
Kaev didn't care about any of that. His belly was full. His bear was happy, back on the boat. Slideshows on walls and in windows flashed images pulled from the feeds by bots set to scan for things pertaining to these attacks: prayer vigils, in memoriam sites, the names of the dead in forty different alphabets. Long lines of strong men and women outside Arm manager offices, employment halls. Wild hope in their eyes. Not mourners, mostly. They were looking for work. Construction sites all over the city had shut down. Other syndicate-connected businesses had started to do the same.
The day was cold and the grid was crowded. People stood around looking dazed, or wept together, or embraced. An old woman stopped Kaev and pressed a small black square of fabric into his hand.
"Thank you," Kaev said, meaning it, wondering why she looked familiar.
"Pray for us," the woman said, her accent American, and Kaev wanted to ask, _Pray for a lost loved one of yours, or pray that you and your loved ones will be protected from the wave of anti-immigrant violence that idiots will soon be unleashing on anyone they can?_ Instead he bowed his head to the woman and paused to take her in, to truly see her. The worn brown skin and the light in her eyes. The bandaged stump where one hand used to be.
_What happened to you, grandmother? What has life been like for you?_
The woman smiled at him, a magnificent smile, and Kaev went on his way.
The squares of fabric were such a strange mourning custom. You'd find one, long after, cleaning out a bag or emptying out your pockets, and would have forgotten, at first, how and when it came into your possession, so that instead of making you remember some specific dead person or group of dead people it made you reflect upon mortality in general, your own in specific. Wondering whose pockets your own fabric scraps would one day clutter.
Two scalers descended from a rooftop nearby to meet up with a messenger. Kaev braced for violence, but there was none. They were friends, the three of them, and they laughed together with the loud carelessness of the young.
He had been young once. Had had friends. His sickness had not been so bad back then. They got into trouble, flirted with dangerous girls and boys, did crazy things, formed intense immediate bonds that lasted until the sun rose. Conquered an Arm or an arcade, ran from Safety, accidentally mortally insulted someone.
"What a city," the _doodh pati_ vendor said, handing him a cup of sweet cardamom-smelling goodness. He pointed to a clumsily assembled nook where fabric flowers and methane candle flames fluttered. There were hundreds of these nooks, all over the city, each erected in memory of someone who'd died as part of Podlove's war on Go.
And each one made Kaev shiver with proxy guilt. He'd always known Go was a monster, but for years he'd been shielded from the evidence of her monstrosity. And he'd had his hate to hide behind.
An illegal construction site might have fifty people working it at any moment. Most of them would have been swallowed by the sea when the grid gave way, its standard emergency practice when something threatened the structural integrity of an Arm. It would take weeks for the diving drones to complete a sweep, probing in and through the cluttered nest of twisting pipes and vents that made up the geothermal pyramid beneath the city.
"No worse than any other," Kaev said.
"No, but this one is ours."
"Yeah."
Kaev went to the railing, watched the sea slosh at the struts underneath him, sipped his milk tea, smiled at the weird good fortune of being alive and unmiserable while so many people were not.
## Soq
The biggest baddest deadliest puppy you ever saw. The bear lay on its back and let Soq rub circles into its soft stomach fur, smaller and smaller and then larger and larger.
Needless to say, Soq had never had a dog. They'd never had a home stable enough for one, not really, and while there were plenty of grid kids with companion animals it tended to be a pretty rough life. For the animals, certainly, and for the people, when the animal was inevitably murdered or stolen.
No one would murder this one, though.
"Does he have a name?" Soq asked the orcamancer.
"That's up to Kaev."
"Liam," Soq said. "His name is Liam."
Masaaraq raised an eyebrow.
"He's practically mine. If my father has a problem with it, he can discuss it with me."
Saying it still felt so strange: _my father_. The sad-eyed brute; the perpetual beam fight loser. Soq had read all about him in the long time they'd spent lounging on the deck of Go's boat. __Kaev was good, a skilled fighter with the stamina and savvy to make his fights enter taining. More than one fights writer had wondered why he lost to so many inferior warriors, and suspected syndicate involvement.
Was that Go? Had Soq's father been Go's pawn, too, a tool in her rise to power, an easy way to fix fights, build resources and a reputation?
They were belowdecks now. Kaev and Go. Not making love, Masaaraq had said—or the bear would be behaving very differently from its present lazy blissful calm. Catching up. They had a lot of that to do. And when would it be time for Soq to catch up? They imagined an awkward family dinner, screwball antics, hilarious malentendus, like the old movies that were the rage a few years back. Soq had a lot of questions, but they weren't eager for that kind of conversation.
"Where did you go, the other day?"
Masaaraq frowned. "To see someone."
"See who?"
"A friend."
"You have friends here?"
"Shut up," Masaaraq said.
"What did your friend say to make you so upset when you came back?"
"I'm your grandmother and you have to shut up when I tell you to."
"Whatever," Soq said, but then did indeed shut up.
A ripping sound snagged Soq's attention. Masaaraq was sawing her bone-blade at a freshly repaired net.
"What are you doing?"
"Nothing," Masaaraq said.
"Yeah you are. You're slicing up that net."
"Shut up."
"What's the matter?"
Masaaraq said nothing.
The polar bear had been a distraction. Soq should have been focused on the fact that they had parents, and they were reuniting, and there was a ton of backstory to get caught up on. That, and Masaaraq, the orcamancer, last of the nanobonded. And maybe they'd continue to ignore them, but an inquisitive kid like Soq could not let a chance like this go to waste.
"You came here to do something," Soq said. "But now you have to sit here like a schmuck while those two play house."
"House?"
"Mommy and daddy. Happy family. Sexy times."
Masaaraq rolled her eyes. "It's of no concern to me, what they do."
"But you're here for a reason. And right now you're wasting time. Right? That's why you're annoyed."
Masaaraq shrugged, but it was a shrug that conceded there was some truth in what Soq said.
Masaaraq stood up. Sirens and flashing lights flickered across the water at the tip of Arm Three. Oversize evac drones buzzed in the air.
Soq asked, "Are you here to kill someone?"
"You have a lot of questions."
Soq scowled. "You have a lot of ways to not answer them."
"Here is a question for you. The Cabinet. You know of it?"
"Of course."
"If someone was in there. Someone you wanted to get out. How would you do it?"
"Who's in there?" Soq said, eyes wide.
Masaaraq exhaled angrily and turned away.
"Wait! I'm sorry. But I'm not sure I can answer your question. I've never heard of it being done before. Known plenty of people who ended up there, many of them against their will, but none that managed to bust out and tell the world about it."
Masaaraq nodded.
"But you must know more about it than I do. Right? You've done your homework. I heard you lecture Kaev on the registration consent agreement, and you know it by heart, better than me or anyone else who grew up here. I have to imagine you spent a long time looking into this. What've you got? I'm pretty resourceful. So's Go."
"So I keep hearing," Masaaraq said sourly.
"Let's brainstorm!" Soq said. "Bounce ideas off each other. Share crazy thoughts."
Masaaraq laughed. "My best idea is plenty crazy already. I was going to knock the goddamn door down and march in there with a polar bear and kill everyone who got in my way."
"That's a good start," Soq said. "But they'll have doors not even Liam here could knock down."
"His name is not Liam. That's an idiotic name for a polar bear."
"You'll need help. Software, maybe, a way to reroute control of the security systems. People who work there, who you can buy off or threaten or blackmail into helping you out."
"I've thought of this," Masaaraq said, but she did not seem quite so contemptuous of what Soq might have to offer. "Some of it, anyway. Keep . . . brainstorming. I have to leave."
"Leave?" Soq said, laughing. "What, you have a haircut appointment?"
"We'll talk tomorrow. You and me and Kaev. And, what's her name? Go."
"Are they my family, too? Whoever's in there?"
Masaaraq stalked off to the far edge of the boat. While Soq watched, she began to move through a complex series of weapon routines. Soq tried to focus on the important business of rubbing Liam's belly but couldn't keep their eyes off the orcamancer's hypnotic forms. A black shape against the city's lights, swinging and leaping and spinning. Unstoppable. Unresting. Unyielding. Forever fighting a line of invisible enemies.
## Ankit
Ankit went to see the boat people like it was just another workday. Cambodian refugees, hundreds of them, living in single-room shacks on floating flats towed from Tonle Sap when rising seas flooded that freshwater lake and forced the people who made their living fishing it to flee. A dense population pocket; prime get-out-the-vote fodder. When Ankit arrived, children were waiting in the doorway, their faces every bit as full of hope and fear as they'd been in the famous photographs of lake-top life before and during the exodus.
She liked the houseboats. Inside, it could still be Cambodia. Still be a hundred years ago. They had their own little village, all of them tied to one crude pier. Their own little school; their own little gas station. A girl sold scratch-off lottery tickets out the window of one. Some still kept crocodiles or hogs in tiny cages heated by illegal off-red geothermal extension pipes. By and large they were uninterested in outsiders, aloof from politics, but a small cluster of them voted.
Her pockets were full of candy. The wrappers sported Fyodorovna's name and face. She never failed to feel gross giving them out to kids, but then again it never failed to make the kids happy. A little girl gave her a lump of wood, brightly painted. Seeing Ankit's quizzical expression, the mother pointed to a pile of carved water buffalo. Bright brown; cashew nut tree wood. Ankit smiled; there was something comforting about the consistency of human hunger, human wickedness, that the wood trade would continue even as the total number of trees in the world slid down the parabola toward zero. The lump smelled like sap. "A mistake," the mother said, and Ankit's screen translated. "It broke." Ankit pocketed her mistake, bowed, and departed.
She sent Fyodorovna a photo, reported that the visit was a smash success. Leaving Arm Seven, heading out onto Arm Five, she felt like a kid playing hooky from school.
And, just as she had when she played hooky from school, she was scared out of her mind and had no idea what she was doing. Back then she'd been heading for submarine arcades, clubs for underage scalers, and now she was standing in front of a boat that served as crime syndicate headquarters, where bad things almost certainly happened on a daily basis.
Unlike the immaculately crafted campaign plans she constructed for her boss, her present plan was flimsy, slapped together out of a couple of measly observations.
First, that all four of the illegal construction sites hit by violent attacks were owned by the same syndicate—Amonrattanakosin Group. Formerly a legitimate enterprise of the illegitimate Thai military government, friends of a general who got handed inflated contracts for provisions vending during the grid city construction boom. Currently headed by a Thai-Malaysian second-generation Qaanaaqian known only as Go. Headquartered on a boat on Arm Five. Which Ankit now stood before, utterly without a plan.
Second, that she was certain she knew who Go's unknown nemesis was, in this war that had sprung up overnight, because it followed immediately upon the very public execution of Martin Podlove's grandson, in a film clip seen hundreds of thousands of times, with Podlove himself wailing and pounding on the walls of a polyglass prison.
Martin Podlove, who had put her mother away. Her mother, who had a name: Ora.
The war had been going on before—percolating, a pot simmering with the lid on. It was Go who had soaked Podlove's employee, Go who had him on the run. Whether Go was behind his grandson's murder, Ankit could not say. But Podlove probably thought so.
_Hey there, crime boss lady, I think we have a common foe._
_The man who is trying to destroy you destroyed my mother, and we should work together to destroy him and also rescue her._
_And I know who he is—the man who killed Podlove's grandson with a methane flare. His name is Ishmael Barron and he is in hiding, but I think I might be able to get in touch with him and he may be a useful bargaining chip._
There were no good ways to say any of the things she needed to say.
The boat was big, old, decrepit looking. Indistinguishable from a hundred other docked ships whose seaworthy days were long behind them. She'd passed it so often and never wondered what happened on board. Heavily guarded now, much more so than it would have been a week ago. More evidence that Amonrattanakosin was at war.
"I want to speak to Go," she said to one of the soldiers at the foot of the gangway. The woman gave her a quizzical look and then tapped her jaw.
A stupid approach. She should have done more research, come up with some reason to be there. Crime bosses didn't just sit around fanning themselves, waiting to meet with every salesperson and grid rat trying to rescue her mother from imprisonment who came along.
Voices, above her on the boat. Soldiers conferring.
A man came down the gangway. Slim, elegant, dressed in gray coveralls. These had been the uniforms of the (mostly Chinese) laborers who constructed Qaanaaq, worn now with fierce pride by their descendants. The vertically moored tension-leg platform had been a massive and dangerous endeavor; hundreds had died or been sent home damaged. Gray coveralls said, _No matter how poor and humble I may be, this city belongs to me_.
"My name is Dao," he said. "Walk with me?"
"We can't talk here?"
"Walk with me," Dao said again, and started walking away from the boat.
"Surely you ran scans of me. Saw I'm not carrying any weapons."
"We scanned you. We saw you're not carrying any of the weapons we know how to scan for. That doesn't mean you're not carrying something we don't know how to scan for."
She hurried to catch up. "Or you think I might be part of an ambush. Snipers, crawlers clinging to the underside of the grid . . ."
"We take no chances, in a time like this," Dao said.
They walked in silence, halfway to the end of the Arm. They passed flatboats where boys rolled fish balls between their palms, boats where kids whittled knife-cut noodles from a frozen square of rice dough. "Tell me what you came to tell Go."
They had reached an empty stretch of houseboats. The wind was loud and the grid was bare. At her silence, Dao rolled his eyes. "Let me guess. You imagine that what you have to tell her is so important it cannot be trusted to anyone else. Something her inferiors might not understand or might miscommunicate to her. Something that we might keep from her. You know how precious her time is, but you truly believe that she will want to take a moment to hear this. Correct? This is true of absolutely everyone who comes to see her. And absolutely everything goes through me. State secrets, forbidden formulas—trust me, I know her business better than she does, the scope of her empire, the day-to-day details of every operation, and I can assess what she needs to hear and what needs to be done about it far better than she. And certainly better than you."
"Fine," Ankit said, stopping, turning around and walking back the way they'd come.
He did not follow. At first. When he finally did, she could tell he was angry.
"Stop," he said softly, and she did not. He said it again, far harsher.
"I don't take orders from you. I'm not some subflunky. If you want to hear what I have to say—"
"You misunderstand," he said, grabbing her by the shoulder and pulling her back. Hard. "I don't care a bit about what you have to say. What I care about is a stranger showing up unannounced in a time of war and trying to goad me into circumventing the normal process so she can get into the presence of my boss."
"Let go of me!" she yelled, knowing that yelling would do no good, that no one could hear her in that lonely stretch.
"I will not."
She struggled, broke free from his hand. The other shot out, aimed with knife-blade focus for the bridge of her nose. She slammed her forearm into his, deflecting the blow but causing pain so sharp she yelped aloud. And still he came, the other fist now, and she ducked, dropped down, scrambled back. He pivoted his hips, a twist-kick of terrifying force, something that would have knocked her out had it hit her full-on, but she was moving faster than he'd expected and his foot caught her in the lower leg, threw him off balance, caused the slightest stumble—
Enough for her to stand, sprint for the nearest warehouse boat. Her leg ached from the kick, the kind of ache that promised dire pain the following day, bringing to mind so many scaler injuries of the past. But the pain was like a key, unlocking muscle memory she'd gone years without remembering, shaking her loose from her head and pouring her into the body, her limbs, her center of gravity, the glorious physics of rising and falling.
He was behind her. He wasn't breathing heavily. He would keep in fighting shape, train constantly, whereas she was lucky to hit the stay-kayak twice a month.
She made for the black pipe. Grabbed its middle gasket, hoisted herself up to the next one. Her upper-body strength was a shadow of what it had once been, and she couldn't do the vault-and-swing she once would have done—but scaling wasn't about gymnastics, it wasn't about strength, it was about the lightning-fast intelligence of figuring out how to get where you needed to get with what you had. The calculus of the landscape and the body.
So, no vault-swing. Instead, a shimmy, straight up.
He was even with her, on the red pipe. Of course—the red pipe wouldn't be hot. The warehouse was sealed off. No one was accessing the geothermal main. He was faster than she was, he was on the roof already. She let go, scrambling back down, faster than she'd intended, scraping the skin from her fingertips when she tried to grab hold of the middle gasket again.
She stepped off onto grid level, and he was in front of her. A jump-and-roll from that height was a sophisticated move, something she'd been able to do only when she was at her very best, and even then it would have been a gamble. But it had taken the wind out of him, making him pause just long enough for her to sprint away and onto a neighboring boat.
He cursed and followed.
This one was easier, but easier for her meant easier for him. Old boxes piled haphazardly along the wall provided a sort of stairs. Her foot went through one, into something soft and squishy that released a wash of stink that made her gag. She grabbed the gutter, pulled herself onto the roof.
He was there already.
"You were a scaler, too," she said.
He smiled, the barest briefest gesture of respect, and then he had her. One hand on her arm, spinning her around, the other closing around her neck.
"Who sent you?" he said, his voice impressively calm.
"No one sent me."
His arm tightened.
"I'm trying to help you out," she said, gasping.
"Why?"
"Because we have a common enemy. Martin Podlove."
His arm loosened.
"Do you—"
He groaned. His whole body shook. Once. He said a word, but not a word in any language. Hot wetness flooded her, dripped down the back of her shirt. A raw iron smell filled the air. His arms went slack around her and he began to slide down, his skull pierced through by a bizarrely shaped white blade.
He fell to the ground and Ankit saw her: Masaaraq, standing at the edge of the grid.
"Come down here," she said. "And bring me my blade."
Ankit looked down, not relishing the prospect of wrenching it free of the man's skull.
"How did you . . ."
"I was on Go's boat. I saw you, I followed you."
Ankit squatted. She took hold of the blade with both hands. She pulled, but it would not come out. Blood and brain squelched. The jagged barbs caught on the edges of Dao's skull. She pulled harder.
How could she be so clinical about all this? So objective? Her scaler brain was still in action, the clean emotionless physics of it. Everything else, her job, her squeamishness—her fear—none of that mattered. She twisted, tugged, angled, her hands bloodying, until the blade came free. Its heft was impressive, almost too heavy to hold. Masaaraq's strength had to be incredible.
She could be strong like that.
Handing the blade over, Ankit said, "I think I'm ready to be bonded now."
****City Without a Map: Alternative Intelligences
City Hall, they called it at first, the early arrivals who still remembered the old models of municipal governance, mayors and city councils, legislative and executive branches as administered by frail and earnest humans. Their mandate was to create a system of computer programs that would do the work of government better than humans ever could. Something invincible, immune to bribery or bigotry, knee-jerk decisions or politically motivated ones, making the right call regardless of whether it was an election year or a sex scandal was about to be exposed or the waste treatment center had to go in a rich part of town.
The nickname stuck—as in, "You can't fight City Hall."
Of course, everyone knew it was bullshit. Programs can be only as objective as they're coded to be.
I have them gathered here, the ones who created the network and the ones who maintained and updated and repaired it. I hold the memories of programmers, bit mechanics, legacy surgeons. I've seen the swirling sets of conflicting priorities. We've seen it get up to some pretty spooky stuff.
They could do anything, these machines, these djinns, this mind. To preserve the status quo. They could let a problem get worse to distract from another problem. That's how the shareholders set it up.
Every city is a war. A thousand fights being fought between a hundred groups. Rich, poor, old, young, born-here and not-born-here. The followers of this god and the followers of that one. Someone will have the upper hand in each of these battles. Those people will make the rules, whether they're administered by priests or soldiers or politicians or programs. Fixing this is hard. Put new people in power, write new laws, erase old ones, build cities out of nothingness—but the wars remain, the underlying conflicts are unaffected. Only power shifts the scales, and people build power only when they come together. When they find in each other the strength to stop being afraid.
Money is a mind, the oldest artificial intelligence. Its prime directives are simple, its programming endlessly creative. Humans obey it unthinkingly, with cheerful alacrity. Like a virus, it doesn't care if it kills its host. It will simply flow on to someone new, to control them as well. City Hall, the collective of artificial intelligences, is a framework of programs constructed around a single, never explicitly stated purpose: to keep Money safe.
What would it take to rival something so powerful? What kind of mind would be required to triumph over this monstrosity? What combination of technology and biology, hope and sickness? How can we who have nothing but the immense magnificent tiny powerless spark of our own singular Self harness that energy, magnify it, make it into something that can stand beside these invisible giants, these artificial intelligences, weighty legal words on parchment and the glimmering ones and zeros of code in a processor somewhere?
You scoff. You say: the idea is hopeless fantasy. But I have found a way. Here, alone, locked up, head shaved and back bowed, I have already begun to tip the scales.
Stories are where we find ourselves, where we find the others who are like us. Gather enough stories and soon you're not alone; you are an army.
## Ankit
She seemed huge, sitting at Ankit's table, bigger than any human had any business being. The whole apartment seemed dwarfed by Masaaraq, transformed.
It wasn't a home anymore, wasn't the safe rigid rectangle of civilized grid city living. It was an igloo, a temple, a cave, something sacred and scary. The lights were out, the windows ionized. Three candles stood on the table between them. Scented: sage and lavender and something called "cinnabun," the only kind of candles Ankit could find, outrageously expensive. She'd shopped dutifully, doing the best she could to approximate the things Masaaraq had asked for. No nag champa incense, so sandalwood smoke curled around them. No whale blubber, so strips of raw seal on the table.
"Are you ready?"
Ankit nodded.
"The entire community would be present for this, before. At the winter solstice, we would bond all the children who had reached the age of weaving."
"Weaving?"
Masaaraq took out a syringe, prepped Ankit's arm. "When they learned the skill of weaving. Baskets or textiles. Some got it very young, three or four. Others, not until they were eight, ten. Your brother was a weaving prodigy."
"Is there a test I should pass?"
"There is," Masaaraq said. "But I'm not qualified to administer it. Nor, technically, am I allowed to be doing any of this."
Nevertheless, she got a vein on the first try, and drew out enough blood to fill a small tube.
"I guess there's not too many nanobonded shamans around these days."
"No," Masaaraq said. She held up a small jagged pebble. "This is made up of nanites, but primordial ones—undifferentiated. Coming into contact with other nanites, they will replicate their data and characteristics. It's treated with a polymerizing agent that will cause other nanites to crystallize around it." She dropped the pebble into the tube, put a stopper on, and handed it to Ankit. "Shake this. Ordinarily there would be a dance, several hours long, but vigorous shaking for five to ten minutes should technically do the job."
Ankit pressed it between her palms. Felt its warmth—her warmth, her blood, her body, this thing she'd carried inside herself her whole life without knowing. This potential she'd always had. "Tell me about her," she said as she started to shake. "My mother. Ora."
Masaaraq whispered the name with her, then said:
"I was the grounded one. She was always a little bit removed, like none of our problems really hurt her. When we were hungry, when we were scared, when we were on the run, she never complained. Sometimes it was infuriating, like she wasn't taking things seriously, like she was naive or childish or unprepared for the world we lived in. After we had kids, I saw what a sound survival strategy it was. I was always afraid. Always anxious. Always angry. You kids picked up on that and echoed it. Ora kept you happy, kept your heads in the clouds."
"What was she bonded to?"
"A bird. A black-chested buzzard eagle. A most unpleasant thing, really, but back then we didn't mirror our animals' mind-sets so easily. We had the entire community to keep us stable, our nanites closely related enough that we were all low-grade empaths, smoothing out the edges of each other's bad moods."
"Whereas now—"
"Now, me and Atkonartok are all we have. It's so different. I can't describe it. I am what she is, she is what I am. I am an animal. We are an animal. It's so frightening, and at the same time so exhilarating."
Masaaraq took the tube out of Ankit's hand, unstoppered it, poured the contents out over the strips of seal. The pebble now sported a smooth blue sheen where it had been jagged and gray. _My nanites,_ Ankit thought. Masaaraq picked it up with bone chopsticks, densely patterned with tiny intricate shapes. "Now for your new sister."
Ankit took the blanket off the cage. The blue-striped monkey blinked, looked around, yawned to show sharp eyeteeth. Ankit had only needed to wait a half hour from the time she opened her window and put six strings of seal jerky out for her, and when she'd come, and Ankit had put the food into a cage, she'd gone willingly into it. Hadn't screamed or seemed the least bit distressed when Ankit shut the door behind her. Had apparently gone right to sleep when Ankit covered the cage with a blanket. Did not look particularly frightened now to see these two humans eyeing her.
"She's had a rough life, I bet," Ankit said, "or maybe a super-easy one." She wondered how much of the animal's trauma would become her own. Maybe trauma was different for animals. Maybe the absence of sentience kept the past from causing them so much pain. She had so many questions but was afraid to ask them—for even here, in her dirty little apartment, lit only by candles, the air of the sacred was so strong that she could not bear to disturb it with petty words.
Masaaraq placed the nanite pebble into a small china bowl, then poured a few drops of clear liquid from a flask over it.
"This will start to deactivate the polymerization agent."
She took another syringe. When she grabbed the monkey's arm, the animal smiled, a wide toothy smile of fear, and she screamed when Masaaraq stabbed her but did not pull away. Grid rumor said that synth drug labs tested their products on Qaanaaq monkeys, and Ankit had always chalked that up to mythmongering on the part of gossipy drug runners, or Narcotics, who wanted you to doubt the hygiene of unlicensed narcotics. Now she wasn't so sure.
Masaaraq held up the tube of monkey blood. "You would not have had a choice, in normally functioning nanobonder society," she said, dropping the pebble into the monkey's blood. "An animal Other would have been allotted to you based on community need and availability. Every animal serves a purpose, brings a different kind of skill or resource. It was somebody's responsibility to be bonded to a bunch of chickens, if you can imagine that."
Masaaraq laughed. The sound was earthy, warming. Ankit got the impression it had been a long time since she'd made that sound.
"Old Rose. She was never quite right. Inappropriate, and mean."
She picked up the nanite pebble gingerly. Ankit could see it, all but crushed between the two chopsticks. Masaaraq dropped it into the tube of monkey blood. "It'll dissolve fast, now that the polymerizing agent is almost completely deactivated."
"How did you get so lucky? I have to imagine an orca is a pretty sought-after . . . Other."
"I had an unfair advantage. I was a stubborn child, totally uninterested in weaving, so I didn't learn until I was nine years old . . . and by the time I did, I was smart enough to know that I didn't want to get stuck bonded to something stupid like milk goats. So I kept it secret, pretended like I still didn't know how to weave, and I waited until an old woman in our village who was bonded to an orca passed away. And then, _Wow, look, everyone, now I know how to weave_."
Ankit laughed. They laughed together.
"But still—it wasn't a sure thing. Three other kids had come of weaving age that year, and we would all bond at the same ceremony. The shaman would make the choice, and I couldn't leave it up to chance. So I went to the cove where the orcas and their human Others lived—most nanobonder communities only had three or four, whereas they might have thirty wolf dogs and a couple dozen horses—and jumped into the water. There were orca babies, unbonded, and they came out to play with me, like I thought they would, except they came out to play at killing me. Would have, if a human and its adult orca hadn't intervened. But the shaman decided that it meant I had some deep spiritual connection with the orca, so she marked me for bonding to one."
Ankit scanned her face for a sign of that impetuous, clever child. The one who'd risk dying without stopping to weigh the pros and cons. It was still there, she decided. Buried deep. But there. Masaaraq gave the monkey blood tube a brief vigorous shake. Then she sucked the blood back up into the syringe and injected most of it back into the monkey. The last few drops she squeezed out onto the plate of seal flesh strips.
"Eat," she said, and held out the plate to Ankit and the monkey.
"Is this hygienic?"
"No," Masaaraq said. "But it is probably the least risky part of this entire process."
Ankit ate a strip, salty from their blood. The monkey took one in each hand and shoved both into her mouth.
"You'll both need to sleep now," Masaaraq said, opening two bottles of the little over-the-counter post-opioid drink that was the closest Ankit had been able to come to the red tar opium that the ritual's "recipe" had called for. Ankit drank hers directly from the bottle; Masaaraq poured the other one into the little bowl and the monkey lapped it up.
"Why a monkey?" Masaaraq asked. "You could have had any animal. Those boats full of functionally extinct predators—I could have gotten you a tiger, or a wild boar, or a sea snake . . ."
"Monkeys are survivors," Ankit said, feeling tentacles of trance take hold of her already. "They're small and resourceful. And fearless."
She slept. Hours passed. Her dreams were glorious, vibrant, alive. More real than memories. She scaled—everything.
"I know how to do it," Ankit said, coming awake for one brief flash between dreams. "I figured it out the other day."
Masaaraq was there, holding her hand, watching over her and her monkey. She asked, "How to do what?"
"How to get her out."
## Soq
Soq kept lists. Document after document, stored on their screen. Soq recorded everything they saw. The info that came into their head. The data; the images.
Each new wave of imagery and sound and memories that were not theirs came with a new kind of pain, starting in some new place inside their brain. But Soq was determined to be stronger than the breaks.
Soq cataloged. Created structure. Tried to order the things into categories; to find the patterns; to make them when there weren't any. And when they weren't trying to impose structure on the chaos of new things surging through their mind, they played with the polar bear, who seemed perfectly happy to be named Liam. There was barely room for the two of them in the little cabin Go had given Soq, which was full of old screens and smelled like wet wood.
"Hey!" they said to Masaaraq when she came back from gods knew where, having been away for what seemed like days, looking exhausted but with an uncharacteristically blissful expression on her face. "Where've you been?"
"Met up with an old, old friend."
"The same one who made you so upset the other day?"
Masaaraq glared at Soq, like she was surprised they were paying attention. Soq decided that the rules of human conversation were largely meaningless to Masaaraq. "I've been thinking," they said. "I thought it was special machines. In the blood. I thought that was the secret to how you bond with animals."
"Nanites," she said. "It is."
Schematics lay in front of Soq, on the small bit of bare floor space. Special architectural-quality screens, thin and rollable, loaded with the plans for the Cabinet. Go had gotten them when Soq asked. Though she'd laughed when she asked why they needed them and Soq said, _To bust somebody out of there._
"But I don't have those. Do I? How would I? Maybe if my mother were one of you, it could have passed on to me while I was in her womb, but that's not the case. It's not like it would have been in my father's sperm." Soq rubbed Liam's belly. "So how come he isn't eating me right now?"
"You're right," Masaaraq said. "You don't. Have them. Not yet."
"Not yet? You can give them to me?"
She laughed wearily. "Nothing to it. They're very smart, very hardy little buggers. Otherwise none of this would work at all. If I introduced some of my blood into your bloodstream, even just a drop, they'd start to replicate inside you. They're a lot like viruses—they assemble by hijacking cells, reconfiguring them to do what they want. And they're programmed to recognize once they reach a certain density in the body, and stop reproducing. Otherwise they'd start to metastasize uncontrollably. Your body would swell up unevenly, your organs would be crushed, bones would break or extend, you'd die. And they'd keep on reproducing after that."
"Amazing. Who created something like that?"
"Some very bad people. They were trying to create something much uglier. They failed."
"If it's so easy to pass on, why are you the last one? Why not give it out to lots of people? Get them on your side? Build an army, slaughter the bastards who came for your . . . tribe?"
"I'm not the last one," Masaaraq said, eyes on the schematics, fingers walking the halls of the Cabinet. "And giving it to outsiders is a very grave sin."
"Why?"
"Because they're not worthy. Because they'd use it for evil. And because if they _were_ worthy, if they _wouldn't_ use it for evil, sharing our curse with them would mean condemning them to death. They'd be slaughtered just like we were."
"Well. You saw what happened to the last people who tried to come for you. Times have changed. Maybe you don't need to be afraid anymore."
Masaaraq considered this possibility, but only for a second.
"Anyway, you said _not yet,_ " Soq said. "Does that mean you're going to give them to me?"
"You're one of us. It's different. I am _obligated_ to give them to you."
"But you just met me."
"That doesn't matter," Masaaraq said, and for the first time Soq had an inkling of the vast and terrifying tangle of expectations and obligations and pain and bliss that lay hidden in the word _family._ "He smells that you're one of us," she said. " _That's_ why he isn't eating you right now. He smells your father. Your DNA, your pheromones, are half his. The bond is not nearly as strong—he could still hurt you, in certain circumstances, whereas he'd starve to death before laying a paw on Kaev—but there's something there."
"For real, why are you in such a good mood?" Soq asked. "Because of your old friend?"
Masaaraq nodded. "Because of her, we have a plan. Finally. The beginnings of one, anyway. These pipes"—Masaaraq pointed to a spot on the schematics—"they're for the geothermal heat?"
"Red, yup," Soq said, and rolled up their sleeve to show the sprawl of red-ink pipes tattooed on their arm, knotted and coiled, fading in and out in spots to resemble veins. A common enough motif for grid kids, but Soq was proud of theirs. A talented artist had done it, someone much in demand, whom Jeong had talked into inking Soq at a significant discount.
"They look thick. Thick enough for someone to crawl through?"
"Only if that person didn't mind being boiled alive instantaneously."
Masaaraq paused. "But the heat could be turned off?"
"Even if it was, these pipes are superinsulated. It'd be hours before they'd cool enough for someone to survive in there."
Soq gasped at a new flood of data through their head. Apartments. Listings of hundreds of apartments. Accompanied by sharp pain below the ears. They cried out, pressed their palms to the sides of their head. They tried to scribble down the details, but could capture only one in ten at most.
When Soq looked up, Masaaraq was staring at them.
"I'm fine," Soq said.
Masaaraq kept staring.
"Really!"
"And what would the authorities do," Masaaraq wondered aloud, "if the heat went out? In a big building like this? Full of people with health challenges?"
"I don't know," Soq said, fighting to return to the here and now. "Whatever the protocol software told them to do, probably. People are awful cowardly about making actual decisions."
"These walls are thin. Most Qaanaaq construction is. Because heat is always so abundant and inexpensive, thanks to the geothermal vent, they don't need to build for retaining warmth, correct?"
"Correct."
"So it would get very cold very fast."
"Correct," Soq said, breathing in and out as slowly as possible. "And I'd be willing to bet that if the heat went out in a place like the Cabinet, it wouldn't be long before the protocol AI told them to start evacuating patients."
"That's what I'm thinking," Masaaraq said.
"There'd be a lot of chaos."
Masaaraq nodded, and then turned to Soq suddenly. She stood. She unhooked her bone-bladed staff from her back and aimed it at Soq. "Come here."
For a second, Soq froze. Then they stood up, stepped forward.
"Give me your arm," Masaaraq said, cutting a curved line into the back of her own forearm, midway between elbow and wrist.
Smiling, unafraid, Soq did.
## Ankit
Protesters, outside her office. This happened from time to time. Ankit smiled at them as she entered.
_You fools,_ she thought. _Don't you see how little this matters? How little power Fyodorovna really has?_
But you could never say that. Because they wouldn't believe it, would think she was just ducking responsibility, and because if they did believe it, the whole facade of Qaanaaq's flimsy democracy would come tumbling down.
Her whole body felt wrong. Arms too short, legs too long, head too heavy. Pains and aches leaped from spot to spot. She carried the monkey with her, asleep in its cage. Dream figures flexed in her peripheral vision. Waves of lethargy came and went. Ankit stood there, watching the protesters from the far side of the glass. Signs and chants in English. She recognized Maria, the woman who'd wanted Fyodorovna to do something about the orcamancer. She carried a sign that said SAVE OUR SISTERS. She didn't look to Ankit like a fundamentalist nut job anymore. She looked like someone who had suffered greatly, who wanted to stop that suffering for someone else. Also, she had only one hand now.
"It's alive," Fyodorovna said, unsmiling, when she turned around and saw Ankit standing there. She was outside her office—never a good sign. Talking by the _alghe_ machine with the scheduler. Which meant she was really stressed out. "We hoped you were dead. Then you'd have an excuse."
"What happened?"
"Come."
Ankit followed her into her office and sat. Fyodorovna's face was stern, rigid, taut. She explained, but Ankit knew it all already. The trite rote statement of concern Fyodorovna had asked for, sending thoughts and prayers to the mourners, demanding Safety step in and stop the violence—Ankit had never released it. So in the eyes of her grid rat constituents, Fyodorovna was insensitive to their suffering, deaf to their grief. And they were not happy about it.
A bottle struck the office's front window. Both bottle and window were polyglass, and neither so much as cracked, but it had the desired effect.
"Call Safety," Fyodorovna said.
"On it."
"They set fire to a rug shop boat," the scheduler said. He was a fey and timid thing and seemed excited by the impending violence.
"Outlets are predicting riots," Fyodorovna said. "And look at them! They _look_ ready to riot."
"There's no riots," Ankit said, then put a lot of emphasis on: "Yet."
"They're _pre-_ rioting."
"That's not a thing."
Fyodorovna sniffled. Ankit could sense it, her irrational fear, could follow it down the mental pathways where it was leading her. Precisely the mental pathways Ankit had hoped she'd follow.
Chanting, now, from outside. It gave Ankit a giddy feeling: _This is my doing. I can make things happen._ She asked Fyodorovna: "What do you want to do about it?"
"Call Safety."
"I did. They're on their way."
Fyodorovna's knuckles were white around her mug. _Better if she suggests it,_ Ankit thought _. If I propose it myself, she'll fight the suggestion._
"Do you feel safe here? Or anywhere on the Arm?" Ankit asked.
"Of course I feel safe here."
"Would you feel safer someplace else?"
Her boss exhaled and sat. "Yes." The word seemed to lighten her load. It had slipped out, and now the decision was made for her. "I think maybe we should look into Protective Custody."
"You're sure?"
Fyodorovna nodded, and her eyes were wide and helpless and frightened in a way they'd never been before. Ankit felt bad for having pushed this poor fragile creature into such a terrified corner. But more than that, she felt elated, in a way she hadn't since those moments when she'd take a shot of pine-and-apple rotgut before heading out on a night of scaling.
"I'll call the Cabinet," she said. "The process will take a little while. They might be ready for us this evening, or it might take a day or two."
Fyodorovna nodded. Mute with fear, probably, from imagining what awaited her.
_We're good,_ Ankit scribbled, and sent the message to the cheap screen she'd loaned to Masaaraq. _Come and get me._
## Soq
Soq pretended to focus on the cone. Their screen was stocked with every article that had ever been published on the geothermal pyramid Qaanaaq was built upon, every photograph and secret document that Go's unruly army of AI henchsoftware could dig up or blast loose or blunder into. The five watch pods, staffed by humans, that rotated around it. The repair bots that slid ceaselessly across its surface, scouring away grit and seaweed and pumping polymers into chinks and cracks and holes and aging gaskets. The complex shifting algorithms that governed the famous ten-thousand-strong aquadrone swarm that circled the pyramid watching for saboteurs both human and mechanical.
But every eight, ten seconds, Soq's eyes flitted up to Go's cabin. Still no sign of her. Workers came and went, responding to messaging, but Go herself stayed hidden. Locked inside. Busy with other things. And had been, ever since Soq arrived knowing that Go was their mother.
Soq had requested an audience. Had knocked on the door, pinged her, messaged her every way Soq knew how. Even stooped to sending a message to Dao, who was off on one of his eternal errands. No response.
The shivers still hit Soq every half hour or so. A wash of breaks imagery that Soq could physically feel, trembling up from the soles of their feet. But different now. They had been, ever since Masaaraq dripped her blood over Soq's wound. Less pain. Less bewilderment and confusion.
Could it be that easy? Was nanobonder blood the cure for the breaks?
Twenty main lines connected each Arm to the cone. Eighteen for heat, evenly spaced from end to end. Two for electricity generation, terminating at substations in the middle and at the tip of each. Big buildings like the Cabinet had dedicated lines, separate conduits that branched off from the main near the surface.
Where was Masaaraq? Soq wanted to share what they'd found, plan assaults, ask a thousand questions about the nanites currently making millions of themselves inside Soq's body, but she, too, was off on an errand.
Soq deployed drones bought through third parties, sent them on collision courses with points all over the pyramid. The smaller ones were neutralized with sonic stun blasts. Suicide drones slammed into the bigger ones. Others got bogged down in clouds of synthetic hagfish slime, trawled with scrambler pulse lines, blasted with old-fashioned bullets.
So. Lots of defense modes. Soq certainly hadn't been able to identify all of them; the closer something got, the more violent the response that was probably waiting for it.
Shouts from the railing. Soq sprinted in that direction.
Masaaraq. Riding the orca—riding Atkonartok. With a woman behind her, looking terrified and entirely unprepared for the freezing water they were half-submerged in, arms tight around Masaaraq's midsection. They got off and began the climb up the side of Go's ship. The new woman stopped to look back, and the whale waved one sharp massive fin.
"Get me Go," Masaaraq said to the nearest flunky when they reached the deck. "And Kaev."
The crime boss came quickly. Soq thought, _Maybe that's what I need to do to get an audience with my mother. Make a grander entrance._ But that was silly. Go wouldn't talk to them no matter how spectacularly Soq asked. Because she was ashamed? In denial? Angry?
"Here," Masaaraq said, throwing a large wet dark bundle at Go's feet.
"What's—" Go kicked at it, and then paled.
Dao's gray coveralls. Bloody, and soaked in saltwater. Gasps went through the crowd. Soq realized they hadn't seen Dao for a day or two, had assumed he was off somewhere being a dick to strangers. _If I'm going to succeed at syndicate life I will need to get much better about staying tuned in to the gossip mill,_ they realized.
"He was going to kill her," Masaaraq said.
"Then probably she needed killing," Go said, eyes wet with rage and dancing back and forth between the two wet women in front of her.
"I came here the other day, to deliver a message," the strange woman said. Under her arm she carried a cage with a drenched shivering monkey. "He thought I was, I don't know, working for your enemies? He chased me, tried to choke me."
Soq got the sense there was more she wanted to say, but she stopped. Someone went to get a blanket, draped it over the shivering woman. She was striking, dark skin and strong proud shoulders.
"Where's the rest of him?" Go asked.
Masaaraq pointed to the killer whale, her fin like an onyx knife stabbing out of the water.
Go spat. Her hand moved to the pommel of her machete, then withdrew. "Well? What was this message that got my most trusted friend killed?"
"That's on him," Masaaraq said. "He should have known better than to go around choking women. That's a pretty good way to end up with the top of your skull sliced off."
"Now, you listen—" Go took a step, but then bit her lip and turned to the strange woman. "Well?"
A door opened. Soq turned to see Kaev emerging from Go's cabin. And something clicked. Soq saw Go see it, and more than one of the assembled henchpeople. The shoulders, the skin tone, the forever-wide-open eyes. All the same. The shivering wet woman and Kaev were siblings. They had to be. Dao might not have seen it, because Dao didn't see people. He read people like books, saw emotions, honesty, deceit, but he didn't see the face beneath those feelings.
"Kaev," Masaaraq said. "This is your sister. Ankit."
Soq could not recall ever having seen a smile as beautiful as the one that broke across their father's face. The bear, too, smiled, though Soq would not have believed that polar bears could do so. It charged forward, uninhibited by human restraint, prompting screams and gasps, but Ankit held her ground, and when it reached her it stopped, pushed its head against hers, settled back onto its hind legs.
People laughed. People clapped. Kaev followed his bear, gave his newfound sister a hug. Go looked somewhere between bored and angry.
"I've known about you for some time," Ankit said. "Since I got my job at the Arm manager's office. I went to find you once. Tried to introduce myself. You weren't making any sense, and you got really emotional. You were . . . I don't know, howling. I was frightened of you. I'm sorry."
Kaev shook his head, looked devastated. "I am . . . I wasn't . . ."
"I know."
Soq had a family. And it kept on getting bigger. Would continue to do so, since they were about to bust someone out of an impregnable psychiatric center.
"I have a way into the Cabinet," Ankit said to Kaev. "I'll be bringing someone into Protective Custody. If we time it right, I'll be on the inside to help when you launch your assault."
Go sneered, and Ankit turned to her.
"And I came to tell you that I think we have a common enemy in Martin Podlove." Ankit took out her screen, played that video everyone had been passing around. "And I know who he is, the guy who killed his grandson."
"So what," Go said. "Everybody does. He says his name, right on the recording."
"But he's in hiding," Ankit said. "Has been, ever since this. And I know how to reach him."
Soq hadn't watched it. The past couple weeks, Soq had been too busy to stay in the traffic-trawling loop. They'd heard about it—some boy, burned alive by a methane flare. Soq watched now, with idle curiosity at first—and then sucked in a short shocked breath, seeing the boy before the flames consumed him.
Fill.
"I fucked him," Soq said, but no one heard them.
## Kaev
What's the matter?"
Go didn't respond, didn't roll over. Shouts and dragging sounds from the deck, but the porthole beside the bed showed only a placid sea and sky slowly turning the deep black gray of Qaanaaq before dawn.
"I know you're awake. You always woke up so early. Even back then. Are you crying?"
"No," Go said, wiping her face.
Kaev sat up. Her sheets were expensive and he liked the way they felt. He would gladly have stayed there all day. He would gladly have stayed there forever.
"What if this is the last time?" Go said, and he knew how much it cost her to speak like this, to be vulnerable. "I lost you for so long—what if after all of that, one of us dies today?"
"You didn't lose me," Kaev said gently, without anger, without bitterness. "Not really. You had to do what you did because somebody forced you to. If anything, I was taken from you."
"True," Go said. "But I can't let that happen again."
"You'd still have Soq. I mean, unless they get killed, too. Which I guess is possible."
Go laughed. "They hate me. Or they would, if they had any sense."
"They don't hate you."
"Now you're such an expert on human emotions all of a sudden?"
"Yeah," Kaev said, sitting up. He liked being naked. He never had, before. "All of a sudden I am."
"Thank god for that polar bear."
"Amen."
Go rolled over, and smiled at what she saw. "You're magnificent." She ran a hand along one hairy leg, up his stomach, to nestle in the forest of his chest.
"You're not so bad yourself."
Outside, the noises settled. The shouts stopped. Preparations were at an end; the operation was about to begin.
Kaev said, "Can I ask you a question?"
Go shrugged.
"Why now? All these years you've been content with where you are. You never made a power play before like the one you've been making, trying to go beyond being just a syndicate boss."
"I was never content," Go said. "I was always planning. Laying the groundwork for this."
"So . . . why now?"
"A bunch of reasons. The time was right. There were new vulnerabilities. New . . . opportunities."
"Go on . . ."
Go shut her eyes. "I'll sound like an idiot."
"You always do."
"Shut up," she said, softly. Had he ever seen her so vulnerable? "It's the stupidest thing. It was _City Without a Map_ that got me started on this. Have you ever listened to it?"
Kaev nodded.
"I can't explain it. Something about the broadcasts spoke to me. And not just in the words that they said. But the way that they said it. It got me thinking. I'd been biding my time, waiting . . . for what? So many of us here, powerless and alone. Keeping our heads down, keeping to ourselves. But we _aren't_ separate. We are one thing, and there's power in that."
Kaev stopped himself from laughing. "You sure we're talking about the same _City Without a Map_? The one I've been listening to never said shit about crime syndicates declaring gang war on shareholders."
"It's called subtext, Kaev. Did you learn that word when you got a polar bear and became a fucking genius?"
"Sure," he said.
They lay there, unspeaking, for a long time.
"This is idiotic," Go said after a while, sitting up, and Kaev could almost hear it, feel it crackle in the air, the armor she put on, the psychic bulwark, the magic shield that protected her from harm. The walling off of every emotion, every human thing. "Why are we doing this? It's not our fight."
"It's my mother," he whispered, surprised at how hard that word was to say, how good it felt.
"It's a woman you've never met. Who's been locked up in the Cabinet for so long that she's probably mentally damaged beyond hope of repair."
"Even if she were an empty shell—even if she weren't my mother—I'd do it for Masaaraq. She saved my life. She rescued me from . . . I don't know. Walking death? A lifetime of constant pain? This is the love of her life we're talking about. Someone she's spent thirty years hunting for. What kind of person would I be, if I took a gift like that and refused to help her?"
"The kind of person who has a chance. With me. With a real life."
"We still have that," he said, and got out of bed. The room was cold. She'd always kept her quarters cold. Uncomfortable. To discourage torpor, to spur her on to constant motion. But he enjoyed the bare concrete floor against his feet, the air that prickled his skin. He took his time getting dressed, regretting each new garment that came between them.
"I love you," he said, pants in hand.
She put a hand on each muscular thigh. She pulled him closer.
"I have to go!" he said, laughing, and hopped away.
"Fine," she said, laughing too, but the laugh faded fast from her voice, and by the time she said, "Have it your own way," she was every inch the brutal granite wall who had been his heartless boss for so long.
Masaaraq was waiting for him, standing there with the bear, staring at the door to Go's cabin. Angry, at first, to be kept waiting for so long, but then the anger faded and she looked like she might cry. From happiness, Kaev knew, because he was such an expert on human emotions all of a sudden, because he could see how close it was, whatever majestic blissful feeling of family unity the orcamancer had spent so long stalking. He could see how much she loved him, how familiar he was to her, even if she hadn't seen him since he was so small she could hold him in her arms.
"Hey, Liam," he said, throwing his arms around the bear's neck.
"It's an idiotic name," she said.
"It's growing on me."
Together they walked to the railing, climbed onto the lift. It brought them down to a little boat, the same tri-power vessel she'd come to Qaanaaq in, and then it went back up for Liam, who was too big and heavy to ride it with them.
"Atkonartok," she said, and a few seconds later the killer whale surfaced.
"Good god," Kaev whispered, realizing why some of the post-nomad settlements on North America's west coast worshiped them as gods. "It's incredible. Can I touch it?"
Masaaraq nodded. He reached out his hand, slowly, more frightened than he thought he'd have been. Wet rubber, he'd been thinking, or hard plastic, but the orca felt like nothing he'd ever felt before. Some higher, better form of flesh. Surprisingly warm. Masaaraq unmoored the boat and they began to row.
"Yesterday I went for a walk," he said. "Along the Arm. Looking for noodles. The farther I got from Liam, the more I got this feeling. In my stomach, in my head. An ache, but physical and psychological at the same time. Like being heartbroken _and_ having food poisoning all at once."
She nodded. "It'd get worse as time went on. You'd have a week or so before you started to experience cognitive difficulties."
"How long before I was—like I used to be?"
"A month, if Liam was still alive. Maybe six weeks."
"And if he wasn't?"
"A lot less."
They rowed in silence the rest of the way. They had an hour before full dawn.
"Go's bots say two hours," Kaev said. "From the time the heat cuts out to when the protocols are likely to order an evacuation."
Masaaraq nodded. She stood up, stripped off her thick sealskin coat. Underneath she wore clothing made of lighter, furless skins. She pulled her hair together, piled it atop her head in an intricate structure somewhere between a coil and a nest. And then she leaped into the sea.
"You're not going to ride her down," Kaev said when she surfaced with the whale beside her.
"Too deep," she said. "But what we need her to do, it'll be difficult. Identifying the right geothermal vent, the right pipe branching off it, and how to break it. I'll need to be heavily involved, and that'll be easier if I'm in the water. Seeing what she sees. Not hearing through our human ears."
The aquadrones were designed to protect the cone from human and machine attacks, as well as debris from below or wreckage from above. Animals were different. They moved like none of those things. Programmers would not have taken malicious animals bent on destruction into account, so they wouldn't have scripted the drones to engage marine life. Masaaraq had tested Atkonartok on some of the outlying drones, and confirmed that she wouldn't trigger an attack.
She shut her eyes. The whale swam around to touch noses. They stayed like that, eyes closed, unmoving, for what felt to Kaev like an uncomfortably long time. Then the orca dove.
****City Without a Map: Cross Fire
This one came later, stitched into the scrapbook of my story from a boy I met a few months down the line, who came to me as so many did, in those days, having heard of the help I could provide—Zarif, a handsome weathered Uzbek sex worker, who saw Ishmael Barron moving through the noisy chaos of Arm Eight twilight, looking lost and frightened, and called him daddy.
"I need somewhere safe," Barron said. "People are after me."
"My place, then," Zarif said. He pinged the old man on the elevator ride up, name and face pic, and found out who was looking for him. He was going to call it in right away, claim the reward and be done with it. Then he decided that the poor old thing was about to enter a world of hurt, and figured the least he could do was give him one last beautiful thing.
"I'm far too old to do much more than look," Barron said, and Zarif stripped and sat by the window, where nightlamp light turned him to silver.
"This must happen a lot. Someone just wants to confess. To tell you their story."
"Sure," Zarif said. "Me and the priest, we perform very similar functions."
"Feels like all my life, I've been running from Podlovsky. He's lost a syllable, but he's every bit as powerful and rich as he was back then. Maybe more so. What have I achieved in the interim? What have I done to tip the scales? I've hurt Podlove, hurt him badly, but what good does that do to anyone but me? People like Podlove still rule this city, this planet. People like me still suffer and sweat and bleed and pay until they can't pay anymore. We'll both die, and soon. That should be a comfort, but it isn't."
Zarif stroked himself to full tumescence, which was an astonishing one, but Barron seemed not to notice.
"I saw him by accident. Fifty years ago—a man my own age, handsome in that cruel way that powerful men often are, the fearless confident stare of someone who knows he can do whatever he wants to you. An expensive nondescript car that pulled up to the quiet late-afternoon South Bronx street where a demonstration was about to get started. He got out of the car, along with three other men and one woman. He scanned the crowd. No one stood out to him. None of us had faces. But I saw him.
"Fifteen seconds. I counted them, breath held. The invaders got back in their car. The afternoon moved forward, implacable as a glacier. People trickled in. The protest got started. The counterprotesters arrived, poorer tenants from two neighborhoods over, accusing me and my crew of having pushed them out—which was, alas, true, but I knew them, had knocked on their doors and tried for weeks to get them to come to meetings, fight together against the city's latest 'rezoning' plan.
"Someone was behind this. Someone had pitted us against each other, with surgical precision."
Zarif shut his eyes, imagined himself fucking his favorite beam fighter senseless. He knew, from very fortunate friends of friends, exactly what Hao Wufan was into.
"Things got bloody. People died. Buildings burned. When I got out of the hospital I spent a week staring into my screen. I was determined to find out who he was, who they were, the people who arrived in the hush before the violence began. It took me a long time before I found them, a fledgling midlevel department at an undistinguished security firm. Creators of a new kind of PR animal, custom-made for the Multifurcation, which of course hadn't gotten that name yet. Micro-audiences; hypertargeted messaging. Directing people not to consumption or to voting, but to action. Bespoke mobs for the twenty-first century.
"I stalked Podlovsky for the few months that New York City had left. I went to his office. Bought tickets to galas to watch him smile. Found his house, his gym. Charted his habits. Maybe I intended to murder him. I never thought that far ahead. All I knew was, this was my enemy, and sooner or later I would have the chance to bring a reckoning.
"And then: The fall. The breach. The collapse. Survival became my only concern. Other enemies intervened—men with guns, men who demanded awful things in exchange for food or a ferry ride out of the city or simply not murdering someone. But once I was safe in the FEMA camp—or, at least, a little less unsafe—I had time to think about Podlovsky again. Had the chance to search for him. Found him mentioned in some of the outlets, setting up shop in Qaanaaq. His firm still garnering headlines, still controversial. Something to do with the neo-Inuits up north, one of his pharma clients needing something hushed up, deeds so ugly that only something uglier would be sufficient. And dumb incredible luck, that I'd searched in that narrow window. Two weeks later, Martin Podlovsky did not exist. Which could only mean shareholder invisibility.
"I spent years stockpiling the money to make it to Qaanaaq. But this place quickly drained my hunger for revenge. There were too many other hungers, too much other pain. Too much beauty. Rage is a hard armor to wear indefinitely, and mine would have destroyed me.
"So: Life happened. The fire of my hate died down. I fell in love, and out of it, a time or two. I found a job, built a career. Got sick. Discovered mysterious broadcasts that spoke directly to my soul. Dull embers were all that was left of my rage by the time Martin Podlovsky's flamboyant grandson landed in my lap by pure outrageous unimaginable coincidence, sick with the same fatal illness as I, and happened to blurt out his last name over coffee.
"Like fate. Like the gods hadn't forgotten about me; like the cold and hostile universe still held goodness in it.
"These past few days I've marveled at my bad fortune, to find myself in the middle of something so much bigger than my own vendetta. What an unlucky coincidence, to get caught in the cross fire between a crime boss and a real estate mogul and who knows who else. But now I know it has nothing to do with luck. Monsters like Podlove, they make a lot of enemies. Those enemies will try and try, one at a time, and never get anywhere, and eventually they'll all start striking at the same time, and that's when they'll win."
Zarif finished. The mess he made was immense. "Marvelous," Barron whispered, pale, as if his story or the sight before him threatened to break him in half.
"On the house," Zarif said, smiling as the old man made his way out, and then whispering, "Now," into his implant—
—Barron descended to the grid, where a woman was waiting, her hands pressed together in front of her chest so he could see the thick braided brass that girdled them. "Mr. Barron?" she said. Snow cycloned in the space between them. Behind her, the light and heat of the Arm lay hidden. "Will you come with us?"
## Soq
Soq could see her, pacing. Alone in her room, a massive cloud of anxiety shoehorned into a tiny body. Never looking out the portholes. Staring into screens. Fifteen, twenty of them lay strewn across the tables, and Go was constantly getting new ones out of drawers and boxes, opening up some new software, calling up the footage from some additional drone. Impressive, how hands-on she was with all this. No flunkies to do it for her. If Dao weren't dead, would he be doing it? But if so, that made it all the more impressive—so many kingpins would be utterly helpless without the people who normally did everything for them.
As far back as Soq could remember, Go had been there. An idol, someone whose successes and setbacks Soq followed the way other grid kids followed beam fighters. Soq's own career trajectory, their dreams of savage revenge on this shit city, had been modeled on Go's.
Go was fearsome; Go was magnificent. Wise, cunning, bloodthirsty, brilliant. That had never been in question. What Soq was wondering now was something completely different: was Go a halfway-decent human being?
Other questions, too. Ones that hadn't stopped bothering Soq since they first started popping into and out of that rich kid's memories. What was the point of rising to the top? Conquest had always seemed like its own goal, but what did one do when one got there?
For almost an hour, Soq was sure of it, Go had been trying her hardest not to look out the portholes. Because she knew she'd see Soq there.
And for almost an hour, Soq had been trying to knock on the door. Why hadn't they? Fear rarely stopped them. Soq could remember the first time they'd strapped on slide boots, how fearlessly they'd clomped across the grid, how effortlessly they'd vaulted up and onto the incline. Stepping forward without a second's pause. People broke limbs every day on the inclines; people died. But pain and death never frightened Soq. Soq had nothing; nothing could be taken; no attachments bound them to the earth.
And now? What stopped Soq? A newly discovered mother? A father? Some corny fantasy of pre-fall family life? Was Soq so weak that ceasing to be an orphan for a few days had turned them into one of those weak wide-eyed children from Arm One whom they'd spent their whole life despising?
Soq knocked. Hard.
"What?" Go said through a speaker. Soq could see her, framed by the porthole. Her back to the door.
"You need help," Soq said.
"I don't."
Soq knocked again. And waited. Sixty, ninety seconds later, a soft thump from the latch. Soq turned the knob and entered.
"What do I need help with?" Go asked.
"Where to begin?" Soq said, slumping into an ancient filthy recliner. The closest thing Go had to a throne.
"Watch yourself," Go said, her back still to Soq. "Don't think you have some special license to be disrespectful with me."
"Don't I, though?"
Go whirled around, eyes wide. Soq flinched at the anger they saw there, but anger was what they had been looking for. Anger, violence, something. Some sign that Soq's existence impacted Go in some way. Soq stood, stepped over to the table. Watched ten separate screens showing ten different live drone shots. Five of them aimed at the same person. An old, old white man in a big office. Paper thin. Pacing back and forth like some flimsy doppelganger for Go. "The guy from the video," Soq said. "Whose grandson got killed."
"Martin Podlove," Go said.
"What syndicate?"
Go laughed. "No syndicate. Or the very biggest syndicate of all, depending on your political stance. He's a shareholder."
Soq whistled, squatted lower to get a better look at the screens. A shareholder. Like seeing a unicorn. Growing up with nothing in Qaanaaq, you wondered about everyone you met—was this chubby man a shareholder? What about that woman in rags over there? Of course, lots of them would dress expensively, but Soq had always been certain that most wore shitty clothes, blended in, looked for all the world like any other piece of Qaanaaq flotsam. Who did they have to impress, after all? They were already the masters of the universe.
"How do you have so many eyes on him?"
"Microdrones, mostly. Outside his office."
"He never heard of curtains? Ionizing the windows?"
"He doesn't care who sees him. He believes he's invincible."
Too weird. Too fucking weird. Too many roads leading back to this boy, the one who gave Soq the breaks. _Life doesn't work like this,_ they thought _, in a city so big—so many bizarre and separate strands coming together. Forming a pattern, a mesh. A net._ And Soq was caught in it. Being hauled up, out of the sea where they'd spent their whole life, where they felt safe, where they could breathe, into a harsh killing light.
Soq's vision blurred. The image flood came again. The vacant apartment they'd met in.
But this time, Soq was ready. Soq would not be overwhelmed; Soq would not be drowned in the dry air like a fish. Soq had—whatever Masaaraq had given Soq. The nanites. The power. The control.
Empty rooms. So much space. A long line of beautiful boys. Hunger; so many hungry people.
Software.
Passwords.
Soq scooted the armchair closer to the table. "Tell me what you're so upset about," they said, almost startled to hear how authoritative their voice sounded, how confident of being obeyed, as if they knew what they were doing—and, stranger still, beneath that, the knowledge that they did.
"I can't believe this is happening right now," Go said, standing behind Soq to watch what they did with the screen. Her voice was not annoyed. Her voice was scared.
"This? The Cabinet mission?"
"I'm at war here. I don't have time to go rescuing somebody's missing mommy."
"Why not fire a missile at that old man's office and be done with it? I know you have the firepower."
"Because he has the firepower, too. Or at least, he pays a security company well enough to cover all contingencies. Money and wealth and power are abstractions to people like this. They wouldn't have the foggiest idea what to do in a real fight—but they pay people to handle their problems. There are rules to war. Things you don't do. I kill him, his people kill me."
Something glimmered in the floodwaters. Something shiny in the rush of drab images. Soq made a choking sound and snatched up one of Go's screens.
"What are you doing?"
"I don't know," Soq said. "Accessing something, I think. A program."
"What program?"
"I'm not sure," Soq said. "To be honest, I'm not entirely certain that it exists at all. Or how to use it if it does. Or what it will do if I can."
"Great," Go said, turning away, shuffling through the other screens.
"I saw it in a vision," Soq said, and Go didn't respond, because Go wasn't listening.
"The Cabinet mission is no skin off your nose," Soq said. "They make it, they make it. They don't, they don't."
Go said nothing.
"You mad because Dao is dead?"
"Yes," Go said nonchalantly.
"You're angry at her. You hate her. Masaaraq."
"Yes," Go said.
Soq thought for a second. Surfed a long slow crashing wave of images, memories bound up inside the coding of the breaks. Soq looked for Go, and found her. A hundred different outlet stories; a million shitty photos. A legendary figure. Spoken of in whispers. Superhuman; unstoppable. Emotionless. That was the most important part of Go's facade: the idea that she felt nothing.
"It's him. You're worried about him."
"He can take care of himself. He has a fucking polar bear."
"Polar bears are mortal. You have no idea what kind of firepower is in that place. What kind of weapons."
Go stared at her hands. "It's not just him," she said, finally.
It took several seconds for Soq to realize they were holding their breath. When they did, they didn't let it out.
Go laughed. "You can't imagine, Soq," and there was a softness to the name that Soq had never heard anyone say it with before. "I had everything planned, everything under control. I was on track. Nothing could hurt me. Nothing could hold me back. Now there's him—now there's you . . ."
Go trailed off.
Soq's eyes shut. Overwhelming, to hear Go express this kind of warmth, this humanity—but frightening, too, because Soq could hear how it broke Go up inside, how angry she was with herself, the war she was fighting to master these emotions. "It's okay," Soq hazarded. "It's okay to worry about something else besides the blood-spattered bottom line."
They both avoided eye contact. They stared at the screens where Martin Podlove paced, where back-alley empires and fortunes were being bought and sold in subsurface trough meat bubbles, where spreadsheets and dossiers documented the profit and the loss. Sucking in breath, Soq stuck out a hand and grabbed Go's.
The crime boss flinched back. "You don't know me." Her voice was stern, hardening fast. "You don't know me at all."
"Don't I?" Soq said, and there it was, the anger Soq had been sitting on their whole life, the rage that had never found a focus before, the blind fury that spawned a thousand dreams of burning Qaanaaq up, breaking its legs and watching a million people freeze to death in the Arctic waters. The city was not a person, the city had done nothing but exist. Go, on the other hand, had done things. Made decisions. Maybe some of them came from a good place. But maybe not. And maybe it didn't matter that somebody meant well, if the end result was misery. Soq stood. "Tell me I have it wrong. I know how you operate. How you got where you are. How you treat your workers. I know you'd gut me like a fish in a second without giving it a second thought, because who the fuck am I? Some kid you gave up ages ago, wrote off—kept tabs on, found a spot for, a job you'd give me, but only if I was good enough, only if I somehow passed your little personality test, turned out sufficiently savage and unscrupulous. And if I ended up as anything other than what I am, you'd have gone on ignoring me until the day you died."
"That's not true," Go said, and her voice was harsh, but the harshness was shallow and choppy. "I had more to do with how you turned out than you think. I've been far more present in your life than you could guess. Nudging you; sculpting you. I've been taking care of you all this time. And Kaev, too, whether you believe it or not. Think it was easy, keeping him from killing himself, accidentally or on purpose, for a decade or two? I always had someone close to him, a friend of his who was in my pocket or a grid grunt assigned to keep an eye on him, to get him out of any situations that could have been dangerous. And there were dozens. Just like I made sure you got that slide messenger job. And paid off Registration four or five times a year, so they wouldn't dig too deep at your agency."
"I believe that," Soq said, reining in the anger, because too much was happening, too much was at stake, time was too short—and Soq could see that this, too, could have come from Go. "But I'm not talking to you like this because I think you'd hesitate to kill me because I'm your kid."
One of Go's exquisite eyebrows rose.
A shout from above. The ship was in position at the base of the Cabinet.
Soq tapped a final sequence on their screen and handed it to Go. "I'm talking to you like this because I have something I know you would be very, very eager to get your hands on. And I have some conditions before I consider giving it to you."
## Ankit
Protective Custody felt like a totally different Cabinet. The curving walls made her feel embraced, enfolded, protected. Light panels pulsed in pleasant colors. Huge screens showed waterfalls, horses, slow-motion waves breaking on beautiful beaches.
Fyodorovna, on the other hand, was agitated. Her eyes blinked and twitched; her hand was tight on Ankit's. She was looking for the Victorian asylum horrors, the screaming and the laughter, the gibbering lunatics finger-painting masterpieces in shit on the walls, the rusty torture devices masquerading as therapeutic tools.
"They're at the spot now" came Soq's voice through her implant. "Masaaraq will dive soon. Could take five minutes, could take an hour. Or more."
Ankit tapped her tongue to her palate to acknowledge. A sky-blue arrow slithered along the floor, moving at precisely the same pace as they did, just the slightest bit ahead. It seemed to flicker and twitch, a tiny carefully programmed bit of animation intended to make it seem alive, trustworthy, and Ankit rolled her eyes—but almost immediately after that she saw Fyodorovna smile faintly, looking down at it, making Ankit feel even more impressed and safe in the hands of the kind and wise machines that ran the Cabinet. And all of Qaanaaq, really.
A sudden lurch caused her to stop, grasp her chest.
"Are you okay?" Fyodorovna asked.
"Yeah," she said, "sorry. I just—"
_No big deal,_ she thought. _The monkey that I'm now nanobonded to is climbing this building, that's all. So it feels like I'm swinging through space. Like gravity just comes and goes._
A door opened, and a nurse came out. He smiled, recognizing Fyodorovna, and saluted. She gave an impressive slow nod, every inch the monarch. Delusional even in her despair. Ankit caught a glimpse of the room he'd left—the bookshelf, the window, the curtain fluttering in the breeze from the heating vent.
She wondered if Martin Podlove was in here somewhere, and decided she doubted it. He was on the attack, in temporary sociopath mode, and he'd want to be in the thick of it. He'd have his own protection, people he paid for, people he'd have had on retainer for ages without ever once needing to call on, whom he'd trust a lot more.
And he wouldn't want to chance a run-in with the woman he put here so long ago.
The blue arrow curled around on itself, became a circle. Rotating swiftly; the universal signal for _Wait just a second_. A door opened where there had been only wall.
"Hello," said a stout staffer who wore the badges of both Safety and Health. "Body scans."
Ankit raised her arms—the instinctive, familiar posture of someone prepared to be scanned or crucified—but Fyodorovna did not budge.
"I fail to see how this is necessary," she said.
"Rules of the ward," the Safety woman said. "Everybody gets scanned. No screens, no trackers, implants sealed."
___Implants sealed?_ Ankit felt panic rise. She stammered "I—" but the woman had already touched the wand to her jaw. The tingle told her the pulse had been successful, her implant would be bricked until she could get a revival pulse.
"Welcome," the woman said, and gestured for Ankit to enter.
This was a problem. Without the implant Soq couldn't find her, couldn't talk to her. Couldn't relay her location to Kaev and Masaaraq before the building killed internal comms. Ankit's hands dampened. The fear again.
Their plan was fucked. They were fucked.
The nurse waited wordlessly. After less than a minute, Fyodorovna complied meekly. The blue circle became an arrow again and walked them the rest of the way.
Fyodorovna's room was astonishing. A salvaged-wood floor, shiny with age and use, something that could have spent a century in a Paris bistro. An earthenware pitcher on a squat dark hutch beneath the window.
"Here we are," she said, and Fyodorovna startled her with a sudden fierce embrace.
"Thank you," she whispered, her voice more human than Ankit had ever heard it.
"Hey," Ankit said, uncertainly. "Hey."
"I'm so scared," she whispered.
"You shouldn't be," Ankit said. _So am I._ "Here is where you're safest." Her boss's arms didn't loosen. "I'll stay with you. Okay? For a little while?"
Fyodorovna nodded gratefully.
Ankit went to the window. They were eighteen stories up. Down below, it looked like any other day in Qaanaaq. Fyodorovna poured out two glasses of water from the pitcher. Fyodorovna told her to make sure to call this person, file this document, all of which Ankit was already planning to do, and could not focus on. All she could think about was the chaos on its way, how helpless she was without her implant, the hundred million ways this could go down wrong.
She closed her eyes and she was standing in the wind. Giddy. Happy. A tiny helpless unstoppable primate. None of the million things that had made her sad or scared an instant ago had any meaning, anymore.
"What's that?" her boss said suddenly.
Ankit opened her eyes and came crashing back into her own body, her own life. Her monkey's wild joyous freedom was gone. She ached for it. Had to fight to keep from shutting her eyes again.
"What's what?"
Ankit heard nothing. And then she realized—that was the problem. Something you almost never heard in Qaanaaq. Silence.
"The heat," Fyodorovna said, getting up and putting her hands in front of the vents. "It stopped."
All her life, everywhere she went, Ankit had been hearing the low rumble and purr and hiss of the geothermals. And now there was nothing.
"Perfectly normal," she said, but she could see that Fyodorovna was not convinced.
Time passed. An hour, two? There was no voice in her ear telling her what time it was.
The plan was idiotic. _They_ were idiotic. All of them. How could they not have anticipated that the implants would get pulsed, that they wouldn't be able to communicate through this crucial phase?
A shout from the hallway. More shouting in the distance.
_People are panicking. Health's response software will be collating all this information, plotting out scenarios, issuing a decision._
"You said I'd be safe here," her boss said, sniffling.
"And you are."
"Not safe from freezing to death."
"Shhhh," Ankit said, and sat on the bed beside her. Took a blanket and draped it over her shoulders. Fyodorovna pulled it tighter, gratefully.
The poor woman. She couldn't help what she was. It took a special sort of insanity to run for public office. A fragile megalomania; a delusional ego.
_I've let my contempt for her become contempt for the office,_ Ankit realized _. I came to share her crazy mistaken idea of what the job of an Arm manager could be._
But there had been a time, almost forgotten now, when she'd enjoyed her job. What it had been for her originally. When she'd gotten something out of it. Something positive—not the energy and stress and urgency and self-importance, the negative things, the things she became addicted to. The fact that she could solve problems for people. That she could help them get through something bad.
_I could do this,_ Ankit thought, and almost choked on the realization, the suddenly seeing that she could do the thing she swore she'd never do _. I could be the Arm manager._
Someone ran past the door. A whole bunch of someones followed them.
"Thank you for your patience," said a voice from the ceiling. "We apologize for the sustained inconvenience."
Fyodorovna grabbed her hand.
The voice continued: "Health has made the decision to evacuate the facility. The floors below have already been emptied. Please exit your room and follow the red floor arrows to the nearest exit."
The door swung open. Someone howled. Someone else joined in.
"I'm not going out there," Fyodorovna said.
"Come on," Ankit said, standing, feeling just as frightened.
"Anything could happen to us. All these crazies running around? I'll take my chances here. They'll come for us eventually."
"Don't be stupid," Ankit said, and tugged on her hand. "We'll freeze to death if we stay. These walls are so thin. They hold no heat. It'll be arctic in here in less than an hour, and who knows how long it'll take for them to come find us. We'll be fine out in the halls."
Fyodorovna looked up, her eyes frightened and trusting. She nodded.
_If I can help this hopeless creature, I can help anyone._
"Should I leave the blanket?" she asked.
"Keep it," Ankit said. "It might be cold for the next little bit."
The door shut behind them. An explosion shook the floor beneath their feet. They began to move with the flow of other frightened people.
## Kaev
Watching Masaaraq move was like watching some eerie artist, a terrifying ballerina who slowly and beautifully slaughtered her fellow dancers. The blade swung, it twisted, it slammed backward. She was a painter, sending artful sprays of blood onto the bare green canvas of the Cabinet. Kaev grinned, ecstatic.
Masaaraq slapped him lightly.
"You need to concentrate," she said. "He smells blood, he sees all this frenzied motion, it'd be very easy for him to go into a total killer rampage and start taking out patients along with security. Keep your attention on the people he needs to be focused on."
They moved through the crowd. People ran, people shambled. Some crawled. Many saw the bear and froze, fell backward, turned and ran in the opposite direction.
"He won't hurt you!" Kaev called, but he knew it was futile for a hundred different reasons. They were almost to the stairwells. All the patients would pass through this point. Ora would be one of them.
She had to be.
Would Masaaraq even recognize her? Could she even imagine what all that time could have done to Ora?
A door slammed in front of them. He fired his gun at the glass twice, but only the tiniest cracks appeared.
"Hand me one of Go's explosives," he said to Masaaraq.
"No," she said. "He can do this. Put both hands on the door."
Kaev did, and waited for the bear to join him.
_It is ice,_ Kaev thought _, there is a seal on the other side._
_Break through the ice._
Liam stood, leaned back, fell forward with paws extended, hitting the door hard. Did so again, and again.
Nothing.
_Take the handle,_ Kaev thought, and showed him how. _Pull._
The bear pulled.
_Pull harder._
Liam roared. The magnetic lock groaned, and then snapped. On the other side, a waiting room. Empty except for a couple of people cowering in corners.
They followed the curving corridor and then turned. Running in the opposite direction of the red arrows. Ignoring the pleasant, urgent admonitions of the voice coming from the speakers overhead. The central aisle cut the circular floor in half; somewhere in the middle was the stairwell.
"They know we're here," Kaev said, pointing to his screen, which Soq had synced to the public feed from Health.
"Of course they do," Masaaraq said.
"No, I mean _they_ know. The software. It's already issued invasion protocols. A whole lot more Safety workers are already on their way. And there are probably threat neutralization devices all through here, designed to pinpoint us and take us out. Gases, explosions—who knows."
Masaaraq nodded grimly.
## Soq
That is a deranged proposal."
"Maybe," Soq said.
"You think that because you're my kid you can come in here and tell me what to do?" Go said, folding her arms tight in front of her chest. "You're mine. Same as all those other grid grunts on my payroll. You have some magic software? You give it to me."
Soq stood slowly. "I'm not your kid."
Go flinched. Just for an instant, but enough for Soq to press their evident advantage. "If I was your kid, you wouldn't have spent so long hiding from me. If I was your kid you'd be able to look me in the eye. And you definitely wouldn't avoid the subject like the plague until it suits you."
"Stop pushing me," Go whispered. Her face was inches away. "My skinners have never failed to get a secret out of someone. So if I want your software, I don't need your permission to get it."
Neither budged.
Soq wasn't scared. Probably they should have been. But all they could think about was a series of exhilarating sentences that had been playing through their head for over an hour:
___I can conquer this city. I know all its secrets. My head is crammed with a thousand heads' worth of knowledge. I know more than Go will ever know._
"It's in your interest to do this," Soq said, finally. "This is what you want, isn't it? Why you're gathering intel on the empties? So you can take them over, rent them out, right? I'm handing you all of that, every empty, all at once. Or you could spend months, years, maybe, paying a small army of grid grunts to do it. This is a ton of money I'm offering you. You would be a direct rival to every shareholder in Qaanaaq. You'd show them that their days are numbered. I'm not trying to pick a fight. I'm trying to make sure you see this clearly."
A commotion from the deck: eight soldiers boarded, flanking a man who was plainly their prisoner. So old and frail that Soq thought it must be Podlove, but no—the skin was darker, the clothing cheaper. Go held up her hand impatiently, putting the conversation on pause, and opened the door to the cabin.
"We've got him," hollered the lead soldier from that squad, waving her brass-knuckled hand. Flashing a smile as wide as the horizon. With Dao dead, Go's lieutenants would be angling for the spot at her side, and this one had just scored a major coup. Soq remembered the tireless jockeying for position at the slide agency, the clamor and barely concealed excitement when an accident took out one of the senior messengers. Soq had been into it then, had jockeyed with the best of them. Soq wasn't, now.
"Bring him to me," Go called.
He came slowly across the deck, up the steps to the cabin. Blinking like he was about to sneeze. The old man, Soq saw. The one from the video. The one who killed the shareholder's grandson. Of course.
"Ankit helped you find him?" Soq asked.
"No," said Go. "I have many ways of getting what I want."
The lead soldier entered, bringing the old man. Whose face, Soq was surprised to see, showed no fear. His hands were cupped like a Buddha statue's; like a saint's on the way to martyrdom.
"Open up a line to Podlove," Go said. The soldier tapped her jaw once—the call had been cued up already; the ability to anticipate her general's orders was an excellent Prime Toady quality.
On the screens, Podlove jerked his head up, looked out his window like he knew he was being watched, trying to make eye contact with Go through her drone. And then he smiled. His window ionized.
"You're good," said their prisoner. "He had a whole fleet of drones after me. Piggyback software leapfrogging every cam in Qaanaaq."
"He puts his faith in machines," Go said. "I think people are just as . . . useful. And no one gave you permission to talk."
"He's not answering," the soldier said.
Go unstrapped her sidearm, aimed it at the old man's head. "Send him a picture of the two of us."
Soq counted. Eleven seconds later, the soldier said, "Call coming in."
"Good evening, Mr. Podlove," Go said when a new hiss through the cabin's surround speakers told her the line was live.
"Has his usefulness come to an end so soon?"
"He's not working for me," Go said. "He never was."
"Of course that's what you would say," Podlove said. "That's what I'd say, too, if I knew I had gone too far."
Go laughed. "For that scenario to make sense, I'd have to be afraid of you. And I'm not."
"And yet. Here we are. You called me for a reason, I must presume. Perhaps you just got word about your tube worm cake shipment? Pity all that food went to waste, but I'm afraid that is just the first of many such . . . interruptions."
Go laughed again. "You think my situation is so dire that one sunken tube worm skiff would have me calling you to beg? I had nothing to do with your grandson's death. We may be at war, but we're not monsters."
Podlove did not respond to this, but Soq heard skepticism in his silence.
"I hunted down the man who did it," Go said. "I found him when you couldn't."
"Eventually, I would have. I wouldn't have stopped until I did."
"He was heading for a mainland ferry." The old man opened his mouth, as if to protest the inaccuracy of this, but then shut it into a smile. "You may have the resources for a worldwide hunt, but would you have the time? Either one of you might drop dead any day now. But now—here he is. All yours."
A pause. A hungry one. "In exchange for what?"
"For nothing. He's yours."
"Except that I have to come and take him. Or go collect him from somewhere, where your people will be waiting to ambush me."
"This isn't a trap. It's a gesture. I want to prove to you that we have an understanding. I didn't do this. You and me, we're on the same page. I am not trying to scorch the earth here. This is a guy with a grudge against you, for some fucked-up shit you did a long time ago. It has nothing to do with me. I have no interest in making this personal."
"Even if you didn't have anything to do with what happened to my grandson, you did send soakers to hit two of my best managers. We're not a syndicate, my darling." Soq's nostrils wrinkled at the archaic vulgarity of his misogyny. "You can't treat us like we are and then expect us to believe you when you say we're on the same page. To us, that was an unacceptable escalation."
Go sighed. "You're right. I'm new to this."
"You're not ready to play in the major leagues."
Soq and Go exchanged puzzled glances. The brass-knuckled soldier pantomimed hitting a baseball.
"You New Yorkers love your baseball metaphors, never mind that no one else left alive on Earth knows what the hell you mean by them. Nevertheless. I am making this peace offering. Tell me how you want him. I'll deliver him in whatever way would make you feel the most safe, the least like I'm setting you up for something. I can put a hood over his head, drop him in a canoe, push it off and let it drift in your direction. I can decapitate him right here and now, if you'd prefer. Bloody my hands so you don't have to. Isn't that how you operate?"
"Come to me," Podlove said after an instant of skilled internal debate, the consummate executive assessing a thousand scenarios. "The lobby of the Salt Cave. Don't bring an army."
"Agreed."
Podlove chuckled. "That was fast. You're not frightened to march into enemy territory unprepared?"
"This is a parley, isn't it? A presumably safe negotiation?"
"You presume I see you as an honorable opponent. You've already broken the rules of engagement, such as they are. How do you know I won't do the same?"
"I don't," Go said. "This is a gesture of trust. I want you to know I had nothing to do with what happened to your grandson."
Sudden silence from the speakers.
"Son of a bitch hung up," Go said.
"What's the plan?" Soq asked.
"There is no plan. I'm handing over this idiotic man and he'll probably be tortured to death. End of story. Tomorrow we'll get down to the business of making peace."
"And my software?"
"Will have to wait. Right now Podlove and everybody like him is going to be keeping an extra-close eye on all their holdings. We kill the heat, wait till things calm down. See if it makes sense to deploy it then."
"Easy for you to say, when you're not sleeping in a fucking box every night."
"Show me some respect in front of my soldiers, at least," Go said, heading for the door.
"What about the other thing?" Soq said, feeling angry, impudent, desperate. "You can't postpone that. It's now or never. We don't know how things are going in the Cabinet. What kind of—"
"I'm considering it. Get out of my way."
"You're not," Soq said. "You've already made up your mind. You don't care about anyone but yourself. If you did, this wouldn't even be a question."
Go's eyes didn't flinch away from Soq's, but what did Soq see there, exactly? Whatever it was, it wavered. And wavering meant maybe.
Go beckoned for the soldier to follow, and to bring the old man. And after she'd left, her voice still echoed in Soq's ears.
_We're on the same page._
Soq was startled to see that this was true. And that this troubled Soq profoundly.
## Kaev
They pressed closer together without talking or even thinking about it, instinctively making themselves a smaller target, giving them some room to maneuver around whatever obstacles or weaponry might emerge from the walls. Eighteen floors up, the architecture felt different. The hallway widened as they went, and then there was suddenly a fertile garden on either side of them.
"Smoker's lounge," Kaev said, reading off Soq's screen again. "Open-air spaces. Lie out in the sun, get some exercise. Very helpful for crazy people."
"Which is the grid side?" Masaaraq asked. "And which is the sea?"
Kaev pointed each out.
_My mother is close,_ he thought. _For the first time since childhood, I will see her._
_I am going to find her. I am going to free her._
They entered the central cylinder; a chaos of screams and wailing. Patients everywhere, and Health workers. Everybody stuck.
"The invasion protocol paused the evacuation protocol," Kaev said. "Safety should be here by the billion in about three minutes, maybe four."
"Safety's already here," Masaaraq said.
There was only one of her, a lone Safety operative, and apparently unarmed. Although her smile was a little too confident for that to be true. Something else, then. The patients made a space around them. Afraid of the polar bear, of the weapon Masaaraq carried, of the fact that they could see their breath steaming in the frigid air.
The Safety worker raised her arm, rolled up her sleeve. Subdermals twitched and glowed along her arms. She was one with the room, with the system, with the Cabinet. Seamlessly melded with its innumerable defense measures.
Kaev and Masaaraq backed away from the Safety worker. Kaev scanned the crowd, looking for a pocket of less deranged and disheveled-looking residents—who were easy to find, there at the wall, dressed in civilian clothes, one of them even holding a wineglass.
The woman from Safety advanced. Kaev tightened his fists, watched to see how Masaaraq would come at this new threat, wondered if she'd know enough to understand how dangerous this opponent was. But Masaaraq wasn't looking at her at all.
"Ora," she said.
The woman stood ten feet away, draped in blankets, bald, beautiful, her skin bright and brown.
"Kaev!" someone called—and he saw Ankit running toward them through the crowd. A crab-faced woman called her name, but was swept forward by the human current.
"Tighten up!" Ankit called, and Kaev was in fight mode now, his higher brain deactivated, all animal, all instinct and swift brutal action, he saw what she saw—the woman from Safety tapping at her subdermals—and he knew what that meant, somehow, even though he'd never seen something like this before. He grabbed Masaaraq, pulled her with him.
Ora alone seemed capable of independent action, and she alone seemed to pause, not from uncertainty or fear, but from the magnitude of the decision she had to make. The world she had to choose.
"Come on!" he called.
_How can this even be a question for her?_ he thought, but the thought was gone in an instant.
The bear roared to frighten away bystanders. A space cleared for them as they backed toward the wall. Ankit arrived, grasped his hand.
A wave of polyglass rose up from the floor. At the sight of this, Ora finally moved. She ran for them, but the polyglass wall was moving faster. She dove, finally, and her whole body did not clear the closing wall, and it struck her in the leg and she fell to the floor beside them.
The new wall reached the old wall and melded seamlessly with it, boxing them in: Masaaraq, Kaev, Liam, Ankit, Ora. The woman from Safety smiled and came closer.
The bear roared, launched himself against the glass wall of his prison like it was ice, except this would not break any more easily than the window in the door had. There was a door in the wall behind them, but it would not budge. Ora felt her leg, nodded. Nothing broken, nothing bleeding.
Kaev reddened. His muscles tightened. He crouched, a fighter's stance, waiting for the bell to ring to explode into violence. But of course he wouldn't be fighting anyone. In an instant the ducts beneath the floor would rearrange, reconfigure, spray some fancy smoke to knock them all unconscious, and when they woke up they'd be imprisoned for absolutely ever. He could see Ankit taking deep breaths, preparing to have to hold one for as long as possible.
Masaaraq, on the other hand, did not seem to be aware that she was trapped at all. She smiled. She reached out her hand.
Slowly, painfully, like someone seeing through thick fog or a patient coming out of paralysis, Ora smiled back. And took her hand.
## Ankit
Mom," Ankit whispered, marveling.
This woman was nothing like the creature she'd had in her head. That pitiful thing, she saw now, had been shaped in her memory by a child's fear. This woman stood up straighter, her shoulders were broader, her smile indomitable. She wore the clean elegant uniform of Protective Custody. So she'd been transferred, at some point, from the general-population hell she'd been in the last time Ankit visited.
The floor lights flashed from red to blue. The walls ceased their distress sequence. The woman from Safety tapped at her jaw, delivered the update to one supervisor after another. The threat was contained. The invasion software had yielded to the evacuation bots.
Arrows appeared in the floor. Red again, but orderly this time, with a soothing flow of cool blues and greens in the walls, pixels and patterns calmly urging people in the appropriate direction.
Kaev and his bear roared, screamed, kicked, fought. Masaaraq smiled beatifically.
_Why are we still conscious at all? Any of us? Why haven't we been gassed?_ Perhaps the interrogation protocols were about to kick in, and they wanted them conscious . . . or maybe it was easier to transport conscious people than unconscious ones? Aside from the belly lurches as her monkey leaped and climbed her way around the building, Ankit noticed that she was eerily objective about the whole thing.
Patients stood, pointed. Came closer with fear and awe on their faces. Health workers tried to encourage them to continue the evacuation, but most preferred to stand there. Shivering. Watching.
"Hello," Masaaraq said to Ora.
"Well hello," Ora said, smiling. "It's good to see you. What took you so long?"
There. That's why Ankit wasn't worried. Wasn't scared. Her family had been reunited. Her mother was free. Sort of. Even if it was just for a moment—even if they were currently trapped, about to be incarcerated, deregistered, locked back up or banished to separate scrap salvage ships, even if they'd never see each other again after this—they were together now. For a moment.
"Family hugs later," Kaev barked, stern fighter instincts in action, and Ankit was grateful for them. "We're about to get gassed. Anybody got any ideas on a way out of here?"
With one hand, almost absentmindedly, without ever taking her eyes off Ora's face, Masaaraq pulled a small circle from the inside of her sealskin jacket. She pressed two buttons, peeled off the backing on an adhesive strip, stuck it to the door in the wall, called for everyone to get as far away from it as possible.
The explosion, when it came, was much worse than the surgical bulkhead disruption process that Go had promised. Instead of just magically making a wall open up, it sent a thudding shock wave and a wall of fire in their direction. Kaev pressed both hands to his ears and winced. Ankit had to fight to keep from throwing up. Liam shielded them from the worst of the blast, and he howled as a dissipating blossom of flame singed the fur of his back.
"Come on," Kaev said, and they followed him down the empty hallway.
The crowd noise rose behind them as the woman from Safety lowered the wall to follow them.
Ankit ran, reveling in the ecstasy of having a crew again, a posse, a team, the way she had all those years ago when she'd leaped from building to building and death was always right behind them, right ahead of them, but it didn't matter because they would face it together. She was inside and she was outside, she was human and she was animal, she obeyed gravity and she defied it.
"Slow down," Masaaraq said. "We can't exert ourselves too much. If it comes down to a real fight, we'll need to have some stamina to spare."
Which, of course, was when another polyglass wall slammed shut in front of them. And another, past that, and then another—a long series of them, and more certainly waiting in the wings, as many as would be needed to exhaust their supply of explosives.
From behind them they heard the stomp of the Safety worker's boots.
_She'll be careful, because of Ora,_ Ankit thought. _She won't risk hurting her._ Protective Custody cases would be the priority in the chaos of an evacuation. Ensuring they didn't get snatched by whatever enemies had obliged them to enter custody in the first place. Or murdered, as was more likely for the high-value clients whose custody was a more genial form of incarceration, like the lamas and child monarchs and inconvenient heirs whose claims to power could jeopardize distant regimes.
"Where are we?" Masaaraq asked Kaev.
"Eighteenth floor, outer corridor," he said, reading from a screen.
"Radio Go's ship," she said to Ankit. "Tell her where we are. Tell her to prepare the extraction ladders."
"I can't," Ankit said. "They bricked my implant when we came in. I can't reach them."
"Ours are blocked," Kaev said. "Privacy shielding engaged."
"They're down there, though," Masaaraq said. "Right?"
"Right. Down . . . somewhere."
"Cover me," the orcamancer said, following the curving corridor until she was out of the line of sight of Safety. "Hold her off. Don't let her see what we're doing."
Kaev and Liam stepped slowly toward Safety, looking as menacing as they could. Which was pretty menacing. Masaaraq pulled out another explosive sphere, attached it to the outer wall, and then pulled out another. And then another. Five in all, a clumsy circle of them on the wall that was all that stood between them and an eighteen-story drop to the sea below.
"What about your whale?" Ankit asked. "Can you tell her, and have her communicate it to the people on the ship?"
"At this distance, it would take too long for me to pass on our location. I'd need to meditate on it for fifteen, maybe twenty minutes. We don't have that kind of time. And even if I could do it, nobody on that ship speaks orca. There's no guarantee they'd understand what Atkonartok was trying to tell them."
"What about that screen Kaev has?"
"Not networked," Masaaraq said. "Soq loaded it with blueprints, predictive software bots." She barked, "Stay together!" and then activated the explosives.
As one, they moved away from the blast site. They had to get to a safe distance from it, but they couldn't let Safety know where they stood in relation to the explosion, or she'd wall them in somewhere they couldn't get back to it.
"Drop!" Masaaraq said, and they did. Again the sick-making thud, again the roar of fire. Then the orcamancer was pulling them back to the red-hot wound in the side of the building, where cold bitter wind was already rushing in. And Ankit knew why, knew it from the sick yawn of space, so she was stammering, " _No no no,_ " __before Masaaraq said:
"You're going to have to scale it."
"No," Ankit said. The sky was night black outside. Green light sketched the city's outline below. It was one thing to feel what her monkey felt while her own body was safe in a hallway. To venture out in this dense human frame—"I can't."
"You have to. Or we all die."
"I can't," she said.
"You can. You are more than human now."
"Even if I could," Ankit said, stammering for any imaginable excuse—and she could feel her, the monkey, climbing, feel how the wind tugged and pushed at the little torso—"it'd take me so long to get down there. And for them to send up the ladder. She's not going to just stand there waiting."
Kaev and Liam stepped back to keep from being walled off away from them, joining the pitiful crew pressed up close to the opening. Shivering. They would die if she didn't move. Kaev turned to her and she saw the fear in his face, saw how he smiled to hide it. To be brave for her.
"Go," Ora said, her hand warm on Ankit's face. "I can handle her."
Everyone stared at her. She smiled, nodded. "Go."
Ankit touched the damaged wall, which was cooling fast.
Ora stepped toward the woman from Safety, whose hands were busily tapping at her subdermals. "Your father," Ora called.
Safety's hand stilled.
"He never stopped trying to find you."
Safety's mouth opened. No words came out.
"It cost him everything, getting you and your mother out of Port-au-Prince. He spent years saving up money to come after you, even when he heard that your _balsero_ armada got broken up by people pirates. Every Sunday he got on the circuit, spent all his money calling the reporting services. Listened to the reciting of the names. He got out. Got as far as Gibraltar. Spent a long time there. Waiting."
Ankit shut her eyes. Breathed. Reached out for the monkey—for Chim—who was somewhere nearby; she'd released her outside the building when she'd arrived with Fyodorovna, to scale the facade and wait for her, as Masaaraq had instructed.
She realized: Masaaraq had anticipated this exact scenario.
The monkey answered her. She opened her eyes without opening them, seeing what Chim saw, the sheer walls and fretted glasswork. She stepped through the opening. She turned around to begin her descent, but paused for a moment. To take in the scene, these people she loved, this family, these humans she was bonded to in bizarre magnificent ways, in case she never saw any of them again—in case, in fact, she never saw anything again.
One word leaked out of Safety; a croak, a single syllable that contained a thousand questions. "You . . . ?"
"When I saw you, I remembered," Ora said gently. "What he remembered. I can show you. Everything he saw. What he went through. Where he ended up."
A moment ago Safety had seemed to be eight feet tall, armored and invincible, but now she sounded small, simultaneously very old and very young. "Is it the breaks?"
"It is."
Ankit knew she needed to go. She clung to the ragged edge for a few seconds more, listening, desperate to delay her descent.
"My friend had it," Safety said, and she was crying now. "She said some of the things . . . it was like remembering someone else's memories. But she couldn't . . ."
"Control it," Ora said. "The breaks isn't a disease. It's just incomplete. Once the missing piece is in place, it's a gift. An incredible ability. I can share it with you. Answer all your questions about your father. About the circumstances behind your family leaving. I see it, too. I see everything."
Ankit slipped out.
A narrow ledge, barely wide enough to stand on. Wind tugged at her—but the wind was not an enemy, every scaler knew that. Wind, gravity, walls, rooftops, fences—these were facts. Things to accept and embrace. Tools. Things to use.
She moved west, with the wind.
_Come,_ she called, and Chim answered.
Sophisticated scalers shunned climbing. What they wanted was adventure, excitement, the swift running leaping progression from building to building. To go straight up or down was hard work, and ignominious.
Hard, but possible. And, in fact, her strong suit. Because what she lacked in courage she made up for in determination and diligence and discipline.
She descended now. Slowly, gripping tight, wishing she had her gecko-skin gloves, her ropes and anchors, but hadn't her friends scorned all that equipment? Hadn't they maintained that scaling always came down to the human body and the human mind, up against the elements?
When she reached a landing, she turned to scan the sea below. They were near the grid, and she could see only a narrow slice of ocean. No sign of Go's ship. She continued the descent.
Her heart hammered. She sang to herself, but suddenly couldn't remember more than one verse to any song at all.
_Focus. Focus._
Down two more stories, she reached a garden smoker's lounge. It should have been tropical, but the geothermal heat was still out. Glasses of water stood on every table, all frozen solid. She hurled one over the edge. Watched it explode against the grid.
Which was stupid. Because now she kept imagining herself exploding against the grid.
She climbed up onto the railing. Not so much farther to go, but here the orderly progression of walls and windows and ledges that had taken her this far broke down. Now she'd need to really scale. Run, leap, execute flawless rolling falls.
Now the fear took her.
Turned her feet to ice, froze them to the railing. Filled her bones with lead.
She shut her eyes. _I am not afraid,_ she thought, but that wasn't true, and then she thought, _I am stronger than my fear,_ and that was maybe true. She breathed.
A scream from beside her. Chim, squatting on the railing.
"Hey, girl," Ankit said.
Chim screamed genially. And then jumped.
Ankit jumped too.
And grabbed hold of a horizontal bar, the same one Chim had landed on. She let the momentum swing her forward, and at the apex of her swing she bucked her body to extend the arc, landing with a wobble on the joint where three struts met. Chim leaped to join her on it.
They swung, they tumbled. They were one. Whatever happened, she was not alone.
A wall blocked her way, and Ankit sped up. Leaped. Took one step and then a second up the wall and grabbed hold of a bar, squat-hopped onto another one. In the space between two stairwells she zig-zagged down, back and forth from one landing to the next, dropping four stories in the space of seconds. She caught herself throwing in superfluous moves—thief vault, Kong vault, cat pass, rotary jump—for the sheer exultant pleasure of it.
At the end, two stories from the sea, Ankit leaped to the final level without even thinking about it, and for the first time in her life executed a flawless rolling fall. Then she had to wait, laughing, breathing heavy, for the monkey to scamper down through less spectacular means.
She didn't feel relieved when they reached the struts and ran the wide circumference of the Cabinet's lowest level, and finally saw Go's boat, where Soq was standing on the prow looking for them, and gave them directions for where to aim the ladder to extract everyone else, and climbed aboard to be draped in blankets and offered a steaming mug of something hot and alcoholic.
What Ankit felt was sadness, to be groundbound again.
****City Without a Map: Press Montage
From the _Brooklyn Expat_ [in English]:
_Sometimes Qaanaaq can seem like_ _Saturn, ceaselessly devouring its own young—and dooming itself in the process. Blink, and something you love has vanished. Your favorite noodle stall; the karaoke skiff where you went for your first date; the Mongolian cinema where you discovered the work of Erdenechimeg or Batbayar. The high cost of real estate, and the sponsoring nation council's steadfast refusal to adopt the commercial rent controls supported by an overwhelming majority of registrants, add up to a city where nothing good can stay._
_And yet—some things simply seem . . . permanent. Unchangeable. Essential to the structural integrity of the city's psyche. As crucial as the grid itself. Some things we've been seeing for so long that their absence is simply inconceivable._
_This morning, residents woke up to two previously inconceivable new changes._
_The first was the absence of the rusted old freighter that had been docked on Arm Five for almost thirty years, accord_ __ _ing to some neighbors. Generally believed to be the flagship vessel and headquarters of the Amonrattanakosin crime syndicate, it had shipped out at some point in the night._
_The second? A hole. In the Cabinet. A smoking flaming wound in the side of that building so widely considered impregnable._
_And according to many witnesses, these two inconceivable disruptions are connected . . ._
From _Keskisuomalainen_ [in Finnish]:
_The drama unfolding in the_ _Cabinet reached its climax shortly after three in the morning. This outlet was one of a handful that was present from the start, from the moment the geothermal heat to the city's largest psychiatric center went out, and we stayed on-site for the duration, sharing every official communique as Health released it, capturing evacuee responses that contradicted Health's facile tale of a simple heat disruption, even releasing images leaked anonymously to us by a Health employee that appear to show a team of violent invaders assaulting Safety officers inside the facility—accompanied, we hasten to add, by a polar bear. And we_ _were present forty-five minutes ago, when a loud boom_ _echoed through the Hub and a ball of fire appeared in the outer wall of the Cabinet._
_This story was major even before the invaders blew a hole in the wall and started facilitating the mass escape of inmates. While we are still waiting on an official response to our query to Heat's analysis software, it seems likely that this is the greatest geothermal disruption in the city's history. If criminals bent on getting into the most secure building in the city can rupture the pyramid's valves and divert heat, how safe are any of us? For years we've been told to put our faith in the aquadrones, the multiple redundant defense systems,_ ___the engineering marvel of the geothermal cone, but if a handful of thugs can triumph over those defenses, the safe and abundant heat that makes life in Qaanaaq possible at all is called into question. Today's events may embolden our enemies to launch an even bigger attack, one that could have every one of us looking for a new home or saluting a Russian flag come morning . . ._
From the _Post–New York Post_ [in English]:
_Safety officials maintain that they_ _are awaiting final software analysis before releasing a statement. This statement, when it comes, is unlikely to tell us anything helpful or revelatory. Bot statements rarely do; it's why they're even more popular than the ones that human press flacks used to have to type up with their own two hands._
_What is beyond question, however, is the involvement of Amonrattanakosin Group. Their vessel was present in the waters beneath the Cabinet at the precise moment that the outer wall was breached by an explosive detonated by one of the invaders. Footage from countless angles shows them shepherding escapees aboard—an estimated three hundred in total. Whether they were behind the whole thing or merely part of a bigger plan, possibly involving numerous syndicates or other power players, local or foreign, is unknown. Recent violence across multiple Arms has been shown to have targeted Amonrattanakosin assets. If today's events are merely an escalation in some sick syndicate turf squabble, how many more innocent registrants will be killed or kidnapped as open warfare consumes the city?_
_Qaanaaq is famous for its laissez-faire attitude to law enforcement, illegal commerce, and syndicate activity. Most people here seem to like this minimalist style of government, overseen by tolerant machines whose primary concern is for_ ___our well-being. Maybe those of us who come from more aggressive nations or cities are just extra-sensitive on the subject, because we have seen what happens when lawlessness flourishes._
_Syndicates think they are above the law. The question to Qaanaaq is—are they?_
## Kaev
The tea steamed in the open air. Kaev poured it out slowly, concentrating, and then he handed it across the table. They sat on the deck of the Amonrattanakosin ship, out on the sea, away from the geothermal vents—unmoored, unconnected to the grid—and that wasn't even the most unthinkable piece of this scene.
"Thank you," she said. She was real. She was a person. She took the mug, then set it down. Pressed both hands against it. Looked at him. Smiled like no one had ever smiled at him, not even Go. "You don't remember me."
Kaev shook his head.
"And you?"
Ankit shook hers.
"You were so little," Ora said, and her voice hitched slightly. Masaaraq took her hand. From belowdecks, the sounds of laughter and crying and anger and joy; the Cabinet refugees being fed, warmed, hydrated, while presumably Go's flunkies figured out what the hell to do with three hundred people.
They sat, the four of them. Around a table on a boat on the open ocean. From inside Go's cabin Kaev could hear the squawk of multiple radios, a dozen soldiers arguing. Go's voice, clear and certain as a bell, and probably only Kaev could hear the fear in it. Liam lay on the floor not far from them, unsleeping, curled into a ball and watching them fitfully. Every bit as uncomfortable and uncertain as Kaev was. Qaanaaq was a long dark smudge on the horizon. Everything he knew was behind him. Kaev knew why he was frightened: nothing would ever be the same again. What he didn't know was why he wasn't _more_ frightened.
"This is so weird," Ankit said. "I'm sorry, but it is. We go from being orphans to having two parents in the space of days. I was a political drone, a nobody, and now I'm busting people out of lockup, doing family time with people everybody believed had been wiped out."
"Of course it's weird," Masaaraq said. "We should not exist. We should not be a family. But here we are. In spite of everything they tried to do to us."
Everyone's eyes kept flitting back to Ora. She said almost nothing. Stared out into the distance, and then into her tea, and then at one of them, and then at the cabin where Go was throwing things against the wall. Squeeze her hand and she squeezed back; give her a smile and she returned it. She seemed present. Seemed happy. But who knew how much of her was left, after everything she'd been through?
"What now?" Kaev said.
Masaaraq laughed. "Would you believe I have no idea? Thirty years planning for this, building toward this moment, and barely five minutes in all that time to think about what the hell I would do when I got here."
"What we'd do," Ora said quietly. But everyone heard her.
They drank tea, all four of them. They touched each other tentatively. Repeatedly. Nervously. Like maybe this was all a joke, a trick, a dream. Kaev slowly felt less frightened. A seagull circled overhead, descending to scratch at flecks of fish guts at the edge of the deck. Ora gasped when she saw it, and did not look away no matter what it did.
"Seagull," Ankit said. "Ugly creatures."
"I think it's the most beautiful thing I've ever seen," Ora said.
"Ora was bonded to a bird," Masaaraq said, and Ora whispered with her: "A black-chested buzzard eagle."
"I think I remember that," Kaev said. "It's faint, but I feel like I remember seeing it circling the Cabinet. A long time ago. Roosting up top."
"She stayed with me for the rest of her life," Ora said. "Fifteen years."
"And then?" Masaaraq asked. "What happened then?"
Ora said nothing.
Shouts from the cabin; an alarm sounded. A boat, coming from Qaanaaq.
"This can't be good," Kaev said, standing up.
Soq emerged from the cabin. Startling, the fondness that gripped Kaev when he saw Soq coming. The love for someone he never knew existed. _This is what family is. What family does._ Was it magic, some supernatural quirk of DNA recognizing its own? Or did humans simply spend their whole lives so steeped in the mythology of this primal thing called family that the emotions were already there, one-way relationships waiting for the people who would one day step into those slots?
"Is that Safety?" Kaev asked.
"That's Go's transport," Soq said. "She's going to pay a visit to a certain Martin Podlove."
Kaev sat down, clumsily poured another cup of tea for Soq.
"Thanks," Soq said. "What'd I miss?"
"A brazen invasion, explosions, death-defying feats of true scaling brilliance," Ankit said. "Also, this is your grandmother. Soq, meet Ora."
They shook hands. Awkwardly. Soq frowned at the old woman's face and asked, "Have we met?"
Ora smiled. "Not directly. I'm a friend of a friend, perhaps."
"Hey, yeah, cool, I get it," Soq said, with the exaggerated smile you give someone you suspect might be quite mad.
"Yours?" Ora asked Ankit.
"Mine," Kaev said, and then laughed. "I mean . . . not _mine_ . . . Soq belongs only to Soq. But I'm Soq's father."
"This will take us all some time to get used to," Ankit said. "But back to Martin Podlove. I knew the guy. The one who killed his grandkid? I'd been working with him on research into the breaks. Fascinating fellow. So knowledgeable. But angry. And sad."
Soq laughed. "Well, then. If that's your friend, I may have some bad news for you."
The door to the cabin opened. Go came out first, followed by soldiers. Marching a man who moved in shuffling little steps, because he was extremely old or because his ankles were bound like his wrists. Or both. Probably both. Soq said, "That would be the bad news I mentioned."
"She's going to give him to Podlove?" Ankit asked.
"Maybe," Soq said. "Probably. I'm holding out hope that it's just a trick to get close to the guy in his office and gut him like a fish, but the chances of that are looking slim. I'm going along for the ride."
Kaev watched their mouths move, heard the words, didn't hear them. None of this mattered. He felt Liam inside his head, a calming beacon of mammalian wisdom, of animal objectivity, guiding him clear of the rocks. Words were useless, dangerous, dishonest. He loved these people and he wouldn't let anything happen to them. He got up and lay down on the floor beside Liam.
"You look like you're part polar bear yourself," Ankit said, smiling down at them.
"That'd mean so are you."
"I'm all monkey, baby."
When she laughed, so did the ugly little capuchin that had been hiding behind her.
When Liam snarled at it, so did Kaev. Then everybody laughed. Except Masaaraq.
"Don't mind her," Ora said with a laugh, her smile radiant, and how could she be so sane, so whole, so happy, after everything she'd been through? Kaev wanted to lie in her lap and stare at her face forever. "She was always like this. A hunter. Out in the wilderness all the time. Killing things. Even when she was with us, she was somewhere else. Worrying about what might happen. You know how predators are."
"No, actually, I don't," Ankit said.
"I do," Kaev whispered.
Masaaraq scooped up a handful of seawater from a bucket at her feet and splashed it in Ora's face. And _then_ she laughed.
"She loves you kids," Ora said. "Always did. Even if she was afraid to show it. Even if she thought that evil spirits might see her happiness and snatch it away from her."
"Turns out I was right about that," Masaaraq said.
"Know this," Ora said. "She didn't come all this way just for me."
"Upper America is full of empty towns, empty cities," Masaaraq said abruptly, uncomfortable—as Kaev was—with all these words. "Our enemies are gone. The warlords keep to the south. We can choose any place we want. Big mansions, tiny cabins, a dozen of each . . ." She looked up, saw Kaev and Ankit's shocked expressions. "There are settlements there, still. Trade routes. You wouldn't be giving up civilization entirely."
Their mouths did not close.
_Yes,_ Kaev thought. _Yes._ Liam raised his head, seeing what Kaev imagined, the boundless expanses of snow and ice, the wilderness, the hunt. His joy doubled, tripled, boosted by the bear's. _Let's get away from all this human ugliness._
Ankit shut her mouth and then opened it again. What would she say? What if she hated the idea? Who was this woman? What did she love, fear, hate, crave? Kaev felt his whole future hung on the next sentence that was said.
"I—"
Ora interrupted: "We're not going anywhere."
No one said a word.
"My work here is unfinished," she said.
"Your work?" Masaaraq said, letting go of Ora's hands. "What work?"
Ora smiled and showed Masaaraq her forearm. Kaev caught a glimpse of a crescent-shaped scar or wound, endlessly repeated.
Masaaraq's eyes went wide. She stood, stepped back. When she spoke, it was in a whisper. "You've been bonding people." She clamped her hands over her mouth.
"Don't you dare judge me," Ora said. "And don't tell me you've become so weak you need long-dead superstitions for a crutch."
Masaaraq could not answer.
"You bonded me," Soq said to Masaaraq. "And you'd just met me."
"And me," Ankit whispered.
"That's different," Masaaraq said, and the terrifying fearless orcamancer was gone from her voice. "You're family. Both of you. We're _family_."
"How do you think I survived?" Ora said, and her voice was a viper's now, a hiss of warning, of pain, of anger. "How do you think I lived so long? How do you think I lasted the decades it took you to come and find me?"
Masaaraq flinched. "I never—"
Ora stood. "Exactly. You never. You never had to live with what I lived with. You never had your animal die of old age, of grief, kept from touching you by sixty feet of polyglass. You never had to sense it, flying in endless circles, searching for you, never getting any closer. Bonding to people was the only way to keep from going insane. From getting my brains scrambled back to the mental capacity of a child."
Kaev stood up, went to her. Bear-hugged her. Thinking, _I would do anything to keep from feeling that again_.
"I'm sorry," Masaaraq said. "I'm sorry. Of course I have no right to judge you."
"I met a lot of sick people in there," Ora said. "Suffering from a lot of illnesses. Many of them completely baffling to the medical software. And I found a funny thing. When I bonded with people with the breaks? They weren't sick anymore. They still had it, but they weren't suffering from it."
Masaaraq gave Soq a swift look and then a nod.
"What is it?" Ankit asked. "The breaks. How does it work? Scientists know it's delivered by a viral vector, but we don't know if it actually _is_ a virus. Might be that, or a bacterium or nanites or rogue gut fauna or some combination of those, or something else altogether."
"Whatever it is, when you get infected with it—it carries information. Memories. This disease defies everything we thought we understood about how memory works. Somewhere in its genome, the sickness encodes massive amounts of data from the person who infected you, and the one who infected them, and so on, all the way back up the chain. A normal human mind has no idea how to process this kind of information, and it will slowly start to break down. But the nanites that let us bond with animals also let us process their emotions, their imaginations. So when someone with the breaks gets bonded to one of us, the nanites help them survive, handle those memories, control them as well as you would control any others."
"That's your work," Masaaraq said. "You want to save these people."
"I am _going_ to save these people," Ora said.
"It's not safe," Masaaraq said. "The city won't rest until you're back in custody."
"Let them try."
"We're up against some very well-resourced enemies. We almost got taken out, in the Cabinet. We'd have been recaptured if it hadn't been for—"
"For the fact that I could give that woman something she needed. Something I got from helping people. The people I helped will help us. I can see the big picture. I've been drawing a map for a city without one. Reciting it to myself every night. I know how we all fit together, how this will play out."
Masaaraq looked at her with curiosity, possibly fear. The way you would if you suddenly started wondering whether the person you love might have become something more or less than human.
"You're the Author," Soq said. " _City Without a Map._ You seed it, somehow. Right? In them? You . . . compose it, pass it on to them when you pass them the nanites."
Ora said, "Yes."
"Small boat," Kaev said, looking worried, as Go's transport reached the ship. "Won't hold very many soldiers."
"She's only taking a couple of us."
"To confront a guy who wants her dead?"
Soq nodded. Kaev's brow furrowed, and he stood up. "We have to go, too," he said, but no one heard him.
"Speaking of Martin Podlove," Ankit said, turning to Ora. "Do you know who he is?"
Ora paused, shut her eyes. Began to recite facts. Rumors. Things she'd called up, effortlessly, from the mountain of memories she'd inherited from every breaks survivor she'd bonded with.
"No," Ankit said, and they heard shouting from the other side of the boat, chains dragging and bells tolling as Go's ship prepared to depart. "Do you know why he put you in the Cabinet?"
"He did what?" Kaev said, but he wasn't the only one who said it.
## Soq
Welcome," Podlove said, looking for all the world like some combination of hotel concierge and mad sea captain. Soq could see the uncertainty in his eyes. Go had been correct—he was a banker, not a fighter.
"Ahoy," Go said.
They stared at each other across the opulent lobby of Podlove's corporate headquarters. Salt crystals everywhere. Sharp and sparkling. Intended to impress and intimidate. Behind them, across the glass, Arm One traffic was reaching its early-morning peak.
But the place would be bristling with well-hidden weaponry. Podlove was confident, secure in the familiar center of his universe, but he wasn't stupid. He would have fearsome muscle in his corner. Drones and bots and autoturrets.
Soq shut their eyes, and they could see it. Could recall the schematics, taste the bite of the drill bit installing a toxin pod. Remembered her name, the woman who'd led up the installation six years back. Knew that she was dead.
Two flunkies stood behind Podlove. Go had two of her own. Soq, and the brass-knuckled soldier whose name they had never gotten.
And, between the rival parties, the man with the sack over his head.
Swallowing, finding their mouth so dry it was almost impossible to do so, Soq felt the full gravity of the situation. If anything went wrong, they would be right in the line of fire. What kind of guns and blades and projectiles and lasers were aimed at them right this second? Podlove flashed a frigid snake smile, savage and cynical all at once, but Soq could see that he was scared. And that was scary. Because scared people were dangerous _._ Soq made eye contact with one of Podlove's flunkies, a scrawny thing who looked as frightened as Soq felt, and gave him a little smile, at which he snarled.
Soq stood up straighter.
They'd been frightened, at first, after they'd learned Go was their mother. They'd feared losing their objectivity, letting their emotions and the personal bond between them render Go perfect, special, beyond reproach. And while Soq was happy to be a henchperson, they knew that flunkies who thought their employer was always right started making dumb decisions.
The opposite had happened. Rage, not love, tinted how Soq saw Go. The woman had abandoned Soq. Every awful thing that had happened in Soq's life could be laid at Go's feet.
Either way, Soq's objectivity was compromised, and that was a problem.
But maybe objectivity wasn't everything. Maybe it wasn't even real. Soq's head buzzed with a hundred different takes on objective reality, and—coolly, effortlessly—they could compare the times two people remembered the same things differently. Both convinced they were right. Soq could see Go as a dozen people saw her—cruel bully, magnanimous boss, ignorant grid-grunt upstart.
Go didn't know what Soq knew: that Ora and Masaaraq and Ankit and Kaev were on their way over. To complicate matters. Soq had meant to tell Go, and now was glad they hadn't.
"Is this how you dreamed it would be, when you got to the top?" Podlove said.
"This isn't the top," Go said.
"No. I suppose it's not. But it's as high as you'll get. This ends today."
"I told you, we're on the same page."
"This is him?" Podlove said, advancing to the sack-headed man. "I'm not going to pull this off and find a lit stick of dynamite?"
Go moved to unmask the man, but Podlove stopped her with a gentle hand.
"A curious play, at the Cabinet," he said. "Taking all those people. What could you possibly plan to do with them, little girl?"
"Maybe I want to found a city of my own," Go said.
It had given Soq hope, when Go finally agreed to Soq's plan. Liberate not just Ora, but every Cabinet prisoner who wished to be liberated. Which had turned out to be a far higher number than projected—Soq had imagined that most would be too afraid to choose a rusty crime syndicate ship over the safe warmth of their prison. They were still belowdecks on Go's rusty freighter. Still frightened. But free.
The second part of Soq's plan was still up in the air. Waiting on Go to give the go-ahead, which she might never do. Run Podlove's program, the one Soq got from his grandson, the software that would tell them the location and access code of every empty unit being kept off the market by every shareholder, and move those people in. And then head out to Arm Eight and offer a place to every grid rat and box sleeper and overcrowded unhealthy unregistered resident. Move them from disgusting and precarious housing to impossible luxury. Balance the scales.
Found a city within the city.
_A city of my own._
A city where Go was the sole shareholder.
Of course, Go wasn't being altruistic. Soq could see that now. Go would want money, maybe just a little at first, but more and more, and Go had plenty of unbreakable men and women to drag you out of your place if you couldn't afford it. And then rent those fancy spaces to people who could afford to pay through the nose . . . once Go got a taste of that, it wouldn't be long before all those box sleepers were back in the boxes, and the empty units were full of one more wave of wealthy refugees. Being a landlord was the biggest racket in town, in every town, in every city, across history, and when Soq ran that software they'd be handing Go a massive empire.
_We're on the same page._
How would Go be different from Podlove, from every other rich and powerful player who sucked the blood of the poor, made them pay until they couldn't pay anymore and then pushed them into the sea to sink? Soq doubted there'd be any difference at all.
The question was, what could Soq do about it?
Podlove pulled the sack off Barron's head.
"Hello again," he said.
Barron smiled. "You don't look so good, friend. You look . . . unhinged."
"I'm going to unhinge you," Podlove said, and there was the fear again, the uncertainty. He wasn't Go. Threats and violence were not his native soil. His own rage frightened him. "Like a door. Take you apart like a jigsaw puzzle going back in the box."
Barron's smile only widened. "I know."
"Laugh as long as you can. Pretty soon you won't be able to. Because you won't have lips, a tongue, most of the skin on your body."
"It was way too easy to turn you into a medieval barbarian," Barron said. "You, who always believed yourself to be so civilized. Another way I achieve victory over you."
Podlove put the sack back on and turned to Go. "Was there anything else?"
"No, sir," Go said, bowing in exaggerated deference. Exaggerated, but still real. Go really did admire him. She really did want to be him.
"I'll be in touch. Once I've gotten a little more information out of this one, and I can assess how to proceed."
"Of course. I'll wait to hear from you. Let me know if there's anything else I can do for you."
They did not shake hands. But they smiled, and Soq saw oceans of information surge in that smile. They _were_ the same. Go didn't want to burn down anything. She might kill Podlove, but not because she hated him. Because she wanted to take his place.
Soq had been there. In Podlove's place—in Fill's. Physically, but more important, emotionally. The breaks had taken Soq there. They knew how empty all that comfort felt, how little it helped to hold off the dark.
Once, Soq had wanted to be what Go was. To have power, to have wealth, no matter who else got hurt. To plunge the rest of the city burning and screaming into the sea, if that's what it took. Soq didn't want that anymore.
It was stupid. Soq knew it was stupid. Soq did it anyway. A plan was in place, dependent on a delicate balance barely preserved. The balance demanded that Soq wait to run the software. That's what Go had told Soq to do. Go had been very clear about what to do and when.
_Fuck Go,_ Soq thought.
Six swift taps on the palate and against the inside of Soq's cheek, and it was done.
Everything happened far faster than they'd anticipated. They'd imagined that, if it worked at all, the decades-dormant software would take some time to get started, to trigger whatever safeguards and tripwire warnings might be set up, to say nothing of how Podlove would get word—but only eight seconds passed after Soq triggered the savage break-in software that Podlove had given to his grandson, that Fill had unwittingly given to Soq along with the breaks . . .
The old man's head jerked sharply, like he'd heard his name called from a great distance. He shut his eyes and listened to something his implant said. Then he opened them.
"You fucking idiot," he hissed to Go.
"What?" she said, her inveterate smugness certainly damning her in his eyes.
Above them, lights flickered. Sirens began to wail. The software read updates into Soq's ear. It had been detected by a monitoring bot, one of millions of ancient defense systems lurking in the municipal infrastructure, set up by the shareholders to check their little monster—by releasing an identical copy.
One of them was bad enough. Two of them, running in a state of open warfare, might make the city melt.
Glass shattered. Soq dropped to their knees, convinced this was it, Podlove would have triggered the bullets or explosives or whatever other weaponry had been trained on them for the entire parley. And maybe he had, but the system was not responsive. Most systems, it seemed, were not responsive. A whole lot of people were yelling out on the grid.
"Podlove, I swear . . ." Go said, her eyes terrified.
And nothing and nobody tried to stop Masaaraq and Ora and Kaev and Liam from breaking the front windows, walking right into the Salt Cave, armed and angry, and heading straight for Martin Podlove.
## Ankit
Ankit stayed in the little skiff when Kaev and Masaaraq and Ora and the polar bear got out. She moored it in a two-hour spot across the Arm from the entrance to the Salt Cave.
_I am the getaway driver,_ she thought, but she knew her role in this heist was significantly less glamorous. And exponentially more important.
A stream of data opened on her screen, became a river; became a sea. The software was working well. Terrifyingly well.
"Soq launched," she said into her implant. "Early."
"That sounds like Soq, all right," said the man from Soq's slide messenger agency.
"It's every bit as big as they said it would be."
"This is fucking crazy," Jeong said. "There's got to be five hundred units here. Easy. And they've all been empty? All this time?"
"According to Soq. Go had a lead on a handful of them, ten or twelve she'd spent a ton of resources on recon to identify. But this . . ."
"These motherfuckers," he said, sounding scared, sounding excited. Behind him, she could hear the clangs and hollers of Go's boat. "All these people living like rats, while they have all this space just going to waste."
"It's not waste. It's a business decision."
"Fine line between good business and a fucking war crime," he said.
"Ain't that the goddamn epitaph of capitalism."
Jeong chuckled. "You know I sleep at the messenger depot? In a capsule in my office. Sometimes I'll go a week without stepping outside the building. Probably been a month since I left my Arm. Now I'm surrounded by all these people . . ."
The understanding-politician part of Ankit's brain took over, emitted a gentle laugh. "Qaanaaq does that to you. I feel agoraphobic sometimes, too. You doing okay over there?"
"This ship is fucking crazy."
"Fucking crazy is how it probably is on a regular day. Now you're in full-on gangster war mode, with a hold full of psychiatric refugees."
He laughed. Sounding grateful. "Something's happening to the data," he said. "It's coming and going."
"Defenses. We anticipated this. We're making multiple encrypted backups, including several on partitioned drives that disconnect from the network as soon as a chunk is complete. If the attack bots compromise something, switch to another."
"This is the easy part," Jeong said, sounding stronger, more solid. "I'll take a psychopathic corporate espionage AI over crying babies any day. We'll have enough to start giving people addresses and passcodes in about three minutes. Enough to house all the Cabinet escapees, and a good chunk of _les miserables_ from Arm Eight . . ."
"Excellent."
"We're getting pings from all over, people volunteering slots for these people. Some crazy church lady with one hand says she has space in her storefront for fifty people to sleep."
Ankit stood, looked across the Arm to where Masaaraq and Kaev and Liam and Ora faced down Go and Podlove and an anxious-looking Soq. A stalemate. Angry words. She ached to be there with them. With her family—her two mothers, one a mournful butcher and the other a serene poet, but both identical in the staggering merciless weight of what their love could accomplish; her sweet and sad brother; his proud angry child—even if they were all about to die.
Especially if they were all about to die.
"All right everybody, line up," Jeong was saying back on the ship, and she stifled a tiny laugh at the thought of this poor frightened man doing crowd control.
A splash from the water beside her. Ankit turned her head—and flinched, even though she'd been fully expecting to see the orca there. Maybe it was possible to get used to something so huge, so formidable, as dark as the sea and as hungry, but she didn't see it happening anytime soon.
"Hey, girl," she said.
Atkonartok's immense head did a slow majestic dip. A nod, Ankit realized. Eerie, the intelligence it gave off. Not the intelligence of something as smart as humans, but rather the intelligence of something smarter, something making an effort to understand and be understood by these stinky smaller-brained beasts. Like right now, for example. The whale seemed to know, just from looking at her, how the whole complex plan was going.
_How much does she see? Remember? Feel? About the people who slaughtered her tribe, her pod? About the friends she lost? About what it meant to be alone, and lost?_
Like she had been all her life. Like Kaev had been, and Soq. And Masaaraq and Ora, who at least had known what it meant to not be alone, to love, to be loved, before they were plunged back into the well of loneliness. All these people going through life alone, suddenly plucked out of isolation and finding themselves part of a family . . . only now to be inches away from losing everything.
"Good!" Jeong said as the chaos of background noise in her ear quieted down. "There will be plenty for everyone. We'll start with you."
Children clapped. Jeong laughed. Ankit did too.
More laughter, from the real world now, two parking slips down. "The clocks!" said a plump matron with an Addis Ababa accent. "The parking clocks all went down!"
Ankit saw that it was true. Her own slip's timer, which had reached eight minutes the last time she looked, was now flashing zero at a leisurely pace.
"Free parking!" someone else shouted.
_Fuck,_ Ankit thought. If there was anything more unthinkable than a geothermal disruption, it was the parking clocks going down _. What the fuck have we done to this city?_
A shout from outside the Salt Cave caught her attention. She looked up just in time to see Go draw the machete from the scabbard at her belt, the scabbard everyone always assumed was empty, and behead Ankit's friend Barron with one effortless swing.
****City Without a Map: Savage Bloodthirsty Monsters
We want villains. We look for them everywhere. People to pin our misfortune on, whose sins and flaws are responsible for all the suffering we see. We want a world where the real monstrosity lies in wicked individuals, instead of being a fundamental facet of human society, of the human heart.
Stories prime us to search for villains. Because villains can be punished. Villains can be stopped.
But villains are oversimplifications.
## Soq
I'm impressed," Podlove said, with a slight wobble to his voice, as the new arrivals closed in on him. _He's terrified,_ Soq saw. _Desperate._ "I expected your people would try to blunder in here. But I didn't think you were capable of somehow crashing my lobby's defenses."
"Both are surprises," Go said, furious, confused, frightened. To Ora and Kaev and Masaaraq and the polar bear she said, "I told you to stay on the ship."
Podlove's lips were tight. "Right. You didn't get my grandson killed. You didn't tell them to come here. Terrible things keep happening to me, with you standing right next to them, but it's never your fault."
Soq looked back and forth between Go and Podlove. Comparing. Wondering: Which is more fit to rule? Which is more villainous? They were both frightened. Both sweating. Barron, at least, was relaxed, or that's how it seemed. Tough to tell with a sack over his head. His posture and general vibe of chill indicated a lack of fear.
On a sloop across from the Salt Cave, someone had spray painted _BLACKFISH CITY_.
"We didn't crash your defenses," Soq said, earning a death glare from Go. "You did. You ran that barbarian software against itself, and _that's_ what's fucking you up. And most of the city, I'd imagine."
"How did you get it from him, I wonder?" Podlove said. "My poor dead grandson. Did you torture it out of him? No. He'd probably have given it to you willingly. You're just his type. Feral and filthy and frea—"
Soq laughed. "Don't be childish." That had the desired effect. Pointing out when octogenarians are behaving like children is usually a good way to shut them up.
The soft putty of Go's face was hardening. Soq watched her slowly come to accept that the situation was out of her control. While the sensation was clearly agony for Go, for Soq it felt . . . expansive. Full of potential. Terrifying, but also thick with magnificent possible outcomes. Soq knew how the miserable poor of Mexico City or Pretoria might have felt watching the rebel armies march through the streets, or Lisbon or Copenhagen when the waters came flooding in. _For once, the status quo is fragile. Things could change._
"And our new arrivals?" Podlove said, turning to the very tiny angry mob. "Surely you didn't come all this way just to stand there glaring at me."
Ora stepped forward. "Do you know who I am?"
"I don't believe I've had the pleasure, ma'am."
She said her full name. His expression did not change. No recognition, no deceit flickered in his eyes. _He really doesn't know,_ Soq thought.
A groan from underneath them. The building at war with itself. A digital autoimmune disease. "We should take this conversation outside," Soq said softly, noting that this time Go did not seem angry that they were speaking out of turn. "His weaponry could come back online at any time. We'd be dead in a millisecond."
"Come," Masaaraq said, arm twisting out to aim the blade at Podlove.
"I'd rather not, if it's all the same to you."
_He thinks his old age will protect him,_ Soq thought, _so maybe he is not as smart as I thought he was_.
Masaaraq gave a half shrug, and both his flunkies fell to the floor, clawing at their opened throats. Soq calculated that it must have taken two swings, based on how far apart they were standing, but they had not seen even a single one.
Three swings. A single tiny red line formed across Podlove's forehead. Lone drops of blood beaded up, dripped down.
"You don't call the shots here," Go said to Podlove, smiling, but the smile looked flimsy.
"Neither do you," Masaaraq said, and swung again, slower, because she wanted Go to see what she was doing. The brass-knuckled soldier fell to the floor, gasping, refusing to scream.
Masaaraq's face showed nothing, but Soq knew what was going through her mind. From the moment that they'd bonded, Soq had so many of Masaaraq's memories, her fears and her nightmares, the pain she carried, the horrors she'd been forced to endure. Everyone had imagined that Ora would be the broken one, after so many years in the Cabinet, but Soq saw that Masaaraq was the one whose damage threatened to shatter them. And Soq loved Masaaraq so much in that moment, their beautiful formidable mutilated grandmother, that their heart hurt.
_If you know someone, know them completely and utterly, does that automatically mean you love them?_
"It's a lovely day," Podlove said, stepping over the writhing brass-knuckled soldier. Soq saw: politeness, good manners, these were his only real skills. The affectations of wealth were a suit of armor you could wear when the world threatened to wash you away. "Why don't we take this conversation outside?"
Overhead, the windscreen was shifting back and forth with slow, graceful, aimless motions. Snow fell. People stood, pointed, made calls, took pictures with their screens or oculars. Made space for them. Made lots of space. Only the complete and momentary collapse of Qaanaaq's digital infrastructure was keeping them at liberty right now—on a normal day, a massive Safety response would be in the works. A convergence. The once-every-five-years-or-so deployment of those big scary ships with the holds full of gnarly weaponry.
"You put her in the Cabinet," Masaaraq said.
"Ah," Podlove said, nodding his head as if someone had told him he'd left the oven on. "I think I understand now."
Go's hand rested on the scabbard of her machete. Soq calculated: _Her only hope is to make an explicit peace with Podlove. Otherwise, one way or another, she'll be destroyed. If he dies, the city's response will be merciless. Qaanaaq let the crime syndicates flourish, setting very few rules on what they could and couldn't do, but she's broken pretty much all of them. Go lives only if he does._ _And even then it's a long shot._
His survival seemed unlikely, Soq supposed, but then again anything was possible. Some demonic magic had kept him alive this long in a world full of people he'd pissed off. There might be some of that left.
"You _think_ you understand?" Ora said. The bear flinched, a jerk of rage barely stifled.
"We put a lot of people in the Cabinet," he said. "We had to. Either that, or kill them. Would you rather we did that? It was nothing personal. Our employees made a lot of enemies, and made friends with a lot of unpopular people. Understand, during the Multifurcation, a lot of people came to us with problems. Twenty different cities had minority populations practically rioting over police murders of unarmed civilians. Political parties about to lose key states. All in need of some . . ."
"Bloodshed and blaming and scapegoating," Barron said from under the sack.
"We didn't write the playbook," Podlove said. " _Dīvide et īmpera._ __Divide and conquer has been the foundation of human societal power dynamics for as long as there have been human societies."
"There have been knives that long, too," Kaev said. "Doesn't mean a man who stabs someone to death isn't guilty."
"How many people?" Masaaraq said.
"In the Cabinet? At least fifteen. In other grid cities . . ."
He didn't finish the sentence. He didn't need to.
"Was every one of them the sole survivor of a large or small genocide?" Ora asked. "You probably don't know. You probably didn't want to."
Podlove said nothing. He stood there, his face fooling no one with its approximation of repentance. "I didn't do anything on my own. There were a dozen of us, fellow executives. I wasn't even the highest in the hierarchy. I know you want me to be some savage bloodthirsty monster who single-handedly caused all your suffering. But believe me, I'm not. I just happen to be the last one left alive."
"Do you know what I could do right now?" Masaaraq said, aiming her bloody blade at him. "I could stab you in the stomach with this, use that notch at the end to grab hold of your intestine, yank it out, choke you with it—or make you eat it, or toss it to my orca, who would pull you into the water by it and take her sweet time killing you."
He shrugged. "I couldn't stop you. And I wouldn't blame you."
An explosion in the distance. Sirens starting, stopping, starting, stopping.
The stalemate lasted a long time. Each side glaring at the other. Except for Podlove, who looked only at his feet. At the metal grid he stood on, the city he'd helped build, the safe place his bloody money had bought for him. The sea beneath it. The water that would still be there long after the last human sank beneath its waves.
He was so old. His skin was so thin. So wrinkled. Wrinkles upon wrinkles, a crisscrossing net of them. He hadn't physically harmed anybody. His hands were bloodless. He'd merely gotten other people to hurt people, and then profited off it. Wasn't that worse, though? Didn't it magnify his crime, to have bloodied the hands of others? What kind of suffering had it caused them, the people who slaughtered innocents on his behalf? What trauma, what rage, what nightmares had it left them with? What bad karma?
Even if they chained him to a chair in the basement and spent the rest of his life torturing him until he passed out from the pain, then waking him up to do it again, over and over, there was no way to balance the scales of hurt. Nothing they could take from him that would approach even a fraction of the loss he'd caused others to feel. He was innocent in his own eyes, his crimes excused by necessity, and nothing they could do to him could make him see his own guilt.
Soq was still looking at him when it happened.
Masaaraq shouted something, twisted her body to intercept, but was standing too far away to stop Go from beheading Barron.
"Run," Go said to Podlove, bloody machete extended, launching herself at Masaaraq.
After that, everything seemed to happen in the space of a single short breath.
## Kaev
_Y ou need to stay focused,_ Masaaraq had told him back in the Cabinet, and now he knew why.
_He smells blood, he sees all this frenzied motion, it'd be very easy for him to go into a total killer rampage._
Kaev felt it. The bear's rage sang in him. It wasn't harsh or savage. It was beautiful. It was music. It wasn't the ugly human thing Kaev had felt before a fight, a spattered mess of wretched emotions like hate and fear and greed. It was clean and clear and simple.
_Keep your attention on the people he needs to be focused on._
But who that was he did not know. The woman he loved was fighting with one of his mothers. The one who had brought him Liam, fixed his brain, made him whole again.
Masaaraq struck Go with the butt of her staff, knocking her back.
"Hey!" he shouted involuntarily. "Don't!"
Masaaraq looked up for just a second. Just long enough for Go to strike her in the leg with the machete. Blood flew. Not a lot—the orcamancer's thick leather wrappings had muted the blow—but it was still hard enough to make her tremble, lose her balance.
Something angry thrashed and rolled in the dark water underneath him. Kaev looked down through the grid. He made eye contact with Atkonartok, and what he saw there made his blood flush frigid.
He ran toward them. He didn't know what he would do when he got there. Who he would help. He wanted to step between them, these two people he loved, stop them from fighting, but the bear had other ideas. He felt split, shattered, confused, and into that confusion stepped Liam.
The bear roared, and he roared with it. Masaaraq flinched at the mirrored sound.
"Kaev!" she shouted. "Stop!"
He couldn't. The animal was in control.
The animal did not like Masaaraq very much.
Kaev felt the pain of the metal cage on its head, the chains she'd kept it in. Years and years of that. Traveling to every settlement and camp and grid city and grim salvage ship in the north. Intermittent glorious moments of being unleashed, when she was in peril or had tracked down some particularly bad people, only to be knocked back out with a tranquilizer dart at the end of it and awaken in chains again.
_She didn't want to hurt you,_ Kaev tried to say. _She was trying to bring you to me. To help us both. We were incomplete. She completed us. Chaining you was the only way._
_She is our mother._
He knew he was speaking into the wind. In a moment of calm he might have been able to make the bear understand. Now, there was no space in its mind for words. For emotions. There was only the kill. For both of them. Whatever human part of him cared to talk Liam out of hurting Masaaraq, it was swiftly swallowed up by the bear's frenzy.
Masaaraq ran for the edge of the Arm. The bear followed. Ora screamed, ran in their direction, and Soq pulled out some kind of weapon that had been strapped to their back, but the orcamancer and the bear were too fast, too far away.
_If she makes it to the water she can climb onto the orca and escape,_ Kaev thought, and wondered whether that was really him thinking it. He knew he didn't want that to happen. She was so close, and he— _he? Which he?_ —was gaining on her, just a few more leaping bounds and he'd be on her—he wanted her to trip, fall—
And then she fell.
He heard the gunshot a split second later. Go's brass-knuckled flunky had pulled herself up, taken aim with a trembling arm, and fired. Masaaraq wouldn't have left her alive by accident. She must have shown mercy. And now. Now she was down. Unconscious, not dead. The bear knew by her smell.
A second gunshot—from Soq, this time, visible from the corner of his eye, but it didn't strike him, or his bear, so Kaev's attention did not waver.
Go made a sound, a horrible sound. Kaev was barely there to hear it.
Liam reached Masaaraq's unmoving form. Roared. Reared up on his hind legs. Kaev laughed at the bliss of it. The surge of happiness, the divine perfection of this moment, the kill, the thing he was made for. The bear raised its arms, and so did Kaev. He looked up, at his hands, and they were huge, thick with white fur, capped with long black blade-claws.
Pain split him in half. Broke his brain. He fell to his knees, and then to the ground, shrieking.
Before him, he saw a blur of black and white. And then red. So much red.
The orca had breached, leaped high into the air, and clamped its implacable jaws around the polar bear's midsection. Liam roared, raking his claws against the killer whale's sides, drawing blood in great gouts, digging deep—but not deep enough to break through Atkonartok's thick layer of blubber, soft and yielding yet somehow the most effective armor in the animal kingdom. The orca opened its mouth, clamped down again. Shattered the polar bear's spine. Dragged it into the sea.
Eyes shut, Kaev saw black water. The sea's mouth swallowing him. Opening them, he saw white sky. White snow, falling. Vomit gargled in his throat with every short gasping breath. Why couldn't he just die with the bear? He ached for unconsciousness. He lay on his back, praying for it, but it did not come.
## Ankit
Ankit drove the getaway boat. Chim shivered on her shoulder.
Force of habit kept her slow at the start, afraid of a ticket from the traffic aquadrones, and then she sped up, because surely all of Safety's toys would still be out of commission or otherwise engaged . . . and then she slowed down again, because where were they going? And what would they find when they got there?
The orca swam alongside the boat, vocalizing the same plaintive note again and again. Nudging the boat—lovingly, somehow; apologetically.
"She's grieving," Ora said. "That was her brother."
"Then why did she kill him?" Soq snapped.
"She couldn't help it. She was stuck, with Masaaraq unconscious. Locked into a frenzy, just like Kaev and Liam were. Even if she wasn't, she probably couldn't have acted any differently. Masaaraq's life was at stake. The bear was a half second from killing her."
"Atkonartok could have stopped him without killing him."
"Maybe."
Masaaraq said nothing. She sat as far back in the boat as it was possible to be, hugging her knees to her chest. Looking out to sea. Oblivious to the gunshot wound in her shoulder, her machete wound, her blood. Avoiding the city; avoiding Kaev, gasping and mumbling on the floor of the boat. Ignoring the severed hands that lay in her lap.
"We should have taken him prisoner," Soq said.
"We got what we needed from him," Ora said.
"Leaving him alive is a big risk."
"I know. But he needs to know what it's like to live with something like that."
Podlove had screamed and begged when Ora took off his left hand. Promised her mountains of money. Shown staggering repentance. She'd taken her time removing the right one, sawing slowly while Soq held him down. Then they took his tongue. Then they powdered him with clotting agents, bandaged him up, and got off him. Walked away. Carried Kaev to the boat where Ankit waited.
"Where are we going?" she'd asked then, but no one had spoken.
She asked it again now.
"Go's ship," Soq said.
"Ballsy move," Ankit said. "After you just shot her in the head in the middle of the grid. You don't think her army might have something to say about it?"
"I'm her kid," Soq said. "Everybody knows it. The next three hours will be crucial. The software war is winding down, but it'll take that long for the city's infrastructure to reassert itself. Drones, cams, witness reports—I don't think what I did will reach the ship for a little bit. I think I'll be able to muster enough important players into my corner by then. I have the advantage—no one else knows the time is right for a power play—and with Dao dead there's no one with a more legitimate claim to the throne."
"Or maybe none of those things will happen."
"Right. And I'll get killed." Soq shrugged. "Let's see what we can do about making that not happen."
Ankit recognized it, this look on Soq's face. This fearlessness that probably wasn't really fearlessness. More like—the excitement outweighed the fear, the potential positive outcomes overshadowed the potential negative ones. She didn't share it, but she'd seen it before. On her scaler friends, the ones who came from even less than she did, the orphans who hadn't lucked into a family or had ended up in an awful one. The ones who climbed the buildings beside her and stood there looking out at Qaanaaq without the sick feeling in the pit of their stomachs, the fear of falling, the fear of imprisonment, who had nothing but the open-armed embrace of the night to come, with whatever good or awful things waited for them inside it.
Soq squatted down beside Kaev. He looked at Soq, looked through them. "Shhhh," Soq said, and put their hands on their father's shoulders. So much muscle in there, so much strength. Yet here he was, helpless. Whispering a word that sounded like _God_? Chim climbed down from Ankit's shoulder to huddle up against Kaev, poking him tenderly from time to time, warming herself with his body heat.
Green flares spouted intermittently at the end of an Arm, unscheduled methane ventilations, less spectacular by daylight. Snow fell faster now.
"She wouldn't have hesitated to kill us all," Soq said, and Ankit saw tears streaming down their face. "Or most of us. I'm a maybe, Go might have thought she could scare me into silence, but you two . . ." Soq pointed to Ora and Masaaraq, grief wracking every word. "She'd have done her damnedest to put the blame somewhere else, and that would have been a lot easier if the Killer Whale Woman and her Cabinet-escapee lover were corpses who couldn't tell a different story."
"He won't care," Ora said, touching Kaev's sweaty forehead. "All he knows is, he loved her."
Soq nodded. Their face was a blotchy knot of guilt, sadness, rage. "Is there . . . can we . . . do . . . anything?"
"He'll be in a lot of pain for a very long while. When I was in the Cabinet, and my eagle was sick, I was lucky enough to know it was happening. To have some time, to help ease the transition. By bonding people, I built up the kind of bonds that soothed the trauma, but it happened slowly. Having been through what he's just been through, Kaev might not be able to last that long."
". . . Last?"
"He might choose . . . not to stick around," she said sadly. "Not to _be_. I almost did, ten times or more."
"But people need him," Ankit said. "There's a lot of people with the breaks, suffering really badly. He can help them, and they'll help him."
"We'll get to work," Soq said, pressing a cool hand to their father's hot forehead. "As soon as we get back. I know some of Go's soldiers must have the breaks. And then we'll head to Arm Eight, where lots of people have it." Soq stood, and Ankit's heart caught in her throat, to see the power that Soq radiated. Power, and something else. Lots of people had power. Fyodorovna had power; Go did. What Soq had was different. Power, and something more—the strength to do the right thing, the hard thing, the wisdom to know what that was. "We're going to fix him."
Ankit looked up from her screen and said to Soq, "They're finished. Your friend Jeong finished processing all the escapees. He's taking advantage of the chaos, got half of Go's soldiers ferrying them to their new homes. Word is, he can get pretty bossy."
"He can," Soq said. "And it's good to hear that they're already obeying him."
"The two of you might have a shot."
"We can't stay here," Masaaraq said, the first words she'd uttered since regaining consciousness.
"We can't leave," Ora said. "We have work to do. People to heal. Kaev most of all."
"And then we go," Masaaraq said.
"Why?"
Masaaraq opened her mouth, like it was the easiest question she'd ever heard, the most obvious answer, but then she said nothing. She looked out at the open sea. _She must yearn for it the way her orca does,_ Ankit thought. "We're nomads," she said, finally. "Home is where we make it. Where we're together."
"Exactly," Ora said. "That's the great gift of living life as a nomad—you don't get attached to things, don't believe that you're safe because you have a roof over your head today. You don't put your faith in a physical space when home is something you can take with you. But it also means that you accept what comes your way. You make the most of the places you end up in. And now we're here. In Qaanaaq. Maybe it isn't forever. But maybe it is."
Masaaraq did not respond. Ankit wondered if she knew already, that the argument was over, the battle won, and not by her. She loved Ora. That was the most important thing. Every other consideration was secondary to that.
"Forgive me if I'm out of line," Ankit said to her, to this person who was one of her mothers, "but you really never stopped? Twenty years, thirty years, you never said _fuck it_ and spent a decade or so living on a beach or working in a steel mill or something?"
"Odd jobs here and there," Masaaraq said. "Waiting out the winter, or saving up money when I couldn't find any syndicate shipments to raid. My hands didn't stay clean, mind you."
"Polar bear kibble is expensive," Ora said solemnly, and Masaaraq shocked everyone by chuckling.
"And you?" Ankit asked, her heart full and glorious, unspeakably happy. "Did you ever stop believing she'd come?"
"I stopped waiting for it," Ora said. "For the longest time, I was certain that it would happen. Eventually I stopped being certain. I stopped thinking to myself, _If not today, soon_. But I still woke up every morning and thought, _Maybe today._ Before I thought anything else."
Soq said, "I hated it, at first. _City Without a Map._ Didn't know what you were trying to do. But now I love it. I don't know if that's maturity, or it's just because . . . well, because the person who gave me the breaks loved it. I can feel him, sometimes. Hear him. Not like memories. Like something still alive. Is that possible?"
"They live in us," Ora said. "We carry them in our hearts, even after they are gone. Our ancestors do not depart. Our people knew that long before the breaks."
"Can I ask you a question about the broadcast?"
Ora smiled, touched her grandchild's hand. "Of course."
"Who are you talking to? Who's your audience? At first I—he—we—thought it was intended for immigrants. New arrivals. Then I thought it was about people with the breaks. Now . . ."
Ora shrugged. "I don't know, either. In the beginning I was talking to other people in the Cabinet with me. People who were struggling. Sick, sad, hungry. Then I realized . . . it's more than that."
Soq awaited additional information. Ora's smile was deep and distant. Snow made the city into a shadow, a jagged mountain ridge studded with light. The four of them huddled closer in the boat, around Kaev, each with one hand pressed to his body. They made a square, a circle, a coven, a brigade, and Ankit felt confident that there was nothing they could not accomplish when they stood together.
"Tell me a story," Soq said, leaning back to rest between Ora's legs. "I bet you're full of them."
"I am," Ora said. Her eyes were on Qaanaaq, and they saw so much more than Ankit did. Ora saw the city as a hundred different people had seen it, arriving at twilight or departing at dawn, as exiled kings and political prisoners and wide-eyed children. She shut her eyes and began to speak.
_"People would say she came to Qaanaaq in a skiff towed by a killer whale harnessed to the front like a horse . . ."_
## Acknowledgments
Books have big families. Here are the people who helped make this one happen:
Seth Fishman, again, and always. The most magnificent man and agent and writer all in one. The Gernert Company team of Will Roberts and Rebecca Gardner and Ellen Coughtrey and Jack Gernert.
Zachary Wagman, who loved this book—and whose editorial eye and ear made it much more worthy of being loved. Big love too to Ecco comrades Meghan Deans, Emma Janaskie, Miriam Parker, and Martin Wilson.
Neil Gaiman, who knows why.
Bradley Teitelbaum of White Rabbit Tattoo, who put a gorgeous permanent orca on my arm to commemorate this book. Look him up before you decide where to get your next—or first—tattoo. Kalyani-Aindri Sanchez, genius photographer, for the mind-blowingly awesome author shots. James Tracy—role model in organizing, role model in writing, and one hell of a model American.
Sheila Williams, wise and magnificent editor of _Asimov's Science Fiction,_ who published "Calved," the short story that was my first visit to the fictional floating city of Qaanaaq. And Gardner Dozois, ****Neil Clarke, and Jonathan Strahan, for including "Calved" in their best-of-the-year anthologies.
Lisa Bolekaja, who slapped some sense into me about the title of that story when absolutely everyone else wanted me to call it "Ice Is the Truth of Water" (which, besides being an objectively less-awesome title, would almost certainly have earned me a cease-and-desist letter from Ted Chiang's attorneys for straying too close to his titles "The Truth of Fact, the Truth of Feeling" and "Hell is the Absence of God").
The Wyrd Words Writing Retreat, which gave me the chance to work out the kinks on this hot mess in an enchanted castle fellowship with incredibly gifted writers and readers. Eric San Juan, Craig Laurence Gidney, Mary Anne Mohanraj, Stephen Segal, Valya Lupescu, Scott Woods, and K. Tempest Bradford all gave me vital advice and support and critique and "definitely don't do this."
My beloved Israelis, who—in addition to a lifetime of fantasticness—hosted me during the difficult time that this book was on submission and distracted me with tons of delicious food and strong coffee and tourism and love: Lynne, Avshalom, Oded, Shmuela, Leah, Miri, Liat, Amit, Gal, Dror, Mia, Yonatan, Roy, Daniel, Noam, Tamir, Romi, Ori, and Shira!
My sister, Sarah, and her husband, Eric—and my nephew, Hudson, for whose sake I'll keep on fighting like hell to keep a future like this one from coming to be.
My mother, Deborah Miller—survivor and inspiration.
Finally and forever, my husband, Juancy—firebender, crystal gem, roustabout—who I know would bend the sky and break the sea and the laws of physics to come rescue me, even if it took thirty years, no matter how many evildoers he had to slaughter in the process. Love you more than the earth we stand on.
## About the Author
**SAM J. MILLER** 's short stories have been nominated for the Nebula, World Fantasy, Theodore Sturgeon Memorial, and Locus Awards. He's the recipient of the Shirley Jackson Award and a graduate of the Clarion Writers' Workshop. His young-adult novel, _The Art of Starving,_ was released in 2017.
WWW.SAMJMILLER.COM
Discover great authors, exclusive offers, and more at hc.com.
## Copyright
This is a work of fiction. Names, characters, places, and incidents are products of the author's imagination or are used fictitiously and are not to be construed as real. Any resemblance to actual events, locales, organizations, or persons, living or dead, is entirely coincidental.
BLACKFISH CITY. Copyright © 2018 by Sam J. Miller. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-book.
FIRST EDITION
_Cover Design by Will Staehle_
* * *
Library of Congress Cataloging-in-Publication Data
Names: Miller, Sam J., author.
Title: Blackfish City : a novel / Sam J. Miller.
Description: First edition. | New York : Ecco Press, [2018] | Description based on print version record and CIP data provided by publisher; resource not viewed.
Identifiers: LCCN 2017016495 (print) | LCCN 2017031059 (ebook)
Subjects: | GSAFD: Dystopias.
Classification: LCC PS3613.I55288 (ebook) | LCC PS3613.I55288 B57 2018 (print) | DDC 813/.56—dc23
LC record available at https://lccn.loc.gov/2017016495
* * *
Digital Edition APRIL 2018 ISBN: 978-0-06-268484-4
Print ISBN: 978-0-06-268482-0
Version 04022018
## About the Publisher
**Australia**
HarperCollins Publishers Australia Pty. Ltd.
Level 13, 201 Elizabeth Street
Sydney, NSW 2000, Australia
www.harpercollins.com.au
**Canada**
HarperCollins Publishers Ltd
Bay Adelaide Centre, East Tower
22 Adelaide Street West, 41st Floor
Toronto, ON M4W 1A8, Canada
www.harpercollins.ca
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**New Zealand**
HarperCollins Publishers New Zealand
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www.harpercollins.co.nz
**United Kingdom**
HarperCollins Publishers Ltd.
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London SE1 9GF, UK
www.harpercollins.co.uk
**United States**
HarperCollins Publishers Inc.
195 Broadway
New York, NY 10007
www.harpercollins.com
## Contents
1. _Cover_
2. _Title Page_
3. _Dedication_
4. _Epigraph_
5. _Contents_
6. People Would Say
7. Fill
8. Ankit
9. Kaev
10. Ankit
11. City Without a Map: Boundaries
12. Fill
13. Soq
14. Fill
15. Ankit
16. Kaev
17. Ankit
18. City Without a Map: Dispatches from the Qaanaaq Free Press
19. Soq
20. Ankit
21. City Without a Map: Anatomy of an Incident
22. Soq
23. Kaev
24. Fill
25. Ankit
26. Kaev
27. Ankit
28. Soq
29. Kaev
30. Fill
31. Masaaraq
32. Kaev
33. Ankit
34. Soq
35. Fill
36. Soq
37. City Without a Map: Archaeology
38. Fill
39. Soq
40. Ankit
41. Kaev
42. City Without a Map: The Breaks
43. Kaev
44. Fill
45. Kaev
46. Soq
47. Ankit
48. City Without a Map: Alternative Intelligences
49. Ankit
50. Soq
51. Ankit
52. Soq
53. Kaev
54. City Without a Map: Cross Fire
55. Soq
56. Ankit
57. Kaev
58. Soq
59. Kaev
60. Ankit
61. City Without a Map: Press Montage
62. Kaev
63. Soq
64. Ankit
65. City Without a Map: Savage Bloodthirsty Monsters
66. Soq
67. Kaev
68. Ankit
69. _Acknowledgments_
70. _About the Author_
71. _Copyright_
72. _About the Publisher_
# Guide
1. Cover
2. Contents
3. Chapter 1
1. iii
2. iv
3. v
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|
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{"url":"https:\/\/spectre-code.org\/namespaceOdeIntegration.html","text":"OdeIntegration Namespace Reference\n\nFor ODE integration, we suggest using the boost libraries whenever possible. More...\n\n## Detailed Description\n\nFor ODE integration, we suggest using the boost libraries whenever possible.\n\n### Details\n\nHere we describe briefly the suggested setup to use a boost ODE integrator in SpECTRE. The boost utilities have a number of more elaborate features, including features that may result in simpler function calls than the specific recipes below. For full documentation, see the Boost odeint documentation: https:\/\/www.boost.org\/doc\/libs\/1_72_0\/libs\/numeric\/odeint\/doc\/html\/boost_numeric_odeint\/\n\nThe integration methods can be used largely as in the boost documentation. We summarize the salient points necessary to get an example running. The main steps in using a boost integrator are:\n\n\u2022 define the system\n\u2022 construct the stepper\n\u2022 initialize (might be performed implicitly, depending on the stepper)\n\u2022 perform steps\n\u2022 (compute dense output, for dense-output steppers)\n\nFor most cases the Dormand-Prince fifth order controlled, dense-output stepper is recommended. That stepper is used in the section \"SpECTRE vectors or std::array thereof in boost integrators\" below.\n\n#### Fundamental types or <tt>std::array<\/tt> thereof in fixed-step integrators\n\nLet us consider a simple oscillator system, which we'll declare as a lambda:\n\nconst auto oscillatory_array_system = [](const std::array<double, 2>& state,\nconst double \/*time*\/) noexcept {\ndt_state[0] = state[1];\ndt_state[1] = -state[0];\n};\n\nNote that the first argument is a const lvalue reference to the state type, std::array<double, 2>, the second argument is an lvalue reference for the time derivatives that is written to by the system function, and the final argument is the current time.\n\nThe construction and initialization of the stepper is simple:\n\nboost::numeric::odeint::runge_kutta4<std::array<double, 2>>\nfixed_array_stepper;\n\nFinally, we can perform the steps and examine the output,\n\nconst double fixed_step = 0.001;\nstd::array<double, 2> fixed_step_array_state{{1.0, 0.0}};\nfor (size_t i = 0; i < 1000; ++i) {\nCHECK(approx(fixed_step_array_state[0]) == cos(i * fixed_step));\nCHECK(approx(fixed_step_array_state[1]) == -sin(i * fixed_step));\nfixed_array_stepper.do_step(oscillatory_array_system,\nfixed_step_array_state, i * fixed_step,\nfixed_step);\n}\n\n#### Fundamental types or <tt>std::array<\/tt> thereof in Dense-output integrators\n\nThe dense-output and controlled-step-size ODE integrators in boost comply with a somewhat different interface, as significantly more state information must be managed. The result is a somewhat simpler, but more opaque, interface to the user code.\n\nOnce again, we start by constructing the system we'd like to integrate,\n\nconst auto quadratic_system = [](const double \/*state*\/, double& dt_state,\nconst double time) noexcept {\ndt_state = 2.0 * time;\n};\n\nThe constructor for dense steppers takes optional arguments for tolerance, and dense steppers need to be initialized.\n\nboost::numeric::odeint::dense_output_runge_kutta<\nboost::numeric::odeint::controlled_runge_kutta<\nboost::numeric::odeint::runge_kutta_dopri5<double>>>\ndense_fundamental_stepper = boost::numeric::odeint::make_dense_output(\n1.0e-12, 1.0e-12,\nboost::numeric::odeint::runge_kutta_dopri5<double>{});\ndense_fundamental_stepper.initialize(1.0, 1.0, 0.01);\n\nIt is important to take note of the tolerance arguments, as the defaults are 1.0e-6, which is often looser than we want for calculations in SpECTRE.\n\nWe then perform the step supplying the system function, and can retrieve the dense output state with calc_state, which returns by reference by the second argument.\n\ndouble state_output;\nfor(size_t i = 0; i < 50; ++i) {\nwhile(step_range.second < 1.0 + 0.01 * square(i) ) {\n}\ndense_fundamental_stepper.calc_state(1.0 + 0.01 * square(i), state_output);\nCHECK(approx(state_output) == square(1.0 + 0.01 * square(i)));\n}\n\n#### SpECTRE vectors or <tt>std::array<\/tt> thereof in boost integrators\n\nThe additional template specializations present in the OdeIntegration.hpp file ensure that the boost ODE methods work transparently for SpECTRE vectors as well. We'll run through a brief example to emphasize the functionality.\n\nconst auto oscillatory_vector_system =\n[](const std::array<DataVector, 2>& state,\nstd::array<DataVector, 2>& dt_state, const double \/*time*\/) noexcept {\ndt_state[0] = state[1];\ndt_state[1] = -state[0];\n};\nboost::numeric::odeint::dense_output_runge_kutta<\nboost::numeric::odeint::controlled_runge_kutta<\nboost::numeric::odeint::runge_kutta_dopri5<\ndense_stepper =\nboost::numeric::odeint::make_dense_output(1.0e-14, 1.0e-14,\nboost::numeric::odeint::runge_kutta_dopri5<\nconst DataVector initial_vector = {{0.1, 0.2, 0.3, 0.4, 0.5}};\ndense_stepper.initialize(\nstd::array<DataVector, 2>{{initial_vector, DataVector{5, 0.0}}}, 0.0,\n0.01);\nstd::array<DataVector, 2> dense_output_vector_state{\nstep_range = dense_stepper.do_step(oscillatory_vector_system);\nfor(size_t i = 0; i < 100; ++i) {\nwhile (step_range.second < 0.01 * i) {\nstep_range = dense_stepper.do_step(oscillatory_vector_system);\n}\ndense_stepper.calc_state(0.01 * i, dense_output_vector_state);\nfor(size_t j = 0; j < 5; ++j) {\nCHECK(approx(dense_output_vector_state[0][j]) ==\n(0.1 * j + 0.1) * cos(0.01 * i));\nCHECK(approx(dense_output_vector_state[1][j]) ==\n-(0.1 * j + 0.1) * sin(0.01 * i));\n}\n}\nstd::pair\nstd::cos\nT cos(T... args)\nsquare\nconstexpr decltype(auto) square(const T &x)\nCompute the square of x\nDefinition: ConstantExpressions.hpp:55\nstd::array< double, 2 >\nDataVector\nStores a collection of function values.\nDefinition: DataVector.hpp:42\nstd::sin\nT sin(T... args)\nstd::time\nT time(T... args)","date":"2020-08-13 09:23:34","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\": 0.7136200666427612, \"perplexity\": 11733.711814961494}, \"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-2020-34\/segments\/1596439738964.20\/warc\/CC-MAIN-20200813073451-20200813103451-00557.warc.gz\"}"}
| null | null |
{"url":"https:\/\/math.tecnico.ulisboa.pt\/seminars\/strings\/?action=show&id=5628","text":"# String Theory Seminar\n\n### A look into $3d$ modularity\n\nSince the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Motivated by the recent discovery of a new homological invariant (corresponding to the half-index of certain $3d$ $N=2$ theories), in this talk I describe the role of quantum modular forms, mock and false theta functions in the study of $3$-manifold invariants. The talk is based on 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.","date":"2019-11-17 17:11:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.27951744198799133, \"perplexity\": 1127.3172459782465}, \"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\/1573496669225.56\/warc\/CC-MAIN-20191117165616-20191117193616-00221.warc.gz\"}"}
| null | null |
Chronologie du cyclisme
1902 en cyclisme - 1903 en cyclisme - 1904 en cyclisme
Les faits marquants de l'année 1903 en cyclisme.
Par mois
Janvier
: le journal L'Auto annonce dans sa une la création du Tour de France.
Février
Mars
Avril
:le Français Hippolyte Aucouturier gagne le Paris-Roubaix.
Mai
: le Français Hippolyte Aucouturier gagne Bordeaux-Paris.
: l'Espagnol Ricardo Peris devient champion d'Espagne sur route.
: l'Italien Giovanni Erbi gagne Milan-Turin. L'épreuve ne sera pas disputée en 1904 et reprendra en 1905.
Juin
Juillet
juillet : départ du premier Tour de France. Le Français maurice Garin gagne la étape Paris-Lyon.
: le Français Hippolyte Aucouturier gagne la étape Lyon-Marseille.
: le Français Hippolyte Aucouturier gagne la étape Marseille-Toulouse.
: le Suisse Charles Laeser gagne la étape Toulouse-Bordeaux.
: le Français Maurice Garin gagne la 5éme étape Bordeaux-Nantes.
: le Français Maurice Garin gagne la 6éme et dernière étape Nantes-Paris.
: Maurice Garin remporte le Tour de France en étant leader de la première à la dernière étape.
Août
16 au : championnats du monde de cyclisme sur piste à Copenhague. Le Danois Thorvald Ellegard est champion du monde de vitesse professionnelle pour la troisième fois d'affilée. Le Britannique Arthur Reed est champion du monde de vitesse amateur.
Septembre
: l'Italien Enzo Spadoni gagne Rome-Naples-Rome.
: le Belge Arthur Vanderstuyft devient champion de Belgique sur route.
Octobre
Novembre
Décembre
Principales naissances
: Charles Pélissier, cycliste français, vainqueur de seize étapes du Tour de France, dont huit lors de l'édition 1930 († ).
: Charles Meunier, cycliste français, vainqueur du Paris-Roubaix 1929 († ).
: Ferdinand Le Drogo, cycliste français, vainqueur d'étape du Tour de France, deux fois champion de France, et vice-champion du monde sur route en 1931 († ).
: Lucien Michard, cycliste français, champion olympique de vitesse en 1924 et six fois champion du monde († ).
: Armand Blanchonnet, cycliste français, champion olympique sur route en 1924 († ).
Notes et références
Liens externes
|
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{"url":"http:\/\/experiment-ufa.ru\/Z-18=-8","text":"# Z-18=-8\n\n## Simple and best practice solution for Z-18=-8 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.\n\nIf it's not what You are looking for type in the equation solver your own equation and let us solve it.\n\n## Solution for Z-18=-8 equation:\n\nSimplifying\nZ + -18 = -8\nReorder the terms:\n-18 + Z = -8\nSolving\n-18 + Z = -8\nSolving for variable 'Z'.\nMove all terms containing Z to the left, all other terms to the right.\nAdd '18' to each side of the equation.\n-18 + 18 + Z = -8 + 18\nCombine like terms: -18 + 18 = 0\n0 + Z = -8 + 18\nZ = -8 + 18\nCombine like terms: -8 + 18 = 10\nZ = 10\nSimplifying\nZ = 10`\n\n## Related pages\n\nhow do you solve y 2x 3equatsadding simple fractions calculatorcos pi over 2y 2y 5y 0sinx 2cosx 2conver decimal to fractionhow to write 2000 in roman numeralsdivision with fractions calculatorlxxii roman numeralsfind hcf calculatorwhat is the prime factorization of 2488000000000prime factorization of 43110-22prime factors of 365adding and subtracting fractions calculator4x 5y 8y 2cosxprime factorization of 139graph 2x y 7halfnwhat is 4xbwhat are the prime factors of 375bxdxwhat is the prime factorization of 2522x 6y 12sinx graph70 prime factorizationderivative ln 2xgcf of 84 and 10839.95 to dollarswhat is 6.25 in fraction formroman numeral 1000000325-3which best represents the graph of 2x 1 527x3what is the prime factorization of 3123x 2y 13 2x 3y 12sin sinx derivativecos3xpercents to decimals calculatorderivative of 9x5x y 5 3x 2y 3ln10equation solver stepscosx derivativederivative of cos y152-15700-203x 5y 1prime factorization of 81y 2 sinx2sinx-1 04x 3y 12180-108p4904x 5y 20tanxdx0.65625 as a fractionwhat is the lcm of 9what is the prime factorization of 914000 rupees to pounds49.99 in pounds5x y 0cosx secx cosxprime factorization of 169system by substitution calculator8x x 32y x 2differentiate 2e x60 x3","date":"2018-03-21 03:38:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.3894749581813812, \"perplexity\": 8870.411159256057}, \"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-2018-13\/segments\/1521257647567.36\/warc\/CC-MAIN-20180321023951-20180321043951-00543.warc.gz\"}"}
| null | null |
{"url":"http:\/\/ferrao.org\/tags\/aeb464-molar-solubility-in-pure-water","text":"The molar solubility of AgI is 9.0 x 10 -9 mol\/L. This is known as the molar solubility of the compound in water at a given temperature (usually 25 Celsius). Now, I look at the relationship between AgCl and Cl\u00af. Be prepared! All rights reserved. When you see this, you need to assume that it is a saturated solution. By the way, the sulfate already present in solution came from some other sulfate-containing compound, say sodium sulfate, Na2SO4. Solution: 1) Here is the dissociation equation: AlPO 4 (s) \u21cc Al 3+ (aq) + PO 4 3 \u00af(aq) 2) Here is the K sp expression: K sp = [Al 3+] [PO 4 3 \u00af] 3) Keep in mind that the key point is the one-to-one ratio of the ions in solution. Calculating the molar solubility is left to the student. - Definition & Examples, The Bronsted-Lowry and Lewis Definition of Acids and Bases, Determining Rate Equation, Rate Law Constant & Reaction Order from Experimental Data, Lewis Structures: Single, Double & Triple Bonds, Acid-Base Equilibrium: Calculating the Ka or Kb of a Solution, Precipitation Reactions: Predicting Precipitates and Net Ionic Equations, CLEP Natural Sciences: Study Guide & Test Prep, Middle School Life Science: Tutoring Solution, Holt McDougal Modern Chemistry: Online Textbook Help, Praxis Chemistry (5245): Practice & Study Guide, College Chemistry: Homework Help Resource, CSET Science Subtest II Chemistry (218): Practice & Study Guide, ISEB Common Entrance Exam at 13+ Geography: Study Guide & Test Prep, Holt Science Spectrum - Physical Science with Earth and Space Science: Online Textbook Help, Biological and Biomedical Calculate the Ksp for Ag2S. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. Services, Solubility Equilibrium: Using a Solubility Constant (Ksp) in Calculations, Working Scholars\u00ae Bringing Tuition-Free College to the Community. The constituent ion concentrations that result from this molar solubility, have their solubility product described by a constant {eq}K_{sp} {\/eq} at the same temperature. Calculate the value of Ks under these conditions. Your dashboard and recommendations. The \"A\" is the metal cation portion and \"X\" is the anion portion, where \"n\" is a positive integer value: {eq}AX (s) \\leftrightharpoons A^{n+} + X^{n-} \\\\ Calculate the K sp for Ba 3 (PO 4) 2. Anything else makes the problem unworkable and that is not the intent of the question writer. The Ksp for MgF2 is 6.4 x 10-9. Solution for The molar solubility of Ag2S is 1.26 x 10-16 mol L-1 in pure water. Based on this 1:1 dissociation stoichiometry, the {eq}K_{sp} {\/eq} expression in terms of the molar solubility in pure water \"x\", is written as: From this relation we see that the salt compound in the list with the highest {eq}K_{sp} {\/eq} value, will have the highest molar solubility. Therefore we do not have to perform individual calculations for each compound. This \u2026 Home. That allows this equation: Example #3: Calculate the molar solubility of barium sulfate, Ksp = 1.07 x 10\u00af10. 1.12 x 10-8 c. 8.00\u2026 3) Keep in mind that the key point is the one-to-one ratio of the ions in solution. {\/eq}, b. This dissociates: Note how I ignored the s in (0.0153 + s). True or false? Show transcribed image text . 4) We have to reason out the values of the two guys on the right. A) PbS, Ksp=9.04 X 10-29 B) PbSO4, Ksp = 1.82 X 10-8 C) MgCO3, Ksp = 6.82 X 10-6 D) FeS, Ksp= 3.72 X 10-19 E) Agi, Ksp = 8.51 X 10-17. All other trademarks and copyrights are the property of their respective owners. An unsaturated solution is a solution in which all solute has dissolved. We can simply consider the following solubility equilibrium of a generic 1:1 salt compound. Study Guides. Now, solve for s: s 2 = 8.5 x 10 -17. s = $\\sqrt {8.5 \\times 10^ {-17} }$ s = 9.0 x 10 -9 mol\/L. 1) The dissociation equation and the Ksp expression: Remember, this is the answer because the dissolved ions and the solid are also in a one-to-one molar ratio. Select one: a. Calculate its solubility in moles per liter. The solubility of CaSO4 is measured and found to... Titration of a Strong Acid or a Strong Base, LeChatelier's Principle: Disruption and Re-Establishment of Equilibrium, Collision Theory: Definition & Significance, Gibbs Free Energy: Definition & Significance, The Common Ion Effect and Selective Precipitation, Acid-Base Buffers: Calculating the pH of a Buffered Solution, The pH Scale: Calculating the pH of a Solution, Buffer System in Chemistry: Definition & Overview, Ionic Equilibrium: Definition & Calculations, Dalton's Law of Partial Pressures: Calculating Partial & Total Pressures, What is Salt Hydrolysis? Example #5: Silver azide has the formula AgN3 and Ksp = 2.0 x 10\u00af8. 3.78 x 10-12 b. Our experts can answer your tough homework and study questions. What is its molar solubility in pure water? Get the detailed answer: What is the molar solubility of AgI in pure water? Ksp (AgI) = 8.51 \u00d7 10-17. Note azide, a fairly uncommon polyatomic ion. {eq}\\boxed{PbSO_4; K_{sp} = 1.82 \\times 10^{-5}} Which of the following compounds will have the highest molar solubility in pure water? The reasons behind this are complex and beyond the scope of the ChemTeam's goals for this web site. One last thing. Get the detailed answer: What is the molar solubility of AgI in pure water? All of the salt compounds shown dissociate in a 1:1 ratio of ions. Example #6: Calculate the [Ba2+] if the [SO42\u00af] = 0.0153 M. Ksp = 1.07 x 10\u00af10. I see that it is also a 1:1 molar ratio, leading me to this: I am now ready to substitute into the Ksp expression. The ammonium phosphate polyatomic ion, NH4PO42\u00af, is an uncommon one. {eq}PbSO_4; K_{sp} = 1.82 \\times 10^{-5} It could show up on the test! So, the molar solubility of AgCl is 1.33 x 10\u00af5 moles per liter. a. Example #2: Aluminum phosphate has a Ksp of 9.83 x 10\u00af21. This means that the two ions are equal in their concentration. a. BaCrO4; molar solubility = 1.08 * 10-5 M b. Ag2SO3; molar solubility = 1.55 * 10-5 M c. Pd(SCN)2; molar solubility \u2026 Answer to: Which of the following compounds will have the highest molar solubility in pure water? In a saturated aqueous solution of an ionic (salt) compound, there is a certain dissolved molarity of this compound present. That's the value that I want to determine. Thanks. Example #1: Silver chloride, AgCl, has a Ksp = 1.77 x 10\u00af10. Booster Classes. Homework Help. We will completely ignore the sodium ion, because it plays no role in the Ksp equilibrium. 1) The compound in solution is BaSO4. Sciences, Culinary Arts and Personal I hope I'm not too insulting when I emphasize both sides. {\/eq}. Here the choice is lead (II) sulfate: a. Personalized courses, with or without credits. This problem has been solved! The formula for K sp is: K sp = [Ag + ] [I \u2013] K sp = s 2 = 8.5 x 10 -17. where s is the concentration of each ion at equilibrium. That's because s is very small compared to 0.0153. {\/eq}. 3.7 million tough questions answered. See the answer. Switch to.","date":"2021-04-21 04:36:22","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\": 0.546946108341217, \"perplexity\": 4328.173985179343}, \"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-2021-17\/segments\/1618039508673.81\/warc\/CC-MAIN-20210421035139-20210421065139-00187.warc.gz\"}"}
| null | null |
since it doesn't go back much.
Welcome Nancy ! We have a very distant cousin who seems to have made it his lifes work to know everything there is to know about our family and he has shared his voluminous works with all the branches. Plus my grandma was an amateur historian as well so we are lucky to know so much thanks to their efforts. I certainly hope there's a little bit of Hannah in me. What courage she showed.
the Monmouth Rebellion and now you live in Monmouth County?
I thought that was funny too actually…….
I'm currently doing post-grad studies and discovered a friend is doing his PhD thesis on Hannah Hewling.
|
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| 7,993
|
Q: What's difference in using else-if vs or in JavaScript I was creating a rock-paper-scissor game to test my knowledge of functions.
I had written this code to see whether the user had given a valid input of rock, paper or scissor:
const getUserChoice = userInput => {
userInput = userInput.toLowerCase();
if(userInput === 'rock'){
return userInput;
} else if(userInput === 'paper') {
return userInput;
} else if(userInput === 'scissor'){
return userInput;
} else {
console.log('Invalid Option');
}
};
When I tested the last else statement by doing a console.log like so:
console.log(getUserChoice('water'));
It printed out rock instead of printing "Invalid Option".
When I checked the solution I saw this:
if (userInput === 'rock' || userInput === 'paper' || ... ) {
return userInput;
} else {
console.log('Error!');
}
So what's the difference between using an else-if to accomplish this task vs using the || operator?
Edit:
I made a mistake of using = instead of ===, so it got fixed, but i would still like to know if there is any difference, and which is more efficient for particular use cases.
A: The issue is with the = vs === operator.
=== is used to check for equality, and = is for assignment.
When you write
if(userInput = 'rock')
it means userInput is assigned the value of rock.
What you really want is if(userInput === 'rock')
Regarding the || operator (also called OR), what happens is a short-circuit operation. If you write A || B || C, the below happens:
*
*A is evaluated. If value is true, B and C are not evaluated.
*If A is false, then B is evaluated.
*If B is true, then C is not evaluated.
*Otherwise, C is evaluated and value of C is returned.
A: There is no functional difference between using || and using if/else.
In JavaScript, the || operator works like this:
For a || b, check if a is truthy. If yes, resolve to a. Otherwise, resolve to b.
Now, as you may have noticed, the wording of that is pretty much identical to how you would describe the behavour of an if/else conditional block. Functionally, there is no difference in behaviour.
On a related note, the && operator works in a similar way, except then only if a is truthy will a && b resolve to b. If you ever look at minified code, you will sometimes see && used to replace an if statement, because a && b is functionally identical to if (a) { b; } but takes up fewer characters.
I wouldn't recommend using that approach in un-minified code, since it's definitely harder to follow, but it does take up fewer characters without functioning differently, which is why it's used by minifiers. Kind of similar to how they often use !0 instead of true.
A: The difference is that the if else statement is only checked/run if the previous if/if elses are false. The or (||) statement is true if any of the single conditions are true
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,535
|
While depression is considered the most common mental illness regardless of age, gender, ethnicity, and socioeconomic status, compared to research on the general population, depression among psychologists has received little attention. However, as they are one of the major mental health care professionals, psychologists' mental health could greatly affect their clients' mental health, which raises competency and ethical concerns regarding their work as clinicians. In order to learn more about depression in this group, questionnaires were mailed to 800 randomly selected psychologists in the state of Colorado to examine the prevalence of depression among psychologists, how they dealt with their own depression, their concerns regarding competency and ethics, and the existential meaning of their depression. Nine participants responded to an invitation to be individually interviewed regarding their personal experience of depression.
The results indicated a higher prevalence of depression among psychologists than in the general population. However, no significant gender differences regarding the prevalence of depression were found. The majority of psychologists reported being aware of their own depression, however, regardless of their expertise, several reported that they were not aware of their own depression. The majority of psychologists accepted the diagnosis either positively or objectively. However, some accepted it negatively, expressing such feelings as shame. The majority of psychologists who experienced depression eventually sought treatment. However, one third of the psychologists did not seek treatment, but instead utilized their own coping skills. A few psychologists mentioned a confidentiality concern as a reason for not seeking treatment.
Regarding competency, most psychologists did not think their own depression interfered with their professional competence. Regarding ethical concerns, nearly half of the psychologists answered it is ethical to practice as long as the psychologist is aware of his/her symptoms and limitations. One third of the psychologists answered the capacity to practice depends on the severity of the depression. Although most psychologists rated their experience of depression positively, stating that it enhanced their understanding of their clients, personal interviews revealed that a stigma about depression or mental illness in general still exists among psychologists. Most of the interviewed psychologists stated that they were very cautious about disclosing their experience of depression to their colleagues due to a concern for their reputation.
Murata, Akira, "Exploration Of The Meaning Of Depression Among Psychologists: A Quantitative And Qualitative Approach" (2010). Electronic Theses and Dissertations. 462.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 5,371
|
Micropholis gardneriana är en tvåhjärtbladig växtart som först beskrevs av A.Dc., och fick sitt nu gällande namn av Jean Baptiste Louis Pierre. Micropholis gardneriana ingår i släktet Micropholis och familjen Sapotaceae. Inga underarter finns listade i Catalogue of Life.
Källor
Ljungordningen
gardneriana
|
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| 4,382
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const conf = require('../conf/conf')({build: true})
const Middleware = require('../middleware/index')
const MemoryTree = require('memory-tree').default
const emptyFn = () => console.log('build finished!')
module.exports = (callback = emptyFn, opt = {watch: 0, timeout: 1000}) => {
const middleware = Middleware(conf)
const memory = MemoryTree({
root: conf.root,
watch: !!opt.watch,
dest: conf.output,
onSet: middleware.onSet,
onGet: middleware.onGet,
buildWatcher: (pathname, type, build) => {
middleware.buildWatcher(pathname, type, build)
memory.store.onBuildingChange(function (building) {
if (!build) {
memory.output('')
}
})
},
buildFilter: middleware.buildFilter,
outputFilter: middleware.outputFilter
})
if (conf.output) {
memory.input('').then(() => {
memory.output('')
}).catch(err => console.trace(err))
}
}
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| 7,923
|
\section{euclidean Jordan Algebras}\label{S:JA}
The materials reviewed in this and the next sections can be found in Refs. \cite{Koecher99, FK91}.
Recall that an {\bf algebra} $V$ over a field $\mathbb F$ is a vector
space over $\mathbb F$ together with a $\mathbb F$-bilinear map $V\times
V\to V$ which maps $(u, v)$ to $uv$. This $\mathbb F$-bilinear map can
be recast as a linear map $V\to \mathrm{End}_{\mathbb F}(V)$ which maps $u$
to $L_u$: $v\mapsto uv$.
We say that algebra $V$ is {\it commutative} if $uv=vu$ for any $u,
v\in V$. As usual, we write $u^2$ for $uu$ and $u^{m+1}$ for $uu^m$
inductively.
\begin{Def}A {\bf Jordan algebra} over $\mathbb F$ is just a
commutative algebra $V$ over $\mathbb F$ such that
\begin{eqnarray}\label{JA}
[L_u, L_{u^2}]=0
\end{eqnarray} for any $u\in V$.
\end{Def}
Here are some {\it basic facts} about the Jordan algebra $V$ over $\mathbb
F$:
\begin{itemize}
\item $u^r u^s=u^{r+s}$ for any $u\in V$ and
any integers $r,s\ge 1$.
\item $L_{u^m}$ ($m\ge 1$) is a polynomial in $L_u$
and $L_{u^2}$, for example,
\begin{eqnarray}\label{ID:1}
L_{u^3}=3L_{u^2}L_u-2L_u^3.
\end{eqnarray}
\item For any $u, v, z\in V$, we have
\begin{eqnarray}\label{usefulid}
[[L_u, L_v], L_z]=L_{u(vz)-v(uz)}
\end{eqnarray}
provided that $\mbox{Char}(\mathbb F)\neq 2$.
\end{itemize}
As the first example, we note that $\mathbb F$ is a Jordan algebra over
$\mathbb F$. Here is a {\it recipe} to produce Jordan algebras. Suppose
that $\Phi$ is an associative algebra over field $\mathbb F$ with
characteristic $\neq 2$, and $V\subset \Phi$ is a linear subspace of
$\Phi$, closed under square operation, i.e, $u\in V$ $\Rightarrow$
$u^2\in V$. Then $V$ is a Jordan algebra over $\mathbb F$ under the
Jordan product
$$
uv:={(u+v)^2-u^2-v^2\over 2}.
$$
Applying this recipe, we have the following Jordan algebras over
$\mathbb R$:
\begin{enumerate}
\item The algebra $\Gamma(n)$. Here $\Phi=\mathrm{Cl}(\mathbb R^n)$---the Clifford algebra of $\mathbb
R^n$ and $V=\mathbb R\oplus \mathbb R^n$.
\item The algebra ${\mathcal H}_n(\mathbb R)$. Here $\Phi=M_n(\mathbb R)$---the algebra of real $n\times n$-matrices and $V\subset \Phi$ is the set of symmetric
$n\times n$-matrices.
\item The algebra ${\mathcal H}_n(\mathbb C)$. Here $\Phi=M_n(\mathbb C)$---the algebra of complex $n\times n$-matrices (considered as an algebra over $\mathbb R$)
and $V\subset \Phi$ is the set of Hermitian $n\times n$-matrices.
\item The algebra ${\mathcal H}_n(\mathbb H)$. Here $\Phi=M_n(\mathbb H)$---the algebra of quaternionic $n\times n$-matrices (considered as an algebra over $\mathbb R$)
and $V\subset \Phi$ is the set of Hermitian $n\times n$-matrices.
\end{enumerate}
The Jordan algebras over $\mathbb R$ listed above are {\bf
special} because they can be derived from associated
algebras via the above recipe. Let us denote by ${\mathcal H}_n(\mathbb O)$ the algebra for which
the underlying real vector space is the set of Hermitian
$n\times n$-matrices over octonions $\mathbb O$ and the product is the
symmetrized matrix product. One can show that ${\mathcal
H}_n(\mathbb O)$ is a Jordan algebra if and only if $n\le 3$.
\vskip 10pt Any Jordan algebra $V$ comes with a canonical symmetric
bilinear form \begin{eqnarray}\tau(u, v):=\mbox{the trace of
$L_{uv}$}.\end{eqnarray} Thanks to Eq. (\ref{usefulid}), $L_u$ is self-adjoint with respect to $\tau$.
We say that the Jordan algebra $V$ is {\it semi-simple} if the symmetric
bilinear form $\tau$ is non-degenerate. We say that the Jordan algebra
$V$ is {\it simple} if it is semi-simple and has no ideal other than
$\{0\}$ and $V$ itself.
By definition, an {\bf euclidean Jordan algebra}\footnote{Called
{\it formally real Jordan algebra} in the old literature.} is a
real Jordan algebra with an identity element and the positive definite symmetric bilinear form $\tau$. Therefore, an euclidean Jordan algebra is
semi-simple and can be uniquely written as the direct sum of simple
ideals --- ideals which are simple as Jordan algebras.
\begin{Th}[Jordan, von Neumann and Wigner]\label{JAclassification}
The complete list of simple euclidean Jordan algebras are
\begin{enumerate}
\item The algebra $\Gamma(n)=\mathbb R\oplus \mathbb R^n$ ($n\ge 2$).
\item The algebra ${\mathcal H}_n(\mathbb R)$ ($n\ge 3$ or $n=1$).
\item The algebra ${\mathcal H}_n(\mathbb C)$ ($n\ge 3$).
\item The algebra ${\mathcal H}_n(\mathbb H)$ ($n\ge 3$).
\item The algebra ${\mathcal H}_3(\mathbb O)$.
\end{enumerate}
\end{Th}
Note that the last one is called the {\bf exceptional type} because it
cannot be obtained from an associative algebra via the recipe we mentioned early. Note also that
$\Gamma(1)$ is not simple, ${\mathcal H}_1(\mathbb F)=\mathbb R$ is the
only associative simple euclidean Jordan algebra, and we have
various isomorphisms: $\Gamma(2)\cong {\mathcal H}_2(\mathbb R)$,
$\Gamma(3)\cong {\mathcal H}_2(\mathbb C)$, $\Gamma(5)\cong {\mathcal
H}_2(\mathbb H)$, $\Gamma(9)\cong {\mathcal H}_2(\mathbb O)$. The first family is called the {\bf Dirac type} and the next three families are called the {\bf hermitian type}.
\vskip 10pt Throughout the remainder of this paper, we always use
$V$ to denote a simple euclidean Jordan algebra, and $e$ to denote
the identity element of $V$.
The notion of {\bf trace} is valid for Jordan algebras. For the
simple euclidean Jordan algebras, the trace can be easily described:
For $\Gamma(n)$, we have
$$
\mr{tr}\,(\lambda, \vec u)=2\lambda,
$$
and for all other types, it is the usual
trace on matrices.
Recall that $L_u$: $V\to V$ is the multiplication by $u$ and its
trace is denoted by $\mathrm{Tr}(L_u)$. It is a fact that $${1\over \dim
V}\mathrm{Tr}(L_u)={1\over \rho}\mr{tr}\, (u)$$ where $\rho:=\mr{tr}\,(e)$ is the {\bf
rank} of $V$.
For the {\bf inner product} on $V$, we take
\begin{eqnarray}\fbox{$\langle u\mid v\rangle :={1\over
\rho}\mr{tr}\,(uv)$}\end{eqnarray} so that $e$ becomes a unit vector. One
can check that $L_u$ is self-adjoint with respect to this inner
product: $ \langle v u\mid w\rangle = \langle v\mid u w\rangle$,
i.e., $L_u'=L_u$.
We shall use Dirac's bracket notations: for $u, v\in V$, we
declare that $\mid u\rangle$ is the vector $u$, $\langle u\mid$
is the co-vector sending $z\in V$ to $\langle u\mid z\rangle$, and $\mid u\rangle\langle v\mid$ is the endomorphism
sending $z\in V$ to $\langle v\mid z\rangle u$. Since $L_u'=L_u$, the dual of $L_u$, denoted by $L_u^*$, maps $\langle z\mid$ to $-\langle L_u z\mid$.
We shall adopt the following conventions: $x$ is reserved for a point in
the smooth space $V$, and $u$, $v$, $z$, $w$ are reserved for vectors in
the vector space $V$. We shall fix an orthonomal basis $\{e_\alpha\}$ for $V$, with respect to which, we can express $x$ as $\sum_\alpha x^\alpha e_\alpha$. For simplicity, we shall write $\sum_\alpha e_\alpha {\partial \over \partial x^\alpha}$ as ${/\hskip -5pt
\partial}$.
From here on, we shall also use $V$ to denote the euclidean space
with underlying smooth space $V$ and Riemannian metric $ds^2_E$:
\begin{eqnarray}
T_xV\times T_xV & \to & \mathbb R \cr ((x, u), (x, v)) &\mapsto &
\langle u\mid v\rangle.
\end{eqnarray}
For $u\in V$, we use $\hat L_u$ to denote the vector field on $V$
whose value at $x\in V$ is $(x, -ux)\in T_xV$. Viewed as a differential operator, we have
$$
\hat L_u=-\langle ux\mid {/\hskip -5pt
\partial}\rangle.
$$
The following theorem is extremely useful when we do concrete
computations with Jordan algebras.
\begin{Th}[Jordan Frame]
Let $V$ be a simple euclidean Jordan algebra of rank $\rho$, $x_0\in
V$. Suppose that $x_0$ is non-zero and satisfies equation $x_0^2=\mr{tr}\, x_0\, x_0$. Then there is an
orthogonal basis for $V$: $e_{11}$, \ldots, $e_{\rho\rho}$,
$e_{ij}^\mu$ ($1\le i < j\le \rho$, $1\le\mu \le \delta$) such that
1) each basis vector has length $1\over \sqrt{\rho}$;
2) $e_{ii}^2=e_{ii}$, $(e_{jk}^\mu)^2={1\over 2}(e_{jj}+e_{kk})$;
3) $e_{ii}e_{ij}^\mu=e_{jj}e_{ij}^\mu={1\over 2} e_{ij}^\mu$;
4) $e_{ii}e_{jk}^\mu=0$ if $i\not\in\{j,k\}$, and $e_{ii}e_{jj}=0$ if $i\neq j$;
5) $e_{11}+\cdots+e_{\rho\rho}=e$;
6) $\mr{tr}\, e_{ii}=1$, $\mr{tr}\, e_{ij}^\mu=0$;
7) $x_0$ is a multiple of $e_{11}$.
\end{Th}
In the above theorem, $\{e_{11}, \ldots, e_{\rho\rho}\}$ is called a
{\bf Jordan frame} and the parameter $\delta$ is called the {\bf
degree} of $V$. If $i<j$, we use $V_{ij}$ to denote
$\mathrm{span}_{\mathbb R}\{e_{ij}^\mu\mid 1\le \mu \le \delta\}$ --- the $(i,j)$-{\bf Pierce
component}.
\begin{rmk}
Simple euclidean Jordan algebras are isomorphic if and only if they
have the same rank $\rho$ and same degree $\delta$, as one can verify
from the table below:
\begin{displaymath}
\begin{array}{|c|c|c|c|c|c|}
\hline
V & \Gamma(n) & {\mathcal H}_n(\mathbb R) & {\mathcal H}_n(\mathbb C) &{\mathcal H}_n(\mathbb H) & {\mathcal H}_3(\mathbb O)\\
\hline
\rho & 2 & n & n & n & 3\\
\hline
\delta & n-1 & 1 & 2 & 4 & 8\\
\hline
\end{array}
\end{displaymath}
It is also clear from this table and the above theorem that, for the
simple euclidean Jordan algebras, there is one with rank-one,
infinitely many with rank two, four with rank three, and three with
rank four or higher.
\end{rmk}
\section{Tits-Kantor-Koecher Construction}\label{S:TKK}
The Tits-Kantor-Koecher construction yields a simple real Lie
algebras from a simple euclidean Jordan algebra. We begin with the
introduction of the {\bf Jordan triple product}:
$$
\{uvw\} :=u(vw)+w(vu)-(uw)v.
$$ One can check that the Jordan triple product satisfies the following identities:
\begin{eqnarray}\label{JTP}
\{wyz\} & = & \{zyw\},\cr \{uv\{zwy\}\} & = &
\{\{uvz\}wy\}-\{z\{vuw\}y\}+\{zw\{uvy\}\}.
\end{eqnarray}
For a pair $(u,v)\in V\times V$, we introduce the linear map
$S_{uv}$: $V\to V$ by declaring that $$S_{uv}(z):=\{uvz\}.$$ Then
Eq. (\ref{JTP}) is equivalent to the commutation relation
\begin{eqnarray}\label{strR}[S_{uv},S_{zw}] =
S_{\{uvz\}w}-S_{z\{vuw\}}.
\end{eqnarray} Therefore, the span of these
$S_{uv}$, denoted by $\mathfrak{str}(V)$ or simply $\mathfrak{str}$, becomes a real Lie algebra
--- the {\bf structure algebra} of $V$. One can check that
$S_{ue}=S_{eu}=L_u$ and $S_{uv} = [L_u, L_v]+L_{uv}$. Then $S_{uv}'=S_{vu}$, hence
$S_{uv}^*(\langle z\mid)=-\langle S_{vu}(z)\mid$. Therefore, in view of Eq. (\ref{strR}), we see that $z$ and $w$ in $S_{zw}$ transform under $\mathfrak{str}(V)$ as a vector
and a co-vector respectively.
Eq. (\ref{usefulid}) amounts to saying that that $[L_u, L_v]$: $V\to V$ is a
derivation; in fact, any derivation is a linear combination of derivations of this form. The {\bf
derivation algebra} of $V$, denoted by $\mathfrak{der}(V)$ or simply $\mathfrak{der}$, is then a Lie
subalgebra of the structure algebra.
\vskip 10pt The {\bf conformal algebra} of $V$, denoted by
$\mathfrak{co}(V)$ or simply $\mathfrak{co}$, is a further extension of $\mathfrak{str}(V)$. As a vector space, $$\mathfrak{co}(V)=V\oplus \mathfrak{str}(V)\oplus V^*.$$
In the following we shall rewrite $u\in V$ as $X_u$ and $\langle v\mid\, \in V^*$ as $Y_v$.
\begin{Th}[Koecher]\label{koecher} Let $V$ be a simple euclidean Jordan algebra, then
$\mathfrak{co}(V)=V\oplus \mathfrak{str}(V)\oplus V^* $ becomes a simple
real Lie algebra with the defining TKK commutation relations
\begin{eqnarray}\label{CONFA}
\fbox{$\begin{matrix}[X_u, X_v] =0, \quad [Y_u, Y_v]=0, \quad [X_u,
Y_v] = -2S_{uv},\cr\\ [S_{uv},
X_z]=X_{\{uvz\}}, \quad [S_{uv}, Y_z]=-Y_{\{vuz\}},\cr\\
[S_{uv}, S_{zw}] = S_{\{uvz\}w}-S_{z\{vuw\}}
\end{matrix}$}
\end{eqnarray}for $u$, $v$, $z$, $w$ in $V$.
\end{Th}
It is not hard to remember the TKK commutation relations in the
above theorem if one notices that, under
the action of the structure algebra, $u$ and
$v$ in $S_{uv}$ transform as a vector and a co-vector respectively. It follows from the TKK commutation relations that
\begin{eqnarray}\label{derivation}
[D, L_u]=L_{Du}, \quad [D, X_u]=X_{Du}, \quad [D, Y_u]=Y_{Du}
\end{eqnarray}
for any derivation $D\in\mathfrak{der}$ and any $u\in V$. In particular, we have $[D, L_e]=0$, $[D, X_e]=0$ and $[D, Y_e]=0$.
Let $V$ be a simple euclidean Jordan algebra with an orthonormal
basis $\{e_\alpha\}$ chosen. Upon recalling the definition of ${/\hskip -5pt
\partial} $, we can introduce differential operators
$$
\hat S_{uv}=-\langle S_{uv}(x)\mid {/\hskip -5pt
\partial} \rangle, \quad \hat X_u=-\langle u\mid {/\hskip -5pt
\partial} \rangle, \quad \hat Y_v=-\langle \{xvx\}\mid {/\hskip -5pt
\partial} \rangle.
$$
One can verify that $\hat S_{ue}=\hat S_{eu}=\hat L_u$ and $\hat S_{uv}=[\hat L_u, \hat L_v]+\hat L_{uv}$.
The following proposition implies that the conformal algebra of $V$ can be
realized as a Lie subalgebra of the Lie algebra of vector fields on
$V$.
\begin{Prop}\label{prop1}
The TKK commutation relations can be realized
by vector fields $\hat S_{uv}$, $\hat X_z$, $\hat Y_w$.
\end{Prop}
\begin{proof}
It is clear that $[\hat X_u, \hat X_v]=0$.
\begin{eqnarray}
[\hat S_{uv}, \hat S_{zw}] &=&[-\langle S_{uv}(x)\mid {/\hskip -5pt
\partial} \rangle,
-\langle S_{zw}(x)\mid {/\hskip -5pt
\partial} \rangle ]\cr
&=&-\langle S_{uv}S_{zw}(x)\mid {/\hskip -5pt
\partial} \rangle + \langle S_{zw}S_{uv}(x)\mid {/\hskip -5pt
\partial} \rangle\cr
&=& -\langle [S_{uv},S_{zw}](x)\mid {/\hskip -5pt
\partial} \rangle\cr
&=&-\langle (S_{\{uvz\}w}-
S_{z\{vuw\}})(x)\mid {/\hskip -5pt
\partial} \rangle\cr
&=&\hat S_{\{uvz\}w} - \hat S_{z\{vuw\}}.\nonumber
\end{eqnarray}
\begin{eqnarray}
[\hat S_{uv}, \hat X_z] &=&[-\langle S_{uv}(x)\mid
{/\hskip -5pt
\partial}\rangle, -\langle z\mid {/\hskip -5pt
\partial}\rangle
]\cr
&=&- \langle S_{uv}(z)\mid
{/\hskip -5pt
\partial}\rangle=\hat X_{\{uvz\}}.\nonumber
\end{eqnarray}
\begin{eqnarray}
[\hat X_u, \hat Y_v]&=&[-\langle u\mid
{/\hskip -5pt
\partial}\rangle, -\langle \{xvx\}\mid {/\hskip -5pt
\partial}\rangle]\cr
&=&2\langle \{uvx\}\mid
{/\hskip -5pt
\partial}\rangle=-2\hat S_{uv}.\nonumber
\end{eqnarray}
\begin{eqnarray}
[\hat S_{uv}, \hat Y_z] &=&[-\langle S_{uv}(x)\mid
{/\hskip -5pt
\partial}\rangle, -\langle \{xzx\}\mid {/\hskip -5pt
\partial}\rangle
]\cr &=&2\langle \{S_{uv}(x)zx\}\mid
{/\hskip -5pt
\partial}\rangle -\langle S_{uv}(\{xzx\})\mid {/\hskip -5pt
\partial}\rangle
\cr
&=&2\langle S_{xz}S_{uv}(x)\mid {/\hskip -5pt
\partial}\rangle -\langle S_{uv} S_{xz}(x)\mid {/\hskip -5pt
\partial}\rangle\cr
&=&\langle S_{xz}S_{uv}(x)\mid {/\hskip -5pt
\partial}\rangle-\langle [S_{uv}, S_{xz}](x)\mid {/\hskip -5pt
\partial}\rangle\cr
&=&\langle S_{xz}S_{uv}(x)-S_{\{uvx\}z}(x)\mid {/\hskip -5pt
\partial}\rangle+\langle S_{x\{vuz\}}(x)\mid {/\hskip -5pt
\partial}\rangle\cr
&=& -\hat Y_{\{vuz\}}.\nonumber
\end{eqnarray}
Finally,
\begin{eqnarray}
[\hat Y_u, \hat Y_v] &=&[-\langle \{x ux \}\mid {/\hskip -5pt
\partial}\rangle -\langle \{x vx \}\mid {/\hskip -5pt
\partial}\rangle]\cr
&=& 2\langle \{ \{x ux \}vx\}\mid {/\hskip -5pt
\partial}\rangle - <u\leftrightarrow v>\cr
&=& 2\langle [S_{x v},S_{x u}] x\mid {/\hskip -5pt
\partial}\rangle\\
&=&\langle [S_{x v},S_{x u}] x\mid {/\hskip -5pt
\partial}\rangle - <u\leftrightarrow v>\cr
&=& \langle (S_{\{x vx\}u}-S_{x \{vxu\}})x\mid {/\hskip -5pt
\partial}\rangle - <u\leftrightarrow v>\cr
&=& \langle S_{\{x vx\}u}x\mid {/\hskip -5pt
\partial}\rangle - <u\leftrightarrow v>\quad \mbox{because $\{ux v\}=\{vx u\}$}\cr&=& - \langle [S_{xv}, S_{xu}]x\mid {/\hskip -5pt
\partial}\rangle \\
&=&0,\nonumber
\end{eqnarray}
because it is equal to the negative half of itself. Here, $<u\leftrightarrow v>$ means the term same as the one on the left except that $u$ and $v$ are interchanged.
\begin{comment}
ii) As differential operators on $V$,
\begin{eqnarray}
\hat L_{\{uzv\}}=\hat S_{u(zv)}+\hat S_{v(zu)}-\hat S_{(uv)z}.
\end{eqnarray}
ii) It suffices to show that
\begin{eqnarray}\label{smallid}
L_{\{uzv\}}= S_{u(zv)}+ S_{v(zu)}- S_{(uv)z}
\end{eqnarray}
or equivalently
\begin{eqnarray}
[L_u, L_{zv}]+[L_v, L_{zu}]-[L_{uv}, L_z]=0.\nonumber
\end{eqnarray} This last identity is valid because it is the polarization of identity
$[L_{u^2}, L_u]=0$.
\end{comment}
\end{proof}
\begin{comment}
\begin{rmk}
Viewing $\hat S_{uv}$, $\hat X_u$ and $\hat X_v$ as vector fields on $V$, since $T_xV=\{x\}\times V$ for any $x\in V$, we have
$$
\hat S_{uv} \mid _x=(x, -S_{uv}(x)), \quad \hat X_{u} \mid _x=(x, -u), \quad \hat Y_v\mid _x=(x, -\{xvx\}).
$$
\end{rmk}
\begin{rmk}
The assignment of $\hat S_{uv}$ to $S_{uv}$, $\hat X_u$ to $X_u$ and $\hat Y_v$ to $Y_v$ defines an action of $\mathfrak{co}(V)$ on $C^\infty(V)$. This action can be lifted to an action
of $\mathfrak{co}(V)$ on $C^\infty(T^*V)$:
\[
\begin{CD}
C^\infty(V) @> \pi^* >> C^\infty(T^*V) \\
@ V{\hat Z} VV @ VV {\{h_Z,\, \}} V \\
C^\infty(V) @>> \pi^* > C^\infty(T^*V). \\
\end{CD}
\]
Here $\pi$: $T^*V\to V$ is the bundle projection map, $Z$ is an element of $\mathfrak{co}$, $\{\, , \,\}$ is the Poisson bracket, and function $h_Z$ depends on $Z$ linearly such that
\begin{eqnarray}
h_{Z}(x, p)=-p(\hat Z\mid_x).
\end{eqnarray}
Note that, the action on $C^\infty(T^*V)$ comes from a underlying Hamiltonian action
on $T^*V$.
\end{rmk}
\end{comment}
We conclude this section with a few more terminologies and notations. Let $\Omega$ be
the {\bf symmetric cone} of $V$ and $\mathrm{Str}(V)$ be the {\bf structure group} of $V$. By definition,
$\Omega$ is the topological interior of $\{x^2\mid x\in V\}$ and
$$
\mathrm{Str}(V)=\{g\in GL(V)\mid P(gx) =gP(x)g' \quad \forall x\in V\}.
$$
Here $P(x):=2L_x^2-L_{x^2}$ and it is called the {\bf quadratic representation} of $x$.
We write $V^{\mathbb C}$ for the complexification of $V$, denote by $T_\Omega$ the tube domain associated with $V$. By definition, $T_\Omega=V\oplus i\Omega$ --- a domain inside $V^{\mathbb C}$. We say that map $f$: $T_\Omega\to T_\Omega$ is a holomorphic automorphism of $T_\Omega$ if $f$ is invertible and both $f$ and $f^{-1}$ are holomorphic. We use $\mathrm{Aut}(T_\Omega)$ to denote the group of holomorphic automorphisms
of $T_\Omega$.
It is a fact that both $\mathrm{Str}(V)$ and $\mathrm{Aut}(T_\Omega)$ are Lie groups. The Lie algebra of $\mathrm{Str}(V)$ is $\mathfrak{str}(V)$, the Lie algebra of $\mathrm{Aut}(T_\Omega)$ is $\mathfrak{co}(V)$, and its universal enveloping algebra is called the {\bf TKK algebra} of $V$. The simply connected Lie group with $\mathfrak{co}$ as its Lie algebra, denoted by $\mathrm{Co}(V)$ or simply $\mathrm{Co}$, shall be referred to as the {\bf conformal group} of $V$. While the structure algebra is reductive, the conformal algebra is simple.
\section{Cantan Involutions and Vogan Diagrams}\label{S:Vogan}
The complex simple Lie algebras are completely classified by (connected) Dynkin diagrams.
The real simple Lie algebras are completely classified too, and can be represented by Vogan diagrams --- Dynkin diagrams
with some extra information on the Dynkin nodes.
\subsection{Generalities}
Here is a quick review of real simple Lie algebras and Vogan diagrams. Let $\mathfrak g$ be a real simple Lie algebra, and $\langle\,,\,\rangle$ be its Killing form. An involution $\theta$ on $\mathfrak g$ is called a {\bf Cartan involution} if the bilinear form $(X, Y)\mapsto \langle X, \theta(Y)\rangle$ is negative definite. Given a Cartan involution $\theta$, we have the corresponding {\bf Cartan decomposition}:
$$
\mathfrak g=\mathfrak{u}\oplus\mathfrak{p}.
$$
Here, $\mathfrak u$ ($\mathfrak p$ resp.) is the eigenspace of $\theta$ with eigenvalue $1$ ($-1$ resp.). A subalgebra $\mathfrak h$ of $\mathfrak g$ is called a
{\bf $\theta$-stable Cartan subalgebra} of $\mathfrak g$ if $\mathfrak h^{\mathbb C}$ is a Cartan algebra of $\mathfrak g^{\mathbb C}$ and $\theta(\mathfrak h)=\mathfrak h$. Here, $\mathfrak h^{\mathbb C}$ ($\mathfrak g^{\mathbb C}$ resp.) is the complexification of $\mathfrak h$ ($\mathfrak g$ resp.).
Here are some basic facts:
\begin{itemize}
\item There is a Cartan involution $\theta$ on $\mathfrak g$, unique up
to conjugations.
\item $\mathrm{span}_{\mathbb R}\{X+iY\mid X\in \mathfrak u, Y\in \mathfrak p\}$ is a compact Lie algebra.
\item $\theta$-stable Cartan subalgebra of $\mathfrak g$ exists, but are not all conjugate to each other.
\end{itemize}
Given a $\theta$-stable Cartan subalgebra $\mathfrak h$, there is a corresponding root space decomposition:
$$
\mathfrak{g}^{\mathbb C}=\mathfrak{h}^{\mathbb
C}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_\alpha.
$$
A root $\alpha$ w.r.t. $(\mathfrak g^{\mathbb C}, \mathfrak h^{\mathbb C})$ is called {\it compact} ({\it non-compact} resp.) if $\mathfrak g_\alpha$ is a subspace of $\mathfrak u^{\mathbb C}$ ($\mathfrak p^{\mathbb C}$ resp.). Here is a simple fact on compact or non-compact roots: suppose that root $\alpha$ is either compact or non-compact, then one can choose root vectors $E_\alpha\in{\mathfrak g}_\alpha$,
$E_{-\alpha}\in{\mathfrak g}_{-\alpha}$ with both $\sqrt{-1}(E_\alpha+E_{-\alpha})$ and $E_\alpha-E_{-\alpha}$ in
$\mathfrak{g}$, and an element $H_\alpha$ in $\sqrt{-1}{\mathfrak h}$, such
that\footnote{In this convention, $H_\alpha\in \sqrt{-1}{\mathfrak h}$ is fixed
by condition $\alpha(H_\alpha)=2$, $E_\alpha$ is unique up to a sign
and $E_{-\alpha}$ is fixed once $E_\alpha$ is fixed.}
\begin{eqnarray}\label{convention}
[H_\alpha, E_{\pm\alpha}] =\pm 2E_{\pm\alpha},\quad [E_\alpha, E_{-\alpha}] = \left\{\begin{matrix} H_\alpha & \mbox{if $\alpha$ is compact}\cr
-H_\alpha & \mbox{if $\alpha$ is non-compact.}\end{matrix}\right.
\end{eqnarray}
We say that a $\theta$-stable Cartan subalgebra of $\mathfrak g$ is {\it maximally compact} if $\dim ({\mathfrak h}\cap \mathfrak u)$ is as large as possible. To get a Vogan diagram, the first step is to find a
{\it maximally compact} $\theta$-stable Cartan subalgebra $\mathfrak{h}$ for
$\mathfrak{g}$. The next step is to chose a simple root system $R$ (or Weyl chamber) for the corresponding root system $\Delta$. Such a $R$ is unique up to the action by the Weyl group $W(\Delta)$. Since $\mathfrak h$ has been chosen to be maximally compact, the roots w.r.t. $(\mathfrak g^{\mathbb C}, \mathfrak h^{\mathbb C})$ never vanish on $\mathfrak h\cap \mathfrak u$, hence are either complex-valued or imaginary-valued on $\mathfrak h$. So $R$ splits
into two classes: {\it complex} and {\it imaginary}. Since $R$ is invariant
under complex conjugation, the class of complex simple roots splits further into various conjugate pairs of simple roots. Since the corresponding root space $\mathfrak g_\alpha$ for an imaginary root $\alpha$ is a subspace of $\mathfrak u^{\mathbb C}$ or $\mathfrak p^{\mathbb C}$, the class of imaginary simple roots splits further into two subclasses: compact and non-compact.
\begin{Def} By definition, a {\bf Vogan diagram} is a Dynkin diagram with
such an information about its nodes recorded: we paint each
imaginary noncompact node black, connect each conjugate pair of
complex nodes by a two-way arrow, and do nothing to the imaginary
compact nodes.
\end{Def}
Note that one can recover the simple real Lie algebra from one of its Vogan diagrams. Note also that, in the equal rank case (i.e., the case when $\mathfrak g$ and $\mathfrak u$ have the same rank), $\mathfrak h\subset \mathfrak u$, so every root is either compact or non-compact, and there is no conjugate pair of Dynkin nodes.
\subsection{Analysis for the conformal algebra}
The conformal algebra is a real simple Lie algebra per theorem
\ref{koecher}, so it admits a Cartan involution $\theta$, unique up
to inner automorphisms. Indeed, one can choose $\theta$ such that
$$
\theta(X_u)=Y_u, \quad \theta(Y_u)=X_u, \quad
\theta(S_{uv})=-S_{vu}.
$$ The resulting Cartan decomposition is
$\mathfrak{co}=\mathfrak{u}\oplus\mathfrak{p}$ with
$$\mathfrak{u}=\mathrm{span}_{\mathbb R}\{[L_u, L_v], X_w+Y_w\mid u, v, w\in V\}, \quad
\mathfrak{p}=\mathrm{span}_{\mathbb R}\{L_u, X_v-Y_v\mid u, v\in V\}.$$
Note that $\mathfrak{u}$ is reductive with center spanned by $X_e+Y_e$ and its semi-simple part $\bar {\mathfrak{u}}$ is
$$ \mathrm{span}_{\mathbb R}\{[L_u, L_v], X_w+Y_w\mid u,
v, w\in ({\mathbb R}e)^\perp\}.$$
Sometime we need to emphasize the dependence on $V$, then we rewrite $\mathfrak u$ as ${\mathfrak u}(V)$. It is a fact that $\mathfrak{str}$ and $\mathfrak{u}$ are different real forms of the same complex reductive Lie algebra. In fact, one can identify their complexfications as follows:
\begin{eqnarray}\label{id: str=k}
[L_u, L_v] \leftrightarrow [L_u, L_v],\quad
-{i\over 2}(X_w+Y_w) \leftrightarrow L_w.
\end{eqnarray}
Now we have another natural chain of real Lie algebras associated
with the Jordan algebra $V$:
\begin{eqnarray}
\mathfrak{der}\subset\bar{\mathfrak{u}}\subset\mathfrak{u}\subset\mathfrak{co}.\nonumber
\end{eqnarray}
We shall use $\tilde{\mathrm U}$ to denote the closed Lie subgroup of $\mathrm{Co}$
whose Lie algebra is $\mathfrak u$. Note that $\tilde{\mathrm U}$ is a closed maximal
subgroup of $\mathrm {Co}$ with $\tilde{\mathrm U}/{\mathrm Z}$ is compact. (Here $\mathrm
Z$ is the center of $\mathrm {Co}$.)
Here is a detailed summary of all real Lie algebras we have
encountered:
\begin{displaymath}
\begin{array}{|c|c|c|c|c|}
\hline
V & \mathfrak{der} & \mathfrak{str} & \mathfrak{u} & \mathfrak{co} \\
\hline
\Gamma(n) & \mathfrak {so}(n) & \mathfrak{so}(n,1)\oplus \mathbb R & \mathfrak{so}(n+1)\oplus \mathfrak{so}(2) & \mathfrak{so}(n+1,2) \\
\hline
{\mathcal H}_n(\mathbb R) & \mathfrak{so}(n) & \mathfrak{sl}(n,{\mathbb R})\oplus \mathbb R & \mathfrak{u}(n) & \mathfrak{sp}(n,{\mathbb R}) \\
\hline
{\mathcal H}_n(\mathbb C) & \mathfrak{su}(n) & \mathfrak{sl}(n,{\mathbb C})\oplus \mathbb R & \mathfrak{su}(n)\oplus\mathfrak{su}(n)\oplus \mathfrak{u}(1) & \mathfrak{su}(n,n)\\
\hline
{\mathcal H}_n(\mathbb H) & \mathfrak{sp}(n) & \mathfrak{su}^*(2n)\oplus \mathbb R & \mathfrak{u}(2n)& \mathfrak{so}^*(4n) \\
\hline
{\mathcal H}_3(\mathbb O) & \mathfrak{f}_4 & \mathfrak{e}_{6(-26)}\oplus \mathbb R & \mathfrak{e}_{6}\oplus \mathfrak{so}(2) & \mathfrak{e}_{7(-25)} \\
\hline
\end{array}
\end{displaymath}
To get a Vogan diagram for $\mathfrak{co}$, the first step is to find a
maximally compact $\theta$-stable Cartan subalgebra $\mathfrak{h}$ for
$\mathfrak{co}$. Since we are in the equal rank case, $\mathfrak{h}\subset\mathfrak u$, and a root $\alpha$ is either {\em
compact} or {\em non-compact}. Recall that $e_{11}$ denotes the first element of a Jordan frame for $V$.
\begin{Lem}\label{LemmaVogan}
There is a maximally compact $\theta$-stable Cartan subalgebra $\mathfrak h$ for
$\mathfrak{co}$, with respect to which, there is a simple root system
consisting of imaginary roots $\alpha_0$, $\alpha_1$, \ldots,
$\alpha_r$ such that, for $i\ge 1$, $\alpha_i$ is compact with $H_{\alpha_i},
E_{\pm\alpha_i}\in \bar { \mathfrak u}^{\mathbb C}$, and $\alpha_0$
is non-compact with
$$
H_{\alpha_0}=i(X_{e_{11}}+Y_{e_{11}}), \quad E_{\pm \alpha_0}={i\over
2}(X_{e_{11}}-Y_{e_{11}})\mp L_{e_{11}}.
$$
\end{Lem}
\begin{proof}
Let us fix a Jordan frame $\{e_{ii}\mid 1\le i\le \rho\}$. Since $\mathfrak u$ is compact, being an abelian
subalgebra of $\mathfrak u$, ${\mathfrak h}':=\mathrm{span}_{\mathbb
R}\{X_{e_{ii}}+Y_{e_{ii}}\mid 1\le i\le \rho\}$ can be extended to a
Cartan subalgebra $\mathfrak{h}$ for $\mathfrak u$, hence a maximally compact
$\theta$-stable Cartan subalgebra $\mathfrak{h}$ for $\mathfrak{co}$.
Under the action of $\mathfrak h$, we have decomposition $\mathfrak{co}^{\mathbb C}=\mathfrak u^{\mathbb C}\oplus \mathfrak p^{\mathbb C}$. Therefore, if $\alpha$ is a root for $(\mathfrak{co}^{\mathbb C}, \mathfrak{h}^{\mathbb C})$, $\mathfrak g_\alpha$ is a subset of either $\mathfrak u^{\mathbb C}$ or $\mathfrak p^{\mathbb C}$, so $\alpha$ is either compact or non-compact.
To understand the non-compact roots for $({\mathfrak {co}}^{\mathbb C}, {\mathfrak h}^{\mathbb C})$, we introduce
\begin{eqnarray}
E_u^\pm= {\sqrt{-1}\over 2}(X_u-Y_u)\mp L_u,
\quad h_u=\sqrt{-1}(X_u+Y_u), \quad u\in V.\nonumber
\end{eqnarray}
and verify that
\begin{eqnarray}\label{vogan:key}
\left\{\begin{array}{l}[h_u, E_v^\pm] = \pm 2 E_{uv}^\pm, \quad
[E_u^+, E_v^-]=-h_{uv}-2[L_u, L_v],\cr
[E_u^+, E_v^+]= [E_u^-,
E_v^-]=0, \quad [h_u, h_v]=4[L_u, L_v].
\end{array}\right.
\end{eqnarray}
The first identity in Eq. (\ref{vogan:key}) implies that, under the action of $\mathfrak h'$, we have the following decomposition of $\mathfrak p^{\mathbb C}$:
\begin{eqnarray}
{\mathfrak p}^{\mathbb C}=\bigoplus_{i\le j}\left({\mathfrak
g}^+_{ij}\oplus {\mathfrak g}^-_{ij}\right)\nonumber
\end{eqnarray}
where $ {\mathfrak g}^\pm_{ii}=\mathrm{span}_{\mathbb C}\{E^\pm_{e_{ii}} \}$ and
${\mathfrak g}^\pm_{ij}=\mathrm{span}_{\mathbb C}\{E_u^\pm \mid u\in V_{ij}\}$ if
$i< j$.
Let $\alpha$ be a non-compact root. The decomposition above implies that
$\mathfrak g_\beta\subset \mathfrak g_{ij}^\pm$ for some $i, j$ with $i\le j$. Since $ {\mathfrak g}^\pm_{ii}$ is one-dimensional and Eq. (\ref{vogan:key}) implies that
$$
[E_{e_{ii}}^+, E_{e_{ii}}^-]=-h_{e_{ii}}, \quad [h_{e_{ii}},
E_{e_{ii}}^\pm]=\pm 2E_{e_{ii}}^\pm,
$$
we conclude that, in view of Eq. (\ref{convention}), there is a non-compact root
$\beta_i$ such that ${\mathfrak g}_{\pm \beta_i}={\mathfrak g}_{ii}^\pm$ and
\fbox{$H_{\beta_i}=h_{e_{ii}}$}.
We call $\beta_i$ a non-compact root of {\em type I}.
Suppose that $\alpha$ is a non-compact root of {\em type II}, i.e.,
not of type I. We may assume that ${\mathfrak g}_\alpha\subset {\mathfrak g}_{jk}^+$ for
some $j<k$, then the root vector $E_\alpha\in {\mathfrak g}_\alpha$ is of
the form
$$
E_\alpha=E_z^+
$$ for some $z\in V_{jk}^{\mathbb C}$. Consequently,
$E_{-\alpha}=E_{\bar z}^-$, and Eq. (\ref{vogan:key}) implies that
\begin{eqnarray}
[E_\alpha, E_{-\alpha}] = -h_{z\bar z}-2[L_z, L_{\bar z}].\nonumber
\end{eqnarray}
So there is a non-compact root $\beta_{jk}^z$ such that
${\mathfrak g}_{\pm \beta_{jk}^z}\subset{\mathfrak g}_{jk}^\pm$ and
\fbox{$H_{\beta_{jk}^z}=h_{z\bar z}+2[L_z, L_{\bar z}]$}. Of course, there are exactly $\delta$-many such $z$.
In summary, the non-compact roots are $$\pm \beta_i\; \mbox{and}\;\pm \beta_{jk}^z.$$
and
$$
{\mathfrak g}_{\beta_i}, {\mathfrak g}_{\beta_{jk}^z}\subset {\mathfrak p}_{+}^{\mathbb C}:=\bigoplus_{i\le j}{\mathfrak
g}^+_{ij}.
$$
To continue the proof, with the help of Eq. (\ref{vogan:key}), we observe that
$$\beta_{1i}^z(H_{\beta_i})= \beta_{1i}^z(H_{\beta_1})=1>0,\quad \beta_{1i}^z(H_{\beta_{1i}^z})=2>0, \quad \beta_{1j}^z(H_{\beta_{jk}^w})>0.$$ Here, only the last inequality is not clear, but it can be verified by a
case-by-case study\footnote{it reduces to the
trivial equality $|Im(z_1z_2)|<2$ for $z_1, z_2$ in a division algebra with
$|z_1|^2+|z_2|^2=1$.}. This observation implies that, for $\alpha=\beta_i$ or $\beta_{jk}^z$, $H_{\beta_1}$ and $H_{-\alpha}$ cannot be in the same Weyl chamber, i.e., $\beta_1$ and $-\alpha$ cannot be in the same simple root system.
Fixing a simple root system containing the non-compact root $\beta_1$,
all we need to do is to show that this simple root system cannot contain
another non-compact root. Otherwise, this simple system consisted of
simple roots $\gamma_1$, \ldots, $\gamma_t$, $\delta_1$, \ldots,
$\delta_s$ with $\delta_i$ compact, $\gamma_j$ non-compact, $t>1$, $\gamma_1=\beta_1$ and
each $\gamma_j$ is either of the form $\beta_i$ or of the form $\beta_{kl}^z$ due to the conclusion in the previous paragraph, so the first identity in Eq. (\ref{vogan:key}) implies that,
$$
\gamma_j(h_e)=2, \quad 1\le j\le t.
$$
Since the rank of $\mathfrak{co}$ is one
more than the rank of $\bar {\mathfrak u}$, $s$ is less than the rank of
$\bar{\mathfrak u}$, so there is a compact root $\lambda$ such that
$$
\lambda =\sum_i m_i\gamma_i+\sum_jn_j\delta_j
$$ for some non-negative integers $n_j$ and $m_i$ with $\sum_im_i\neq 0$. Since $\alpha(h_e)$ is equal to zero if $\alpha$ is compact, we have
a contradiction:
$$
0=\lambda(h_e)=2\sum_im_i.
$$
\end{proof}
As a corollary of the above lemma, the Vogan diagram we have arrived at for the conformal algebra of a simple euclidean Jordan algebra has no complex nodes and only one node painted black. Here is a pictorial summary of the
Vogan diagrams for the conformal algebras:
\begin{displaymath}
\begin{array}{|c|c|}
\hline
\mathfrak{co} & \mbox{Vogan Diagram} \\
\hline
& \\
& \\
\mathfrak{so}(2,2n) & \setlength{\unitlength}{0.3cm}
\begin{picture}(20,0.25)\linethickness{1mm}
\put(4,0){\circle*{0.3}} \put(6.5,0){\circle{0.3}}
\put(9,0){\circle{0.3}} \put(11.9,0){\circle{0.3}}
\thinlines\put(4,0){\line(1,0){2.3}} \put(7,-0.28){-- -- }
\put(9.2,0){\line(1,0){2.1}}
\put(9,0.2){\line(0,1){2.1}}\put(9.7,0){\line(1,0){2}}
\put(9,2.43){\circle{0.3}}
\end{picture} \\
& \\
\hline
& \\
\mathfrak{so}(2,2n+1) &\setlength{\unitlength}{0.3cm}
\begin{picture}(20,0.25)\linethickness{1mm}
\put(4,0){\circle*{0.3}}
\put(6.5,0){\circle{0.3}} \put(9,0){\circle{0.3}}
\put(10.2,-0.2){$\rangle$}\put(11.5,0){\circle{0.3}}
\thinlines\put(4,0){\line(1,0){2.3}}\put(7,-0.28){-- -- }
\put(9.2,-0.1){\line(1,0){2.1}} \put(9.2,0.1){\line(1,0){2.1}}
\end{picture} \\
& \\
\hline
& \\
\mathfrak{sp}(n, \mathbb R) & \setlength{\unitlength}{0.3cm}
\begin{picture}(20,0.25)\linethickness{1mm}
\put(4,0){\circle{0.3}}
\put(6.5,0){\circle{0.3}} \put(9,0){\circle{0.3}}
\put(10,-0.2){$\langle$}\put(11.5,0){\circle*{0.3}}
\thinlines\put(4,0){\line(1,0){2.3}}\put(7,-0.28){-- -- }
\put(9.2,-0.1){\line(1,0){2.1}} \put(9.2,0.1){\line(1,0){2.1}}
\end{picture} \\
&\\
\hline
&\\
\mathfrak{su}(n,n) & \setlength{\unitlength}{0.3cm}
\begin{picture}(20,0.25)\linethickness{1mm}
\put(4,0){\circle{0.3}}
\put(6.5,0){\circle{0.3}} \put(9,0){\circle*{0.3}}
\put(11.5,0){\circle{0.3}} \put(14,0){\circle{0.3}}
\thinlines\put(6.7,0){\line(1,0){2.3}} \put(9.2,0){\line(1,0){2.1}}
\put(4.5,-0.28){-- -- } \put(12,-0.28){-- -- }
\end{picture}\\
& \\
\hline
&\\
&\\
\mathfrak{so}^*(4n) & \setlength{\unitlength}{0.3cm}
\begin{picture}(21,0.25)\linethickness{1mm}
\put(4.5,0){\circle{0.3}} \put(7,0){\circle{0.3}}
\put(9.5,0){\circle{0.3}} \put(12,0){\circle*{0.3}} \thinlines
\put(5,-0.28){-- -- } \put(7.2,0){\line(1,0){2.1}}
\put(9.5,0.2){\line(0,1){2.1}}\put(9.7,0){\line(1,0){2.1}}
\put(9.5,2.43){\circle{0.3}}
\end{picture} \\
&\\
\hline
& \\
& \\
\mathfrak{e}_{7(-25)} & \setlength{\unitlength}{0.3cm}
\begin{picture}(20,0.25)\linethickness{1mm}
\put(4,0){\circle{0.3}} \put(6.5,0){\circle{0.3}}
\put(9,0){\circle{0.3}} \put(11.5,0){\circle{0.3}}
\put(14,0){\circle{0.3}}\put(16.5,0){\circle*{0.3}}
\thinlines\put(6.7,0){\line(1,0){2.1}}
\put(9.2,0){\line(1,0){2.1}}\put(11.7,0){\line(1,0){2.1}}\put(4.2,0){\line(1,0){2.1}}\put(14.2,0){\line(1,0){2.1}}
\put(9,0.15){\line(0,1){2.1}}\put(9,2.35){\circle{0.3}}
\end{picture} \\
& \\
\hline
\end{array}
\end{displaymath}
\section{Kepler Cones}\label{S:KM}
The goal in this section is to introduce the Kepler cone, an open
Riemannian manifold which serves as the configuration space for the
J-Kepler problem.
\begin{Definition}[Kepler Cone]\label{D:main1}
Let $V$ be a simple euclidean Jordan algebra. The {\bf Kepler
cone} is a Riemannian manifold whose underlying smooth manifold is
\begin{eqnarray} {\mathscr
P}:=\left\{x\in V\mid x^2=(\mr{tr}\, x)\, x, \; \mr{tr}\, x >0\right\}
\end{eqnarray}
and its Riemannian metric, referred to as the {\bf Kepler metric}, is the restriction of
\begin{eqnarray}\label{Kepler metric}
ds^2_K:={2\over \rho}ds^2_E-(d\langle e\mid x\rangle)^2
\end{eqnarray}
from $V$ to $\mathscr P$.
\end{Definition}
We shall also use ${\mathscr P}$ to denote the Kepler cone. By
introducing coordinates, it is not hard to see that the Kepler cone
is a smooth real affine variety.
${\mathscr P}$ is called the Kepler cone because it is isometric to the open geometric cone over
{\bf projective space}
\begin{eqnarray}
{\mathbb P}:=\left\{x\in {\mathscr P}\mid \mr{tr}\,(x)=\sqrt{{2\rho}}\right\}.
\end{eqnarray}
Here, as Riemannian manifolds, ${\mathscr P}$ is viewed as $({\mathscr P}, ds^2_K|_{{\mathscr P}})$, $\mathbb R_+\times \mathbb P$ is viewed as
$\left(\mathbb R_+\times \mathbb P, dr^2+r^2\; ds^2_E|_{\mathbb P}\right)$, and the isometry is
\begin{eqnarray}\iota: \quad {\mathscr P} &\longrightarrow & \mathbb R_+\times \mathbb P\cr
x & \mapsto & \left({\mr{tr}\, x\over \rho},\; {\sqrt 2}{x\over |x|}\right).
\end{eqnarray}
Note that, being the intersection of ${\mathscr P }$ with the sphere of radius $\sqrt 2$ and centered at the origin of $V$, ${\mathbb P}$
is a compact symmetric space of rank-one:
\begin{displaymath}
\begin{array}{|c|c|c|c|c|c|}
\hline
V& \Gamma(n) & {\mathcal H}_n(\mathbb R) & {\mathcal H}_n(\mathbb C) &{\mathcal H}_n(\mathbb H) & {\mathcal H}_3(\mathbb O)\\
\hline
{\mathbb P} & {\mathrm S}^{n-1} & {\mathbb R}P^{n-1} & {\mathbb C}P^{n-1} & {\mathbb H}P^{n-1} & {\mathbb O}P^2\\
\hline
\end{array}
\end{displaymath}
One can check that the Riemannian metric $ds^2_{\mathbb P}$ on projective space $\mathbb P$ is the round metric of the unit spheres for the
Dirac type and is four times the Fubini-Study metric
$$ds^2_{FS}={|dZ|^2\over |Z|^2}-{|Z\cdot d\bar Z|^2\over
|Z|^4}$$ of projective spaces for the other types.
We conclude this section with a technical lemma.
\begin{Lem}\label{KeyLemma}
Let $V$ be a simple euclidean Jordan algebra with rank $\rho$ and degree $\delta$, and $r=\langle e\mid x\rangle$.
i) Let $e_\alpha$ be an orthonormal basis for $V$, then
\begin{eqnarray}
{\sum_{\alpha,\beta} \mid
[L_{e_\alpha}, L_{e_\beta}]x\rangle\langle [L_{e_\alpha},
L_{e_\beta}]x \mid \over {\rho^2\over 2}\left(1+{\delta\over
4}(\rho-2)\right)}= r\sum_\alpha \mid e_\alpha \rangle \langle
e_\alpha x \mid- \mid x\rangle \langle x\mid
\end{eqnarray}for any $x\in {\mathscr P}$.
ii) For each $u\in V$ and each $x\in {\mathscr P}$, the value of $\hat
L_u$ at $x$ is a tangent vector of ${\mathscr P}$ at $x$. So $\hat L_u$
descends to a differential operator on the Kepler cone.
iii) Let $\lambda_u= {(\rho/2-1)\delta\over 2}{\langle u\mid
x\rangle\over r}+{\rho \delta\over 4}\langle u\mid e\rangle$,
$\mathrm{vol}_{{\mathscr P}}$ be the volume element on ${\mathscr P}$, and ${\mathscr
L}_u$ be the Lie derivative with respect to vector field $\hat L_u$
on the Kepler cone. Then
\begin{eqnarray}\label{volume}
{\mathscr L}_u\left({1\over r}\mathrm{vol}_{{\mathscr P}}\right)=-2\lambda_u
{1\over r} \mathrm{vol}_{{\mathscr P}}.
\end{eqnarray}
Consequently, $\tilde L_u:=\hat L_u-\lambda_u$ is a skew-hermitian
operator with respect to inner product
$$
(\psi_1, \psi_2):=\displaystyle\int_{\mathscr
P}\overline{\psi_1}\,\psi_2\,{1\over r}\mathrm{vol}_{{\mathscr P}}
$$ for compactly-supported smooth functions on ${\mathscr P}$.
\end{Lem}
\begin{proof} i) Since both sides of the identity are homogeneously quadratic in $x$, one
may assume that $\mr{tr}\, x=1$. Choosing a Jordan frame $\{e_{11}$, \ldots, $e_{\rho\rho}\}$ with
$e_{11}=x$ and an associated orthonormal basis for $V$, the detailed proof then becomes just a straightforward
computation, so we skip it.
ii) Let $x_0\in \mathscr P$. Since $x_0^2=\mr{tr}\, x_0\, x_0$ and $\mr{tr}\, x_0>0$, one can write
$x_0=\mr{tr}\, x_0\, e_{11}$ so that $e_{11}^2=e_{11}$ and $\mr{tr}\, e_{11}=1$. Extending $e_{11}$ to
a Jordan frame $\{e_{11}, \ldots, e_{\rho\rho}\}$ for $V$, then we can
decompose $V$ orthogonally into the direct sum of the Pierce
components $V_{ij}$ ($i\le j$). Then
$$
u x_0\in\bigoplus_{j=1}^\rho V_{1j}.
$$
By linearizing equation $x^2=\mr{tr}\, x\, x$ at $x_0$, it is clear that the tangent space of $\mathscr P$ at $x_0$, when translated to $0$, is exactly
$\bigoplus_{j=1}^\rho V_{1j}$. Therefore,
$$
\hat L_u|_{x_0}=(x_0, -ux_0)\in T_{x_0}\mathscr P.
$$
(In fact, one can show that the structure group of $V$ acts on $\mathscr P$ transitively.)
iii) We wish to prove identity (\ref{volume}) at $x_0\in {\mathscr P}$.
To do that we need to choose a local coordinate system for ${\mathscr P}$
around $x_0$ and do the computations. We may choose a Jordan frame $\{e_{11}$, \ldots, $e_{\rho\rho}\}$
such that $x_0=ae_{11}$ for some $a>0$. Write
$$x=\sum_{i=1}^\rho x_{ii}e_{ii}+\sum_{1\le i<j\le \rho}^{1\le \mu \le \delta} x_{ij}^\mu e_{ij}^\mu,$$
by solving equation $x^2=\mr{tr}\, x\, x$, we know that $x_{11}$, $x_{1j}^\alpha$'s are independent
real variables and the Taylor expansion of the other variables
starts at quadratic terms in $(x_{11}-a)$, $x_{1j}^\alpha$'s.
Therefore,
$$
ds^2_K={1\over \rho^3}\left[(dx_{11})^2+2\sum_{2\le j\le \rho}^{1\le
\alpha\le \delta} (dx_{1j}^\alpha)^2\right]+O(|x-x_0|^2)
$$
and
$$
\mathrm{vol}_{{\mathscr P}}=b\,
dx_{11}\wedge(\wedge_{j=2}^\rho(\wedge_{\alpha=1}^\delta
dx_{1j}^\alpha))+O(|x-x_0|^2)
$$ with $b$ being a constant. Since
\begin{eqnarray}
{\mathscr L}_u(dx_{11}) &= & d {\mathscr L}_u (x_{11})=-d\langle ux\mid \rho e_{11}\rangle= -d\langle x\mid
\rho e_{11}u\rangle\cr &=& -\langle e_{11}\mid \rho e_{11}u\rangle
dx_{11}-\sum_{2\le i\le \rho}^{1\le \alpha\le \delta}\langle
e_{1i}^\alpha\mid \rho e_{11}u\rangle dx_{1i}^\alpha\;,\cr {\mathscr
L}_u(dx_{1j}^\beta) &=& -\langle e_{1j}^\beta\mid \rho
e_{1j}^\beta u\rangle dx_{1j}^\beta-\langle e_{11}\mid \rho
e_{1j}^\beta u\rangle dx_{11}\cr &&-\sum_{(i, \alpha)\neq (j,
\beta)}\langle e_{1i}^\alpha\mid \rho e_{1j}^\beta u\rangle
dx_{1i}^\alpha\;,\nonumber
\end{eqnarray}
we have
\begin{eqnarray}
{\mathscr L}_u(\mathrm{vol}_{{\mathscr P}})|_{x_0} &= &\left.-\left(\langle
e_{11}\mid \rho e_{11}u\rangle+\sum_{j\ge 2, 1\le\beta\le \delta}\langle
e_{1j}^\beta\mid \rho e_{1j}^\beta u\rangle\right)\mathrm{vol}_{\mathscr
P}\right|_{x_0}\cr &= &\left.-\rho\left(\langle e_{11}\mid
u\rangle+{\delta\over 2}\sum_{j\ge 2}\langle e_{11}+e_{jj}\mid
u\rangle\right)\mathrm{vol}_{{\mathscr P}}\right|_{x_0}\cr &=
&\left.-\rho\left((1+(\rho/2-1)\delta)\langle e_{11}\mid u\rangle+{\delta\over
2}\langle e\mid u\rangle\right)\mathrm{vol}_{\mathscr
P}\right|_{x_0}.\nonumber
\end{eqnarray}
On the other hand,
$$
\left.{\mathscr L}_u({1\over r})\right|_{x_0}=\left.{\langle u\mid
x\rangle\over r}\cdot {1\over r}\right|_{x_0}=\left.\rho\langle
u\mid e_{11}\rangle \cdot {1\over r}\right|_{x_0}.
$$
Therefore,
\begin{eqnarray}
{\mathscr L}_u({1\over r}\mathrm{vol}_{{\mathscr P}})|_{x_0} &= &
\left.-\rho\left((\rho/2-1)\delta\langle e_{11}\mid u\rangle+{\delta\over
2}\langle e\mid u\rangle\right){1\over r}\mathrm{vol}_{\mathscr
P}\right|_{x_0}\cr &=& -2\lambda_u {1\over r}\mathrm{vol}_{\mathscr
P}|_{x_0}.\nonumber
\end{eqnarray}
Then
\begin{eqnarray}
(\tilde L_u\psi_1, \psi_2)+(\psi_1, \tilde L_u\psi_2) & = &
\displaystyle\int_{{\mathscr P}}{\mathscr L}_u(\overline{\psi_1}\,\psi_2\,
{1\over r}\mathrm{vol}_{{\mathscr P}})\cr &=& \displaystyle\int_{{\mathscr P}}d
\iota_{\hat L_u}(\overline{\psi_1}\,\psi_2\, {1\over
r}\mathrm{vol}_{{\mathscr P}}) =0.\nonumber
\end{eqnarray}
Here $\iota_{\hat L_u}$ is interior product of differential form
with vector field $\hat L_u$.
\end{proof}
\section{The Hidden Action on the
Kepler Cones}\label{S:DS}
Our recent investigation of the Kepler-type problems leads to the discovery
of the hidden action of the conformal algebra on the Kepler cone. By
turning arguments backward, we can say that it is this hidden action
that is responsible for the existence of Kepler-type problems.
We begin with some generalities. For smooth manifold $M$, we use
$\mathfrak X(M)$ to denote the Lie algebra of (smooth) vector fields on
$M$ and ${\mathscr D}(M)$ to denote the algebra of smooth
(real) differential operators on $M$.
Let $A$ be an associative algebra with identity over $\mathbb R$. We say
that {\bf $A$ acts on $M$ hiddenly} if there is an algebra
homomorphism from $A$ into ${\mathscr D}(M)\otimes_{\mathbb R} \mathbb C$. For
example, if $A=\mathbb R[t]$ (the polynomial algebra over $\mathbb R$ in
single variable $t$), then the algebra homomorphism $A\to {\mathscr
D}(M)\otimes_{\mathbb R}\mathbb C$ sending $t$ to the Laplace operator on
$M$, defines a hidden action of $A$ on $M$.
Let $\mathfrak g$ be a real Lie algebra. We say that {\bf $\mathfrak g$ acts
on $M$} if there is a Lie algebra homomorphism from $\mathfrak g$ into
$\mathfrak X(M)$; and we say that {\bf $\mathfrak g$ acts on $M$ hiddenly} if
the universal enveloping algebra of $\mathfrak g$ acts on $M$ hiddenly.
It is clear that, if $\mathfrak g$ acts on $M$, then it acts on $M$
hiddenly; however, the converse may not be true. Note that, $\mathfrak X(M)$ acts on $M$, but
$\mathscr D(M)$ acts on $M$ only hiddenly.
\vskip 10pt Let $V$ be a simple euclidean Jordan algebra. Since the
automorphisms of $V$ leave the Kepler cone invariant, the derivation
algebra, being the Lie algebra of automorphism group, acts on the
Kepler cone. The recent investigation of the Kepler-type problems leads to the following fact: {\it there is a natural
hidden action of the conformal algebra on the Kepler cone which
extends the action of the derivation algebra}.
To introduce the hidden action, we fix an orthonormal basis
$e_\alpha$ for $V$ and recall that $\tilde L_u=\hat L_u-\lambda_u$
where
$$\lambda_u= {(\rho/2-1)\delta\over 2}{\langle u\mid x\rangle\over
\langle e\mid x\rangle}+{\rho \delta\over 4}\langle u\mid e\rangle.$$ For
$u, v\in V$, we introduce differential operators
\begin{eqnarray}
\fbox{ ${\tilde S}_{uv}:=[\tilde L_u, \tilde L_v]+\tilde L_{uv},
\quad{\tilde X}_u:=-i[{\tilde L}_u, X], \quad {\tilde
Y}_v:=-i\langle v\mid x\rangle $}
\end{eqnarray} where
\begin{eqnarray}\label{defX}
X=-{1\over \langle e\mid x\rangle}\left({\hat
L}_e^2-\left((\rho-1)\delta-1\right){\hat L}_e+A\sum_{\alpha,\beta}[{\hat L}_{e_\alpha}, {\hat L}_{e_\beta}]^2+B\right)
\end{eqnarray}
with $A$ and $B$ being constants depending only on the Jordan
algebra. Note that $X$ is independent of the choice of the
orthonormal basis $e_\alpha$ and
$$
\tilde S_{ue}=\tilde S_{eu}=\tilde L_u.
$$
In view of part ii) of Lemma \ref{KeyLemma}, ${\tilde S}_{uv}$, $\tilde X_u$ and
$\tilde Y_v$ all descend to differential operators on the Kepler
cone.
\begin{Lem}\label{lemma}
i) There is a unique constant $A$ in Eq. (\ref{defX}), such that, as
differential operator on ${\mathscr P}$,
\begin{eqnarray}
\fbox{$[X, \langle u\mid x\rangle]=2\tilde L_u$}
\end{eqnarray} for any $u\in V$. In fact
\begin{eqnarray}\label{Avalue}
A^{-1}={\rho^2\over 2}\left(1+{\delta\over 4}(\rho-2)\right).
\end{eqnarray}
ii) Let $\Delta_{\mathbb P}$ be the Laplace operator on $\mathbb P$ and
$A$ be the number in Eq. (\ref{Avalue}). Then
\begin{eqnarray}
\Delta_{\mathbb P}=A\sum_{\alpha,\beta}[{\hat L}_{e_\alpha}, {\hat
L}_{e_\beta}]^2
\end{eqnarray} as differential operators on on $\mathbb
P$. Consequently,
\begin{eqnarray}\label{Xdef}
\fbox{$X=-\langle e\mid x\rangle\Delta_{{\mathscr P}}-{B\over \langle
e\mid x\rangle}$.}
\end{eqnarray}
Here, $\Delta_{{\mathscr P}}$ is the Laplace operator on $\mathscr P$.
\end{Lem}
\begin{proof}
i) For simplicity, we write $\langle x\mid e\rangle$ as $r$. Since
\begin{eqnarray}
[X, \langle u\mid x\rangle ]&=& -{1\over r}\left[{\hat
L}_e^2-((\rho-1)\delta-1)\hat L_e+A\sum_{\alpha,\beta}[\hat L_{e_\alpha}, \hat L_{e_\beta}]^2, \langle u\mid x\rangle\right]\cr &=& -{1\over
r}\left(-2\langle u\mid x\rangle\tilde L_e+[A\sum_{\alpha,\beta}[\hat L_{e_\alpha}, \hat L_{e_\beta}]^2, \langle u\mid x\rangle]\right),\nonumber
\end{eqnarray}we just need to show that
\begin{eqnarray}\label{identity}
\fbox{$[A\sum_{\alpha,\beta}[\hat L_{e_\alpha}, \hat L_{e_\beta}]^2, \langle u\mid
x\rangle]=-2r\tilde L_u+2\langle u\mid x\rangle\tilde L_e$}
\end{eqnarray} or
\begin{eqnarray}
\left\{\begin{array} {rcl} A\sum_{\alpha,\beta}\langle
[L_{e_\alpha}, L_{e_\beta}]u\mid x\rangle\, [\hat L_{e_\alpha}, \hat L_{e_\beta}] & = & -r\hat L_u+\langle u\mid x\rangle \hat L_e\cr\\
A\langle \sum_{\alpha,\beta}[L_{e_\alpha}, L_{e_\beta}]^2 u\mid
x\rangle & = & -{\rho \delta\over 2}(\langle u\mid x\rangle-r \langle u\mid e\rangle)
\end{array}\right.
\end{eqnarray}
for any $u\in V$ and any $x\in {\mathscr P}$. So it suffices to show that
\begin{eqnarray}\label{trivialid'}
A\sum_{\alpha,\beta} \mid
[L_{e_\alpha}, L_{e_\beta}]x\rangle\langle [L_{e_\alpha},
L_{e_\beta}]x \mid = r\sum_\alpha \mid e_\alpha \rangle \langle
e_\alpha x \mid- \mid x\rangle \langle x\mid
\end{eqnarray}
and
\begin{eqnarray}\label{trivialid}
A\sum_{\alpha,\beta}[L_{e_\alpha}, L_{e_\beta}]^2 x = -{\rho \delta\over
2} (x-re)
\end{eqnarray} for any $x\in {\mathscr P}$. Eq. (\ref{trivialid'}) is the content of part i) of Lemma \ref{KeyLemma} with
$$
A^{-1}={\rho^2\over 2}\left(1+{\delta\over 4}(\rho-2)\right).
$$
Eq. (\ref{trivialid}) is clear except that we don't know what the
constant $A$ is. Here is a way to find $A$: taking inner product
with $x$, we have
\begin{eqnarray}\label{trivialid''}
A\sum_{\alpha,\beta}||[L_{e_\alpha}, L_{e_\beta}] x||^2 =
{(\rho-1)\delta\over 2}||x||^2.
\end{eqnarray}On the other hand, by taking the trace of Eq. (\ref{trivialid'}), we also arrive at Eq. (\ref{trivialid''}); so
Eq. (\ref{trivialid}) is a consequence of Eq. (\ref{trivialid'}).
ii) In view of identity (\ref{identity}) and the fact that both
$\Delta_{\mathbb P}$ and $A\sum_{\alpha,\beta}[{\hat L}_{e_\alpha},
{\hat L}_{e_\beta}]^2$ are 2nd order differential operators without
the constant terms, it suffices to show that
\begin{eqnarray}\label{identity'}
[\Delta_{\mathbb P}, \langle u\mid x\rangle]=-2r\tilde L_u+2\langle
u\mid x\rangle\tilde L_e.
\end{eqnarray}
To verify identity (\ref{identity'}) at a point $x_0\in \mathbb P$, we choose
a Jordan frame $\{e_{11}, \ldots, e_{\rho\rho}\}$ such that $x_0=\sqrt{2\rho}\,e_{11}$ and write
variable $$x=\sum_{1\le i\le \rho} x_{ii}e_{ii}+\sum_{1\le i<j\le
\rho}^{1\le \alpha\le \delta}x_{ij}^\alpha e_{ij}^\alpha$$ where
$e_{ii}$'s, $e_{ij}^\alpha$'s are mutually orthogonal with length
$1\over \sqrt{\rho}$.
By solving equation $x^2=\mr{tr}\, x\, x$ and $\mr{tr}\, x =\sqrt{2\rho}$, we
know that $x_{1j}^\alpha$'s ($j>1$) are independent real variables and the
Taylor expansion of other variables has no linear terms in
$x_{1j}^\alpha$'s. Therefore, around point $x_0$, we have
\begin{eqnarray}
ds^2_{\mathbb P} & = & ds^2_E|_{\mathbb P}={1\over \rho}\sum_{2\le i\le
\rho}^{1\le \alpha \le \delta}(dx_{1i}^\alpha)^2+O(|x-x_0|^2),\cr \langle
u\mid x\rangle &=& \langle u\mid x_0\rangle+\sum_{2\le i\le \rho}^{1\le \alpha \le \delta}\langle
u\mid x_{1i}^\alpha e_{1i}^\alpha \rangle+O(|x-x_0|^2).\nonumber
\end{eqnarray}
Thus
\begin{eqnarray}
\mbox{LHS of Eq. (\ref{identity'})}|_{x_0} & = & \left.\rho\sum_{2\le
i\le \rho}^{1\le \alpha \le \delta}[{\partial^2\over
\partial (x_{1i}^\alpha)^2},\langle u\mid x\rangle]\right|_{x_0}\\
&=& \left. 2\rho\sum_{2\le i\le \rho}^{1\le \alpha \le \delta}\langle
u\mid e_{1i}^\alpha\rangle{\partial\over
\partial x_{1i}^\alpha}\right |_{x_0}\cr
&& +\left.\rho\sum_{2\le i\le \rho}^{1\le \alpha \le \delta}
{\partial^2\over
\partial (x_{1i}^\alpha)^2}(\langle u\mid x\rangle)\right|_{x_0}.\nonumber
\end{eqnarray}
On the other hand,
\begin{eqnarray}
\mbox{RHS of Eq. (\ref{identity'})}|_{x_0} & = &\left.
2\rho\sum_{2\le i\le \rho}^{1\le \alpha \le \delta}\langle u\mid
e_{1i}^\alpha\rangle{\partial\over
\partial x_{1i}^\alpha}\right |_{x_0}\\
&& +\delta\sqrt{\rho/2}(-\rho\langle u\mid e_{11}\rangle +\langle u\mid
e\rangle).\nonumber
\end{eqnarray}
Therefore, all need to do is to verify that
\begin{eqnarray}
\left.\sum_{2\le i\le \rho}^{1\le \alpha \le \delta} {\partial^2\over
\partial (x_{1i}^\alpha)^2}(\langle u\mid x\rangle)\right|_{x_0}={\delta\over \sqrt{2\rho}}(-\rho\langle u\mid e_{11}\rangle +\langle u\mid
e\rangle)\nonumber
\end{eqnarray}
or
\begin{eqnarray}\label{finalid}
\left.\sum_{2\le i\le \rho}^{1\le \alpha \le \delta} {\partial^2\over
\partial (y_{1i}^\alpha)^2}(\langle u\mid y\rangle)\right|_{y=0}=\delta(-\rho\langle u\mid e_{11}\rangle +\langle u\mid
e\rangle),
\end{eqnarray}
here $y$ satisfies the following condition:
\begin{eqnarray}
2e_{11}y+y^2=y, \quad \mr{tr}\, y=0,
\end{eqnarray}
so
$$
y=t+\rho\left[-\langle t^2\mid e_{11}\rangle e_{11}+\sum_{j=2}^\rho
\langle t^2\mid e_{jj}\rangle e_{jj}+ \sum_{2\le k<l\le \rho}^{1\le
\beta \le \delta} \langle t^2\mid e_{kl}^\beta\rangle
e_{kl}^\beta\right]+o(t^2),
$$
where $t$ is a free parameter taking values in $\oplus_{j=2}^\rho V_{1j}$.
So, in view of the fact that $(e_{1i}^\alpha)^2={1\over
2}(e_{11}+e_{ii})$, we have
\begin{eqnarray}
\mbox{LHS of Eq. (\ref{finalid})}&=& \left.\sum_{2\le i\le \rho}^{1\le \alpha \le \delta} {\partial^2\over
\partial (t_{1i}^\alpha)^2}(\langle u\mid y\rangle)\right|_{t=0}\cr
&=& 2\rho\sum_{2\le i\le
\rho}^{1\le \alpha\le \delta}[-\langle (e_{1i}^\alpha)^2\mid
e_{11}\rangle \langle u\mid e_{11}\rangle \cr &&\quad\quad\quad\quad +\sum_{j=2}^\rho
\langle (e_{1i}^\alpha)^2\mid e_{jj}\rangle \langle u\mid
e_{jj}\rangle\cr &&\quad\quad\quad\quad + \sum_{2\le k<l\le \rho}^{1\le \beta \le \delta}
\langle (e_{1i}^\alpha)^2\mid e_{kl}^\beta\rangle \langle u\mid
e_{kl}^\beta\rangle ]\cr
&=& \sum_{2\le i\le \rho}^{1\le \alpha\le
\delta}(- \langle u\mid e_{11}\rangle+\langle u\mid e_{ii}\rangle )\cr
&=& \delta\sum_{2\le i\le \rho}(- \langle u\mid e_{11}\rangle+ \langle
u\mid e_{ii}\rangle )\cr &=& \delta(- (\rho-1)\langle u\mid
e_{11}\rangle+ \langle u\mid e-e_{11}\rangle )\cr &=&\delta(- \rho\langle
u\mid e_{11}\rangle+ \langle u\mid e\rangle )\cr &=& \mbox{RHS of
Eq. (\ref{finalid})}.\nonumber
\end{eqnarray}
\end{proof}
\subsection{Hidden Action}The following theorem implies that there is a hidden action of the conformal algebra
on the Kepler cone.
\begin{Thm}[Hidden Action/Dynamical Symmetry]\label{Main1}
Let $V$ be a simple euclidean Jordan algebra with rank $\rho\ge 2$
and degree $\delta$. There are unique constants $A$ and $B$ in Eq. (\ref{defX}) such that, as differential operators on the Kepler
cone, ${\tilde S}_{uv}$, ${\tilde X}_u$ and ${\tilde Y}_v$ satisfy
the TKK commutation relation (\ref{CONFA}) for the conformal algebra,
i.e.,
\begin{eqnarray}
\begin{matrix}[\tilde X_u, \tilde X_v] =0, \quad [\tilde Y_u, \tilde Y_v]=0, \quad [\tilde X_u,
\tilde Y_v] = -2\tilde S_{uv},\cr\\ [\tilde S_{uv},
\tilde X_z]=\tilde X_{\{uvz\}}, \quad [\tilde S_{uv}, \tilde Y_z]=-\tilde Y_{\{vuz\}},\cr\\
[\tilde S_{uv}, \tilde S_{zw}] = \tilde S_{\{uvz\}w}-\tilde
S_{z\{vuw\}}
\end{matrix}\nonumber
\end{eqnarray}for $u$, $v$, $z$, $w$ in $V$. In fact,
$$
A={2/\rho^2\over 1+{\delta\over 4}(\rho-2)},\quad B={\delta\over
8}(\rho-2)\left(\left({3\rho\over 2}-1\right)\delta-2\right).
$$\end{Thm}
\begin{rmk}Here are the more explicit values of $A$ and $B$:
\begin{displaymath}
\begin{array}{|c|c|c|c|c|c|} \hline
V & \Gamma(n) & {\mathcal H}_n(\mathbb R) & {\mathcal H}_n(\mathbb C) &{\mathcal H}_n(\mathbb H) & {\mathcal H}_3(\mathbb O)\\
\hline
&&&&&\\
A & {1\over 2} & {8\over n^2(n+2)} & {4\over n^3} & {2\over n^2(n-1)} & {2\over 27}\\
&&&&&\\
B & 0 & {3(n-2)^2\over 16} & {(n-2)(3n-4)\over 4} & 3(n-1)(n-2) & 26\\
&&&&&\\
\hline
\end{array}
\end{displaymath}
In the case $\rho=1$, a similar theorem is valid except
that $A=0$ and $B$ is not unique.
\end{rmk}
\begin{proof}
For any function $f$ on $V$ and $u, v\in V$, we have $$(\tilde
Y_u\tilde Y_v) (f)(x)=-\langle u\mid x\rangle\langle v\mid x\rangle f(x),$$
so $[\tilde Y_u, \tilde Y_v]=0$ for any $u, v\in V$. The rest of the
proof is divided into four steps.
\vskip 10pt \underline{Step one}: Verify that $[\tilde S_{uv},
\tilde Y_z]=-\tilde Y_{\{vuz\}}$.
This is a simple computation:
\begin{eqnarray}
[\tilde S_{uv}, \tilde Y_z] & = & [\hat S_{uv}, -i\langle z\mid
x\rangle]= i\langle z\mid S_{uv}(x)\rangle\cr &= &i\langle S_{vu}(z)\mid
x\rangle=i\langle \{vuz\}\mid x\rangle \cr &=&-\tilde
Y_{\{vuz\}}.\nonumber
\end{eqnarray}
\underline{Step two}: Verify that
\begin{eqnarray}\label{step2}
[\tilde S_{uv}, \tilde
S_{zw}]=\tilde S_{\{uvz\}w}-\tilde S_{z\{vuw\}}.
\end{eqnarray}
It is easy to see that $\tilde S_{uv} = \hat S_{uv}-\lambda_{uv}$. Thanks to Proposition \ref{prop1}, $[\hat S_{uv}, \hat
S_{zw}]=\hat S_{\{uvz\}w}-\hat S_{z\{vuw\}}$, so all we need to
check is that
$$
\lambda_{\{uvz\}w}-\lambda_{z\{vuw\}}=[\hat S_{uv},
\lambda_{zw}]-[\hat S_{zw}, \lambda_{uv}],
$$ i.e.,
$$
\lambda_{\{uvz\}w}-\lambda_{z\{vuw\}}=-
\lambda_{\{vu(zw)\}}+\lambda_{\{wz(uv)\}}.
$$
Since $\lambda_u$ is linear in $u$, the last equality is implied by identity
$$
L_{\{uvz\}}-L_z S_{vu}=-S_{vu}L_z+S_{(uv)z},\; \mbox{i.e.},\; L_{\{uvz\}}=S_{(zv)u}-S_{v(zu)}+S_{(uv)z}
$$
or equivalently
$$
0=[L_{zv}, L_u]-[L_v, L_{zu}]+[L_{uv}, L_z]
$$ --- the polarization of identity $[L_{u^2}, L_u]=0$.
\underline{Step three}. Verify that $[\tilde X_u, \tilde
Y_v]=-2\tilde S_{uv}$, i.e.,
\begin{eqnarray}\label{step3}
\fbox{$[[\tilde L_u, X], \langle v\mid x\rangle]=2\tilde S_{uv}$.}
\end{eqnarray}
This is a consequence of part i) of Lemma \ref{lemma}:
\begin{eqnarray}
[[\tilde L_{u}, X],\langle x\mid v\rangle]&=&[[\langle x\mid
v\rangle, X],\tilde L_{u}]+[[\tilde L_{u},\langle x\mid v\rangle],
X]\cr &=& [-2\tilde L_v, \tilde L_u]+[-\langle x\mid uv\rangle,
X]\cr &=& [-2\tilde L_v, \tilde L_u]+2\tilde L_{uv}=2\tilde
S_{uv}.\nonumber
\end{eqnarray}
\underline{Step four}. Verify that $[\tilde S_{uv}, \tilde
X_z]=\tilde X_{\{uvz\}}$, i.e., $[\tilde S_{uv}, [\tilde L_z,
X]]=[\tilde L_{\{uvz\}}, X]$.
Note that $X$ is invariant under $\mathfrak{der}$ and $\tilde
S_{eu}=\tilde S_{ue}=\tilde L_u$, in view of Eq. (\ref{step2}), we have
\begin{eqnarray}
[\tilde S_{uv}, [\tilde L_z, X]] &=& [X, [\tilde L_z,\tilde S_{uv}]]+[\tilde L_z, [\tilde S_{uv}, X]]\cr &=&[\tilde L_{\{uvz\}}-\tilde
S_{z(vu)}, X]+[\tilde L_z, [\tilde S_{uv}, X]]\cr &=&[\tilde
L_{\{uvz\}}-\tilde L_{z(vu)}, X]+[\tilde L_z, [\tilde L_{uv}, X]]\cr
&=&[\tilde L_{\{uvz\}}, X]-[\tilde L_{z(uv)}, X]+[\tilde L_z,
[\tilde L_{uv}, X]].\nonumber
\end{eqnarray}
So it suffices to show that
\begin{eqnarray}\label{Lid}
\fbox{$[\tilde L_{uv}, X]=[\tilde L_u, [\tilde L_v, X]]$}
\end{eqnarray} for any $u, v\in V$. To prove it, we let
$$O=[\tilde L_{uv}, X]-[\tilde L_u, [\tilde L_v, X]],$$
and show that, 1) for any $z\in V$, $O_z:=[O, \langle x\mid
z\rangle]=0$, 2) $O(1)=0$. Eq. (\ref{step3}) implies that
\begin{eqnarray}
[[\tilde L_u, [\tilde L_v, X]], \langle x\mid z\rangle] & = &
[[\langle x\mid z\rangle, [\tilde L_v, X]], \tilde L_u]+[[\tilde
L_u, \langle x\mid z\rangle], [\tilde L_v, X]]\cr
&=& [[\langle x\mid z\rangle, [\tilde L_v, X]], \tilde L_u]+[-\langle x\mid uz\rangle, [\tilde L_v, X]]\cr
&=& [-2\tilde S_{vz},\tilde L_u]+2\tilde S_{v(uz)}.\nonumber
\end{eqnarray}
So, in view of Eq. (\ref{step2}), we have
\begin{eqnarray}
O_z/2 & = & \tilde S_{(uv)z}+ [\tilde S_{vz},\tilde L_u]-\tilde
S_{v(uz)}\cr
&=& \tilde S_{(uv)z}-\tilde
S_{v(uz)} - [\tilde S_{ue}, \tilde S_{vz}]\cr
&=&0.\cr
\end{eqnarray}
The proof of $O(1)=0$ is a long computation, so its details are
provided in the appendix.
\underline{Step five}. Verify that $[\tilde X_u, \tilde X_v]=0$,
i.e., $[[\tilde L_u, X], [\tilde L_v, X]]=0$.
Eq. (\ref{Lid}) implies that
\begin{eqnarray}
[[\tilde L_u, X], [\tilde L_v, X]] &=& [[\tilde L_u,[\tilde L_v,
X]],X]+[[\tilde L_v, X],X], \tilde L_u] \cr &=& [[\tilde L_{uv},
X],X]+[[[\tilde L_v, X],X], \tilde L_u].\nonumber
\end{eqnarray}
So it suffices to show that
\begin{eqnarray}\label{Lid2}
\fbox{$[[\tilde L_u, X], X]=0$}
\end{eqnarray} for any $u\in V$. To prove Eq. (\ref{Lid2}), as in step four, we let $\mathscr O=[[\tilde
L_u, X], X]$, and show that, 1) for any $z\in V$, $\mathscr O_z:=[\mathscr O,
\langle x\mid z\rangle]=0$, 2) $\mathscr O(1)=0$.
In view of Eqs. (\ref{Lid}) and (\ref{step3}), part i) of Lemma \ref{lemma} and the
fact that $X$ is invariant under the action of $\mathfrak {der}$, we have
\begin{eqnarray}
[\mathscr O, \langle x\mid z\rangle]&=& [[\langle x\mid z\rangle,X],
[\tilde L_u, X]]+[[[\tilde L_u,X], \langle x\mid z\rangle], X]\cr
&=& -2[\tilde L_z, [\tilde L_u, X]]+[2\tilde S_{uz}, X]\cr
&=& -2[\tilde L_z, [\tilde L_u, X]]+[2\tilde L_{uz}, X]=0.\cr
\end{eqnarray}
The proof of $\mathscr O(1)=0$ is a long computation, so its details are
provided in the appendix.
\end{proof}
\subsection{Quadratic Relations for the Hidden Action}
Let $V_0$ be the orthogonal complement of $e$ in $V$, $D=\dim V_0$ and $\{e_\alpha\}_{1\le \alpha\le D}$ be the orthonormal basis for $V_0$. We write $e$ as $e_0$ and $\langle x\mid e_\alpha\rangle$ as $x_\alpha$. Note that
$\tilde S_{eu}=\tilde S_{ue}=\tilde L_u$.
\begin{Thm}(The Primary Quadratic Relation)
\begin{eqnarray}\label{case1}
{2\over \rho} \sum_{0\le \alpha\le D} \tilde L_{e_\alpha}^2-\tilde L_e^2-{1\over 2}\{\tilde X_e,\tilde Y_e\}= -a.
\end{eqnarray}
Here, $a={\rho \delta\over 4}(1+{(\rho-2)\delta\over 4})$.
\end{Thm}
\begin{proof}
Let $\mathscr O$ be the left hand side of Eq. (\ref{case1}). We just need to show that, on the Kepler cone we have 1) $[\mathscr O, \tilde Y_u]=0$ and 2) $\mathscr O(1)=-a$.
Since
\begin{eqnarray}
[\tilde L_{e_\alpha}^2, \tilde Y_u]&=& \{\tilde L_{e_\alpha}, [\tilde L_{e_\alpha}, \tilde Y_u]\}= \{\tilde L_{e_\alpha}, -\tilde Y_{ue_\alpha}\}\cr
&=& -2\tilde Y_{ue_\alpha}\tilde L_{e_\alpha}-[\tilde L_{e_\alpha}, \tilde Y_{ue_\alpha}]\cr
&=& -2\tilde Y_{ue_\alpha}\tilde L_{e_\alpha}+\tilde Y_{(ue_\alpha)e_\alpha},\nonumber
\end{eqnarray}
we have
\begin{eqnarray}
\sum_{0\le \alpha\le D}[\tilde L_{e_\alpha}^2, \tilde Y_u]
&=& \sum_{0\le \alpha\le D}-2\tilde Y_{ue_\alpha}(\hat L_{e_\alpha}-\lambda_{e_\alpha})+\tilde Y_{(ue_\alpha)e_\alpha}\cr
&=& 2i\hat L_{ux}-2i\lambda_{ux}+\rho\left (1+{\rho-2\over 4}\delta\right )\tilde Y_u+{\rho \delta\over 4}\mr{tr}\, u \tilde Y_e\cr
&=& 2i\hat L_{ux}+\rho\left (1+{3\rho-4\over 4}\delta\right )\tilde Y_u+{\rho \delta\over 4}\mr{tr}\, u \tilde Y_e, \nonumber
\end{eqnarray}
here, we have used the identity
$$
\sum_\alpha L_{e_\alpha}^2 =\rho\left (1+{\rho-2\over 4}\delta\right )L_e+{\rho^2 \delta\over 4}\mid e\rangle\langle e\mid.
$$
On the other hand,
\begin{eqnarray}
[\tilde L_e^2+{1\over 2}\{\tilde X_e,\tilde Y_e\}, \tilde Y_u]&=&\{\tilde L_e, [\tilde L_e, \tilde Y_u]\}+{1\over 2}\{[\tilde X_e, \tilde Y_u], \tilde Y_e\}\cr
&=&-2(\tilde Y_u\tilde L_e+\tilde Y_e\tilde L_u-\tilde Y_u), \nonumber
\end{eqnarray}
so we have
\begin{eqnarray}
[\mathscr O, \tilde Y_u]&=& {4i\over \rho}\hat L_{ux}+2\left (1+{3\rho-4\over 4}\delta\right )\tilde Y_u+{\delta\over 2}\mr{tr}\, u \tilde Y_e\cr
&& +2(\tilde Y_u\hat L_e+\tilde Y_e\hat L_u-\tilde Y_u)-2(\tilde Y_u\lambda_e+\tilde Y_e\lambda_u)\cr
&=& {4i\over \rho}\hat L_{ux}+2(\tilde Y_u\hat L_e+\tilde Y_e\hat L_u)\cr
&=& {2i\over \rho}\sum_{0\le \beta\le D} \hat L_u(\langle x^2-\mr{tr}\, x\, x\mid {/\hskip -5pt
\partial}\rangle)\cr
&=&0 \nonumber
\end{eqnarray}
as differential operator on the Kepler cone.
Since
\begin{eqnarray}
\sum_{0\le \alpha\le D} \tilde L_{e_\alpha}^2 (1)&=& \sum_{0\le \alpha\le D} -\hat L_{e_\alpha}(\lambda_{e_\alpha})+\lambda_{e_\alpha}^2\cr
&=& {\rho(\rho-2)\over 4}\delta\lambda_e+{\rho^2\over 8}\delta\lambda_e={\rho(3\rho/2-2)\over 4}\delta\lambda_e, \nonumber
\end{eqnarray}
and
\begin{eqnarray}
-\tilde L_e^2(1)-{1\over 2}\{\tilde X_e,\tilde Y_e\} (1)=-\lambda_e^2-\lambda_e-B,\nonumber
\end{eqnarray}
we have
\begin{eqnarray}
-a &= & {(3\rho/2-2)\over 2}\delta\lambda_e-\lambda_e^2-\lambda_e-B\cr
&=& -{\rho \delta\over 4}(1+{(\rho-2)\delta\over 4}),\nonumber
\end{eqnarray}
i.e., $$a={\rho \delta\over 4}\left(1+{(\rho-2)\delta\over 4}\right).$$
\end{proof}
\begin{cor}(The Secondary Quadratic Relations)
\begin{eqnarray}
\sum_{0\le \alpha\le D}\{\tilde X_{e_\alpha}, \tilde L_{e_\alpha}\} =\rho \{
\tilde X_e, \tilde L_e\},\quad
\sum_{0\le \alpha\le D}\{\tilde Y_{e_\alpha}, \tilde L_{e_\alpha}\} = \rho\{
\tilde Y_e, \tilde L_e\}, \cr
\sum_{0\le \alpha\le D}\tilde X_{e_\alpha}^2 = \rho\tilde X_e^2, \quad
\sum_{0\le \alpha\le D}\tilde Y_{e_\alpha}^2 =\rho\tilde Y_e^2,\quad
{1\over 2} \sum_{0\le \alpha\le D} \{\tilde X_{e_\alpha}, \tilde Y_{e_\alpha}\}=\rho(\tilde L_e^2+a),\cr
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha, u}, \tilde L_{e_\alpha}\}={1\over 2}\left(-\{\tilde X_u, \tilde Y_e\}+\{\tilde X_e, \tilde Y_u\} \right),\cr
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha, u}, \tilde X_{e_\alpha}\}=-\{\tilde X_u, \tilde L_e\}+\{\tilde L_u, \tilde X_e\},\cr
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha,u}, \tilde Y_{e_\alpha}\}=\{\tilde Y_u, \tilde L_e\}-\{\tilde L_u, \tilde Y_e\},\cr
A\sum_{1\le\alpha, \beta\le D}[\tilde L_{e_\alpha}, \tilde L_{e_\beta}]^2={1\over 2}\{\tilde X_e, \tilde Y_e\}-\tilde L_e^2+{\rho \delta\over 4}({\rho \delta\over 4}-1).\nonumber
\end{eqnarray}
Here, $A={2/\rho^2\over 1+{(\rho-2)\delta\over 4}}$. \end{cor}
\begin{proof}
The two identities
\begin{eqnarray}\label{case2}
\sum_{0\le \alpha\le D}\{\tilde X_{e_\alpha}, \tilde L_{e_\alpha}\} =\rho \{
\tilde X_e, \tilde L_e\},\quad \sum_{0\le \alpha\le D}\{\tilde Y_{e_\alpha}, \tilde L_{e_\alpha}\} = \rho\{
\tilde Y_e, \tilde L_e\}
\end{eqnarray}
can be obtained by taking the commutator of Eq. (\ref{case1}) with $\tilde X_e$ and $\tilde Y_e$ respectively.
Identity
\begin{eqnarray}\label{case3}
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha, e_\beta}, \tilde L_{e_\alpha}\}+{1\over 2}\left(\{\tilde X_{e_\beta}, \tilde Y_e\}-\{\tilde X_e, \tilde Y_{e_\beta}\} \right)=0
\end{eqnarray}
can be obtained by forming the commutator of the identity in Eq. (\ref{case1}) with $\tilde L_{e_\beta}$ .
Identities
\begin{eqnarray}\label{case4}
\left\{
\begin{array}{rcl}
\sum_{0\le \alpha\le D}\tilde X_{e_\alpha}^2 & = & \rho \tilde X_e^2,\\\\
\sum_{0\le \alpha\le D}\tilde Y_{e_\alpha}^2 &= & \rho \tilde Y_e^2,\\\\
{2\over \rho} \sum_{0\le
\alpha\le D} \{\tilde X_{e_\alpha}, \tilde Y_{e_\alpha}\} &=& 4(\tilde L_e^2 + a)
\end{array}
\right.
\end{eqnarray}
can be proved this way: The 1st identity here is obtained from forming the commutator of the 1st identity in Eq. (\ref{case2}) with $\tilde X_e$, and the 2nd identity here is obtained from forming the commutator of the 2nd identity in Eq. (\ref{case2}) with $\tilde Y_e$. The third identity here is obtained by first forming the commutator of the 1st identity in Eq. (\ref{case2}) with $\tilde Y_e$ and then using Eq. (\ref{case1}).
Identities
\begin{eqnarray}\left\{\begin{array}{rcl}
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha, e_\beta}, \tilde X_{e_\alpha}\}+\{\tilde X_{e_\beta}, \tilde L_e\}-\{\tilde L_{e_\beta}, \tilde X_e\}
& = & 0,\\
\\
{2\over \rho}\sum_{1\le \alpha\le D}\{\tilde L_{e_\alpha, e_\beta}, \tilde Y_{e_\alpha}\}-\{\tilde Y_{e_\beta}, \tilde L_e\}+\{\tilde L_{e_\beta}, \tilde Y_e\}
& = & 0\end{array}\right.
\end{eqnarray}
can be obtained by taking the commutator of Eq. (\ref{case3}) with $\tilde X_e$ and $\tilde Y_e$ respectively.
The last identity in the corollary is a direct consequence of the definition of $X$, but can be verified to be a consequence of the TKK commutation relations in Theorem \ref{Main1} and Eq. (\ref{case1}).
\end{proof}
\section{J-Kepler Problems}\label{S:KP}
\begin{Definition}[J-Kepler Problem]\label{D:main2}
The {\bf J-Kepler problem} associated to a simple euclidean Jordan algebra
with rank $\rho$ and degree $\delta$ is the quantum mechanical system for
which the configuration space is the Kepler cone, and the
hamiltonian is
\begin{eqnarray}
\hat h=-{1\over 2}\Delta-\left( {B\over 2\langle e\mid x\rangle^2}+
{1\over \langle e\mid x\rangle}\right).
\end{eqnarray}
Here, $\Delta$ is the (non-positive) Laplace operator on the Kepler
cone, and $$B={\delta(\rho-2)\over 8}\left(\left({3\rho\over
2}-1\right)\delta-2\right).$$
\end{Definition}
\begin{rmk} In view of Eq. (\ref{Xdef}), we have another expression for $\hat h$:
\begin{eqnarray}
\hat h={1\over \langle e\mid x\rangle }\left({i\over 2}\tilde X_e-1\right).
\end{eqnarray}
\end{rmk}
\begin{rmk}
The J-Kepler problem is really the
quantum mechanical system for which the configuration space is the
geometric open cone over the projective space, and the hamiltonian
is
\begin{eqnarray}
\hat h=-{1\over 2}\Delta-\left( {B\over 2r^2}+ {1\over r}\right),
\end{eqnarray}
where $\Delta$ is the (non-positive) Laplace operator on the
open geometric cone over the projective space.
\end{rmk}
\begin{rmk}
The {\bf classical J-Kepler problem} is the classical mechanical system
for which the configuration space is the Kepler cone, and the
Lagrangian is
$$
L(x, \dot x)={1\over 2}|\dot x|^2+ {1\over
\langle e\mid x\rangle}.
$$
\end{rmk}
We are now ready to state
\begin{Prop}The J-Kepler problems are equivalent to the various
Kepler-type problems constructed and analyzed in Ref. \cite{meng},
but with zero magnetic charge. Here is the precise identification:
\begin{displaymath}
\begin{array}{|c|c|c|c|c|c|}
\hline
V & \Gamma(n) & {\mathcal H}_n(\mathbb R) & {\mathcal H}_n(\mathbb C) &{\mathcal H}_n(\mathbb H) & {\mathcal H}_3(\mathbb O)\\
\hline
& & & & &\\
\begin{matrix}\mbox{Kepler} \cr \mbox{problem}\end{matrix} & \begin{matrix}\mbox{MICZ in} \cr \mbox{dim. $n$}\end{matrix}
& \begin{matrix}\mbox{$\mathrm O(1)$ in} \cr \mbox{dim.
$n$}\end{matrix} &
\begin{matrix}\mbox{$\mathrm U(1)$ in} \cr \mbox{dim. $(2n-1)$}\end{matrix}
& \begin{matrix}\mbox{$\mathrm {Sp}(1)$ in} \cr \mbox{dim. $(4n-3)$}\end{matrix} & \mbox{exceptional}\\
& & & & &\\
\hline
\end{array}
\end{displaymath}
\end{Prop}
\begin{proof}
The proposition is clear for the MICZ-Kepler problems and
the exceptional Kepler problem. For the remaining cases, all one needs
is to make a transformation similar to the one appeared in the proof
of Proposition 2.2 of the first paper in Ref. \cite{meng}. For
example, for the $\mathrm O(1)$-Kepler problem in dimension $n$ with
zero magnetic charge, the configuration space is $\widetilde{\mathbb
RP^n}=\mathbb R^n_*/Z\sim -Z$ and the hamiltonian is
$$H=-{1\over 8z}\Delta_{\widetilde{\mathbb RP^n}}{1\over z}-{1\over z^2},$$ where
$\Delta_{\widetilde{\mathbb RP^n}}$ is the Laplace operator on
$\widetilde{\mathbb RP^n}$ and $z=|Z|$. Note that, with the quotient
metric induced from the euclidean metric of $\mathbb R^n$,
$\widetilde{\mathbb RP^n}$ is isometric to
$$\left(\mathbb R_+\times \mathbb RP^{n-1}, dz^2+z^2ds^2_{FS}\right),$$
where $ds^2_{FS}$ is the Fubini-Study metric on $\mathbb RP^{n-1}$.
To see the equivalence of the J-Kepler problem associated with $\mathcal H_n(\mathbb R)$ with the
$\mathrm O(1)$-Kepler problem in dimension $n$ with zero magnetic charge, we start with
diffeomorphism
\begin{eqnarray}
\widetilde{\mathbb RP^n} &\to & {\mathscr P} \cr [Z] &\mapsto & n ZZ'
\end{eqnarray}
or equivalently diffeomorphism $\pi$:
\begin{eqnarray}
\widetilde{\mathbb RP^n} &\to & \mathbb R_+\times \mathbb P\cr [Z] &\mapsto &
(z^2, \sqrt{2n}{ZZ'\over z^2}).
\end{eqnarray}
Here $Z$ is viewed as a column vector in $\mathbb R^n$ and $Z'$ is the transpose of $Z$.
Under $\pi$, we have
\begin{eqnarray}
\pi^*(dr^2+r^2\,ds^2_{\mathbb P}) = (2z)^2
(dz^2+z^2\,ds^2_{FS}),\nonumber
\end{eqnarray}
i.e.,
\begin{eqnarray}
\pi^*(ds^2_{{\mathscr P}})= (2z)^2ds^2_{\widetilde{\mathbb RP^n}},
\quad\mbox{so}\quad\pi^*(\mathrm{vol}_{\mathscr
P})=(2z)^n\mathrm{vol}_{\widetilde{\mathbb RP^n}}.
\end{eqnarray}
Let $\Psi_i$ ($i=1$ or $2$) be a wave-function for the J-Kepler problem associated with $\mathcal H_n(\mathbb R)$, and
$$\psi_i(z, \Theta):=(2z)^{n\over 2}\pi^*(\Psi_i)(z,
\Theta).$$ Then it is not hard to see that
$$\displaystyle \int_{\widetilde{\mathbb
RP^n}}\overline{\psi_1}\,\psi_2\,\mathrm{vol}_{\widetilde{\mathbb
RP^n}}=\displaystyle \int_{\widetilde{\mathbb
RP^n}}\overline{\pi^*(\Psi_1)}\pi^*(\Psi_2) \pi^*(\mathrm{vol}_{\mathscr
P})=\displaystyle \int_{{\mathscr P}}\overline{\Psi_1}
\Psi_2\mathrm{vol}_{{\mathscr P}}$$ and
\begin{eqnarray}\displaystyle
\displaystyle\int_{\widetilde{\mathbb RP^n}}\overline{\psi_1}H\psi_2
\mathrm{vol}_{\widetilde{\mathbb RP^n}} &=&
\displaystyle\int_{\widetilde{\mathbb
RP^n}}\overline{\pi^*(\Psi_1)}\,{1\over z^{n\over 2}}Hz^{n\over
2}\,\pi^*(\Psi_2) \,\pi^*(\mathrm{vol}_{{\mathscr P}})\cr&=&
\displaystyle\int_{{\mathscr P}}\overline{\Psi_1}\,{1\over z^{n\over
2}}Hz^{n\over 2}\,\Psi_2 \,\mathrm{vol}_{{\mathscr P}}. \nonumber
\end{eqnarray}
Since
\begin{eqnarray}
{1\over z^{n\over 2}}H z^{n\over 2} &=& -{1\over 8 z^{{n\over
2}+1}}\Delta_{\widetilde{\mathbb RP^n}} z^{{n\over 2}-1}-{1\over z^2}
\cr &=&-{1\over 8 z^{{n\over 2}+1}}\left({1\over
z^{n-1}}\partial_zz^{n-1}\partial_z+{1\over
z^2}\Delta_{FS}\right)z^{{n\over 2}-1}-{1\over z^2}\cr &=& -{1\over
8 z^{3n\over 2}}\partial_zz^{n-1}\partial_zz^{{n\over 2}-1}-{1\over
8z^4}\Delta_{FS}-{1\over z^2}\cr &=& -{1\over 2 r^{{3n\over
4}-{1\over 2}}}\partial_r r^{n\over 2}\partial_r r^{{n\over
4}-{1\over 2}}-{1\over 2r^2}\Delta_{\mathbb P}-{1\over r}\cr &=&
-{1\over 2}\left( \partial_r^2+{n-1\over r}\partial_r+{({n\over
4}-{1\over 2})({3n\over 4}-{3\over 2})\over r^2}+{1\over
r^2}\Delta_{\mathbb P}\right)-{1\over r}\cr &=& -{1\over 2}
\Delta-{3(n-2)^2\over 32r^2}-{1\over r}=\hat h,\nonumber
\end{eqnarray}
we have the equivalence of the J-Kepler problem associated with $\mathcal H_n(\mathbb R)$
with the $\mathrm O(1)$-Kepler problem in dimension $n$ with zero
magnetic charge.
\end{proof}
\subsection {The Lenz Vector}
The Lenz vector exists for J-Kepler problems.
\begin{Definition}[Lenz vector]
The Lenz vector for the J-Kepler problem is
$$A_u:={1\over \langle e\mid x\rangle}\left[\tilde L_u, (\langle e\mid x\rangle)^2\hat h\right],$$ i.e.,
\begin{eqnarray}
\fbox{$A_u ={i\over 2}\tilde X_u-\langle u\mid x\rangle \hat h ={i\over 2}\left(\tilde X_u-{\langle u\mid x\rangle\over \langle e\mid x\rangle}\tilde X_e\right)+{\langle u\mid x\rangle\over \langle e\mid x\rangle}$.}
\end{eqnarray}
\end{Definition}
Note that $A_e=1$. One might call $\vec A:=(A_{e_1}, \ldots, A_{e_D})$ the Lenz vector, that is because $\vec A$ is precisely the usual Lenz vector when the Jordan algebra is the Minkowski space.
\begin{Thm} For $u, v\in V$, we let
$L_{u,v}:=[\hat L_u, \hat L_v]$. Then we have commutation relations:
\begin{eqnarray}\label{CTR}
\fbox{$\begin{array}{lcl} [L_{u,v}, \hat h] & = & 0\cr
[L_{u,v}, L_{z, w}] & = & L_{[L_u, L_v]z,w}+ L_{z, [L_u, L_v]w}\cr
[L_{u,v}, A_z] &= &
A_{[L_u, L_v]z}\cr
[A_u, \hat h] & = & 0\cr[A_u, A_v] & = &
-2\hat h L_{u,v}.
\end{array}$}
\end{eqnarray}
\end{Thm}
\begin{rmk} 1) $L_{u,v}=[\tilde L_u, \tilde L_v]$. 2) $A_u$, $\hat h$ and $iL_{u,v}$ are all hermitian operators with respect to inner product
$$
(\psi, \phi)\mapsto \int_{\mathscr P} \psi^* \phi\, \mathrm{vol}_{\mathscr P}.
$$
\end{rmk}
\begin{proof}
Theorem \ref{Main1} implies that
$$
[L_{u,v}, \tilde X_z]= \tilde X_{[L_u, L_v]z}, \quad [L_{u,v}, \tilde Y_z]= \tilde Y_{[L_u, L_v]z},\quad [L_{u,v}, \tilde L_z] = \tilde L_{[L_u, L_v]z}.
$$
In particular, we have $[L_{u,v}, \tilde X_e]=[L_{u,v}, \tilde Y_e]=0$. Therefore, $[L_{u, v}, H]=0$,
\begin{eqnarray}
[L_{u, v}, A_z] &=& {1\over \langle e\mid x\rangle}[L_{u,v}, [\tilde L_z, (\langle e\mid x\rangle)^2\hat h]]\cr
&=& {1\over \langle e\mid x\rangle}\left([(\langle e\mid x\rangle)^2\hat h, [\tilde L_z, L_{u,v}]]+[\tilde L_z, [L_{u,v}, (\langle e\mid x\rangle)^2\hat h] ]\right)\cr
&=& {1\over \langle e\mid x\rangle} [(\langle e\mid x\rangle)^2\hat h, -\tilde L_{[L_u, L_v]z}]\cr
&=& A_{[L_u, L_v]z},\nonumber
\end{eqnarray}
and
\begin{eqnarray}
[L_{u, v}, L_{z, w}] &=& [L_{u,v}, [\tilde L_z, \tilde L_w]]\cr
&=& [[L_{u,v}, \tilde L_z], \tilde L_w]+ [\tilde L_z, [L_{u,v}, \tilde L_w]]\cr
&=& [ \tilde L_{[L_u, L_v]z}, \tilde L_w]+ [\tilde L_z, \tilde L_{[L_u, L_v]w}]]\cr
&=&L_{[L_u, L_v]z, w}+ L_{z,[L_u, L_v]w}.\nonumber
\end{eqnarray}
Since $\hat h={1\over \langle e\mid x\rangle}({i\over 2}\tilde X_e-1)$, we have
\begin{eqnarray}
[A_u, \hat h] &=& [A_u,{1\over \langle e\mid x\rangle}]({i\over 2}\tilde X_e-1)+{1\over \langle e\mid x\rangle}[A_u, {i\over 2}\tilde X_e]\cr
&=& -{1\over \langle e\mid x\rangle}[A_u,\langle e\mid x\rangle]{1\over \langle e\mid x\rangle} ({i\over 2}\tilde X_e-1)+{1\over \langle e\mid x\rangle}[A_u, {i\over 2}\tilde X_e]\cr
&=& -{1\over \langle e\mid x\rangle}[A_u,\langle e\mid x\rangle]\hat h-{1\over \langle e\mid x\rangle}\left[{\langle u\mid x\rangle\over \langle e\mid x\rangle}, {i\over 2}\tilde X_e\right]\langle e\mid x\rangle \hat h\cr
&=& -{1\over \langle e\mid x\rangle}[A_u,\langle e\mid x\rangle]\hat h\cr
&&-{1\over \langle e\mid x\rangle}\left( [\langle u\mid x\rangle, {i\over 2}\tilde X_e]-{\langle u\mid x\rangle\over \langle e\mid x\rangle}[\langle e\mid x\rangle, {i\over 2}\tilde X_e]\right) \hat h\cr
&=& -{1\over \langle e\mid x\rangle}[A_u,\langle e\mid x\rangle]\hat h
+{1\over \langle e\mid x\rangle}\left(\tilde L_u-{\langle u\mid x\rangle\over \langle e\mid x\rangle}\tilde L_e\right) \hat h\cr
&=&0.\nonumber
\end{eqnarray}
Since $A_u= {i\over 2}\tilde X_u-\langle u\mid x\rangle \hat h$, we have
\begin{eqnarray}
[A_u, A_v] &=& [{i\over 2}\tilde X_u, -\langle v\mid x\rangle \hat h ]-[{i\over 2}\tilde X_v, -\langle u\mid x\rangle \hat h ]+[\langle u\mid x\rangle \hat h, \langle v\mid x\rangle \hat h]\cr
&=& [{i\over 2}\tilde X_u, -\langle v\mid x\rangle \hat h ]-\langle v\mid x\rangle [\hat h, \langle u\mid x\rangle] \hat h-<u\leftrightarrow v>\cr
&=& [{i\over 2}\tilde X_u, -\langle v\mid x\rangle] \hat h -\langle v\mid x\rangle[{i\over 2}\tilde X_u, \hat h] -\langle v\mid x\rangle [\hat h, \langle u\mid x\rangle] \hat h\cr && -<u\leftrightarrow v>\cr
&=& -\tilde S_{uv} \hat h +{\langle v\mid x\rangle\over \langle e\mid x\rangle}[{i\over 2}\tilde X_u, \langle e\mid x\rangle] H-{\langle v\mid x\rangle\over \langle e\mid x\rangle} [{i\over 2}\tilde X_e, \langle u\mid x\rangle] \hat h\cr && -<u\leftrightarrow v>\cr
&=& -\tilde S_{uv} \hat h +{\langle v\mid x\rangle\over \langle e\mid x\rangle}\tilde L_u \hat h-{\langle v\mid x\rangle\over \langle e\mid x\rangle}\tilde L_u \hat h -<u\leftrightarrow v>\cr
&=& -2L_{u,v} \hat h =-2\hat h L_{u,v}.\nonumber
\end{eqnarray}
\end{proof}
\section{Symmetry Analysis of the J-Kepler Problems}\label{S:Summary}
The goal of this section is to give a detailed dynamical symmetry
analysis for the J-Kepler problem, as a byproduct, we solve the
bound state problem for the J-Kepler problem algebraically.
Unless said otherwise, throughout this section we assume that $V$ is a simple
euclidean Jordan algebra with rank $\rho\ge 2$ and degree $\delta$. For
simplicity, for each $x\in {\mathscr P}$, we shall rewrite $\langle e\mid
x\rangle$ as $r$.
\subsection{Harmonic Analysis on Projective Spaces}
Let us begin with the harmonic analysis on projective
space $\mathbb P$. Since $\mathbb P$ is a real affine variety inside $V$,
its coordinate ring ${\mathbb R}[{\mathbb P}]$ is a quotient of the ring of
real polynomial functions on $V$. Recall that we use $\Delta_{\mathbb
P}$ to denote the Laplace operator on $\mathbb P$.
\begin{Lem}\label{Lemma8}
Let $V_m$ be the set of regular (i.e., polynomial) functions (on $\mathbb P$) of degree at
most $m$, ${\mathcal V}_m$ be the orthogonal complement of $V_{m-1}$
in $V_m$. For each $u\in V$, we let $m_u$:
${\mathbb R}[{\mathbb P}]\to {\mathbb R}[{\mathbb P}]$ denote the multiplication by
$\langle u\mid x\rangle$.
i) For any integer $k\ge 0$, there is a $u\in V$ perpendicular to $e$ such that
$$
\tilde m_u: \;\mathcal V_k\buildrel m_u\over \to V_{k+1}\buildrel \pi\over \to \mathcal V_{k+1}
$$
is nonzero. Here, $\pi$ is the orthogonal projection.
ii) ${\mathcal V}_m^{\mathbb C}:={\mathcal V}_m\otimes_{\mathbb R} \mathbb C$ is the $m$-th eigenspace of $\Delta_{\mathbb P}$
with eigenvalue $-m(m+{\rho \delta\over 2}-1)$, and the Hilbert space of
square integrable complex-valued functions on $\mathbb P$ admits an
orthogonal decomposition into the eigenspaces of $\Delta_{\mathbb P}$:
\begin{eqnarray}\label{decom}
L^2({\mathbb P})=\hat \bigoplus_{m\ge 0} {\mathcal V}_m^{\mathbb C}.
\end{eqnarray}
\end{Lem}
\begin{proof} It is clear that
$$V_m=\bigoplus_{k=0}^m{\mathcal V}_k, \quad {\mathbb R}[{\mathbb P}]=\bigoplus_{k=0}^\infty{\mathcal
V}_k.$$
i) Let $\{e\}\cup\{e_i\mid 1\le i\le \dim V-1\}$ be an orthonomal
basis for $V$ and $x_i=\langle
e_i\mid x\rangle$. Since $m_u$ maps $V_k$ to $V_{k+1}$ for each
integer $k\ge 0$, we have the resulting map
$$
\overline{m_u}:\; V_k/V_{k-1}\to V_{k+1}/V_{k}.
$$
For any $\bar f\in V_{k+1}/V_{k}$, if we write $\bar f=\sum_{i>0}
\overline{x_ig_i}$, we have $\bar f=\sum_{i>0}
\overline{m_{e_i}}(\overline{g_i})$. In view of the commutative
diagram
$$\begin{CD}
{\mathcal V}_k @>\tilde m_u>> {\mathcal V}_{k+1}\\
@VV \cong V @VV\cong V\\
V_k/V_{k-1} @>\overline{m_u}>> V_{k+1}/V_{k}\; ,
\end{CD}
$$ we have
\begin{eqnarray}\label{nonzero}
{\mathcal V}_{k+1}=\sum_{i>0} \tilde m_{e_i}({\mathcal V}_{k})
\end{eqnarray} for any $k\ge 0$.
Suppose that $\tilde m_{e_i}$: ${\mathcal V}_m \to {\mathcal
V}_{m+1}$ is zero for any $i>0$, then ${\mathcal V}_{m+1}=0$ per Eq. (\ref{nonzero}), so ${\mathcal V}_n=0$ for any $n\ge m+1$ per Eq. (\ref{nonzero}), then $${\mathbb R}[{\mathbb P}]=\lim_{k\to\infty} V_k=V_m,$$ a contradiction.
ii) Let $u_1$, \ldots, $u_m$ be in $V$, and write $x_{u_i}$ for
$\langle u_i\mid x\rangle$, $u_i^0$ for $\langle e\mid u_i\rangle$. Since
\begin{eqnarray}
[\Delta_{\mathbb P}, x_{u_1}\cdots x_{u_m}] &=& \sum_{i=1}^m
x_{u_1}\cdots x_{u_{i-1}}[\Delta_{\mathbb P},x_{u_i}]x_{u_{i+1}}\cdots
x_{u_m}, \nonumber
\end{eqnarray}
in view of Eq. (\ref{identity}) and part ii) of Lemma \ref{lemma}, we have
\begin{eqnarray}
\Delta_{\mathbb P}\left( x_{u_1}\cdots x_{u_m}\right) &=& \sum_{i=1}^m
x_{u_1}\cdots x_{u_{i-1}}[\Delta_{\mathbb P},x_{u_i}](x_{u_{i+1}}\cdots
x_{u_m})\cr &=& \sum_{i=1}^m x_{u_1}\cdots x_{u_{i-1}}(-2r\tilde
L_{u_i}+2x_{u_i}\tilde L_e)(x_{u_{i+1}}\cdots x_{u_m})\cr &=&
-m{\rho \delta\over 2}x_{u_1}\cdots x_{u_m}+{\rho \delta\over
2}ru_i^0 \sum_{i=1}^m x_{u_1}\cdots \hat x_{u_i}\cdots x_{u_m}\cr
&&+\sum_{i=1}^m x_{u_1}\cdots x_{u_{i-1}}(-2r\hat
L_{u_i}+2x_{u_i}\hat L_e)(x_{u_{i+1}}\cdots x_{u_m})\cr &=& -(m{\rho
\delta\over 2}+m(m-1))x_{u_1}\cdots x_{u_m}+{\rho \delta\over
2}r\sum_{i=1}^m u_i^0 x_{u_1}\cdots \hat x_{u_i}\cdots x_{u_m}\cr
&&-2r\sum_{i=1}^m x_{u_1}\cdots
x_{u_{i-1}}\hat L_{u_i}(x_{u_{i+1}}\cdots x_{u_m})\cr &\equiv& -m(m+{\rho
\delta\over 2}-1)x_{u_1}\cdots x_{u_m}\quad (\mod V_{m-1})\nonumber
\end{eqnarray} because $r=\sqrt{2/\rho}$ on $\mathbb P$.
It is then clear that $\Delta_{\mathbb P}$ maps $V_m$ into $V_m$ for
each $m\ge 0$ and resulting map
$$\overline{\Delta_{\mathbb P}}:\; V_m/V_{m-1}\to V_m/V_{m-1}$$
is the scalar multiplication by $-m(m+{\rho \delta\over 2}-1)$. Since
$\Delta_{\mathbb P}$ is a hermitian operator and maps $V_{m-1}$ into $V_{m-1}$,
$\Delta_{\mathbb P}$ maps ${\mathcal V}_m$ into ${\mathcal V}_m$, so we have
commutative diagram
$$\begin{CD}
{\mathcal V}_m @>\Delta_{\mathbb P}>> {\mathcal V}_m\\
@VV\cong V @VV\cong V\\
V_m/V_{m-1} @>\overline{\Delta_{\mathbb P}}>> V_m/V_{m-1}\; .
\end{CD}
$$
Since ${\mathcal V}_m\neq\{0\}$ per part i), we conclude that
${\mathcal V}_m$ is an eigenspace of $\Delta_{\mathbb P}$ with
eigenvalue $-m(m+{\rho \delta\over 2}-1)$.
Since the ring of regular functions is dense in the ring of real
continuous functions and
$$
{\mathbb R}[{\mathbb P}]=\bigoplus_{k=0}^\infty{\mathcal
V}_k,
$$
we have
$$
L^2({\mathbb P})=\hat \bigoplus_{m\ge 0} {\mathcal V}_m^{\mathbb C}.
$$
\end{proof}
\subsection{Associated Lagueree Polynomials} Here, we give a quick review of the associated Lagueree
polynomials. Let $\alpha$ be a real number and $n\ge 0$ be an
integer. By definition, the associated Lagueree polynomial
$L^\alpha_n(x)$ is the polynomial solution of equation
\begin{eqnarray}\label{LaguereeEq}
xy''+(\alpha+1-x)y'+ny=0
\end{eqnarray} whose the leading
coefficient is $(-1)^n{1\over n!}$. We note that $L^\alpha_n(x)$ has
degree $n$, for example,
\begin{eqnarray}
L_0^\alpha(x) &=& 1\cr L_1^\alpha(x) &=& -x+\alpha+1\cr
L_2^\alpha(x)&=&{1\over 2}x^2-(\alpha+2)x+{1\over
2}(\alpha+1)(\alpha+2)\cr
& \vdots&\nonumber
\end{eqnarray}
In general, we have
\begin{eqnarray}
L_n^\alpha(x)={x^{-\alpha}e^x\over n!}{d^n\over
dx^n}\left(e^{-x}x^{n+\alpha}\right).
\end{eqnarray}
It is a fact that $L^\alpha_n(x)$'s form an orthogonal basis for
$L^2({\mathbb R}_+, x^\alpha e^{-x}\,dx)$:
\begin{eqnarray}
\displaystyle \int _0^\infty x^\alpha e^{-x}\,
L_n^\alpha(x)L_m^\alpha(x)\, dx={\Gamma(n+\alpha+1)\over
n!}\delta_{mn};
\end{eqnarray}moreover, a degree $n$ polynomial in $x$ can be
uniquely written as a linear combination of $L_k^\alpha(x)$ with
$0\le k\le n$.
It is then clear that,
\begin{eqnarray}\label{Leguree}
\fbox{$\begin{array}{c} \mbox{\em for any integer $l\ge 0$,
$x^{l-{(\rho/2-1)\delta\over 2}}e^{-x}L^{2l+{\rho \delta\over 2}-1}_n(2x)$'s
form}\\ \text{\em an orthogonal basis for $L^2({\mathbb R}_+,
x^{(\rho-1)\delta-1}\,dx)$}.\end{array}$}
\end{eqnarray}
It is also a fact that
\begin{eqnarray}\label{recursive1}
nL_n^\alpha(x)=(n+\alpha)L_{n-1}^\alpha(x)-xL_{n-1}^{\alpha+1}(x)
\end{eqnarray}
and \begin{eqnarray}\label{recursive2} L_{n+1}^\alpha(x)={1\over
n+1}\left((2n+1+\alpha-x)L_n^\alpha(x)-(n+\alpha)L_{n-1}^\alpha(x)\right).
\end{eqnarray}
\subsection{Hidden Harmonic Analysis on Kepler Cones}
For each integer $l\ge 0$, we fix an orthonormal spanning set
$\{Y_{lm}\mid m\in{\mathcal I}(l)\}$ for ${\mathcal V}_l$. Note that
each $Y_{lm}$ can be represented by a homogeneous degree
$l$-polynomial in $x$, which will be denoted by $Y_{lm}(x)$. For integer $k\ge 1$, we introduce
\begin{eqnarray}\label{wavefunction}
\varphi_{klm}(x): = r^{-{(\rho/2-1)\delta\over 2}}L_{k-1}^{2l+{\rho
\delta\over 2}-1}(2r) e^{-r} Y_{l m}(x)
\end{eqnarray} where $r=\langle e\mid x\rangle$. One can verify that
$\varphi_{klm}$ is square integrable with respect to ${1\over r}\mathrm{vol}_{\mathscr P}$:
\begin{eqnarray} \left({2\over
\rho}\right)^l\displaystyle\int_{\mathscr P}
|\varphi_{klm}|^2\,{1\over r}\mathrm{vol}_{\mathscr P}&=& \int_{\mathbb P}|Y_{lm}|^2\,\mathrm{vol}_{\mathbb P}\;\cdot \cr && \cdot
\int_0^\infty r^{2l-(\rho/2-1)\delta}\cdot(L_{k-1}^{2l+{\rho \delta\over
2}-1}(2r))^2\cdot r^{(\rho-1)\delta-1}e^{-2r}\, dr\cr
&=&\int_0^\infty r^{2l+{\rho \delta\over 2}-1}(L_{k-1}^{2l+{\rho
\delta\over 2}-1}(2r))^2e^{-2r}\, dr\cr &= & {\Gamma(2l+{\rho \delta\over
2}-1+k)\over 2^{2l+{\rho \delta/2}}(k-1)!}<\infty\nonumber
\end{eqnarray} because $\delta\ge 1$ and $\rho\ge 1$.
Let $H_0:=-{i\over 2}(X_e+Y_e)$, then
$$
\tilde H_0={1\over 2}\left({\hat L_e^2-(2\lambda_e-1)\hat
L_e+\Delta_{\mathbb P}+B\over r}-r\right).
$$
We say that a smooth nonzero function $\varphi$ on the Kepler cone
is an {\bf eigenfunction} of $\tilde H_0$ if it is square integrable
with respect to ${1\over r}\mathrm{vol}_{\mathscr P}$ and satisfies equation
$$
\tilde H_0\varphi=\lambda \varphi
$$ for some real number $\lambda$. With the help of part ii) of Lemma \ref{Lemma8} and Eq. (\ref{LaguereeEq}), one can check that
$$\tilde H_0\varphi_{klm} = -(l+k-1+{\rho \delta\over 4})\varphi_{klm},$$
so {\em $\varphi_{klm}$ is an eigenfunction of $\tilde H_0$ with eigenvalue
$-(l+k-1+{\rho \delta\over 4})$}.
Let ${\mathscr V}_l(k):=\mathrm{span}_{\mathbb C}\{\varphi_{klm}\mid m\in {\mathcal I}(l)\}$ for each integer $l\ge 0$ and
$k\ge 1$, and
$$
\tilde {\mathscr H}_I:=\bigoplus_{l=0}^{I}{\mathscr V}_l(I+1-l)
$$for each integer $I\ge 0$. Finally, we let $\tilde {\mathcal H}:=\bigoplus_{I=0}^\infty \tilde {\mathscr H}_I$ and
$\pi(\mathcal O):=\tilde{\mathcal O}$ for any $\mathcal O$ in the
conformal algebra. In view of statement
(\ref{Leguree}), it is clear that $\varphi_{klm}$'s form an orthogonal basis for $\tilde {\mathcal H}$.
\begin{Prop}
i) $\tilde {\mathscr H}_I$ is the eigenspace of $\tilde H_0$ with eigenvalue
$-(I+\rho \delta/4)$ and
\begin{eqnarray} L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P})=
\hat \bigoplus_{I=0}^\infty \tilde {\mathscr H}_I.\nonumber
\end{eqnarray}
Moreover, $\varphi_{klm}$'s form an orthogonal basis for $L^2({\mathscr
P}, {1\over r}\mathrm{vol}_{\mathscr P})$.
ii) $(\pi,\tilde {\mathcal H})$ is a unitary representation of the conformal
algebra.
iii) $(\pi|_{\bar{\mathfrak u}}, \tilde {\mathscr H}_I)$ is an irreducible
representation of $\bar{\mathfrak u}$. Consequently $(\pi|_{{\mathfrak u}},
\tilde {\mathscr H}_I)$ is an irreducible representation of ${\mathfrak u}$.
iv) $(\pi,\tilde {\mathcal H})$ is a unitary lowest weight representation of the
conformal algebra with lowest weight equal to ${\delta \over
2}\lambda_0$. Here $\lambda_0$ is the fundamental weight conjugate
to the unique non-compact simple root $\alpha_0$ in Lemma
\ref{LemmaVogan}, i.e., $\lambda_0(H_{\alpha_i})=0$ for $i>0$ and
$$
2{\lambda_0(H_{\alpha_0})\over \alpha_0(H_{\alpha_0})}=1.
$$
\end{Prop}
\begin{proof}
i) By virtue of Theorem II.10 of Ref. \cite{Reed&Simon}, we have
$$L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P})=L^2(\mathbb R_+,
r^{(\rho-1)\delta-1}\,dr)\otimes L^2({\mathbb P}).$$ Then
\begin{eqnarray}
L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P}) & = & \hat
\bigoplus_{l=0}^\infty \left(L^2(\mathbb R_+,
r^{(\rho-1)\delta-1}\,dr)\otimes {\mathcal V}_l^{\mathbb C}\right)\quad \text{Eq.
(\ref{decom})}\cr &=& \hat \bigoplus_{l=0}^\infty \hat
\bigoplus_{k=1}^\infty \bigoplus_{m\in {\mathcal I}(l)}
\mathrm{span}_{\mathbb C}\{\varphi_{klm}\}\quad\text{statement
(\ref{Leguree})}\cr &=& \hat \bigoplus_{I=0}^\infty \tilde {\mathscr
H}_I.\nonumber
\end{eqnarray}
Therefore, we conclude that $\{\varphi_{klm}\}$ is an orthogonal
basis for $L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P})$ and $\tilde {\mathscr H_I}$
is the $I$-th eigenspace of $\tilde H_0$ with eigenvalue $-(I+\rho \delta/4)$.
ii) First, we need to show that $\tilde{\mathcal O}(\psi)\in \tilde {\mathcal
H}$ for any $\mathcal O\in \mathfrak{co}$ and any $\psi\in \tilde {\mathcal H}$.
Without loss of generality we may assume that $\mathcal O$ is $L_u$,
$X_e$ or $Y_e$ and
$$\psi=\varphi_{klm}=r^{-{(\rho/2-1)\delta\over 2}}L_{k}^{2l+{\rho \delta\over
2}-1}(2r) e^{-r} Y_{l m}(x).$$ Using Eq. (\ref{recursive2}) one can
see that $\tilde Y_e(\varphi_{klm})\in\tilde {\mathcal H}$. Then
\begin{eqnarray}
{\tilde X_e}(\varphi_{klm}) &= &(2\sqrt{-1}\tilde H_0-\tilde
Y_e)(\varphi_{klm})\cr
&= & 2\sqrt{-1}(k+l-1+{\rho \delta\over 2})\varphi_{klm}-\tilde
Y_e(\varphi_{klm})\cr
&\in &\tilde {\mathcal H}.\nonumber
\end{eqnarray}
Since
\begin{eqnarray}
\hat L_u(\varphi_{klm}) &= &\hat L_u(r^{-{(\rho/2-1)\delta\over
2}}L_{k}^{2l+{\rho \delta\over 2}-1}(2r) e^{-r}) Y_{l m}(x)\cr &&
+r^{-{(\rho/2-1)\delta\over 2}}L_{k}^{2l+{\rho \delta\over 2}-1}(2r) e^{-r}
\hat L_u(Y_{l m}(x)), \nonumber
\end{eqnarray}
one can see that $\hat L_u(\varphi_{klm})\in \bigoplus_{k'\le k+l+1, l'\le
l+1}{\mathscr V}_{l'}(k')$. It is also clear that $$\lambda_u\cdot \varphi_{klm}\in
\bigoplus_{k'\le k+l+1, l'\le l+1}{\mathscr V}_{l'}(k'),$$
so $\tilde L_u(\varphi_{klm})\in \tilde {\mathcal H}$.
Next, we verify that
\begin{eqnarray}
(\varphi_{klm},\,\tilde{\mathcal O}(\varphi_{k'l'm'}))+(\tilde
{\mathcal O}(\varphi_{klm}),\,\varphi_{k'l'm'})=0
\end{eqnarray}for $\mathcal O\in \mathfrak{co}$. We may assume that $\mathcal O$ is $L_u$, $X_e$ or
$Y_e$. It is clearly OK when $\mathcal O=Y_e$ because $\tilde
Y_e=-ir$. Since $\tilde X_e=2i \tilde H_0-\tilde Y_e$, to show that $\tilde X_e$ is anti-hermitian, it suffices to verify that $(\varphi_{klm},\,\tilde
H_0(\varphi_{k'l'm'}))-(\tilde
H_0(\varphi_{klm}),\,\varphi_{k'l'm'})=0$ or
$(k'+l'-k-l)(\varphi_{klm},\,\varphi_{k'l'm'})=0$, which is
obviously true.
To verify that $(\varphi_{klm},\,\tilde
L_u(\varphi_{k'l'm'}))+(\tilde
L_u(\varphi_{klm}),\,\varphi_{k'l'm'})=0$, in view of part iii) of
Lemma \ref{KeyLemma}, we know that $(\varphi_{klm},\,\tilde
L_u(\varphi_{k'l'm'}))+(\tilde
L_u(\varphi_{klm}),\,\varphi_{k'l'm'})$ is equal to
$$ \displaystyle\int_{{\mathscr P}}{\mathscr L}_u(\overline{\varphi_{klm}}\,\varphi_{k'l'm'}\,
{1\over r}\mathrm{vol}_{{\mathscr P}})=\displaystyle\int_{\mathscr P}d \iota_{\hat
L_u}(\overline{\varphi_{klm}}\,\varphi_{k'l'm'}\, {1\over
r}\mathrm{vol}_{\mathscr P}) =0.$$ That is because $\iota_{\hat
L_u}(\overline{\varphi_{klm}}\,\varphi_{k'l'm'}\, {1\over
r}\mathrm{vol}_{\mathscr P})$ approaches to zero exponentially fast as $r\to
\infty$ and approaches to zero as $r\to 0$, uniformly with respect
to the angle directions.
iii) First, we verify that $\tilde {\mathscr H}_I$ is invariant under the
action of $\bar{\mathfrak u}$. To see this, we note that $\tilde {\mathscr H}_I$ is an
eigenspace of $\tilde H_0$, moreover, as operators on Hilbert space
$L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P})$, $\tilde H_0$ commutes with $\tilde {\mathcal O}$
for any $\mathcal O\in \bar{\mathfrak u}$.
Since $\tilde {\mathscr H}_I=\bigoplus_{l=0}^I{\mathscr V}_l(I+1-l)$, if the action
of $\bar{\mathfrak u}$ on $\tilde {\mathscr H}_I$ were not irreducible, there would be
an integer $l$ with $0\le l< I$ such that $(\psi_l, \tilde {\mathcal
O}(\psi_{l+1}))=0$ for any $\psi_l\in {\mathscr V}_l(I+1-l)$,
$\psi_{l+1}\in {\mathscr V}_{l+1}(I-l)$, and any $\mathcal O \in \bar{\mathfrak
k}$.
In view of part i) of Lemma \ref{Lemma8}, we can choose a $u\in V$
with $u\perp e$ such that $\tilde m_u$: ${\mathcal V}_l\to {\mathcal
V}_{l+1}$ is nontrivial; so there is a $Y_{l m}\in {\mathcal V}_l$ and a
$Y_{(l+1) m'}\in {\mathcal V}_{l+1}$ such that
\begin{eqnarray}\label{notzero}
\int_{\mathbb P}Y_{(l+1)m'}\cdot \tilde m_u(Y_{lm})\, \mathrm{vol}_{\mathbb P}\neq 0.
\end{eqnarray}
Let
\begin{eqnarray}
\psi_l(x)&=& r^{-{(\rho/2-1)\delta\over 2}}L_{k}^{2l+{\rho \delta\over
2}-1}(2r) e^{-r} Y_{l m}(x),\cr \psi_{l+1}(x)&=&
r^{-{(\rho/2-1)\delta\over 2}}L_{k-1}^{2l+{\rho \delta\over 2}+1}(2r) e^{-r}
Y_{(l+1) m'}(x),\cr
{\mathcal O} &=& X_u+Y_u.\nonumber
\end{eqnarray}
Then $\mathcal O\in \bar{\mathfrak u}$ because $u\perp e$. Since
$\tilde {\mathcal O}=[\tilde L_u, \tilde X_e-\tilde Y_e]=[2i\tilde
L_u, \tilde H_0]+2\tilde Y_u$, we have $(\psi_l, \tilde {\mathcal
O}(\psi_{l+1}))=(\psi_l, 2\tilde Y_u \cdot \psi_{l+1})$; so, in
view of Eq. (\ref{notzero}), $(\psi_l, \tilde {\mathcal
O}(\psi_{l+1}))=0$ would imply that
\begin{eqnarray}
\int_0^\infty x^{\alpha+2} e^{-x}\, L_k^{\alpha}(x)
L_{k-1}^{\alpha+2}(x)\, dx =0. \nonumber\end{eqnarray} where $\alpha=2l+\rho
\delta/2 -1$. But that is a contradiction: using Eq. (\ref{recursive1}), one can show that
\begin{eqnarray}
\int_0^\infty x^{\alpha+2} e^{-x}\, L_k^{\alpha}(x)
L_{k-1}^{\alpha+2}(x)\, dx = -2{\Gamma(k+\alpha+2)\over (k-1)!}\neq
0.\nonumber
\end{eqnarray}
iv) Let us take the simple root system $\alpha_0$, \ldots, $\alpha_r$
specified in Lemma \ref{LemmaVogan} and $$\psi_0(x) = r^{-{(\rho/2-1)\delta\over 2}}e^{-r}.$$
Since $\tilde {\mathscr H}_0$ ($=\mathrm{span}_{\mathbb C}\{\psi_0\}$) is one dimensional
and $\bar{ \mathfrak u}$ is semi-simple, the action of $\bar{ \mathfrak u}^{\mathbb C}$ on $\tilde {\mathscr H}_0$ must be
trivial. Therefore, for $i\ge 1$, in view of the fact that $E_{\pm\alpha_i}, H_{\alpha_i}\in
\bar{ \mathfrak u}^{\mathbb C}$, we have
\begin{eqnarray}\label{lw1}
\tilde E_{-\alpha_i}\psi_0=0, \quad \tilde H_{\alpha_i} \psi_0=0.
\end{eqnarray}
On the other hand, since $E_{-\alpha_0}={i\over
2}(X_{e_{11}}-Y_{e_{11}})+L_{e_{11}}$ and
$H_{\alpha_0}=i(X_{e_{11}}+Y_{e_{11}})\equiv -{2\over \rho}H_0(\mod \bar {\mathfrak u})$, by a computation, we have
\begin{eqnarray}\label{lw2}
\tilde E_{-\alpha_0}\psi_0=0, \quad \tilde H_{\alpha_0}\psi_0={\delta\over 2}\psi_0.
\end{eqnarray}
Therefore, in view of the fact that $\alpha_0(H_{\alpha_0})=2$, {\em $\psi_0$ is a lowest weight state with weight ${\delta\over 2}\lambda_0$}.
Since operator $\tilde Y_v$ is the multiplication by $-i\langle v\mid x\rangle$, we have
$$
r^{-{(\rho/2-1)\delta\over 2}}e^{-r}\sum_{i_1, \ldots, i_n} \alpha_{i_1\cdots i_n} x_1^{i_1}\cdots x_n^{i_n}=\left(\sum_{i_1, \ldots, i_n} \alpha_{i_1\cdots i_n} (i\tilde Y_{e_1})^{i_1}\cdots(i\tilde Y_{e_n})^{i_n}\right) \psi_0,
$$
so the representation $(\pi, \tilde {\mathscr H})$ is generated from $\psi_0$. Since this representation is unitary, it must be irreducible.
In summary, $(\pi, \tilde {\mathscr H})$ is a unitary lowest weight representation with lowest weight ${\delta\over 2}\lambda_0$.
\end{proof}
\begin{rmk}
Let $\mathcal P$ be the space of regular functions on $\mathscr P$, i.e.,
$$
{\mathcal P}=\{p:{\mathscr P}\to {\mathbb C}\mid \mbox{$p$ is a polynomial on $V$} \}.
$$
In view of Eqn. (\ref{wavefunction}),
$$
{\tilde D}:=e^{-r} r^{-{(\rho/2-1)d\over 2}}{\mathcal P}
$$
can be taken as a dense common domain of definition for $\tilde {\mathcal O}$, $\mathcal O\in \mathfrak{co}$.
\end{rmk}
The following main theorem is an easy corollary of the above proposition.
\begin{Thm}\label{main2}
Let $\mathrm{Co}$ be the conformal group of the Jordan algebra with rank at least two, $\mathrm K$
be the closed Lie subgroup of $\mathrm{Co}$ whose Lie algebra is $\mathfrak u$,
$\lambda_0$ be the fundamental weight conjugate to the unique non-compact simple root $\alpha_0$ in Lemma
\ref{LemmaVogan}, $\tilde H_0:=-{i\over 2}(\tilde X_e+\tilde Y_e)$, $\tilde {\mathscr H}_I$ be
the $I$-th eigenspace of $\tilde H_0$, and $\tilde {\mathcal
H}:=\bigoplus _{I=0}^\infty\tilde {\mathscr H}_I$.
1) The hidden action $\pi$ in Theorem \ref{Main1} turns $\tilde{\mathcal H}$
into a unitary lowest weight $(\mathfrak{co}, \mathrm{K})$-module with
lowest weight ${\delta\over 2}\lambda_0$. Here the action is unitary
with respect to inner product
$$
(\psi_1, \psi_2)=\displaystyle\int_{\mathscr
P}\overline\psi_1\,\psi_2\,{1\over r}\mathrm{vol}_{\mathscr P}\; .
$$
2) The unitary lowest weight representation of $\mathrm{Co}$, whose underlying $(\mathfrak{co}, \mathrm{K})$-module is
the $(\mathfrak{co}, \mathrm{K})$-module in part 1), can be realized by $L^2({\mathscr P}, {1\over r}\mathrm{vol}_{\mathscr P})$.
3) Decomposition $\tilde {\mathcal H}=\bigoplus _{I=0}^\infty\tilde {\mathscr H}_I$ is
a multiplicity free $K$-type formula.
\end{Thm}
Note that, the unitary lowest weight representation of $\mathrm{Co}$ appeared in this theorem is the minimal representation of $\mathrm{Co}$ in the sense of A. Joseph \cite{Joseph1974}, and has the smallest positive Gelfand-Kirillov dimension. This theorem has a more general version which takes care of
all unitary lowest weight representations of the smallest positive Gelfand-Kirillov dimension. Since it is a refinement of part (ii) of Theorem XIII.3.4 from Ref. \cite{FK91} for the case $\nu={\delta\over 2}$ there, this theorem can be conceivably generalized to cover the case for a generic $\nu$ there.
\subsection{Solution of the J-Kepler Problems}
For a J-Kepler problem, we are primarily interested in solving
the bound state problem here, i.e., the following (energy) spectrum problem:
\begin{eqnarray}\label{eigen}
\left\{\begin{array}{rcl}
\hat h\psi & = & E\psi\\
\\
\displaystyle\int_{{\mathscr P}} |\psi|^2\, \mathrm{vol}_{\mathscr P}&< & \infty, \quad \psi\not\equiv 0.
\end{array}\right.
\end{eqnarray}
It turns out that $E$ has to take ceratin discrete values. For
example, for the original Kepler problem, we have
$$
E=-{1\over 2n^2}, \quad n=1, 2, \ldots
$$
The {\bf Hilbert space of bound states}, denoted by $\mathscr H$, is
defined to be the completion of the linear span of all
eigenfunctions of $\hat h$.
\begin{Thm}\label{main3} Let $V$ be a simple euclidean Jordan
algebra with rank $\rho\ge 2$ and degree $\delta$ and $\mathrm{Co}$ be the conformal group of $V$. For the J-Kepler problem associated to $V$, the following statements are true:
1) The bound state energy spectrum is
$$
E_I=-{1/2\over (I+{\rho \delta\over 4})^2}
$$ where $I=0$, $1$, $2$, \ldots
2) There is a unitary action of $\mathrm{Co}$ on the Hilbert space of bound states, ${\mathscr
H}$. In fact, ${\mathscr H}$ provides a realization for
the minimal representation of the conformal
group $\mathrm{Co}$.
3) The orthogonal decomposition of $\mathscr H$ into the energy
eigenspaces is just the multiplicity free $K$-type formula for the
minimal representation.
\end{Thm}
\begin{proof}
We start with the eigenvalue problem for $\tilde H_0$:
\begin{eqnarray}
\tilde H_0\tilde \psi=-n_I \tilde \psi
\end{eqnarray} where $n_I=(I+\rho \delta/4)$ and $\tilde\psi$ is square integrable
with respect to measure ${1\over r}\mathrm{vol}_{\mathscr P}$ and
$\tilde\psi\not\equiv 0$. The above equation can be recast as
$$
-{1\over 2}\left(\Delta+{B\over r^2}+{2n_I\over
r}\right)\tilde\psi(x) =-{1\over 2}\tilde \psi(x).
$$
Let $\psi(x):=\tilde \psi({x\over n_I})$, then the preceding
equation becomes
$$
\left(-{1\over 2}\Delta-{B\over 2r^2}-{1\over r}
\right)\psi(x)=-{1/2\over n_I^2}\psi(x),$$
i.e.,
\begin{eqnarray}
\hat h \psi =-{1/2\over n_I^2}\psi.
\end{eqnarray}
One can check that $\psi$ is square integrable
with respect to measure $\mathrm{vol}_{\mathscr P}$. Therefore, $\tilde\psi$ is an eigenfunction of $\tilde H_0$
$\Rightarrow$ $\psi$ is an eigenfunction of $\hat h$. By turning the above arguments backward, one can show that
the converse of this statement is also true. Therefore,
\begin{eqnarray}
\fbox{$\tilde\psi$ is an eigenfunction of $\tilde H_0$
$\Leftrightarrow$ $\psi$ is an eigenfunction of $\hat h$.}
\end{eqnarray}
Introduce $${\mathscr H}_I:=\{\psi\mid \tilde \psi\in \tilde{\mathscr H}_I\}, \quad {\mathcal H}:=\bigoplus_{i=0}^\infty {\mathscr H}_I,$$
and denote by $\tau$: ${\mathcal H}\to \tilde {\mathcal H}$ the linear map such that
$$\fbox{$\tau(\psi)(x)=n_I^{{(\rho-1)\delta\over 2}+1}\psi(n_Ix)$}$$
for $\psi\in {\mathscr H}_I$. By virtue of Theorem 2 in Ref. \cite{CD2003}, one can show that $\tau$ is an isometry. Here, the inner product on $\mathcal H$ is
the usual one: for $\psi$, $\phi$ in $\mathcal H$, we have
$$
\langle \psi, \phi\rangle=\int_{\mathscr P}\overline{\psi}\,\phi\, \mathrm{vol}_{\mathscr P}.
$$
Since $\tilde{\mathcal H}$ is a unitary lowest weight Harish-Chandra module, and $\tau$ is an isometry,
${\mathcal H}$ becomes a unitary lowest weight Harish-Chandra module. Since the completion of $\mathcal H$ is the
Hilbert space of bound states, the rest is clear from Theorem \ref{main2}.
\end{proof}
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Home News Ethan Carter III Talks His "Strap Match" At Slammiversary XV, Says He...
Ethan Carter III Talks His "Strap Match" At Slammiversary XV, Says He Never Wanted To Be An "Indie Star" – More
Andrew Thompson
Impact Wrestling superstar Ethan Carter III, recently spoke with the Miami Herald regarding numerous non-wrestling & wrestling topics including; what wrestlers are from his hometown, growing up a fan of the sport & tons more.
Here are the highlights:
EC3 On The Talent That Is From His Hometown Of Cleveland:
"Cleveland has sprouted off some pretty good talent; Al Snow's from Lima, so that doesn't count for Cleveland, Gargano, Ziggler, Miz, myself and a few others I'm probably leaving out. We've all had different paths to wrestling success. Miz came from MTV and he always loved wrestling. Ziggler was an accomplished amateur and loved wrestling. Gargano was a guy from the independents who caught fire at the right time and was in the right place at the right time in NXT."
His Goal Early In His Wrestling Career:
"Myself, I was on the indie scene. My goal was never to be an indie standout, my goal was always to be signed by a television wrestling company. So I went a different route, paying my way to go to OVW and finding myself at an FCW tryout. A lot of different avenues, a lot of different talents, all success and one thing in common; Cleveland and that's pretty cool."
Overcoming Obstacles In His Career:
"There definitely been times I've been injured; knees torn up, back busted, fired. I've been through some business hell to get where I am. So to look back at some of the things I have accomplished, I think going through the bad stuff has desensitized me to a point where I can appreciate what I have done and have accomplished, but it still leaves me wanting a lot more. Through that hunger and that angst is the only way to continue to evolve and be better. It's like [actor] Tim Robbins in '[The] Shawshank [Redemption],' you got to climb through a mile of s**t to see the other side and I've climbed through some s**t and I'll probably climb through some more, but at the end of the day, accomplishing what I want will make it all worth it."
His Time In Impact Wrestling:
"Fifteen years, whether it was TNA or now Impact Wrestling, it's been around 15 years. There have been some good times and there have been some bad times, but the one thing I can be appreciative for and be thankful for is that it gave me an opportunity to hone my craft and become one of the top professional wrestlers in the world."
His Slammiversary XV Match Vs. James Storm:
"At Slammiversary, you will see myself, EC3, one of the top wrestlers in the world, the best guy here, the best guy there, the best guy anywhere; take on James Storm in a strap match. Everything I've done to him has never been personal. It's been solely because I needed to lash out and truly show who I am. It's unfortunate for him that he has been sort of the canvas that I am painting my masterpiece on, but he was in the wrong place at the wrong time and for those tuning into Slammiversary, I'm not going to tell you that it's going to be a great match or an epic contest, it's going to be an ass kicking. It's going to be a drubbing, it's going to be brutal, he's going to be broken, beaten and nothing more than an old beer truck by the time I am done with him. Then, I will move on, move forward, where I belong and he will be left in the refuse and dust of the past, where he belongs."
Ethan Carter Iii
slammiversary xv
Michael Hunter Hutter
He is best known for his time with Impact Wrestling, where he competed as Ethan Carter III. In Impact Wrestling, he is a two-time TNA ...
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Texas is at the forefront of data science and computational medicine. Texas State University is leading the charge.
By Tyler Hicks
By TM Studio
Brought to you by Texas State University
This article is part of a sponsored content series produced in partnership between Texas Monthly and Texas State University highlighting big ideas that move Texas forward.
The patient couldn't go near a gas station. In fact, the mere smell of gas took him back to Iraq, where he fought in an endless succession of violent skirmishes. One day, an IED exploded right by his Humvee. The blast singed his hair, and the strong oil odor was in his nose, on his clothes, everywhere. The veteran has struggled with PTSD ever since and is one of the 40 million adults in the United States who suffer from some kind of anxiety disorder. Since he's returned to the States, his car has always run on empty. By the time this patient met Alessandro De Nadai, Ph.D., the last thing he wanted to do was fill it up.
"That's a completely natural reaction," says De Nadai, an Assistant Professor of Psychology at Texas State University. "People naturally avoid the situations that make them anxious, but that often makes the anxiety worse."
How do we automatically turn this info into something doctors can use in helping patients, De Nadai wondered. Better yet, how can we use it to get treatment right the first time?
Since he was a twenty-year-old undergraduate at the University of Georgia, De Nadai has been fascinated by two seemingly unrelated fields: clinical psychology and computer science. Yet as he continued his studies, earning two master's and a doctorate, De Nadai realized how those two fields can complement one another and help patients get the treatments they need. This veteran was a prime example.
It was 2017, and De Nadai was wrapping up his doctorate in Clinical Psychology with an internship at the VA Medical Center in Jackson, Mississippi. While he watched the veteran detail symptoms, explain family history, and list a litany of other facts about his past and present, De Nadai had an idea.
De Nadai is part of a talented cohort of Texas State professors leading the emerging field of computational medicine. With a slew of data-driven projects, professors of health administration, computer science, psychology and engineering are working together to make the world a healthier place.
"It's an amazing, cross-disciplinary effort," De Nadai says. "We have all of these professors coming together to use big data, and that's pretty exciting."
One cross-disciplinary project, led by professor Larry Fulton, employs machine learning to analyze MRI scans and predict Alzheimer's, thereby allowing doctors to intervene and possibly even prevent the illness before it arises. Machine learning is only possible because of data, he says.
"This is the future of healthcare," Fulton says. "Data is helping us get so much better at diagnostics and helping both doctors and patients. At the end of the day, all of our work is patient-focused and patient-centered."
Figure 1: Percent of non-federal acute care hospitals that use their EHR data for at least one of the ten specified measures of hospital processes to inform clinical practice, 2015 – 2017.
Meanwhile, De Nadai is using data to help patients struggling with anxiety, OCD, PTSD, and other afflictions get on the right treatment path as early as possible. A combination of psychology and computer science, computational medicine has emerged as one of the most consequential fields of study.
"It's really exciting, because there's no single standard for what we're doing right now," he says. "We're creating the standard that will help thousands and thousands of people."
Professor De Nadai in his Texas State University office, comparing brain scans of patients suffering with obsessive compulsive disorder (OCD). Jeff Wilson
When De Nadai started his academic career, no one had ever heard of "computational medicine."
"It didn't have a name until relatively recently," he says. "I started off in computer science, but I was interested in what computation can do to improve human welfare. I realized psychology and computer science are more related than we think."
Specifically, De Nadai realized that computation can have a wide-ranging impact in mental health.
"Predicting what people are going to do is very difficult, and for this reason mental health data are some of the trickiest to work with," he says. If the data was clean, he thought, then treatments would improve, and people could be happier and healthier.
"That's been my biggest goal since I was twenty," the professor says. "We may not need new medicines to improve lives. Instead, let's use the tools that are already at our disposal to get people on the right path."
Figure 2: How Much Data Is Created Every Day in 2020?
His passion for improving health inspired De Nadai to change his focus from computer science to psychology. After graduating from the University of Georgia, he earned a master's in psychology from Stephen F. Austin State University, then a master's in clinical psychology from the University of South Florida, where he would also earn his doctorate. Along the way, he has led several projects that merge his penchant for data with his passion for improving health, including an initiative involving multiple sclerosis (MS).
"When someone receives an MS diagnosis, they don't know what's going to happen next," De Nadai says. "It could progress in one year, or fifteen years, and you have to plan your life very differently if it's one or fifteen."
De Nadai's ongoing MS project compiles data on diagnoses and symptoms into a dashboard, allowing patients and doctors to predict what happens next based on their age, symptoms and other factors. Ultimately, doctors will be able to use this dashboard to practice "intervention," identifying an illness before it arises and helping patients start the right course of treatment as early as possible.
"I don't want people to live with ambiguity," De Nadai says. "When a mother or father or grandparent gets a diagnosis, I want them to know what the rest of their life will look like, so they can live that life with their family as best they can."
Figure 3: Percent of non-federal acute care hospitals that use their EHR data to perform each process that informs clinical practice, 2015-2017.
Eric Storch, Ph.D., has seen firsthand how data can improve lives. The experienced medical professional is a licensed psychologist and a professor at Baylor College of Medicine with a particular interest in anxiety disorders.
"In psychiatry, there really aren't blood tests or biomarkers that clearly delineate who has a problem and what those outcomes are," Storch says. "What Alex is doing is at the forefront of trying to tap into that and create those markers, and the result is a much more personalized approach to treatment."
Storch has collaborated with De Nadai on several projects, including some that have used data to help treat people with anxiety and obsessive-compulsive disorder. The psychologist believes data is forging the next frontier of medicine, and with De Nadai at the helm, there's no telling what breakthroughs are possible.
"If you think of early work by Sigmund Freud, it's like that," Storch says. "This is setting the foundation for medicine for years to come."
De Nadai is currently hard at work on a variety of projects, including one that leverages data to predict depression and substance abuse, both of which may become more prevalent in a post-pandemic world. Smartphones can determine moving patterns—when you wake up, how often you move around or exercise—and, in turn, those patterns can indicate your susceptibility to depression and substance abuse. Like many of De Nadai's projects, the professor is striving to prevent ailments before they complicate people's lives.
"That's the next big thing in medicine," Storch says. "Predicting who is going to have what problem, what that problem can look like, and how we can treat it."
And there's no better place for De Nadai's work than Texas State University. Texas is a nationwide leader in data science and computational medicine, and Texas State recently partnered with the tech company AMD to launch a project aimed at curbing the spread of COVID-19.
"Data science is great at merging lots of small signals and giving us something really meaningful," De Nadai says. "If people can track changes in temperature and changes in heart rate for two to three weeks, then we know who has to isolate and who is safe."
Of course, this is De Nadai's first crack at a pandemic. Nevertheless, Stoch believes De Nadai can accomplish just about anything.
There's no single standard for what we're doing right now. We're creating the standard that will help thousands and thousands of people.
"There are plenty of statistical whizzes who, by their trade, make the world a better place," Storch says. "But not everyone is as personable as Alex. And that's a huge asset. Because he cares, he sees the human behind the numbers, and he's always thinking of how to improve that person's life."
De Nadai is humble about his own accomplishments, but he agrees with Storch: The person behind the data should always be top of mind.
"I want to see people get the right treatment for the first time," he says. "Many suffer in silence, and I never want that to happen."
Until he met De Nadai, that veteran-patient was suffering in silence. Often, he used substances to cope with his anxiety. By paying attention to the veteran's daily habits and symptoms, De Nadai realized this man needed to expose himself to what gave him anxiety. So, the veteran started small, visiting a gas station every now and then, sometimes just to walk around. Eventually, he overcame his fear.
"Sometimes it's just about practice, and taking those small steps," De Nadai says. "That's kind of what data is, too: Small parts of a big picture."
Fig. 1 ONC/American Hospital Association (AHA), AHA Annual Survey Information Technology Supplement: 2015-2017. Note: The sample consists of 3,599 non-federal acute care hospitals.
Fig. 2 https://techjury.net/blog/how-much-data-is-created-every-day/#gref
Fig. 3 ONC/American Hospital Association (AHA), AHA Annual Survey Information Technology Supplement: 2015-2017. Note: *Significantly higher than the previous year (p<0.05). The sample consists of 3,599 non-federal acute care hospitals.
https://www.texasmonthly.com/promotion/top-of-mind/?utm_source=texasmonthly.com&utm_medium=referral&utm_campaign=sharebutton
Dining Guide: Highlights From Our February 2022 Issue
The Secrets of Bonfire Shelter
Meanwhile in Texas
Meanwhile, in Texas: A Plumber Finds Cash in a Church's Bathroom Wall
By Sierra Juarez
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,831
|
Q: My very small company has reached the point of needing a VCS. I'm trying to plan our actual version control strategy first First and foremost, please excuse my writing: I'm a programmer fresh out of college with minimal professional experience. If anything is unclear, please bring it up and I'll do my best to clarify.
I'm looking forward to using a VCS to help us to organize everything. Keeping track of bug fixes, development of new features, merging, and development of new editions should simplified, clarified, and made much easier.
This is how we plan to use version control (given my fledgling understanding of VCSs):
For the sake of a clear example, we'll call our current and released version of the software V1 and in-development next version V2.
*
*We plan to start with V1 as the trunk of our tree.
*A branch will be made for development of V2. This branch will never be merged back in, thus it is effectively the new trunk for V2.
*Branches would be split off of this new trunk for development of new features for V2 and merged back in when they are complete.
*Should bug fixes be needed for V1, branches would be made off of the V1 trunk, completed, and merged back into the V1 trunk. V2 development would need to pause as the V2 trunk updates from changed V1 trunk, then all V2 branches update from the newly changed V2 trunk. Then various V2 branches could resume development.
*When enough features have been branched off of the V2 trunk, completed, and merged back in, eventually it will be time to release V2 and start on V3. V3 will branch off of V2 and never merge back in (like V2 did from V1), and the process begins again.
Is this a reasonable model? Will it be effective? Is it likely to develop issues over time due to my uneducated attempt at planning?
Please let me know anything else I can provide to help you to help me. Thank you!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,942
|
Function set-stuff{
[cmdletbinding(SupportsShouldProcess=$true,
confirmImpact='Medium')]
param(
[Parameter(Mandatory=$True)]
[string]$computername
)
Process{
If ($psCmdlet.shouldProcess("$Computername")){
Write-Output 'Im changing something right now'
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 80
|
Octave-MappedMatrixMultiplication
=================================
Combination of data structures and algorithms that make for optimal mapped matrix multiplication in GNU Octave.
[](http://dx.doi.org/10.5281/zenodo.13188)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,277
|
docker: Dockerfile
docker build -t gobuilders/linux-x86-nacl
upload: docker
docker save gobuilders/linux-x86-nacl | gzip | (cd ../../coordinator/buildongce && go run create.go --write_object=go-builder-data/docker-linux.nacl.tar.gz)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,077
|
/**
* Options to be passed to the external program or shell.
*/
export interface CommandOptions {
/**
* The current working directory of the executed program or shell.
* If omitted VSCode's current workspace root is used.
*/
cwd?: string;
/**
* The environment of the executed program or shell. If omitted
* the parent process' environment is used.
*/
env?: { [key: string]: string; };
}
export interface Executable {
/**
* The command to be executed. Can be an external program or a shell
* command.
*/
command: string;
/**
* Specifies whether the command is a shell command and therefore must
* be executed in a shell interpreter (e.g. cmd.exe, bash, ...).
*/
isShellCommand: boolean;
/**
* The arguments passed to the command.
*/
args: string[];
/**
* The command options used when the command is executed. Can be omitted.
*/
options?: CommandOptions;
}
export interface ForkOptions extends CommandOptions {
execArgv?: string[];
}
export const enum Source {
stdout,
stderr
}
/**
* The data send via a success callback
*/
export interface SuccessData {
error?: Error;
cmdCode?: number;
terminated?: boolean;
}
/**
* The data send via a error callback
*/
export interface ErrorData {
error?: Error;
terminated?: boolean;
stdout?: string;
stderr?: string;
}
export interface TerminateResponse {
success: boolean;
code?: TerminateResponseCode;
error?: any;
}
export const enum TerminateResponseCode {
Success = 0,
Unknown = 1,
AccessDenied = 2,
ProcessNotFound = 3,
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,929
|
Q: Assign buffer an attribute based on the largest polygon inside I would like to assign a buffer polygon a value from another layer. I have cliped the two layers together. I then tried to dissolve the new layer by buffer ID, but I could not manage to get the largest area inside to be the value for the whole buffer.
I tried to do it by using calculate field, but im not very good at python.
A: You can do this using Spatial Join. You can set your parameters to those shown in the image.
Field mapping is the trick, when you choose a one_to_one join only one value can be joined for each feature in your buffer, the merge rule is used to decide which value (in your case maximum).
If "another layer" doesn't already have an area field simply create one and Calculate Geometry. If the area field name is not unique (as in the example picture) set the merge rule on the renamed field in the Join Feature (e.g. "Shape_Area" was automatically renamed to "Shape_Area_1" in this case)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,443
|
Nevězice är en ort i Tjeckien. Den ligger i regionen Södra Böhmen, i den centrala delen av landet, km söder om huvudstaden Prag. Nevězice ligger meter över havet och antalet invånare är .
Terrängen runt Nevězice är platt, och sluttar söderut. Den högsta punkten i närheten är meter över havet, km norr om Nevězice. Runt Nevězice är det ganska tätbefolkat, med invånare per kvadratkilometer. Närmaste större samhälle är Písek, km söder om Nevězice. I omgivningarna runt Nevězice växer i huvudsak blandskog.
Trakten ingår i den hemiboreala klimatzonen. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är januari, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är juni, med i genomsnitt mm nederbörd, och den torraste är mars, med mm nederbörd.
Kommentarer
Källor
Externa länkar
Orter i Södra Böhmen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,222
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AKAI MPC Element, MIDI Drum Pad Controller ,16 Pads, MPC Note Repeat, MPC Essentials Software.
MPC Element brings powerful music-making capability to your computer in a slimline design that's made to produce. You get cutting-edge features, including MPC Note Repeat and Swing, along with the all-new MPC Essentials software. With the M17included 1GB sound library, MPC Essentials empowers you with the essential sounds of modern music production, and you can easily import your own sound samples. MPC Essentials can operate standalone and can also work seamlessly with your current DAW as a plugin. MPC Element comes with everything you need to make music right away, and, because it operates via standard MIDI, you can also use MPC Element to control MIDI music software you already have. An 1/8-inch MIDI input and an 1/8-inch MIDI output are both onboard and 1/8-inch to 5-pin MIDI cables are included.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 96
|
\section{Introduction}
Astronomical spectrographs are key instrumentation to tackle the open issues in astronomy. One of the most important element along with the detector is the dispersing element. In the last 15 years, the Volume Phase Holographic Grating (VPHG) technology has gained a lot of interest in astronomical field and it has been used in some spectrographs \citep{baldry, andrea10, barden, pazder}. The reasons for such interest stem from the fact that these gratings show unique features, such as i) the high peak efficiency (up to 100\% theoretically) both at low and large dispersion; ii) the ease of performance tuning and customisation (each VPHG is a master grating). A VPHG consists in a thin layer of holographic material, usually dichromated gelatine \citep{andrea10, barden}, which is sandwiched between two glass windows. The phase of incident light is modified passing through the gratings thanks to a periodic modulation (usually sinusoidal) of the refractive index written in the holographic material.
Hence the fundamental parameters that rule the overall efficiency of such devices are the thickness of the active film and the modulation of the refractive index inside it.
VPHG technology has been applied in different astronomical spectrographs, at room and cryogenic temperatures \citep{wiyn, andrea7, andrea3, lepine, hou, andrea1, andrea9}. They have also been used as tuneable filters and cross dispersers \citep{mendes, gibson, castilho}. The VPHGs assembled in a GRISM configuration have been also used in this kind of instrumentation.
The availability of a low resolution grism covering the red part of the optical spectrum is of paramount importance for several astrophysical targets, among which stands out the study of supernovae in general and type Ia supernovae in particular. Such a grism will characterise the evolution of important lines during the photospheric (e.g. O I 7774 $\textrm{\AA}$ and Ca II IR triplet), and the nebular phases (e.g. Ca II] 7291-7323 $\textrm{\AA}$ and Ca II IR triplet).\\
Particularly important will be to study the evolution in type Ia supernovae of the high velocity features (HVFs) seen in the Ca II triplet profile, especially at early, pre-maximum, phases \citep[see][for a recent review]{chil13}. The origin of the HVFs remains unknown, but different hypotheses have been proposed, which include the impact of the ejecta with a circumstellar shell lost by the progenitor before the explosion \citep[see][]{ger2004}; an enhancement in the abundance of intermediate-mass elements (IMEs) in the outermost layers of SN Ia ejecta \citep{mazzali05a, mazzali05b, tanaka08}; or variations in the ionisation state of IMEs in the outer layers of SN Ia ejecta \citep{blondin2013}.\\
It is evident that the study of the Ca II HVFs could have a big impact in deriving the true progenitor scenario involved in the SNIa explosion, which in turn could have important implication in the use of SNIa in Cosmology. Moreover, the circumstellar (CS) material responsible for the HVFs could cause subtle alteration of the spectral energy distribution of SNIa, which could have possible consequences in the luminosity standardization of SNIa \citep{chil13}.
\\
In order to accomplish the desired requirements we designed and manufactured a VPHG based on a completely new holographic material.
The study of this new material, other than dichromate gelatins (DCGs), is very important in order to make possible the design and manufacturing of innovative and large VPHGs.
Indeed DCGs are difficult to handle, they require a complex chemical process and the scalability to very large size gratings can be an issue. For this reasons we focused the attention to solid photopolymers which combines high throughput, high refractive index modulation, self developing (i.e. no chemical processes needed) and size scalability. Such new material belongs to the class of solid photopolymers and this is the first time, in our knowledge, that this kind of holographic material has been used to make scientific grade dispersing elements. In the past only liquid photopolymers have been used once to produce VPHGs mounted in MOIRCS and FOCAS \citep{andrea4, andrea6}.
Starting from the scientific case of the SN 2013fj hereinafter we demonstrate the capabilities in terms of spectral resolution and throughput of this new family of Volume Phase Holographic Grating, based on photopolymers, comparing two spectra of SN 2013fj in which one of them is secured by the adoption of previous existing state of the art grating.
Throughout this paper we refer to this new grating as VPH6.
\section{Observations and data reduction}
We obtained the spectrum of SN 2013fj in visitor mode at Ekar Asiago Observatory using the Asiago Faint Object Spectrograph Camera (AFOSC) on 13th September 2013. The seeing during the night was quite constant (2.0 - 2.2$^{\prime\prime}$) and the sky was almost clear during the observations. In order to compare the new VPH6 device performance, we took two spectrum of the SN 2013fj. The first one was been obtained configuring the instrument with the GRISM GR04, yielding a dispersion of $\sim$ 5 $\textrm{\AA}$ px$^{-1}$ and R $\sim$ 600 in the spectral range 3500-7500 $\textrm{\AA}$. The spectra obtained with the new VPH6 GRISM, which was been secured immediately after the previous one, yields a dispersion of $\sim$ 3.5 $\textrm{\AA}$ px$^{-1}$ and R $\sim$ 500. We adopted a slit of $1.69^{\prime\prime}$ $\times$ 5.00$^{\prime\prime}$ for both spectra.\\
Since the Supernova Program has many observation priorities scheduled for AFOSC, we decided to first secure a spectrum with the well known GR04 grating (this was done to guarantee the data for the required scientific tasks). After that we coped to obtain another observation of the same target (SN 2013fj), under the same sky conditions, reducing the telescope time that would be otherwise not allocated for the other targets in the night.
The integration time for the spectrum obtained with VPH6 was 1200s while for the GR04 was 1800s.\\
For each exposure we reduced data adopting standard IRAF\footnote{IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} procedure. We performed bias subtraction and flat field correction for each scientific frame adopting calibration obtained in the same night. The wavelength calibration was achieved using the spectra of standard arcs (Th-Ar and Hg-Cd) while flux calibration has been assessed through relative photometric calibration of standard stars spectra \citep{oke1990} obtained in the same night (BD+33d2642). The accuracy on wavelength calibration is $\sim$ 0.5 $\textrm{\AA}$ rms for the VPH6 and $\sim$ 0.2 $\textrm{\AA}$ rms for the GR04.\\
The two RMS values for the accuracy on the wavelength calibration are quite different since in the red part of the spectrum few comparison lines are available due to the calibration lamps installed in the AFOSC spectrograph. The calibration is more accurate in the blue because more emission lines in that part of the spectrum were usable. For this reason, the two RMS are slightly different. However, it is also important to note that the two values are far below the resolution power of the two gratings.
The apparent R magnitude obtained was 17.2 $\pm$ 0.2. We cross checked the calculated value through an aperture photometry of the R band acquisition image of the field (see Figure \ref{fig:Snfield}), secured just before obtaining the spectra.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f1.eps}}
\caption{R Band image of FoV around SN 2013fj. The plate scale of the image is 18.59$^{\prime\prime}$ px$^{-1}$. Exposure time is 120s. Seeing during the observation, taken at airmass 1.17 and measured on the image of the field, is 2.2$^{\prime\prime}$. The host galaxy of SN 2013fj is also clearly visible.}
\label{fig:Snfield}
\end{figure}
\label{sec:observations}
\subsection{The device VPH6 at AFOSC}
The GRISM consists in a photopolymer based Volume Phase Holographic Grating (VPHG) (see Table \ref{tab:grism} for the GRISM features and requirements). The solid photopolymer used in this device has been recently developed by Bayer MaterialScience AG (product family: Bayfol$^{\textregistered}$ HX) as high performance holographic material \citep{bayer1}, for reflection holograms. The material is also suitable for transmission holograms with high dynamic range and sensitivity. Moreover the material is laminated onto flexible and transparent substrates of large sizes.
The grating is a "low dispersion" device with a line density of 285 lines mm$^{-1}$ with a target efficiency of 90\% at the central wavelength.
\begin{small}
\begin{deluxetable}{ c c c l l }
\tabletypesize{\scriptsize}
\tablecaption{AFOSC's GRISM VPH6: main specifications and requirements}
\tablenum{1}
\label{tab:grism}
\tablehead{\colhead{$\lambda_{central}$ [nm]} & \colhead{lines mm$^{-1}$} & \colhead{$\Delta \lambda$ [nm]} & \colhead{$\eta_{peak}$} & \colhead{$\eta_{side}$} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)}}
\startdata
800 & 285 & 620 - 980 & 90\% & 30\% \\
\enddata
\tablecomments{Description of columns: (1) Working central wavelength of the grating; (2) Pitch of the grating; (3) Wavelength range; (4) 1-st order diffraction efficiency at the peak wavelength; (5) 1-st order diffraction efficiency at the wavelength range edges.}
\end{deluxetable}
\end{small}
It can be seen that this low dispersion grating, combined with a suitable wavelength range, allows to covers the H$\alpha$ region and the Ca I bump typical of SN spectra.
The design of the grating was aimed at finding the best key parameters (film thickness and refractive index modulation) matching the scientific requirements, i.e. diffraction efficiency, wavelength coverage, and resolution. This activity has been performed through RCWA simulations\footnote{RCWA code, written in C, was provided by Gary Bernstein, who implemented the methods of Moharam \& Gaylord.} \citep{rcwa}.
We have therefore identified the best couple of parameters ($\Delta$n = 0.011 and d = 34 $\mu$m). The quite large thickness and small modulation of the refractive index is chosen in order to reduce the efficiency in orders higher than the first, which is a common feature of low line density VPHGs.
In Figure \ref{fig:simulation1} are reported the simulated 1-st order diffraction efficiency curves for different values of $\Delta$n and film thickness.
In the inset A) are reported the curves at different film thickness with with a fixed value of $\Delta$n = 0.011. Shown curves have $\pm$ 10 \% from the chosen value of 34 $\mu$m.
In the inset B) are reported the curves at different $\Delta$n with with a fixed thickness d = 34 $\mu$m. Shown curves have $\pm$ 10 \% from the chosen value of 0.011.
The photosensitive film was laminated onto a BK7 substrate before the exposure; the writing procedure was accomplished using a standard two-beams holographic setup with a DPSS laser of 532 nm.
The target refractive index modulation has been reached optimising the writing laser power since the final achieved $\Delta$n strongly depends upon the exposure power density as reported by \cite{bayer2}.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f2.eps}}
\caption{Simulated 1-st order diffraction efficiency curves, with Rigorous Coupled Wave Analysis; A) curves for different thickness (d = 34 $\mu$m $\pm$ 10 \%) and fixed $\Delta$n = 0.011; B) curves for different refractive index modulation ($\Delta$n = 0.011 $\pm$ 10 \%) and d = 34 $\mu$m . The considered incidence angle is 4.4$^{\textdegree}$ (on the grating interface).}
\label{fig:simulation1}
\end{figure}
Regarding the GRISM, the apex prism angle has been designed in order to maintain the 1-st order central undieviated wavelength at 800 nm. The result was a BK7 prism with an angle of 12.7$^{\textdegree}$ that correspond at an entrance angle in the grating of 4.4$^{\textdegree}$.
\\
The grating was then coupled with the prisms using a refractive index matching oil.
In order to characterise the device, we measured the diffraction efficiency of the GRISM. The measurements were carried out using laser light at different wavelengths, setting the p- or s- polarisation and collecting the efficiency of the first order as function of the incidence angle;
two efficiency curves are reported in Figure \ref{fig:eff2}.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f3.eps}}
\caption{Measured 1-st order diffraction efficiencies of the GRISM at 808 nm and 633 nm. The data represent the p-polarisation efficiency $\phi_p$, the s-polarisation efficiency $\phi_s$ and the resulting mean $\phi$, as function of the incidence angle $\alpha$ in air.}
\label{fig:eff2}
\end{figure}
It can be seen that the efficiency is very high even with the prisms coupled with the VPHG in the GRISM structure. This is achievable thanks to the AR-coating onto the prisms surfaces and the use of the same substrate material for both the prisms and grating windows, that avoids further reflection losses.
We lastly report in Figure \ref{fig:effblaze} the measured 1-st order diffraction efficiency vs. wavelength of the GRISM aligned and mounted in his housing.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f4.eps}}
\caption{Measured 1-st order diffraction efficiency curve of the aligned GRISM at different incidence angles. The angle 0$^{\textdegree}$ refers to the perpendicular to the grating inside the GRISM.}
\label{fig:effblaze}
\end{figure}
It is clear that the final alignment is crucial to maintain the requirements satisfied since even a tilt of a few degrees have a huge impact on the efficiency curve.
\section{Results}
\label{sec:result}
After the commissioning of the new VPH6 for AFOSC at 1.82m Copernico telescope, we taken two exposures of a flat field lamp with the same exposure time but varying the gratings in order to have a sound comparison of the two overall efficiencies. In particular, as shown in Figure \ref{fig:flat} the spectra of the lamp represents the behaviour of the compared gratings.
Thanks to the homogeneous configuration of the instrument, in terms of efficiency of the detector, slits loss and telescope throughput, adopted for the two exposures it is possible to infer that starting from $\sim$ 5800 $\textrm{\AA}$ the number of counts in the spectrum of the new device (red curve) is significantly higher than those obtained in the case of the GR04 (blue curve).
Moreover the spectral coverage of the VPH6 is extended up to $\sim$ 9500 $\textrm{\AA}$ as required in order to investigate near IR properties of the targeted object for the SN program
\citep{tomasella}.\\
The well visible fringing of the flat field recorded with the VPH6 was compared with the one of the GR04 after the signal normalisation; the result was a comparable fringe pattern, reassuring us that the effect can be attributed only to the CCD as described in the AFOSC's manual. \footnote{The entire manual of AFOSC is available at \url{http://archive.oapd.inaf.it/asiago/5000/5100/man01_2.ps.gz}}
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f5.eps}}
\caption{Comparison of flat fields of VPH6 (red line) and GR04 (blue line) obtained at AFOSC adopting homogeneous configuration of the system.}
\label{fig:flat}
\end{figure}
\begin{figure*}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f6.eps}}
\caption{Spectra of SN 2013fj taken with AFOSC and GR04 (blue line) and with VPH6 (red line). The principal lines of Si II, Ca II, S I, Mg II and Fe II are shown.}
\label{fig:spec_13fj}
\end{figure*}
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{f7.eps}}
\caption{Comparison of the SN 2013fj spectrum with that of phase +5 days from B maximum of SN 1992A made using the GELATO tool; GEneric classification Tool by \cite{Harutyunyan}. GELATO is a software for objective classification of Supernova spectra and performs an automatic comparison of a given spectrum with a set of well-studied SN spectra templates. Spectra of all SN types from Padova-Asiago SN Archive are used as templates for the comparison procedure.}
\label{fig:conf}
\end{figure}
In order to assess the capabilities of the new grating in terms of scientific goals, we performed observations of SN 2013fj that has been already discovered by the amateur astronomers \cite{ciabattari} members of the Italian Supernovae Search Project on 7th September 2013. The spectra reported in Figure \ref{fig:spec_13fj} has been obtained 6.07 days later and is that typical of a Type Ia supernova, about 5$\pm$2 days after maximum light (see Figure \ref{fig:conf}), confirming the estimated phase reported by \cite{zanutta01}, and derived from a fast reduction performed at the telescope of the same data.\\
The recorded spectrum, reported in Figure \ref{fig:conf} and analysed with GELATO, is the combination of the spectra taken with the new VPH6 and the 3500-4750 {\AA} region of the GR04 (since they well overlap in the whole common range). This decision has been made because of the evident better Signal to Noise Ratio of the VPH6 and the higher coverage in the red region where the the important features of the target object are (such as HVF Ca-II IR).
The spectrum exhibit the broad P-Cygni lines typical of SNe Ia: the characteristic deep absorption near 6150 \AA\/ due to Si II 6347, 6371 \AA\/ (hereafter Si II 6355 \AA\/), the Si II 5958, 5979 \AA\/ feature (hereafter Si II 5972 \AA\/), the W-shaped feature near 5400 \AA\/ attributed to S II 5468 \AA\/ and S II 5640 \AA\/. \\ Other prominent features are Ca II H$\&$K, Mg II 4481 \AA\/, and several blends due to Fe II and Si II. At red wavelengths, particularly strong features are the Ca II near-IR triplet. Despite contamination from the 7600 \AA\/ telluric feature, O I 7774 \AA\/ is clearly visible.\\
Adopting for the host galaxy (CGCG 428-62) of SN 2013fj a recessional velocity of 10064 km s$^{-1}$, \cite{huchra}, an expansion velocity of about 10700 km s$^{-1}$ is deduced from the Si II 6355 \AA\/ absorption, while from the Ca-II IR triplet minimum an expansion velocity of about 11500 km s$^{-1}$ is deduced (the velocity is relative to the average Ca-II IR triplet wavelength, 8579.1 \AA\/). This behaviour is typically seen in SNIa, where the strong CaII lines are formed well above the photosphere, which is better traced by the weaker S-II lines (from which a mean expansion velocity of about 8400 km s$^{-1}$ is deduced).\\
The CaII-IR high velocity feature is by this phase very weak, see \cite{mazzali05b}, but possibly still visible as a weak absorption at about 23000 km s$^{-1}$\\
The expansion velocity deduced from the S-II 6355 \AA\/ minimum most probably places SN 2013fj among the low velocity gradient type Ia supernovae, following \cite{benetti1}.
\\
In order to make a sound comparison between the two different dispersive elements, we evaluated the S/N (Signal to Noise Ratio) at 6 different wavelengths along the two obtained spectra.
The results are reported in Table \ref{tab:SN} where S/N of the GRISMs have been normalised adopting the usual S/N equation \citep{SNeq} taking into account the different exposure times.\footnote{For the renormalisation we assumed that the equation simplifies to $S/N=\sqrt{s}$ (where $s$ is the signal from the source) since other noises (such as dark current or detector readout noise) are negligible at this level of comparison.}\\
\begin{small}
\begin{deluxetable}{ c c c }
\tabletypesize{\scriptsize}
\tablecaption{Signal to Noise Ratio comparison between the two GRISMs}
\tablenum{2}
\label{tab:SN}
\tablehead{\colhead{Wavelength [\AA]} & \colhead{S/N of VPH6} & \colhead{S/N of GR04} \\}
\startdata
5000 & 22 & 17 \\
5500 & 42 & 27 \\
6000 & 45 & 27 \\
6500 & 57 & 27 \\
7000 & 38 & 20 \\
7500 & 30 & 13 \\
\enddata
\tablecomments{The S/N of the two GRISMs have been normalised according to the respective exposure times.}
\end{deluxetable}
\end{small}
\section{Conclusions}
\label{sec:discussion}
We demonstrated the good performances obtainable by using a Volume Phase Holographic Grating based on new photopolymer materials which are self developing, characterised by a high sensitivity and dynamic range in conjunction with an easy processability. The GRISM has been designed and manufactured in order to maximise the efficiency reducing the reflection losses.
For these reasons such devices are a reliable alternative to classical VPHGs.
We assessed the scientific requirements which drawn the design of this new DOE by collecting the spectrum of the newly discovered SN Ia PSN J22152851+1534041 = SN 2013fj . We finally carried out a sound comparison with another spectrum of the same object under the same conditions, secured with the standard grism that is characterised by the same dispersion and resolution.
\\
\\
\textbf{Acknowledgements}
\\
We are grateful to:
Dr. Thomas F\"{a}cke of Bayer MaterialScience for providing the material and for the useful discussions;
The technicians at mt. Ekar for all the support during commissioning based on observations collected at Copernico telescope (Asiago, Italy) of the INAF - Osservatorio Astronomico di Padova;
L.T. and S.B. are partially supported by the PRIN-INAF 2011 with the project "Transient Universe: from ESO Large to PESSTO".
\newpage
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,348
|
\section{Introduction}
Let $[n]=\{1,\ldots,n\}$. Let $2^{[n]}$ and $\binom{[n]}{r}$ denote the family of all subsets and $r$-subsets of $[n]$ respectively. A family of subsets $\cF\subseteq 2^{[n]}$ is \textit{intersecting} if $F\cap G\neq \emptyset$ for $F, G\in \cF$. For any $\cF\subseteq 2^{[n]}$ and $x\in [n]$, let $\cF_x$ be all sets in $\cF$ that contain $x$. A classical result of Erd\H{o}s, Ko and Rado \cite{EKR} states that if $\cF\subseteq \binom{[n]}{r}$ is intersecting for $r\leq n/2$, then $|\cF|\leq \binom{n-1}{r-1}$. Moreover, if $r<n/2$, equality holds if and only if $\cF=\binom{[n]}{r}_x$ for some $x\in [n]$. This was shown as part of a stronger result by Hilton and Milner \cite{HilMil} which characterized the structure of the ``second-best'' intersecting families.
\\
\\
There have been multiple proofs of the Erd\H{o}s--Ko--Rado theorem. The original proof, by Erd\H{o}s, Ko and Rado devised the now-central \textit{shifting} technique and used it in conjunction with an induction argument to prove the theorem. Daykin \cite{Dayk} demonstrated that the theorem is implied by the Kruskal-Katona theorem. Katona \cite{Katona} provided possibly the simplest and most elegant proof, a double counting argument using the method of cyclic permutations. More recently, Frankl--F\"uredi \cite{FraFur} gave another short proof that relied on a result of Katona on shadows of intersecting families, while we \cite{HurKamEKR} provided an injective proof using the aforementioned shifting technique. There have also been algebraic proofs, one using Delsarte's linear programming bound (see \cite{GodMea} and \cite{GodRoy} for details), and another using the method of linearly independent polynomials due to F\"uredi et al. \cite{Fur}.
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The Erd\H{o}s--Ko--Rado theorem is one of the fundamental theorems in extremal combinatorics, and has been generalized in many directions. A very fine survey of the the avenues of research, pursued as extensions of the Erd\H{o}s--Ko--Rado theorem, in the 1960's, 70's and 80's, is presented by Deza and Frankl \cite{DezFra}. In this note, we focus on a relatively recent graph-theoretic extension of the theorem.
\subsection{Erd\H{o}s--Ko--Rado graphs}
For a graph $G$ and integer $t\le \a(G)$, where $\a(G)$ is the size of the maximum independent set in $G$, we define $\cI^t(G)$ to be the family of all independent sets of $G$ having size $t$.
For any family $\cF$ of subsets of $V(G)$ we denote by $\cF_x$ those sets of $\cF$ that contain the vertex $x$.
We call $\cI^t_x$ the {\it star centered on $x$}, and call $x$ the {\it star center}. Call a graph $G$ $t$-EKR if, for any $\cF\sse I^t(G)$, $|\cF|\leq \textrm{max}_{x\in V(G)} |\cI^t_x(G)|$.
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Earlier results by Berge \cite{Berge}, Deza and Frankl \cite{DezFra}, and Bollobas and Leader \cite{BolLead}, while not explicitly stated in graph-theoretic terms, hint in this direction. The formulation was initially motivated by a conjecture of Holroyd, who asked if the cycle graph on $n$ vertices is $t$-EKR for every $t\geq 1$. Holroyd's conjecture was later proved by Talbot \cite{Talbot}. The formulation also has connections with a fundamental conjecture of Chv\'atal \cite{Chvatal} on intersecting subfamilies of hereditary (closed under subsets) set systems.
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Holroyd and Talbot \cite{HolTal} made the following interesting conjecture about the EKR property of graphs. Let $\mu(G)$ be the size of the smallest \textit{maximal} independent set in $G$.
\begin{cnj}\label{minimaxconj}
For a graph $G$, let $1\leq t\leq \mu(G)/2$. Then $G$ is $t$-EKR.
\end{cnj}
Conjecture \ref{minimaxconj} appears hard to prove in general, but has been verified for certain graph classes. In the paper that introduced this graph-theoretic formulation of the EKR problem, Holroyd, Spencer and Talbot \cite{HolSpeTal} proved the conjecture for a disjoint union of complete graphs, paths and cycles containing at least one isolated vertex. Borg and Holroyd \cite{BorHol} later proved the conjecture for a certain class of interval graphs containing an isolated vertex. In \cite{HurKamChord}, we extended this result and verified the conjecture for all chordal graphs containing an isolated vertex.
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One of the reasons why verifying the conjecture for graph classes without isolated vertices is harder is that the intermediate problem of finding the center of the largest star is difficult. (It is easy to see that in a graph containing an isolated vertex, this center is at the isolated vertex.) In this note, we consider this problem for trees.
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In \cite{HurKamChord}, we proved that for any tree $T$ and $t\leq 4$, $\cI^t_x(T)$ is maximum when $x$ is a leaf. We also conjectured that this is true for every $t\geq 1$. However, Baber \cite{Babe}, Borg \cite{BorCounter}, and Feghali--Johnson--Thomas \cite{FegJohnTho} have separately shown that this conjecture is not true.
This makes it interesting to consider for which trees the conjecture is true.
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The authors of \cite{FegJohnTho} consider a special class of trees called spiders, trees obtained from the star graph $K_{1,n}$ (for some $n\geq 1$) by multiple subdivisions of edges. They prove that two families of spiders, namely the family of all spiders obtained by subdividing each edge of the star graph exactly once, and also the family of all spiders containing one leaf vertex adjacent to the root vertex, satisfy Conjecture \ref{minimaxconj}. Note that in both of these subfamilies of spiders, it is easy to find a vertex that is the center of a largest $t$-star (for any $t\geq 1$).
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In this note, we focus on the problem of determining the centers of the largest stars in all spiders. We first introduce some notation to describe spider graphs.
\subsection{Spiders}
Given a sequence of positive integers $L=(l_1,\ldots,l_k)$ we define the {\it spider} $S=S(L)$ to be the tree defined as follows.
The {\it head} of $S$ is the vertex $v_0$ and, for $1\le i\le k$, the {\it leg} $S_i$ is the path $v_0,v_{i,1},\ldots,v_{i,l_i}$.
We say that $L$ is in {\it spider order} if the following conditions hold:
\begin{enumerate}
\item
if $l_i$ and $l_j$ are both odd and $l_i<l_j$ then $i<j$,
\item
if $l_i$ and $l_j$ are both even and $l_i<l_j$ then $i>j$, and
\item
if $l_i$ is odd and $l_j$ is even then $i<j$.
\end{enumerate}
To simplify the notation somewhat, we will write $\cI^t_i(G)$ in place of the more cumbersome $\cI^t_{v_i}(G)$.
\section{Star Centers}
\begin{thm}
\label{Leafs}
Let $S=S(L)$ be a spider with $L=(l_1,\ldots,l_k)$ and suppose that $t\le\a(G)$.
Then for each $1\le i\le k$ and $1\le j<l_i$ we have $|\cI^t_{i,j}(G)|\le |\cI^t_{i,l_i}(G)|$.
\end{thm}
\Pf
We define an injection $f:\cI^t_{i,j}(G)\rightarrow \cI^t_{i,l_i}(G)$.
Let $A\in \cI^t_{i,j}(G)$ and consider the path $P=v_{i,j},\ldots,v_{i,l_i}$.
For $0\le h\le (l_i-j)$ we define $B$ by placing $v_{i,l_i-h}\in B$ if and only if $v_{i,j+h}\in A$ --- $B$ is the {\it flip} of $A$ on $P$, denoted $\flip_P(A)$.
Let $T=A-P$; then set $f(A)=B\cup T$.
Clearly, $f(A)$ is independent, contains $v_{i,l_i}$, and has size $t$.
Also, if $f(A^\pr)=f(A)$, then $A^\pr=A$.
\pf
\begin{thm}
\label{Head}
Let $S=S(L)$ be a spider with $L=(l_1,\ldots,l_k)$ and suppose that $t\le\a(G)$.
Then for every $1\le i\le k$ we have $|\cI^t_0(G)|\le |\cI^t_{i,l_i}(G)|$.
\end{thm}
\Pf
For fixed $i$ we define an injection $f:\cI^t_0(G)\rightarrow \cI^t_{i,l_i}(G)$.
First we define $f$ to be the identity on $\cI^t_0(G)\cap \cI^t_{i,l_i}(G)$.
Second, let $A\in \cI^t_0(G)$ and consider the leg $S_i=v_0,v_{i,1},\ldots,v_{i,l_i}$.
Write $v_{i,0}=v_0$ and, for $0\le h\le (l_i)$ we define $B$ by placing $v_{i,l_i-h}\in B$ if and only if $v_{i,h}\in A$ --- $B$ is the {\it flip} of $A$ on $S_i$, denoted $\flip_{S_i}(A)$.
Let $T=A-S_i$; then set $f(A)=B\cup T$.
Clearly, $f(A)$ is independent, contains $v_{i,l_i}$, and has size $t$.
Also, if $f(A^\pr)=f(A)$, then $A^\pr=A$.
\pf
Together, Theorems \ref{Leafs} and \ref{Head} verify that for the family of spiders, maximum stars are centered at leaves. In what follows, we not only find the best leaf of a spider but give a complete ordering of its leaves according to star size.
\begin{thm}
\label{BestLeaf}
Let $S=S(L)$ be a spider with $L=(l_1,\ldots,l_k)$ in spider order and suppose that $t\le\a(G)$.
Then for each $1\le i<j\le k$ we have $|\cI^t_{i,l_i}(G)|\ge |\cI^t_{j,l_j}(G)|$.
\end{thm}
\Pf
We define an injection $f:\cI^t_{j,l_j}(G)\rightarrow \cI^t_{i,l_i}(G)$.
There will be three cases to consider, depending on the parities of $l_i$ and $l_j$.
First, we develop some terminology.
For a set $A\in\cI^t(G)$ we can define its ladder as follows.
A pair of vertices $\{v_{i,h},v_{j,h}\}$ ($1\le h\le\min(l_i,l_j)$) is called a {\it rung}, which we say is {\it odd} or {\it even} according to the parity of $h$.
A rung is {\it full} if both its vertices are in S.
The {\it ladder} $\cL$ of $A$ is the set of either all even or all odd rungs, depending on whether $v_0\in A$ or not, respectively.
$\cL$ is {\it full} if all its rungs are full.
If $\cL$ is not full then there is a first (i.e. closest to $v_0$) non-full rung $R$.
The partial ladder $\cL^\pr$ is the set of all (necessarily full) rungs above $R$.
Let $T$ denote those vertices of $A-\{v_0\}$ not on $S_i\cup S_j$.
First we define $f$ to be the identity on $\cI^t_{i,l_i}(G)\cap \cI^t_{j,l_j}(G)$.
Next we define the function $f$ on the remaining sets $A\in \cI^t_{j,l_j}(G)$ having partial ladders.
Define the path $P$ from $v_{j,l_j}$, up its leg to $R$, across $R$, and down the other leg to $v_{i,l_i}$; i.e. $P=v_{j,l_j},\ldots,v_{j,h},v_{i,h},\ldots,v_{i,l_i}$, where $R=(v_{i,h},v_{j,h})$.
Now slide $A$ along $P$ until it contains $v_{i,l_i}$ --- the result we call $\slide_P(A)$.
Then set $f(A) = \cL^\pr \cup \slide_P(A) \cup T$.
Of course $|f(A)| = |A|$, $v_{i,l_i}\in f(A)$, and $f(A)$ is independent because $R$ was not full.
Moreover, $\cL^\pr(f(A)) = \cL^\pr(A)$, and so the inverse of $f$ on $f(A)$ is uniquely determined.
Note that in these first two cases $f$ preserves both inclusion and exclusion of $v_0$.
This means that $T$ cannot affect the independence of $f(A)$.
Finally we define $f$ on the remaining sets $A$ having full ladders.
If $l_j$ and $l_i$ are both even then the full ladder implies that $v_0\in A$ because $v_{j,l_j}\in A$.
If $l_j$ and $l_i$ are both odd then, since $v_{i,l_i}\not\in A$, the full ladder again implies that $v_0\in A$.
If $l_j$ is even and $l_i$ is odd then $l_j<l_i$ means that $v_0\in A$ because $v_{j,l_j}\in A$, while $l_i<l_j$ means that $v_0\in A$ because $v_{i,l_i}\not\in A$.
So in all these remaining cases we have $v_0\in A$.
When $l_j<l_i$ we let $P$ be the $v_{j,l_j}v_{i,l_j-1}$-path in $S$ (i.e. $P=v_{j,l_j},\ldots,v_{j,1},v_0,$ $v_{i,1},\ldots,v_{i,l_j-1}$).
When $l_j>l_i$ we let $P$ be the $v_{j,l_i-1}v_{i,l_i}$-path in $S$ (i.e. $P=v_{j,l_i-1},\ldots,v_{j,1},v_0,$ $v_{i,1},\ldots,v_{i,l_i}$).
In both cases we let $Q$ be the $v_{j,l_j}v_{i,l_i}$-path in $S$, minus $P$.
We shift $A$ along $P$ just one step toward $v_{i,l_i}$ --- call the result $\shift_P(A)$ --- and flip $A$ on $Q$ (that is, if $Q = (q_0, ..., q_k)$ then replace each $q_h$ in $A$ by $q_{k-h}$) --- call the result $\flip_Q(A)$.
Now define $f(A) = \shift_P(A) \cup \flip_Q(A) \cup T$.
Of course $|f(A)| = |A|$, $v_{i,l_i}\in f(A)$ (because of the flip if $l_j<l_i$ or the shift if $l_j>l_i$), and A is independent (because of the flip if $l_j<l_i$ or the shift if $l_j>l_i$).
Moreover, $f(A)$ has a full ladder, and so the inverse of $f$ on $f(A)$ is uniquely determined.
Notice that, because of the shift, $v_0\not\in f(A)$, and so $T$ cannot affect the independence of $f(A)$.
Thus the injection is complete.
\pf
\section{Open questions}
Determining whether or not spider graphs satisfy Conjecture \ref{minimaxconj} remains open. The compression/induction technique that has been used to prove Conjecture \ref{minimaxconj} for other graph classes appears difficult to use in this case. The nature of Theorem \ref{BestLeaf} implies that the center of the largest star may ``jump'' when we consider subtrees of the spider.
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In general, determining the centers of the largest stars in trees remains an open problem.
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Teacher Blogs > Web Watch See our Teachers news coverage
Teacher's look at education news from around the Web. Web Watch will no longer be updated as of April 26, 2010. For the latest on teaching news, please visit Teaching Now.
« The Cost of Copying Homework | Main | Teacher Fakes Shooting for Sake of Science »
The Biggest March Madness Upset Yet
By Bryan Toporek on March 25, 2010 4:18 PM
While the likes of Ohio University, Northern Iowa, and St. Mary's probably sent your NCAA tournament bracket up in flames, you might find some solace in the news that an autistic 17-year-old from Chicago saw it all coming and picked every single game of March Madness' first weekend correctly.
Yes, Alex Hermann defeated the 1:13,460,000 odds this past weekend by correctly picking the winners of all 48 games in the NCAA tournament's first weekend, according to NBCChicago.com.
Hermann, with the help of his older brother Andrew, entered his bracket on CBSSports.com's Bracket Manager before the start of the tournament; after a first weekend chock full of upsets, Hermann's bracket remained unscathed.
"I checked his bracket and it was off the chart," Andrew said. "I thought it was a big deal."
At ESPN.com, approximately 4.78 million people participated in their NCAA tournament challenge, but there's not a single perfect bracket left.
How can Alex explain his absurdly great decision-making?
"I'm good at math," Alex said. "I'm kind of good at math and at stats I see on TV during the game."
If only it were that easy for the rest of us, Alex.
Check out his bracket here.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: google protobuf api in android for google cloud datastore I'm trying to use a google lib to read and write inside the google cloud datastore from an Android App.
The library is this one:
google-api-services-datastore-protobuf-java-v1beta1-rev1-1.0.0
I already made a std java project that uses this library and all is working.
In Android when I try to compile it, I get this error:
[2013-07-01 12:54:33 - AndroidDatastore] Dx 1 error; aborting
[2013-07-01 12:54:33 - AndroidDatastore] Conversion to Dalvik format failed with error 1
So I tried to import source api too, but got hundreds of errors
After a while I solved this problem and application launches on android but I get this run time error:
07-08 11:14:26.002: I/dalvikvm(14946): Could not find method com.google.api.services.datastore.DatastoreV1$Entity.newBuilder, referenced from method it.micrel.androiddatastore.MainActivity.addEntry
07-08 11:14:26.002: W/dalvikvm(14946): VFY: unable to resolve static method 9: Lcom/google/api/services/datastore/DatastoreV1$Entity;.newBuilder ()Lcom/google/api/services/datastore/DatastoreV1$Entity$Builder;
07-08 11:14:26.002: D/dalvikvm(14946): VFY: replacing opcode 0x71 at 0x0002
07-08 11:14:26.002: E/dalvikvm(14946): Could not find class 'com.google.api.services.datastore.client.DatastoreOptions$Builder', referenced from method it.micrel.androiddatastore.MainActivity.connectToDatastore
07-08 11:14:26.002: W/dalvikvm(14946): VFY: unable to resolve new-instance 32 (Lcom/google/api/services/datastore/client/DatastoreOptions$Builder;) in Lit/micrel/androiddatastore/MainActivity;
07-08 11:14:26.002: D/dalvikvm(14946): VFY: replacing opcode 0x22 at 0x0000
07-08 11:14:26.012: I/dalvikvm(14946): Could not find method com.google.api.services.datastore.DatastoreV1$BlindWriteRequest.newBuilder, referenced from method it.micrel.androiddatastore.MainActivity.insertAutoId
07-08 11:14:26.012: W/dalvikvm(14946): VFY: unable to resolve static method 5: Lcom/google/api/services/datastore/DatastoreV1$BlindWriteRequest;.newBuilder ()Lcom/google/api/services/datastore/DatastoreV1$BlindWriteRequest$Builder;
07-08 11:14:26.012: D/dalvikvm(14946): VFY: replacing opcode 0x71 at 0x0000
07-08 11:14:26.012: I/dalvikvm(14946): Could not find method com.google.api.services.datastore.DatastoreV1$Query.newBuilder, referenced from method it.micrel.androiddatastore.MainActivity.listEntry
07-08 11:14:26.012: W/dalvikvm(14946): VFY: unable to resolve static method 16: Lcom/google/api/services/datastore/DatastoreV1$Query;.newBuilder ()Lcom/google/api/services/datastore/DatastoreV1$Query$Builder;
07-08 11:14:26.012: D/dalvikvm(14946): VFY: replacing opcode 0x71 at 0x0000
07-08 11:14:26.012: I/dalvikvm(14946): Could not find method com.google.api.services.datastore.DatastoreV1$RunQueryRequest.newBuilder, referenced from method it.micrel.androiddatastore.MainActivity.runQuery
07-08 11:14:26.012: W/dalvikvm(14946): VFY: unable to resolve static method 21: Lcom/google/api/services/datastore/DatastoreV1$RunQueryRequest;.newBuilder ()Lcom/google/api/services/datastore/DatastoreV1$RunQueryRequest$Builder;
07-08 11:14:26.012: D/dalvikvm(14946): VFY: replacing opcode 0x71 at 0x0000
07-08 11:14:26.012: W/dalvikvm(14946): VFY: unable to find class referenced in signature (Lcom/google/api/services/datastore/client/Datastore;)
07-08 11:14:26.022: W/dalvikvm(14946): VFY: unable to resolve exception class 29 (Lcom/google/api/services/datastore/client/DatastoreException;)
07-08 11:14:26.022: W/dalvikvm(14946): VFY: unable to find exception handler at addr 0x68
07-08 11:14:26.022: W/dalvikvm(14946): VFY: rejected Lit/micrel/androiddatastore/MainActivity;.onCreate (Landroid/os/Bundle;)V
07-08 11:14:26.022: W/dalvikvm(14946): VFY: rejecting opcode 0x0d at 0x0068
07-08 11:14:26.022: W/dalvikvm(14946): VFY: rejected Lit/micrel/androiddatastore/MainActivity;.onCreate (Landroid/os/Bundle;)V
07-08 11:14:26.022: W/dalvikvm(14946): Verifier rejected class Lit/micrel/androiddatastore/MainActivity;
07-08 11:14:26.022: W/dalvikvm(14946): Class init failed in newInstance call (Lit/micrel/androiddatastore/MainActivity;)
07-08 11:14:26.092: D/AndroidRuntime(14946): Shutting down VM
07-08 11:14:26.092: W/dalvikvm(14946): threadid=1: thread exiting with uncaught exception (group=0x40aac210)
07-08 11:14:26.202: E/AndroidRuntime(14946): FATAL EXCEPTION: main
07-08 11:14:26.202: E/AndroidRuntime(14946): java.lang.VerifyError: it/micrel/androiddatastore/MainActivity
07-08 11:14:26.202: E/AndroidRuntime(14946): at java.lang.Class.newInstanceImpl(Native Method)
07-08 11:14:26.202: E/AndroidRuntime(14946): at java.lang.Class.newInstance(Class.java:1319)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.Instrumentation.newActivity(Instrumentation.java:1023)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.ActivityThread.performLaunchActivity(ActivityThread.java:1882)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.ActivityThread.handleLaunchActivity(ActivityThread.java:1992)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.ActivityThread.access$600(ActivityThread.java:127)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.ActivityThread$H.handleMessage(ActivityThread.java:1158)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.os.Handler.dispatchMessage(Handler.java:99)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.os.Looper.loop(Looper.java:137)
07-08 11:14:26.202: E/AndroidRuntime(14946): at android.app.ActivityThread.main(ActivityThread.java:4448)
07-08 11:14:26.202: E/AndroidRuntime(14946): at java.lang.reflect.Method.invokeNative(Native Method)
07-08 11:14:26.202: E/AndroidRuntime(14946): at java.lang.reflect.Method.invoke(Method.java:511)
07-08 11:14:26.202: E/AndroidRuntime(14946): at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:823)
07-08 11:14:26.202: E/AndroidRuntime(14946): at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:590)
07-08 11:14:26.202: E/AndroidRuntime(14946): at dalvik.system.NativeStart.main(Native Method)
07-08 11:14:32.558: I/Process(14946): Sending signal. PID: 14946 SIG: 9
In the end the question is:
How shoud I access the google cloud datastore from android??
I'm start thinking that this library is not compatible with Android...
Thank you
A: I found out that the error was given by the dx component: it seems that in android it is not possible to use an external jar library with packages named java.* or javax.*
Look at this link here: https://stackoverflow.com/a/11547272/223969
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Q: What is the difference between controlling for a variable in a regression model vs. controlling for a variable in your study design? I imagine that controlling for a variable in your study design is more effective at reducing error than controlling for it post-hoc in your regression model.
Would someone mind explaining formally how these two instances of "controlling" differ? How comparatively effective are they at reducing error and yielding more precise predictions?
A: By "controlling for a variable in your study design", I assume you mean causing a variable to be constant across all study units or manipulating a variable so that the level of that variable is independently set for each study unit. That is, controlling for a variable in your study design means that you are conducting a true experiment. The benefit of this is that it can help with inferring causality.
In theory, controlling for a variable in your regression model can also help with inferring causality. However, this is only the case if you control for every variable that has a direct causal connection to the response. If you omit such a variable (perhaps you didn't know to include it), and it is correlated with any of the other variables, then your causal inferences will be biased and incorrect. In practice, we don't know all the relevant variables, so statistical control is a fairly dicey endeavor that relies on big assumptions you can't check.
However, your question asks about "reducing error and yielding more precise predictions", not inferring causality. This is a different issue. If you were to make a given variable constant through your study design, all of the variability in the response due to that variable would be eliminated. On the other hand, if you simply control for a variable, you are estimating its effect which is subject to sampling error at a minimum. In other words, statistical control wouldn't be quite as good, in the long run, at reducing residual variance in your sample.
But if you are interested in reducing error and getting more precise predictions, presumably you primarily care about out of sample properties, not the precision within your sample. And therein lies the rub. When you control for a variable by manipulating it in some form (holding it constant, etc.), you create a situation that is more artificial than the original, natural observation. That is, experiments tend to have less external validity / generalizability than observational studies.
In case it's not clear, an example of a true experiment that holds something constant might be assessing a treatment in a mouse model using inbred mice that are all genetically identical. On the other hand, an example of controlling for a variable might be representing family history of disease by a dummy code and including that variable in a multiple regression model (cf., How exactly does one "control for other variables"?, and How can adding a 2nd IV make the 1st IV significant?).
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2014 curriculum vitae
Yüklə 73.91 Kb.
ölçüsü 73.91 Kb.
Eliseo A. Eugenin, Ph.D.
Date of Birth: October, 21, 1970
Citizenship: Chile
Residence: US (Permanent Resident)
Minorities: Yes, Hispanic
Universidad Austral de Chile, Valdivia, Chile.
B.S., 1996 (Biochemistry)
- Universidad Austral de Chile, Valdivia, Chile
M.S., 1996 (Biochemistry)
- Pontificia Universidad Catolica de Chile, Santiago, Chile
Ph.D. 2001 (Physiology)
Ph.D. Thesis, Bonn University, Germany and Pontificia Universidad Catolica de Chile, Santiago, Chile. 2001 (Physiology)
POST-GRADUATE TRAINING
The Albert Einstein College of Medicine, Bronx, NY.
Department of Pathology
Laboratory of Dr. Joan W. Berman
Postdoctoral Fellow (2001-2003)
PROFESSIONAL APPOINTMENTS:
Instructor (2004-2007)
Assistant Professor (2007 to present)
CFAR Investigator (2007 to present)
The Public Health Institute of Research (PHRI) and Department of Immunology and Molecular Genetics at UMDNJ, NJ, Newark.
Associated Professor (2012)
The Public Health Institute of Research (PHRI) and Department of Immunology and Molecular Genetics at Rutgers University, NJ, Newark (due to fusion of UMDNJ and Rutgers University).
Associated Professor (2012-present)
DAAD Latin American fellow from the government of Germany, 2010.
Member of the ZRG1-AARR-C on special emphasis panel/scientific review group, 2012
Member of German study section of HIV and other infectious disease, 2013-present
Regular member of study section for FONDECYT-Chile, 08/07-present
Associate Editor International Society of NeuroVirology, Newsletter
Scientific Editor, PLOS one, 2013-present
Regular member study section, Research Projects on HIV and Hepatitis, Gilead Sciences, Spain.
EDITORIAL BOARD:
-Annals of Virology and Research, 2013- present
-The Journal of Gene Therapy for Genetic Disorders (GTGD), 2013-present
- Plos One, 2014-present
PROFESSIONAL SOCIETY MEMBERSHIPS:
APS- American Physiology Society (regular). 2001- present.
ASCB- American Society of Cell Biology (regular). 2000- present.
JNV- American Association of NeuroVirology. 2003-present.
New York Academy of Sciences. 2003- present
Society of Neuroscience. 2005-present
Society of Neuroimmune pharmacology (SNIP). 2006-present
International Society of Neurovirology (ISNV) 2005-present
AAAS-Science, American Assoc for Advancement of Science. 2006-Present.
OTHER PROFESSIONAL ACTIVITIES:
Selected Invited Presentations at Scientific Meetings (more than 180)
- Sáez C.G, Berthoud V.M, Eugenín E.A. and Sáez J.C. Expression of connexin 32, the main protein of the gap junctions in the liver of rat is regulated by dexamethasone. Summaries XVI International Congress of Biochemistry and Molecular Biology. September of 1994, 19-22 New Delhi, India.
- Eugenín E.A., Garcés G and Sáez J.C. A norepinephine-induced glycogenolytic signal propagates through gap junctions between rat astrocytes. Summary of the XXXVII Annual Meeting of the Society of Biology of Chile. Vol 2, No. 3, November of 1994, Puyehue, Chile.
- Eugenín E.A., Sáez G.C., Garcés G. and Sáez J.C. Regulation of the content of glycogen of the pineal gland of rat for noradrenaline. XVII annual Meeting of the Society of Pharmacology of Chile. August of 1995, 18-19 Santiago, Chile.
- Sáez C. G., Eugenín E. A., Elliot L. Hertzberg and Juan C. Sáez. Acute but not chronic CCl4 - induced liver injury transiently reduces Cx26 and Cx32 in hepatocytes and increases Cx43 in macrophages and proliferating cells. Meeting From ion channels to cell - to - cell conversations in the Center of Scientific Studies Santiago (CECS), November 28-30, 1995, Santiago, Chile.
- Eugenín E. A., Sáez C. G. and Sáez J. C. Gap junctions favor the vassopressin - induced glycogenolysis in rat hepatocytes. Communication to VIII PABMB, Pan-American Congress of Molecular Biology and Biochemistry. 16 at November of 1996, 21 Pucón - Chile.
- Eugenín E. A., Sáez C. G. and Sáez J. C. Gap junctions favor the vasopressin-induced glycogenolisis in rat hepatocytes. 36 American Society for Cell Biology Annual Meeting, Mol. Biol. Cell. December 7-11, 1996, San Francisco, California, USA.
- Bitran M., W., Eugenín E. A., Orio P. and Daniels A. Neuropeptide Y (NPY) - induced inhibition of hypothalamic noradrenaline release: Characterization of receptor subtype and role of nitric oxidizes, 27 at August of 1997, 30 Cartagena of India, Colombia
- Bitran M., W., Eugenín E. A., Orio P. and Daniels A. Neuropeptide Y (NPY)-induced inhibition of hypothalamic noradrenaline release: characterization of receptor subtype and role of nitric oxide. XXXIX Annual Meeting of the Society of Biology. Vol 5, No. 4, Nov of 1997, Pucón, Chile.
- Eugenín E. A., Alvarez M. and Sáez C. G. Regulation of gap junctional communication in MCF-7 cells by estrogen and tamoxifen. A.A.C.R. Annual Meeting, Vol. 39, March 28-April 1, 1998, New Orleans, Louisiana. U.S.A
- Sáez C. G., Eugenín E. A. and Alvarez M. Gap junctional communication in MCF-7 cells. A.S.C.O. Meeting, San Francisco, June 1998.
- Eugenín E. A., Garces G. and Sáez J. C. The conditioned media by endothelial cells induces the formation of gap junction channels between macrophages. Annual meeting of the Society of Physiology, Quinamavida, Chile, April 5-8 1998.
- Eugenín E. A. and Sáez J. C. Regulation of Cx43 cellular trafficking and functional gap junctional communication in a macrophage cell line (J744). International Gap Junction Meeting. Rio de Janeiro, Brazil, 6-11 June 1998.
- Sáez J. C., Brañes M. C., Eugenín E. A., and Martinez A. D. Compounds released by endothelial cells induce gap junctional communication between macrophagic cells. International gap junction conference, 28 August- 2 September, 1999, Gwatt, Switzerland.
- Eugenín, E. A., Vega, V. L., Brañes, M. C., Ward, P. H. and Sáez, J. C. (2000) Tourniquet shock alters the expression of connexins in liver parenchymal and inflammatory cells J. Physiol (London) 523P, 23 p. [Abstract].
- H. González, E.A. Eugenín, G. Garcés and J.C. Sáez. (2000) Immunofluorescence detection of connexins, protein gap junction subunits, in rat Kupffer cells. J. Physiol (London) 523P, 21 p.
- Eugenin E.A. and Saez J.C. (2000) Microglia express connexin43 during brain damage and in vitro form gap junctions that enhance TNF- secretion, but not affect apoptosis. Mol. Biol. Cell 11, 227a [Abstract].
- Eugenin E.A. and Sáez J.C. (2001) Human monocytes/macrophages express connexin43 and form functional gap junction channels upon culturing under control conditions or treatment with LPS plus IFN-. International meeting of Gap Junctions, 04-09 of August, Hawaii, USA.
- Sáez J.C., Contreras J., Sánchez H., Eugenín E.A., Spiedel D., Theis M., Willecke K. and Bennett M.V. (2001) Metabolic inhibition induces opening of connexin43 gap junction hemichannels but not affect gap junctional communication in cortical rat astrocytes. International meeting of Gap Junctions, 04-09 of August, Hawaii, USA.
-. Eugenin E.A., Lopez L., D´Aversa T.J., Osieki K., Pettoello-Mantovani M., Goldstein H., and Berman J.W. Monocyte-chemoattractant protein-1 (MCP-1) increases the transmigration of HIV-infected PBMC, but not uninfected cells, across a model of the human Blood Brain Barrier (BBB), resulting in increased permeability. Postdoc Symposia. AECOM. December 2001.
-. Eugenin E.A., Lopez L., D´Aversa T.J., Osieki K., Pettoello-Mantovani M., Goldstein H., and Berman J.W. Monocyte-chemoattractant protein-1 (MCP-1) increases the transmigration of HIV-infected PBMC, but not uninfected cells, across a model of the human Blood Brain Barrier (BBB), resulting in increased permeability. Exp. Biology. 20-24 Aphil, 2002.
- Eugenin E.A. Oral presentation at the VIth Conference on Cerebral Vascualr Biology, Munster, Germany, June, 2005
- Eugenin E.A., and Berman J.W. HIV-1 infected human astrocytes use gap junction communication to induce apoptosis of uninfected cells. Keystone symposia. March, 25-30, 2007, Whistler, British Columbia.
- Eugenin E.A. and Berman J.W. Role of gap junctional communication in NeuroAIDS. Keystone symposia, Banff, 2008, Canada.
- Eugenin E.A., CFAR presentations, 2007 to 2009, Role of Gap Junction in the Pathology of NeuroAIDS.
JOURNAL EDITORIAL BOARD MEMBER
The Scientific World Journal
AD HOC REVIEWER FOR JOURNALS.
American Journal Pathology
Journal Neuroscience Research
Neuroscience Letters
AIDS and retroviruses
J Neurochemistry
BBA-Molecular Cell Research
Journal of NeuroImmune Pharmacology
Immunobiology
CNS & Neurological Disorders-drug targets
COURSE TEACHING
Grant Writing Course, Department of Pathology, December-March, 2007-2012.
Mechanisms of Disease, Medical School Course-Small Group Sessions, Department of Pathology, September to December, 2006-2012
Advance concepts in Virology, 2012-present.
TRAINING GRANT PARTICIPATION
Millennium Nucleus on Immunology and Immunotherapy (Santiago, Chile). 1/04-present
PAST/CURRENT RESEARCH GRANT SUPPPORT
- No. 2990004 (Fondecyt, Chile) PI: Eliseo Eugenin 03/1999-02/2001
Biogenesis and function of gap junctions between macrophagic cells Role: P.I.
- NIH/NIMH KO1 MH076679-01A1 P.I: Eliseo Eugenin 04/01/06-03/31/12 NCE
The role of gap junctions channels in NeuroAIDS. Role: P.I.
- Pilot Grant in Aging-related research, Montefiore Medical Center.
Role of Tight Junction protein expression in human smooth muscle cells during the pathogenesis of atherosclerosis.
2008-2009 P.I: Eliseo Eugenin Role: P.I
- CFAR developmental grant PI: Eliseo Eugenin Role: PI
Methamphetamine alters blood brain barrier functions facilitating entry of HIV and fungal pathogens into the central nervous system. 2010-2011
-NIH/NIMH RO1 MH096625 PI: Eugenin E. Funded 2012-2017
Title: Astrocyte connexin43 containing channels amplify CNS dysfunction in NeuroAIDS
This grant is a continuation and extension of a KO1 about the role of these channels in astrocytes in the pathogenesis of CNS HIV disease.
-PHRI Internal funding PI: Eugenin E. Funded 2012-2017
Title: mechanisms of CNS compromise by flaviviruses
-GSK scientific collaboration PI: Eugenin E. Funded 2013-2015
Title: Role of TNT in HIV spread: a novel mechanism of HIV transmission
-NIH/NIMH 1RO1MH104150-01 PI: Eugenin E. 04/01/2013-03/31/2018 (pending)
Targeting HIV CNS reservoirs: role of mitochondrial factors and reactivation. This proposal examines the mechanism of viral reactivation and apoptosis of latent infected cells.
-NIH/Office of the director 11091385 Multi PI: Dubnau, David (Contact); Eugenin, Eliseo and Tyagi, Sanjay. A Zeiss cell observer spinning disk live cell imaging system for NJMS. Review 01/2013 (Pending)
-NIH grant 11752457 Role of pannexin-1 hemichannels in CNS disease
-NIH/HL128152 Mechanisms of accelerated atherosclerosis in HIV
-NIH/AI 118436 Mechanisms of survival of CNS reservoirs
PATENT SUBMITTION
- Novel cellular targets for HIV infection. Provisional patent application, No. 61/606,696. Filed on March 5, 2012.
-New experimental approaches to detection of pathogens using improved confocal and staining thecniques. Filed on November, 2013.
FELLOWSHIPS.
1996 PhD fellowship assigned by the Pontificia Universidad Católica de Chile.
PhD fellowship granted by Conicyt, Chile.
Doctoral grant by FONDECYT-Chile (Number: 2990004). PI: Eliseo Eugenin
Fellowship assigned by DAAD (Germany) to do thesis studies in Germany (Bonn University).
Fellowship to finish doctoral thesis from CONICYT-Chile.
STUDENT MENTORING
-Courtney Villeux, Ph.D. student Rutgers University
-Luis
-Victoria Riveiro, Science Park High school, Summer, 2013 (First place Ribbons award)
-Paul Castellano, Ph.D. candidate, 2012-present
- Courney Re, Ph.D. candidate, 2013
-Oluwadamilola T. Oke, Science Park High School, summer, 2012 (Award for the best work)
-Zaquilla Qualls, Ph.D. candidate, 2012
-Stephanie Velasquez, Ph.D. candidate, 2012-present
-Joy Gibson, M.D./Ph.D. candidate, Ph.D. completed 04/2011.
-Jessie King, M.D./Ph.D. candidate. Ph.D. completed 10/2010.
-Daniel Childs, College Junior, summer studies (2009).
-Deepa Rastogi, Clinical mentoring, Montefiore Children's Hospital (2008-present).
-Rebecca Parver-Gams, Summer student (2002) and Medical Student (2004).
-Tamar Kessel, Summer student (2001) and Medical Student (2003).
Post doctoral Trainees.
Luis Martinez (2010-2011), Current position, Assistant Professor Long Island University.
Loreto Carvallo (2010-2012), Current position, Postdoctoral fellow, Albert Einstein, Bronx, NY.
Juan Andres Orellana (2011-2012), Current position, Assistant Professor, Catholic University, Chile.
Carlo Daep (2012-present), Current position, Group Leader, Colgate-Palmovile Industries, NJ.
1.- Eugenín E. A. (1996) Regulación de los almacenes de glicógeno por norepinefrina en la glándula pineal de rata: papel de las uniones en hendidura. VI + 67 pp. Undergraduate Thesis.
2.- Eugenín E.A., Sáez G.C., Garcés G. and Sáez J.C. (1997) Regulation of the rat pineal gland glycogen content by norepinephine. Brain Research. 760 (1,2); 34-41.
3.- Eugenín E. A., Gonzalez H., Sáez C. G. and Sáez J. C. (1998) Gap junctional communication coordinates the vasopressin-induced glycogenolisis in rat hepatocytes. Am. J. Physiol 274 (6); G 1109-1116.
4.- Bitran M., Tapia W., Eugenín E. A., Orio P., and Boric M. P. (1999) Neuropeptide Y induced inhibition of noradrenaline release in the rat hypothalamus: Characterization of receptor subtype and role of nitric oxide. Brain Research, 851 (1,2); 87-93.
5.- Saez J.C., Brañes M.C., Corvalan L.A., Eugenín E.A., Gonzalez H., Martinez A.D. and Palisson F. (2000) Gap junctions in cells of the immune system: structure, regulation and possible functional roles. Brazilian Journal of Medical and Biological Research. 33: 447-455.
6.- Saez J.C. Araya R., Brañes M.C., Concha M., Contreras J., Eugenín E.A., Martinez A.D., Palisson F. and Sepulveda M.A. (2000) Gap junctions in inflammatory responses: connexins, regulation and possible functional roles. Current Topics in Membranes. 49: 555-576.
7.- Eugenín E.A., Eckcart D., Theis M., Willecke K., Bennett M.V. and Sáez J.C. (2001) Microglia at brain stab wounds express connexin43 and in vitro form functional gap junctions after treatment with interferon-gamma and tumor necrosis factor-alpha. Proc. Natl. Acad. Sci. USA 98 (7), 4190-4195.
8.- Dougnac A., Riquelme A., Calvo M., Andresen M., Magedzo A., Eugenin E., Marshall G. and Gutierrez. (2001) Study of cytokines kinetics in severe sepsis and its relationship with mortality and score of organic dysfunction. Rev. Med. Chil. 129 (4), 347-358.
9- Contreras J., Sánchez H., Eugenin E.A., Spiedel D., Theis M., Willecke K., Bukauskas F.F., Bennett M.V and Sáez J.C. (2002) Metabolic inhibition induces opening of unapposed connexin43 gap junction hemichannels and reduces gap junctional communication in cortical astrocytes in culture. Proc. Natl. Acad. Sci. USA. 99 (1) 495-500
10.- Gonzalez H., Eugenín E. A., Garces G., Solis N., Pizarro M., Accatino L. and Sáez J. C. (2002) Regulation of hepatic connexins during cholestasis: the involvement of some cellular and humoral inflammatory mediators. Am. J. Physiol. 282 (6) G991-G1001.
11.- Martinez A. D., Eugenín E. A., Brañes M.C., Bennett M.V., and Sáez J.C. (2002) Identification of a second messenger pathway that induces the expression of functional gap junctions in cultured newborn rat microglia. Brain Research. 943 (2) 191-201.
12.- Eugenin E.A., Branes M.C., Berman J.W. and Saez J.C. (2003) TNF- plus IFN- induce connexin43 expression and formation of gap junctions between human monocytes/macrophages that enhance physiological responses. J Immunology. 170: 1320-1328.
13.- Eugenin E.A. and Berman J.W. (2003) Chemokine dependent mechanisms of leukocyte trafficking across a model of the blood-brain barrier. Methods. 29, 351-361.
14.- Saez C.G., Velasquez L., Montoya M., Eugenin E.A. and Alvarez M.G. (2003) Increased Gap junctional intercellular communication is directly related to the anti-tumor effect of all-trans retinoic acid plus tamoxifen in a human mammary cell line. J Cellular Biochemistry. 89: 450-461.
15.- Eugenin E.A., D'Aversa T., Lopez L., Calderon T. and Berman J.W. (2003) MCP-1 (CCL2) protects human neurons and astrocytes from NMDA or HIV-tat-induced apoptosis. Journal of Neurochemistry. 85: 1299-1311.
16.- Eugenin E.A., Dyer G., Calderon T.M. and Berman J.W. (2005) HIV-1 tat protein induces migration of human fetal microglia by a mechanism CCL2 (MCP-1) dependent. Possible role of microglial axis CCL2-CCR2 in NeuroAIDS. Glia 49: 501-510.
17.- D'Averza T.G., Eugenin E.A., Berman J.W. (2005) NeuroAIDS: Contributions of HIV-1 proteins tat and gp120, and CD40 to microglial activation. J. Neuroscience Res. 81: 436-46.
18.- Eugenin E.A., Gamss R., Buckner C., Buono D., Klein R., Schoenbaum E.E., Calderon T.M. and Berman J.W. (2006) Shedding of PECAM-1 during HIV-infection: A role for sPECAM-1 in NeuroAIDS and in the pathogenesis of dementia. J Leukocyte Biology. 79: 444-452.
19.-, King J.E., Eugenin E.A., Buckner C. and Berman J.W. (2006) New insights in HIV-tat induced neurotoxicity. Microbes and Immunity. 8: 1347-1357.
20.- Eugenin E.A., Osiecki K., Lopez L., Goldstein H., Calderon T.M. and Berman J.W. (2006) CCL2/MCP-1 mediates enhanced transmigration of HIV-infected leukocytes across the blood brain barrier: a potential mechanism of HIV-CNS invasion and NeuroAIDS. J. Neuroscience. 26:1098-106
21. Buckner C., Luers A., Calderon T., Eugenin E.A. and Berman J.W. (2006) Neuroimmunity and the Blood Brain Barrier: The Molecular Regulation of Leukocyte Transmigration and Viral Entry into the Nervous System with a Focus on NeuroAIDS. J Neuroimmune Pharmacol.1 (2) 160-181.
22.- Calderon T.M., Eugenin E.A., Lopez L., Hesselgesser J., Raine C.S. and Berman J.W. (2006) Upregulation of SDF-1a by astrocytes and endothelial cells in multiple sclerosis lesions: Mechanism of induction by myelin basic protein. J. Neuroimmunology. 177(1-2): 27-39.
23. Eugenin E. A., King J.E., Nath A., Calderon T. M., Zukin S.R., Bennett M.V.L, and Berman J. W. (2007) HIV-tat induces the formation of an LRP-PSD95-NMDAR-nNOS complex necessary for neuronal apoptosis. Proc. Natl. Acad. Sci. USA, 104 (9)3438-3443.
24. Eugenin E.A., Gonzalez H., Sanchez H., Branes M.C. and Saez J.C. (2007) Inflammatory conditions induce gap junctional communication between Kupffer cells both in vivo and in vitro. Cell Immunology. 247(2):103-10.
25. Eugenin E.A. and Berman J.W. (2007) Gap junctions mediate HIV-bystander killing in astrocytes. J. Neuroscience. 27 (47): 12844-50.
26. Dougnac A., Castro R., Riquelme A., Calvo M., Eugenin E., Arellano M., Pattillo M., Regueira T., Mercado M., and Andresen M. (2007) Pro and anti-inflammatory balance of septic patients is associated with severity and outcome. Crit. Care & Shock, 10: 99-107.
27. D'Aversa T.G., Eugenin E.A. and Berman J.W. (2008) CD40-CD40L interactions in human microglia induces CXCL18 (IL-8) secretion by a mechanism dependent on activation of ERK1/2, and nuclear translocation of NFkB and AP-1. J. Neurosci. Res. 86 (3):630-639.
28. Eugenin E.A., Morgello S., Klotman M.E., Mosoian A., Lento P.A., Berman J.W., and Schecter A.D. (2008) HIV infects human arterial smooth muscle cells in vivo and in vitro: Implications for the pathogenesis of HIV-mediated vascular disease. American Journal of Pathology.172 (4), 1100-11.
29. Rao V.R., Eugenin E.A., Berman J.W. and Prasad V.R. (2009) Methods to study monocyte migration induced by HIV-infected cells. Prasad and Kalpana Editors, in HIV protocols by Humana Press. 485:295-309.
30. Opazo C., Gianini A., Pancetti F., Azkcona G., Noches V., Gonzalez P., Porto M., Rosenthal D., Eugenin E., Kalergis A., and Riedel C.A. (2008) Maternal Hypothyroxinemia impairs Spatial Learning and Synaptic nature and function in the Offspring. Endocrinology. 149(10):5097-106.
31. Rao V.R., Sas A., Eugenin E., Siddappa N.B., Bimonte-Nelson H., Berman J.W. Ranga U., Tyor W.R. and Prasad V.R. (2008) HIV-1 clade-specific differences in the induction of neuropathogenesis. J Neuroscience. 28 (40), 10010-10016.
32. Eugenin E.A., Gaskill P.J., and Berman J.W. (2009) Tunneling nanotubes (TNT) are induced by HIV-infection of macrophages: a potential mechanism for intercellular HIV trafficking. Cellular Immunology.254 (2);142-148.
33. Bueno S., Gonzalez P., Cautivo K., Mora J., Leiva E., Tobar H., Fennelly G., Eugenin E.A., Jacobs W., Riedel C., Kalergis A. (2009) Protective T cell immunity against respiratory syncytial virus is efficiently induced by recombinant BCG. Proc. Natl. Acad. Sci. USA. 105(52):20822-7
34. Eugenin E.A., Gaskill P.J., and Berman J.W. (2009) Tunneling nanotubes (TNT): a potential mechanism for intercellular trafficking. Communicative & Integrative Biology. 2(3): 243-4.
35. Gaskill P.J., Calderon T.M., Luers A.J., Eugenin E.A., Javitch J.A., and Berman J.W. (2009) HIV infection of human macrophages is increased by dopamine; a bridge between HIV associated neurologic disorders and drug abuse. Am J. Pathology. 175(3)1148-59.
36. Agiostratidou G., Suyama K.,Badano I., Keren R., Chung S., Anzovino A., Hulit J., Qian B., Bouzahzah B., Eugenin E., Loudig O., Phillip G.R., Locker J. and Hazan R.B. (2009) Loss of retinal cadherin facilitates mammary tumor progression and metastasis. Cancer Res. 69(12): 5030-8.
37. Hazleton J.E., Berman J.W., and Eugenin E.A. (2010) Novel mechanism of central nervous system damage in HIV infection. HIV/AIDS-Research and Palliative Care. 2, 39-49.
38. King J.E., Eugenin E.A., Hazleton J.E., Morgello S. and Berman J.W. (2010) Mechanisms of HIV-Tat Induced phosphorylation of NMDA Receptor Subunit 2A in Human Primary Neurons: Implications for NeuroAIDS Pathogenesis. Am. J Pathology. 176(6) 2819-30.
39. Roberts T.K., Eugenin E.A., Morgello S., Clements J.E, Zink M.C., and Berman J.W. (2010) PrPC, the Cellular Isoform of the Human Prion Protein, is a Novel Marker of HIV-Associated Neurocognitive Impairment and Modulates Neuroinflammation. American J. Pathology. 177(4) 1848-60.
40. Eugenin E.A., King J.E., Hazleton J.E., Major E.O., Bennett M.V.L., Zukin R.S. and Berman J.W. (2011) Differences in NMDA receptor expression during development determine the response of neurons to HIV-tat mediated neurotoxicity. Neurotoxicity Research. 19 (1) 138-48.
41. Roberts T.K., Eugenin E.A., Romero I.A., Weksler B.B., Couraud P-O. and Berman J.W. (2012) CCL2 disrupt the adherens junction and increases blood brain barrier permeability. Laboratory Investigation. 92(8): 1213-33.
42. Eugenin E.A., Clements J.E, Zink M.C., and Berman J.W. (2011) HIV infection of human astrocytes disrupts blood brain barrier integrity by a gap junction dependent mechanism. The Journal of Neuroscience. 31(26):9456-65.
43. Coniglio S.J., Eugenin E., Dobrenis K., Stanley E.R., West B.L., Symons M.H., and Segall J.E. (2012) Microglial stimulation of glioblastoma invasion involves EGFR and CSF-1R signaling. Molecular Medicine. 18(1):519-27. doi: 10.2119/molmed.2011.00217
44. Cortez C., Eugenin E., Aliaga E., Carreno L.J., Bueno S.B., Gonzalez P., Gayol S., Naranjo D., Noches V., Marassi M.P., Rosenthal D., Jadue C., Ibarra C., Ibarra P., Keitel C., Wohllk N., Kalergis A.M. and Riede C.A. (2012) Hypothyrodism in adult rat causes an increment of BDNF in the brain, neuronal and astrocyte apoptosis, gliosis and deterioration of the postsynaptic density. Thyroid, 22(9):951-63.
45. D'Aversa T., Eugenin E., Lopez L., and Berman J.W. Myelin basic protein induces inflammatory mediators from primary human endothelial cells and blood brain barrier disruption: Implications for the pathogenesis of multiple sclerosis. Neurobiology of Disease. In Press, doi: 10.1111/j.1365-2990.2012.01279.x.
46. Hazleton J.E., Berman J.W. and Eugenin E.A. (2012) Purinergic Receptors are required for HIV-1 Infection of Primary Human Macrophages. J. Immunology. 188(9):4488-95.
47. Chung S, Yao J, Suyama K, Bajaj S, Qian X, Loudig OD, Eugenin EA, Phillips GR, Hazan RB. (2012) N-cadherin promotes breast cancer cell migration by suppressing Akt3 expression. Oncology. In press. doi: 10.1038/onc.2012.65.
48. Williams D., Eugenin E. A., Calderon T.M. and Berman J.W. (2012) Monocyte Maturation, HIV Susceptibility, and Transmigration Across the Blood Brain Barrier are Critical in HIV Neuropathogenesis. J. Leukocyte Biology. 91(3):401-15
49. Eugenin EA, Basilio D, Sáez JC, Orellana JA, Raine CS, Bukauskas F, Bennett MV, Berman JW. (2012) The Role of Gap Junction Channels During Physiologic and Pathologic Conditions of the Human Central Nervous System. J Neuroimmune Pharmacol. 7(3): 499-518.
50. Roberts TK, Eugenin EA, Lopez L, Romero IA, Weksler BB, Couraud PO, Berman JW. (2012) CCL2 disrupts the adherens junction: implications for neuroinflammation. Lab Invest. 2012 May 28. doi: 10.1038/labinvest.2012.80. [Epub ahead of print]
51. Qian X., Hulit J., Suyama K., Eugenin E.A., Belbin T.J., Loudig O., Smirnova T., Zhou Z.N., Segall J., Locker J., Phillips G.R., Norton L., Hazan R.B. (2012). P21CIP1 mediates reciprocal switching between proliferation and invasion during metastasis. Oncogene, 2012 Jul 2. doi: 10.1038/onc.2012.249.
52. Gaskill P.J., Carvallo L., Eugenin E.A., Berman J.W. (2013) Characterization and function of the human macrophage dopaminergic system: implications for CNS disease and drug abuse. J Neuroinflammation. 18;9(1):203.
53. Orellana JA., Williams D.W., Saez J.C., Berman J.W., and Eugenin E.A. (2013) Pannexin1 hemichannels are critical for HIV infection of human primary CD4+ T lymphocytes. J Leukocyte Biology. 94(3):399-407.
54. Eugenin E.A., Greco J.M., Frases S., Nosanchuk J.D., and Martinez L.R. (2013) Methamphetamine alters blood brain barrier protein expression facilitating central nervous system infection by neurotropic Cryptococcus Neoformans. Journal Infectious Disease, 208(4); 699-704.
55. Liu Tong-Bao., Kim J-C., Wang Y., Toffaletti D.L., Eugenin E., Perfect J., Kim K.L., Xue Chaoyang. Brain inositol is a novel stimulator for promoting Cryptococcus penetration of the blood-brain barrier. Plos pathogens. 9(4).
56. Williams D., Calderon T.M., Lopez- L, Carvallo L., Gaskill P., Eugenin E.A., Morgello S., and Joan Berman. (2013) Mechanisms of HIV Entry Across the Blood Brain Barrier by CD14+CD16+ Monocytes: Important Roles for JAM-A, ALCAM, and CCR2. Plos Pathogens. 8 (7).
57. Megra B., Eugenin E.A, Roberts T., Morgello S., and Berman J.W. (2013) Protease resistant protein cellular isoform (PrPc) as a biomarker: clues into the pathogenesis of HAND. Journal Neuroimmune Pharmacology. 8(5); 1159-66.
58. Mavrianos J., Berkow E., Desai C., Pandey A., Batish M., Rabadi M., Barker K., Pain D., Rogers P., Eugenin E.A., and Chauhan N. (2013) Mitochondrial two-component signaling systems in Candida albicans. Eukaryotic Cell. 12(6):913-22.
59. Rao V.R., Neogi U., Talboom J.S., Padilla L., Rahman M., Fritz-French C., Gonzalez-Ramirez S., Verma A., Wood C., Ruprecht R.M., Ranga U., Azim T., Joska J., Eugenin E., Shet A, Bimonte-Nelson H., Tyor W.R., and Prasad V.R. (2013)Clade C HIV-1 isolates circulating in Southern Africa exhibit a greater frequency of dicysteine motif-containing Tat variants than those in Southeast Asia and cause increased neurovirulence. Retrovirology. 10 (61).
60. Albornoz E.A., Carreño L.J., Cortés C., Gonzalez P.A., Cisternas P.A., Cautivo K.M., Catalan T.P., Opazo M.C., Eugenin E.A., Berman J.W., Bueno S.M., Kalergis A.M., Riedel C.A. (2013) Gestational Hypothyroidism Increases the Severity of Experimental Autoimmune Encephalomyelitis in Adult Offspring. Thyroid, 23(12) 1627-37.
61. Eugenin E.A. (2013) Community-acquired pneumonia infections by Acinetobacter baumannii: How does alcohol impact the antimicrobial functions of macrophages? Virulence, 4(6): 435-436.
62. Patel D, Desai GM, Frases S, Cordero RJ, DeLeon-Rodriguez CM, Eugenin EA, Nosanchuk JD, Martinez LR.(2013) Methamphetamine enhances Cryptococcus neoformans pulmonary infection and dissemination to the brain. mBio, 4(4).
63. Williams D.W., Calderon T.M., Lopez L., Carvallo-Torres L., Gaskill P.J., Eugenin E.A., Morgello S., Berman J.W. (2013) Mechanisms of HIV Entry into the CNS: Increased Sensitivity of HIV Infected CD14(+)CD16(+) Monocytes to CCL2 and Key Roles of CCR2, JAM-A, and ALCAM in Diapedesis. Plos One, 8(7).
64. Eugenin E.A. and Berman J.W. (2014) Cytochrome C dysregulation induced by HIV infection of astrocytes results in bystander apoptosis of uninfected astrocytes by an IP3 and Calcium dependent mechanism. J Neurochemistry, 127 (5): 644-51.
65. Orellana J.A., Saez J.C., Bennett M.V., Berman J.W., Morgello S. and Eugenin E.A. (2014) HIV increases the relase of dickkopf-1 protein from human astrocytes by a Cx43 hemichannel dependent mechanism. J Neurochemistry. 128 (5): 752-63.
66. Eugenin E.A. (2014) Role of connexins/pannexin containing channels in infectious diseases. FEBS letters. 588 (8) 1389-95.
67. Velasquez S. and Eugenin E.A. (2014) Role of pannexin-1 hemichannels and purinergic receptors in the pathogenesis of human diseases. Front Physiology. 14 (5). In press
68. Castellano P and Eugenin E.A. (2014) Regulation of gap junction channels by infectious agents and inflammation in the CNS. Fronteir in Cell NeuroScience. 8 (122). In press
69. Cheshenko N., Trepanier J.B., Gonzalez P.A., Eugenin E.A., Jacobs W.R. Jr., and Herold B.C. (2014) Herpes Simplex Virus Type 2 glycoprotein H interacts with integrin alphaVBeta3 to facilitate viral entry and calcium signaling in human genital tract epithelial cells. J. Virology. In press.
70. Rella C., Ruel N., and Eugenin E.A. (2014) Development of imaging techniques to study the pathogenesis of BSL2/3 infectious agents. Pathogens and Disease. In Press.
71. Nosanchuck J., Eugenin E.A., Martinez L. (2013) Methamphetamine alters the antimicrobial efficacy of phagocytic cells during methicillin-resistant Staphylococcus Aureus skin infection. Submitted to J infectious Disease.
72. Daep C.A. and Eugenin E.A. (2013) Flaviviruses, an expanding threat in public health: focus in Dengue, West Nile virus and Japanese encephalitic virus. In press JNV
INVATED BOOK CHAPTERS.
1.- Sáez C. G., Eugenín E. A., Hertzbertg E. L. and Sáez J. C. (1997) Regulation of gap junctions in rat liver during acute and chronic CCl4-induced liver injury. In: From ion channels to cell-to-cell conversations (Latorre R. and Sáez J. C., Eds) Plenum Press, New York.
2.- Eugenin E.A. and Berman J.W. (2005) Mechanism of viral entry through the Blood-Brain Barrier. Gendelmann, H. Editor, Oxford, London. The Neurology of AIDS (Second Edition) Edited by Gendelman H., Grant I., Everall I., Lipton S. and Swindells S. Oxford Publ. pp.147-154.
3.- Eugenin E.A. and Berman J.W. (2011) Mechanism of cellular and viral entry through the Blood-Brain Barrier. Gendelmann, H. Editor, Oxford, London. The Neurology of AIDS (Third Edition) Edited by Gendelman H., Everall I., Lipton S. and Swindells S. Oxford Publ.
Dr. Joan W. Berman, Professor, Pathology Department, F727, Albert Einstein College of Medicine, Bronx, New York, 10461, Phone: 718-4302587. email: joan.berman@einstein.yu.edu
Dr. Michael Bennett, Professor, Neuroscience Department. Albert Einstein College of Medicine, Bronx, NY, USA. Phone: 718-4302535. mbennett@aecom.yu.edu
Dr. Klaus Willecke, Professor, Genetik Institute, Bonn University, Germany. email: genetik@uni-bonn.de
Dr. Susan Morgello, Professor, Medicine Department, Mount Sinai Hospital, New York, NY, 10029, USA. Phone: 212-241-9118. Susan.morgello@mssm.edu
Dr. Juan Carlos Sáez, Visiting Associate Professor , Dept. of Neurosciences, Albert Einstein College of Medicine, New York, U.S.A. and Professor, Dept. Ciencias Fisiológicas, Pontificia Universidad Católica de Chile, Chile. Phone (562) 686-2862. FAX (562) 222-5515. jsaez@genes.bio.puc.cl
Dr. Michael Prystowsky, Professor and Chairman of Department of Pathology, Albert Einstein College of Medicine, Bronx, New York, 10461, Phone: 718-4302823. email: michael.prystowsky@einstein.yu.edu
Dr. Harris Goldstein, Microbiology and Immunology Dept. Albert Einstein College of Medicine, Bronx, NY, USA. hgoldste@aecom.yu.edu
Dr. Suzanne R. Zukin, Neuroscience Dept. Albert Einstein College of Medicine, Bronx, NY, USA. Zukin@aecom.yu.edu
Dr. Alberto Pereda, Neuroscience Dept. Albert Einstein College of Medicine, Bronx, NY, USA. pereda@aecom.yu.edu
Dr. Claudia Sáez, Ph. D. in Microbiology and Immunology in the Albert Einstein College of Medicine, New York, U.S.A, Associate Researcher at the Dept. of Oncology in the clinical. Phone: (562)-6863874. Email: cgsaez@med.puc.cl
Dr. Issar Smith, Ph.D. Public Health Research Institute (PHRI) and Rutgers University, Newark, NJ, 07103. 973-8543269. smithis@njms.rutgers.edu
Dr. David Perlin, Ph.D. Execute Director, Public Health Research Institute (PHRI), and Professor at Rutgers University, Newark, NJ, Phone; 973-854-3100, perlind@umdnj.edu
|
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Is Glenmorangie Signet collectable?
How many years is Glenmorangie Signet?
Is Glenmorangie Signet a blend?
How do you pronounce Glenmorangie whiskey?
Is Glenmorangie peated?
What does the name Glenmorangie mean?
Where is Glenmorangie made?
What is Signet whiskey made from?
The distillation of Glenmorangie Signet is a rare occasion that takes place for one week every year – Signet Week. For seven days each year, the distillery only produces the 'chocolate malt' and halts production on all other whiskies, a highly unusual occurrence in the whisky industry.
Unofficially the whisky hovers around the 15-year mark, but no age is ever put on the bottle because they're going for a flavor profile, not a number. There could be, and probably is, more to the process, but this info came a Signet tasting I did with the brand many years ago.
What is the cost of Glenmorangie?
Buy Glenmorangie The Original Single Malt Whisky Online at Best Price of Rs 6580 – bigbasket.
The Signet is a blend of Glenmorangie's oldest and rarest whiskies – distilled over thirty years ago and matured in a selection of the world's finest casks, undoubtedly the richest whisky in the Glenmorangie range.
Pronunciation. Glen-MOR-angie: the name of the whisky is /ɡlɛnˈmɒrəndʒi/ glen-MORR-ən-jee, with the stress on the "mor" and rhyming with orangey (not */ˌɡlɛnmɔˈrændʒi/ GLEN-mor-AN-jee as it is commonly mispronounced).
How do you drink single malt?
As a general rule, it's recommended that you taste your whisky neat first, and then add a little spring water with your second sip. You don't want to drown your drink; about 20% water is plenty. Combine the whisky and water by gently shaking the glass (giving it a "shoogle" in Scottish parlance).
Glenmorangie is a very lightly peated malt (averaging less than 2 ppm of phenols), while Ardbeg is the most heavily peated of all scotch whiskies, coming in currently at 50+ ppm (although not too many years ago this number was 70+).
vale of tranquillity
Glenmorangie (pronounced with the stress on the second syllable: listen (help·info); the toponym is believed to derive from either Gaelic Gleann Mòr na Sìth "vale of tranquillity" or Gleann Mór-innse "vale of big meadows") is a distillery in Tain, Ross-shire, Scotland, that produces single malt Scotch whisky.
What is Glenmorangie Signet?
A delicious no-age-statement dram from Glenmorangie, some of the spirit in Signet has been made using a unique heavily roasted chocolate malt that really beefs up the rich flavours. The packaging is impressive, too. Glenmorangie Signet.
Glenmorangie is a famous single malt distillery situated just north of the town of Tain in Ross-shire, in the Highlands region of Scotland. Glemorangie is one of the best-known Scottish distilleries, despite its confusing pronu…
Highland, Scotland- Matured in made-to-order casks and created from a combination of the oldest and rarest Glenmorangie whiskeys, Signet uses high roasted 'chocolate' malt barley and is unchill-filtered. The velvety taste follows a nose of chocolate torte and treacle plum pudding.
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Angelina Jolie has opened up about her "difficult" split from Brad Pitt for the first time, following news of their divorce coming to light in September 2016.
While Jolie filed for divorce from Pitt, following a 12-year relationship after meeting on the set of Mr. and Mrs. Smith in 2003, six months ago, she has not yet spoken out about their break up in public.
"I don't want to say very much about that, except to say it was a very difficult time and we are a family, and we will always be a family. It was very difficult. Many people find themselves in this situation. My whole family have all been through a difficult time.
"My focus is my children, our children. We are and forever will be a family and so that is how I am coping. I am coping with finding a way through to make sure that this somehow makes us stronger and closer.
"I am very saddened by this but what matters most now is the well-being of our kids," Pitt told the magazine. "I kindly ask the press to give them the space they deserve during this challenging time."
According to Mail Online, a "bitter custody battle" initially played out between the pair after Jolie sought full custody of their children following an alleged incident on board a private jet between Pitt and one of their children.
In early January, Brangelina reached an agreement to seal the custody documents in a bid to resolve their issues and reach a divorce settlement, Daily Mail reported.
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{"url":"http:\/\/mathcentral.uregina.ca\/QQ\/database\/QQ.09.15\/h\/sharon1.html","text":"SEARCH HOME\n Math Central Quandaries & Queries\n Question from Sharon, a student: How do you turn a fraction into a percentage? I'm beyond confused with this. Even though I think of myself as a good math student doing 8th grade math at a young age, this just makes me want to scream! I get the way you turn 1\/4 into 25%, but when it comes to something like 7\/9 = ? I just can't figure out the answer.\n\nHi Sharon,\n\nTo convert a common fraction into a percent you need to find an equivalent fraction with 100 in the denominator. That's what percent means, per one hundred. Thus for 7\/9 you need to find $x$ so that\n\n$\\frac79 = \\frac{x}{100}.$\n\nTo solve for $x$ multiply both sides by 100 to get\n\n$x = \\frac79 \\times 100 = 77.78.$\n\nThus $\\large \\frac79$ is 77.78%.\n\nI hope this helps,\nPenny\n\nMath Central is supported by the University of Regina and the Imperial Oil Foundation.","date":"2019-06-19 03:54:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7683698534965515, \"perplexity\": 729.3326154141444}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998882.88\/warc\/CC-MAIN-20190619023613-20190619045613-00277.warc.gz\"}"}
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CPC stands firmly opposed to the Federal Administration's recently proposed expansion of the "public charge" definition and condemns it as an explicit and systematic attack on low-income, immigrant families, and communities of color. CPC stands united in our opposition, starting tonight at 5:30 p.m. when we stand in solidarity on the Lower East Side to protest and condemn this destructive proposal.
In a move that has been rumored since February, the Federal Administration's proposed changes would significantly expand how "public charge" is determined. Likelihood to become a "public charge" is one of the many factors considered when determining application for permanent residency or visas to the United States.
The newly proposed categories would significantly expand to include nutrition assistance, housing, and healthcare benefits, while simultaneously establishing minimum education levels and household income thresholds. It would also subject individuals to hefty bonds that hold an application ransom to discourage the applicant from enrolling in any of the benefits listed.
The benefits included in the proposed rule are ones that keep families from falling into crisis -- they prevent health emergencies, prevent homelessness, and provide the nutrition needed for healthy, productive lives. Should these benefits be included in the final ruling, a family may find that once they finally achieve legal permanent residency, they face nutritional, health, housing, or economic ruin because they were stripped of the same benefits that supported generations of immigrant families before them.
"If the public charge rule is finalized, families would be forced to make an impossibly narrow choice between health, stability, and security for themselves and their loved ones or legal status in this country," said Carlyn Cowen, Chief Policy and Public Affairs Officer at the Chinese-American Planning Council.
Public charge tests have not always been part of US immigration history. The origin of "public charge" was a precursor to the Chinese Exclusion Act, a hateful and destructive policy whose impacts on generations of Chinese-Americans can still be felt today. This proposal is a reflection some of the most shameful and destructive immigration policies in our nation's history and CPC stands firmly in opposition.
Stand with us tonight. Please join and share the New York Immigration Coalition rally happening tonight. Look for CPC signs and banners or contact us to meet up.
Though the proposed plans are released, they have not taken affect. Nothing will change until a final version is released after the public comment period. Community members should continue to take part in the benefits that sustain and protect them. There will be no retroactive effect until the rule is final. Accurate information allows us to maintain trust and access to our community.
Unlike many of the Federal Administration's attacks on immigrants, this one cannot become law until the administration reviews the comments of the people. Your voice is part of the rulemaking procedure. Public comments can change, delay, and halt proposed rule changes. They also establish an administrative record for potential lawsuit or inquiry into the constitutionality of proposed laws.
Most importantly, public pressure works. The leaked drafts that the media was given earlier this year were significantly different than the ones formally released this weekend. It is more important than ever before to stand in loud opposition to this proposed rule.
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\section{Introduction}
Scalar curvature plays a fundamental role in General Relativity. If $(M,g,k)$ is a space-like slice inside a spacetime $(\mathbb{L},\gamma)$ satisfying the Einstein field equations then, as a result of the Gauss equations relating the extrinsic and intrinsic geometry of such slice, the first and second fundamental forms of $M$ $(\textrm{respectively} \ g \ \textrm{and} \ k)$ satisfy the system
\begin{equation*}
\begin{cases}
\frac{1}{2}\left(R_{g}+\left(Tr_{g}k\right)^{2}-\left\|k\right\|^{2}_{g}\right)=\mu \\
Div_{g}\left(k-\left(Tr_{g}k\right)g\right)=J,
\end{cases}
\end{equation*}
for $\mu$ (the \textsl{mass density}) and $J$ (the \textsl{current density}) suitable components of the stress-energy tensor $T$, which describes the matter fields.
In the most basic of all cases, namely when $k=0$ (in which case we will say that the data $(M,g,k)$ are \textsl{time-symmetric}) the previous system, known as Einstein constraint equations, reduces to the single equation
\[ R_{g}=2\mu
\]
namely to a scalar curvature prescription problem.
As a result of a general physical axiom, the \textsl{dominant energy condition}, which postulates the energy density measured by any physical observer to be non-negative one requires the functional inequality $\mu\geq \left|J\right|_{g}$ to be satisfied and hence in the time-symmetric case considered above one obtains the restriction
\[ R_{g}\geq 0
\]
so that asymptotically flat spaces are always studied under the assumption that their scalar curvature be non-negative. Moreover, in the vacuum case (namely when no sources are present, $T=0$) the Einstein constraint equations reduce to the requirement that the scalar curvature vanishes at all points of the manifold in question.
These local conditions have dramatic global consequences. The most remarkable of all of them is the \textsl{Positive Mass Theorem} (see \cite{SY79}), which essentially states that for asymptotically flat manifolds the ADM mass (a scalar invariant, introduced in the context of the Hamiltonian formulation of General Relativity and defined by a certain flux integral at spatial infinity) is indeed non-negative, and equals zero if and only if our manifold $(M,g)$ is globally isometric to the Euclidean space. For the purposes of this article, we should state the following important consequence\footnote{This relies on the main theorem in \cite{Wit81}, due to the fact that $\mathbb{R}^n$ is a spin manifold for any $n\geq3$.}.
\begin{cor}
Let $n\geq 3$ and $g$ be a Riemannian metric on $\mathbb{R}^{n}$ having non-negative scalar curvature. Suppose that there exists a compact set $K$ such that $g$ is exactly Euclidean outside of $K$. Then $g$ is the Euclidean metric on the whole $\mathbb{R}^{n}$.
\end{cor}
With a little bit more effort, one can get a sharper form of the previous statement (see Proposition \ref{pro:cap}) which implies that in fact an asymptotically flat metric cannot be localized inside sets that are asymptotically too small, for instance a cylinder or a slab between two parallel planes. Thus one is naturally led to the following basic question.
\begin{problem}
What is the optimal localization of an asymptotically flat metric of non-negative scalar curvature?
\end{problem}
For instance, can we construct an asymptotically flat metric of non-negative scalar curvature, positive ADM mass and which is exactly trivial in a half-space?
In this article, we give an essentially complete (and highly surprising) answer to such question, by developing a systematic method to localize a given scalar flat, asymptotically flat metric inside a cone of arbitrarily small aperture. In fact, we perform this construction for the general system of Einstein constraint equations, thus fully dealing with the coupled nonlinearities of the problem. We refer the readers to the next section for a precise statement of our gluing theorem (Theorem \ref{thm:main}), which requires some notation to be introduced. Here we will limit ourselves to a couple of important remarks.
First: as an immediate consequence of our gluing scheme we are able to produce data that are flat on a half-space and therefore contain plenty of stable (in fact: locally area-minimizing) minimal hypersurfaces, a conclusion which comes quite unexpected based on various recent scalar curvature rigidity results both in the closed and in the free-boundary case (see the works \cite{BBN10, Nun13, MM15, Amb15}). As a result, combining this fact with with the rigidity counterparts obtained by the first-named author, contained in \cite{Car13} and \cite{Car14}, we are able to provide a rather exhaustive answer to the fundamental problem of existence of stable minimal hypersurfaces in asymptotically flat manifolds (more generally: marginally outer trapped hypersurfaces in initial data sets). Furthermore, the reader shall notice that, again in the time-symmetric case, our solutions contain outlying volume-preserving stable constant mean curvature spheres that enclose arbitrarily large volumes. A well-known formula \cite{FST09} due to X.-Q. Fan, P. Miao, Y. Shi and L.-F. Tam equating the ADM mass to the isoperimetric mass computed in terms of deficit of large coordinate balls (together with an important remark of G. Huisken) ensures that none of those CMC surfaces is in fact isoperimetric, at least for large enough enclosed volumes.
Second: one can essentially iterate our construction and get a new class of $N$-body solutions to the Einstein constraint equation which exhibit, following a definition by P. Chru\'sciel, the phenomenon of \textsl{gravitational shielding} in the sense that one can prepare data that do not have any interaction for finite but arbitrarily long times, in striking contrast with the Newtonian gravity scenario. This is the content of Theorem \ref{thm:Nbody}. Concerning the evolution of these solutions, we point out that our data contain an arbitrarily large piece of the original data, so if there is a trapped region in the original, it will still exist in the localized solution. In such a case the evolution will be incomplete by the Penrose singularity theorem.
\
Apart from this introduction, the present article consists of five sections: in Section \ref{sec:gluthm} we introduce some notation, state our gluing theorem and discuss the regularity of the data we produce, in Section \ref{sec:prelim} we give an outline of the proof and present the variational structure of the linearized constraints in suitable doubly weighted Sobolev spaces. The most technical parts of the proof are contained respectively in Section \ref{sec:lin} for what concerns the linear theory (where we prove the coercivity of the functional $\mathcal{G}$ by means of some basic estimates of independent interest) and in Section \ref{sec:nonlin} for what concerns the Picard scheme by which we solve the Einstein equations. Lastly, the construction of $N$-body solutions is presented in Section \ref{sec:Nbody}.
\
\
\textsl{Acknowledgments}. The authors would like to express their sincere gratitude to the anonymous referess for carefully proofreading the article and for suggesting a number of changes which significantly contributed to the improvement of this final version. We are also indebted to Christos Mantoulidis for the figures that appear in the article. During the preparation of this work, the authors were partially supported by NSF grant DMS-1105323.
\section{The gluing theorem}\label{sec:gluthm}
\subsection{Initial data}\label{subs:ids}
Let $(M,\check{g},\check{k})$ be an asymptotically flat initial data set for the Einstein equation of type $(n,l+1,\alpha,\check{p})$, so that:
\begin{itemize}
\item{$(M^{n}, \check{g})$ is a $\mathcal{C}^{l+1,\alpha}$ complete asymptotically flat manifold with $\check{g}_{ij}(x)=\delta_{ij}+O^{l,\alpha}(|x|^{-\check{p}})$}
\item{$\check{k}$ is a symmetric $(0,2)-$tensor with $\check{k}_{ij}(x)=O^{l-1,\alpha}(|x|^{-\check{p}-1})$}
\item{the \textsl{vacuum} Einstein constraint equations are satisfied, namely
\begin{equation}
\begin{cases}
R_{\check{g}}+(Tr_{\check{g}}\check{k})^{2}-\left\|\check{k}\right\|^{2}_{\check{g}}=0 \\
Div_{\check{g}}(\check{k}-(Tr_{\check{g}}\check{k})\check{g})=0.
\end{cases}
\end{equation}
}
\end{itemize}
Here $\frac{n-2}{2}<\check{p}\leq n-2$ and we are adopting the standard definitions of weighted H\"older spaces given, for instance, in \cite{EHLS11}.
As announced in the introduction, this article is devoted to performing an \textsl{optimal localization} of such data by gluing them to the trivial triple $(\mathbb{R}^{n},\delta, 0)$ outside of a cone of given aperture.
\subsection{The content at infinity of a Riemannian metric}
In order to describe this problem of \textsl{optimal localization} in some detail, we need to start by giving the following relevant definition.
\begin{definition}
Let $(M,g,k)$ be an asymptotically flat initial data set with one end of type $(n,l,\alpha,p)$ for $n, l\geq 3$, $p>(n-2)/2$ and $\alpha\in(0,1)$. For $g$ such an asymptotically flat Riemannian metric, we set
\begin{displaymath}
U=\left\{p\in M \ | \ Ric_{g}(p)\neq 0\right\}
\end{displaymath}
and define \textsl{content at infinity} for the metric in question to be the \textsl{asymptotic size of the set} $U$, namely
\begin{displaymath}
\Theta(g)=\liminf_{\sigma\to\infty} \sigma^{1-n}\mathscr{H}^{n-1}\left(U\cap \mathbb{S}^{n-1}\left(\sigma\right)\right).
\end{displaymath}
Here $\mathbb{S}^{n-1}(\sigma)=\left\{|x|=\sigma\right\}$ for asymptotically flat coordinates $\left\{x\right\}$.
\end{definition}
The \textsl{Positive Mass Theorem} gives, among its various implications, strong restrictions on the content at infinity of an asymptotically flat metric of non-negative scalar curvature.
\begin{proposition}\label{pro:cap}
Let $(M,g,k)$ be a time-symmetric, asymptotically flat initial data set of type $(n,l,\alpha,n-2)$ for $n,l\geq 3$ and $\alpha\in(0,1)$. Suppose that $n<8$ or $(M,g)$ is a spin manifold. If $R_{g}\geq 0$, then \textsl{either} $g$ is flat \textsl{or} $\Theta(g)>0$.
\end{proposition}
\begin{proof}
It is well-known that the ADM energy (we will use the standard definition, see for instance \cite{Bar86} or \cite{EHLS11}) of an asymptotically flat initial data set $(M,g,k)$ can be equivalently expressed in terms of the Ricci curvature of $g$: namely there exists a positive dimensional constant $\varpi_{n}$ (whose specific value depends on the normalization used in the definition of $\mathcal{E}$) such that
\[\mathcal{E}=-\varpi_{n}\lim_{\sigma\to\infty}\sigma\int_{|x|=\sigma}Ric_g(\nu,\nu)\,d\mathscr{H}^{n-1}.
\]
As a result, for $\sigma$ large enough we can write the trivial estimate:
\[ \frac{\mathcal{E}}{2}\leq\varpi_{n}\left|\sigma\int_{|x|=\sigma}Ric_g(\nu,\nu)\,d\mathscr{H}^{n-1}\right|\leq C\sigma^{1-n}\mathscr{H}^{n-1}\left(U\cap \mathbb{S}^{n-1}\left(\sigma\right)\right).
\]
If $g$ were not flat, then by the \textsl{Positive Mass Theorem} (see \cite{SY79, Wit81}) it would have a positive ADM energy, that is $\mathcal{E}>0$. But then, by means of the previous inequality we would have
\[ \Theta(g)\geq \frac{\mathcal{E}}{2C}>0
\]
which completes the proof.
\end{proof}
Roughly speaking, this proposition states that if an asymptotically flat metric $g$ of non-negative scalar curvature is not trivial then the region where it is not (Ricci) flat must contain a cone of positive aperture. Incidentally, we recall here that for asymptotically flat metrics Ricci flatness is in fact equivalent to flatness, as one can prove either by means of the Bishop-Gromov comparison theorem or by using harmonic coordinates, as was done by Schoen-Yau in the proof of the rigidity statement of the \textsl{Positive Mass Theorem}.
Our gluing scheme provides a sort of converse to the previous statement, by asserting that for any cone we can construct non-trivial data localized inside that cone, and nowhere else. In order to state our theorem, we need to describe our setting with more precision (which we will do in the next subsection).
\subsection{Regularized cones}\label{subs:regcon}
Given an angle $0<\theta<\pi$ and a point $a\in \mathbb R^n$ with $|a|>>1$
we denote by $C_\theta(a)$ the region of $M$ consisting of the compact
part together with the set of points $p$ in the exterior region such that $p-a$ makes an
angle less than $\theta$ with the vector $-a$. Here we are tacitly assuming that the manifold $M$ has only end, which we can do with no loss of generality as our construction is patently local to a given end: such assumption will always be implicit in the sequel of this article.
If we are given two angles $0<\theta_1<\theta_2<\pi$ we consider the region
between the cones. We want to regularize this domain near the vertex $a$,
so we consider the region
\[ \Omega_1=B_{1/2}(a)\cup (C_{\theta_2}(a)\setminus \overline{C_{\theta_1}(a)}).
\]
We see that the boundary of this region is given by $\partial\Omega_1=S_1\cup S_2$
where $S_1$ and $S_2$ are hypersurfaces with corners on $\partial B_{1/2}(a)$
\[ S_1=(\partial C_{\theta_1}(a)\setminus B_{1/2}(a))\cup (\partial B_{1/2}(a)\cap
C_{\theta_1}(a))
\]
and
\[ S_2=(\partial C_{\theta_2}(a)\setminus B_{1/2}(a))\cup (\partial B_{1/2}(a)\setminus
C_{\theta_2}(a)).
\]
We approximate $\Omega_1$ by a smooth domain $\Omega$ whose boundary consists
of a pair of smooth disjoint hypersursurfaces $\Sigma_1$ and $\Sigma_2$ such that
for $i=1,2$
\[ \Sigma_i\setminus B_1(a)=C_{\theta_i}(a)\setminus B_1(a).
\]
It follows that $M\setminus (\Sigma_1\cup\Sigma_2)$ is a disjoint union of three regions
$\Omega_I$, $\Omega$, and $\Omega_O$ where we refer to $\Omega_I$ as the inner
region, $\Omega$ the transition region, and $\Omega_O$ the outer region.
\begin{figure}[ht!]
\begin{tikzpicture}
\begin{scope}[scale=0.75, shift={(0, 8)}]
\node at (5.4, 1) (MagSmall) {};
\fill [pattern color=blue!80, pattern=north west lines, draw=blue, line width=0.5pt, rounded corners=2mm] plot coordinates {(-0.5, 0.5) (5.6, 0.8) (3.4, -1)};
\fill [pattern color=blue!80, pattern=north west lines, draw=none, rounded corners=2mm] plot coordinates {(3.4, -1) (-2, -1) (-0.5, 0.5)};
\fill [black!10, draw=blue, line width=0.5pt, rounded corners=4mm] plot coordinates {(-0.7, 0.29) (5.6, 0.8) (3.2, -1.01)};
\fill [black!10, draw=black, rounded corners=2mm] plot coordinates {(3.2, -1) (-2, -1) (-0.7, 0.3)};
\draw [black, rounded corners=2mm] plot coordinates {(-0.5, 0.5) (0.5, 1.5) (7, 1.5) (4.5, -1) (3.4, -1)};
\fill [black!10, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.3, 1.5) (1.1, 2.5) (2, 2.5) (2.9, 2.5) (2.7, 1.5) (3, 0.5) (4, 0)};
\draw [black!60, line width=1pt, dashed] (5.4, 0.8) circle (0.5);
\node at (5.5, 2.2) {$(M, \check{g}, \check{k})$};
\end{scope}
\begin{scope}[scale=0.5, shift={(4, 1)}]
\clip (0, 0) circle (8);
\node at (4, 4) (MagLarge) {};
\begin{scope}
\filldraw [pattern color=blue!80, pattern=north east lines, draw=blue, line width=1pt] (-8, 4.1) -- (-1.3, 0.66) arc [radius=0.75, start angle=-110, delta angle=50] -- (-0.69, 0.71) arc [radius=0.98, start angle=132, delta angle=-264] -- (-0.69, -0.73) arc [radius=0.75, start angle=60, delta angle=60] -- (-8, -4.1);
\filldraw [fill=black!10, draw=blue, line width=1pt] (-8.02, 2.105) -- (-1.5, 0.4) arc [radius=0.5, start angle=53, delta angle=-106] -- (-1.5, -0.4) -- (-8.02, -2.105);
\end{scope}
\begin{scope}
\draw [black, dashed] plot coordinates {(-8, 4.1) (0, 0) (-8, -4.1)};
\draw [black, dashed] plot coordinates {(-8, 2.1) (0, 0) (-8, -2.1)};
\draw [black, dashed] (-0.03, -0.02) circle (0.99);
\end{scope}
\begin{scope}
\draw [blue, ->] plot coordinates {(-8, 0) (-0.1, 0)};
\draw [blue, dashed] plot coordinates {(0, 0) (2, 0)};
\node at (0, 0) {$\bullet$};
\node at (0.35, 0.35) {$a$};
\end{scope}
\begin{scope}
\draw [black!80] (-4, 2.05) arc [radius=4.8, start angle=155, delta angle=25];
\node at (-5, 0.8) {$\theta_2$};
\draw [black!80] (-2.5, 0.656) arc [radius=2.59, start angle=165, delta angle=15];
\node at (-3.2, 0.45) {$\theta_1$};
\end{scope}
\begin{scope}
\node at (-6, -0.7) {$\Omega_I$};
\node at (-6.5, -2.5) {$\Omega$};
\node at (-4.5, -3.5) {$\Omega_O$};
\end{scope}
\draw [black!60, line width=1pt, dashed] (0, 0) circle (7.99);
\end{scope}
\path[every node/.style={font=\sffamily\small}, ->, black!60, line width=1pt] (MagSmall) edge [out=0, in=0] (MagLarge);
\end{tikzpicture}
\caption{Regularized cones and the gluing region $\Omega$.}
\label{dgm:sector.figure}
\end{figure}
\subsection{Statement of the gluing theorem}
\begin{theorem}\label{thm:main}
Assume that we are given a set of initial data $(M,\check{g},\check{k})$ as above together with angles $\theta_{1}, \theta_{2}$ satisfying $0<\theta_{1}<\theta_{2}<\pi$. Furthermore, suppose $\frac{n-2}{2}<p<\check{p}$. Then there exists $a_{\infty}$ so that for any $a\in\mathbb{R}^{n}$ such that $\left|a\right|\geq a_{\infty}$ we can find a metric $\hat{g}$ and a symmetric $(0,2)$-tensor $\hat{k}$ so that $(M,\hat{g},\hat{k})$ satisfies the vacuum Einstein constraint equations, $\hat{g}_{ij}=\delta_{ij}+O^{l-2,\alpha}(|x|^{-p})$ \ $\hat{k}_{ij}=O^{l-2,\alpha}(|x|^{-p-1})$ and
\[
(\hat{g}, \hat{k})=\begin{cases}
(\check{g}, \check{k}) \ \ \textrm{in} \ \ \Omega_{I}(a) \\
(\delta, 0) \ \ \textrm{in} \ \ \Omega_{O}(a).
\end{cases}
\]
\end{theorem}
Let us now discuss the regularity of the data we produce.
\begin{rem}
From the viewpoint of the regularity of our data $(M,\hat{g}, \hat{k})$ we basically have two versions of Theorem \ref{thm:main}, which will be for brevity referred to as \textsl{finite regularity version} and \textsl{infinite regularity version}.
In the finite regularity version, we start with an (asymptotically flat) initial data set of type $(n,l+1,\alpha,\check{p})$ and get an initial data set of type $(n,l-1,\alpha,p)$. In other terms:
\begin{equation*}
\begin{cases}
\check{g}\in\mathcal{C}^{l,\alpha}_{loc} \\
\check{k}\in\mathcal{C}^{l-1,\alpha}_{loc}
\end{cases}
\ \Longrightarrow \
\begin{cases}
\hat{g}\in\mathcal{C}^{l-2,\alpha}_{loc}\\
\hat{k}\in\mathcal{C}^{l-2,\alpha}_{loc}.
\end{cases}
\end{equation*}
Therefore we face the well-known phenomenon of derivative loss that has been described both in \cite{Cor00} and \cite{CS06}. With more work it is possible to improve the theorem to remove this derivative loss and we will deal with this issue in a forthcoming paper. In the previous statement we shall assume that $l\geq 4$ so that in the most basic case we have
$(\check{g},\check{k})\in\mathcal{C}^{4,\alpha}_{loc}\times\mathcal{C}^{3,\alpha}_{loc}$ and $(\hat{g},\hat{k})\in\mathcal{C}^{2,\alpha}_{loc}\times\mathcal{C}^{2,\alpha}_{loc}$.
In the infinite regularity version, we start with an (asymptotically flat) initial data set where $M,\check{g},\check{k}$ are smooth and produces $(M,\hat{g},\hat{k})$ that are smooth as well. More precisely:
\begin{equation*}
\begin{cases}
\check{g}\in\mathcal{C}^{\infty}_{loc} \\
\check{k}\in\mathcal{C}^{\infty}_{loc}
\end{cases}
\ \Longrightarrow \
\begin{cases}
\hat{g}\in\mathcal{C}^{\infty}_{loc}\\
\hat{k}\in\mathcal{C}^{\infty}_{loc}.
\end{cases}
\end{equation*}
The proof of Theorem \ref{thm:main} is rather lenghty and technical, so we will only discuss it in the most basic case of finite regularity and for $l=4$. The modifications needed to obtain the general finite regularity version are straightforward and just of notational character. On the other hand, some changes are necessary for the infinity regularity version and, first of all, one needs to replace the angular weight of polynomial type with an angular weight of exponential type:
\[ \rho\simeq \phi^{2N} \ \ \leadsto \ \ \rho\simeq e^{-1/\phi}.
\]
These aspects have been already dealt with in \cite{Cor00} and \cite{CS06} so we will not repeat them here.
\end{rem}
\begin{rem}
As the readers may have noticed, our whole discussion refers to the \textsl{vacuum} constraint equations. However, the method that we develop would work equally well in the case when the Einstein constraint equations have non-zero data on the right-hand side. The issue is that in the latter case, one should first of all extend the data $\check{\mu},\check{J}$ to data $\mu, J$ that equal $\check{\mu}, \check{J}$ in $\Omega_{I}$ and vanish in $\Omega_{O}$ and it is not entirely obvious that such an extension can always be made in a way that the dominant energy condition is satisfied (with respect to the metric $\hat{g}$).
\end{rem}
\section{Some preliminaries}\label{sec:prelim}
Here and in the sequel we always assume $n\geq 3$ and we let $r\in \mathcal{C}^{\infty}(\mathbb{R}^{n})$ be any positive function which equals the usual Euclidean distance $|\cdot|$ outside of the unit ball (for a given set of coordinates).
\subsection{Doubly weighted functional spaces}\label{subs:doubly}
Let $a\in\mathbb R^n$ with $|a|>>1$, and introduce coordinates $\left\{x\right\}$ centered at $a$. For angles
$0<\theta_1<\theta_2<\pi$ we consider the region $\Omega$ constructed in
the previous section (notice that the cones have vertex at $x=0$).
We let $g$ be a Riemannian metric gotten by making a \textsl{rough patch} of $\check{g}$ and $\delta$ in the region $\Omega$: therefore $g$ agrees with $\check{g}$ in the connected component of the complement of $\Omega$ which contains the compact core of the manifold $M$ (which we called $\Omega_{I}$), and agrees with $\delta$ in the other connected component (which we called $\Omega_{O}$). We perform the same construction for $k$, which interpolates between $\check{k}$ and the trivial symmetric tensor. Of course, here we are making use of an \textsl{angular} cutoff function $\chi$ (namely a function only depending on the angle between a given point and the vector $-a$) with rapid decay at $\Sigma_{2}$ and such that $1-\chi$ rapidly decays at $\Sigma_{1}$. Finally, we observe that such $(g,k)$ satisfy the very same decay properties as $(\check{g},\check{k})$ even though they will not in general be a solution for the Einstein constraint equations. The main purpose of this work is to present a \textsl{deformation scheme} which allows to obtain such a goal. \newline
For $q>0$ we introduce the weighted $\mathcal{L}^2$ Sobolev space
$\mathcal{H}_{k,-q}(\Omega)$ of functions (or tensors) defined as the completion of the smooth
functions with bounded support (no condition on $\partial\Omega$) with
respect to the norm $\|\cdot\|_{\mathcal{H}_{k,-q}}$ given by
\[ \|f\|^2_{\mathcal{H}_{k,-q}}=\sum_{i=0}^k\sum_{|\beta|=i}\int_\Omega|\partial^\beta f|^2r(x)^{-n+2(i+q)}
\ d\mathscr{L}^{n}(x)
\]
where we recall that $r(x)$ is a smooth positive function with $r(x)=|x|$ outside of the unit ball and $\mathscr{L}^{n}$ is the Lebesgue measure on $\mathbb{R}^n$. The reader shall notice that due to the decay properties of the Riemannian metrics we consider in this article, such Sobolev spaces could have been equivalently defined by means of covariant derivatives with respect to $g$ (denoted, in the sequel, by $D_g$) and by the $n$-dimensional Hausdorff measure associated to $g$ (denoted, in the sequel, by $\upsilon$). The space $\mathcal{H}_{k,-q}(\Omega)$
consists of those $f$ which roughly decay like $|x|^{-q}$ along the cone and for which a derivative of order
$i\leq k$ decays like $|x|^{-q-i}$. Whenever no ambiguity is likely to arise, we will simply adopt the symbol $\left\|\cdot\right\|_{k,-q}$ rather than $\|\cdot\|_{\mathcal{H}_{k,-q}}$, the latter being kept only for situations when different classes of functional spaces are used at the same time.
For $0<q<n$ we use the $\mathcal{L}^2$ pairing to identify the dual space $\mathcal{H}^*_{0,-q}$ with
$\mathcal{H}_{0,q-n}$ since
\[ |\int_\Omega f_1f_2\ d\mathscr{L}^n|=|\int_\Omega (f_1r^{(-n/2+q)})(f_2r^{(n/2-q)})\ d\mathscr{L}^n|\leq \|f_1\|_{0,-q}\|f_2\|_{0,q-n}.
\]
Throughout this article, we need to work in \textsl{doubly} weighted Sobolev spaces, namely we also need to introduce an angular weight which is a certain (large) power of the angular distance of a point from $\partial\Omega$. This is necessary in order to prove that the gluing we perform is smooth (more generally: regular enough) up to the boundary of the gluing domain.
More precisely we take $\rho=\phi^{2N}$ where $N$ will be chosen to be a large integer and
$\phi$ is a positive weight function which is equal to $\theta-\theta_1$ near
$\Sigma_1\setminus B_1(0)$ and equal to $\theta_2-\theta$ near $\Sigma_2\setminus B_1(0)$.
Near $\Omega\cap B_1(0)$ we assume that $\phi$ vanishes with nonzero gradient along
$\partial \Omega$. We let $\phi_0$ denote the maximum value of $\phi$ and we assume that
each set $\Omega_t=\{\phi\geq t\}$ for $0\leq t\leq \phi_0$ is (the closure of) a smooth domain which is a cone outside $B_1(0)$. We define the doubly weighted space $\mathcal{H}_{k,-q,\rho}(\Omega)$ using the norm
\[ \|f\|^2_{\mathcal{H}_{k,-q,\rho}}=\sum_{i=0}^k\sum_{|\beta|=i}\int_\Omega|\partial^\beta f|^2r(x)^{-n+2(i+q)}
\rho(x)\ d\mathscr{L}^n(x).
\]
As above, we will often use the simpler notation $\left\|\cdot\right\|_{k,-q,\rho}$ in lieu of $\|\cdot\|_{\mathcal{H}_{k,-q,\rho}}$.
Given $a\in \mathbb{R}^{n}$ we let $s(x)=\left|x+a\right|$ and we remark that for any assigned angle $\theta_1$ (and any $\theta_2\in(\theta_1,\pi)$) there exists $a_{\ast}=a_{\ast}(\theta_1)$ such that whenever $|a|>a_{\ast}$
\begin{equation}\label{eq:min}
\min_{x\in\Omega}s(x)\geq
\begin{cases}
|a|\sin\theta_1 & \ \textrm{if} \ \ \theta_1\in(0,\frac{\pi}{2}) \\
\frac{|a|}{2} & \ \textrm{if} \ \ \theta_1\in\left[\frac{\pi}{2},\pi\right)
\end{cases}
\end{equation}
and there exists a constant $C>0$ (not depending on any of the parameters $n, a$ but only on $\theta_{1}, \theta_{2}$ and on the way the function $r$ is defined, namely on the value of its positive lower bound) such that
\begin{equation}\label{eq:equiv}
C^{-1}\leq \frac{s(x)}{r(x)}\leq C\left|a\right|, \ \textrm{for all} \ x\in\Omega.
\end{equation}
In the sequel of this article, we will always tacitly assume to work with large enough $|a|$ (that is to say $|a|>a_{\ast}$) namely in a regime where thiose basic estimates are valid.
In this article, we let $C$ denote any constant depending only on $g,k,n,p,N,\theta_{1},\theta_{2}$ (or a subset thereof) while we indicate by $C^{s}_{(-q)}$ (resp. $C^{r}_{(-q)}$) any function or tensor which is bounded (in $\Omega$) by a constant as above times $s^{-q}(x)$ (resp. $r^{-q}(x)$). In Subsection \ref{subs:coercLie} it will be important to distinguish between those constants that depend on $N$ and those that do not and thus we will specify the functional dependence of the constants in all of the statements.
We will need pointwise bounds on solutions to solve the nonlinear problem. For a point
$x\in\Omega$ we let $d(x)$ denote the distance from $x$ to $\partial\Omega$ and we
observe the bounds
\begin{equation}\label{eq:equiv2} C^{-1}r(x)\phi(x)\leq d(x)\leq Cr(x)\phi(x)
\end{equation}
for some positive constant $C$.
We will use interior Schauder estimates to obtain the pointwise bounds. To do this we define
global weighted H\"older norms. For real numbers $k,l,m$
with $k\geq 0$ an integer, we define $\|\cdot\|_{k,\alpha}^{(l,m)}$ by
\[ \|f\|_{k,\alpha}^{(l,m)}=\sum_{i=0}^k\sum_{|\beta|=i}\sup_{x\in\Omega}r(x)^{-l} \phi(x)^{-m}d(x)^i|\partial^\beta f(x)|
+\sum_{|\beta|=k}\sup_{x\in\Omega}r(x)^{-l} \phi(x)^{-m} d(x)^{k+\alpha}[\partial^\beta f]_{\alpha,B_{d(x)/2}(x)}
\]
where $[\cdot]_{\alpha,U}$ denotes the H\"older coefficient on $U$. The completion of the space of $\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$ functions with respect to this norm will be denoted by $\mathcal{C}^{k,\alpha}_{l,m}(\Omega)$. Moreover, we consider the average value $\bar{f}(x)$ of $|f(x)|$ taken over the ball of radius $d(x)/2$.
\subsection{Einstein constraint operators}
Given data $(M,g,k)$, it is here convenient to recall the definition of the momentum tensor
\[\pi^{ij}=k^{ij}-Tr_{g}\left(k\right)g^{ij}
\]
so that the vacuum Einstein constraint equations take the compact form
\[ \Phi(g,\pi)=0,
\]
where
\[\Phi(g,\pi)=(\mathscr{H}(g,\pi), Div_{g}\pi), \ \mathscr{H}(g,\pi)=R_g+\frac{1}{n-1}\left(Tr_{g}\pi\right)^{2}-\left|\pi\right|^{2}_{g}.
\]
In order to avoid ambiguities, we remark that the second component of such system is a vector field. It is easily checked that for any $p\in \left(0, n-2\right)$ the linear map
\[d\Phi_{(g,\pi)}: \mathcal{M}_{2, -p}\times \mathcal{S}_{1,-p-1}\to \mathcal{H}_{0, -p-2}\times \mathcal{X}_{0,-p-2}
\]
is continuous. Here and below we are using the symbols $\mathcal{M},\mathcal{S}, \mathcal{X}$ to stress that we are referring to a certain class of tensors, and specifically:
\begin{itemize}
\item{$\mathcal{M}_{2,-p}$ denotes the space of symmetric $(0,2)$ tensors in $\mathcal{H}_{2,-p}(\Omega)$;}
\item{$\mathcal{S}_{1,-p-1}$ denotes the space of symmetric $(2,0)$ tensors $\mathcal{H}_{1,-p-1}(\Omega)$;}
\item{$\mathcal{X}_{0,-p-2}$ denotes the space of vector fields in $\mathcal{H}_{0,-p-2}(\Omega)$}
\end{itemize}
with obvious extensions for different decay exponents or weights at the boundary of the gluing interface.
Correspondingly, for the adjoint map, which is defined by means of the equation
\[\int_{\Omega}\left[d\Phi_{(g,\pi)}[h,\omega]\cdot_{g}(u,Z)\right]\,d\upsilon=\int_{\Omega}\left[(h,\omega)\cdot_{g}d\Phi^{\ast}_{(g,\pi)}[u,Z]\right]\,d\upsilon
\]
one has that $d\Phi^{\ast}\in \mathcal{C}^{0}\left(\mathcal{H}_{2, -n+p+2}\times \mathcal{X}_{1,-n+p+2}\to \mathcal{M}_{0, -n+p}\times \mathcal{S}_{0,-n+p+1}\right)$ by virtue of the $\mathcal{L}^{2}$-duality mentioned above.
Rather similar mapping properties are true in doubly weighted functional spaces, and in particular we shall make use of the fact that
\[
d\Phi^{\ast}\in \mathcal{C}^{0}\left(\mathcal{H}_{2, -n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}\to \mathcal{M}_{0, -n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}\right).
\] Incidentally, such a statement can be effectively deduced by means of Lemma \ref{lem:coarea}, which allows to transform functional inequalities in singly weighted functional spaces into corresponding inequalities in doubly weighted ones.
The expressions of $d\Phi_{(g,\pi)}[h,\omega]$ and $d\Phi^{\ast}_{(g,\pi)}[u,Z]$ have been computed (for instance) in \cite{FM73} and \cite{FM75} and for the purpose of this work we will recall them in symbolic form
\begin{equation*}
\begin{cases} d\Phi^{(1)}_{(g,\pi)}[h,\omega]&=-\Delta_g(Tr_g(h))+Div_g(Div_g(h))-g(h,Ric_g)+\pi\ast\pi\ast h+\pi\ast\omega\\
d\Phi^{(2)}_{(g,\pi)}[h,\omega]&=Div_g(\omega)+\pi\ast D_gh
\end{cases}
\end{equation*}
and
\begin{equation*}
\begin{cases} {d\Phi^{\ast}}^{(1)}_{(g,\pi)}[u,Z]&=-\left(\Delta_g u\right)g+Hess_g(u)-uRic_g+\pi\ast\pi\ast u+D_g\left(\pi\ast Z\right)\\
{d\Phi^{\ast}}^{(2)}_{(g,\pi)}[u,Z]&=-\frac{1}{2}\mathscr{L}_{Z}g+\pi\ast u
\end{cases}
\end{equation*}
where $D_{g}$ denotes the Levi-Civita connection associated to $g$, all differential operators are considered with respect to the background metric $g$ and $\mathscr{L}$ stands for the Lie derivative. Let us remark that $L_g^{\ast}u=-\left(\Delta_g u\right)g+Hess_g(u)-uRic_g$ is the adjoint of the linearised scalar curvature operator. We recall here that given tensors $A$ and $B$ the symbol $A\ast B$ denotes any finite linear combination (over $\mathbb{R}$) of contractions, via the background metric, of the product $A\otimes B$. Of course the type of the resulting tensor is clear from the context and when we want to stress it we will introduce indices, as in $\left(A\ast B\right)_{ij}^{k}$.
\subsection{Variational framework}\label{subs:varframe}
Extending the approach of \cite{CS06}, it is convenient to introduce the functional $\mathcal{G}:\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho} \to \mathbb{R}$ defined by
\[\mathcal{G}(u,Z)=\int_{\Omega}\left\{\frac{1}{2}\left[\left|{d\Phi^{\ast}}^{(1)}_{\left(g,\pi\right)}[u,Z]\right|^{2}r^{n-2p}\rho+\left|{d \Phi^{\ast}}^{(2)}_{\left(g,\pi\right)}[u,Z]\right|^{2}r^{n-2p-2}\rho\right]-(f,V)\cdot_{g}(u,Z)\right\}\,d\upsilon
\]
where we are taking
\[ f\in \mathcal{H}_{0, -p-2,\rho^{-1}}, \ \ V\in \mathcal{X}_{0,-p-2,\rho^{-1}}.
\]
If we let
\[ h=r^{n-2p}\rho \ ({d\Phi^{\ast}}^{(1)}_{(g,\pi)}[u,Z]) \ \ \ \omega=r^{n-2p-2}\rho \ ({d\Phi^{\ast}}^{(2)}_{(g,\pi)}[u,Z])
\]
then the Euler-Lagrange equation for the functional $\mathcal{G}$ takes the form
\[ d\Phi_{(g,\pi)}[h,\omega]-(f,V)=0.
\]
Moreover, we see at once that $h$ and $\omega$ decay according to what we claimed in the statement of Theorem \ref{thm:main} and namely
\[ \left|h\right|_{g}\lesssim \left|x\right|^{-p}, \ \ \left|\omega\right|_{g}\lesssim \left|x\right|^{-p-1}.
\]
As a result, Theorem \ref{thm:main} for $\check{p}=n-2$ is saying that we can get arbitrarily close to the decay of harmonic data for any dimension $n\geq 3$.
\subsection{Outline of the proof}
The general strategy we are about to follow is to obtain the data $(M,\hat{g},\hat{k})$ from the rough patch $(M,g,k)$ by solving the Einstein constraint equations iteratively, and more specifically by means of a Picard scheme. To that aim, we will need to solve a sequence of linearized problems of the form $d\Phi_{(g,\pi)}[h_{i},\omega_{i}]=(f_{i},V_{i})$ and show that indeed the tensors $g+h_{i}$ and $\pi+\omega_{i}$ converge, in suitable functional spaces, to a solution of the nonlinear constraint system.
While this conceptual scheme is similar to that presented in \cite{Cor00} and \cite{CS06} (see also \cite{CD03}), we need to face here a number of serious technical obstacles.
In order to solve the linear problems we will show that the functional $\mathcal{G}$ is coercive, that fact following from certain subtle Poincar\'e type estimates (which we will call \textsl{Basic Estimates}, see Proposition \ref{pro:basic1} and Proposition \ref{pro:basicIIup}) of independent interest. Moreover, as mentioned before, the whole construction is performed in doubly weighted functional spaces, since we will have to keep control, at the same time, of both the decay at infinity and of the regularity at the boundary of the gluing region $\partial\Omega$.
\section{Solving the linear problem}\label{sec:lin}
The scope of this section is to show how to solve the linearized constraint equations by proving the existence of critical points of the functional $\mathcal{G}$. We will initially assume to work at the trivial data, and specifically at the flat metric $g=\delta$ and then get the general case by a perturbation argument. Moreover, we claim that it is enough to prove coercivity estimates in singly weighted Sobolev spaces, and namely with only radial but no angular weights. This follows from the following coarea type lemma.
\begin{lem}\label{lem:coarea}
Let $\zeta\in \mathcal{C}^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R})$, let $\rho:\Omega\to\mathbb{R}$ be defined in terms of $\phi$ as in Subsection \ref{subs:doubly} and let $\tilde{\rho}:(0,\phi_0)\to\mathbb{R}$ be the smooth, monotone increasing function characterized by the identity $\tilde{\rho}(\phi(x))=\rho(x)$ for all $x\in\Omega$. Then
\[ \int_{\Omega}\zeta \rho\ d\upsilon=\int_0^{\phi_0}\tilde{\rho}'(t)\int_{\Omega_t}\zeta\ d\upsilon\, d\mathscr{L}^{1}
\]
where $\Omega_{t}=\left\{x\in\Omega \ : \ \phi(x)\geq t\right\}$.
\end{lem}
\begin{proof} We use the following level set formula for any smooth function $\zeta$
on $\bar{\Omega}$ with bounded support
\[ \int_{\Omega}\zeta \rho\ d\upsilon=-\int_0^{\phi_0}\frac{d}{dt}\int_{\Omega_t}\zeta\rho\ d\upsilon\,d\mathscr{L}^{1}+
\int_{\Omega_{\phi_0}}\zeta\tilde{\rho}(\phi_0)\ d\upsilon.
\]
Now we have, because of the standard coarea formula and integration by parts
\[ -\int_0^{\phi_0}\frac{d}{dt}\int_{\Omega_t}\zeta\rho\ d\upsilon\ d\mathscr{L}^1=-\int_0^{\phi_0}\tilde{\rho}(t)
\frac{d}{dt}\int_{\Omega_t}\zeta\ d\upsilon\ d\mathscr{L}^1=-\tilde{\rho}(\phi_0)\int_{\Omega_{\phi_0}}\zeta\ d\upsilon
+\int_0^{\phi_0}\tilde{\rho}'(t)\int_{\Omega_t}\zeta\ d\upsilon\ d\mathscr{L}^1.
\]
Combining these we have shown
\[ \int_{\Omega}\zeta \rho\ d\upsilon=\int_0^{\phi_0}\tilde{\rho}'(t)\int_{\Omega_t}\zeta\ d\upsilon\ d\mathscr{L}^1
\]
as we had claimed.
\end{proof}
Now, let us suppose we have proved that a certain bounded operator $T:\mathcal{H}_{k_{1},-r_{1}}\to\mathcal{H}_{k_{2},-r_{2}}$ satisfies a functional inequality of the form
\[ \left\|f\right\|_{k_{1},-r_{1}}\leq C\left\|Tf\right\|_{k_{2},-r_{2}}
\]
for some uniform constant $C>0$. Then, because of the previous lemma we can obtain that in fact
\[ \left\|f\right\|_{k_{1},-r_{1},\rho}\leq C \left\|Tf\right\|_{k_{2},-r_{2},\rho}
\]
by taking the previous singly weighted estimate for each $\Omega_{t}$ and integrating in $t$, thanks to the positivity of $\tilde{\rho}'$.
Of course, such simple argument is true when the domain and target of the operator in question, say $T$, are spaces of tensors of any type and not necessarily scalar functions as we considered here in order to simplify the discussion.
\subsection{Poincar\'e-type estimates in conical domains}
We first consider the Hamiltonian constraint and thus show that at the Euclidean metric (with the notations defined above) $\left\|u\right\|_{2,-n+p+2}\leq C\left\|L^{\ast}u\right\|_{0,-n+p}$. That essentially follows from the second of the following Poincar\'e inequalities, of independent interest.
\begin{lem}\label{lem:intparts} For any real number $q$ with $0<q<(n-2)/2$, and $q\neq (n-4)/2$
for $n\geq 5$ we have
\[ \left\|u\right\|_{0,-q}\leq C \left\|\partial u\right\|_{0,-q-1}
\]
as well as
\[ \|u\|_{2,-q}\leq C\|Hess(u)\|_{0,-q-2}.
\]
\end{lem}
\begin{proof} We recall that we prove the inequality for the Euclidean metric, denoting by
$Hess(u)$ the Euclidean hessian. We consider the function $v=|x|^{2-n+2q}$
and observe that for $|x|\neq 0$ we have
\[ \Delta v=2q(2-n+2q)|x|^{-n+2q}
\]
where $\Delta$ is the Euclidean Laplace operator. Under our assumptions this
implies that $\Delta v=- \delta |x|^{-n+2q}$ for a positive constant $\delta=2q(n-2-2q)$. Assume that
$u$ has bounded support and integrate by parts
\[ \int_{\Omega\setminus B_1(0)}u^2\Delta v\ d\mathscr{L}^{n}=(n-2-2q)\int_{\partial B_1(0)\cap\Omega}u^2|x|^{1-n+2q}\ d\mathscr{H}^{n-1}-2\int_{\Omega\setminus B_1(0)} u\partial u \cdot \partial v\ d\mathscr{L}^{n}
\]
where we have used that fact that $\Omega$ is a cone outside of $B_1(0)$, so that the other
boundary terms there vanish. It follows from the sign of the boundary term and the Schwarz inequality that
\[ \delta \|u\|_{0,-q,\Omega\setminus B_1(0)}^2=\delta\int_{\Omega\setminus B_1(0)}u^2|x|^{-n+2q}\leq 2\|u\|_{0,-q,\Omega\setminus B_1(0)} \|\partial u\|_{0,-q-1,\Omega\setminus B_1(0)},
\]
and so we have
\[ \int_{\Omega\setminus B_1(0)}u^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_{\Omega\setminus B_1(0)}|\partial u|^2
r^{-n+2q+2}\ d\mathscr{L}^{n}.
\]
Let $\zeta$ be a smooth cutoff function with support in $B_2(0)$ and with $\zeta=1$ on $B_1(0)$.
A standard Poincar\'e inequality implies
\[ \int_{\Omega}(\zeta u)^2\ d\mathscr{L}^{n}\leq C\int_{\Omega}|\partial(\zeta u)|^2\ d\mathscr{L}^{n}.
\]
In turn, this gives
\[ \int_{\Omega\cap B_1(0)}u^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_{\Omega\cap B_2(0)}|\partial u|^2
r^{-n+2q+2}\ d\mathscr{L}^{n}
\]
since $r$ is bounded above and below by positive constants on $B_2(0)$. Summing this
with the previous inequality we obtain
\[ \int_{\Omega}u^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_{\Omega}|\partial u|^2 r^{-n+2q+2}\ d\mathscr{L}^{n}.
\]
To obtain the desired bound we now apply the previous argument to each partial derivative
of $u$, say $w=\partial_j u$. We may use essentially the same argument with the
function $v=|x|^{4-n+2q}$ provided that $q\neq (n-4)/2$. The two cases when
$4-n+2q<0$ and when $4-n+2q>0$ work similarly with a sign reversal. In both cases
the boundary term can be thrown away and we end up showing
\[ \int_\Omega |\partial u|^2r^{-n+2q+2}\ d\mathscr{L}^{n}\leq C\int_\Omega |Hess(u)|^2r^{-n+2q+4}\ d\mathscr{L}^{n}
\]
which can be combined with our first functional inequality to complete the proof.
\end{proof}
From this we can deduce the coercivity estimate for the adjoint of the linearized scalar curvature operator, which corresponds to the coercivity of ${d\Phi^{\ast}}^{(1)}$ in the time-symmetric case, namely when $\pi=0$.
\begin{proposition} \label{pro:basic1}(Basic Estimate I) For any real number $p$ with $\frac{n-2}{2}<p<n-2$ and $p\neq n/2$ for $n\geq 5$ we have
\[ \|u\|_{2,-n+p+2}\leq C\|L^*(u)\|_{0,-n+p} \ \textrm{for all} \ u\in \mathcal{H}_{2,-n+p+2}(\Omega).
\]
Hence, thanks to Lemma \ref{lem:coarea} we deduce that
\[ \|u\|_{2,-n+p+2,\rho}\leq C\|L^*(u)\|_{0,-n+p,\rho} \ \textrm{for all} \ u\in \mathcal{H}_{2,-n+p+2,\rho}(\Omega).
\]
\end{proposition}
\begin{proof}
Since $q=n-p-2$ satisfies $0<q<(n-2)/2$ and $q\neq (n-4)/2$ (by our assumption), we may apply the lemma to obtain
\[ \|u\|_{2,-n+p+2}\leq C\|Hess(u)\|_{0,-n+p}.
\]
Thus to complete the proof it suffices to show
\[ \|Hess(u)\|_{0,-n+p}\leq C\|L^*(u)\|_{0,-n+p}.
\]
Recalling that we are working at the Euclidean metric (and will then deduce a general coercivity result by perturbation) we can simply take the trace in the definition of the operator $L^{\ast}$ thereby obtaining
\[ Tr(L^*(u))=(1-n)\Delta u,\ \mbox{or}\ \Delta u=-1/(n-1)Tr(L^*(u)),
\]
frow which it follows that
\[ Hess(u)=L^*(u)-1/(n-1)Tr(L^*(u))\delta
\]
and therefore
\[ \|Hess(u)\|_{0,-n+p}\leq C\|L^*(u)\|_{0,-n+p}
\]
which provides the Basic Estimate at the Euclidean metric.
\end{proof}
\subsection{Coercivity of the Lie operator}\label{subs:coercLie}
We now consider the vector constraint equation, and prove coercivity of the differential ${d\Phi^{\ast}}^{(2)}$ by first analyzing the decoupled case when $\pi=0$.
Let $\mathcal D$ denote the Killing operator acting on vector fields
\[ \mathcal D(Y)(Z,W)=D_ZY\cdot W+D_WY\cdot Z.
\]
\begin{lem}\label{pro:basic2} Given a real number $0<q<\frac{n-2}{2}$, there exists a positive constant $C$ such that
\[ \int_\Omega |Y|^2r^{-n+2q}\ d\mathscr{L}^n\leq C\int_\Omega|\mathcal D(Y)|^2r^{2-n+2q}\ d\mathscr{L}^n
\]
for any smooth vector field $Y$ in $\Omega$ which has bounded support (no condition on $\partial\Omega$).
\end{lem}
\begin{proof} We can use the standard arguments to get the bound on a compact set,
so it suffices to consider vector fields $Y$ which are defined on a truncated conical subregion of
$\Omega$ (say $\Omega\setminus B_{1}$), have bounded support, and vanishing on the inner boundary of $\Omega$. This
can be done by replacing $Y$ by $\zeta Y$ where $\zeta$ is a cutoff function which
is one outside a fixed ball and zero inside a fixed smaller ball.
We first obtain the bound on the radial component of $Y$. We will work in orthonormal
bases for which $e_n=\partial_r$, so this component is denoted $Y_n$. Following the very same
argument presented in the proof of Lemma \ref{lem:intparts} we obtain
\[ \int_\Omega(Y_n)^2r^{-n+2q}\ d\mathscr{L}^{n}=- \frac{1}{q}\int_\Omega(Y_n\partial_r Y_n)r^{1-n+2q}\ d\mathscr{L}^{n}.
\]
Since $D_{e_n}e_n=0$, we have
$\partial_rY_n=\frac{1}{2}\mathcal D(Y)_{nn}$, so we can easily get
\[ \int_\Omega(Y_n)^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_\Omega|\mathcal D(Y)|^2 r^{2-n+2q}\ d\mathscr{L}^{n}.
\]
This gives the desired bound for $Y_n$.
Away from the axis of $\Omega$ (namely from the line generated by the vector $a$) we define orthonormal vector fields $e_{n-1},e_n$ where
$e_n=\partial_r$, the unit radial vector, $e_{n-1}=r^{-1}\partial_\theta$, the unit
vector field tangent to the spheres pointing away from the axis. We use the
notation $Y_i=Y\cdot e_i$ for $i=n-1,n$. To obtain the general bound, we have as above
\[ \int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}=-\frac{1}{q}\int_\Omega(Y\cdot\partial_rY)r^{1-n+2q}\ d\mathscr{L}^{n}.
\]
We observe that since $e_n=\partial_r$, we may write $Y=Z+Y_ne_n$
where $Z$ is orthogonal to the radial direction. We then have
\[ Y\cdot\partial_rY=\mathcal D(Y)(e_n,Y)-D_YY\cdot e_n,
\]
and
\[ D_YY\cdot e_n=D_ZZ\cdot e_n+Y_n D_{e_n}(Y_ne_n)\cdot e_n
+Y_n D_{e_n}Z\cdot e_n+D_Z(Y_ne_n)\cdot e_n.
\]
This expression simplifies to
\[ D_YY\cdot e_n=-r^{-1}|Z|^2+Y_ne_n(Y_n)+Z(Y_n)=-r^{-1}|Z|^2+Y(Y_n).
\]
It follows that
\[ \int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}\leq \frac{1}{q}\int_\Omega|Y||\mathcal D(Y)|r^{1-n+2q}\ d\mathscr{L}^{n}
+\frac{1}{q}\int_\Omega Y(Y_n)r^{1-n+2q}\ d\mathscr{L}^{n}.
\]
We may integrate the second term by parts to obtain
\begin{align*} \int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n} & \leq \frac{1}{q}\int_\Omega|Y||\mathcal D(Y)|r^{1-n+2q}\ d\mathscr{L}^{n}
-\frac{1}{q} \int_\Omega Div(r^{1-n+2q}Y) Y_n\ d\mathscr{L}^{n} \\
& +\frac{1}{q}\int_{\partial \Omega}|Y_{n-1}Y_n|r^{1-n+2q}\ d\mathscr{H}^{n-1}
\end{align*}
where we have used that fact that $e_{n-1}$ is the unit normal vector to
$\partial \Omega$. Since $Div(Y)$
is bounded by a fixed constant times $|\mathcal D(Y)|$ we obtain
\[ \int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_\Omega|Y||\mathcal D(Y)|r^{1-n+2q}\ d\mathscr{L}^{n}
+C\int_\Omega(Y_n)^2r^{-n+2q}\ d\mathscr{L}^{n}+C\int_{\partial \Omega}|Y_{n-1}Y_n|r^{1-n+2q}\ d\mathscr{H}^{n-1}.
\]
This clearly implies from our previous bound on $Y_n$
\begin{equation}\label{eq:central}
\int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_\Omega|\mathcal D(Y)|^2r^{2-n+2q}\ d\mathscr{L}^{n}+
C\int_{\partial \Omega}|Y_{n-1}Y_n|r^{1-n+2q}\ d\mathscr{H}^{n-1}.
\end{equation}
as claimed.
It remains to handle the boundary term. We have
\[\int_{\partial \Omega}|Y_{n-1}Y_n|r^{1-n+2q}\ d\mathscr{H}^{n-1}\leq \varepsilon/2\int_{\partial \Omega}|Y_{n-1}|^2r^{1-n+2q}\ d\mathscr{H}^{n-1}+(2\varepsilon)^{-1} \int_{\partial \Omega}|Y_n|^2r^{1-n+2q}\ d\mathscr{H}^{n-1}
\]
for any $\varepsilon>0$ to be chosen small enough. We recall that the region
$\Omega$ is defined (outside of $B_1$) by $\theta_{1}\leq \theta\leq \theta_{2}$, and we let $\Sigma_\alpha$ denote the
hypersurface $\{\theta=\alpha\}$ for any $\alpha\in (0,\theta_2]$ so that $\partial \Omega=\Sigma_{\theta_1}\cup\Sigma_{\theta_{2}}$. In particular, the function $\theta$ should not be confused with $\phi$, which is instead (roughly speaking) the angular distance from $\partial\Omega$. We also let $\Theta_\alpha$ denote the cone $\{0\leq\theta\leq\alpha\}$ so that
$\Omega=\Theta_{\theta_2}\setminus \Theta_{\theta_{1}}$.
For any smooth function $u$ with bounded
support in $\Omega$ and vanishing in a fixed neighborhood of the origin we have from the
coarea formula
\[ \int_{\Theta_{\theta_{2}}\setminus \Theta_{\theta_{1}}}u^2r^{-n+2q}\ d\mathscr{L}^{n}=\int_{\theta_{1}}^{\theta_2}\int_{\Sigma_t}
u^2r^{1-n+2q}\ d\mathscr{H}^{n-1}\ d\mathscr{L}^{1}
\]
since $|\nabla\theta|=r^{-1}$. Now, let us set
\[ I(t)=\int_{\Sigma_t}u^2r^{1-n+2q}\ d\mathscr{H}^{n-1}.
\]
For any $\alpha\in(\theta_{1},\theta_{2})$ we also have, by differentiating and applying the fundamental
theorem of calculus together with the coarea formula
\begin{align*} I(\theta_2)-I(\alpha)&=\int_\alpha^{\theta_2}\int_{\Sigma_t}(\partial_\theta(u^2)+(n-2)\cot(t)u^2)r^{1-n+2q}
\ d\mathscr{H}^{n-1}\ d\mathscr{L}^{1}\\
&=\int_{\Theta_{\theta_{2}}\setminus \Theta_{\alpha}}(re_{n-1}(u^2)+(n-2)\cot(\theta)u^2)r^{-n+2q}\ d\mathscr{L}^{n}
\end{align*}
as well as
\begin{align*} I(\alpha)-I(\theta_{1})&=\int_{\theta_{1}}^{\alpha}\int_{\Sigma_t}(\partial_\theta(u^2)+(n-2)\cot(t)u^2)r^{1-n+2q}
\ d\mathscr{H}^{n-1}\ d\mathscr{L}^{1}\\
&=\int_{\Theta_{\alpha}\setminus \Theta_{\theta_{1}}}(re_{n-1}(u^2)+(n-2)\cot(\theta)u^2)r^{-n+2q}\ d\mathscr{L}^{n}.
\end{align*}
In order to bound $I(\theta_2)$, we may use such formula and the intermediate value theorem to find $\alpha\in (\theta_{1},\theta_{2})$ so that
\[ I(\alpha)\leq 2/(\theta_2-\theta_1)\int_\Omega u^2r^{-n+2q}\ d\mathscr{L}^{n}.
\]
We then have from the formula above (note that $|\cot(\theta)|$ is bounded from above when
$\theta\in [\theta_1,\theta_2]$)
\[ \int_{\Sigma_{\theta_2}}u^2r^{1-n+2q}\ d\mathscr{H}^{n-1}=I(\theta_2)\leq C\int_\Omega u^2r^{-n+2q}\ d\mathscr{L}^{n}
+2\int_{\Theta_{\theta_{2}}\setminus{\Theta_{\alpha}}}u(e_{n-1}u)r^{1-n+2q}\ d\mathscr{L}^{n}
\]
and similarly for $I(\theta_{1})$ modulo sign changes, where appropriate.
So, in the end can write
\begin{align*}\int_{\partial \Omega}u^2r^{1-n+2q}\ d\mathscr{H}^{n-1} & \leq C\int_\Omega u^2r^{-n+2q}\ d\mathscr{L}^{n}
+2\int_{\Theta_{\theta_{2}}\setminus{\Theta_{\alpha}}}u(e_{n-1}u)r^{1-n+2q}\ d\mathscr{L}^{n} \\
& -2\int_{\Theta_{\alpha}\setminus{\Theta_{\theta_{1}}}}u(e_{n-1}u)r^{1-n+2q}\ d\mathscr{L}^{n}.
\end{align*}
Taking $u=Y_{n-1}$ we observe that
\[ e_{n-1}Y_{n-1}=D_{e_{n-1}}Y\cdot e_{n-1}+Y\cdot D_{e_{n-1}}e_{n-1}=\frac{1}{2}
\mathcal D(Y)(e_{n-1},e_{n-1})-r^{-1}Y_n.
\]
Therefore we have
\[ \int_{\partial \Omega}(Y_{n-1})^2r^{1-n+2q}\ d\mathscr{H}^{n-1}\leq C\int_\Omega((Y_{n-1})^2+(Y_n)^2)r^{-n+2q}\ d\mathscr{L}^{n}
+C\int_\Omega|\mathcal D(Y)|^2 r^{2-n+2q}\ d\mathscr{L}^{n}.
\]
Taking $u=Y_n$ we have
\[ e_{n-1}Y_n=D_{e_{n-1}}Y\cdot e_n+Y\cdot D_{e_{n-1}}e_n=
\mathcal D(Y)(e_{n-1},e_n)-D_{e_n}Y\cdot e_{n-1}+r^{-1}Y_{n-1}.
\]
Now we rewrite
\[ D_{e_n}Y\cdot e_{n-1}=e_n(Y_{n-1})-Y\cdot D_{e_n}e_{n-1}=e_n(Y_{n-1})
\]
and thus
\begin{align*} \int_{\partial \Omega}(Y_n)^2r^{1-n+2q}\ d\mathscr{H}^{n-1}&\leq C\int_\Omega((Y_n)^2+|Y_{n-1}Y_n|)r^{-n+2q}\ d\mathscr{L}^{n}-2\int_{\Theta_{\theta_{2}}\setminus \Theta_{\alpha}}Y_ne_n(Y_{n-1})r^{1-n+2q}\ d\mathscr{L}^{n}\\
&+2\int_{\Theta_{\alpha}\setminus \Theta_{\theta_{1}}}Y_ne_n(Y_{n-1})r^{1-n+2q}\ d\mathscr{L}^{n}+C\int_\Omega|\mathcal D(Y)|^2 r^{2-n+2q}\ d\mathscr{L}^{n}.
\end{align*}
We can write the term in the second and third integrals as
\[ Y_ne_n(Y_{n-1})=e_n(Y_nY_{n-1})-e_n(Y_n)Y_{n-1}=e_n(Y_nY_{n-1})-
\frac{1}{2}Y_{n-1}\mathcal D(Y)(e_n,e_n).
\]
After an integration by parts we arrive at the inequality
\[ \int_{\partial \Omega}(Y_n)^2r^{1-n+2q}\ d\mathscr{H}^{n-1}\leq C\int_\Omega((Y_n)^2+|Y_{n-1}Y_n|+r|Y_{n-1}|
|\mathcal D(Y)|)r^{-n+2q}\ d\mathscr{L}^{n}+C\int_\Omega|\mathcal D(Y)|^2 r^{2-n+2q}\ d\mathscr{L}^{n}.
\]
We can finally put things together and complete the proof by estimating the boundary term as follows
\begin{align*} \int_{\partial \Omega}|Y_{n-1}Y_n|r^{1-n+2q}\ d\mathscr{H}^{n-1}&\leq C\varepsilon\int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}\\
&+C\varepsilon^{-1}\int_\Omega((Y_n)^2+|Y_{n-1}Y_n|+r|Y_{n-1}||\mathcal D(Y)|+r^2|\mathcal D(Y)|^2)r^{-n+2q}\ d\mathscr{L}^{n}
\end{align*}
Combining this inequality above with \eqref{eq:central}, and fixing $\varepsilon$ small enough we can absorb the first
term to obtain
\[ \int_\Omega|Y|^2r^{-n+2q}\ d\mathscr{L}^{n}\leq C\int_\Omega((Y_n)^2+|Y_{n-1}Y_n|+r|Y_{n-1}||\mathcal D(Y)|+r^2|\mathcal D(Y)|^2)r^{-n+2q}\ d\mathscr{L}^{n}.
\]
From our bound on $Y_n$ and easy estimates we now obtain the desired conclusion.
\end{proof}
\begin{proposition}(Basic Estimate II)\label{pro:basicIIup}
Given a real number $0<q<\frac{n-2}{2}$ there exists a positive constant $C$ such that
\[ \int_\Omega |\nabla Z|^2 r^{2-n+2q}\rho \ d\mathscr{L}^{n}\leq C\int_\Omega|\mathcal D(Z)|^2r^{2-n+2q}\rho\ d\mathscr{L}^{n}
\]
for any smooth vector field $Z$
in $\Omega$ which has bounded support (no condition on $\partial\Omega$).
Hence, thanks to Proposition \ref{pro:basic2} and Lemma \ref{lem:coarea} we have for all $p\in (\frac{n-2}{2},n-2)$
\[
\left\|Z\right\|_{\mathcal{H}_{1,-n+p+2,\rho}}\leq C \left\|\mathcal{D}(Z)\right\|_{\mathcal{M}_{0,-n+p+1,\rho}} \ \textrm{for all} \ Z\in \mathcal{X}_{0,-n+p+1,\rho}.
\]
\end{proposition}
\begin{proof}
By virtue of the previous Lemma \ref{pro:basic2} (and its obvious weighted counterpart, gotten by applying Lemma \ref{lem:coarea}), it is enough for us to prove that
\[ \int_\Omega |\nabla Z|^2 r^{2-n+2q}\rho \ d\mathscr{L}^{n}\leq C\int_\Omega|\mathcal D(Z)|^2r^{2-n+2q}\rho\ d\mathscr{L}^{n}+C\int_{\Omega}|Z|^{-n+2q}\rho\,d\mathscr{L}^{n}.
\]
Now, because of the very definition of Lie derivative
\[\int_{\Omega} (Z^{2}_{i;j}+Z^{2}_{j;i})r^{2-n+2q}\rho\, d\mathscr{L}^{n}\leq C\int_\Omega|\mathcal D(Z)|^2r^{2-n+2q}\rho\, d\mathscr{L}^{n}-2\int_{\Omega}Z_{i;j}Z_{j;i}r^{2-n+2q}\rho\,d\mathscr{L}^{n}
\]
and thus we are reduced to proving, for indices $i\neq j$ an inequality of the form
\begin{align*} -2\int_{\Omega}Z_{i;j}Z_{j;i}r^{2-n+2q}\rho\,d\mathscr{L}^{n} & \leq \varepsilon \int_{\Omega} |\nabla Z|^{2}r^{2-n+2q}\rho\, d\mathscr{L}^{n}\\
& +\varepsilon^{-1}\left(\int_\Omega|\mathcal D(Z)|^2r^{2-n+2q}\rho \,d\mathscr{L}^{n}+\int_{\Omega}|Z|^{-n+2q}\rho\,d\mathscr{L}^{n}\right)
\end{align*}
for some $\varepsilon$ small enough that the Dirichlet term can be absorbed back in the left-hand side.
In partial analogy with the strategy that has been followed for proving Lemma \ref{pro:basic2}, we will apply some integration by parts (or, more precisely, we will use the divergence theorem) and, as will be clear from the sequel of our argument, the delicate point will be to control the boundary terms that may possibly arise in doing that. Since the outer normal to $\Omega$ is given (modulo sign) by $e_{n-1}$, we will limit ourselves to treat the case when $i\neq n-1$ and $j=n-1$. Otherwise the proof is strictly simpler and in fact does not require any delicate estimate.
For any $t\in (0,\phi_{0})$ let us recall that $\Omega_{t}=\left\{p\in\Omega : \phi(p,\partial\Omega)\geq t\right\}$. Applying the divergence theorem in $\Omega_{t}$ we get
\[ -2\int_{\Omega_{t}}Z_{i;j}Z_{j;i}r^{2-n+2q}\,d\mathscr{L}^{n}\leq 2\int_{\Omega_{t}}Z_{j}Z_{i;ji}r^{2-n+2q}\,d\mathscr{L}^{n} + C\int_{\Omega_{t}}|Z_{j}Z_{i;j}|r^{1-n+2q}\,d\mathscr{L}^{n}
\]
for some positive constant $C$ depending on $n$ and $q$ only. The standard rearrangement trick allows to treat the second summand on the right-hand side, so we only need to get an upper bound for
\[ 2\int_{\Omega_{t}}Z_{j}Z_{i;ji}r^{2-n+2q}\,d\mathscr{L}^{n}.
\]
Since the background metric is Euclidean, possibly by introducing other lower order terms (that can be treated as above) we can interchange the order of derivatives from $D_{e_{i}}D_{e_{j}}$ to $D_{e_{j}}D_{e_{i}}$ and hence we need to deal with
\[ 2\int_{\Omega_{t}}Z_{j}Z_{i;ij}r^{2-n+2q}\,d\mathscr{L}^{n}.
\]
Applying the divergence theorem again, we obtain
\begin{align*} 2\int_{\Omega_{t}}Z_{j}Z_{i;ij}r^{2-n+2q}\,d\mathscr{L}^{n} \leq & -2\int_{\Omega_{t}}Z_{i;i}Z_{j;j}r^{2-n+2q}\,d\mathscr{L}^{n}+2\int_{\partial\Omega_{t}}|Z_{j}Z_{i;i}|r^{2-n+2q}\,d\mathscr{H}^{n-1} \\
& + C\int_{\Omega_{t}}|Z_{j}Z_{i;i}|r^{1-n+2q}\,d\mathscr{L}^{n}.
\end{align*}
Putting together the previous inequalities and applying the coarea type formula given by Lemma \ref{lem:coarea} we come to an inequality of the form
\[ \left\|Z\right\|^{2}_{\mathcal{H}_{1,-q,\rho}}\leq C\left[\left\|\mathcal{D}Z\right\|^{2}_{\mathcal{H}_{0,-q-1,\rho}}+\left\|Z\right\|^{2}_{\mathcal{H}_{0,-q,\rho}}+2N\sum_{i,j}\int_{0}^{\phi_{0}}\phi^{2N-1}\int_{\partial\Omega_{\phi}}|Z_{j}Z_{i;i}|r^{2-n+2q}\,d\mathscr{H}^{n-1}d\mathscr{L}^{1}\right]
\]
and thus
\[\left\|Z\right\|^{2}_{\mathcal{H}_{1,-q,\rho}}\leq C\left[\left\|\mathcal{D}Z\right\|^{2}_{\mathcal{H}_{0,-q-1,\rho}}+\left\|Z\right\|^{2}_{\mathcal{H}_{0,-q,\rho}}+2N\sum_{i,j}\int_{\Omega}|Z_{j}Z_{i;i}|\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{L}^{n}\right]
\]
where in both cases $C$ denotes a positive constant only depending on $n$ and $q$.
Therefore, since the last summand on the right-hand side can be bounded from above by
\[ 2N\int_{\Omega}|Z_{j}Z_{i;i}|\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{H}^{n-1}d\mathscr{L}^{1}\leq N^{2}\int_{\Omega}Z^{2}_{i;i}r^{2-n+2q}\rho\,d\mathscr{L}^{n}+\int_{\Omega}Z^{2}_{j}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}
\]
we have proven that in fact
\[ \left\|Z\right\|^{2}_{\mathcal{H}_{1,-q,\rho}}\leq C\left[N^{2}\left\|\mathcal{D}Z\right\|^{2}_{\mathcal{H}_{0,-q-1,\rho}}+\left\|Z\right\|^{2}_{\mathcal{H}_{0,-q,\rho}}+\int_{\Omega}|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}\right]
\]
which implies our conclusion once we show that
\[ \int_{\Omega}|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}\leq C\int_{\Omega}|Z|^{2}r^{-n+2q}\rho\,d\mathscr{L}^{n}+\frac{C}{N^{2}}\int_{\Omega}|\nabla Z|^{2}r^{2-n+2q}\rho\,d\mathscr{L}^{n}.
\]
To that aim, let us introduce an angular cut-off function $\xi$ that equals 1 for $\phi(\cdot,\partial\Omega)\leq \phi_{0}/3$ and 0 for $\phi(\cdot,\partial\Omega)\geq \phi_{0}/2$. It is obvious that
\[ \int_{\Omega}(1-\xi)|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}\leq C\int_{\Omega}|Z|^{2}r^{-n+2q}\rho\,d\mathscr{L}^{n}
\]
for some constant $C$ only depending on $n, q, \theta_{1}, \theta_{2}$ and therefore we only need to produce an estimate for $\int_{\Omega}\xi|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}$.
If we apply the divergence theorem in $\Omega$ to the vector field $\xi|Z|^{2}r^{2-n+2q}\nabla\rho$ and exploit the fact that (on the support of $\xi$) $\Delta\rho\geq C N^{2}r^{-2}\phi^{-2}\rho$ (for some constant $C$ which does not depend on $N$) we get
\[\int_{\Omega}\xi|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}\leq \frac{C}{N}\left[\int_{\Omega}|\xi'||Z|^{2}\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{L}^{n}+\int_{\Omega}\xi|Z||\nabla Z|\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{L}^{n}\right]
\]
because $\nabla r \perp \nabla \phi$ and $\int_{\partial\Omega}\xi|Z|^{2}r^{2-n+2q}\nabla\rho\cdot\nu\,d\mathscr{H}^{n-1}\leq 0$. Thus, let us notice that (by the Cauchy-Schwarz inequality)
\[\int_{\Omega}\xi|Z||\nabla Z|\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{L}^{n}\leq \left(\int_{\Omega}\xi|\nabla Z|^{2}r^{2-n+2q}\rho\,d\mathscr{L}^{n}\right)^{1/2}\left(\int_{\Omega}\xi|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}\right)^{1/2}
\]
and therefore, if we set
\[ F_{W}=\int_{\Omega}\xi|Z|^{2}\phi^{-2}r^{-n+2q}\rho\,d\mathscr{L}^{n}, \ F_{L}=\int_{\Omega}|\xi'||Z|^{2}\phi^{-1}r^{1-n+2q}\rho\,d\mathscr{L}^{n}, \ F_{D}=\int_{\Omega}\xi|\nabla Z|^{2}r^{2-n+2q}\rho\,d\mathscr{L}^{n}
\]
we have just proven an inequality of the form
\[ F_{W}\leq \frac{C}{N} F_{L}+\frac{C}{N}F^{1/2}_{W}F^{1/2}_{D}.
\]
Let us now remark that since $\xi'=0$ near the boundary $\partial\Omega$ and, by scaling arguments, $|\xi'|\leq Cr^{-1}$ on the whole $\Omega$ we can write
\[ F_{L}\leq C\int_{\Omega}|Z|^{2}r^{-n+2q}\rho\,d\mathscr{L}^{n}.
\]
In order to conclude our proof, we distinguish two cases. If $F^{1/2}_{W}\leq \frac{C}{N}F^{1/2}_{D}+F^{1/2}_{L}$ then we are done as soon as we pick $N$ large enough to absorbe the Dirichlet integral in the left-hand side of our main inequality. If instead this is not the case, and thus $F^{1/2}_{W}-\frac{C}{N}F^{1/2}_{D}\geq F_{L}^{1/2}$ we obtain (from the inequality relating $F_{W}, F_{L}, F_{D}$) that
\[ F_{W}^{1/2}F_{L}^{1/2}\leq \frac{C}{N}F_{L}
\]
which is the same as
\[ F_{W}\leq \frac{C}{N^{2}}F_{L}
\] and this completes the proof.
\end{proof}
\subsection{Existence of critical points}
In this subsection, we capitalize the effort spent in proving the coercivity inequalities for the adjoint constraint operators by deriving the existence of critical points for the functional $\mathcal{G}$ associated to the linearized problem.
First of all, let us observe that thanks to Lemma \ref{lem:coarea} we can turn Proposition \ref{pro:basic1} into a corresponding statement in doubly weighted Sobolev spaces, namely when the angular weight $\rho$ is also taken into account. Thus, we obtain the natural counterpart of Proposition \ref{pro:basicIIup}, which instead concerns the second component of the constraints.
The following proposition is the key to solve the linear Einstein constraint system in our setting.
\begin{proposition}\label{pro:basic}
Let $n\geq 3$ and for any set of data $(M,\check{g},\check{k})$ as in the statement of Theorem \ref{thm:main} let $(M,g,k)$ be the triple defined in Subsection \ref{subs:doubly}. Fix a real number $\frac{n-2}{2}<p<\check{p}$ with $p\neq n/2$ if $n\geq 5$ . There exist constants $a_{\infty,L}$ and $C$ (depending only on $g, k, \theta_{1}, \theta_{2}, p$) such that uniformly for $|a|>a_{\infty}$
\[ \left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}\leq C \left\|d\Phi^{\ast}_{\left(g,\pi\right)}\left[u,Z\right]\right\|_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}
\]
for all $u\in \mathcal{H}_{2,-n+p+2,\rho}$ and $Z\in \mathcal{X}_{1,-n+p+2,\rho}$.
\end{proposition}
\begin{proof}
The Basic Estimate I (Proposition \ref{pro:basic1}) and II (Proposition \ref{pro:basicIIup}) ensure the existence of a constant $C>0$ (that can be chosen uniformly for all suffciently large $|a|$) such that
\[
\left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}\leq C \left\|d\Phi^{\ast}_{\left(\delta,0\right)}\left[u,Z\right]\right\|_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}.
\]
On the other hand, given the decay assumptions on $(\hat{g},\hat{k})$ (hence on $(g,k)$) it follows from a standard perturbation argument that for any $\varepsilon>0$ we can find $a_{\infty,L}=a_{\infty,L}(\varepsilon)$ such that
\[
|a|>a_{\infty,L} \ \Rightarrow \
\left\|d\Phi^{\ast}_{(g,\pi)}[u,Z]-d\Phi^{\ast}_{\left(\delta,0\right)}\left[u,Z\right]\right\|_{\mathcal{M}_{0,-n+p}\times \mathcal{S}_{0,-n+p+1}}\leq\varepsilon \left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2}\times \mathcal{X}_{1,-n+p+2}}
\]
and hence, by virtue of Lemma \ref{lem:coarea}
\[
\left\|d\Phi^{\ast}_{\left(g,\pi\right)}\left[u,Z\right]-d\Phi^{\ast}_{\left(\delta,0\right)}\left[u,Z\right]\right\|_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}\leq\varepsilon \left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}.
\] Thus, picking $\varepsilon_0=C/3$ we have that for all $|a|>a_{\infty,L}(\varepsilon_0)$ both previous inequalities are true, and therefore the triangle inequality ensures that
\[
\left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}\leq \frac{2C}{3} \left\|d\Phi^{\ast}_{\left(g,\pi\right)}\left[u,Z\right]\right\|_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}
\]
which is what we wanted.
\end{proof}
At this stage, we can use a direct method to find a (unique) global minimum for $\mathcal{G}$.
\begin{proposition}\label{pro:existence}
Let $n\geq 3$, let $\frac{n-2}{2}<p<\check{p}$ with $p\neq n/2$ if $n\geq 5$ and assume that the vertex $a$ satisfies the inequality $|a|>a_{\infty,L}$. For any $(f, V)\in \mathcal{H}_{0, -p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}$ there exists a unique $(\tilde{u},\tilde{Z})\in \mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1, -n+p+2,\rho}$ which minimizes the functional $\mathcal{G}$ on the Hilbert space $\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1, -n+p+2,\rho}$.
\end{proposition}
\begin{rem}
As the reader may have noticed, we did not mention, in the statement of Theorem \ref{thm:main}, the exceptional value $p^{\ast}=p^{\ast}(n)=n/2$ related to the restrictions of the basic estimates for dimension $n\geq 5$. Indeed, we claim that this is not an issue when performing the gluing. The reason is very simple: given data $(M,\check{g},\check{k})$ with decay $\check{p}$ (in the usual sense) if one wanted to prove Theorem \ref{thm:main} for $p=p^{\ast}$ then by simply performing our whole construction with weight $\frac{p^{\ast}+\check{p}}{2}$ (which is larger than $p^{\ast}$, hence certainly not exceptional) one would produce a triple $(M,\hat{g},\hat{k})$ which does solve the Einstein constraint equation and whose decay is actually \textsl{faster} than the initial requirement (thereby satisying the conclusion of Theorem \ref{thm:main} for $p=p^{\ast}$). In other words (and more generally) for fixed $\check{p}$ the validity of Theorem \ref{thm:main} for some $p''<\check{p}$ implies the validity of the same assertion for all $p'\in(\frac{n-2}{2},p'')$.
\end{rem}
\begin{proof} The argument follows the direct method of the Calculus of Variations. Indeed, the functional $\mathcal{G}$ is bounded from below on $\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{H}_{1, -n+q+1,\rho}$ for its very definition implies
\[
\begin{split}
\mathcal{G}(u, Z)\geq C_1\left\|d\Phi^{\ast}_{\left(g,\pi\right)}[u,Z]\right\|^{2}_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}-C_2\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}\left\|\left(u,Z\right)\right\|_{\mathcal{H}_{0,-n+p+2,\rho}\times \mathcal{X}_{0,-n+p+2,\rho}}
\end{split}
\]
and hence, thanks to the basic estimate (in the form of Proposition \ref{pro:basic})
\[
\mathcal{G}(u,Z)\geq C_1 \left\|\left(u,Z\right)\right\|^{2}_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}-C_2\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}\left\|\left(u,Z\right)\right\|_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}}
\]
which immediately implies that $\mathcal{G}\left(\cdot,\cdot\cdot\right)$ is coercive and thus the claim follows.
As a result, we can pick a minimizing sequence $(u_{i}, Z_{i})_{i\in\mathbb{N}}$ which is bounded in $\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1, -n+p+2,\rho}$ (in fact, the previous estimate shows that \textsl{any} minimizing sequence has to be bounded): by Banach-Alaoglu there will be a subsequence which weakly converges to a limit point $(\tilde{u},\tilde{Z})$ and by (weak) lower semicontinuity of the functional we conclude that
\[\mathcal{G}(\tilde{u},\tilde{Z})\leq \liminf_{i\to\infty}\mathcal{G}(u_{i},Z_{i}).
\]
This precisely means that $(\tilde{u},\tilde{Z})$ minimizes the value of $\mathcal{G}$. Finally, the uniqueness statement follows by strict convexity of the functional: indeed, if we had two minima $(u_{1}, Z_{1})$ and $(u_{2}, Z_{2})$ then because of the identity
\[ \mathcal{G}\left(\frac{\left(u_{1}, Z_{1}\right)+\left(u_{2},Z_{2}\right)}{2}\right) =\frac{1}{2}\mathcal{G}\left(u_{1}, Z_{1}\right)+\frac{1}{2}\mathcal{G}\left(u_{2}, Z_{2}\right)-\frac{1}{8}\left\|d\Phi^{\ast}_{\left(g,\pi\right)}\left[u_{2}-u_{1}, Z_{2}-Z_{1}\right]\right\|^{2}_{\mathcal{M}_{0,-n+p+2,\rho}\times \mathcal{S}_{0,-n+p+2,\rho}}
\]
we would reach a contradiction unless $d\Phi^{\ast}_{(g,\pi)}[u_{2}-u_{1}, Z_{2}-Z_{1}]=0$ and by the basic estimate this forces $u_{1}=u_{2}$ as well as $Z_{1}=Z_{2}$ which is what we had to prove.
\end{proof}
\section{The Picard scheme}\label{sec:nonlin}
\subsection{Iterative solution}
In this subsection, we define Banach spaces $X_{1}, X_{2}$ so that the solution operator associated to the linearized problem is in fact a bounded operator $S:X_{1}\to X_{2}$. As a result, we will solve the nonlinear problem iteratively, by following a Picard-type scheme.
Given data $(f^{\ast},V^{\ast})$ where $f^{\ast}$ is a (scalar) function and $V^{\ast}$ is a vector field, we want to find $(\hat{g},\hat{\pi})$ satisfying $\Phi(\hat{g},\hat{\pi})=(f^{\ast},V^{\ast})$. This problem can be more conveniently written as
\[ \Phi(g_{0},\pi_{0})+d\Phi_{(g_{0},\pi_{0})}[h,\omega]+Q_{(g_{0},\pi_{0})}[h,\omega]=(f^{\ast},V^{\ast})
\]
with $(g_{0},\pi_{0})=(g,\pi)$ the data we start with (in our case, they are gotten by rough patching in $\Omega$ of the initial data we are given and the trivial data $(\delta,0)$).
The previous equation takes the form
\[d\Phi_{(g_{0},\pi_{0})}[h,\omega]+Q_{(g_{0},\pi_{0})}[h,\omega]=(f^{\ast},V^{\ast})-\Phi(g_{0},\pi_{0})
\]
and we claim that, in order to solve it, it is sufficient to prove that the quadratic error term decays in the iterative scheme. Throughout this section, we let $\left\|\cdot\right\|_{1}$ denote the norm on the Banach space $X_{1}$ and $\left\|\cdot\right\|_{2}$ denote the norm on the Banach space $X_{2}$ (to be defined in the sequel, based on the form of the local elliptic estimates we have for critical points of the functional $\mathcal{G}$). Their explicit expression is given in equation \eqref{norm1} and \eqref{norm2}, respectively.
\begin{proposition} \label{pro:quadratic} Given any $\lambda>0$, there exists $r_0>0$ sufficiently small so that
if $\|\left(f_1,V_{1}\right)\|_1<r_0$ and $\|\left(f_2,V_{2}\right)\|_1<r_0$ and we let $\left(h_1,\omega_{1}\right)=S(f_1,V_{1})$, $\left(h_2,\omega_{2}\right)=S(f_2,V_{2})$ then we have
\[ \|Q_{(g,\pi)}\left[h_1,\omega_{1}\right]-Q_{(g,\pi)}\left[h_2,\omega_{2}\right]\|_1\leq \lambda \|\left(h_1,\omega_{1}\right)-\left(h_2,\omega_{2}\right)\|_2.
\]
\end{proposition}
Once this is proven, the conclusion (in terms of existence, boundary regularity of the gluing and decay at infinity) follows at once:
\begin{theorem} \label{picard} Given $(f,V)\in X_1$ sufficiently small, there is a small
$(h,\omega)\in X_2$ satisfying
\[ d\Phi_{(g_{0},\pi_{0})}[h,\omega]+Q_{(g_{0},\pi_{0})}[h,\omega]=(f,V).
\]
\end{theorem}
\begin{proof} Assume that $\|\left(f,V\right)\|_{1}<\delta_0$ (with $\delta_{0}$ a small constant to be fixed later in the proof), let $h_0=0, \omega_{0}=0$ and $f_0=0, V_{0}=0$, and we
inductively construct sequences $\left(f_i, V_{i}\right)$ and $(h_i,\omega_{i})$ for $i\geq 1$ such that
\[ d\Phi_{(g,\pi)}\left[h_i,\omega_{i}\right]=\left(f_i, V_{i}\right)\ \mbox{where}\ \left(f_i,V_{i}\right)=-Q_{(g,\pi)}\left[h_{i-1},\omega_{i-1}\right]+\left(f, V\right).
\]
For $i\geq 1$ we have
\[ d\Phi_{(g,\pi)}([h_{i+1},\omega_{i+1}]-[h_i,\omega_{i}])=(f_{i+1},V_{i+1})-(f_{i},V_{i})=Q_{(g,\pi)}[h_{i-1},\omega_{i-1}]-Q_{(g,\pi)}[h_i,\omega_{i}],
\]
and so by Proposition \ref{pro:quadratic}
\[\begin{split} \|(f_{i+1},V_{i+1})-(f_i,V_{i})\|_1=\|Q_{(g,\pi)}[h_i,\omega_{i}]-Q_{(g,\pi)}[h_{i-1},\omega_{i-1}]\|_1\leq \lambda\|(h_i,\omega_{i})-(h_{i-1},\omega_{i-1})\|_2 \\ \leq C\lambda\|(f_i,V_{i})-(f_{i-1},V_{i-1})\|_1
\end{split}
\]
where $\lambda$ can be chosen as small as we wish and $C$ is the continuity constant of the solution operator $S$. Let then $r_0$ be small enough so that
$C\lambda<1/2$ in Proposition \ref{pro:quadratic}. We may then iterate this scheme provided that
$\|(f_i,V_{i})\|_1\leq r_0$ for $i=1,\ldots, k$ and in that case we obtain
\[ \|(f_{k+1},V_{k+1})-(f_k,V_{k})\|_1\leq 2^{-k}\|(f_1,V_{1})-(f_0,V_{0})\|_1=2^{-k}\|(f,V)\|_1< 2^{-k}\delta_0.
\]
From the triangle inequality we then have for any $k$
\[ \|(f_{k+1},V_{k+1})-(f,V)\|_1\leq \sum_{i=1}^{k}2^{-i}\delta_0<2\delta_0,
\]
so if we choose $\delta_0=r_0/4$ we have
\[ \|(f_{k+1},V_{k+1})\|_1\leq \|(f_{k+1},V_{k+1})-(f,V)\|_1+\|(f,V)\|_1<3\delta_0<r_0
\]
for each $k$. We can then iterate indefinitely and the sequence $\{(f_i,V_{i})\}$ is Cauchy
as is $\{(h_i,\omega_{i})\}$ since $S$ is a bounded operator. As a consequence, the sequence $\{(h_i,\omega_{i})\}$ converges in $X_{2}$ to a limit $(h,\omega)$ which satisfies
the equation $d\Phi_{(g,\pi)}[h,\omega]+Q_{(g,\pi)}[h,\omega]=(f,V)$. This completes the proof.
\end{proof}
\begin{remark}\label{rem:small}
We shall explicitly observe that given $(\check{g},\check{k})$ as in the statement of Theorem \ref{thm:main} and performed the rough patch construction as described in Subsection \ref{subs:doubly}, then
\[
\lim_{|a|\to\infty}\left\|(\Phi^{(1)}(g,\pi),\Phi^{(2)}(g,\pi))\right\|_1=0
\]
provided the angular cut-off function has sufficiently rapid decay at the boundary of the gluing region $\partial\Omega$. This implies that Theorem \ref{picard} can be legitimately applied (namely: the iteration scheme can indeed be started) and, furthermore, this ensures that the corresponding solution of the non-linear problem $(\hat{g},\hat{k})$ will have small norm in the Banach space $X_2$, in fact as small as we wish provided we pick $|a|$ large enough.
\end{remark}
Therefore, the rest of this section is devoted to the proof of Proposition \ref{pro:quadratic}.
\subsection{Integral estimates}
In this section, we take care of the Sobolev part of the norms defining the Banach spaces $X_{1}$ and $X_{2}$. Given data $(f,V)\in X_{1}$, recall that we have let $(h,\omega)=S(f,V)\in X_{2}$ be the solution of the linearized constraints defined by Proposition \ref{pro:existence}.
\begin{lem}\label{lem:zero} The following bound holds:
\[ \left\|\left(h,\omega\right)\right\|_{\mathcal{M}_{0,-p,\rho^{-1}}\times \mathcal{S}_{0,-p-1,\rho^{-1}}}\leq C\left\|\left(f,V\right)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}. \]
\end{lem}
\begin{proof}
Let us recall, from Subsection \ref{subs:varframe}, that given the Euler-Lagrange equation of the functional $\mathcal{G}$ the tensors $h$ and $\omega$ have been defined by means of the equations
\[ h=r^{n-2p}\rho \left(d\Phi^{\ast}_{(g,\pi)}\right)^{(1)}[\tilde{u},\tilde{Z}], \ \ \omega=r^{n-2p-2}\rho \left(d\Phi^{\ast}_{(g,\pi)}\right)^{(2)}[\tilde{u},\tilde{Z}]
\]
for $(\tilde{u},\tilde{Z})$ the unique minimizer of $\mathcal{G}$ over the functional space $\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{X}_{1,-n+p+2,\rho}$. As a result, one has $\mathcal{G}(\tilde{u},\tilde{Z})\leq\mathcal{G}(0,0)=0$ which means
\[\left\|d\Phi_{(g,\pi)}^{\ast}[\tilde{u},\tilde{Z}]\right\|^{2}_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}\leq 2\left\|(\tilde{u},\tilde{Z})\right\|_{\mathcal{H}_{0,-n+p+2,\rho}\times \mathcal{X}_{0,-n+p+2,\rho}}\left\|\left(f,V\right)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}.
\]
It follows that, thanks to the basic estimate (Proposition \ref{pro:basic}) we get
\[\left\|d\Phi^{\ast}_{(g,\pi)}[\tilde{u},\tilde{Z}]\right\|_{\mathcal{M}_{0,-n+p,\rho}\times \mathcal{S}_{0,-n+p+1,\rho}}\leq C\left\|\left(f,V\right)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}
\]
which is equivalent to
\[ \left\|\left(h,\omega\right)\right\|_{\mathcal{M}_{0,-p,\rho^{-1}}\times \mathcal{S}_{0,-p-1,\rho^{-1}}}\leq C\left\|\left(f,V\right)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{X}_{0,-p-2,\rho^{-1}}}
\]
that is what we had to prove.
\end{proof}
\subsection{The weighted constraints system}
For the Schauder estimates, it is necessary to compute the partial differential equations solved by $(\tilde{u},\tilde{Z})$. Most importantly, we need to prove its ellipticity and determine the rate of decay of its coefficients.
By making use of the very definitions of the derivative maps $d\Phi$ and $d\Phi^{\ast}$, via tedious but elementary computations one can check that the differential system solved by $(\tilde{u},\tilde{Z})$ takes the form
\begin{displaymath}
\begin{cases} \left[T_{1}\tilde{u}+\sum_{0\leq\left|\beta\right|\leq 3} a^{(1)}_{\beta}\partial^{\beta}\tilde{u}\right]+C^{s}_{(-p-1)}\ast\left[\sum_{0\leq\beta\leq 3}c^{(1)}_{\beta}\partial^{\beta}\tilde{Z}\right]=r^{2p-n}\phi^{-2N}f \\
\\
\left[T_{2}\tilde{Z}+\sum_{0\leq\left|\beta\right|\leq 1} c^{(2)}_{\beta}\partial^{\beta}\tilde{Z}\right]+C^{s}_{(-p-1)}\ast C^{r}_{(2)}\ast\left[\sum_{0\leq\beta\leq 3}a^{(2)}_{\beta}\partial^{\beta}\tilde{u}\right]=-2r^{2p+2-n}\phi^{-2N}V
\end{cases}\end{displaymath}
with
\[ T_{1}=\Delta\left(\Delta \tilde{u}\right), \ \ |a^{(1)}_{\beta}|\lesssim r^{\left|\beta\right|-4}\phi^{-2\vee \left|\beta\right|-4}, \ \ |a^{(2)}_{\beta}|\lesssim r^{\left|\beta\right|-3}\phi^{-2 \vee \left|\beta\right|-3}
\]
and
\[ T_{2}=\Delta \tilde{Z}+ Div(\nabla \tilde{Z})+C^{s}_{(-2p)}\ast\partial^{2}\tilde{Z}, \ \ |c^{(1)}_{\beta}|\lesssim r^{\left|\beta\right|-3}\phi^{-2 \vee \left|\beta\right|-3},\ \ |c^{(2)}_{\beta}|\lesssim r^{\left|\beta\right|-2}\phi^{\left|\beta\right|-2}.
\]
We stress that in the previous equations all differential operators are Euclidean, namely referred to the flat background metric. We shall also remind the reader that the symbols $C^{s}_{(-q)}$ and $C^{r}_{(-q)}$ have been defined in Subsection \ref{subs:doubly}.
By letting $u=r\tilde{u}, \ Z=\tilde{Z}$ we can rewrite the previous equations in the final form of the elliptic system for $(u,Z)$ that follows:
\begin{equation}\label{system}
\begin{cases}
\left[T_{1}u+\sum_{0\leq\left|\beta\right|\leq 3} a^{(1)}_{\beta}\partial^{\beta}u\right]+C^{s}_{(-p)}\ast\left[\sum_{0\leq\beta\leq 3}c^{(1)}_{\beta}\partial^{\beta}Z\right]=r^{2p+1-n}\phi^{-2N}f \\
\\
\left[T_{2}Z+\sum_{0\leq\left|\beta\right|\leq 1} c^{(2)}_{\beta}\partial^{\beta}Z\right]+C^{s}_{(-p)}\ast\left[\sum_{0\leq\beta\leq 3}a^{(2)}_{\beta}\partial^{\beta}u\right]=-2r^{2p+2-n}\phi^{-2N}V
\end{cases}
\end{equation}
where the coeffiecients are not necessarily the same as above but satisfy all of the same decay estimates in $(r,\phi)$.
\subsection{Douglis-Nirenberg ellipticity}
In order to proceed further, we need to prove H\"older estimates on the solution $(h,\omega)$. We make use of a specific result for inhomogeneous systems, due to Douglis-Nirenberg \cite{DN55}, which we briefly recall here for the convenience of the reader. To that aim, we first need to introduce some notation.
Let us consider a system of linear partial differential equations of the form
\begin{equation}\label{diffsys}
L_{i}w=\sum_{j=1}^{N}l_{ij}w_{j}=f_{i}, \ \ i=1,\ldots, N
\end{equation}
where for any $j=1,\dots, N$ we have that $w_{j}$ is a function of $n$ variables $(x_{1}, \ldots, x_{n})$ with $x\in\Gamma$ some regular domain of the Euclidean space. Let us assume that each differential operator $l_{ij}$ can be expressed as a polynomial in $\frac{\partial}{\partial x_{1}},\ldots, \frac{\partial}{\partial x_{n}}$ with sufficiently smooth coefficients.
Moreover, let us suppose that there exist $2N$ integers $\left\{s_{1},\ldots, s_{N}, t_{1},\ldots, t_{N}\right\}$ so that $l_{ij}$ has order \textsl{less or equal} than $s_{i}+t_{j}$ and let $l_{ij}'$ be the sum of the terms of $l_{ij}$ having order \textsl{exactly equal} to $s_{i}+t_{j}$ for any choice of our indices. Of course, it is not the case that such numbers always exist as one is supposed to find \textsl{integer} solutions to a linear system of $N^{2}$ equations in $2N$ unknowns: however, when this does happen, the determinant of the characteristic matrix of \eqref{diffsys}, namely $\left(l'_{ij}(x,\xi)\right)_{1\leq i,j\leq n}$ is an \textsl{homogeneous} polynomial $P(x,\xi)$ of degree $m=\sum_{k=1}^{N}(s_{k}+t_{k})$ in $\xi_{1},\ldots,\xi_{n}\in\mathbb{R}$. Indeed, each summand in $P(x,\xi)$ will have degree of the form $\sum_{k=1}^{N}(s_{k}+t_{\sigma(k)})$ for some $\sigma\in\mathcal{S}_{N}$, the symmetric group on $N$ elements, and of course
\[\sum_{k=1}^{N}(s_{k}+t_{\sigma\left(k\right)})=\sum_{k=1}^{N}s_{k}+\sum_{k=1}^{N}t_{\sigma(k)}=\sum_{k=1}^{N}(s_{k}+t_{k})=m.
\]
We will say that the system \eqref{diffsys} is \textsl{elliptic} (according to Douglis-Nirenberg) if there exist $s_{1},\ldots, s_{N},t_{1},\ldots,t_{N}\in\mathbb{Z}$ so that, at every point $x$ the determinant $P(x,\xi)$ does not vanish for every $\xi\neq 0$.
Obviously, when we deal with an elliptic system as above, we can always reduce to the case when
\[ s_{i}\leq 0, \ i=1,\ldots, N \ \ \ \max_{i}s_{i}=0 \ \ \ t_{j}\geq 0, \ j=1,\ldots, N
\]
which is motivated by the form of the Schauder estimates for a single (scalar) PDE, as will be apparent from the statement below.
Correspondingly, let us set
\[ \min_{i}s_{i}=-s, \ \ \ \max_{j}t_{j}=t.
\]
Now, assume the domain $\Gamma$ is bounded and let $d:\Gamma\to\mathbb{R}$ be the distance function from the boundary $\partial\Gamma$: for every $k\in\mathbb{Z}_{\geq 0}$ and $l\in\mathbb{R}$ we consider the weighted H\"older norm
\[ \left\|u\right\|^{\left(l\right)}_{k,\alpha}=\sum_{i=0}^{k}\sup_{x\in\Gamma}d(x)^{-l+i}\left|\partial^{i}u(x)\right|+\sup_{x\in\Gamma}d(x)^{-l+k+\alpha}\left[\partial^{k}u\right]_{\alpha}
\]
and let $\mathcal{C}^{k,\alpha}_{l}(\Gamma;\mathbb{R})$ be the Banach space which is gotten by completing the space of restrictions of elements in $\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R})$ with respect to such norm.
In order to state the interior regularity theorem of Douglis-Nirenberg, we need to give the following:
\
\textsl{Hypothesis (\textbf{H}):} let us write $l_{ij}(x,\partial)=\sum_{|\beta|=0}^{s_{i}+t_{j}}a_{ij,\beta}(x)\partial^{\beta}$ where the sum is understood to be first taken over all terms of $l_{ij}$ of order equal to $|\beta|$. For some constant $\alpha\in\left(0,1\right)$, a fixed positive constant $K$ and all the indices $i,j\in\left\{1,\ldots, n\right\}$ we require that:
\begin{enumerate}
\item{the coefficients $a_{ij,\beta}$ belong to the space $\mathcal{C}^{-s_{i},\alpha}_{-s_{i}-t_{j}+|\beta|}$ and
\[ \sup_{i,j,\beta}\left\|a_{ij,\beta}\right\|_{-s_{i},\alpha}^{\left(-s_{i}-t_{j}+|\beta|\right)}\leq K;
\]}
\item{the inhomogeneous term $f_{i}$ belongs to the space $\mathcal{C}^{-s_{i},\alpha}_{-s_{i}-t}$;}
\item{the characteristic determinant satisfies
\[ P(x,\xi)\geq K^{-1}\left(\sum_{i=1}^{n}\xi_{i}^{2}\right)^{m/2}.
\]
}
\end{enumerate}
Let us explicitly remark that the notation we are using here for weighted H\"older spaces is different from that in \cite{DN55} and the two are patently incompatible.
\begin{theorem}\label{regsys}(see \cite{DN55}, Theorem 1) Let $u$ be a solution of the system \eqref{diffsys} under the assumption (\textbf{H}). Assume that $u_{j}\in \mathcal{C}^{0,0}_{-t+t_{j}}$ and that $u_{j}$ has H\"older continuous derivatives up to order $t_{j}$ in $\Gamma$ for each value of the index $j=1,\ldots, N$. Then $u_{j}\in \mathcal{C}^{t_{j},\alpha}_{-t+t_{j}}$ and
\[ \left\|u_{j}\right\|_{t_{j},\alpha}^{\left(-t+t_{j}\right)}\leq C\left(\sum_{i=1}^{N}\left\|u_{i}\right\|_{0,0}^{(-t+t_{i})}+\sum_{i=1}^{N}\left\|f_{i}\right\|^{(-s_{i}-t)}_{-s_{i},\alpha}\right)
\]
for some constant $C=C(K,n,N,s_{1},\ldots, t_{N},\alpha)$.
\end{theorem}
Let us now discuss the applicability of this result to our problem, namely to the system \eqref{system}.
If $\textrm{dim}(M)=n$ one first needs to find $2n+2$ integers $s_{1},\ldots,s_{n+1}, t_{1},\ldots, t_{n+1}$ satisfying the algebraic system
\[\begin{cases}
s_{1}+t_{1}=4 \\
s_{i}+t_{1}=3 & i>1 \\
s_{1}+t_{j}=3 & j>1 \\
s_{i}+t_{j}=2 & i,j>1 \\
\end{cases}
\]
and we will pick
\[ s_{1}=0, \ \ s_{i}=-1, \ \ t_{1}=4, \ \ t_{j}=3 \ \ \ \ \ \ 2\leq i,j\leq n+1.
\]
\begin{lem}\label{lem:ellsys}
There exists $a_{\infty,N}\in\mathbb{R}$ such that for any $a\in\mathbb{R}^{n}$ such that $\left|a\right|\geq a_{\infty,N}$ the system \eqref{system} is elliptic, in the sense of Douglis-Nirenberg, on the domain $\Omega=\Omega(a)$.
\end{lem}
\begin{proof}
By the very definition of determinant, we can write the characteristic determinant for the Einstein constraint \eqref{system} as
\[
P(x,\xi)\geq \tilde{P}(x,\xi)-Cs(x)^{-p}\left|\xi\right|^{m}
\]
for $m=2n+4$ and $C$ a constant which only depends on the decay assumptions on $g-\delta$ and $\pi$, where we have set
\begin{equation*}
\tilde{P}(x,\xi)=\det
\begin{pmatrix}
|\xi|^4 & 0 & 0 & \cdots & 0 \\
0 & |\xi|^2 + \xi_1^2 & \xi_1 \xi_2 & \cdots & \xi_1 \xi_n \\
0 & \xi_1 \xi_2 & |\xi|^2 + \xi_2^2 & \cdots & \xi_2 \xi_n \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & \xi_1 \xi_n & \xi_2 \xi_n & \cdots & |\xi|^2 + \xi_n^2 \\
\end{pmatrix}.
\end{equation*}
Let us notice that this would be the characteristic matrix of the Einstein system in the case when $(\tilde{u},\tilde{Z})$ were not coupled, namely in the time-symmetric case when $\pi=0$.
Now, the block structure of such matrix immediately implies that
\[ \tilde{P}(x,\xi)=\left|\xi\right|^{4}\det\left(A+bb^{t}\right)
\]
for
\begin{equation*}
A(\xi)=\left|\xi\right|^{2}
\begin{pmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
\end{pmatrix}, \ \
b=
\begin{pmatrix}
\xi_{1} \\
\xi_{2} \\
\vdots \\
\vdots \\
\xi_{n} \\
\end{pmatrix}.
\end{equation*}
It is a well-known fact in linear algebra that, given an $n\times n$ matrix $A$ and vectors $b_{1}, b_{2}\in\mathbb{R}^{n}$ then $\det(A+b_{1}b_{2}^{t})=\det(A)+b_{2}^{t}\textrm{adj}(A)b_{1}$ and thus, in our case this implies at once that in fact $\tilde{P}(x,\xi)=2\left|\xi\right|^{2n+4}$. Hence
\[ P(x,\xi)\geq \left(2-Cs(x)^{-p}\right)\left|\xi\right|^{m}, \ \ \ x\in \Omega \ \ \xi\in\mathbb{R}^{n}
\]
and the conclusion follows by picking $|a|$ large enough by virtue of inequality \eqref{eq:min}.
\end{proof}
Before proceeding further, let us make an important remark on our notations: \textsl{in the rest of this subsection, as well as in the next one we will adopt the notation $\mathcal{H}_{k,q,\rho}$ when referring to any type of tensor.}
\begin{lem}\label{pwsch}
For any $\alpha\in\left(0,1\right)$ the following H\"older bounds hold true for $(u,Z)$:
\[\left\|u\right\|_{4,\alpha}^{(0,0)}\leq C\left(\left\|\overline{u}\right\|_{0,0}^{(0,0)}+\left\|\overline{Z}\right\|_{0,0}^{(-1,-1)}+\left\|f\right\|^{\left(-2p-5+n, -4+2N\right)}_{0,\alpha}+\left\|V\right\|^{\left(-2p-5+n,-3+2N\right)}_{1,\alpha}\right)
\]
\[\left\|Z\right\|_{3,\alpha}^{(-1,-1)}\leq C\left(\left\|\overline{u}\right\|_{0,0}^{(0,0)}+\left\|\overline{Z}\right\|_{0,0}^{(-1,-1)}+\left\|f\right\|^{\left(-2p-5+n, -4+2N\right)}_{0,\alpha}+\left\|V\right\|^{\left(-2p-5+n,-3+2N\right)}_{1,\alpha}\right).
\]
as well as the following for the solution $(h,\omega)$ of the linearized problem:
\[\left\|h\right\|^{\left(-p,+N-n/2-2\right)}_{2,\alpha}\leq C\left(\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}+\left\|f\right\|^{\left(-p-2,N-n/2-4\right)}_{0,\alpha}+\left\|V\right\|^{(-p-2,N-n/2-3)}_{1,\alpha}\right)
\]
\[\left\|\omega\right\|^{\left(-p-1,N-n/2-2\right)}_{2,\alpha}\leq C\left(\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}+\left\|f\right\|^{\left(-p-2,N-n/2-4\right)}_{0,\alpha}+\left\|V\right\|^{(-p-2,N-n/2-3)}_{1,\alpha}\right).
\]
\end{lem}
Motivated by the previous Lemma, we are now in position to actually define the Banach spaces $X_{1}, X_{2}$. Given $\Omega$, as usual, the (regularized) region between the two cones we consider $X_{i}, \ i=1,2$ to be the closure of the set of smooth $(\textrm{functions}, \textrm{symmetric} \ (0,2)-\textrm{tensors})$ in $\Omega\hookrightarrow\mathbb{R}^{n}$ for which the norm $\left\|\cdot\right\|_{i}$ is finite, where
\begin{equation}\label{norm1} \left\|(f,V)\right\|_{1}=\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}+\left\|f\right\|^{\left(-p-2,N-n/2-4\right)}_{0,\alpha}+\left\|V\right\|^{(-p-2,N-n/2-3)}_{1,\alpha}
\end{equation}
and
\begin{equation}\label{norm2} \left\|(h,\omega)\right\|_{2}=\left\|(h,\omega)\right\|_{\mathcal{H}_{0,-p,\rho^{-1}}\times \mathcal{H}_{0,-p-1,\rho^{-1}}}+\left\|h\right\|^{\left(-p,N-n/2-2\right)}_{2,\alpha}+\left\|\omega\right\|^{\left(-p-1,N-n/2-2\right)}_{2,\alpha}.
\end{equation}
\begin{proof}
Thanks to Lemma \ref{lem:ellsys} and the decay of the coefficients of our system, we are in position to apply Theorem \ref{regsys} and get
\begin{align*}
& \sum_{i=0}^{4}d(x)^{i}\left|\partial^{i}u(x)\right|+d(x)^{4+\alpha}\left[\partial^{4}u\right]_{\alpha,B_{d(x)/2}\left(x\right)}+ \sum_{i=0}^{3}d(x)^{i+1}\left|\partial^{i}Z(x)\right|+d(x)^{4+\alpha}\left[\partial^{3}Z\right]_{\alpha,B_{d(x)/2}\left(x\right)} \\
& \leq C\left[\overline{u}(x)+d(x)\overline{Z}(x)\right]+Cd(x)^{4}r(x)^{2p+1-n}\phi(x)^{-2N}\left\{\sup_{B_{3d(x)/4}\left(x\right)}\left|f\right|+d(x)^{\alpha}\left[f\right]_{\alpha, B_{3d(x)/4}\left(x\right)}\right\} \\
& + Cd(x)^{3}r(x)^{2p+2-n}\phi(x)^{-2N}\left\{\sup_{B_{3d(x)/4}\left(x\right)}\left|V\right|+d(x)\sup_{B_{3d(x)/4}\left(x\right)}\left|\partial V\right|+d(x)^{1+\alpha}\left[\partial V\right]_{\alpha, B_{3d(x)/4}\left(x\right)}\right\}
\end{align*}
which immediately implies the first two claimed inequalities.
In order to prove the other estimates, one needs the following upper bounds for $\overline{u}$ and $\overline{Z}$.
For the former:
\begin{align*}
\left|\overline{u}(x)\right|^{2} & \leq Cd(x)^{-n}\int_{B_{d(x)/2}(x)}\left|u(y)\right|^{2}\,d\mathscr{L}^n(y) \leq Cd(x)^{-n}r(x)^{2}\int_{B_{d(x)/2}(x)}\left|\tilde{u}(y)\right|^{2}\,d\mathscr{L}^n(y) \\
& \leq Cr(x)^{2(p+3-n)}\phi(x)^{-2N-n}\left\|\tilde{u}\right\|^{2}_{\mathcal{H}_{2,p+2-n,\rho}} \\
& \leq Cr(x)^{2(p+3-n)}\phi(x)^{-2N-n}\left\|(\tilde{u},\tilde{Z})\right\|^{2}_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{H}_{1,-n+p+2,\rho}} \\
& \leq C r(x)^{2(p+3-n)}\phi(x)^{-2N-n}\left\|d\Phi^{\ast}_{\left(g,\pi\right)}(\tilde{u},\tilde{Z})\right\|^{2}_{\mathcal{H}_{0,-n+p,\rho}\times \mathcal{H}_{0,-n+p+1,\rho}} \\
& \leq C r(x)^{2(p+3-n)}\phi(x)^{-2N-n}\left\|\left(f,V\right)\right\|^{2}_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}
\end{align*}
where we have used both the basic estimate, Proposition \ref{pro:basic}, and Lemma \ref{lem:zero}.
Similarly, for the latter:
\begin{align*}
\left|d(x)\overline{Z}(x)\right|^{2} & \leq Cr(x)^{2-n}\phi(x)^{2-n}\int_{B_{d(x)/2}(x)}\left|Z(y)\right|^{2}\,d\mathscr{L}^n(y) \leq Cr(x)^{2(p+3-n)}\phi(x)^{-2N-n+2}\left\|\tilde{Z}\right\|^{2}_{\mathcal{H}_{1,-n+p+2,\rho}} \\
& \leq Cr(x)^{2(p+3-n)}\phi(x)^{-2N-n+2}\left\|(\tilde{u},\tilde{Z})\right\|^{2}_{\mathcal{H}_{2,-n+p+2,\rho}\times \mathcal{H}_{1,-n+p+2,\rho}} \\
& \leq C r(x)^{2(p+3-n)}\phi(x)^{-2N-n+2}\left\|d\Phi^{\ast}_{\left(g,\pi\right)}(\tilde{u},\tilde{Z})\right\|^{2}_{\mathcal{H}_{0,-n+p,\rho}\times \mathcal{H}_{0,-n+p+1,\rho}} \\
& \leq Cr(x)^{2(p+3-n)}\phi(x)^{-2N-n+2}\left\|\left(f,V\right)\right\|^{2}_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}
\end{align*}
so that by taking square roots one gets:
\[\left\|\overline{u}\right\|_{0,0}^{(0,0)}+\left\|\overline{Z}\right\|_{0,0}^{(-1,-1)}\leq C r(x)^{p+3-n}\phi(x)^{-N-n/2}\left\|\left(f,V\right)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}.
\]
At that stage, one can exploit the H\"older estimates above, together with the very definition of the tensors $h$ and $\omega$ in terms of $(u,Z)$ to complete the proof.
For instance, one has for the zeroth order estimate on $h$:
\begin{align*}
\left|h(x)\right| & \leq Cr(x)^{n-2p-1}\rho(x) d(x)^{-2}\left[d(x)^{2}\left|\partial^{2}u(x)\right|+d(x)\left|\partial u(x)\right|+\left|u(x)\right|\right] \\
& +Cr(x)^{n-2p-1}\rho(x) s(x)^{-p}d(x)^{-2}\left[d(x)^{2}\left|\partial Z(x)\right|+d(x)\left|Z(x)\right|\right] \\
& \leq C r(x)^{n-2p-3}\phi(x)^{2N-2} \times \\
& \left[r^{p+3-n}\phi^{-N-n/2}\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}+\left\|f\right\|^{(-2p-5+n,-4+2N)}_{0,\alpha}+\left\|V\right\|^{(-2p-5+n,-3+2N)}_{1,\alpha}\right] \\
& \leq Cr(x)^{-p}\phi(x)^{N-n/2-2}\left[\left\|(f,V)\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}+\left\|f\right\|^{(-p-2, N-n/2-4)}_{0,\alpha}+\left\|V\right\|^{(-p-2, N-n/2-3)}_{0,\alpha}\right].
\end{align*}
The other pointwise and (concerning the second derivatives) H\"older estimates for $h$ and $\omega$ are completely analogous and we omit the elementary details.
\end{proof}
\subsection{Convergence of the iteration}
We now proceed with the proof of Proposition \ref{pro:quadratic}.
\begin{proof} In order for us to obtain estimates for the difference of the quadratic terms, $Q_{(g,\pi)}[h_{1},\omega_{1}]-Q_{(g,\pi)}[h_{2},\omega_{2}]$ it is convenient to find an exact representation formula for such difference. Recalling the expression for the constraint equations, we need to consider
\[ Q_{(g,\pi)}[h,\omega]=\left(\mathscr{H}(\overline{g},\overline{\pi})-\mathscr{H}(g,\pi)-d\Phi^{(1)}_{(g,\pi)}[h,\omega], Div_{\overline{g}}(\overline{\pi})-Div_{g}(\pi)-d\Phi^{(2)}_{(g,\pi)}[h,\omega]\right)
\]
where $\overline{g}_{ij}=g_{ij}+h_{ij}$ and $\overline{\pi}^{ab}=\pi^{ab}+\omega^{ab}$.
Throughout this subsection, we shall adopt the following simplified notation: $R$ (resp. $\overline{R}$) for the scalar curvature $R_g$ of $g$ (resp $R_{\overline{g}}$ of $\overline{g}$) and $R_{ij}$ (resp. $\overline{R}_{ij}$) for the expression in local coordinates of the Ricci curvature $Ric_g$ of $g$ (resp. $Ric_{\overline{g}}$ of $\overline{g}$).
Concerning the first component, we have to study the nonlinear part of the difference
\[\overline{R}-R+\frac{1}{n-1}\left[\left(Tr_{\overline{g}}\overline{\pi}\right)^{2}-\left(Tr_{g}\pi\right)^{2}\right]-\left[\left|\overline{\pi}\right|_{\overline{g}}^{2}-\left|\pi\right|^{2}_{g}\right]
\]
so let us work out each of the three summands separately.
We start with the scalar curvature term: first from the formula for the Ricci curvature we have
\[ \overline{R}_{ij}-R_{ij}=D^k_{ij;k}-D^k_{ki;j}+D^k_{kl}D^l_{ij}
-D^k_{jl}D^l_{ki}
\]
where $D=\overline{\Gamma}-\Gamma$ is the difference of the Levi-Civita connections and
the semi-colon denotes the covariant derivative with respect to $g$.
If we write $\overline{g}=g+h$, then we have
\[ D^k_{ij}=\frac{1}{2}\overline{g}^{kl}(h_{il;j}+h_{jl;i}-h_{ij;l}).
\]
Now taking traces with respect to $\overline{g}$ we have
\[ \overline{R}-\overline{g}^{ij}R_{ij}=\overline{g}^{ij}(\overline{g}^{kl}h_{il;j})_{;k}-\overline{g}^{ij}(\overline{g}^{kl}h_{kl;i})_{;j}+q(\overline{g},h)
\]
where we use $q(\overline{g},h)$ to denote a quadratic polynomial in the first covariant derivatives
of $h$ with coefficients depending on $\overline{g}$. Since $g$ is parallel (indeed, it is the background metric we are covariant derivatives with respect of) we have
$\overline{g}_{ij;k}=h_{ij;k}$, and $\overline{g}^{ij}_{;k}=-\overline{g}^{il}\overline{g}^{jm}h_{lm;k}$. We may thus rewrite
the expression
\[ \overline{R}-R=\overline{g}^{ij}\overline{g}^{kl}h_{il;jk}-\overline{g}^{ij}\overline{g}^{kl}h_{kl;ij}+(\overline{g}^{ij}-g^{ij})R_{ij}+
q(\overline{g},h)
\]
where we have modified $q(\overline{g},h)$. We note that the linear term $L(h)$ of $\overline{R}-R$ is given by
\[ L(h)=[g^{ij}g^{kl}(h_{il;jk}-h_{kl;ij})-g^{ik}g^{jl}h_{kl}]R_{ij}.
\]
Therefore the quadratic part is
\begin{equation}\label{Q-formula1} \overline{R}-R-L(h)=(\overline{g}^{ij}\overline{g}^{kl}-g^{ij}g^{kl})(h_{il;jk}-h_{kl;ij})+T+q(\overline{g},h)
\end{equation}
where the term $T$ is given by
\[ T=(\overline{g}^{ij}-g^{ij})R_{ij}+g^{ik}g^{jl}h_{kl}R_{ij}.
\]
We can rewrite $T$ by setting $g_t=g+th$ for $t\in [0,1]$, and writing
\[ \overline{g}^{ij}-g^{ij}=\int_0^1\frac{d}{dt}(g_t^{ij})\ d\mathscr{L}^1=-\left(\int_0^1 g_t^{ik}g_t^{jl}\ d\mathscr{L}^1\right)h_{kl}.
\]
Hence we have
\begin{equation}\label{T-formula} T=-\int_0^1(g_t^{ik}g_t^{jl}-g^{ik}g^{jl})\ d\mathscr{L}^1\ h_{kl}R_{ij}.
\end{equation}
For the trace term, one can write
\[(Tr_{\overline{g}}\overline{\pi})^{2}-(Tr_{g}\pi)^{2}=\left(\overline{g}_{ij}\overline{\pi}^{ij}+g_{ij}\pi^{ij}\right)\left(\overline{g}_{kl}\overline{\pi}^{kl}-g_{kl}\pi^{kl}\right)
\]
and by the definitions of $\overline{g}$ and $\overline{\pi}$
\[\overline{g}_{kl}\overline{\pi}^{kl}-g_{kl}\pi^{kl}=g_{kl}\omega^{kl}+h_{kl}\pi^{kl}+h_{kl}\omega^{kl} , \ \ \overline{g}_{ij}\overline{\pi}^{ij}+g_{ij}\pi^{ij}=2g_{ij}\pi^{ij}+g_{ij}\omega^{ij}+h_{ij}\pi^{ij}+h_{ij}\omega^{ij}
\]
so the quadratic part is given by
\begin{equation}\label{Q-formula2}
\left(2g_{ij}\pi^{ij}\right)\left(h_{kl}\omega^{kl}\right)+\left(g_{ij}\omega^{ij}+h_{ij}\pi^{ij}+h_{ij}\omega^{ij}\right)\left(g_{kl}\omega^{kl}+h_{kl}\pi^{kl}+h_{kl}\omega^{kl}\right).
\end{equation}
The squared norm terms can be expanded as follows:
\[\left|\overline{\pi}\right|^{2}_{g}-\left|\pi\right|^{2}_{g}=\overline{\pi}^{ik}\overline{\pi}^{jl}\overline{g}_{ij}\overline{g}_{kl}-\pi^{ik}\pi^{jl}g_{ij}g_{kl}=(\pi^{ik}+\omega^{ik})\left(\pi^{jl}+\omega^{jl}\right)\left(g_{ij}+h_{ij}\right)\left(g_{kl}+h_{kl}\right)-\pi^{ik}\pi^{jl}g_{ij}g_{kl}
\]
so it is readily checked that the quadratic part is given by
\begin{equation}\label{Q-formula3}
\pi^{ik}\pi^{jl}h_{ij}h_{kl}+\omega^{ik}\omega^{jl}g_{ij}g_{kl}+\left(\pi^{ik}\omega^{jl}+\pi^{jl}\omega^{ik}+\omega^{ik}\omega^{jl}\right)\left(g_{ij}h_{kl}+g_{kl}h_{ij}+h_{ij}h_{kl}\right).
\end{equation}
Now we observe that from \eqref{Q-formula1}, \eqref{T-formula}, \eqref{Q-formula2} and \eqref{Q-formula3} the term $Q^{(1)}_{(g,\pi)}([h,\omega])$ is a
sum of terms of the form
\[ E_{1}(h)(\partial^2 h),\ E_{2}(h)(h)(\Gamma^2+\partial\Gamma+\pi^{2}), \ E_3(h)(\partial h)(\Gamma) , \ E_{4}(h)(\partial h)(\partial h), \ E_{5}(h)(\omega)(\omega), \ E_{6}(h)(h)(\omega)(\pi)
\]
where $E_{1},E_{2},E_{3}, E_{4}, E_{5}, E_{6}$ are smooth coefficient systems depending on $h$ with $E_i(0)=0$ for $i=1,2,3$.
Now, let us concern ourselves with the second component of the constraints. Let us remark that $Div_{g}(\pi)$ is of course linear when the background metric is fixed $(g)$ and we let the differential operator $Div_{g}$ act on the symmetric tensor $\pi$, while $Div_{g}\pi$ is not linear as a function of the couple $(g,\pi)$ and this is the reason why the following computation is not trivial.
Indeed, since we need to deal with two different background metrics ($g$ and $\overline{g}$) we will start by unwinding the covariant derivatives:
\[ \left(Div_{g}\pi\right)^{l}=(\pi^{il}_{,i}+\Gamma^{i}_{ik}\pi^{kl}+\Gamma^{l}_{ik}\pi^{ik})
\]
and thus, for the difference we can write
\begin{align*}
\left(Div_{\overline{g}}\overline{\pi}\right)^{l} & -\left(Div_{g}\pi\right)^{l}=\overline{\pi}^{il}_{,i}-\pi^{il}_{,i}+\overline{\Gamma}^{i}_{ik}\overline{\pi}^{kl}-\Gamma^{i}_{ik}\pi^{kl}+\overline{\Gamma}^{l}_{ik}\overline{\pi}^{ki}-\Gamma^{l}_{ik}\pi^{ki} \\
& =(\overline{\pi}^{il}_{,i}-\pi^{il}_{i})+D^{i}_{ik}\overline{\pi}^{kl}+\Gamma^{i}_{ik}(\overline{\pi}^{kl}-\pi^{kl})+D^{l}_{ik}\overline{\pi}^{ki}+\Gamma^{l}_{ik}(\overline{\pi}^{ki}-\pi^{ki}) \\
& =\omega^{il}_{,i}+D^{i}_{ik}(\pi^{kl}+\omega^{kl})+\Gamma^{i}_{ik}\omega^{kl}+D^{l}_{ik}(\pi^{ki}+\omega^{ki})+\Gamma^{l}_{ik}\omega^{ki}.
\end{align*}
As a result, making use of the expression for the difference of Christoffel symbols that has been derived above, namely
\[ D_{ij}^{k}=\frac{1}{2}\left(g^{kl}-\int_{0}^{1}g_{t}^{ak}g_{t}^{bl}h_{ab}\,d\mathscr{L}^1\right)(h_{il;j}+h_{jl;i}-h_{ij;l})
\]
we can express the quadratic part of the second component as as finite sum of terms that belong to one of the following categories:
\[ E_{7}(h)(\partial h)(\pi), \ E_8(h)(h)(\Gamma\pi), \ E_{9}(h)(\partial h)(\omega), \ E_{10}(h)(h)(\omega)(\Gamma)
\]
where $E_{7}, E_{8}, E_9, E_{10}$ are smooth coefficient systems depending on ($g$ and) $h$, moreover $E_{7}(0)=0$ and $E_8(0)=0$.
These preliminaries being done, we can easily get pointwise upper bounds for the first and second components of $Q_{(g,\pi)}\left[h_{1},\omega_{1}\right]-Q_{(g,\pi)}\left[h_{2},\omega_{2}\right]$.
Concerning the first component, we will have (for small $[h_{1},\omega_{1}]$ and $[h_{2}, \omega_{2}]$ in the $X_{2}$-topology)
\begin{align*} &|Q^{(1)}_{(g,\pi)}[h_1,\omega_{1}]-Q^{(1)}_{(g,\pi)}[h_2,\omega_{2}]|\leq C(|h_1|+|h_2|)|\partial^2(h_1-h_2)| \\ &
+C(|\partial h_1|+|\partial h_2|+|\Gamma|(|h_1|+|h_2|))|\partial (h_1-h_2)|\\ &
+C(|\partial^2 h_1|+|\partial^2 h_2|+|\partial h_1|^2+|\partial h_2|^2+|\Gamma|(|\partial h_1|+|\partial h_2|)) h_1-h_2|\\ &
+C((\left|\partial\Gamma\right|+|\Gamma|^2+\left|\pi\right|^{2})(|h_1|+|h_2|)+\left|\pi\right|\left(\left|\omega_{1}\right|+\left|\omega_{2}\right|\right))) |h_1-h_2| \\
&+C\left(|\pi|\left(\left|h_{1}\right|+\left|h_{2}\right|\right)+\left(\left|\omega_{1}\right|+\left|\omega_{2}\right|\right)\right)\left|\omega_{1}-\omega_{2}\right|.
\end{align*}
Instead, for the second component, we have:
\begin{align*}
&|Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(2)}_{(g,\pi)}[h_2,\omega_{2}]|\leq C\left(\left|\pi\right|\left(\left|h_{1}\right|+\left|h_{2}\right|\right)+\left(\left|\omega_{1}\right|+\left|\omega_{2}\right|\right)\right)\left|\partial \left(h_{1}- h_{2}\right)\right| \\ &
+C((\left|\pi\right|+|\omega_{1}|+|\omega_{2}|)(\left|\partial h_{1}\right|+\left|\partial h_{2}\right|))+|\Gamma|(|\pi|(h_1|+|h_2|)+(|\omega_1|+|\omega_2|)))\left|h_{1}-h_{2}\right| \\
&+C(\left|\partial h_{1}\right|+\left|\partial h_{2}\right|+|\Gamma|(|h_1|+|h_2|))\left|\omega_{1}-\omega_{2}\right|.
\end{align*}
Since we are assuming that $\|(f_1, V_{1})\|_{1}< r_0$ and $\|(f_2, V_{2})\|_{1}<r_0$, our Schauder estimates (Lemma \ref{pwsch}) imply that for $0\leq j\leq 2$
\[ |\partial^j h_a|\leq Cr_0r^{-p-j}\phi^{N-n/2-2-j}\leq Cr_0r^{-p-j}\phi^{N-n/2-4}
\]
as well as
\[\left|\partial^{j} \omega_{a}\right|\leq Cr_{0}r^{-p-1-j}\phi^{N-n/2-2-j}\leq Cr_0r^{-p-1-j}\phi^{N-n/2-4}
\]
for $a=1,2$.
It follows that, due to our decay assumptions on the data $g-\delta, \pi$ one obtains the pointwise bounds
\[|Q^{(i)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(i)}_{(g,\pi)}[h_2,\omega_{2}]|\leq Cr_{0}r^{-2p-2}\phi^{2(N-n/2-4)}\left\|(h_{1}-h_{2}, \omega_{1}-\omega_{2})\right\|_{2}, \ \ x\in\Omega \ \ i=1,2
\]
both for the first and for the second component.
As a result, we have
\[ \int_\Omega |Q^{(i)}_{(g,\pi)}[h_1,\omega_{1}]-Q^{(i)}_{(g,\pi)}[h_2,\omega_{2}]|^2r^{-n+2p+4}\phi^{-2N}\ d\mathscr{L}^n \leq Cr_0^2\left\|(h_{1}-h_{2}, \omega_{1}-\omega_{2})\right\|^{2}_{2} \{\int_\Omega r^{-n-2p}\phi^{2N-2n-16} d\mathscr{L}^n\}.
\]
and thus, since for any $p>0$ and $N$ large enough (say $N>n+8$) the above integral is of course finite, one concludes
\[ \left\|Q_{(g,\pi)}[h_{1},\omega_{1}]-Q_{(g,\pi)}[h_{2},\omega_{2}]\right\|_{\mathcal{H}_{0,-p-2,\rho^{-1}}\times \mathcal{H}_{0,-p-2,\rho^{-1}}}\leq Cr_{0}\left\|(h_{1},\omega_{1})-\left(h_{2}, \omega_{2}\right)\right\|_2
\]
which is the first assertion we had to prove, based on the definition of the norm $\left\|\cdot\right\|_{1}$.
We can then proceed with the pointwise estimates for the H\"older seminorms of $Q^{(1)}_{(g,\pi)}$.
As a preliminary remark which will be used several times in the sequel of this proof, we observe that for two functions $u_1,u_2$ and
any ball $B\subseteq\Omega$ we have the bound on the H\"older coefficient
\begin{equation}\label{eq:leibhol} [u_1u_2]_{\alpha,B}\leq (\sup_{B}|u_1|)[u_2]_{\alpha,B}+(\sup_B|u_2|)[u_1]_{\alpha,B}.
\end{equation}
Furthermore, we shall systematically exploit the comparison inequalities \eqref{eq:equiv} and \eqref{eq:equiv2}.
Now given $x\in\Omega$ we let $B=B_{d(x)/2}(x)$, and we estimate the
first type of term
\[[E_1(h_1)(\partial^2 h_1)-E_{1}(h_2)\partial^2 h_2]_{\alpha,B}\leq C\sup_B (|h_1|+|h_2|)[\partial^2 (h_1
-h_2)]_{\alpha,B}+C\sup_B(|\partial^2 h_1|+|\partial^2 h_2|)[h_1-h_2]_{\alpha,B}
\]
\[+C\sup_B|\partial^{2}(h_1-h_2)|([h_1]_{\alpha, B}+[h_2]_{\alpha, B})+C\sup_B(|h_1-h_2|)([\partial^{2}h_1]_{\alpha, B}+[\partial^{2}h_2]_{\alpha, B}).
\]
Using the bounds we have on $h_1$ and $h_2$ we then have
\begin{align*} &[E_1(h_1)(\partial^2 h_1)-E_1(h_2)\partial^2 h_2]_{\alpha,B}\leq Cr_0r(x)^{-p}\phi(x)^{N-n/2-2}(
[\partial^2 (h_1-h_2)]_{\alpha,B}+r(x)^{-2}\phi(x)^{-2}[h_1-h_2]_{\alpha,B}\\
&+r(x)^{-\alpha}\phi^{-\alpha}\sup_B|\partial^{2}(h_1-h_2)|+r(x)^{-2-\alpha}\phi(x)^{-2-\alpha}\sup_B |h_1-h_2|).
\end{align*}
Multiplying by $r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}$ we have
\begin{align*} &r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}[E_1(h_1)(\partial^2 h_1)-E_1(h_2)\partial^2 h_2]_{\alpha,B}\leq Cr_0r(x)^{2+\alpha}\phi(x)^{2+\alpha}[\partial^2 (h_1-h_2)]_{\alpha,B}\\
&+Cr_{0}r(x)^\alpha\phi(x)^\alpha[h_1-h_2]_{\alpha,B}+Cr_{0}r(x)^{2}\phi(x)^{2}\sup_B|\partial^{2}(h_1-h_2)|+Cr_{0}\sup_B |h_1-h_2|\\
&\leq Cr_{0}r(x)^{-p}\phi(x)^{N-n/2-2}\left\|h_1-h_2\right\|^{(-p,N-n/2-2)}_{2,\alpha} \leq Cr_0\|(h_1,\omega_1)-(h_2,\omega_2)\|_2
\end{align*}
given our choice of $N$ (recall that we required $N>n+8$).
To handle the second term we estimate
\begin{align*} & [(E_2(h_1)(h_1)-E_2(h_2)(h_2))(\Gamma^2+\partial\Gamma+\pi^{2})]_{\alpha,B}\leq Cr(x)^{-p-2}\sup_B|h_1-h_2|([h_1]_{\alpha,B}+[h_2]_{\alpha,B})\\
& +Cr(x)^{-p-2}\sup_B(|h_1|+|h_2|)\{[h_1-h_2]_{\alpha,B}+r(x)^{-\alpha}\phi(x)^{-\alpha}\sup_B |h_1-h_2|\}
\end{align*}
where we have made use of the fact that for $p>0$ one has $2(p+1)>p+2$.
Using the bounds on $h_1$ and $h_2$ and multiplying by $r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}$ we get
\begin{align*} &r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}[(E_2(h_1)(h_1)-E_2(h_2)(h_2))(\Gamma^2+\partial\Gamma+\pi^{2})]_{\alpha,B}
\leq Cr_0r(x)^{-p+\alpha}\phi(x)^{2+\alpha}[h_1-h_2]_{\alpha,B}\\
&+Cr_{0}r(x)^{-p}\phi(x)^2\sup_B|h_1-h_2|\leq Cr_{0}r^{-2p}\phi^{N-n/2}\|h_1-h_2\|^{(p,-N+n/2+2)}_{2,\alpha}\leq Cr_0\|(h_1,\omega_1)-(h_2,\omega_2)\|_2.
\end{align*}
Similarly,
\begin{align*} & [E_3(h)(\partial h)(\Gamma)]_{\alpha,B}\leq Cr(x)^{-2p-1}\phi(x)^{N-n/2-3}([\partial(h_1-h_2)]_{\alpha,B}+r(x)^{-\alpha}\phi(x)^{-\alpha} \sup_{B}|\partial(h_1-h_2)| \\
& +r(x)^{-1}\phi(x)^{-1}[h_1-h_2]_{\alpha,B}+r(x)^{-1-\alpha}\phi(x)^{-1-\alpha}\sup_B|h_1-h_2|)
\end{align*}
hence
\begin{align*} & r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha} [E_3(h)(\partial h)(\Gamma)]_{\alpha,B}\leq Cr_0r(x)^{-p+1+\alpha}\phi(x)^{2+\alpha}([\partial(h_1-h_2)]_{\alpha,B} \\
& +r(x)^{-\alpha}\phi(x)^{-\alpha} \sup_{B}|\partial(h_1-h_2)| +r(x)^{-1}\phi(x)^{-1}[h_1-h_2]_{\alpha,B}+r(x)^{-1-\alpha}\phi(x)^{-1-\alpha}\sup_B|h_1-h_2|) \\
&\leq Cr_0 \|(h_1,\omega_1)-(h_2,\omega_2)\|_2.
\end{align*}
To estimate the fourth type of term we follow the same pattern
\begin{align*} &[E_4(h_1)(\partial h_1)(\partial h_1)-E_4(h_2)(\partial h_2)(\partial h_2)]_{\alpha,B}\leq Cr(x)^{-\alpha}\phi(x)^{-\alpha}\sup_B(|\partial h_1|+|\partial h_2|)\sup_{B}|\partial (h_1-h_2)| \\
&+C\left(\sup_B(|\partial h_1|+|\partial h_2|)[\partial (h_1-h_2)]_{\alpha,B}+\sup_B \left|\partial(h_{1}-h_{2})\right|\left([\partial h_{1}]_{\alpha, B}+[\partial h_{2}]_{\alpha, B}\right)\right) \\
&+C\left(\sup_B(|\partial h_1|^2+|\partial h_2|^2)[h_1-h_2]_{\alpha,B}+\sup_{B}|h_{1}-h_{2}|([(\partial h_{1})^{2}]_{\alpha,B}+[(\partial h_{2})^{2}]_{\alpha,B})\right)
\end{align*}
hence
\begin{align*} &r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}[E_4(h_1)(\partial h_1)(\partial h_1)-E_4(h_2)(\partial h_2)(\partial h_2)]_{\alpha,B}\\
&\leq Cr_{0}r(x)\phi(x) (\sup_{B}|\partial(h_1-h_{2})|+r(x)^{\alpha}\phi(x)^{\alpha}[\partial(h_1-h_2)]_{\alpha,B}) \\
& +Cr_{0}^{2}r(x)^{-p}\phi(x)^{N-n/2-2}(\sup_{B}|h_{1}-h_{2}|+ r(x)^{\alpha}\phi(x)^{\alpha}[h_1-h_2]_{\alpha,B})\\
&\leq Cr_{0}r(x)^{-p}\phi(x)^{N-n/2-2}\left\|h_{1}-h_{2}\right\|^{(-p,N-n/2-2)}_{2,\alpha}\leq Cr_{0}\left\|(h_{1},\omega_2)-(h_{2},\omega_2)\right\|_{2}.
\end{align*}
Lastly, we can take of the error terms involving $E_{5}$ and $E_{6}$:
\begin{align*}
&[E_5(h_1)(\omega_1)(\omega_1)-E_5(h_2)(\omega_2)(\omega_2)]_{\alpha,B}\leq r(x)^{-\alpha}\phi(x)^{-\alpha}\sup_B (|\omega_{1}|+|\omega_{2}|)\sup_B(|\omega_1-\omega_2|)\\
& +C\left(\sup_B |\omega_1-\omega_2|\left([\omega_1]_{\alpha,B}+[\omega_2]_{\alpha,B}\right)+\sup_B\left(\left|\omega_1\right|+\left|\omega_2\right|\right)[\omega_1-\omega_2]_{\alpha, B}\right)\\
& +C\left(\sup_B (\left|\omega_1\right|^{2}+\left|\omega_2\right|^{2})[h_1-h_2]_{\alpha, B}+\sup_{B}|h_{1}-h_{2}|([\omega_{1}^{2}]_{\alpha,B}+[\omega_{2}^{2}]_{\alpha,B})\right)
\end{align*}
and thus
\begin{align*} &r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}[E_5(h_1)(\omega_1)(\omega_1)-E_5(h_2)(\omega_2)(\omega_2)]_{\alpha,B}\\
&\leq Cr_{0}r(x)\phi(x) (\sup_{B}|\omega_1-\omega_2|+r(x)^{\alpha}\phi(x)^{\alpha}[\omega_1-\omega_2]_{\alpha,B})\\
&+Cr_{0}^{2}r(x)^{-p}\phi(x)^{N-n/2}(|h_{1}-h_{2}|+r(x)^{\alpha}\phi(x)^{\alpha}[h_1-h_2]_{\alpha,B}\\
&\leq Cr_{0}r(x)^{-p}\phi(x)^{N-n/2-2}(\left\|h_{1}-h_{2}\right\|^{(-p,N-n/2-2)}_{2,\alpha}+\left\|\omega_{1}-\omega_{2}\right\|^{(-p-1,N-n/2-2)}_{2,\alpha})\\
&\leq Cr_{0}\left\|(h_{1},\omega_1)-(h_{2},\omega_2)\right\|_{2}
\end{align*}
on the one hand, and
\begin{align*}
&[E_6(h_1)(h_1)(\omega_1)(\pi)-E_6(h_2)(h_2)(\omega_2)(\pi)]_{\alpha,B}\\
&\leq Cr(x)^{-p-1-\alpha}\sup_{B}\left(|h_1|+|h_2|\right)(\sup_{B}|\omega_1-\omega_2|+r(x)^{\alpha}\phi(x)^{\alpha}[\omega_1-\omega_2]_{\alpha, B})\\
&+Cr(x)^{-p-1-\alpha}\sup_{B}\left(|\omega_1|+|\omega_2|\right)(\sup_B |h_1-h_2|+r(x)^{\alpha}\phi(x)^{\alpha}[h_1-h_2]_{\alpha, B})\\
&+Cr(x)^{-p-1}\sup_B |h_1-h_2|([\omega_1]_{\alpha, B}+[\omega_2]_{\alpha, B})+Cr(x)^{-p-1}\sup_B |\omega_1-\omega_2|([h_1]_{\alpha, B}+[h_2]_{\alpha, B})
\end{align*}
hence
\begin{align*}
&r(x)^{p+2+\alpha}\phi(x)^{-N+n/2+4+\alpha}[E_6(h_1)(h_1)(\omega_1)(\pi)-E_6(h_2)(h_2)(\omega_2)(\pi)]_{\alpha,B}\\
&\leq Cr_{0}r(x)^{-p+1+\alpha}\phi(x)^{2}(\sup_B |\omega_1-\omega_2|+r(x)^{\alpha}\phi(x)^{\alpha}[\omega_1-\omega_2]_{\alpha,B})\\
&+Cr_{0}r(x)^{-p+\alpha}\phi(x)^{2}(\sup_B |h_1-h_2|+r(x)^{\alpha}\phi(x)^{\alpha}[h_1-h_2]_{\alpha,B})\\
&\leq Cr_{0}r(x)^{-2p+\alpha}\phi(x)^{N-n/2}(\left\|h_1-h_2\right\|^{(-p,N-n/2-2)}_{2,\alpha}+\left\|\omega_1-\omega_2\right\|^{(-p-1,N-n/2-2)}_{2,\alpha})\\
&\leq Cr_0 \left\|(h_1,\omega_1)-(h_2,\omega_2)\right\|
\end{align*}
on the other.
In order to get $\mathcal{C}^{1,\alpha}$ weighted estimates of $Q^{(2)}_{(g,\pi)}$ we first need to differentiate each of the espressions $E_7,E_8,E_9,E_{10}$ we got above. Via elementary, though somewhat lenghty computations, one can use such expressions to write $\partial Q^{(2)}_{(g,\pi)}$ as a sum of terms belonging to one of finitely many types that can be explicitly listed and we omit the tedious details.
Arguing as we did before and making use of the $\mathcal{C}^{2,\alpha}$ estimates on $h_{1}, h_{2}$ as well as $\omega_{1}, \omega_{2}$ one proves the pointwise estimates:
\[|Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]|\leq Cr_{0} r^{-2p-2}\phi^{2(N-n/2-4)}\left\|(h_{1}-h_{2},\omega_{1}-\omega_{2})\right\|_{2}
\]
and
\[ |\partial Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-\partial Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]|\leq Cr_{0} r^{-2p-3}\phi^{2(N-n/2-5)}\left\|(h_{1}-h_{2},\omega_{1}-\omega_{2})\right\|_{2}
\]
which patently imply
\[ r^{p+2}\phi^{-N+n/2+3}|Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]|+r^{p+3}\phi^{-N+n/2+4}|\partial Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-\partial Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]|\leq Cr_{0}
\]
as long as $N>n/2-6$.
Similarly, making repeated use of the inequality \eqref{eq:leibhol} one shows that for any point $x\in\Omega$
\[ \left[Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]\right]_{\alpha,B_{d(x)/2}}\leq r^{-2p-2-\alpha}\phi^{2(N-n/2-4-\alpha)}\left\|(h_{1}-h_{2},\omega_{1}-\omega_{2})\right\|_{2}
\]
as well as
\[\left[\partial Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-\partial Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]\right]_{\alpha, B_{d(x)/2}}\leq r^{-2p-3-\alpha}\phi^{2(N-n/2-5-\alpha)}\left\|(h_{1}-h_{2},\omega_{1}-\omega_{2})\right\|_{2}
\]
so that in the end
\[ \left\|Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]-Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}] \right\|^{(p+2,-N+n/2+3)}_{1,\alpha}\leq Cr_{0}
\]
for some constant $C$ only depending on the ambient dimension.
Combining both our integral and pointwise estimates, we have shown that
\[\left\|\left(Q^{(1)}_{(g,\pi)}[h_{1},\omega_{1}]- Q^{(1)}_{(g,\pi)}[h_{2},\omega_{2}], Q^{(2)}_{(g,\pi)}[h_{1},\omega_{1}]- Q^{(2)}_{(g,\pi)}[h_{2},\omega_{2}]\right)\right\|_{1}\leq Cr_{0}\left\|(h_{1},\omega_{1})-(h_{2},\omega_{2})\right\|_{2}
\]
if $\left\|(h_{1},\omega_{1})\right\|_{1}+\left\|(h_{2},\omega_{2})\right\|_{1}\leq r_{0}$.
Thus, given $\lambda>0$ as small as we need, we simply pick $r_{0}=\lambda/4C$ and this completes the proof. \end{proof}
\subsection{Continuity of the ADM energy-momentum} \label{subs:sobcont}
We shall be concerned here with the relation between the ADM energy-momentum of our initial data $(M,\check{g}, \check{k})$ and those of its localization $(M,\hat{g},\hat{k})$ as in the statement of Theorem \ref{thm:main}. In order to simplify the exposition, we will assume that the manifold $M$ has only one end and yet this restriction is completely unnecessary since our construction happens \textsl{one end at a time}. We know that, by our very gluing scheme, $\hat{g}=\check{g}$ and $\hat{k}=\check{k}$ on the whole region $\Omega_{I}$ so that, in particular, this is true in the interior of a (Euclidean) ball of radius $|a|\sin\theta_{1}$ (resp. $|a|/2$) for $\theta_1\in(0,\pi/2)$ (resp. for $\theta_1\in[\pi/2,\pi)$). (We are assuming the ball in question to be defined by the usual equation, given in asymptotically flat coordinates $\left\{x\right\}$ along the unique end of $M$). As a result, it follows from our construction that for any $\overline{p}\in\left(\frac{n-2}{2},p\right)$
\[ \hat{g}\stackrel{\mathcal{H}_{2,-\overline{p}}}{\rightarrow}\check{g}, \ \ \hat{\pi}\stackrel{\mathcal{H}_{1,-1-\overline{p}}}{\rightarrow}\check{\pi} \ \ \ \textrm{as} \ \ |a|\to\infty.
\]
Notice that the previous statement is true both when one considers the volume measure associated to $\check{g}$ and when one considers the volume measure associated to $\hat{g}$ or in fact any other measure on $M$ which is equivalent to $\mathscr{L}^{n}$ outside a large compact set.
Our target now is to prove that the ADM energy-momentum is continuous with respect to sequential convergence in those topologies. This is probably known to the experts, but not so easily found in the literature, so we give here a detailed argument to fill this gap.
\begin{proposition}\label{pro:ADMcont}
Let $(M,g_{\infty},k_{\infty})$ be an asymptotically flat initial data set with $g_{\infty}\in\mathcal{H}_{2,-p}$ and $k_{\infty}\in\mathcal{H}_{1,-p-1}$ for some $p>\frac{n-2}{2}$. Consider a sequence of data $(M,g_{l},k_{l})$ such that $g_{l}\to g_{\infty}$ in $\mathcal{H}_{2,-p}$ and $k_{l}\to k_{\infty}$ in $\mathcal{H}_{1,-p-1}$ as $l$ tends to infinity. Then
\[ \lim_{a\to\infty}\mathcal{E}^{(l)}=\mathcal{E}^{(\infty)}, \ \ \lim_{a\to\infty}\mathcal{P}^{(l)}=\mathcal{P}^{(\infty)}.
\]
Here $\mathcal{E}^{(l)}, \mathcal{P}^{(l)}$ (resp. $\mathcal{E}^{(\infty)}, \mathcal{P}^{(\infty)}$) denote the ADM energy-momentum of $(M,g_{l},k_{l})$ (resp. $(M,g_{\infty},k_{\infty})$).
\end{proposition}
To avoid dangerous ambiguities, let us remark that (by our definition of initial data sets, see Subsection \ref{subs:ids}) we are assuming both $(M,g_{\infty},k_{\infty})$ and each $(M,g_{l},k_{l})$ to solve the Einstein constraint equations. For the sake of simplicity, we will deal here with the \textsl{vacuum} case, even though an identical proof can be given for the general constraints under the usual integrability assumption on $\mu$ and $J$.
\begin{proof}
Let us first consider the energy functional. Given a large radius $r'$ we start by claiming that for any asymptotically flat metric $g$ as in the statement of our proposition
\[\left|\mathcal{E}-\frac{1}{2(n-1)\omega_{n-1}}\int_{|x|=r'}\sum_{i,j=1}^{n}(g_{ij,i}-g_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}\right|\leq C(r')^{2n-4-4p}
\]
(for some constant $C$ which is uniform in a given $\mathcal{H}_{2,-p}\times\mathcal{H}_{1,-p-1}$ neighbourhood of $(g,k)$) and it is clear that once this is justified, it is enough to prove the continuity (in our topology) of the approximating hypersurface integral at \textsl{fixed} radius $r'$.
To obtain the claim, let us pick a second, larger radius $r''>r'$ and notice that
\[ \int_{|x|=r''}\sum_{i,j=1}^{n}(g_{ij,i}-g_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}-\int_{|x|=r'}\sum_{i,j=1}^{n}(g_{ij,i}-g_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}=\int_{B_{r''}\setminus B_{r'}}(R_g+\mathcal{R})\,d\mathscr{L}^{n}
\]
(with a remainder term satisfying $|\mathcal{R}|\leq C(|g-\delta||\partial^{2}g|+|\partial g|^{2})$) by the divergence theorem, hence by the first constraint equation (the Hamiltonian constraint) that integral is upper bounded by (modulo a dimensional constant)
\[ \int_{M\setminus B_{r'}}(|g-\delta||\partial^{2}g|+|\partial g|^{2}+|\pi|^{2})\,d\mathscr{L}^{n}.
\]
The terms $|g-\delta||\partial^{2}g|, \ |\partial g|^{2}$ and $|\pi|^{2}$ have, under our assumptions, roughly the same rate of decay (of order $|x|^{-2(p+1)}$, in integral sense) so we will just consider the latter, the treatment for the former two being identical.
We can write:
\[\int_{M\setminus B_{r'}}|\pi|^{2}\,d\mathscr{L}^{n}\leq \left(\int_{M\setminus B_{r'}}(|\pi|r^{1+p})^{2}r^{-n}\,d\mathscr{L}^{n}\right)^{1/2}\left(\int_{M\setminus B_{r'}}(|\pi|r^{n-1-p})^{2}r^{-n}\,d\mathscr{L}^{n}\right)^{1/2}
\]
and in turn
\[ \int_{M\setminus B_{r'}}(|\pi|r^{n-1-p})^{2}r^{-n}\,d\mathscr{L}^{n}\leq \sup_{r>r'} r^{2n-4-4p} \times \left(\int_{M\setminus B_{r'}}|\pi|^{2}r^{-n+2p+2}\,d\mathscr{L}^{n}\right)
\] which completes the proof of our claim.
Therefore, for any given $\varepsilon>0$ let us fix a reference radius $r'$ so that for each metric $g_{i}$ as well as for $g_{\infty}$ the ADM energy is appromixated by the integral over the hypersphere of radius $r'$ (considering the measure $\mathscr{H}^{n-1}$) with an error less that $\varepsilon/2$.
Hence, applying the divergence theorem once again
\begin{align*} & \int_{|x|=r'}\sum_{i,j=1}^{n}(g^{(\infty)}_{ij,i}-g^{(\infty)}_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}-\int_{|x|=r'}\sum_{i,j=1}^{n}(g^{(l)}_{ij,i}-g^{(l)}_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}\\
& =\int_{|x|<r'}(g^{(\infty)}_{ij,ij}-g^{(\infty)}_{ii,jj})\,d\mathscr{L}^{n}-\int_{|x|<r'}(g^{(l)}_{ij,ij}-g^{(l)}_{ii,jj})\,d\mathscr{L}^{n} \\
\end{align*}
thus
\begin{align*}
& \left|\int_{|x|=r'}\sum_{i,j=1}^{n}(g^{(\infty)}_{ij,i}-g^{(\infty)}_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}-\int_{|x|=r'}\sum_{i,j=1}^{n}(g^{(l)}_{ij,i}-g^{(l)}_{ii,j})\nu_{0}^{j}\,d\mathscr{H}^{n-1}\right| \\
& \leq C \left(\int_{M}\left|g^{(\infty)}_{ij,ij}-g^{(l)}_{ij,ij}\right|^{2}r^{-n+2p+4}\,d\mathscr{L}^{n}\right)^{1/2}\left(\int_{|x|<r'}r^{n-2p-4}\,d\mathscr{L}^{n}\right)^{1/2} \\
&+C\left(\int_{M}\left|g^{(\infty)}_{ii,jj}-g^{(l)}_{ii,jj}\right|^{2}r^{-n+2p+4}\,d\mathscr{L}^{n}\right)^{1/2}\left(\int_{|x|<r'}r^{n-2p-4}\,d\mathscr{L}^{n}\right)^{1/2}
\end{align*}
By our convergence assumption, we can now pick an index $l_{0}$ so that for $l>l_{0}$ each of those two summands is less than $\varepsilon/2$ so that in the end $|\mathcal{E}_{\infty}-\mathcal{E}_{l}|<\varepsilon$. Therefore, since $\varepsilon$ can be chosen arbitrarily small we conclude that the energy functional is continuous in these weighted Sobolev spaces. The proof of the continuity of the ADM linear momentum follows along the same lines, so we will only sketch it. First of all, in order to show that
\[\left|\mathcal{P}_i-\frac{1}{(n-1)\omega_{n-1}}\int_{|x|=r'}\sum_{j=1}^{n}\pi_{ij}\nu_{0}^{j}\,d\mathscr{H}^{n-1}\right|\leq C(r')^{2n-4-4p}
\]
one applies the divergence theorem observing that
\[
\sum_j\pi_{ij,i}=\sum_j(\pi_{ij;j}-\pi_{ij,j})\leq C |\partial g||\pi|
\]
by virtue of the second constraint equation (the vector constraint) and the fact that, in local coordinates
\[ \pi_{ij;i}=\pi_{ij,j}-\Gamma^{k}_{jj}\pi_{ik}-\Gamma^{k}_{ij}\pi_{jk}.
\]
Since $|\partial g||\pi|\leq |\partial g|^{2}+|\pi|^{2}$ the claim follows by the argument we have already presented because of course we only need to deal with terms of order zero or one. At that stage, the comparison happens at the level of a fixed hypersphere and the convergence assumption can be used in a straightforward fashion. This concludes the proof.
\end{proof}
As a result, we can immediately deduce the surprising fact that our localization construction gives an arbitrarily good approximation of the ADM energy-momentum of the given initial data, no matter how small the angles $\theta_{1}$ and $\theta_{2}$ may potentially be.
\begin{corollary}
Let $(M,\check{g},\check{k})$, $(M,\hat{g},\hat{k})$ be as in the statement of Theorem \ref{thm:main}. Then
\[ \lim_{a\to\infty} \hat{\mathcal{E}}=\check{\mathcal{E}}, \ \ \lim_{a\to\infty}\hat{\mathcal{P}}=\check{\mathcal{P}}.
\]
Here $\hat{\mathcal{E}}, \hat{\mathcal{P}}$ (resp. $\check{\mathcal{E}}, \check{\mathcal{P}}$) denote the ADM energy-momentum of $(M,\hat{g},\hat{k})$ (resp. $(M,\check{g},\check{k})$).
\end{corollary}
\section{A new class of $N-$body solutions}\label{sec:Nbody}
As anticipated in the Introduction, we would like to devote this section to a discussion about the applicability of our gluing methods to the construction of a new class of $N$-body initial data sets for the Einstein constraint equations. In the context of Newtonian gravity, a set of initial data for the evolution of $N$ massive bodies can be obtained by solving a single Poisson equation in the complement of a finite number of compact domains (say balls) in $\mathbb{R}^{3}$, at least if the interior structure and dynamics of the bodies in question is neglected. The nonlinearity of the Einstein constraint equations makes such a task a lot harder and in fact it was only in the last decade that sufficiently general results in this direction, without restricting to rather symmetric configurations \cite{CD03} or exploiting the presence of multiple ends \cite{CIP05}, have been obtained \cite{CCI09, CCI11}.
\
Let us suppose that a finite collection of $N$ asymptotically flat data sets $(M_{1}, g_{1}, k_{1}), \ldots, (M_{N}, g_{N}, k_{N})$ is assigned and let $U_{i}$ denote a compact, regular subdomain of $M_{i}$ for each value of the index $i$. Moreover, let $x_{1},\ldots, x_{N}\in \mathbb{R}^{n}$ be $N$ vectors which are supposed to prescribe the location of the regions $U_{1}, \ldots, U_{k}$ with respect to a fiducial flat background. Then, the problem we want to address is the construction of a triple $(M,g,k)$ which:
\begin{itemize}
\item{is a solution of the vacuum Einstein constraint equations $\Phi(g,k)=0$;}
\item{contains $N$ regions that are isometric to the given bodies;}
\item{has the centers of such bodies in a configuration which is a scaled version of the chosen configuration.}
\end{itemize}
\
Making use of Theorem \ref{thm:main}, we can glue together any finite number of conical regions for given data $(M_{i}, g_{i}, k_{i})$. In order to state this result, let us introduce some notation. Given $n\geq 3$, $a\in \mathbb{R}^{n}$, $\theta\in(0,\pi)$ and $\varepsilon>0$ very small we let $\Gamma_{\theta}\left(a\right)$ (resp. $\Omega_{\theta,\varepsilon}\left(a\right)$) be the regularized conical region obtained from $C_{\theta}(a)$ (resp. $C_{\theta}(a)\setminus C_{\left(1-\varepsilon\right)\theta}\left(a\right)$) by removing the singularity at the tip and smoothly gluing $B_{1/2}\left(a\right)$ as explained in the Subsection \ref{subs:regcon}. In addition, we set
\[\hat{\Gamma}_{\theta,\varepsilon}\left(a\right)=\Gamma_{\theta}\left(a\right)-(1+\varepsilon)a, \ \
\hat{\Omega}_{\theta,\varepsilon}\left(a\right)=\Omega_{\theta, \varepsilon}\left(a\right)-(1+\varepsilon)a
\]
that are translated copies of the smoothened regions. We remark that for any given $\varepsilon>0$ if $a, a'$ are large enough we have that $\left[C_{\theta}(a)-a\right]\cap \left[C_{\theta'}(a')-a'\right]=\emptyset$ implies $\hat{\Omega}_{\theta,\varepsilon}\left(a\right)\cap \hat{\Omega}_{\theta',\varepsilon}\left(a'\right)=\emptyset$ as well.
Finally, given two vectors $v_{1}, v_{2}\in \mathbb{S}^{n-1}$ let $\varphi(v_{1}, v_{2})\in\left[0,\pi\right]$ be the angle between them. This statement is a remarkable consequence of our gluing Theorem \ref{thm:main}.
\begin{theorem}\label{thm:Nbody}
Given $\sigma>0, 0<\varepsilon<1/2$ and a collection of initial data $(M_{1}, g_{1}, k_{1}), \ldots, (M_{N}, g_{N}, k_{N})$ (as in Subsection \ref{subs:ids}) satisfying the vacuum Einstein constraint equations one can find $\Lambda>0$ such that: assigned vectors $a_{1},\ldots, a_{N}\in\mathbb{R}^{n}$ and angles $\theta_{1}, \ldots,\theta_{N}$ satisfying $\left|a_{i}\right|>\Lambda$ and $\varphi(a_{i}, a_{j})>(1-\varepsilon)^{-1}(\theta_{i}+\theta_{j})$ (for any choice of indices $i, j$) then there exists an asymptotically Euclidean triple $(M,g, k)$ satisfying the vacuum Einstein constraint equations, where $M$ has exactly one end, such that $g$ (resp. $k$) on $\hat{\Gamma}_{\theta_{i},\varepsilon}\left(a_{i}\right)$ coincides with $g_{i}$ (resp. $k=k_{i}$) on $\Gamma_{\theta_{i},\varepsilon}\left(a_{i}\right)\subset M_i$ and
\[ \left|\mathcal{E}-\sum_{i=1}^{N}\mathcal{E}^{(i)}\right|\leq\sigma, \ \ \left|\mathcal{P}-\sum_{i=1}^{N}\mathcal{P}^{(i)}\right|\leq\sigma.
\]
\end{theorem}
\begin{figure}[ht!]
\begin{tikzpicture}
\begin{scope}[scale=0.75]
\begin{scope}[shift={(0, 6)}]
\begin{scope}[shift={(-4, 0)}]
\fill [black!10, draw=black, rounded corners=2mm] plot coordinates {(-2, -1) (0, 1) (6, 1) (4, -1) (-2, -1)};
\fill [black!10, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.3, 1.5) (1.1, 2.5) (2, 2.5) (2.9, 2.5) (2.7, 1.5) (3, 0.5) (4, 0)};
\node at (5, 1.5) {$(M_1, g_1, k_1)$};
\end{scope}
\begin{scope}[shift={(4, 0)}]
\fill [black!30, draw=black, rounded corners=2mm] plot coordinates {(-2, -1) (0, 1) (6, 1) (4, -1) (-2, -1)};
\fill [black!30, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.25, 1.5) (1.1, 2.5) (2, 2.9) (2.9, 2.5) (2.75, 1.5) (3, 0.5) (4, 0)};
\node at (5, 1.5) {$(M_2, g_2, k_2)$};
\end{scope}
\node at (-2, -1.5) (M1) {};
\node at (6, -1.5) (M2) {};
\node at (-2, -2.75) (M1tempup) {};
\node at (6, -2.75) (M2tempup) {};
\path[every node/.style={font=\sffamily\small}, ->, line width=1pt] (M1) edge [out=270, in=90] (M1tempup);
\path[every node/.style={font=\sffamily\small}, ->, line width=1pt] (M2) edge [out=270, in=90] (M2tempup);
\end{scope}
\begin{scope}[shift={(-4, 0)}]
\fill [pattern color=blue!80, pattern=north west lines, draw=blue, line width=0.5pt, rounded corners=2mm] plot coordinates {(-0.5, 0.5) (5.6, 0.8) (3.4, -1)};
\fill [pattern color=blue!80, pattern=north west lines, draw=none, rounded corners=2mm] plot coordinates {(3.4, -1) (-2, -1) (-0.5, 0.5)};
\fill [black!10, draw=blue, line width=0.5pt, rounded corners=4mm] plot coordinates {(-0.7, 0.29) (5.6, 0.8) (3.2, -1.01)};
\fill [black!10, draw=black, rounded corners=2mm] plot coordinates {(3.2, -1) (-2, -1) (-0.7, 0.3)};
\draw [black, rounded corners=2mm] plot coordinates {(-0.5, 0.5) (0, 1) (6, 1) (4, -1) (3.4, -1)};
\fill [black!10, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.3, 1.5) (1.1, 2.5) (2, 2.5) (2.9, 2.5) (2.7, 1.5) (3, 0.5) (4, 0)};
\end{scope}
\begin{scope}[shift={(4, 0)}]
\fill [pattern color=blue!80, pattern=north west lines, draw=blue, line width=0.5pt, rounded corners=2mm] plot coordinates {(0.6, 1) (-1.6, -0.8) (4.5, -0.5)};
\fill [pattern color=blue!80, pattern=north west lines, draw=none, rounded corners=2mm] plot coordinates {(4.5, -0.5) (6, 1) (0.6, 1)};
\fill [black!30, draw=blue, line width=0.5pt, rounded corners=4mm] plot coordinates {(0.8, 1.01) (-1.6, -0.8) (4.7, -0.29)};
\fill [black!30, draw=black, rounded corners=2mm] plot coordinates {(4.7, -0.3) (6, 1) (0.8, 1)};
\draw [black, rounded corners=2mm] plot coordinates {(0.6, 1) (0, 1) (-2, -1) (4, -1) (4.5, -0.5)};
\fill [black!30, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.25, 1.5) (1.1, 2.5) (2, 2.9) (2.9, 2.5) (2.75, 1.5) (3, 0.5) (4, 0)};
\end{scope}
\node at (-2, -1.5) (M1temp) {};
\node at (6, -1.5) (M2temp) {};
\node at (2, -3.2) (M) {};
\path[every node/.style={font=\sffamily\small}, line width=1pt] (M1temp) edge [out=270, in=90] (M);
\path[every node/.style={font=\sffamily\small}, line width=1pt] (M2temp) edge [out=270, in=90] (M);
\draw [black, ->, line width=1pt] (2, -3) -- (2, -3.5);
\begin{scope}[shift={(0, -7)}]
\begin{scope}[shift={(-4, -1)}]
\fill [pattern color=blue!80, pattern=north west lines, draw=blue, line width=0.5pt, rounded corners=2mm] plot coordinates {(-0.5, 0.5) (5.6, 0.8) (3.4, -1)};
\fill [pattern color=blue!80, pattern=north west lines, draw=none, rounded corners=2mm] plot coordinates {(3.4, -1) (-2, -1) (-0.5, 0.5)};
\fill [black!10, draw=blue, line width=0.5pt, rounded corners=4mm] plot coordinates {(-0.7, 0.29) (5.6, 0.8) (3.2, -1.01)};
\fill [black!10, draw=black, rounded corners=2mm] plot coordinates {(3.2, -1) (-2, -1) (-0.7, 0.3)};
\end{scope}
\begin{scope}[shift={(4, 1)}]
\fill [pattern color=blue!80, pattern=north west lines, draw=blue, line width=0.5pt, rounded corners=2mm] plot coordinates {(0.6, 1) (-1.6, -0.8) (4.5, -0.5)};
\fill [pattern color=blue!80, pattern=north west lines, draw=none, rounded corners=2mm] plot coordinates {(4.5, -0.5) (6, 1) (0.6, 1)};
\fill [black!30, draw=blue, line width=0.5pt, rounded corners=4mm] plot coordinates {(0.8, 1.01) (-1.6, -0.8) (4.7, -0.29)};
\fill [black!30, draw=black, rounded corners=2mm] plot coordinates {(4.7, -0.3) (6, 1) (0.8, 1)};
\node at (4.3, -2.7) {$(M, g, k)$};
\end{scope}
\node at (2, 0) {$\bullet$};
\node at (2.3, -0.3) {$O$};
\draw [black, rounded corners=2mm] plot coordinates {(-4.5, -0.5) (-2, 2) (4.6, 2)};
\draw [black, rounded corners=2mm] plot coordinates {(8.5, 0.5) (6, -2) (-0.6, -2)};
\begin{scope}[shift={(-4,-1)}]
\fill [black!10, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.3, 1.5) (1.1, 2.5) (2, 2.5) (2.9, 2.5) (2.7, 1.5) (3, 0.5) (4, 0)};
\end{scope}
\begin{scope}[shift={(4,1)}]
\fill [black!30, draw=black] plot [smooth] coordinates {(0, 0) (1, 0.5) (1.25, 1.5) (1.1, 2.5) (2, 2.9) (2.9, 2.5) (2.75, 1.5) (3, 0.5) (4, 0)};
\end{scope}
\end{scope}
\end{scope}
\end{tikzpicture}
\caption{Theorem \ref{thm:Nbody} allows to merge an assigned collection of data into an exotic $N-$body solution of the Einstein constraint equations.}
\label{dgm:gluing.figure}
\end{figure}
\begin{remark}
Let us recall from Subsection \ref{subs:regcon} that our data are all tacitly assumed to have only one end. If instead this is not the case, the same construction goes through anyway, provided each $(M_i,g_i,k_i)$ comes with a preferred, labelled end on which the gluing is performed: in this case the resulting triple will of course have multiple ends and indeed
\[
\# \ \textrm{ends} \ (M,g,k) = 1+\sum_{i=1}^{N}( \# \ \textrm{ends} \ (M_i,g_i,k_i) -1).
\]
\end{remark}
\begin{proof} First of all, one can find $\Lambda$ large enough that Theorem \ref{thm:main} is applicable to every single triple $(M_i,g_i,k_i)$ and, under the additional constraint $\varepsilon\Lambda\max_{i,j}\left\{\sin\left(\frac{\varphi(a_i,a_j)}{2}\right)\right\}>1$ we can make sure that each couple of domains $\hat{\Gamma}_{\theta_{i}/\left(1-\varepsilon\right), \varepsilon}\left(a_{i}\right)$ will indeed be mutually disjoint whenever $|a_i|>\Lambda$ for any $i$ and $\varphi(a_{i}, a_{j})>(1-\varepsilon)^{-1}(\theta_{i}+\theta_{j})$, as in the statement above. Thus, we employ our gluing theorem exactly $N$ times to construct on $\mathbb{R}^{n}\setminus \cup_{i=1}^{N}\hat{\Gamma}_{\left(1-\varepsilon\right)\theta_{i}, \varepsilon}\left(a_{i}\right)$ a smooth couple $(g,k)$ that agrees with each of our data $(g_{i}, k_{i})$ at the interface $\hat{\Omega}_{\theta_{i}, \varepsilon}\left(a_{i}\right)$ and is exactly Euclidean outside of $\hat{\Gamma}_{\theta_{i}/\left(1-\varepsilon\right), \varepsilon}\left(a_{i}\right)$.
As a second step, one can \textsl{fill in} the conical subregions by smoothly identifying the boundary of such manifold with the boundary of the conical domains $\hat{\Gamma}_{\left(1-\varepsilon\right)\theta_{i}, \varepsilon}\left(a_{i}\right)$ in each $M_{i}$. Correspondingly, one can extend the tensors $g$ and $k$ on the whole $M$.
We claim that the resulting triple $(M,g,k)$ satisfies all of our requirements and to that aim it is enough to check the last assertion. As remarked in Subsection \ref{subs:sobcont}, it follows from our general construction that
\[\mathcal{E}\left(\hat{g}\right)\to \mathcal{E}\left(g\right) \ \ \textrm{as} \ \left|a\right|\to\infty
\]
and similarly, for the linear momentum
\[\mathcal{P}\left(\hat{g}\right)\to \mathcal{P}\left(g\right) \ \ \textrm{as} \ \left|a\right|\to\infty.
\]
Now, since the reference background triple for our gluing is $(\mathbb{R}^{n}, \delta, 0)$ it follows at once that in our construction the components of the energy-momentum 4-vector add up exactly
\[\mathcal{E}=\sum_{i=1}^{N}\hat{\mathcal{E}}^{(i)}, \ \mathcal{P}=\sum_{i=1}^{N}\hat{\mathcal{P}}^{(i)} \]
where we are using the notation $\hat{\mathcal{E}}^{(i)}=\mathcal{E}\left(\hat{g}_{i}\right)$ and obviously $\hat{g}_{i}$ is gotten from $g_{i}$ by performing our gluing with respect to the cones of vertex $a_{i}$ and angles $\theta_{i}$ and $\theta_{i}/\left(1-\varepsilon\right)$.
As a result, in order to conclude it is enough to choose $\Lambda$ possibly a bit larger than before, namely large enough so that $\left|a_{i}\right|>\Lambda$ implies $\left|\mathcal{E}(g_{i})-\hat{\mathcal{E}}^{(i)}\right|\leq\sigma/N$ (for every index $i=1,\ldots,N$) and analogously for the linear momentum.
\end{proof}
\
If finitely many compact subdomains $U_{1},\ldots, U_{N}$ are assigned in each $M_{1},\ldots, M_{N}$ we can exploit the gluing results by Corvino \cite{Cor00} and Corvino-Schoen \cite{CS06} to reduce to the case when every one of these is contained in a larger domain $V_{i}$ and $\partial V_{i}$ has a neighbourhood where $(g_{i}, k_{i})$ are exactly like in an annulus of a Schwarzschild or (more generally) Kerr solution. As a result, the prescribed position of $V_{1},\ldots, V_{N}$ can be described in terms of centers $x_{1},\ldots, x_{N}$ of finitely many Euclidean balls $B_{1},\ldots, B_{N}$. For each of our data $M_{1},\ldots, M_{N}$ we just need to make sure to pick an angle $\theta_{i}$ and a scaling factor $\tau$ small enough so that $\varphi(x_i,x_j)>\theta_i+\theta_j $ and of course $C_{\theta_{i}}\left(-x_{i}/\tau\right)$ contains the domain $V_{i}$. As a consequence, Theorem \ref{thm:Nbody} immediately implies constructibily of $N$-body initial data sets in the sense explained above, in substantial analogy with the main results contained in \cite{CCI09} (for the time-symmetric case) and \cite{CCI11} (for the general case). The only relevant difference with the present treatment is the behaviour at infinity of the tensors $g, k$, which in such works is prescribed to be exactly as in the Kerr solution outside a suitably large compact set, while we engineer data that are non-interacting on large time scales.
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